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--- author: - | Alexandre Popoff\ al.popoff@free.fr\ France title: Towards A Categorical Approach of Transformational Music Theory --- *Abstract:*\ Transformational music theory mainly deals with group and group actions on sets, which are usually constituted by chords. For example, neo-Riemannian theory uses the dihedral group $D_{24}$ to study transformations between major and minor triads, the building blocks of classical and romantic harmony. Since the developments of neo-Riemannian theory, many developments and generalizations have been proposed, based on other sets of chords, other groups, etc. However music theory also face problems for example when defining transformations between chords of different cardinalities, or for transformations that are not necessarily invertible. This paper introduces a categorical construction of musical transformations based on category extensions using groupoids. This can be seen as a generalization of a previous work which aimed at building generalized neo-Riemannian groups of transformations based on group extensions. The categorical extension construction allows the definition of partial transformations between different set-classes. Moreover, it can be shown that the typical wreath products groups of transformations can be recovered from the category extensions by “packaging” operators and considering their composition. Introduction ============ After the pionneering work of David Lewin [@lewin], music theory has seen developments which have relied heavily on the group structure, wherein group elements are seen as operations between some set elements which usually represent chords. In neo-Riemannian theory, the classical set of elements was originally constituted by the major and minor chords, and the typical corresponding group of transformations is isomorphic to the dihedral group $D_{24}$ of 24 elements, whether it acts through the famous L, R and P operations or through the transpositions and inversions operators [@cohn1; @cohn2; @cohn3; @capuzzo], or many others (see for example the Schritt-Wechsel group) [@douthett]. Following its application to major/minor triads, generalizations have been actively researched. For example, transformational theory has also been applied to other sets of chords [@straus]. Different groups of transformations than the dihedral one have been proposed. Julian Hook’s UTT group is a much larger group of order 288 and has at its a core a wreath product construction [@hook1; @hook2]. Wreath products were also studied by Robert Peck in a more general setting [@peck1]. More recently, Robert Peck also introduced imaginary transformations [@peck2], in which quaternion groups, dicyclic groups and other extraspecial groups appear. A different approach has been undertaken in [@popoff], in an attempt to unify all these different groups, in which generalized neo-Riemannian groups of musical transformations are built as extensions. However, the current group-based transformational theories raise multiple issues. One of them is that they sometimes fail to provide interesting groups of transformations for some sets of chords (an example will be given below). A second one is that transformational theories have also failed to provide a solution to the cardinality problem, namely finding transformations between chords of different cardinalities. While Childs [@childs] studied neo-Riemannian theory applied to seventh chords, his model does not include triads. Hook [@hook3] introduced another approach, namely cross-type transformations, to circumvent this problem. In this paper, we introduce a categorical approach to musical transformations with the aim of generalizing existing constructions. This work can be viewed as a generalization of the previous work on group extensions, by using groupoids instead of groups, and by building the corresponding groupoid extensions. Note that a categorical approach to music theory has been heavily investigated in the book *The Topos of Music* by G. Mazzola [@mazzola]. Mazzola deplores in particular that *“Although the theory of categories has been around since the early 1940s and is even recognized by computer scientists, no attempt is visible in AST (Atonal Set Theory) to deal with morphisms between pcsets, for example”*. While this paper is rather technical and more mathematically- than musically-oriented, we nevertheless hope that it will provide useful leads for application to music analysis. The first section highlights some of the limitations of current transformational theories based on particular examples. The second part introduces a categorical construction for musical transformations. Finally, the third part explores the relation between the categories constructed in section 2 and the more familiar groups of musical transformations, showing in particular how wreath products are naturally recovered from the category extensions. On some limitations of transformational theories ================================================ Groups of transformations acting on three set-classes ----------------------------------------------------- Consider the pitch-class sets \[0,4,7\], \[0,2,5\] and \[0,4,5\], as represented in Figure \[fig:setClasses\]. In the rest of this paper, we will label these sets as M, $\alpha$ and $\beta$ respectively. These set-classes have a well-defined root, which can therefore take any value in $\mathbb{Z}_{12}$. In this paper we will denote by $n_t$ a chord of root $n$ and of type $t$. By analogy with the action of the $T/I$ group on the set of major and minor triads, transposition operators $T_i$ can be defined for M, $\alpha$ and $\beta$. The action of these transposition operators is straightforward as $T_i$ takes a chord $n_t$ to $(n+i)_t$ (all operations are understood modulo 12). There also exists voice-leading transformations $VL$ between these set-classes. For example, if one represents a chord as an ordered set $(x,y,z)$, where $x$ is the root, we can define the $VL$ transformation as $$VL: \left( \begin{array}{lll}x\\y\\z\end{array} \right) \longmapsto \left( \begin{array}{lll}z+2\\x-1\\y-2\end{array} \right)$$ Using the notation $n_t$ for chords, this transformation is then defined as : $$VL: \left( \begin{array}{lll}n_M\\n_\alpha\\n_\beta\end{array} \right) \longmapsto \left( \begin{array}{lll}(n-3)_\alpha\\(n-5)_\beta\\(n-5)_M\end{array} \right)$$ We can define another voice-leading transformation $VL'$ with a similar action as $$VL': \left( \begin{array}{lll}x\\y\\z\end{array} \right) \longmapsto \left( \begin{array}{lll}z+4\\x+1\\y\end{array} \right)$$ or equivalently : $$VL': \left( \begin{array}{lll}n_M\\n_\alpha\\n_\beta\end{array} \right) \longmapsto \left( \begin{array}{lll}(n+1)_\alpha\\(n-3)_\beta\\(n-3)_M\end{array} \right)$$ ![The action of the voice-leading $VL$ operation on set-classes M, $\alpha$ and $\beta$[]{data-label="fig:setClassesVoiceLeading"}](MAlphaBetaVoiceLeading.pdf) We can notice that $VL^{-3}=VL'^{21}=T_1$. The $VL$ and $VL'$ operations are clearly contextual [@kochavi] since their action on the root depends on the type of the chord on which they act. Since their action switches the type of the chords, they can be seen as “generalized inversions” similar to the $I$ transformations of the $T/I$ group, or the $P$, $L$ or $R$ operations of the $PLR$ group. If we wish to build a group which includes both the transposition operators and these generalized inversions, we will obtain that $\langle T_i,VL\rangle = \langle T_i,VL'\rangle \cong \mathbb{Z}_{36}$, as can be checked with any computational group theory software such as GAP. The construction introduced in [@popoff] aims at building generalized neo-Riemannian groups of musical transformations which include both transposition and inversion operators. These groups $G$ are built as extensions of $Z$ by $H$, where $Z$ is the group of transpositions and $H$ can be seen as a group of “formal inversions”. In the present case, $Z$ would be isomorphic to $\mathbb{Z}_{12}$ whereas $H$ would be isomorphic to $\mathbb{Z}_3$ to reflect the inversions between the three different pitch-class sets. If one tries to apply this construction to build a group extension $G$ of simply transitive musical transformations as $$1 \to \mathbb{Z}_{12} \to G \to \mathbb{Z}_{3} \to 1$$ one ends up with only two abelian groups, namely $G=\mathbb{Z}_{12} \times \mathbb{Z}_{3}$ or $G=\mathbb{Z}_{36}$. The reason for this is that $\mathbb{Z}_{12}$ has too few automorphisms (remember that $Aut(\mathbb{Z}_{12}) \cong \mathbb{Z}_{2} \times \mathbb{Z}_{2}$) and therefore there can be no action of $\mathbb{Z}_{3}$ on $\mathbb{Z}_{12}$ except for the trivial one. We thus see that group structures such as semidirect products, as is the case for the dihedral group $D_{24}$ used in neo-Riemannian theory, cannot exist for sets containing three different types of chords. The group extension structure determines a specific group operation based both on the action by automorphism of $H$ on $Z$, and on a 2-cocycle $H \times H \to Z$. As discussed in [@popoff], this group operation directly determines whether left or right actions are contextual. Here, the case $G=\mathbb{Z}_{12} \times \mathbb{Z}_{3}$ corresponds to the trivial direct product, i.e the trivial 2-cocycle, hence both actions are non-contextual. There also exists a non-trivial 2-cocycle which leads to $G=\mathbb{Z}_{36}$. As shown in [@popoff], non-trivial 2-cocycles give rise to contextual group actions on chords, the $VL$ and $VL'$ operations being such examples. However, since the group is abelian (i.e the 2-cocycle is symmetric) the left and right actions of these transformations coincide, and are thus both contextuals. The case of semidirect products is particularly interesting since left actions are non-contextual whereas right actions can be. An important part of the litterature about neo-Riemannian theory has focused on the duality between these left and right actions [@fiore1; @fiore2; @fiore3], and in particular their commuting property. However the present case does not allow for such richness. One could circumvent this problem by considering group extensions of the form : $$1 \to \mathbb{Z}_{3} \to G \to \mathbb{Z}_{12} \to 1$$ but in this case, the transpositions operators would not be well-defined anymore, since $\mathbb{Z}_{12}$ would no longer be a normal subgroup of $G$ in the general case. Moreover, the consideration of group extensions of the form $1 \to \mathbb{Z}_{12} \to G \to \mathbb{Z}_{3} \to 1$ limits the contextual and/or voice-leading transformations that can be applied to this set of chords, even when the 2-cocycle is non-trivial. Consider for example the following transformations : $$I_{M \leftrightarrow \alpha}: \left( \begin{array}{lll}x\\y\\z\end{array} \right) \longmapsto \left( \begin{array}{lll}x\\(2x-3)-y\\(2x-3)-z\end{array} \right), i.e \left( \begin{array}{lll}n_M\\n_\alpha\end{array} \right) \longmapsto \left( \begin{array}{lll}n_\alpha\\n_M\end{array} \right)$$ $$I_{M \leftrightarrow \beta}: \left( \begin{array}{lll}x\\y\\z\end{array} \right) \longmapsto \left( \begin{array}{lll} (2z+4)-y \\ (2z+4)-x \\ z\end{array} \right) i.e \left( \begin{array}{lll}n_M\\n_\beta\end{array} \right) \longmapsto \left( \begin{array}{lll}n+2_\beta\\n-2_M\end{array} \right)$$ $$I_{\alpha \leftrightarrow \beta}: \left( \begin{array}{lll}x\\y\\z\end{array} \right) \longmapsto \left( \begin{array}{lll} (2y-1)-z \\ y \\ (2y-1)-x\end{array} \right) i.e \left( \begin{array}{lll}n_\alpha\\n_\beta\end{array} \right) \longmapsto \left( \begin{array}{lll}n-2_\beta\\n+2_\alpha\end{array} \right)$$ These inversion-like transformations are represented in Figure \[fig:contxtMAlphaBeta\]. Each one of them is an involution, just as the $L$, $R$ and $P$ operations are. However, they can only be applied to the indicated pair of set-classes. In other terms, these operations are partial and cannot form a group of transformations since the closure condition would not be satisfied. We propose a way to unify these transformations in the next section. Transformations between chords of different cardinalities --------------------------------------------------------- The work of Childs [@childs] has shown that neo-Riemannian constructions can be applied to seventh chords. In view of [@popoff] and since seventh chords have a well-defined root, it is indeed possible to envision a group extension acting on seventh chords. However Childs’ work does not include triads. Since $Aut(\mathbb{Z}_{12}) \cong \mathbb{Z}_{2} \times \mathbb{Z}_{2}$, one could consider that there is enough room for transformations of a set of four set-classes and their transpositions. However Hook [@hook3] (see footnote 9 p.5) has argued against putting all set-classes in a single set on which transformations could be applied because (we paraphrase): 1. [A transformation may not have the same meaning as its inverse, especially for transformations between different set-classes.]{} 2. [Transformations should be well-defined on the whole set of chords (this is the totality requirement for groups).]{} 3. [Some transformations may not have inverses at all.]{} 4. [Different sets of chords may not have the same cardinality and defining transformations between them would be problematic if not impossible. (Note that we differentiate between the cardinality of a set of chords, i.e the number of chords of the same set-class that constitutes the set, and the cardinality of a set-class or chord, i.e the number of pitch-classes that constitutes it.]{} We have seen in the previous part examples of transformations which do not apply on the whole set of M, $\alpha$ and $\beta$ chords, i.e closure is lost. On another level, Cohn’s model of triadic progression involves the transformation between major or minor chords, in each case a set of 12 elements, to augmented triads, a set of 4 elements. In that case these transformations are surjective and therefore have no formal inverse, although in Cohn’s model one can freely choose the major/minor image of a given augmented triad. On the other hand, if we push the reasoning behind Hook’s objections one step further, we could wonder why major and minor chords are considered as a single set. Consider for example the usual neo-Riemannian $P$ operation: if one views this operation as an inversion, it is then an involution, i.e $P^2=1$ meaning that this operation is formally equal to its inverse. However if one considers this operation as a voice-leading transformation, it then corresponds to : 1. [A pitch down in the major-to-minor way.]{} 2. [A pitch up in the minor-to-major way.]{} In this view, the $P$ operation cannot be said to be equal to its inverse. In order to restore coherence in this point of view, one has to consider two different transformations, one from the set of major triads to the set of minor ones, the other one from the set of minor triads to the set of major ones. Notice however we can only do so at the expense of closure: the transformations thus defined only acts on a given set of chords, or in other terms these are partial transformations. The $P$ operation can thus be viewed as a “package” of two partial transformations. Notice that this is not a unique case : the $L$ and $R$ operations can also be viewed as “packaged operators”. Moreover, the usual transposition operators in neo-Riemannian theories actually represents two partial transposition operators which apply respectively to the major and minor chords. In Hook’s notation of UTT (Uniform Triadic Transformations), these correspond to the two operations $<+,1,0>$ and $<+,0,1>$. Since they act very similarly, it is conceivable to package them into a single transposition operator $T_1$. The last section of this paper will provide a link between the construction we introduce next and packaged operators. A categorical construction for musical transformations ====================================================== Construction of the category of transformations by extension ------------------------------------------------------------ In this section, we introduce a construction of musical transformations based on categories rather than groups. Notice that groups are themselves a particular case of categories as they can be viewed as single-object categories, where the morphisms are group elements under the usual composition. The construction we will use is based on a generalization of the construction of group extensions that was introduced in [@popoff]. Recall first that the construction of generalized neo-Riemannian groups of transformations as extensions $$1 \to Z \to G \to H \to 1$$ involves a base-group $Z$ and a shape group $H$. Notice that the $T/I$ group or the $PLR$ group are both isomorphic to the dihedral group $D_{24}$, which is an extension (and more precisely a semidirect product) of $\mathbb{Z}_{12}$ by $\mathbb{Z}_2$. Inspired by this case, we can consider $Z$ as a group of “generalized transpositions”, whereas $H$ can be considered as a group of “formal inversions” between different pitch-class sets. Instead of considering $H$ as a group, we now replace it with a groupoid $\mathcal{H}$. Recall that a groupoid is a category in which every morphism is invertible. Groupoids can be viewed as generalizations of groups in which closure (or totality) has been left out. Indeed, morphisms $g_{XY}$ of the groupoid can be seen as partial transformations between objects $X$ and $Y$. In the rest of the paper, the maps $s$ and $t$ will refer to the source and target maps, i.e $s(g_{XY})=X$ and $t(g_{XY})=Y$. It has been suggested that groupoids are in some cases superior to groups in describing symmetries of objects. For a gentle introduction to groupoids, the reader is invited to refer to [@weinstein] and [@guay]. In our case, the objects of the groupoid $\mathcal{H}$ are the different pitch-class sets and the morphisms are the different formal transformations between these set-classes. Actual transformations of chords usually involve transpositions of the root as is the case for the $L$ or $R$ operation, which we will introduce below. By definition, these transformations are partial and the composition of two morphisms $h_2 \cdot h_1$ is only possible if the codomain of $h_1$ matches the domain of $h_2$. Figure \[fig:cat-H\] give two examples of such groupoids, corresponding respectively to transformations between major (M) and minor (m) chords, and M, $\alpha$, and $\beta$ chords. R We now introduce the definition of a category extension, following the work of Hoff [@hoff1; @hoff2; @hoff3] : Definition : [ ]{} This definition closely follows the group extension one, and the third condition is actually similar to the $Im(I)=Ker(P)$ condition. Hoff has shown that if $\mathcal{H}$ is a category extension as defined above, then $\mathcal{Z}$ is a disjoint union of groups indexed by the objects of $\mathcal{H}$. The category $\mathcal{Z}$ thus plays the role of the transposition operators. We will assume in our case that each pitch-class set can be transposed in the same way, i.e there is a simply transitive group action of $\mathbb{Z}_{n}$ on each set of chords of the same type. The category $\mathcal{Z}$ is therefore built as such 1. [The objects of $\mathcal{Z}$ are the same as in $\mathcal{H}$ and represent the pitch-class sets.]{} 2. [For each object $X \in \mathcal{Z}$, we have $Hom(X,X) \cong \mathbb{Z}_{n}$. We denote a morphism of $X$ as $z^p_{X}$. These morphisms represent transpositions of the individual pitch-class sets.]{} 3. [For any two different objects $X$ and $Y$ of $\mathcal{Z}$, $Hom(X,Y) = \emptyset$.]{} With this knowledge, we see that the third condition in the definition of a category extension has a very concrete meaning from a musical point of view. Consider for example the $L$ and $R$ operations acting on the C major triad. It is clear that the images of this triad under $L$ and $R$ differ by a unique transposition. We thus axiomatize this fact by considering a construction, as a category extension, in which any two switching transformation, or partial inversion transformation, differ only by a unique transposition in the target pitch-class set. The role of the functor $I$ is to introduce the transposition operators in the category $\mathcal{H}$, whereas the functor $P$ classifies in $\mathcal{H}$ the morphisms of $\mathcal{G}$ as transpositions or partial inversions. To sum up, the functors $I$ and $P$ are defined as : 1. [$I$ and $P$ map objects in the natural way.]{} 2. [$I$ maps morphisms $z_{X}^p$ of $\mathcal{Z}$ to equivalent transposition morphisms $z_{X}^p$ in $\mathcal{G}$. By an abuse of terminology, $z_{X}^p$ will designate from now on a transposition morphism of $\mathcal{Z}$ or of $\mathcal{G}$ indifferently.]{} 3. [$P$ maps morphisms $z_{X}^p$ in $\mathcal{G}$ to $id_X$ in $\mathcal{H}$, and morphisms $m_{XY}$ in $\mathcal{G}$ to morphisms $h_{XY}$ in $\mathcal{H}$.]{} As shown by Hoff, the extension construction of $\mathcal{G}$ brings more structure with regards to morphism composition, which will allow us to define actions of $\mathcal{G}$ on sets of objects. Indeed, it can be proved that when the groups $Hom(X,X)$ of $\mathcal{Z}$ are abelian, all category extensions $1 \to \mathcal{Z} \to \mathcal{G} \to \mathcal{H} \to 1$ can be constructed as such : 1. [$\mathcal{G}$ has the same objects as $\mathcal{H}$ or $\mathcal{Z}$.]{} 2. [Morphisms of $\mathcal{G}$ are of the form $(z,h)$, i.e they are indexed by the morphisms from $\mathcal{H}$ or $\mathcal{Z}$, with $z$ being a transposition of the codomain of $h$. ]{} 3. [Composition of two morphisms $g_1=(z_1,h_1)$ and $g_2=(z_2,h_2)$, whenever they are compatible (i.e $s(g_2)=t(g_1)$) is given by the law : $$(z_2,h_2) \cdot (z_1,h_1) = (z_2 \cdot \phi_{h_2}(z_1) \cdot \zeta(h_2,h_1), h_2 \cdot h_1)$$ where $\phi$ is an action of the category $\mathcal{H}$ on $\mathcal{Z}$, and $\zeta$ is a 2-cocycle. ]{} The cohomology theory built by Hoff allows to classify all category extensions based on the second cohomology group $H^2(\mathcal{H},\mathcal{Z})$, with the corresponding 1- and 2-cocycles. We now give the definitions for the terms involved in the composition law of morphism. An action $\phi$ of $\mathcal{H}$ on $\mathcal{Z}$ is a functor $\phi : \mathcal{H} \to \mathbf{Grp}$ where the images of the objects of $\mathcal{H}$ are the groups associated to the corresponding objects in $\mathcal{Z}$, i.e for any object $X \in \mathcal{H}$, $\phi(X) \cong Hom_{\mathcal{Z}}(X,X)$. In other terms, this functor defines homomorphisms between the groups of $\mathcal{Z}$ which are compatible with composition of morphisms in $\mathcal{H}$. Some examples of actions in the case of major and minor chords, or M, $\alpha$ and $\beta$ chords are given in Figure \[fig:HZAction\]. Whereas figure \[subfig:HZAction-Mm\] is reminiscent of the typical transformations which are used in neo-Riemannian theory, Figure \[subfig:HZAction-MAlphaBeta1\] and Figure \[subfig:HZAction-MAlphaBeta2\] show new structures. A 2-cocycle is a function $\zeta: \mathcal{H} \times \mathcal{H} \to \mathcal{Z}$ between two morphisms of $\mathcal{H}$ which outputs a morphism from the appropriate object of $\mathcal{Z}$ such that : $$\phi_{h_3}(\zeta(h_2,h_1)) \cdot \zeta(h_3,h_2 \cdot h_1) = \zeta(h_3,h_2) \cdot \zeta(h_3Ê\cdot h_2, h_1)$$ whenever $h_1$, $h_2$ and $h_3$ are compatible. We see that the terminology used for category extensions is very close to the one used for group extensions. In a similar approach, we will now define the actions of $\mathcal{G}$. Construction of partial actions ------------------------------- A left (right) action of a group $G$ (considered as a single-object category) on a set of chords can be described as a covariant (contravariant) functor $F: G \to \mathbf{Set}$. In particular, it is known that simply transitive left (right) group actions are equivalent to representable functors $F: G \to \mathbf{Set}$, i.e functors which are naturally isomorphic to $Hom(\bullet,-)$ (or $Hom(-,\bullet)$) where $\bullet$ represent the single object of $G$. Recall that in such a case, set elements can be put in bijection with group elements after a particular element has been identified to the identity element in the group. As shown in [@popoff], this allows the determination of group actions, and this also determines a Generalized Interval System (GIS), since Kolman [@kolman] has shown that GIS are equivalent to simply transitive group actions. By analogy, we can build actions of $\mathcal{G}$ on the different sets of chords by using a representable functor $F: \mathcal{G} \to \mathbf{Set}$ and the composition law of morphisms in $\mathcal{G}$. Notice that such a functor has multiple images in $\mathbf{Set}$ (one for each object of $\mathcal{G}$), instead of just one in the case of a group. Therefore, we are actually building partial actions between sets of chords. We now show how to recover the partial actions described in Section 2. The category $\mathcal{Z}$ we use has three objects $M$, $\alpha$ and $\beta$, with $Hom(M,M)=Hom(\alpha,\alpha)=Hom(\beta,\beta)=\mathbb{Z}_{12}$. The category $\mathcal{H}$ has the same objects with the formal inversions $h_{M\alpha}$, $h_{M\beta}$ and $h_{\alpha\beta}$. We build the category extension $\mathcal{G}$ with only an action $\phi$ of $\mathcal{H}$ on $\mathcal{Z}$ and no 2-cocycle. This action is depicted in Figure \[fig:HZAction-Build\]. ![The action of $\mathcal{H}$ on $\mathcal{Z}$ used for building the partial transformations between pitch-class sets M, $\alpha$ and $\beta$. We show here the images of the functor $\phi : \mathcal{H} \to \mathbf{Grp}$. The homomorphisms between groups are represented by their multiplicative action.[]{data-label="fig:HZAction-Build"}](HZAction-Build.pdf) The partial transformations between pitch-class sets $\alpha$ and $\beta$, and $M$ and $\beta$ defined in Section 2 are contextual, and therefore we need a contravariant representable functor. We consider the functor $Hom(-,M): \mathcal{G} \to \mathbf{Set}$. This functor sends 1. [the object $M$ to the set of morphisms $\{ (z_M^n,id_M) \}$ which are identified bijectively with the chords $n_M$.]{} 2. [the object $\alpha$ to the set of morphisms $\{ (z_M^n,h_{\alpha M}) \}$ which are identified bijectively with the chords $n_\alpha$.]{} 3. [the object $\beta$ to the set of morphisms $\{ (z_M^n,h_{\beta M}) \}$ which are identified bijectively with the chords $n_\beta$.]{} To compute the action of a morphism $g \in \mathcal{G}$ on a chord, we thus identify the morphism corresponding to the chord, compose with $m$ on the right and identify the chord of the resulting morphism. For example, the action of $(id_M,h_{\alpha M})$ on a chord $n_m$ results in $$(z_M^n,id_M).(id_M,h_{\alpha M}) = (z_M^n \cdot \phi_{id_M}(id_M) , h_{\alpha M}) = (z_M^n , h_{\alpha M})$$ which corresponds to the chord $n_\alpha$. Similarly, the action of $(id_\alpha,h_{M \alpha})$ on a chord $n_\alpha$ results in $$(z_M^n,h_{\alpha M}).(id_\alpha,h_{M \alpha}) = (z_M^n \cdot \phi_{h_{\alpha M}}(id_\alpha) , id_M) = (z_M^n , id_M)$$ which corresponds to the chord $n_M$. We thus recover the partial action between pitch-class sets $M$ and $\alpha$ described previously. If we consider similarly the action of $(z_M^2,h_{\beta M})$ on a chord $n_M$, we obtain $$(z_M^n,id_M).(z_M^2,h_{\beta M}) = (z_M^n \cdot \phi_{id_M}(z_M^2) , h_{\beta M}) = (z_M^{n+2} , h_{\beta M})$$ which corresponds to the chord $(n+2)_\beta$. If we consider now the action of $(z_\beta^2,h_{M \beta})$ on a chord $n_\beta$, we obtain $$(z_M^n,h_{\beta M}).(z_\beta^2,h_{M \beta}) = (z_M^n \cdot \phi_{h_{\beta M}}(z_\beta^2) , id_M) = (z_M^{n-2} , id_M)$$ since $\phi_{h_{\beta M}}(z_\beta^2) = z_M^{10}$, which corresponds to the chord $(n-2)_\beta$ and we thus recover the partial contextual action between pitch-class sets $M$ and $\beta$. The partial contextual action between pitch-class sets $\alpha$ and $\beta$ can be computed in a similar way. We see here that considering groupoids and their extensions allow for much richer structure than the group extension structure does. In particular, the interplay of group homomorphisms between set-classes, as shown in Figure \[fig:HZAction-Build\] is a way to circumvent the limitations of group extensions when considering the automorphisms of the only group $\mathbb{Z}_{12}$. Forming groups of transformations from category extensions ========================================================== Starting from a groupoid $\mathcal{G}$ of musical transformations defined as a full extension, it is possible to revert back to a group-theoretical description by “packaging” partial operations. Definition : [*A packaged operator is a set of morphisms $O=\{\phi_1,...,\phi_n\}$ from $\mathcal{G}$ (n being the number of objects of $\mathcal{G}$) such that for all objects $i$, $i$ appear only once as the domain of a morphism from $O$, and only once as the codomain of a morphism from $O$.* ]{} Packaged operators can be composed according to : Definition : [*The composition $O_1 \cdot O_2$ of two packaged operators $O_1=\{\phi_1,...,\phi_n\}$, $O_2=\{\phi'_1,...,\phi'_n\}$ is the set of morphisms $\{\phi''_1,...,\phi''_n\}$ obtained by composing all morphisms $\phi_x \cdot \phi'_y$ from $O_1$ and $O_2$ whenever their domain and codomain are compatible. It can be verified that $O_1 \cdot O_2$ is also a packaged operator.* ]{} We then have : Proposition : [*Packaged operators form a group under composition.*]{} Proof : The identity packaged operator is the set of identity morphisms of each object. Closure is given by definition. Associativity is inherited from the category structure. Finally, since $\mathcal{H}$ is a groupoid it is always possible to find inverses for each morphism of a packaged operator, thus giving the inverse packaged operator. $\square$ For example, one can define a packaged transposition operator of the form $T_X=\{z_{X}, id_Y\}$ with $z_X \in \mathcal{G}$, for all objects $Y \ne X$. If we have inversion operators $g_{XY}$ in $\mathcal{G}$, we can also form a packaged inversion operator of the form $I_{XY}=\{g_{XY},g_{YX},id_Z\}$ for all objects $Z \ne X, Z \ne Y$, with $g_{YX}$ being the inverse of $g_{XY}$. The next proposition makes the link between such the group generated by such packaged operators and the wreath products that appeared in the work of Hook and Peck. We assume here that the groups associated to each object of $\mathcal{Z}$ are isomorphic to $\mathbb{Z}_n$. Consider on one hand the set $N$ of all packaged transposition operators $\{T_X\}$, for all objects $X \in \mathcal{G}$. It can readily be seen that $N$ is a group under the composition law defined above, and that it is isomorphic to a direct product of $m$ copies of $\mathbb{Z}_n$, where $m$ is the number of objects of $\mathcal{G}$. Consider on the other hand the set of all packaged inversion operators $K=\{I_{XY}\}$, for all pairs of objects $X$ and $Y$ in $\mathcal{G}$, along with the identity element. It can also be seen that $K$ is isomorphic to the symmetric group $S_m$, since $I_{XY}^2=id$, $I_{XY}I_{WZ}=I_{WZ}I_{XY}$ and $I_{XY} \cdot I_{YZ} \cdot I_{XY} = I_{YZ} \cdot I_{XY} \cdot I_{YZ}$. We then have the following result Proposition : [ *The group G generated by the set $\{N,K\}$ of packaged operators is isomorphic to the wreath product $\mathbb{Z}_n \wr S_m$*]{} Proof : We first show that $N$ is normal in $G$. Let $T_X^p$ be an element of $N$ for some object $X$. If $(g \in G)$ is an element of the form $T_Y^q$ then it is obvious that $T_Y^q.T_X^p.T_Y^{-q} \in N$. If $(g \in G)$ is of the form $I_{YZ}$ for some pair $(Y,Z)$ of objects of $\mathcal{G}$ with $Y \ne X$, $Z \ne X$ then we also have immediately $I_{YZ} \cdot T_X^p \cdot I_{YZ}^{-1} \in N$. In the case $Y=X$, we have $$I_{XZ} \cdot T_X^p \cdot I_{XZ}^{-1} = \{g_{XZ} \cdot z_{X}^p \cdot g_{ZX}, id...\}$$ which also belongs to $N$ since $g_{XZ} \cdot z_{X}^p \cdot g_{ZX} = z_X^q$ for some $q$. If $g \in G$ is a composite element of packaged transpositions and inversions, the relation $g.n.g^{-1} \in N$ holds with the previous results. We also have $NK=G$ by definition and $N \cap K = \{id_1,..., id_m\}$. Since $G$ is not abelian (consider for example $T_X \cdot I_{XY}$ and $I_{XY} \cdot T_X$) this shows that $G$ is a semidirect product of $N$ by $K$. $\square$ It is not necessary to include all transpositions operators for each object $X$, as the composition of one $T_X$ with the packaged inversions will lead to the others. One can check for example that the packaged partial operations defined on M, $\alpha$ and $\beta$ chords in section 2.1, along with the packaged transposition operator $\{n_M \to n+1_M, n_\alpha \to n_\alpha, n_\beta \to n_\beta\}$, generate a group of order 10368 which is isomorphic to $\mathbb{Z}_{12} \wr S_3$. Conclusions =========== We have introduced in this paper a categorical construction for musical transformations based on groupoids which extend the precedent construction based on group extensions. It overcome its inherent limitations, in particular the limited choice of automorphims in the pc-set group. More importantly, this construction allows to define compatible set of partial transformations between pair of set-classes. We also saw how groups of transformations can be recovered from category extensions, based on packaged operators and their composition. While this paper is more mathematical than musical, we hope it will provide foundations for building appropriate groups of transformations in musically-relevant domains. This could be applied for example to cardinality changes between chords (ex. major/minor to seventh chords), a very important problem in music theory as of now. In this paper, we only considered the case of groupoids, and in particular the groupoid $\mathcal{H}$ which assume that partial and reversible transformations between set-classes always exist. As seen in the work of Cohn regarding major/minor and augmented triads, there are cases in music theory where partial transformations may not be reversible at all. It would therefore be interesting to consider category extensions in which $\mathcal{H}$ is a more general category. As well, it could also be interesting to investigate non-abelian category extensions, i.e in which the groups of $\mathcal{Z}$ are non-abelian. \[lastpage\] D. Lewin, [*Generalized Musical Intervals and Transformations*]{}, Yale University Press, New Haven, CT, 1987 R. Cohn, [*An Introduction to Neo-Riemannian Theory: A Survey and Historical Perspective*]{}, Journal of Music Theory 42/2 (1998), pp. 167-180 R. Cohn, [*Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late-Romantic Triadic Progressions*]{}, Music Analysis 15/1 (1996), pp. 9-40 R. Cohn, [*Neo-Riemannian Operations, Parsimonious Trichords, and Their Tonnetz Representations*]{}, Journal of Music Theory 41, pp. 1Ð66 G. Capuzzo, [*Neo-Riemannian Theory and the Analysis of Pop-Rock Music*]{}, Music Theory Spectrum 26/2 (2004), pp. 177Ð200 J. Douthett, [*Flip-flop Circles and Their Groups*]{}, in [*Music Theory and Mathematics: Chords, Collections, and Transformations*]{}, pp. 23-49, Eastman Studies in Music, J. Douthett, M. Hyde, and C. Smith. , eds., University of Rochester Press, 2008. J.N. Straus, [*Contextual-Inversion Spaces*]{}, Journal of Music Theory 55/1 (2011), pp. 43-89 J. Hook, [*Uniform Triadic Transformation*]{}, Journal of Music Theory 46/1-2 (2002), pp. 57-126 J. Hook, [*Signature Transformations*]{}, in [*Music Theory and Mathematics: Chords, Collections, and Transformations*]{}, pp. 137-161, Eastman Studies in Music, J. Douthett, M. Hyde, and C. Smith. , eds., University of Rochester Press, 2008. R. Peck, [*Wreath Products in Transformational Music Theory*]{}, Perspectives of New Music 47/1 (2009), pp. 193-211 R. Peck, [*Imaginary Transformations*]{}, Journal of Mathematics and Music 4/3 (2010), pp. 157-171 A. Popoff, [*Building Generalized Neo-Riemannian Groups of Musical Transformations as Extensions*]{}, submitted A. Childs, [*Moving beyond Neo-Riemannian Triads: Exploring a Transformational Model for Seventh Chords*]{}, Journal of Music Theory 42/2 (1998), pp. 181-193 J. Hook, [*Cross-Type Transformations and the Path Consistency Condition*]{}, Music Theory Spectrum 29/1 (2007), pp. 1-40 G. Mazzola, [*The Topos of Music*]{}, BirkhŠuser, Basel (2002), pp. 257 J. Kochavi, [*Some Structural Features of Contextually-Defined Inversion Operators*]{}, Journal of Music Theory 42 (1998), pp. 307-320 T.M. Fiore, R. Satyendra, [*Generalized Contextual Groups*]{}, Music Theory Online 11(3) (2005). A.S. Crans, T.M. Fiore, R. Satyendra, [*Musical Action of Dihedral Groups*]{}, American Monthly Mathematical, June/July 2009. T.M. Fiore, T. Noll, [*Commuting Groups and the Topos of Triads*]{}, Mathematics and Computation in Music, Third International Conference, MCM 2011, C. Agon, M. Andreatta, G. Assayag, E. Amiot, J. Bresson, J. Mandereau. eds., Springer Lecture Notes in ArtiÞcial Intelligence, 6726 (2011), pp 69-83. A. Weinstein, [*Groupoids: Unifying Internal and External Symmetry*]{}, Notices of the AMS 43/7 (1996), pp. 744-752. A. Guay, B. Hepburn, [*Symmetry and Its Formalism: Mathematical Aspects*]{}, Philosophy of Science 76/2 (2009), pp. 160-178. G. Hoff, [*On the Cohomology of Categories*]{}, Rend. Mat. 7 (1974), pp. 169-192. G. Hoff, [*Extensions Multiples de Categories*]{}, Rend. Mat. 26 (2006), pp. 351-365. G. Hoff, [*Cohomologies et Extensions de Categories*]{}, Math. Scand. 74 (1994), pp. 191-207.

--- abstract: 'The Gao-Wald theorem related to time delay [@gaowald] assumes that the Null Energy Condition and the Null Generic Condition are satisfied, and that the underlying gravity theory is General Relativity. In the present work it is shown that the Gao-Wald theorem is true if the space time is null geodesically complete, if the curvature satisfy some reasonable properties stated along the text, and if every null geodesic contains at least two conjugate points. This result may apply to modified theory of gravities and to violating Null Energy Condition models as well.' author: - 'Juliana Osorio Morales [^1] and Osvaldo P. Santillán [^2]' title: Variations of a theorem due to Gao and Wald --- Introduction ============ Since the introduction of the *Alcubierre bubble* [@alcubierre] or the *Krasnikov tube* [@krasnikov], there has been a growing interest in the concept of time delay in General Relativity as well as in modified theories of gravity. The definition of time delay is indeed very subtle [@olum]. The Alcubierre bubble is a space time in which it is possible to make a round trip from two stars $A$ and $B$ separated by a proper distance $D$ in such a way that a fixed observer at the star $A$ measures the proper time for the trip as less than $2D/c$. In fact, this time can be made arbitrary small. This fact does not indicate that the observers travel faster than light, as they are traveling inside their light cone. The Alcubierre constructions employ the fact that, for two comoving observers in an expanding universe, the rate of change of the proper distance to the proper time may be larger than $c$ or much more smaller, if there is contraction instead of expansion. The Alcubierre space time is Minkowski almost everywhere, except at a bubble around the traveler which endures only for a finite time, designed for making the round trip proper time measured by an observer at the star $A$ as small as possible. Details can be found in [@alcubierre]. The examples given above are interesting, however a more careful definition of time delay was introduced in [@olum]. In this reference, a space time which appears to allow time advance was constructed, but it was proven that it is in fact the flat Minkowski metric in unusual coordinates. This suggests that to analyze time advance by simple inspection of the metric may be misleading. However, given a space time that is Minkowski outside a tube or a bubble such that the Alcubierre or Krasnikov space times, the notion of time delay is well defined. By use of some results due to Tipler and Hawking [@tipler1]-[@tipler3], it can be shown that all these examples violate the Null Energy Conditions at least in some region of the manifold. Further issues related to time delay and quantum gravity can be found in the works [@otras]-[@otras4] and references therein. Recall that the Null Energy Condition states that the matter content energy momentum tensor satisfies $T_{\mu\nu}k^\mu k^\nu\geq 0$ for every null vector $k^\mu$ tangent to any null geodesic $\gamma$. This implies, in the context of General Relativity, that $R_{\mu\nu}k^\mu k^\nu\geq 0$ [@Wald]. On the other hand, the Null Generic Condition means that $k_{[\alpha} R_{\beta]\sigma\delta[\epsilon}k_{\gamma]}k^\sigma k^\delta\neq 0$ for some point in the geodesic $\gamma$. Both conditions automatically imply that any null geodesic $\gamma(\lambda)$ possesses at least a pair of conjugate points $p$ and $q$, if it is past and future inextendible, see [@Wald Proposition 9.3.7]. These results hold in the context of General Relativity, and should not be extrapolated to modified gravity theories without further analysis. The results just described raise the question of whether time delay could hold in theories which do not violate the Null Energy Conditions. In this context, a theorem due to Gao and Wald [@gaowald] may be relevant. Its statement is the following.\ *Gao-Wald theorem:* Consider a null geodesically complete space time ($M$, $g_{\mu\nu}$) such that the Null Energy and Null Generic Conditions are satisfied. Then, given a compact region $K$, there exists a compact $K'$ containing $K$ such that for any pairs of points $p, q\notin K'$ and $q$ belonging to $J_+(p)-I_+(p)$, no causal curve $\gamma$ connecting both points intersects $K$.\ The Gao-Wald theorem stated above is related to time advance hypothesis as follows. If there were possible to deform the geometry in a region $K$, similar perhaps to a bubble, in order to produce a time advance, then a fastest null geodesic would enter in the region $K$ in order to minimize this time. The theorem states that this is not possible if the Null Energy Conditions and Null Generic Conditions are satisfied in the space time in consideration. This may constitute a no go theorem. However, there is no control over the size of the region $K'$, thus this theorem should be considered only as a weak version of a time advance hypothesis. The aim of the present work is two folded. The first purpose is to show that the Null Energy and Null Generic conditions are not mandatory, neither is to work in the context of General Relativity, for the Gao-Wald theorem to be true. It will be shown that the Gao-Wald theorem holds when the following three requirements are satisfied.\ - *First requirement:* The space time ($M$, $g_{\mu\nu}$) is null geodesically complete.\ - *Second requirement:* Every null geodesic possesses at least two conjugate points.\ -*Third requirement:* Consider the set $S$ of pairs $\Lambda_0=$($p_0$, $k_0^\mu$) with $p$ a point in $M$ and $k_\mu$ a null vector in $TM_{p_0}$ properly normalized (see formula (\[norma\]) below) and defining a null geodesic $\gamma_0$. Then there exists an open set $O$ in $S$ containing $\Lambda_0$ for which the following two properties hold. For every pair $\Lambda=$($p$, $k^\mu$) in $O$, the corresponding geodesic $\gamma_\Lambda(\gamma)$ will posses a conjugate point $q$ to $p$, $q \in J_+(p)-I_+(p)$. Furthermore the map $h: O\to M$ such that $h(\Lambda)=q$ is continuous at $\Lambda_0$.\ The two properties described in the *third requirement* look a bit technical, but the intuition behind is the following. The first implies that, for a geodesic with two conjugate points $p_0$ and $q_0$, there exist an open set around $p_0$ such all the points $p$ in the open set will have a conjugate point $q$ with respect to some null geodesic emanating from them. The second part states that the conjugate point $q$ to $p$ will be very close to $q_0$ when $p$ is close to $p_0$ and when the geodesics are, in a very rough sense, “pointing in similar directions”. The second and main purpose of the present work is to prove that the *second requirement* implies the *third one* under some more or less reasonable hypotheses about the curvature of the space time. We feel that this statement may be relevant for extending the Gao-Wald results to more general gravity theories or to models violating the Null Energy Conditions. The organization of the present work is as follows. In section 2 some generalities about conjugate points in generic space times are discussed. In addition, certain topological issues related to the light cones in space times are also presented. The presentation is not exhaustive, but focused in the aspects more relevant for our purposes. At the end, a proof of the *third requirement* when the underlying model is General Relativity with Null Energy and Null Generic Conditions outlined. This is included by completeness, as this is one of the results to be generalized here. In section 3, some properties for the curvature of the space time are presented, which are not related neither to General Relativity nor to the weak and strong energy conditions, ensuring that the *second requirement* implies the *third requirement*. In section 4 the aforementioned implication is proved explicitly by the means of some propositions described in section 3. This section is rather technical. In section 5, the modified Gao-Wald theorem is proved explicitly, and the possible application of the obtained results is discussed. The *third requirement* in GR with Null Energy and Null Generic Conditions ========================================================================== As discussed above, the Gao-Wald theorem relies on the notion of conjugate points. Thus, it is convenient to recall some basic but important concepts about them, taking into account some standard references [@Wald]-[@penrose]. In addition, at the end of this section, a sketch of the proof of the *third requirement* in the context of GR and with Null Energy and Null Generic Conditions [@gaowald] is included. The next sections are devoted to generalize this proof to more general gravity models. Null geodesics and conjugate points ----------------------------------- In the present discussion, the space time ($M$, $g_{\mu\nu}$) is assumed to be null geodesically complete such that there exists a globally defined time like future pointing vector $t_\mu$ on it. Given a point $p$ in ($M$, $g_{\mu\nu}$), a point $q$ in $J_+(p)-I_+(p)$ is said to be conjugated to $p$ if the following holds. Consider a null geodesic $\gamma(\lambda)$ emanating from $p$, together with the associated differential equation =-R\^\_k\^k\^A\^\_, supplemented with the following initial conditions $$A^\mu_\nu|_p=0,\qquad \frac{dA^\mu_\nu}{d\lambda}\bigg|_p=\delta^\mu_\nu.$$ Here $\lambda$ is the affine parameter describing $\gamma(\lambda)$ and $k^\mu$ is a vector tangent to the curve $\gamma(\lambda)$, normalized by the following conditions k\^k\_=0, k\^t\_=-1. The point $q=\gamma(\lambda_0)$ is said to be conjugated to $p$ if and only if $$A_\mu^\nu(\lambda_0)=0.$$ The matrix $A^\mu_\nu(\lambda)$ has the following interpretation: the number $A^{\mu}_{\nu}$ are the coefficients of the Jacobi field $\eta^{\mu}$ along $\gamma$, i.e, $$\eta^\mu(\lambda)=A^\mu_\nu(\lambda) \frac{d\eta^\nu}{d\lambda}\bigg|_0,\qquad \eta(0)|_p=0,$$ then (\[smile\]) implies that $\eta(\lambda)$ satisfies the Jacobi equation (hence the name) on $\gamma$ given by =-R\^\_k\^k\^\^. The classical definition of a conjugate point $q$ to $p$ is the existence of a solution $\eta^\mu(\lambda)$ of the Jacobi equation such that $\eta^\mu(0)=0$ and $\eta^\mu(q)=0$. Clearly, the fact that $A^\mu_\nu(\lambda_0)=0$ implies that $\eta^\mu(q)=0$, thus $q$ is a conjugate point to $p$ in the usual sense. For further details see [@Wald Section 9.3]. There is no warrantee that there exists a point $q$ conjugate to a generic point $p$ for a given space time ($M$, $g_{\mu\nu}$). In addition, there might exist two or more different points $q$ and $s$ conjugate to $p$, joined to $p$ by different geodesics. The study of conjugate points has been proven to have many applications in Riemannian and Minkowski geometry. It is well known that, in Riemannian geometry, a geodesic $\gamma(\lambda)$ starting at a point $p=\gamma(0)$ and ending at a point $r=\gamma(\lambda_0)$ is not necessarily length minimizing if there is a conjugate point $q=\gamma(\lambda_1)$ to $p$ such that $\lambda_1<\lambda_0$. The presence of a conjugate point in the middle usually spoil the minimizing property. For time like geodesics in Minkowski geometries, the proper time elapsed to travel between $p$ and $r$ is not maximal if there is a conjugate point in the middle. For null geodesics, there is an important result which will be used below, see [@Wald Theorem 9.3.8]. Let $\gamma$ a smooth causal curve and let $p, r\in \gamma$. Then there does not exist a smooth one parameter family of causal curves $\gamma_s$ connecting both points, such that $\gamma_0=\gamma$ and such that $\gamma_s$ are time like for $s>0$ if and only if there is no conjugate point $q$ to $p$ in $\gamma$. By reading this statement as a positive affirmation, it is found that if a null curve connecting $p$ and $r$ can be deformed to a time like curve, then there is a pair of conjugate points in between and, conversely, if there is such pair, the curve can be deformed to a time like one. The matrix $A_\mu^\nu(\lambda)$ defined by equation (\[smile\]) takes values which depend on the choice of the null geodesic $\gamma$. For this reason it may be convenient to denote it as $(A_\gamma)^\mu_\nu$. The same follows for the quantity $$G_\gamma(\lambda)=\sqrt{\det A_\gamma(\lambda)},$$ which also vanish at both $p$ and $q$. Note that the initial conditions below (\[smile\]) imply that $\det A_\gamma>0$ until the point $q$ is reached, thus the square root in this definition does not pose a problem. The equation (\[smile\]) implies that $G_\gamma(\lambda)$ satisfies the following second order equation [@gaowald] =-\[\_\^+R\_ k\^k\^\]G\_, and that $G_\gamma(0)=0$ and $G_\gamma(\lambda_0)=0$. These two values correspond to the points $p$ and $q$. Here $\sigma_{\mu\nu}$ is the shear of the null geodesics emanating from $p$. The last is an equation of the form $$\frac{d^2 G_\gamma}{d\lambda^2}=-p_\gamma(\lambda) G_\gamma.$$ As near the point $p$ the initial conditions in (\[smile\]) imply that $A_\mu^\nu\sim \delta_\mu^\nu \lambda$ it follows that, at $\lambda=0$ one has that $$G_\gamma(0)=0, \qquad \frac{dG_\gamma(0)}{d\lambda}=0.$$ Then, if $p_\gamma(\lambda)$ is $C^{\infty}$, by taking derivatives of equation (\[smile2\]) with respect to $\lambda$ it may be shown that $$\frac{d^n G_\gamma(0)}{d\lambda^n}=0.$$ This suggest that $G_\gamma(\lambda)$ may not analytical at the point $\lambda=0$. Another typical equation appearing in the literature [@marolf] is given in terms of the expansion parameter $\theta_\gamma(\lambda)$, which is related to $G_\gamma(\lambda)$ by the formula G\_()=G\_i\_[\_i]{}\^\_() d, with $G_i=G(\lambda_i)$ the value of $\sqrt{\det A_\gamma(\lambda)}$ at generic parameter value $\lambda_i>0$. In terms of $\theta_\gamma$ the equation (\[smile2\]) becomes the well known Raychaudhuri equation +=-\_\^-R\_k\^k\^. The definition (\[folio\]) implies that \_= Thus $\theta_\gamma(\lambda)\to-\infty$ when $\lambda\to \lambda_0$, since $G_\gamma(\lambda)$ approaches to zero from positive values at $q$. Analogously, $\theta_\gamma(\lambda)\to\infty$ when $\lambda\to 0$, since $G_\gamma(\lambda)$ grows from the zero value when starting at $p$. On the other hand, the fact that $\theta_\gamma\to-\infty$ at $q$ itself does not imply that $G_\gamma(\lambda)\to 0$ when $\lambda\to\lambda_0$. This can be seen from (\[folio\]), as the integral of the divergent quantity $\theta_\gamma$ may be still convergent. By an elementary analysis of improper integrals it follows that, at the conjugate point $q=\gamma(\lambda_0)$, the expansion parameter $\theta_\gamma(\lambda)$ is divergent with degree \_()\~,0, up to multiplicative constant. The behavior (\[asin\]) will play an important role in the next sections. Some further remarks are in order. In a generic case, there is a possibility that $\theta=G'/2G$ may be divergent at a point $r$ non conjugate to $p$. In this case $G\neq 0$ but then $G'$ should be divergent. However, the equation (\[smile2\]) implies that $G''$ exists unless there is a singularity of $p_\gamma(\lambda)$. As an example, this may happen due to a some sort of singularity of the scalar $R_{\mu\nu}k^\mu k^\nu$ at $r$. The space times considered in the present work are assumed to be free of these pathologies. This means that $\theta_\gamma(\lambda)$ is well defined everywhere except at $p$ and $q$, where it takes the values $\pm \infty$. In other words, between the points $p$ and $q$ or, what is the same, when $\lambda$ varies in the interval $(0,\;\lambda_0)$, the expansion parameter $\theta_\gamma(\lambda)$ takes every real value. If instead $p$ does not have a conjugate point along $\gamma$, then $\theta_\gamma(\lambda)$ is expected to be finite and continuous for every finite value of $\lambda$. In addition, note that the quantity (\[folio\]) is not well defined when $\lambda_i \to 0$, that is, when the initial point is $p$. This reflects the expansion parameter is singular at $p$. Each of the equations (\[raychaudhuri\]) and (\[smile2\]) have their own advantages. In the following, both versions will play an important role, and will be employed in each situation by convenience. Future light cones in curved space times ---------------------------------------- In addition to conjugate points, another important concept is the future light cone emanating from a point $p$ in the space time ($M$, $g_{\mu\nu}$). Given the point $p$ one has to consider all the future directed null vectors $k^\mu$ in $TM_p$ which satisfy the normalization (\[norma\]). Far away from $p$ it is likely that these geodesics may form a congruence $\gamma_\sigma(\lambda)$, but for $\lambda=0$, the congruence is singular since $\gamma_\sigma(0)=p$ for every value of $\sigma$. In other words, $p$ is the tip of the cone. Close to the point $p$ there is an open set $U$ composed by points $p'$, with their respective set of future directed null vectors $k'^\mu$ in $TM_{p'}$ which satisfy the normalization (\[norma\]). When comparing geodesics emanating from different points $p$ and $p'$, one should compare not only both points but also the corresponding null vectors $k_\mu$ and $k'_\mu$. In some vague sense, two null geodesics $\gamma$ and $\gamma'$ are “close’ when $p$ and $p'$ are at close and the corresponding vectors $k_\mu$ and $k'_\mu$ ”point in similar directions". In order to put this comparison in more formal terms, it is convenient to introduce the set $S$ defined as follows [@gaowald] $$S=\{\Lambda=(p, k^\mu)~|~ p \in M, \quad k^\mu \in TM_p,\quad k^\mu k_\mu=0, \quad k^\mu t_\mu=-1\}.$$ This set has an appropriate topology which allows to compare a pair $\Lambda=(p, k^\mu)$ with another one $\Lambda'=(p', k'^\mu)$ and to determine if they are “close”. The definition implies that the vectors $k^\mu$ are all null and satisfying the normalization (\[norma\]). The null geodesic corresponding to the element $\Lambda=(p, k^\mu)$ will be denoted as $\gamma_\Lambda(\lambda)$ in the following. All the quantities depending on this curve such as $G_\gamma(\lambda)$ will be subsequently denoted as $G_\Lambda(\lambda)$ and so on. The reason for this notational change is the desire to study continuity properties of these quantities as functions on $S$. Proof of the *third requirement* for GR with Null Conditions ------------------------------------------------------------ The proof for the *third requirement* when the underlying theory is GR and the Null Energy and Null Generic Conditions are fulfilled is given in [@gaowald]. As this is one of the theorems to be generalized here, it is convenient to sketch the original argument. Consider a null geodesic $\gamma_0(\lambda)$ with $p_0=\gamma_0(0)$ and $q_0=\gamma_0(\lambda_0)$ conjugate points along it, with $\lambda_0>0$. Then $G_0(0)=G_0(\lambda_0)=0$ and $G_0(\lambda)>0$ for all $\lambda$ in the interval $0<\lambda<\lambda_0$. The Null Energy Condition $T_{\mu\nu}k^\mu k^\nu\geq 0$ implies, in the context of General Relativity, that $R_{\mu\nu}k^\mu k^\nu\geq 0$. This, together with (\[smile2\]) shows that $G_0''(\lambda)<0$ in the interval $0<\lambda<\lambda_0$. The mean value theorem applied to $G_0$ shows that $G_0'(\lambda_1)=-C^2$ for some value $\lambda_1$ in the interval and furthermore $G_0'(\lambda_1)<-C^2$ for $\lambda_1<\lambda<\lambda_0$, with $C^2$ a positive constant. By choosing $\lambda_0-\delta<\lambda<\lambda_0$ one has that $$\frac{G_0(\lambda_1)}{|G'_0(\lambda_1)|}<\delta,$$ since $|G_0'(\lambda)|$ is larger than $C^2$ if $\delta$ is small enough. Consider now a small open $O\subset S$ around the point $\Lambda_0=(p_0, k^\mu)$ generating $\gamma_0(\lambda)$. As $G_\Lambda(\lambda)$ and its derivatives are continuous when moving in this open, then $G'_\Lambda(\lambda)<0$ and $$\frac{G_\Lambda(\lambda_1)}{|G'_\Lambda(\lambda_1)|}<\delta,$$ for all the $\Lambda=(p, k^\mu)\in O$ if $O$ is small enough. As $G_\Lambda(\lambda)>0$ and $G_\Lambda''(\lambda)>0$ due to the Null Energy Condition, it can be shown that for all the points $\Lambda=(p, k^\mu)$ in $O$ one has $G_\Lambda(\lambda')=0$ for some point $\lambda'$ such that $|\lambda'-\lambda_0|<\delta$, see [@gaowald] for further details. This shows that there exists a conjugate point $q$ to $p$, which is close to $q_0$ when $O$ is small enough. This is basically the statement of the *third requirement*. For the purposes of the present work however, the Null Energy and Null Generic conditions are not assumed to hold. This means that it can not be assumed that $G_\Lambda''(\lambda)<0$ neither that mean value theorem applied to $G_\Lambda(\lambda)$, in the form presented above, is true. The following part is devoted to sort out the technical complications arising by relaxing these two important conditions. This effort may be useful, as it may allow to extend the Gao-Wald theorem to models with Averaged Null Energy Conditions or Quantum Null Energy Conditions [@averaged1]-[@averaged25] or to modified gravity theories [@odintsov]-[@odintsov3]. The interest in these conditions arises when considering quantum effects in gravitational models. The assumed properties for the space time $(M,\;g_{\mu\nu})$ ============================================================ The next step is to specify a new set of properties for the null geodesically complete space time ($M$, $g_{\mu\nu}$) which, together with the *second requirement*, lead directly to the *third requirement*. The implication is shown in the next sections. The postulated properties are the following.\ *Property 1:* $(M, \; g_{\mu\nu})$ is null geodesically complete. Every null geodesic $\gamma_\Lambda(\lambda)$ in $(M, \; g_{\mu\nu})$ will contain at least a pair of conjugate points $p$ and $q$, with $q$ in $J_+(p)-I_+(p)$.\ *Property 2:* For any constant $c>0$ the integral I\_(\_i)=\_ \_[\_i]{}\^e\^[-c ]{} \[R\_k\^k\^+\_\^\]\_()d, corresponding to a generic geodesic $\gamma_\Lambda$, is always finite, for every finite value of the initial affine parameter $\lambda_i$. In addition, the integrand is continuous and derivable everywhere in the space time $(M, \;g_{\mu\nu})$.\ The *Property 1* is equivalent to the *first* and *second requirements* stated in the introduction. Note that space time manifold ($M$, $g_{\mu\nu}$) is *not* assumed to be a solution of General Relativity neither matter is assumed to satisfy the Null Energy and Null Generic Conditions. It should be emphasized that these properties are not necessarily the unique properties leading to a Gao-Wald theorem. The task of finding variation may be a relevant one. The *Property 2* looks a bit technical, but it may clarified as follows. It basically suggests that a light traveller measures the quantity $[R_{\mu\nu}k^\mu k^\nu+\sigma_{\mu\nu}\sigma^{\mu\nu}]_\Lambda(\lambda)$, which involves the curvature, and does not find an asymptotic grow of an exponential type, with negative sign. In other words, the behavior $$[R_{\mu\nu}k^\mu k^\nu+\sigma_{\mu\nu}\sigma^{\mu\nu}]_\Lambda(\lambda) \sim- e^{c\lambda},$$ when $\lambda\to \infty$ is forbidden. This is a ad hoc hypothesis, but in authors’s opinion, it is physically reasonable and may cover part of the spectrum of null energy violating models. The main task is to show that properties 1 and 2 imply that ($M$, $g_{\mu\nu}$) satisfy the three requirements described in the previous section and consequently that they lead to a Gao-Wald theorem. There is another reason for considering the quantity (\[tang\]), which is related to a mathematical property of the Raychaudhuri equation. This property is summarized in the propositions of the next subsection. Some mathematical consequences of the Properties 1 and 2 -------------------------------------------------------- The following proposition was used extensively in the works [@galloway1]-[@galloway2]. For the origins of the relation between Riccati inequality and Raychaudhuri equation see [@erlich]. \[prop1\] Given a non linear differential equation of the Riccati form +=p\_(), with an initial condition $\theta(0)=\theta_i$ then, if there exists a constant $c_\Lambda>0$ such that \_i+&lt; \_ \_[0]{}\^e\^[-c\_]{} p\_() d, it follows that a solution $\theta_\Lambda$ does not extend beyond a finite value of the affine parameter $\lambda=\lambda_e$. The equation (\[u\]) may be converted, by the change of variables y()=-(\_+c\_)e\^[-c\_]{},into the following one =+r(),y(0)=y\_0. Here r()=e\^[-c\_]{}(p\_()+),q()=e\^[-c\_]{},and $y_0=-\theta_i-c_\Lambda$. From the definition of $q(\lambda)$ given in (\[erre\]) it is clear that \_[+]{}\_0\^+. Assume that the initial condition is such that \_[+]{} \^\_0 r()d&gt;-y\_0, and that $y(\lambda)$ extend to the whole interval $[0,\infty)$. This will imply a contradiction which will show that this affirmation is false thus, $y(\lambda)$ does not extend beyond a finite affine parameter value $\lambda=\lambda_e$. The same observation will hold for $\theta_\Lambda$, as it is defined in terms of $y(\lambda)$ by (\[y\]). In order to show the aforementioned contradiction, note that (\[cudo\]) implies the existence of affine parameter $\lambda_1$ for which $$\int^\lambda_0 r(\xi)d\xi>-y_0, \qquad \textrm{for}\qquad \lambda>\lambda_1.$$ By integrating the equation (\[teor\]) and taking into account the last inequality, it follows that y()=\_0\^ d+\_0\^r() d+y\_0&gt;\_0\^ d. It is convenient to introduce the quantity given by R()=\_0\^ d. As $q(\lambda)$ is positive and $\lambda>0$, it can directly be seen that that $R(\lambda)\geq 0$, the equality holds only for $\lambda=0$. This definition and the inequality (\[teor2\]) shows that &lt;=, for $\lambda>\lambda_1$. From here it is concluded, for every $\lambda_2>\lambda_1$, that $$\int_{\lambda_2}^{\lambda}\frac{d\xi}{q(\xi)}<\int_{\lambda_2}^{\lambda}\frac{1}{R^2}\frac{dR}{d\lambda}d\xi=\frac{1}{R(\lambda_2)}-\frac{1}{R(\lambda)}<\frac{1}{R(\lambda_2)}.$$ However, by condition (\[cudon\]) it follows that the left hand is not bounded when $\lambda\to \infty$. Thus, the last inequality makes sense only for times $\lambda<\lambda_e$, with $\lambda_e$ a fixed time. This shows that $y(\lambda)$ can not extend to the whole interval $ [0,\infty)$, but only to an interval inside $[0, \lambda_e)$. The same applies for the quantity $\theta_\Lambda$ related to $y(\lambda)$ by $y(\lambda)=-(\theta_\Lambda+c_\Lambda)e^{-c_\Lambda\lambda}$. In terms of $\theta_\Lambda$ the condition (\[cudo\]) becomes $$\theta_0+c_\Lambda< \lim_{\lambda\to\infty} \textrm{inf}\int^\lambda_0 e^{-c_\Lambda\lambda}\bigg(p_\Lambda(\lambda)+\frac{c_\Lambda^2}{2}\bigg)d\xi<\lim_{\lambda\to\infty} \textrm{inf}\int^\lambda_0 e^{-c_\Lambda\lambda}p_\Lambda(\lambda)d\xi+\frac{c_\Lambda}{2},$$ or, equivalently $$\theta_0+\frac{c_\Lambda}{2}<\lim_{\lambda\to\infty} \textrm{inf}\int^\lambda_0 e^{-c_\Lambda\lambda}p_\Lambda(\lambda)d\xi.$$ This is precisely the condition (\[tang2\]), which shows the desired result. Proposition \[prop1\] may apply to the Raychaudhuri equation for $\theta_\Lambda$ with $$p_\Lambda(\lambda)=-[R_{\mu\nu}k^\mu k^\nu+\sigma_{\mu\nu}\sigma^{\mu\nu}]_\Lambda(\lambda).$$ In fact, the right hand of (\[tang2\]) becomes (\[tang\]) under this identification. This point will be elaborated in more detail below. \[theta\_i\] Consider again a null geodesically complete space-time $(M,g_{\mu\nu})$ satisfying Properties 1 and 2. There are a pair of conjugate points along any null geodesic $\gamma_\Lambda(\lambda)$, generically denoted as $p$ and $q$. Then for any positive real number $c_\Lambda>0$, there exists a point $s=\gamma_\Lambda(\lambda_i)$ such that $\theta_i:=\theta_\Lambda(\lambda_i)$ satisfies the following inequality \_i+&lt; \_[+]{}\_[\_i]{}\^e\^[-c\_]{} p\_(). d, Here $p=\gamma_\Lambda(0)$, the value of $\lambda_i$ in the integral in (\[tang3\]) is such that $0<\lambda_{i}<\lambda_{e}$, where $\lambda_e$ is the parameter defined by $q=\gamma_\Lambda(\lambda_e)$. This remark is a direct consequence of the fact that $\theta_{\Lambda}((0,\lambda_{e}))=\mathbb R$, the existence of such $\lambda_{i}$ is guaranteed by continuity. This can be seen as follows. Recall that the scalar expansion $\theta_\Lambda$, is such that $\theta_\Lambda(\lambda)\to -\infty$ when $\lambda\to \lambda_e$, which corresponds to the point $q$. On the other hand, as the integrand function in is continuous and by the Property 2 it is seen that all the integrals $$I(\lambda_i)=\liminf_{\lambda\to+\infty} \int_{\lambda_i}^\lambda e^{-c_\Lambda\xi} [R_{\mu\nu}k^\mu k^\nu+\sigma_{\mu\nu}\sigma^{\mu\nu}]_\Lambda(\xi) d\xi,$$ are bounded from below for every $c_\Lambda>0$ and $\lambda_i$. By considering $I(\lambda_{i})$ as a function of $\lambda_{i}$ with $c_\Lambda>0$ fixed, it is seen that $I(\lambda_{i})$ attains a global minima $I_M$ when $\lambda_i$ varies in the compact interval $[0,\lambda_e]$. This follows from the fact that the function $I(\lambda_i)$ by Property 2 is bounded and the interval just defined is compact. Choose a special value of the parameter $\lambda_i$ such that $$\theta_\Lambda(\lambda_i)+\frac{c_\Lambda}{2}<I_M.$$ This parameter exists as $\theta_\Lambda(\lambda)$ takes every real value continuously in $(0, \lambda_e)$. By the minimality of $I_M$ it is seen that this condition implies (\[tang3\]), and this clarifies the Remark \[theta\_i\].[^3] The next task is to apply the content of Proposition \[prop1\] to the study of conjugate points in the null geodesically complete space time ($M$, $g_{\mu\nu}$). \[prop 2\] Let $(M,g_{\mu\nu})$ be a null geodesically complete space time satisfying the Property 1 and Property 2. Consider a geodesic $\gamma_\Lambda(\lambda)$. Let $q$ be conjugate to $p$ through $\gamma_{\Lambda}$ such that $q\in J_+(p)-I_+(p)$. Then the value of the affine parameter $\lambda_e$ such that $\gamma_\Lambda(\lambda_e)=q$ is given implicitly by the following formula: =\_[\_1]{}\^[\_e]{}{1+}\^2e\^[c]{}d. Here $\lambda_1$ is any value of the affine parameter such that $\lambda_i<\lambda_1<\lambda_e$ and the quantity $R_\Lambda(\lambda)$ is defined in (\[dji\]). The definition of $R_\Lambda(\lambda)$ involves $c_\Lambda$ but the formula (\[explota\]) is universal, that is, does not depends on the choice of $c_\Lambda$, neither on the choice of $\lambda_1$. Without loss of generality, it may be assumed that $\gamma_\Lambda(0)=p$ and $\gamma_\Lambda(\lambda_e)=q$. First, it is convenient to re-write the Raychaudhuri equation in the form . This gives $$y(\lambda)=\int_{\lambda_i}^\lambda \frac{y^2(\xi)}{q(\xi)} d\xi +\int_{\lambda_1}^\lambda r(\xi) d\xi+y_0,$$ with $y(\lambda)$ defined in (\[y\]). From here it is seen that $$\frac{e^{c_\Lambda \lambda}y^2(\lambda)}{2}=\bigg[\int_{\lambda_i}^\lambda \frac{y^2(\xi)}{q(\xi)} d\xi +I(\lambda)\bigg]^2\frac{ e^{c_\Lambda \lambda}}{2} ,$$ with $I(\lambda)=\int_{\lambda_i}^\lambda r(\xi) d\xi+y_0$. The last equation can be expressed in terms of the quantity $R_\Lambda(\lambda)$ defined in (\[dji\]), the result is $$\frac{dR}{d\lambda}=[R_\Lambda(\lambda)+I(\lambda)]^2\frac{e^{c_\Lambda \lambda}}{2}.$$ By dividing this result by $R^2_\Lambda(\lambda)$ and by integrating with respect to $\lambda$ leads to $$\frac{1}{R_\Lambda(\lambda_1)}-\frac{1}{R_\Lambda(\lambda_2)}=\int_{\lambda_1}^{\lambda_2}\bigg[1+\frac{I(\xi)}{R_\Lambda(\xi)}\bigg]^2\frac{e^{c_\Lambda \xi}}{2}d\xi.$$ Here $\lambda_1<\lambda_2$ and both values are in $(0, \lambda_e)$ but otherwise arbitrary. Now, under the working hypothesis, the value $\theta_\Lambda(\lambda)\to-\infty$ as $\lambda\to \lambda_e$ signals the presence of a conjugate point. Then the asymptotic condition (\[asin\]) applies and it is seen from (\[dji\]) that $R_\Lambda(\lambda)\to \infty$ when $\lambda\to \lambda_e$. Then, at $\lambda_2=\lambda_e$, the last expression becomes $$\frac{1}{R_\Lambda(\lambda_1)}=\int_{\lambda_1}^{\lambda_e}\bigg[1+\frac{I(\xi)}{R_\Lambda(\xi)}\bigg]^2\frac{e^{c_\Lambda \xi}}{2}d\xi.$$ Here $\lambda_i<\lambda_1<\lambda_e$ is arbitrary. By expressing the last formula in terms of $\theta_\Lambda(\lambda)$ from (\[y\]) and by taking into account the first (\[erre\]) the expression (\[explota\]) is obtained. Proof of the *third requirement* as a consequence of Properties 1 and 2 ======================================================================= The purpose of this section is to show that Property 1 and Property 2 given above imply the *third requirement*. This leads to the conclusion that the Gao-Wald theorem holds for space times satisfying these properties. Before making formal statements, it is worth to give the intuition behind the proof of *third requirement* given below. One considers a reference null geodesic $\gamma_0(\lambda)$ which, by Property 1, has two conjugate points let’s say $p_0$ and $q_0$. Remark \[theta\_i\] shows that (\[tang2\]) is satisfied for some point in the middle $s_0$. That is, for this point $s_0=\gamma_0(\lambda_i)$ one has $$\theta_0(\lambda_i)+\frac{c_\Lambda}{2}< \lim_{\lambda\to\infty} \text{inf}\int_{\lambda_i}^\lambda e^{-c_\Lambda \xi} p_0(\xi) d\xi.$$ The strategy is to show that, for some geodesics emanating from points $p$ close to $p_0$, the condition (\[tang2\]) is also satisfied for the points $s=\gamma_\Lambda(\lambda_i)$, which are close to $s_0$. That is $$\theta_\Lambda(\lambda_i)+\frac{c_\Lambda}{2}< \lim_{\lambda\to\infty} \text{inf}\int_{\lambda_i}^\lambda e^{-c_\Lambda \xi} p_\Lambda(\xi) d\xi,$$ for all these geodesics. The Proposition \[prop1\] will allow to conclude that $\theta_\Lambda(\lambda)$ also is going to tend to $-\infty$ at a finite value of the affine parameter $\lambda$. Therefore, these points $p$ will have a conjugate point $q$ joined by the null geodesic $\gamma_\Lambda(\lambda)$. This leads to the first part of the *third requirement*. The second part is more subtle. The point is that, by assumption, the reference geodesic $\gamma_0(\lambda)$ has two conjugate points $p_0$ and $q_0$. At these points the determinant $G_0(\lambda)$ defined in (\[smile2\]) vanishes. Thus $G_0(0)=0$ and $G_0(\lambda_e)=0$. By continuity in $S$, one may work by analogy with section 2.3 and try to show that $|G_\Lambda(\lambda_e)|<\epsilon$ if $\Lambda\in O$ with $O$ an open in $S$ small enough. But even taking into account that $G_\Lambda(\lambda)$ has a very small modulus, this does not ensure that it is going to vanish for $|\lambda-\lambda_e|<\delta$. In the present case, as the Null Energy Condition is not assumed valid, it follows that $G''_\Lambda(\lambda)$ is not always negative, and the mean value result of section 2.3 does not work. It may be the case that the value of $G_\Lambda(\lambda)$ is close to zero at $\lambda_e$, then grows very rapidly and only at a value $\lambda=\lambda'_e$ which is very far from $\lambda_e$ will vanish. This would spoil the continuity property of the *third requirement*. Below, it will be proven that this is not the case. The proof relies heavily on the formula (\[explota\]) for the explosion value of the affine parameter $\lambda=\lambda_e$, by considering how it varies when moving along “close geodesics”. In view of this discussion, it is convenient to divide the proof of the *third requirement* in two parts. Proof of the first part of the third requirement ------------------------------------------------ Consider an arbitrary null reference geodesic $\gamma_0$ in the space time ($M$, $g_{\mu\nu}$). The *first requirement* implies that it has two conjugate points $p_0$ and $q_0$, where $q_0$ belongs to $J_+(p_0)-I_+(p_0)$. Without losing generality one may assume that the affine parameter is such that $p_0=\gamma_0(0)$, the value at $q_0$ will be denoted as $\lambda=\lambda_0$. The expansion parameter $\theta_0\to \infty$ at $\lambda\to 0^+$ and $\theta_0\to -\infty$ at $q_0$. As stated before, this leads to the important conclusion that this expansion parameter $\theta_0$ takes every real value when $\lambda$ moves in the interval ($0$, $\lambda_0$), if $q$ is the first conjugate point to $p$. The reference null geodesic $\gamma_0$ is generated by a point $\Lambda_0=$($p_0$, $k_0^\mu$) in $S$. At a given value $\lambda_i$ such $0<\lambda_i<\lambda_0$, a generic value of $\theta(\lambda_i)=\theta_i$ is achieved. Denote the corresponding point in the space time by $s_0=\gamma_0(\lambda_i)$. Consider the map given by[^4] $H(\Lambda, \lambda)=\gamma_{\Lambda}(\lambda)$. For any small enough neighborhood $U$ in $M$ containing $s_0$ there exists an open $O$ in $S$ containing $\Lambda_0$ such that $H(\Lambda, \lambda)$ belongs to $U$ for every point $\Lambda=(p, \;k^\mu)$ in $O$ if $|\lambda-\lambda_i|<\delta$. Here $\gamma_\Lambda(0)=p$. In these terms, one may show the following proposition.\ \[1stpart3rdrequire\] Given a null geodesically complete space time ($M$, $g_{\mu\nu}$) with the Property 1 and 2, consider a geodesic $\gamma_0(\lambda)$ with pair of conjugate points $p_0$ and $q_0$, corresponding to a point $\Lambda_0=(p_0$, $k_0^\mu)$ in $S$. Then there exists an open $O$ in $S$ containing $\Lambda_0$ such that every geodesic $\gamma_\Lambda(\lambda)$ generated by points $\Lambda=(p$, $k^\mu)$ in $O$ possess a conjugate point $q$ to $p$. Before going to the proof, note that the Property 1 already states that every null geodesic contains a pair of conjugate points. But it does not specify where these points are located along the geodesic. The new information this proposition gives is that, once $p_0$ has a conjugate point, then all the points $p$ in a neighbourhood of $p_0$ will have a conjugate point along to some geodesic emanating from them. Choose a compact set $O'$ containing the point $\Lambda_0$ defined in the previous paragraph. Consider a curve generated by a point $\Lambda$ inside the compact set $O'$. By Property 2, the integral $$I(\lambda_s)=\lim_{\lambda\to\infty}\text{inf}\int_{\lambda_s}^\lambda e^{-c_\Lambda\xi} [R_{\mu\nu}k^\mu k^\nu+\sigma_{\mu\nu}\sigma^{\mu\nu}]_\Lambda(\xi) d\xi,$$ is bounded from below for any fixed value $\lambda_s$. Vary the value of $\lambda_s$ in the interval $[0,\;\lambda_0]$, which is compact, and find the smallest value $I_i$, which will be achieved for a specific value $\lambda^\ast_s$. At this point, this procedure mimics the proof of Proposition 2. The strategy however is not doing this with one particular null geodesic, but with all the geodesics $\gamma_\Lambda(\lambda)$ generated by the points $\Lambda$ in $O'$. The resulting minimum for a given geodesic, denoted by $I_{\Lambda}$, is not necessarily a continuous function in $O'$ but, by Property 2, is bounded by below. As $O'$ is compact, there will exists an infimum value at a point $\Lambda_m\in O'$, with its corresponding smallest value $I_{\Lambda_m}$. By construction this value is smaller or equal than the corresponding to any other generic point $\Lambda$ in $O'$. Below, for notational simplicity, this value will be denoted by $I_m$ instead of $I_{\Lambda_m}$. Once the value $I_m$ has ben found, choose a value $\lambda_i$ such that $\theta_0(\lambda_i)+c/2<I_m-\eta$ with $\eta> C>0$. The required value of $\lambda_i$ in $(0,\lambda_0)$ exists since, as discussed above, the expansion parameter $\theta_0(\lambda)$ for the reference geodesic $\gamma_0$ takes every real value when $\lambda$ varies in that closed interval. Then, from the minimality of $I_m$ it is clear that $\theta_0(\lambda_i)+c/2<I_m-\eta$ implies that $$\theta_0(\lambda_i)+\frac{c}{2}< \lim_{\lambda\to\infty} \text{inf}\int_{\lambda_i}^\lambda e^{-c \xi} [R_{\mu\nu}k^\mu k^\nu+\sigma_{\mu\nu}\sigma^{\mu\nu}]_0(\xi) d\xi.$$ In these terms, one may choose an open $O$ inside $O'$ containing $\Lambda_0$, such that for every $\theta_\Lambda(\lambda)$ determined by a point $\Lambda$ in $O$ the inequality $|\theta_\Lambda(\lambda_i)-\theta_0(\lambda_i)|<\epsilon$ holds. The validity of this statement may be seen from the continuity of $G(\lambda, \Lambda)$ and $G'(\lambda, \Lambda)$ in $O$, which implies the continuity of $\theta_\Lambda(\lambda)=G'(\Lambda,\lambda)/G(\Lambda, \lambda)$ with respect to $\Lambda$ if $G(\lambda, \Lambda)\neq 0$, that is, outside a conjugate point. This continuity property follows from the fact that $(A_\Lambda)_\mu^\nu$ satisfies the ordinary equation (\[smile\]), and thus $(A_\Lambda)_\mu^\nu$ and $G_\Lambda(\lambda)$ vary continuously with respect to $\Lambda$ and $\lambda$. Now, for $\epsilon<\eta$, the minimality of $I_m$, together with the fact that $|\theta_\Lambda(\lambda_i)-\theta_0(\lambda_i)|<\epsilon$ and $\theta_0(\lambda_i)+c/2<I_m-\eta$, imply that \_(\_i)+&lt; \_ \_[\_i]{}\^e\^[-c ]{} \[R\_k\^k\^+\_\^\]\_()d, for all the points $\Lambda$ of $O\subset S$. A direct application of this fact and the Proposition 1 implies that for any point $\Lambda=$($p$, $k^\mu$) in the open set $O$, if $\gamma_\Lambda(0)=p$, the solution $\theta_\Lambda(\lambda)$ does not extend beyond a finite value $\lambda_e$ if $\epsilon<\eta$. As discussed below formula (\[asin\]), for the space time in consideration the expansion parameter $\theta_\Lambda(\lambda)$ is continuous everywhere except at the conjugate point. Therefore, the only possibility is that $\theta_\Lambda(\lambda)\to -\infty$ when $\lambda\to \lambda_e$. This implies that at $\lambda_e$ there appears a conjugate point $q$ to $p$. This concludes the proof. Proof of the second part of the third requirement ------------------------------------------------- Before proving the second part, the following observation is needed. In the proof of Proposition \[1stpart3rdrequire\], the initial value $\theta_0(\lambda_i)+c/2<I_m-\eta$ with $\eta> C>0$ has been selected. Choose $\eta>c/2+\epsilon$. From the minimality $I_m$ and from (\[plat\]) it follows that \_(\_i)+c&lt;\_[\_i]{}\^e\^[-c ]{} \[R\_k\^k\^+\_\^\]\_()d, for every value of $\lambda\geq \lambda_i$ and $\Lambda$ in $O$. This particular choice will be useful below. In these terms, the proposition to be proved is the following. Consider a generic point $\Lambda=(p, k^\mu)$ in an open $O\subset S$ containing $\Lambda_0=(p_0, k_0^\mu)$. Under the conditions of the Proposition \[1stpart3rdrequire\], the map $h: O\to M$ defined by $h(\Lambda)=q$, with $q$ the first conjugate point to $p$, is continuous at $\Lambda_0$. As in the proof given before, the geodesic $\Lambda_0$ joins two conjugate points $p_0$ and $q_0$ and, by Proposition \[1stpart3rdrequire\], there is an open $O$ in $S$ containing $\Lambda_0$ such that, for all $\Lambda=(p, k^\mu)$ in $O$, there is a conjugate point $q$. By the proof of Proposition \[1stpart3rdrequire\] and the discussion above, one may find a value $\lambda_i$ such that for all the geodesics defined by every $\Lambda$ in $O$, the inequality (\[inicial\]) holds. Note that the value $\lambda_i$ is strictly larger than zero if the origin is defined such that $\gamma_\Lambda(0)=p$. The length parameter $\lambda_e$ that defines the conjugate point $q_0$ in the geodesic $\gamma_0$ is given by (\[explota\]) which, adapted to the present situation, is given by =\_[\_1]{}\^[\_e]{}{1+}\^2e\^[c]{}d. Here $\lambda_i<\lambda_1<\lambda_e$ is an initial parameter, and p\_0()=\[R\_k\^k\^+\_\^\](),R\_0(\_1)=\_[\_i]{}\^[\_1]{}e\^[c]{}(\_0()+c)\^2 d. The length parameter defining the conjugate point $q$ for another geodesic $\gamma_\Lambda(\lambda)$ is $\lambda_e+\Delta\lambda_e$, and is given by =\_[\_1]{}\^[\_e+\_e]{}{1+}\^2e\^[c]{}d, with p\_()=\[R\_k\^k\^+\_\^\]\_(),R\_(\_1)=\_[\_i]{}\^[\_1]{}e\^[c]{}(\_()+c)\^2 d. Note that $\Delta \lambda_e$ depends on the choice of the geodesic, that is, $\Delta \lambda_e=f(\Lambda)$. This dependence would be implicitly understood in the following reasoning. The task is to show that $|\Delta \lambda_e|=|f(\Lambda))|<\epsilon$ when $\Lambda$ is in an open $O$ of $S$ small enough, containing $\Lambda_0$. A point that might cause confusion is that, in principle, $\lambda_e+\Delta\lambda_e$ may be such that $\lambda_e+\Delta \lambda_e<\lambda_1$, as $\Delta\lambda_e$ may be negative. But the formula (\[mat2\]) is true for the opposite case. Thus, $\lambda_1$ should not be chosen so arbitrary. However, it has been mentioned that, from the continuity of $\theta_\Lambda(\lambda)$ in $S$ outside a conjugate point, one has that $|\theta_\Lambda(\lambda_i)-\theta_0(\lambda_i)|<\epsilon$ if $O$ is small enough. In fact $|\theta_\Lambda(\lambda)-\theta_0(\lambda_i)|<\epsilon$ if $O$ is small and $|\lambda-\lambda_i|\leq \delta$. This means that, up to a point $\lambda=\lambda_i+\delta'$ with $\delta'<\delta$, the value of $\theta_\Lambda(\lambda)$ does not explode. Choose $\lambda_i<\lambda_1<\lambda_i+\delta'$, then $\lambda_e+\Delta\lambda_e>\lambda_1$ for all the curves parameterized by $\Lambda$. Then the mentioned problem does not arise. The actual value of $\lambda_1$ is not known, but it exists, and this is the only thing needed in the following reasoning. The subtraction of both expressions (\[mat\]) and (\[mat2\]) obtained above gives that $$\frac{1}{R_\Lambda(\lambda_1)}-\frac{1}{R_0(\lambda_1)}=\int_{\lambda_1}^{\lambda_e+\Delta \lambda_e}\bigg\{1+\frac{1}{R_\Lambda(\beta)}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \beta})+\int_{\lambda_1}^\beta e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]\bigg\}^2e^{c\beta}d\beta$$ $$-\int_{\lambda_1}^{\lambda_e}\bigg\{1+\frac{1}{R_0(\beta)}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \beta})+\int_{\lambda_i}^\beta e^{-c\xi} p_0(\xi)\bigg] \bigg\}^2e^{c\beta}d\beta.$$ The last expression can be cast in the following form $$\frac{1}{R_\Lambda(\lambda_1)}-\frac{1}{R_0(\lambda_1)}=\int_{\lambda_1}^{\lambda_e}\bigg\{1+\frac{1}{R_\Lambda(\beta)}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \beta})+\int_{\lambda_i}^\beta e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]\bigg\}^2e^{c\beta}d\beta$$ $$-\int_{\lambda_1}^{\lambda_e}\bigg\{1+\frac{1}{R_0(\beta)}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \beta})+\int_{\lambda_i}^\beta e^{-c\xi} p_0(\xi) \bigg]\bigg\}^2e^{c\beta}d\beta$$ +\_e {1+}\^2e\^[c’]{}, where in the last integral the mean value theorem has been employed and $\lambda'$ is some value in the interval $[\lambda_e, \; \lambda_e+\Delta \lambda_e]$. The last formula (\[artdeco\]) already gives an intuition about the intended proof. The condition (\[inicial\]) implies that the term multiplying $\Delta \lambda_e$ in is strictly positive (as $R_\Lambda>0$). It vanish for $\lambda=\lambda_i$ but $\lambda'>\lambda_1>\lambda_i$. Thus, $R_\Lambda(\lambda')$ is never vanishing. On the other hand, one may show by use of analysis methods that the remaining terms are as small as possible by restricting $O$ to be is small enough. Thus $\Delta \lambda_e$ will be also very small, and this will prove the continuity property stated in the Proposition. Roughly speaking, it will imply that a point $q$ conjugated to $p$ is close to a point $q_0$ conjugated to $p_0$. A method to prove this intuition goes as follows. First note that, from the definitions (\[prima\]) and (\[prima2\]) and by use of a mean value theorem, one has that $$\bigg|\frac{1}{R_\Lambda(\lambda_1)}-\frac{1}{R_0(\lambda_1)}\bigg|=\bigg|\frac{R_0(\lambda_1)-R_\Lambda(\lambda_1)}{R_\Lambda(\lambda_1)R_0(\lambda_1)}\bigg|=\frac{1}{R_\Lambda(\lambda_1)R_0(\lambda_1)}\bigg|\int_{\lambda_{i}}^{\lambda_{1}}e^{c\xi}[(\theta_0(\xi)+c)^2-(\theta_\Lambda(\xi)+c)^2]\,d\xi\bigg|$$ =|\_0(’)-\_(’)||\_0(’)+\_(’)+2c|(\_1-\_i). where in the last expression the mean value theorem was applied and the resulting expression was factored. Here the value $\lambda'$ depends on the geodesic, that is, $\lambda'=g(\Lambda)$. The quantity (\[cito\]) can be as small as possible since $\delta\theta(\lambda')=\theta_0(\lambda')-\theta_\Lambda(\lambda')$ goes to zero and the other quantities are under control. To see this clearly, given the open $O'$ in $S$ consider a compact $O_c\subset O'$. Find the minimum value of $R_\Lambda(\lambda_1)$ in this compact, denoted as $R_m$, and the maximum value of $\theta_\Lambda(\lambda)$ in $O_c\times [\lambda_1,\lambda_i]$, denoted as $\theta_m$. Then choose another open $O\subset O_c$. The last expression implies that |-| |\_m||2\_m+2c|(\_1-\_i), for any $\Lambda$ in $O$. Here $\delta\theta_m$ is the maximum value of $\delta \theta(\lambda')$ in $O_c\times [\lambda_i,\lambda_1]$. As this set is compact, the Cantor-Heine theorem allows to conclude that $|\theta_0(\lambda)-\theta_\Lambda(\lambda')|$ is small when $|\lambda-\lambda'|<\delta$ and $O_c$ is small enough, independently on the value of $\lambda'(\Lambda)$. This implies in particular that, by making $O'$ and consequently $O_c$ and $O$ small enough one may chose $\delta \theta_m$ such that $$|\delta \theta_m|< \frac{R_m^2\epsilon}{(\lambda_1-\lambda_i)|2\theta_m+2c|}.$$ in $O$, which implies (\[socia\]) that |-| , for all $\Lambda \in O$. This prove that the left side of (\[artdeco\]) can be made arbitrarily small. On the other hand, the absolute value of the sum of those terms of the right side of (\[artdeco\]) which are independent on $\Delta \lambda_e$ can be converted by use of a mean value theorem into $$I=\bigg|\int_{\lambda_1}^{\lambda_e}\bigg\{1+\frac{1}{R_\Lambda(\beta)}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \beta})+\int_{\lambda_i}^\beta e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]\bigg\}^2e^{c\beta}d\beta$$ $$-\int_{\lambda_1}^{\lambda_e}\bigg\{1+\frac{1}{R_0(\beta)}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \beta})+\int_{\lambda_i}^\beta e^{-c\xi} p_0(\xi) \bigg]\bigg\}^2e^{c\beta}d\beta\bigg|$$ $$=\bigg|(\lambda_e-\lambda_1)\bigg\{1+\frac{1}{R_\Lambda(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]\bigg\}^2e^{c\lambda'}$$ $$-(\lambda_e-\lambda_1)\bigg\{1+\frac{1}{R_0(\lambda')}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_0(\xi) \bigg]\bigg\}^2e^{c\lambda'}\bigg|$$ $$=(\lambda_e-\lambda_1)e^{2c\lambda'}\bigg|\bigg\{\frac{1}{R_\Lambda(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]$$ $$-\frac{1}{R_0(\lambda')}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_0(\xi) \bigg]\bigg\}$$ $$\times \bigg\{2+\frac{1}{R_\Lambda(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]$$ $$+\frac{1}{R_0(\lambda')}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_0(\xi) \bigg]\bigg\}\bigg|.$$ $$\leq (\lambda_e-\lambda_1)e^{2c\lambda_e} M \bigg|\bigg\{\frac{1}{R_\Lambda(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]$$ -}| In the last expression $\lambda'$ is some value in the interval $[\lambda_1,\lambda_e]$, which comes from the mean value theorem for integrals, and $M$ denotes the maximum defined by $$M=\text{Max}\bigg|\bigg\{2+\frac{1}{R_\Lambda(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]$$ $$+\frac{1}{R_0(\lambda')}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_0(\xi) \bigg]\bigg\}\bigg|_{O_1},$$ in an initial compact set $O_1\times [\lambda_1,\lambda_e]$. This maximum exists since the function $R_\lambda(\lambda)$ is never zero for $\lambda$ in $[\lambda_1,\lambda_e]$ and the integrals are convergent since the space time $(M, g_{\mu\nu})$ in consideration is such that $p_\Lambda(\lambda)=[R_{\mu\nu}k^\mu k^\nu+\sigma_{\mu\nu}\sigma^{\mu\nu}]_\Lambda(\lambda)$ is not divergent at finite values of $\lambda$. The expression inside the brackets in the last step in (\[last\]) is then $$D_\Lambda=\frac{1}{R_\Lambda(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]$$ $$-\frac{1}{R_0(\lambda')}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_0(\xi) \bigg],$$ where the denomination $D_\Lambda$ is chosen in order to emphasize that it represents a difference. Write this quantity as $$D_\Lambda=\bigg(\frac{1}{R_\Lambda(\lambda')}-\frac{1}{R_0(\lambda')}\bigg)\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]$$ $$+\frac{1}{R_0(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]$$ $$-\frac{1}{R_0(\lambda')}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_0(\xi) \bigg]$$ By taking into account (\[andros\]), it is already seen that the last quantity is bounded since the integral is finite. In any case, by simplifying some terms, the last expression becomes $$D_\Lambda=\frac{1}{R_0(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)+\theta_0(\lambda_i)+\int_{\lambda_1}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi-\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_0(\xi) \bigg]\bigg\}$$ $$+\bigg(\frac{1}{R_\Lambda(\lambda')}-\frac{1}{R_0(\lambda')}\bigg)\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg],$$ From the continuity of $\theta_\Lambda(\lambda)$ it follows that $|\theta_\Lambda(\lambda_i)-\theta_0(\lambda_i)|\leq \epsilon'/4$ by choosing a suitable open $O\subset O_1$ containing $\Lambda_0$. In addition, one has that $$\bigg|\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi-\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_0(\xi)d\xi \bigg|\leq (\lambda_e-\lambda_i)\delta p^M_\Lambda,$$ where $\delta p^M_\Lambda$ denote the maximum value of $\delta p_\Lambda=e^{-c\lambda_i}(p_\Lambda(\lambda)-p_0(\lambda))$. Again, by a Cantor-Heine argument it can be seen that this maximum can be made arbitrarily small by making $O_1$ small enough. In particular, it can be made smaller than $\epsilon'/4(\lambda_e-\lambda_1)$. The minimum value of $R_0(\lambda')$ is $R_0(\lambda_1)$, as (\[prima2\]) shows that this quantity is monotone increasingly with $\lambda$. Thus by selecting $\epsilon'=\epsilon R_0(\lambda_1)$ one has that |D\_|+|(-)|. As the integrand in the last expression is finite for every curve $\gamma_\Lambda(\lambda)$ then it has a maximum $M'$ in the factors in parenthesis in the right side in the compact $C=O_1\times[\lambda_1,\lambda_e] $. By choosing $O_1$ and consequently $O$ small enough one may use (\[andros\]) in order to prove that $$\bigg|\frac{1}{R_\Lambda(\lambda_1)}-\frac{1}{R_0(\lambda_1)}\bigg| \leq\widetilde{\epsilon}.$$ By choosing $\widetilde{\epsilon}=\epsilon/2 M'$ it is seen from (\[last2\]) that |D\_|, for $\Lambda$ inside $O$. Now, the inequality (\[last\]) implies that $$I\leq (\lambda_e-\lambda_1)e^{2c\lambda_e} M |D_\Lambda|.$$ By use of (\[epsi\]) it is clear that, by choosing $O_1$ and thus $O$ small enough then $|D_\Lambda|\leq \epsilon/M(\lambda_e-\lambda_1)e^{2c\lambda_e} $, and this implies that I. In view of this discussion, consider again (\[artdeco\]). It is clear that it can be written as $$\frac{1}{R_\Lambda(\lambda_1)}-\frac{1}{R_0(\lambda_1)}=\pm I+\Delta \lambda_e \bigg\{1+\frac{1}{R_\Lambda(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\lambda')d\xi\bigg]\bigg\}^2e^{c\lambda'},$$ where the $\pm$ is to indicate that in the definition (\[last\]) of $I$, the absolute value has been taken. The sign of the quantity inside the modulus is not known, thus the identity holds with a plus or a minus sign, which will be not relevant in the following discussion. From the last expression and by taking into account (\[ep\]) and (\[andros\]), it is concluded that, by choosing a very small open $O$ around $\Lambda_0$, one has $$\bigg|\Delta \lambda_e \bigg\{1+\frac{1}{R_\Lambda(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\lambda')d\xi\bigg]\bigg\}^2e^{c\lambda'}\bigg|\leq \epsilon_1+\epsilon_2,$$ where both $\epsilon_i$ comes either from $I$ in (\[ep\]) or either from (\[andros\]). The quantity multiplying $\Delta \lambda_e$ is ensured to be positive by the choice (\[inicial\]) employed. In addition, it has a maximum in a compact $O'$ inside $O$ containing $\Lambda_0$, called $M_\Delta$. Find an open $O_2$ inside $O'$ containing $\Lambda_0$ such that $\epsilon_i\leq \epsilon/2 M_\Delta e^{c\lambda_e}$. Then $$|\Delta \lambda_e(\Lambda)|\leq \epsilon,$$ for every $\Lambda$ in $O_2$. Thus $\Delta \lambda_e(\Lambda)\to 0$ continuously as $\Lambda\to \Lambda_0$, which shows the desired result. A modified Gao-Wald theorem =========================== After proving that the Properties 1 and 2 described in section 3 imply that the *second requirement* and the *third requirement* stated in section 1 are satisfied, the next task is to show that the Gao-Wald theorem [@gaowald] is true when these properties are satisfied. Let $(M,g_{\mu\nu})$ be a null geodesically complete space time such that the Property 1 and Property 2 (given in section 3) are satisfied. Then, given a compact region $K$ in $M$ there exists a compact $K'$ containing $K$ such that, for any two points $p, q\notin K'$ and $q$ belonging to $J_+(p)-I_+(p)$, no causal curve $\gamma$ joining $p$ with $q$ can intersect $K$. As the space time manifold $M$ is assumed to be paracompact, it can be made into a Riemannian manifold with Riemannian metric $q_{\mu\nu}$. This metric can be assumed to be complete by multiplying it by a conformal factor if necessary [@Hicks]. Fix a point $r \in M$ and let $d_{r}:M\to\mathbb R$, $d_{r}(s)$ denotes the geodesic distance between $r$ and $s$ using the metric $q_{\mu\nu}$. This function is continuos in $M$ and for all $R>0$ the set $B_{R}=\{p\in M:~ d_{r}(p)\leq R\}$ is compact (see [@Hicks Theorem 15]). In these terms, given $\Lambda \in S$ let $\gamma_{\Lambda}$ a null geodesic determined by $\Lambda$, let’s define the function $f:S\to \mathbb R$ by: $$\begin{aligned} f(\Lambda) =& \inf_{R}\{B_{R}\text{ contains a connected segment of $\gamma_{\Lambda}$ that includes the initial point }\\ &\quad \quad \text{determined by $\Lambda$ together with a pair of conjugate points of $\gamma_{\Lambda}$}\}\end{aligned}$$ The function $f(\Lambda)$ is upper semicontinuous [@gaowald], when the *second requirement* and the *third requirement* are satisfied, the proof has been given in the reference [@gaowald]. Let $K\subset M$ be a compact set. Let $S_{K}=\{(p,k^{\mu}\in S \text{ with } p \in K)\}$, since the tangent bundle has the product topology, $K$ is compact and $k^{\mu}$ is of bounded norm, then $S_{K}$ is compact. Since $f$ is upper-semicontinuous, it must achieve a maximum in $S_{K}$, let’s denote it by $\bar R$. Let $K' = B_{\bar R}$. Let $p,q\notin K'$ and $q\in J_+(p)-I_+(p)$ and let $\gamma$ a causal curve joining $p$ with $q$, then $\gamma$ must be a null geodesic since $q\in J_+(p)-I_+(p)$. However, the Proposition \[temporal\] insures that $\gamma$ does not contain a pair of conjugate points between $p$ and $q$. If $\gamma\cap K\neq \emptyset$ then by the definition of $K'$, $\gamma$ must have a pair of conjugate points lying in $K'$ and in between $p$ and $q$. This contradiction completes the proof. Note that the theorem given above is true if Properties 1 and 2 are replaced by the *first*, *second* and *third requirements* given in the introduction, as the proof would be unchanged. In brief, in the present work it has been shown that the Gao-Wald theorem holds when the space time ($M$, $g_{\mu\nu}$) is null geodesically complete, every null geodesic posses at least a pair of conjugate points, and the curvature is such that a the quantity $R_{\mu\nu}k^\mu k^\nu\neq -e^{c\lambda}$ for large $\lambda$ values, when evaluated on a null geodesic. Therefore this result may apply to models which violate the Null Energy Condition [@averaged1]-[@averaged25] or to modified gravity theories such as the ones described in [@odintsov]-[@odintsov3]. This deserves further attention. In our opinion, the curvature condition introduced in the text may hold for several interesting solutions for these models. 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[^2]: Instituto de Matemática Luis Santaló (IMAS), UBA CONICET, Buenos Aires, Argentina firenzecita@hotmail.com and osantil@dm.uba.ar. [^3]: Note that the remark \[theta\_i\] is in agreement with Proposition \[prop1\], as $\theta_\Lambda(\lambda)$ is expected to explode at affine parameter value $\lambda_e$. [^4]: This map is defined such that $H: O\times [0,\lambda_0]\to M$ where $O$ is an open in $S$.

--- abstract: 'We investigate the relaxation process of ferromagnetic domains in 2D subjected to the influence of both, static disorder of variable strength and weak interactions. The domains are represented by a two species bosonic mixture of $^{87}$Rb ultracold atoms, such that initially each specie lies on left and right halves of a square lattice. The dynamics of the double domain is followed by describing the two-component superfluid, at mean field level, through the time dependent Gross-Pitaevskii coupled equations, considering values of the intra and inter-species interaction, reachable in current experimental setups, that guaranty miscibility of the components. A robust analysis for several values inter-species interaction leads us to conclude that the presence of structural disorder leads to slowdown the relaxation process of the initial ferromagnetic order. As shown by our numerical experiments, magnetization is maintained up to 60 percent of its initial value for the largest disorder amplitude.' author: - 'C. Madroñero' - 'G.A. Domínguez-Castro' - 'L. A. González-García' - 'R. Paredes' title: Relaxation of ferromagnetic domains in a disordered lattice in 2D --- Introduction {#intro} ============ As much of the many-body problems within the condensed matter field, magnetism and particularly the dynamics of microscopic spins lying in ultrathin films in definite regions of space, the so-called magnetic domains dynamics, remain until now as an open question [@Himpsel; @Kim; @Thomson]. The origin of this dynamics can be attributed to several factors, for instance, the presence of external drivings as magnetic or electric fields, the existence of spin-polarized currents inducing the transference of momentum to the domain wall [@Thiaville; @Franke], or the inner dynamics associated with both, the interactions between the microscopic constituents as well as the energetic landscape where the constituents move. In the present investigation, we concentrate on analyzing the dynamics of the magnetic domains arising from the last causes. In particular, we focus on the effects of energy disorder in preventing the motion of spin domains. Motivated by the notable control achieved with large conglomerates of atoms in its quantum degenerate state, and particularly the production of mixtures composed of either, Bose condensates in different hyperfine states [@Myatt], or different atomic species [@Modugno; @Thalhammer; @Lercher; @McCarron; @Wang], confined in particular geometries [@Hinds; @Grimm; @Lewenstein; @Gross; @LaRooji], we propose here the design of an [*ultracold atom device*]{} to quantum simulate the decay of magnetization in magnetic domains in disordered square lattices in 2D. Our proposal is based on various experimental situations, previously performed with $^{87}$Rb atoms, intended to explore the many-body localization phenomenon [@Choi; @IBloch]. In particular, in [@Choi], the initial state prepared is a Bose condensate composed of about one hundred atoms confined in a 2D square lattice in its Mott equilibrium state, and then allowed to evolve in a disordered potential under its own dynamics after suddenly changing an external parameter. Such a quantum quench protocol planned to track the effects of disorder on the atom flux moving across the 2D lattice, together with the possibility of spatially separating different hyperfine components, are the basis of our proposal to study the dynamics of the ferromagnetic domains, particularly its magnetization decay. As we describe below, in this work we shall consider a two-species Bose condensate as the analog of a double spin domain in which each hyperfine component lies in the halves of an inhomogeneous square lattice, thus setting initial configuration that will evolve in a disordered media (see Fig. \[Figure1\]). This arrangement together with a recent study, performed at mean field level, where the effect of disorder is to induce the emergence of spatially localized densities, as a function of the disorder magnitude [@Gonzalez], are our starting point to study the dynamics of the double spin domain. The idea here presented can be extended to other geometrical configurations, like those considered in [@Gonzalez] that can be useful for practical purposes as for instance the design of spin based magnetic protocols with complex configurations. Here we present the results of an extensive set of numerical calculations performed, at the mean-field level through the coupled Gross-Pitaevskii (GP) equations, to describe the evolution in time of the hyperfine spin components spatially separated at $t=0$, and then allowed to evolve under the influence of non-correlated static disorder. Working within the superfluid regime, which is considering values of the intra-species interaction coupling for which the system is far from the Mott insulating phases (MI), we analyze the evolution of the initial state for different values of the ration among intra and inter-species interaction strengths. This work is organized as follows. In section 2, we present the model that we use to describe the relaxation of the ferromagnetic domains under the influence of disorder. Furthermore, we briefly explain the construction of the initial state from which the evolution in time is followed. In section 3 we show the results of our numerical study about the relaxation process of the ferromagnetic domains, as a function of the disorder amplitude and different interaction strengths. Finally, in section 4, we summarize our findings. Model and initial state preparation {#section2} =================================== The model here proposed to study the persistence of magnetization in definite regions of space, is based, as described previously, on a series of experimental designs created with ultracold $^{87}$Rb atoms confined in 2D optical lattices, and their remarkable attribute of generating localized states as a result of both, disorder and two-body interactions. Here we concentrate on weakly interacting systems subjected to disorder. The system under study consists of a mixture of two hyperfine spin components, $|\uparrow \rangle= |F=1,m_F=-1\rangle$ and $|\downarrow \rangle=|F=2,m_F=-2\rangle$, lying in a 2D inhomogeneous square lattice, represented by $V_{ \mathrm{ext}}\left(\vec {r}\right)$. Within the mean-field formalism the wave functions $\Psi_{\uparrow,\downarrow}$ of the two species $|\uparrow \rangle$ and $|\downarrow \rangle$ obey the following effective coupled GP equations: $$\begin{aligned} i\hbar \frac { \partial \Psi _{\uparrow} (\vec {r},t)}{ \partial t } =\left[ H_0(\vec {r}) + g_{\uparrow\uparrow}|\Psi_{\uparrow}|^{2} + g_{\uparrow\downarrow}|\Psi_{\downarrow}|^{2} \right] \Psi_{\uparrow}(\vec{r} ,t)\cr i\hbar \frac { \partial \Psi _{\downarrow} (\vec {r},t)}{ \partial t } =\left[ H_0(\vec {r}) + g_{\downarrow\downarrow}|\Psi_{\downarrow}|^{2} + g_{\downarrow\uparrow}|\Psi_{\uparrow}|^{2} \right] \Psi_{\downarrow}(\vec{r} ,t), \label{coupledGP}\end{aligned}$$ where $H_0(\vec {r})= -\frac { \hbar^ 2 }{2m} \nabla_{\perp}^ 2 +V_{ \mathrm{ext}}\left(\vec {r}\right)$ with $\nabla_\perp^2=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}$ is the Laplacian operator in $2$D and $m$ the equal mass of the two spin components. The external potential in 2D has the following form: $$\begin{aligned} V_{ \mathrm{ext}}\left(\vec {r}\right)= \frac{1}{2} m \omega_r^2 r^2+V_{0}^\delta \Bigg[ \sin^2 \left({\frac{\pi x}{a}}\right)+ \sin^2 \left({\frac{\pi y}{a}}\right) \Bigg],\end{aligned}$$ being $\vec {r}= x \hat i +y \hat j$, $\omega_{r}$ the radial harmonic frequency, which is fixed to a common value used in current experiments $\omega_{r} = 2\pi\times 50$ Hz, $a$ is the lattice constant and $V_{0}^\delta= V_{0}(1+{\epsilon_\delta(x,y)})$, the potential depth at each point $(x,y)$. The function $\epsilon_{\delta} (x, y)$ represents a non-correlated disorder spanned across space and takes random values in the interval $\epsilon_{\delta} (x, y) \in [-\delta,\delta]$, being $\delta$ the disorder amplitude $\delta \in [0,1]$. The random depth $V_{0}^\delta$ mimics the disordered environment introduced by speckle patterns [@Bouyer] and is scaled in units of the recoil energy $E_R= \frac{\hbar^2 k^2}{2m}$, with $k=\pi / a$. Thus, besides the contribution of the harmonic confinement, the potential depth at each point $(x,y)$ is the result of adding/subtracting a random number $\epsilon_{\delta}(x,y)$ to the amplitude of the potential defining the square lattice at zero disorder. Several previous studies have shown that the mean-field approximation describes the main effects of weakly interacting disordered systems [@Ray; @Schulte; @Adhikari; @Kobayashi; @Gonzalez]. The values of the effective interaction couplings $g_{\sigma\sigma'}$ with $\sigma, \sigma' = \{ \uparrow, \downarrow \}$ are written in terms of the s-wave scattering length $a_{\sigma, \sigma'}$ as, $g_{\sigma\sigma'}= 4\pi N\hbar^{2}a_{\sigma\sigma'}/m$, being $N$ the number of particles in the condensate. We should point here that these interaction coefficients must be substituted by effective interaction couplings that take into account that the atom collision processes occur in 2D [@U2D; @Salasnich; @Mateo; @Bao; @Trallero; @Zamora; @U2D2]. The effective scattering length in the plane $x-y$ becomes $a_{\sigma\sigma'} \rightarrow a_{\sigma\sigma'}/\sqrt{2 \pi} l_z$, with $l_z=\sqrt{\hbar/m \omega_z}$, being $\omega_z$ a typical frequency of condensates confined in 2D [@Hadzibabic; @Lung-Hung]. In typical experiments the values of the coupling constants $g_{\sigma\sigma'}$, can be varied via Feshbach resonances, and thus adjusted to have either, equal or different values of the intra and inter-species interactions, that is $g_{\uparrow \uparrow}= g_{\downarrow \downarrow}= g_{\uparrow \downarrow}$, or $g_{\uparrow \uparrow}=g_{\downarrow \downarrow} \neq g_{\uparrow \downarrow}$. In the present investigation we consider $g_{\uparrow \uparrow}= g_{\downarrow \downarrow}$ and $g_{\uparrow \downarrow}= g_{\downarrow \uparrow}$. As stated in [@Wang; @Papp], miscibility of the two component mixture is determined by the relation between $g_{\uparrow \uparrow}$, $g_{\downarrow \downarrow}$ and $g_{\uparrow \downarrow}$, as a matter of fact, the separation of the hyperfine components happens when the condition $g_{\uparrow \downarrow} > \sqrt{g_{\uparrow \uparrow} g_{\downarrow \downarrow}}$ is satisfied. In our analysis we shall consider values of the intra and inter-species interaction that guaranty miscibility of the hyperfine components. ![Schematic form of the density profile prepared as the initial state. Left and right sides represent the superfluid density associated with the hyperfine components $\uparrow$ and $\downarrow$. Such profiles correspond to densities at zero disorder amplitude and a given value of the intra and inter-species interactions amplitudes $g_{\uparrow \uparrow}= g_{\downarrow \downarrow}$, and $g_{\uparrow \downarrow}$.[]{data-label="Figure1"}](Figure1.pdf){width="8cm" height="4cm"} Initial ferromagnetic state --------------------------- To set the initial state from which we shall follow the dynamics to track the demagnetization process, we first solve the coupled equations (\[coupledGP\]) for an optical lattice with lattice spacing $a = 532$ nm and a depth $V_{0}/E_R = 4$ without disorder, that is $\delta = 0$ and determine the stationary state. For this purpose free energy minimization is performed by means of imaginary time evolution $\tau\rightarrow it$ [@Calculations; @Dum; @Zhang]. After this procedure, we manually remove the particles having spin component $\sigma =\uparrow$ from the left half layer, while particles with $\sigma=\downarrow$ from the right half layer, see Fig.\[Figure1\]. This removal of particles mimics experimental procedures in which a digital mirror device is used to optically remove the particles at specific positions [@Choi]. Another route to achieve ferromagnetic domains is by means of a magnetic field [@Weld]. Notice that in our case similar ground state densities of different hyperfine states remain at each half in the 2D lattice. This pattern manually created, that is the two ferromagnetic domains, is our starting point to study its time evolution under the influence of static non-correlated disorder. We should note here that such an initial state is non-stationary, and consequently it evolves under their own dynamics. Interestingly, this kind of states from which a system evolves under its own dynamics are the so called quantum quenches created in the laboratory. Our particular interest is to investigate how the local magnetization of the ferromagnetic domains degrades when the weakly interacting 2D Bose mixture evolves in the absence of other external fields, except the one produced from the combination of a speckle pattern and the square lattice. It is important to mention that the effective coupling interaction coefficients must be rescaled for $t>0$ since half of the population is removed to have the magnetic domains, that is $N \rightarrow N/2$. Results: Demagnetization vs. disorder {#section3} ===================================== With the purpose of establishing how magnetic domains demagnetize, we follow the time dynamics of an initial state prepared as described in the previous section, namely, spatially separated populations of the hyperfine components $\downarrow$ and $\uparrow$ lying in left and right sides of the 2D square lattice respectively. In Fig.\[Figure1\] we show a schematic plot of the initial state. In our simulations we consider $N=300$ giving rise to interaction strengths $g_{\uparrow\uparrow}= g_{\downarrow\downarrow} = 10$, and lattices having $\sim 30 \times 30$ occupied sites. As stated in section \[section2\], the non-correlated disorder is introduced across the whole lattice through the function $\epsilon_{\delta}(x,y)$. To perform a reliable analysis of the physical quantities and have meaningful predictions, we take the average over an ensemble of 200 realizations for each value of the disorder amplitude $\delta$, and given values of the ratio among intra and inter-species interactions $g_{\uparrow \uparrow}/g_{\downarrow\uparrow}$. The purpose behind considering multiple realizations of disorder for a given value of the disorder strength, is to recreate the situation of experiments in which, typically, a single random realization could be not sufficient to represent the usual behavior of multiple scattering events due to uncorrelated disorder. We should notice that the way in which the disorder has been simulated, warrants that although the lattice symmetry is altered, the underlying structure is preserved, that is, the square geometry and the harmonic confinement prevail. Also, we must point out that the initial state depends on the particular values that the intra and inter-species interactions have. The observables to be studied in our analysis are the magnetization in left and right sides $m_L$ and $m_R$ as a function of time. These quantities are defined in terms of the local magnetization $m(x,y;t) = \rho_{\uparrow}(x,y;t)-\rho_{\downarrow}(x,y;t)$, where $\rho_{\uparrow}(x,y;t)$ and $\rho_{\downarrow}(x,y;t)$ are the densities associated with the components $\uparrow$ and $\downarrow$ respectively. Thus, magnetization in left and right sides are, $$\begin{aligned} m_L=\int \int_{\Omega_L} dx \ dy \> m(x,y;t)\cr m_R=\int \int_{\Omega_R} dx \ dy \> m(x,y;t),\end{aligned}$$ where $\Omega_R$ and $\Omega_R$ are the left and right halves of the system, respectively. Because of the particular election of the initial state we have that $m_L(t=0) =-0.5$ and $m_R(t=0)=0.5$. For our analysis, besides the set of random realizations $\epsilon_{\delta}(x,y)$ for a given disorder magnitude, we shall consider three different values of ratio $g_{\uparrow \uparrow}/ g_{\downarrow\uparrow}$. We shall identify the dimensionless time in all of our calculations as $\tau= E_R t/\hbar$. The time dynamics was followed for a period of time such that at zero disorder and a given value of the coupling interactions the magnetization in left and right sides become null. As we shall see, the elapsed time during which the time dynamics is studied depends on the values of the inter- and intra-particle interaction strengths. That is, the relaxation of magnetization at zero disorder is different for each ratio $g_{\uparrow \downarrow}/g_{\uparrow \uparrow}$ considered. It is important to mention here that all of our numerical calculations were performed ensuring that changing $\tau \rightarrow -\tau$, at any temporal step along the time dynamics, allow us to recover the initial state. Furthermore, we found that the high precision used in our numerical simulations allowed us to observe a kind of macroscopic spin oscillation phenomenon, namely, that in the absence of disorder, the domains oscillate back and forth for extremely long evolution times. ![Snapshots of the local magnetization in the square lattice for three different values of the disorder amplitude. Left, center and right columns correspond to three different values of disorder amplitude $\delta$ as indicated in the figure. Upper, middle and bottom rows are associated to $\tau=0$, $\tau=180$ and $\tau=360$ respectively. The ratio of the intra and inter-species interaction is, $g_{\uparrow \downarrow}= 0.9 g_{\uparrow \uparrow}$.[]{data-label="Fig_density"}](Figure2.pdf){width="10.5cm" height="8cm"} Since the prepared initial state is non-stationary, the hyperfine spin populations will evolve under the influence of both, disorder and interactions. Previous analysis of a single BEC component confined in a disordered square in 2D have shown that, in the weakly interacting regime, the net effect of the disorder is to localize the condensate density in bounded regions [@Gonzalez]. As a matter of fact, the size of those bounded regions become shorter and shorter as the amplitude of the disorder strength is increased. Therefore, what we expect in the case of the two component condensate is to have spatially localized densities of the condensate as the disorder magnitude grows, and thus preservation of magnetic domains. In Fig. \[Fig\_density\] we show snapshots of the local magnetization for three different values of the disorder amplitude, $\delta=0$, $\delta= 0.5$ and $\delta=0.9$ (left, center and right columns respectively), and three different times along the dynamics, $\tau=0$, $\tau=180$ and $\tau=360$. Each plot is a snapshot associated to a given realization of disorder $\epsilon_{\delta}(x,y)$ and fixed value of the intra and inter-species interaction ratio, $g_{\uparrow \downarrow}/g_{\uparrow \uparrow} = 0.9$. As one can see from this figure, at zero disorder amplitude, $ \uparrow$ and $\downarrow$ density configurations remain exactly opposite, while showing an asymmetric behavior for $\delta \neq 0$. We also observe how larger values of the disorder amplitude leads to a slow down the dynamics of the magnetic domains. That is, the initial state persists for larger times, thus showing a persistence of the magnetic domains during the time evolution. Fig. \[Fig\_density\] is representative of the magnetization behavior observed at different times as disorder is increased for the ensemble of disorder realizations. In the next paragraphs we outline the findings of our numerical experiments. ![Magnetization on the right side as a function of time for different values of disorder amplitude $\delta$, colors indicate the size of such a disorder. The ratio of the intra and inter-species interaction is $g_{\downarrow\uparrow}/g_{\uparrow \uparrow} = 0.9$. Magnetization in the left side is shown in the inset. Each point in these curves is the result of the average over 200 realizations of disorder for a given value $\delta$. The shadow area around each curve corresponds to the root mean square deviation.[]{data-label="Figure2"}](Figure3.pdf){width="7cm" height="7cm"} ![Magnetization in the right side as a function of time for different values of disorder amplitude $\delta$, colors indicate the size of such a disorder. The ratio of the intra and inter-species interaction is $g_{\downarrow\uparrow}=0.8 g_{\uparrow \uparrow}$. Magnetization in the left side is shown in the inset. Each point in these curves is the result of the average over 200 realizations of disorder for a given value $\delta$. The shadow area around each curve corresponds to the root mean square deviation.[]{data-label="Figure3"}](Figure4.pdf){width="7cm" height="7cm"} ![Magnetization on the right side as a function of time for different values of disorder amplitude $\delta$, colors indicate the size of such a disorder. The ratio of the intra and inter-species interaction is $g_{\downarrow\uparrow}=0.7 g_{\uparrow \uparrow}$. Magnetization in the left side is shown in the inset. Each point in these curves is the result of the average over 200 realizations of disorder for a given value $\delta$. The shadow area around each curve corresponds to the root mean square deviation.[]{data-label="Figure4"}](Figure5.pdf){width="7cm" height="7cm"} As described above, to determine the influence of both, disorder and interactions, we track the evolution of the magnetization for an ensemble of realizations for a given value of the disorder amplitude. We restate that the values of the intra and inter-species interaction considered in our study are such that the miscibility of hyperfine components can occur. In Figures \[Figure2\], \[Figure3\] and \[Figure4\] we summarize the results of our analysis. Each plot correspond to the average of the magnetization on the right (main plot) and left (inset) sides as a function of time for different values of disorder amplitude $\delta$. The specific values of $\delta$ are indicated in the figures. The values of the intra and inter-species interaction of figures \[Figure2\], \[Figure3\] and \[Figure4\] are $g_{\downarrow\uparrow}=0.9 g_{\uparrow \uparrow}$, $g_{\downarrow\uparrow}=0.8 g_{\uparrow \uparrow}$ and $g_{\downarrow\uparrow}=0.7 g_{\uparrow \uparrow}$ respectively. Different curves in each plot are the average over the realizations for a given values of $\delta$, being the shadow area around each curve associated with the root mean square deviation. From these figures, one can observe that for short times $\tau \lesssim 50$, the general relaxation behavior of the magnetization takes quite similar values, independent of the disorder strength. However, for later times $\tau \gtrsim 50$, each magnetization curve departs from each other, thus revealing the effects of the disordered media. Furthermore, as the disorder amplitude is increased, the ferromagnetic order in each domain is preserved against the relaxation process. This slow relaxation dynamics induced by disorder enhances memory-like effects in the coupled magnetic domains. In particular, for the largest value of the disorder amplitude considered, that is $\delta = 1$, the value of the magnetization in left and right sides remains unaltered around 60%. One can also notice two main outcomes associated with the value of the ratio $g_{\downarrow \uparrow}/g_{\uparrow \uparrow}$. The first is that at zero disorder the relaxation time is decreased as the ratio $g_{\downarrow \uparrow}/g_{\uparrow \uparrow}$ is diminished. The second one is that as this ratio is increased, the magnetization becomes progressively worse with respect to its initial value. The time dependence of the left and right magnetizations are well described by a power-law ansatz $m_R(\tau)\propto b(\delta)\tau^{\gamma (\delta)}$. We fit the curves of figures \[Figure2\], \[Figure3\] and \[Figure4\] with such a power-law ansatz at intermediate times scales, where one neglects the transient behavior at short times. We should notice that the coefficients $\gamma$ and $b$ also depend upon the ratio among $g_{\downarrow\uparrow}$ and $g_{\uparrow \uparrow}$. In Figure \[Figure5\] we plot the behavior of $\gamma$ as a function of $\delta$. From this figure one can notice how the value of the characteristic exponent $\gamma$ is modified as the ratio $g_{\downarrow\uparrow}/g_{\uparrow \uparrow}$ is varied. Here it is important to stress that $\gamma$ is also influenced by the presence of the harmonic confinement. The value of this exponent is reduced as $\delta$ grows, and thus the relaxation process becomes slower in time. Quite remarkable is the fact that for strong disorder strengths $\delta$, the $\gamma$ parameters take very similar values, almost independent of the ratio of the inter and intra-species interaction coupling. This can indicate that the disorder has become the dominant contribution during the elapsed evolution. The inset in Fig. \[Figure5\] shows the error of the magnetization fit. ![Power-law fit $m_{R}(\tau)\propto b(\delta)\tau^{\gamma (\delta)}$ for the magnetization on the right side as a function of $\delta$ for three different values of the ratio of the intra and inter-species interaction.[]{data-label="Figure5"}](Figure6.pdf){width="8cm" height="5.5cm"} Final remarks {#section5} ============= We have studied the time dynamics of initially localized ferromagnetic domains evolving under the influence of both, disordered confinement and contact interactions. The purpose of such an investigation was to establish the persistence of ferromagnetic order in the domains, namely spatial regions with definite magnetization, when the competition of structural disorder and interactions could lead the system, evolving under their inner dynamics, to nullify such an initial magnetic pattern. To study such a magnetization relaxation process as a function of time, we proposed a model system simulating a double ferromagnetic domain evolving under static disorder. The model consisted of a two-species $^{87}$Rb Bose-Einstein condensate, whose components labeled as $\uparrow$ and $\downarrow$ states, were placed spatially separated, lying each one in the halves of a 2D potential resulting from the superposition of a harmonic potential and a square lattice. The description of the dynamics was addressed within the mean field Gross-Pitaevskii approach, by solving the coupled equations associated to different hyperfine components. To have a reliable analysis of the evolution in time of the magnetization under the presence of disorder, our analysis consisted of an extensive set of numerical calculations over different realizations of non-correlated disorder having a given amplitude $\delta$, and constant values of the intra and inter-species interactions $g_{\downarrow\uparrow}$ and $g_{\uparrow \uparrow}$ respectively. Regarding the magnitude of the intra and inter-species interaction $g_{\downarrow\uparrow}$ and $g_{\uparrow \uparrow}$, we worked in the regime in which the ratio between these coefficients $g_{\downarrow\uparrow}/g_{\uparrow \uparrow}$ guaranty miscibility of hyperfine components, and also considering appropriate coupling interaction strengths away from the strong interaction effects. Our main conclusion is that the relaxation process of a double ferromagnetic domain, that is loss of magnetization in definite regions of space, becomes slower and slower as the structural disorder is increased, while in contrast, increasing the ratio between inter and intra particle contact interactions $g_{\downarrow \uparrow}/g_{\uparrow \uparrow}$ tends to degrade the initial state. We reach these result from a robust study of the time evolution of the right and left magnetization of a quantum system described above. Textures or local magnetization, as referred in current literature, are suitable observables to track the effects of the disorder media in systems having more that one component and, also, are accessible physical quantities with single spin resolution techniques [@Weitenberg; @Boll] used in current experimental setups. The manuscript here presented sets a platform for the design of specific protocols appropriate to study demagnetization processes or frustration effects associated to geometry and energy disorder [@Pierce; @Windsor; @Korzhovska; @Reinld]. Also, our study aims for the investigation of the dynamics induced by measurement in the sense that sources of disorder can be either internal as those here considered, or external as those associated to reservoirs in contact with assessable quantum systems [@Hilary]. We expect that our work will trigger further theoretical analysis as for instance, the long range character proper of the dipole-dipole magnetic interactions, as well as the homogeneous environment where the elemental constituents move. Those aspects still remain as open questions to be addressed. 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--- abstract: 'We consider a family of manifolds with a class of degenerating warped product metrics $g_\epsilon=\rho(\epsilon,t)^{2a}dt^2 +\rho(\epsilon,t)^{2b}ds_M^2$, with $M$ compact, $\rho$ homogeneous degree one, $a \le -1$ and $b > 0$. We study the Laplace operator acting on $L^{2}$ differential $p$-forms and give sharp accumulation rates for eigenvalues near the bottom of the essential spectrum of the limit manifold with metric $g_{0}$.' author: - Jeffrey McGowan title: Bounds on Accumulation Rates of Eigenvalues on Manifolds with Degenerating Metrics --- Introduction ============ There are many examples of non-compact manifolds which can be thought of as a ‘limit’ of a sequence of compact manifolds. Particularly nice examples are hyperbolic manifolds in dimensions 2 and 3; the cusp closing theorem of Thurston [@Thurston] then says that every complete, non-compact manifold $M_0$ is the limit of a sequence of hyperbolic manifolds $M_k \to M_0$. Since the Laplacian on $M_0$ has continuous spectrum, one expects the eigenvalues of $M_k$ to accumulate. In dimension 2, Ji, Zworski, and Wolpert ([@Ji; @JiZworski; @Wolpert1; @Wolpert2]) have given bounds for the accumulation rate of eigenvalues near the bottom of the essential spectrum in the hyperbolic case, while in dimension 3 analogous results were obtained by Chavel and Dodziuk ([@ChavelDodziuk]). Dodziuk and McGowan obtained similar results for the Laplacian acting on differential forms ([@DM]). Colbois and Courtois considered convergence of eigenvalues below the bottom of the essential spectrum in a much more general setting [@CC]. The accumulation rate for eigenvalues of the Laplacian on functions for manifolds $N=\tilde{N}\cup (M^n\times I)$ with ’pseudo-hyperbolic’ metrics on $(M^n\times I)$ was given by Judge [@Judge]. Judge also computes the essential spectrum for a more general class of degenerating metrics, and investigates the convergence of eigenfunctions. We will consider manifolds $N_{\epsilon}=\tilde{N}\cup (M^n\times I)$, $\tilde{N}$ and $M^n$ compact, with $n=dim(M)$, and a family of metrics $$\label{metric}g_\epsilon=\rho(\epsilon,t)^{2a}dt^2 +\rho(\epsilon,t)^{2b}ds_M^2$$ on $M^n \times I$. Here $\rho = c_1\epsilon +c_2t$, $c_1,c_2>0$, $t\in I=[0,1]$, $a\leq -1$, $b>0$, and $ds_M^2$ is the metric on $M^n$. We identify the boundary of $\tilde{N}$ with $M^{n}\times {1}$. These are the metrics discussed by Melrose in [@Melrose] and considered by Judge in [@Judge]. We consider only non-negative values of $t$ with $t \in [0,1]$, which simplifies the statements of the results, although we must consider manifolds with boundary. The condition $a \leq -1$ means that the limiting manifold $N_{0}$ is complete. We study the accumulation rate for eigenvalues near the bottom of the essential spectrum of the Laplacian acting on both functions and differential forms. Our main results are \[th2\] Suppose $N_\epsilon=\tilde{N}\cup (M^n\times I)$, $\tilde{N}$ and $M^n$ compact, with metric $$g_\epsilon=\rho(\epsilon,t)^{2a}dt^2 +\rho(\epsilon,t)^{2b}ds_M^2$$ on $M^n \times I$, with $\rho$ as above. Let $$R=\int_0^1\rho(\epsilon,s)^a\,ds$$ be the geodesic distance from the boundary ${0} \times M^{n})$ of $N_{\epsilon}$ to $\tilde{N}$. Let $\Xi_\epsilon(x^2)$ be the number of eigenvalues of the Laplacian acting on coexact $p$-forms (satisfying absolute boundary conditions on the boundary of $N_{\epsilon}$) in $[\sigma,\sigma + x^2)$ where $\sigma$ is the bottom of the essential spectrum for coexact forms of degree $p$ and $0 < p < n$. Then $$\Xi_\epsilon(x^2) =\frac{dxR}{\pi} + O_x(1)$$ where $d$ is the dimension of the space of harmonic forms of degree $p$ on $M$. This agrees with the results of Judge [@Judge], Chavel and Dodziuk [@ChavelDodziuk] and Dodziuk and McGowan [@DM] in the special cases they considered. \[th1\] Suppose $N_\epsilon$ is as in Theorem \[th1\]. Then the essential spectrum of the Laplacian acting on coexact $p$-forms, $0 \leq p \leq n$ on $N_0$ is $$\begin{array}{lcr} \left[\left(\frac{n-2p}{2}\right)^2c_2^2b^2,\infty\right)&\qquad&a=-1\\\\ \left[0,\infty\right)&\qquad&a<-1 \end{array}$$ Note that this agrees with Judges results ([@Judge]) for functions when $p=0$, and with Mazzeo and Phillips results for the essential spectrum on geometrically finite hyperbolic manifolds ([@MazzeoPhillips], with $c_2=b=1$ and $a=-1$). We have recently learned that these results for the essential spectrum have been obtained independently by Antoci (\[Antoci\]). This paper is organized as follows. In Section \[geom\] we discuss the geometry of the manifolds under consideration, and rewrite the metric (\[metric\]) in a way which makes the geometry more evident. In Section \[functions\] we illustrate our techniques by computing the essential spectrum and accumulation rates for eigenvalues as $\epsilon \to 0$ in the case of functions ($p=0$). In Section \[upperforms\] we compute the essential spectrum and give lower bounds on the accumulation rate in the $p\neq 0$ case. Finally, in Section \[lowerforms\] we give upper bounds on the accumulation rate for $p \neq 0$, completing the proof of Theorem \[th1\]. We wish to thank Józef Dodziuk for many helpful conversations. The Geometry {#geom} ============ Metrics of the type (\[metric\]) are discussed by Melrose in [@Melrose]. When $a \leq -1$ such metrics are complete on the limit manifold $N_{0}$. Melrose classifies metrics where $a=-1$, $b=1$ as ’hc’, or hyperbolic cusp metrics, and metrics where $a=-1$, $b=0$ as ’boundary’, or metrics with cylindrical end. Since we will consider metrics where $a \leq -1$, $b > 0$ we rewrite the metric to make the geometry more evident. Let $\tau$ be the geodesic distance from $t=0$ to $t=1$, in other words the geodesic distance from the a point $(0,p \in M)$ to $\tilde{N}$. Then $$\label{taudef} \tau = \int_0^t{\rho(\epsilon,s)^a\,ds}= \int_0^t{(c_1\epsilon + c_2s)^a\,ds}$$ and we have two distinct cases, $$\begin{array}{lcc} \tau =\frac{1}{c_2}\left( \ln\left(\frac{{c_1\epsilon +c_2t}}{c_1\epsilon}\right)\right)&\qquad&a=-1\\ \tau = \frac{1}{c_2}\left(\frac{(c_1\epsilon+c_2t)^{a+1}-(c_1\epsilon)^{a+1}}{c_2(a+1)}\right)&\qquad&a<-1 \end{array}$$ Solving for $t$ and substituting into the metric (\[metric\]) we get $$\label{mymetrics} \begin{array}{lcc} ds^2=d\tau^2+(c_1\epsilon)^{2b}e^{2bc_2\tau}ds_M^2&\qquad&a=-1\\ ds^2=d\tau^2+(c_2(a+1)\tau+(c_1\epsilon)^{a+1})^\frac{2b}{a+1}ds_M^2&\qquad&a<-1 \end{array}$$ which is of the form $ds^2=d\tau^2+f_\epsilon(\tau)ds_M^2$ in both cases. As $\epsilon \to 0$, $\tau\to \infty$, and we have a warped product $I \times_{f_\epsilon} M$, with the length of the interval given by $$\label{rcalc} \tau(1)=\Bigg\{\begin{array}{lr}R = \frac{1}{c_2}\ln\left(\frac{c_1\epsilon+c_2}{c_1\epsilon}\right)&a=-1\\R = \frac{(c_1\epsilon+c_2)^{a+1}-(c_1\epsilon)^{a+1}}{c_2(a+1)}&a<-1\end{array}$$ When $a=-1$, $f_e(\tau)$ gives an essentially hyperbolic metric; one thinks of pinching off a closed geodesic, with $\epsilon$ the length of that geodesic. When $a < -1$, the cross sections $M$ shrink at a slower rate as one recedes from $\tilde{N}$, and the warped product $I \times_{f_\epsilon} M$ is intermediate between a hyperbolic cusp and a flat cylinder. Clearly, for any fixed $\epsilon$ and $\tau$, the cross section $\{\tau\}\times f_\epsilon(\tau)M$ has injectivity radius bounded below by some constant. Moreover, $f_\epsilon(\tau)$ is an increasing function of $\tau$ for all $a \le -1$. Since $M$ is compact,a scaling argument shows that the first non-zero eigenvalue of the Laplacian acting on coexact forms of degree $p$ on $M_{\epsilon,\tau}$, say $\nu_{p,\epsilon}(\tau)$, is a decreasing function of $\tau$. Hence, as the geodesic distance of a given cross section from $\tilde{N}$ increases, $\nu_{p,\epsilon}(\tau)$ increases. This allows us, for technical reasons, to restructure our decomposition of $N$ as follows;$$N=\tilde{N}'\cup\left(M\times [0,R-r_0+1]\right),$$ where $\tilde{N}' = \tilde{N} \cup M \times [R-r_0,R]$. $r_0$ will be chosen so that $\nu_{p,\epsilon}(r_0)$ is relatively large. Functions ========= We follow essentially the argument in [@DM]. First, we choose a function $f$ whose restriction to $\tilde{N}'$ is orthogonal to a basis of eigenfunctions with eigenvalues less than or equal to $\sigma + x^2$. This will only change the counting function $N_\epsilon(x^2)$ by a bounded amount which can be absorbed into the $O_x(1)$ term ([@ChavelDodziuk Lemma 3.6]). Next, we decompose $f$ on $M\times [0,R-r_0]$ as $f=\bar{f}+\bar{\bar{f}}$, where $\bar{f}$ depends only on $\tau$ and $\bar{\bar{f}}$ is orthogonal to constants on $M$. $\bar{f}$ is computed by averaging over each cross section. Now, if we choose $r_0$ so that $\nu_{p,\epsilon}(r_0)>\sigma + x^2$, then $\bar{\bar{f}}$ does not contribute to the counting function $N_\epsilon(x^2)$. Concentrating on $\bar{f}$, a straightforward calculation shows that $$\label{funSL} \Delta\bar{f}=*d*d\bar{f}=\frac{1}{f_\epsilon^{\frac{n}{2}(\tau)}}\frac{d}{d\tau}\left(\frac{d\bar{f}}{d\tau}f_\epsilon^\frac{n}{2}(\tau)\right)$$ is a classical Sturm-Liouville problem, and we can convert to the form (see [@CourantHilbert1]) $$u''-ru=\lambda u$$ with $$\begin{aligned} u& = & f_\epsilon^\frac{n}{4}(\tau)\bar{f} \\ r & = &\frac{( f_\epsilon^\frac{n}{4}(\tau))''}{f_\epsilon^\frac{n}{4}(\tau)}\end{aligned}$$ When $a=-1$, $$r=\left(\frac{nbc_2}{2}\right)^2,$$ and (\[funSL\]) becomes $$u''=\left(\lambda + \left(\frac{nbc_2}{2}\right)^2\right)u.$$ We get \[fun1\] Suppose $N_\epsilon=\tilde{N}\cup (M^n\times I)$, $\tilde{N}$ and $M^n$ compact, with metric $$g_\epsilon=\rho(\epsilon,t)^{-2}dt^2 +\rho(\epsilon,t)^{2b}ds_M^2$$ on $M^n \times I$, with $\rho = c_1\epsilon + c_2 t$. Then the essential spectrum of the Laplacian acting on functions on $N_0$ is $$\left[\left(\frac{nc_2b}{2}\right)^2=\sigma,\infty\right).$$ Let $R$ be as in (\[rcalc\]) and let $N_\epsilon(x^2)$ be the number of eigenvalues of the Laplacian acting on function in $[\sigma,\sigma + x^2)$. Then $$N_\epsilon(x^2) = \frac{xR}{\pi}+O_x(1).$$ This is as in [@Judge], with slightly different notation. When $a<-1$, $$r=\frac{(a+1)bn(bn-2a-2)c_{2}^{2}}{2(c_{2}(a_+1)\tau+(c_{1}\epsilon)^{a+1})^{2}}.\label{integrable}$$ The potential (\[integrable\]) is integrable, and [@ChavelDodziuk Theorem 4.1] tells us the counting function for the corresponding Sturm-Liouville problem has the same asymptotics as if the potential were identically 0. Hence, Suppose $N$ is as in Proposition \[fun1\] with metric $$g_\epsilon=\rho(\epsilon,t)^{2a}dt^2 +\rho(\epsilon,t)^{2b}ds_M^2$$ on $M^n \times I$, where $a < -1$ and $\rho$ as above. Then the essential spectrum of the Laplacian acting on functions on $N_0$ is $ \left[0,\infty\right).$ Let $R$ be as in (\[rcalc\]) and let $N_\epsilon(x^2)$ be the number of eigenvalues of the Laplacian acting on function in $[0,x^2)$. Then $$\label{Nfun}N_\epsilon(x^2) = \frac{xR}{\pi}+O_x(1).$$ The essential spectrum in this case was given by Judge ([@Judge]). The accumulation estimate (\[Nfun\]) can also be obtained using Judge’s techniques ([@Judge2]). Upper eigenvalue bounds for forms {#upperforms} ================================= We consider the sequence of eigenvalues of the Laplacian acting on coexact forms of degree $p$, $0 < \nu_{1} \le \nu_{2}\le \cdots \to \infty$. If we can give an upper bound $y\ge \nu_{j}$ for some $j$, we will obtain a lower bound for the counting function $\Xi_{\epsilon}(y) \le j$. We work in the space $\mathcal{E}$ of $C^{\infty}$ coexact forms of degree $p$ on $N_{\epsilon}$ with support contained in $\{x|1 \le d(x,\tilde{N})\le R\}$, with coefficients which depend only on $\tau$. Any form $\omega \in \mathcal{E}$ is zero on $\tilde{N}$, and we choose forms $$\label{omega}\omega=\sum_{i=1}^{d}b_{i}d\tau\wedge H_{i}$$ where $d$ is the dimension of the space of harmonic $p$ forms on the cross section $M$ and $H_{i},i=1,2,\ldots,d$ is a basis of harmonic $p$ forms on $M$. Using Courant’s min-max principle the eigenvalues $\nu_{j}$ are no greater than the critical values of the Rayleigh-Ritz quotient $(\Delta \omega,\omega)/(\omega,\omega))$ with $\omega \in \mathcal{E}$. Since $\omega$ is coexact, we have $$\frac{(\Delta \omega,\omega)}{(\omega,\omega))}=\frac{(d\omega,d\omega)}{(\omega,\omega)}$$ Since the $b_{i}$ depend only on $\tau$, and an application of $d$ to the sum in (\[omega\]) involves only derivatives with respect to other basis elements, we compute $$d{\omega} = \sum_{i=1}^{d}b_{i}'d\tau\wedge H_{i}$$ where the prime indicates differentiation with respect to $\tau$. Computing the respective $L^{2}$ norms we have $$\begin{aligned} (\omega,\omega)&=&C_M\sum_{i=1}^{d}\int_{1}^{R}b_{i}^{2}f_{\epsilon}^{\frac{n-2p}{2}}(\tau)\,d\tau\label{bottom}\\ (d\omega,d\omega)&=&C_M\sum_{i=1}^{d}\int_{1}^{R}(b_{i}')^{2}f_{\epsilon}^{\frac{n-2p}{2}}(\tau)\,d\tau \label{top}\end{aligned}$$ $C_{M}$ can be computed by integrating the basis elements of the cross section $M$. Using integration by parts in the numerator of the Rayleigh-Ritz quotient, we get $d$ copies of a Sturm-Liuoville problem very similar to the one in section \[functions\],$$-\frac{1}{f_{\epsilon}^{\frac{n-2p}{2}}}\left(b_{i}'f_{\epsilon}^{\frac{n-2p}{2}}\right)'=\lambda b_{i}.$$ As in Section \[functions\], we reduce to the form $$u''-ru=\lambda u.$$ In the pseudo-hyperbolic case, when $a=-1$, we have $$r=\left(\frac{n-2p}{2}\right)^{2}c_{2}^{2}b^{2}$$ and we get \[lowerform\] Suppose $N_\epsilon=\tilde{N}\cup (M^n\times I)$, $\tilde{N}$ and $M^n$ compact, with metric $$g_\epsilon=\rho(\epsilon,t)^{-2}dt^2 +\rho(\epsilon,t)^{2b}ds_M^2$$ on $M^n \times I$, with $\rho=c_{1}\epsilon+c_{2}t$. If $N_\epsilon(x^2)$ is the number of eigenvalues of the Laplacian acting on coexact $p$-forms in $[\left(\frac{n-2p}{2}\right)^{2}c_{2}^{2}b^{2},\left(\frac{n-2p}{2}\right)^{2}c_{2}^{2}b^{2}+ x^2)$ , then $$N_\epsilon(x^2) \ge \frac{dxR}{\pi} + O_x(1)$$ where $d$ is the dimension of the space of harmonic forms of degree $p$ on $M$. In this case, $R = \frac{1}{c_2}\ln\left(\frac{c_1\epsilon+c_2}{c_1\epsilon}\right)$, and letting $\epsilon \to 0$, we see that the essential spectrum of the Laplacian acting on coexact $p$-forms, $0 \leq p \leq n$ on $N_0$ is $\left[\left(\frac{n-2p}{2}\right)^2c_2^2b^2,\infty\right)$ if $d \ne 0$. When $a < -1$ a messy but straightforward calculation give $$r=\frac{c_{2}^{2}b(n-2p)(\frac{b(n-2p)}{2}-1}{2(c_{2}(a+1)\tau + (c_{1}\epsilon)^{a+1})^{2}}.$$ This is an integrable potential, and we get \[lowerform2\] Suppose $N_\epsilon=\tilde{N}\cup (M^n\times I)$, $\tilde{N}$ and $M^n$ compact, with metric $$g_\epsilon=\rho(\epsilon,t)^{2a}dt^2 +\rho(\epsilon,t)^{2b}ds_M^2$$ on $M^n \times I$, with $\rho=c_{1}\epsilon+c_{2}t$. If $N_\epsilon(x^2)$ is the number of eigenvalues of the Laplacian acting on coexact $p$-forms in $[0, x^2)$ , then $$N_\epsilon(x^2) \ge \frac{dxR}{\pi} + O_x(1)$$ where $d$ is the dimension of the space of harmonic forms of degree $p$ on $M$. In this case, $R = \frac{(c_1\epsilon+c_2)^{a+1}-(c_1\epsilon)^{a+1}}{c_2(a+1)}$, and letting $\epsilon \to 0$, we see that the essential spectrum of the Laplacian acting on coexact $p$-forms, $0 \leq p \leq n$ on $N_0$ is $\left[0,\infty\right)$ if $d \ne 0$. Lower eigenvalue bounds for forms {#lowerforms} ================================= We will use the method of [@McGowan Lemma ?] to get global lower eigenvalue bounds for forms on $N_{\epsilon}$ based on lower eigenvalue bounds on local eigenvalue bounds on (overlapping) pieces of $N_{\epsilon}$. In particular, we use the idea of constructing a globally defined form while keeping control of the Rayleigh-Ritz quotient as in [@DM]. For details on the underlying Čech-de Rham formalism see [@BottTu Chapter 2]. The pieces we will consider might have mildly singular boundaries, but all the familiar results of Hodge theory hold ([@Cheeger1; @Cheeger2], see also [@McGowan Section ?] and [@DM Section 4]). We omit many details here, but refer the reader especially to [@DM] if they wish to fill in the blanks. First, we pick a simple open cover of $N_{\epsilon}$ consisting of two pieces; $U_{1}=\tilde{N}'\setminus\partial\tilde{N}'$, and $U_{2}=M\times [0,R-r_{0}+1]$. Recall that $\tilde{N}' = \tilde{N} \cup M \times [R-r_0,R]$, so $U_{1}$ and $U_{2}$ overlap, with $U_{1}\cap U_{2}=M\times [R-r_{0},R-r_{0}+1]$. Next, we choose a coexact $p$ form $\phi$ so that the restriction $\phi|_{U_{1}}=\phi_{1}$ is orthogonal to the finite dimensional space of exact eigenforms (on $U_{1}$) of degree $p+1$ with eigenvalue less than or equal to $y^{2}$. This is possible using [@DM Proposition 5.1]; the proof must be modified somewhat to account for the more general setting here, but the modifications are simple if messy. We will specify values for $r_{0}$ and $y^{2}$ later. Now, since $\phi_{1}$ is assumed to be exact with eigenvalue greater than or equal to $y^{2}$, there exists a unique coexact form $\psi_{1}$ of degree $p$ on $U_{1}$ with $d\psi_{1}=\phi_{1}$ and $$\frac{(\phi_{1},\phi_{1})}{(\psi_{1},\psi_{1})}=\frac{(d\psi_{1},d\psi_{1})}{(\psi_{1},\psi_{1})}\ge y^{2}.$$ Likewise, by exactness, there exists a unique coexact form $\psi_{2}$ on $U_{2}$ with $d\psi_{2}=\phi_{2}$, but we do not yet have any bounds on the Rayleigh-Ritz quotient (and hence on eigenvalues) $$\label{RR}\frac{(\phi_{2},\phi_{2})}{(\psi_{2},\psi_{2})}=\frac{(d\psi_{2},d\psi_{2})}{(\psi_{2},\psi_{2})}.$$ Next, we wish to decompose $\phi_{i}$, $i=1,2$ on $M\times [0,R]$ in a similar fashion to our decomposition for functions at the beginning of Section \[functions\]. We will model ourselves on the argument in [@DM], but since the cross section ${\tau} \times M^{n}$ is arbitrary here, we cannot just average coefficients. Rather, we use a harmonic projection. First, decompose $\phi_{i} = \alpha \wedge d\tau + \beta$, where $\beta$ does not contain $d\tau$. Next, use harmonic projection on $\alpha$ to get, $$\alpha = \sum_{i=1}^{d}a_{i}d\tau\wedge H_{i} + \gamma$$ where $d$ is the dimension of the space of harmonic $p$ forms on $M$, $H_{i}$ is a basis of harmonic $p$ forms on $M$, and the $a_{i}$ depend only on $\tau$. Now, we can write $\phi_{i}=\bar{\phi_{i}}+\bar{\bar{\phi_{i}}}$, with $$\begin{aligned} \label{decomp} \bar{\phi_{i}} &=& \sum_{i=1}^{d}a_{i}d\tau\wedge H_{i}\\ \bar{\bar{\phi_{i}}} &=& \phi_{i}-\bar{\phi_{i}} = \beta+\gamma\end{aligned}$$ We do the same for $\psi_{i}$. By construction, the coefficients of $\bar{\phi_{i}}$ and $\bar{\psi_{i}}$ depend only on $\tau$. A straightforward calculation (see, for example, [@Dodziuk]), shows that as the metric on the cross sections scales by a factor $f_{\epsilon}(\tau)$, the Rayleigh-Ritz quotient scales as $$\frac{(d\bar{\bar{\phi_{i}}},d\bar{\bar{\phi_{i}}})|_{g_{_{\epsilon}}}} {(\bar{\bar{\phi_{i}}},\bar{\bar{\phi_{i}}})|_{g_{_{\epsilon}}}} = \frac{1}{f_{\epsilon}(\tau)}\left(\frac{(d\bar{\bar{\phi_{i}}},d\bar{\bar{\phi_{i}}})|_{g_{_{1}}}}{(\bar{\bar{\phi_{i}}},\bar{\bar{\phi_{i}}})|_{g_{_{1}}}}\right).$$ For small $\tau$, $f_{\epsilon}(\tau)$ is small, and thus if $r_{0}$ is chosen appropriately, $\bar{\bar{\phi_{i}}}$ will not contribute to any accumulation of eigenvalues. So far, we have put only a finite number of conditions, depending only on $x$, on our original selection of $\phi$. These conditions guarantee that $\phi_{1}$ is orthogonal to the finite dimensional space of exact eigenforms (on $U_{1}$) of degree $p+1$ with eigenvalue less than or equal to $y^{2}$. We still need to determine how many additional choices we must make to gain control of the Rayleigh-Ritz quotient (\[RR\]). By construction, we can write $$\bar{\phi_{2}}=\sum_{i=1}^{d}a_{i}d\tau\wedge H_{i}$$ where $d$ is the dimension of the space of harmonic $p$ forms on $M$, $H_{i}$ is a basis of harmonic $p$ forms on $M$, and the $a_{i}$ depend only on $\tau$. Consequently, we can write $$\bar{\psi_{2}}=\sum_{i=1}^{\zeta}f_{i}\,d\tau\wedge\alpha_{i,p-1}+\sum_{i=1}^{d}b_{i}H_{i}$$ with $d\bar{\psi_{2}}=\bar{\phi_{2}}$, $a_{i}=b_{i}'$, and the prime denoting differentiation with respect to $\tau$. To evaluate the Rayleigh-Ritz quotient on $U_{2}$, we use $$\begin{aligned} (\bar{\psi_{2}},\bar{\psi_{2}})&=&C_M\sum_{i=1}^{d}\int_{0}^{R-r_{0}}b_{i}^{2}f_{\epsilon}^{\frac{n-2p}{2}}(\tau)\,d\tau\\ (\bar{\phi_{2}},\bar{\phi_{2}})&=&C_M\sum_{i=1}^{d}\int_{0}^{R-r_{0}}(b_{i}')^{2}f_{\epsilon}^{\frac{n-2p}{2}}(\tau)\,d\tau\end{aligned}$$ and we again have $d$ copies of $$-\frac{1}{f_{\epsilon}^{\frac{n-2p}{2}}}\left(b_{i}'f_{\epsilon}^{\frac{n-2p}{2}}\right)'=\lambda b_{i}.$$ Letting $\sigma$ be the bottom of the essential spectrum for $N_{0}$ we see that the number of eigenvalues in the interval $[\sigma,\sigma+x^{2})$ for the equation $\Delta_{p}\psi_{2}= \nu \psi_{2}$ is given by $\frac{dxR}{\pi}+O_{x}(1)$. Thus, we can choose $\phi$ in such a way that $\psi_{2}$ is orthogonal in $L^{2}$ to the basis of eigenforms with eigenvalues less than $\sigma + x^{2}$ on $U_{2}$ by imposing $\frac{dxR}{\pi}+O_{x}(1)$ conditions. The number of conditions imposed on the choice of $\phi$ depends only on $x$. We have chosen $\phi$ in such a way that we have control over the relevant Rayleigh-Ritz quotients on both $U_{1}$ and $U_{2}$, but it is not the case that $\psi_{1}$ and $\psi_{2}$ must match on $U_{1}\cap U_{2}$. Since $d\psi_{1}=\phi|_{U_{1}}$ and $d\psi_{2}=\phi|_{U_{2}}$ it is clear that the difference $\psi_{2}-\psi_{1}$ must be exact, and we use the method of [@McGowan Section ?] to build a globally defined form $\psi$ with $d\psi=\phi$ and with control over the Rayleigh-Ritz quotient. Since our open cover has only two pieces, this produces no difficulties. If we choose $y^{2}$ above in such a way that summing the relevant Rayleigh-Ritz inequalities gives the correct lower eigenvalue bound, we have Suppose $N_\epsilon=\tilde{N}\cup (M^n\times I)$, $\tilde{N}$ and $M^n$ compact, with metric $$g_\epsilon=\rho(\epsilon,t)^{2a}dt^2 +\rho(\epsilon,t)^{2b}ds_M^2$$ on $M^n \times I$, with $\rho$ as above, and $d$ the dimension of the space of harmonic $p$-forms on $M^{n}$. Let $$R=\int_0^1\rho(\epsilon,s)^a\,ds.$$ Let $N_\epsilon(x^2)$ be the number of eigenvalues of the Laplacian acting on coexact $p$-forms in $[\sigma,\sigma + x^2)$ where $\sigma$ is the bottom of the essential spectrum for coexact forms of degree $p$ and $0 < p < n$. Then $$N_\epsilon(x^2) =\frac{dxR}{\pi} + O_x(1)$$ where $d$ is as in theorem \[th1\]. In the special case when $d=0$, i.e. when there are no harmonic forms on the cross section, we have the following corollary, Suppose $N$ is as above, with $d=0$ for some $p$. Then the essential spectrum of the Laplacian acting on exact forms of degree $p$ is empty. [9]{} F. Antoci *On the spectrum of the Laplace-Beltrami operator for $p$-forms for a class of warped product metrics.* to apear, Advances in Mathematics. R. Bott and L. W. Tu. *Differential forms in algebraic topology*, Graduate Texts in Math., no. 82 Springer Verlag, New York, 1982. I. Chavel and J. Dódziuk. *The spectrum of degenerating hyperbolic 3-manifolds.* J. Diff. Geometry **39** (1994), 123–137. J. Cheeger. *On the Hodge theory of Riemannian pseudomanifolds,* Geometry of the Laplace Operator (R. Osserman and A. Weinstein, eds.), Proc. Sympos. Pure Math., vol. 36, Amer. Math. Soc., Providence, RI, 1980, pp. 91–146. J. Cheeger. *Spectral geometry of singular Riemannian spaces.* J. Diff. Geometry **18** (1983), 575-657. B. Colbois and G. Courtois. *Convergence de variŽtŽs et convergence du spectre du Laplacien.* Ann. Sci. École Norm. Sup. **24**, (4) (1991),507–518 . R. Courant and D. Hilbert. *Methods of Mathematical Physics Volume I.* John Wiley Sons, New York, 1937. J. Dódziuk *Eigenvalues of the Laplacian on forms.* Proc. Amer. Math Soc. **85** (1982), 438-443. J. Dódziuk and J.  McGowan. *The spectrum of the Hodge Laplacian for a degenerating family of hyperbolic three manifolds.* Trans. Amer. Math Soc. **347**, 6 (1995), 1981–1995. L. Ji *Spectral degeneration of hyperbolic surfaces* J. Diff. Geom. **38**, (1993), 263–313 L. Ji and M. Zworski *The remainder estimate in spectral accumulation for degenerating surfaces.* J. Functional Anal. **38**, (1993), 263–313 C. Judge. *Tracking eigenvalues to the frontiers of moduli space, I.* J. Functional Anal. **184**, (2001),273–290 . C. Judge. *Private correspondence*. R.  Mazzeo and R. Phillips. *Hodge theory on hyperbolic manifolds.* Duke Math. J. **60**, (1990), 509-559. J.  McGowan. *The p-spectrum of compact hyperbolic three manifolds.* Math. Ann.  **279**, (1993), 725–745. R. Melrose. *Geometric scattering theory.* Cambridge Univ. Press, Cambridge, 1995. W. Thurston. *Three-dimensional geometry and topology. Vol. 1. .* Edited by Silvio Levy. Princeton Mathematical Series, **35**. Princeton University Press, Princeton, NJ, 1997. E. Titchmarsh. *Eigenfunction expansions associated with second order differential equations, 1.* Cambridge Univ. Press, London, 1946. S. Wolpert. *Asymptotics of the spectrum and the Selberg zeta function on the space of Riemann surfaces.* Comm. Math. Phys. **112**, (1987),283–315 . S. Wolpert. *Spectral limits for hyperbolic surfaces, II.* Invent. Math.  **108**, (1992),91-129 .

--- abstract: 'Investigating properties of two-dimensional Dirac operators coupled to an electric and a magnetic field (perpendicular to the plane) requires in general unbounded (vector-) potentials. If the system has a certain symmetry, the fields can be described by one-dimensional potentials $V$ and $A$. Assuming that $|A|<|V|$ outside some arbitrary large ball, we show that absolutely continuous states of the effective Dirac operators spread ballistically. These results are based on well-known methods in spectral dynamics together with certain new Hilbert-Schmidt bounds. We use Lorentz boosts to derive these new estimates.' address: - | Josef Mehringer\ Mathematisches Institut\ Ludwig-Maximilians-Universität\ Theresienstra[ß]{}e 39\ D-80333 München, Germany. - | Edgardo Stockmeyer\ Instituto de Física\ Pontificia Universidad Católica de Chile\ Vicuña Mackenna 4860\ Santiago 7820436, Chile. author: - Josef Mehringer - Edgardo Stockmeyer title: 'Ballistic dynamics of Dirac particles in electro-magnetic fields' --- Introduction {#introduc} ============ It is well known that Dirac particles suffer from a phenomenon called Klein tunneling. In dimension one, it can be roughly described as follows : If one considers a step potential, for instance $V(x)=V_0$ for $x{\geqslant}0$ and zero otherwise, then massless Dirac particles coming from the left will tunnel through the barrier independently of their energy. As opposed to the classical quantum tunneling there is no exponential damping factor diminishing the probability of finding the particle on the right side of the barrier [@Klein1929; @Thaller]. More generally, one-dimensional massless Dirac particles spread as free particles in the presence of electric fields. This effect has attracted renewed attention due to the isolation of graphene in 2003 (see [@Novoselov2004]), since the low-energy charge carriers of this material can be described by the two-dimensional massless Dirac equation [@castro2009electronic; @F2012; @Fefferman2014]. Indeed, experiments have been carried out to observe Klein tunneling in graphene confirming some theoretical predictions [@PhysRevLett.98.236803; @PhysRevLett.102.026807; @young2009quantum]. Consider the massless one-dimensional Dirac equation $$\begin{aligned} -{\mathrm{i\,}}\sigma_1\partial_1 +V \quad \mathrm{on}\quad L^2({\mathbb{R}},{\mathbb{C}}^2),\end{aligned}$$ with an electric potential $V\in L^1_{\rm loc}({\mathbb{R}})$, where $\sigma_1$ is the first Pauli matrix. In this case the Klein tunnel effect is not very surprising from the mathematical point of view since $ -{\mathrm{i\,}}\sigma_1\partial_1 +V $ is unitarily equivalent to the free Dirac operator $-{\mathrm{i\,}}\sigma_1\partial_1$ by means of the transformation $$\begin{aligned} \label{intro1} \exp \left({\mathrm{i\,}}\sigma_1\int_0^xV(s)\mathrm{d}s\right).\end{aligned}$$ However, in the presence of magnetic fields the situation is different. In dimension two it is known that magnetic fields tend to localise Dirac particles, very much like as in the Schrödinger case (see [@Thaller]). In a previous article we considered the combined electromagnetic effect from a spectral theoretical point of view [@MehringerStockmeyer2014]. In the present work we investigate this further but focusing on the wave package spreading. Consider a two-dimensional Dirac operator coupled to an electro-magnetic field described by electric and magnetic potentials $V$ and ${\bf A}$. If the field has translational or rotational symmetry the problem can be reduced to the study of a family of Dirac operators on the line or on the half-line, respectively. (Here the fields may be expressed through one-dimensional potential functions $V$ and $A$.) Denote by $h$ one of the members of these families. Our results roughly state the following: Assuming that the function $\psi\not=0$ is of finite energy and that it belongs to the absolutely continuous spectral subspace of $h$ we obtain a lower bound on the Cèsaro mean of the time evolution of the $p$-th moment ($p>0$), i.e. there is a constant $C(\psi,p)>0$ such that $$\begin{aligned} \label{d1} \frac{1}{T}\int_0^T \||x|^{p/2} e^{-{\mathrm{i\,}}t h}\psi\|^2\,{\mathrm{d}}t{\geqslant}C(\psi,p)T^p.\end{aligned}$$ Besides certain regularity conditions the above inequality holds provided $|{A}|<|V|$ outside some arbitrary large ball (see Theorems \[lastmainthm1\] and \[lastmainthm2\]). As a consequence of the causal behaviour of Dirac particles (see [@Thaller Theorem 8.5]) one has an upper bound of the same type, yielding altogether ballistic dynamics. The consequences of inequalities of type for two-dimensional Dirac operators with symmetries are summarised in Corollaries \[appl1\] and \[appl2\]. We remark that if $V$ grows regularly at infinity the spectrum of $h$ is purely absolutely continuous (see the discussion after Remark \[barry\]). The latter is in stark contrast to the behaviour of non-relativistic particles. An important example is when the electric and magnetic fields are asymptotically uniform, in which case $V$ and $A$ grow linearly in the space coordinate. The proof of the bounds of type are based upon the ideas of [@Guarneri1989], [@Combes1993] and [@Last1996]. These results say roughly the following: Let $K\subset {\mathbb{R}}$ be a compact set and $\mathbbm{1}_K$ be the characteristic function supported in $K$. Then, the inequality holds if the function $\psi\in \mathbbm{1}_{K}(h) L^2$ belongs to the absolutely continuous subspace of $h$ provided a certain Hilbert-Schmidt bound is verified. This latter condition demands the following for the product of characteristic functions in space and energy: There is a constant $C>0$ such that for all $I\subset {\mathbb{R}}$ compact $$\begin{aligned} \label{intro2} \left\| \mathbbm{1}_{I}(x) \mathbbm{1}_{K}(h)\right\|_{\rm{HS}} {\leqslant}C_K\sqrt{|I|}.\end{aligned}$$ It is easy to check that the required bound is satisfied for the free Dirac operator. For Schrödinger operator with potentials bounds like are obtained using semigroup properties combined with perturbation theory [@Simon1982]. However, in our case there is no proper semi-group theory and, in addition, when the potentials are allowed to grow at infinity, naive (resolvent) perturbation theory gives estimates where the scaling in $|I|$ depends on the growth rate of $A$ and $V$; that would eventually not deliver . This is not surprising since in this case $A$ and $V$ should not be treated as perturbations to the free Hamiltonian. In the case $A=0$ one easily sees that the transformation solves this problem. In this work we provide new estimates of the type for the general case $V, A\not=0$ as long as $A$ is dominated by $V$ in certain sense. Our approach is to use Lorentz boosts (of non-constant speed) to transform the Hamiltonian to another operator with a magnetic vector potential that vanishes at infinity. We remark that the transformed operator is not going to be symmetric (see Section \[loro\]) since Lorentz boosts are not represented through unitary maps in $L^2$ but only through invertible transformation (see [@Thaller p. 70]). The relation between the original Hamiltonian and the Lorentz transformed operator is made precise through certain resolvent identities. In the case of operators defined on the real line bounds like are very much a corollary of and the proof of [@Last1996 Theorem 6.2]. However, for Dirac operator defined on the half-line one should proceed more carefully due to their singularities at zero (c.f., Remark \[singatzero\] and the discussion at the beginning of Section \[last\]).\ \ \ [*This article is organised as follows:* ]{} In the next section we state precisely our main results. The definition and basic properties of the one-dimensional Dirac operators used here can be found in Section \[basic\]. In Section \[loro\] we discuss the behaviour of Dirac operators under certain Lorentz boosts of non-constant speed and establish resolvent identities between original and transformed operators. We then apply the insight of Section \[loro\] to prove Theorems \[mainlemma\] and \[hshk\] in Section \[proofhilbert\]. The dynamical bounds for the half-line operators (Theorem \[lastmainthm2\]) are proven in Section \[last\] where we also establish a local compactness property suitable for our regularity assumptions. In Appendix \[s.a.\] we collect some technical facts concerning self-adjointness. We compute the resolvent kernel for a half-line operator in Appendix \[expres\]. Finally, in Appendix \[proofappl\] we prove Corollaries \[appl1\] and \[appl2\] about the consequences for two-dimensional Dirac operator with symmetries. Main results and its applications {#mr} ================================= The massless two-dimensional Dirac operator in an electromagnetic field described by an electric potential $V$ and a magnetic field $B$ (perpendicular to the plane) is given by $$\begin{aligned} \label{hamiltonian} H = {\boldsymbol \sigma}\cdot (-{\mathrm{i\,}}\nabla- {\bf A}) +V \quad \mathrm{on} \quad {\mathcal H}:= {L^2({\mathbb{R}}^2,{\mathbb{C}}^2)}.\end{aligned}$$ Here ${\bf A}:{\mathbb{R}}^2\to{\mathbb{R}}^2$ is a magnetic vector potential satisfying $B= {{\rm curl\,}{\bf A}}:=\partial_1 A_2-\partial_2 A_1$ and $\boldsymbol{\sigma} = (\sigma_1, \sigma_2)$ denote the Pauli matrices $$\begin{aligned} \sigma_1 = \begin{pmatrix} 0&1 \\ 1 &0 \end{pmatrix}, \qquad \sigma_2 = \begin{pmatrix} 0&-{\mathrm{i\,}}\\ {\mathrm{i\,}}&0 \end{pmatrix}.\end{aligned}$$ For a rotational symmetric magnetic field $B$ one can always choose the rotational gauge $$\begin{aligned} \label{rotationalgauge} {\bf A} ({{\bf x}}) = \frac{1}{r^2} \int_0^r B(s) s\,{\rm d} s \begin{pmatrix} -x_2 \\ x_1 \end{pmatrix} =: \frac{A(r)}{r} \begin{pmatrix} -x_2 \\ x_1 \end{pmatrix},\end{aligned}$$ where $r = |{{\bf x}}|$. Thus, if $B$ and $V$ are rotationally symmetric we can decompose $H$ as a direct sum operators defined on the half-line, i.e. $$\begin{aligned} \label{mausi} H \cong \bigoplus_{k \in \, {\mathbb{Z}}+\frac{1}{2}} h_k,\end{aligned}$$ where $$\begin{aligned} \label{haensel} h_k := - {\mathrm{i\,}}\sigma_1 \partial_r + \sigma_2 \left( \frac{k}{r}- A(r)\right) + V(r) \quad \mathrm{on} \quad L^2({\mathbb{R}}^+,{\mathbb{C}}^2).\end{aligned}$$ Here in a slight abuse of notation we write $ V(|{{\bf x}}|) = V({{\bf x}})$. On the other hand, if $B$ is translational symmetric, say, in the $x_2$-direction, we can choose the Landau gauge ${\bf A}({{\bf x}})=(0,A(x_1))$, where $$\begin{aligned} \label{landaugauge} A(x_1) = \int_0^{x_1}B(s){\rm d}s.\end{aligned}$$ Thus, if in addition $V$ has the same symmetry, the Hamiltonian $H$ can be represented as a direct integral of one-dimensional fiber hamiltonians $$\begin{aligned} \label{mausi2} H \cong \int_{\mathbb{R}}^\oplus h(\xi) \, \mathrm{d}\xi,\end{aligned}$$ with $$\begin{aligned} \label{gretel} h(\xi) = -{\mathrm{i\,}}\sigma_1\partial_1 + \sigma_2(\xi-A) +V \quad \mathrm{on}\quad L^2({\mathbb{R}},{\mathbb{C}}^2).\end{aligned}$$ One-dimensional Dirac operators of type and are the object of the next four theorems. Let us fix some notation: Throughout this article we denote by $\mathbbm{1}_K$ the characteristic function on a set $K\subset {\mathbb{R}}$. We write $P_{ac}(H)$ for the projection onto the absolutely continuous subspace associated to a self-adjoint operator $H$. We will make use of standard notation for norms: $\|\cdot\|_p$ denotes de $L^p$-norm, $\|\cdot\|_T$ is the graph norm with respect to an operator $T$, and $\|\cdot\|_{\rm HS}$ stands for the Hilbert-Schmidt norm. In order to perform the afore mentioned Lorentz boosts (essentially of velocity $A/V$) we introduce the following classes of electromagnetic potentials: $A, V \in L^p_{\rm loc} ({\mathbb{R}}, {\mathbb{R}})$ with $p {\geqslant}2 $ such that $A = A_1 + A_2$, $V= V_1+V_2$, where $A_1,V_1$ have compact support and $A_2, V_2 \in C^1({\mathbb{R}}, {\mathbb{R}})$ fulfill 1. $V_2$ is supported away from $0$ and $ {\rm supp}(A_2) \subset {\rm supp}(V_2)$, 2. $\|A_2/V_2\|_\infty <1$, 3. the derivative $(A_2/V_2)'$ is bounded on ${\mathbb{R}}$. $A, V \in L^p_{\rm loc} ([0,\infty), {\mathbb{R}})$ with $p>2$ such that $A = A_1 + A_2$, $V= V_1+V_2$, where $A_1,V_1$ have compact support and $A_2, V_2 \in C^1({\mathbb{R}}^+, {\mathbb{R}})$ fulfill 1. $ V_2$ are supported away from $0$ and $ {\rm supp}(A_2) \subset {\rm supp}(V_2)$, 2. $\|A_2/V_2\|_\infty <1$, 3. the derivative $(A_2/V_2)'$ is bounded on $[0, \infty)$. \[mainlemma\] For $\xi \in {\mathbb{R}}$ let $h(\xi)$ be given as in with $A,V$ satisfying Hypothesis [(H1)]{}. Then there is a constant $C_\xi>0$ such that for any compact intervall $I\subset{\mathbb{R}}$ we have $$\begin{aligned} \left\| \mathbbm{1}_{I}(h(\xi)-{\mathrm{i\,}})^{-1} \right\|_{\rm{HS}} {\leqslant}C_\xi\sqrt{|I|}.\end{aligned}$$ A direct consequence of this HS-bound is the following theorem whose proof is the same as the one of Theorem 6.2 of [@Last1996]. \[lastmainthm1\] Consider the operator $h(\xi)$, $\xi\in {\mathbb{R}}$, with $A,V$ satisfying Hypothesis [(H1)]{}. Let $\Delta\subset {\mathbb{R}}$ be a bounded energy interval and $\psi \in P_{ac}(h(\xi))\mathbbm{1}_{\Delta}(h(\xi)) L^2({\mathbb{R}}, {\mathbb{C}}^2)$ be non-zero. Then, for each $p>0$, there is a constant $C_\xi (\psi, \Delta ,p)$ such that $$\begin{aligned} \label{theineq1} \langle \|x^{p/2} e^{-{\mathrm{i\,}}t h(\xi)} \psi\|^2\rangle_T {\geqslant}C_\xi(\psi, \Delta ,p) T^{p} \end{aligned}$$ for all $T>0$. In the case of the half-line operators $h_k$ we obtain similar results: \[hshk\] For $k \in {\mathbb{Z}}+\tfrac{1}{2}$ let $h_k$ be given as in , with $A,V$ satisfying Hypothesis [(H2)]{}. Then there is a constant $C_k>0$ such that for any compact intervall $I\subset [1,\infty)$ we have $$\begin{aligned} \left\| \mathbbm{1}_{I}(h_k-{\mathrm{i\,}})^{-1} \right\|_{\rm{HS}} {\leqslant}C_k\sqrt{|I|}.\end{aligned}$$ \[singatzero\] Note that the operators $h_k$ have a $k/x$-singularity at zero. We do not need boundary conditions to define them. To deduce the HS-bounds for $h_k$ we compare them with an auxiliar operator, which is regular at zero, and satisfies certain boundary conditions. Because of that, we obtain the HS-bounds only for intervalls $I\subset {\mathbb{R}}$ supported away from zero. A consequence of the latter result is the following: \[lastmainthm2\] Consider the operator $h_k$, $k\in {\mathbb{Z}}+\tfrac{1}{2}$ with $A,V$ satisfying Hypothesis [(H2)]{}. Let $\Delta\subset {\mathbb{R}}$ be a bounded energy intervall and $\psi \in P_{ac}(h_k)\mathbbm{1}_{\Delta}(h_k) L^2({\mathbb{R}}^+, {\mathbb{C}}^2)$ be non-zero. Then, for each $p>0$, there is a constant $C_k (\psi,\Delta, p)$ such that $$\begin{aligned} \label{theineq} \langle \|x^{p/2} e^{-{\mathrm{i\,}}t h_k} \psi\|^2\rangle_T {\geqslant}C_k (\psi,\Delta, p)\, T^{p} \end{aligned}$$ for all $T>0$. This statement is proven in Section \[last\]. We have to modify the argument of [@Last1996] since Theorem \[hshk\] is only valid for intervals with non-vanishing distance to zero. As an additional ingredient we use the local compactness of $h_k$ (also proven in Section \[last\]) for $A,V \in L_{\rm loc}^p([0,\infty),{\mathbb{R}})$ with $p>2$. \[barry\] We note that theorems \[mainlemma\] - \[lastmainthm2\] also hold for massive Dirac operators, i.e. operators of the form $h(\xi) + m\sigma_3$ or $h_k + m\sigma_3$ with a constant $m$. Here $\sigma_3 = -{\mathrm{i\,}}\sigma_1\sigma_2$ is the third Pauli matrix. Since Theorems \[lastmainthm1\] and \[lastmainthm2\] apply only if we have some absolutely continuous spectrum we mention some interesting examples: Let $A,V$ satisfy [(H1)]{} and, in addition, - $(A_2/V_2)'$ is integrable at $\pm \infty$, - $|V_2 (x)| \to \infty$ as $|x| \to \infty$, then $h(\xi)$, $\xi\in{\mathbb{R}}$, has purely absolutely continuous spectrum with $\sigma_{\rm ac}(h(\xi)) ={\mathbb{R}}$. Similarly, if the potentials $A,V$ satisfy [(H2)]{} and - $(A_2/V_2)'$ is integrable at $\infty$, - $|V_2 (x)| \to \infty$ as $x \to \infty$, then $h_k$, $k \in {\mathbb{Z}}+\frac{1}{2}$, has purely absolutely continuous spectrum with $\sigma_{\rm ac}(h_k) ={\mathbb{R}}$ (see [@KMSYamada], Propositions 1 and 2) . Finally we illustrate how one can harness Theorems \[lastmainthm1\] and \[lastmainthm2\] to Dirac operators in dimension two. \[appl1\] Let $H$ be given as in with translational symmetric $B$ and $V$. Let $A(x):=\int_0^x B(s)ds$ and $V$ satisfy [(H1)]{}. Let $\Delta \subset {\mathbb{R}}$ be bounded and $p>0$. Then for any $\psi \in \mathbbm{1}_\Delta(H) \mathcal{H}$ such that the set $$\begin{aligned} \label{condappl1} \big\{\xi \in {\mathbb{R}}\,|\, \widehat\psi (\, \cdot \, , \xi) \neq 0, \ \widehat\psi (\, \cdot \, , \xi) \in P_{ac}(h(\xi)) L^2({\mathbb{R}},{\mathbb{C}}^2)\big\},\end{aligned}$$ has non-trivial Lebesgue measure, there exist a constant $C(\psi, \Delta, p)>0$ such that $$\begin{aligned} \langle\| |x_1|^{p/2}e^{-{\mathrm{i\,}}tH } \psi\|^2\rangle_T {\geqslant}C(\psi, \Delta, p) \,T^{p}\end{aligned}$$ for all $T>0$. Here $\widehat \psi $ denotes the Fourier-transform of $\psi$ in the $x_2$-variable, i.e. we use the notation $\widehat \psi (x_1, \,\cdot\,) = \mathcal{F}_{x_2} \psi(x_1,\,\cdot\,)$. \[appl2\] Let $H$ be given as in with rotational symmetric $B$ and $V$. Let $A(x):=x^{-1}\int_0^x B(s)s {\mathrm{d}}s$ and $V$ satisfy [(H2)]{}. Let $\Delta \subset {\mathbb{R}}$ be bounded and $p>0$. Then for $\psi \in P_{ac}(H)\mathcal{H} \cap \mathbbm{1}_\Delta(H) \mathcal{H}$, with $\psi \neq 0$, there exist a constant $C(\psi, \Delta, p) >0$ such that $$\begin{aligned} \langle\| |{{\bf x}}|^{p/2} e^{-{\mathrm{i\,}}tH}\psi \|^2\rangle_T {\geqslant}C(\psi, \Delta, p) \,T^{p}\end{aligned}$$ for all $T>0$. Basic properties of Dirac operators in dimension one {#basic} ==================================================== In this article we basically work with two different types of one-dimensional Dirac operators. The first type is given by $$\begin{aligned} \label{charlotte} h = \sigma_1 (-{\mathrm{i\,}}\partial_x) - \sigma_2 A + V \quad \mbox{on} \quad L^2({\mathbb{R}}, {\mathbb{C}}^2), \end{aligned}$$ with $A,V \in L^2_{\rm{loc}}({\mathbb{R}}, {\mathbb{R}})$. We note that $h$ is in the limit point case at $\pm \infty$ [@Weidmann2 Korollar 15.21]. Then according to [@Weidmann2 Chapter 15] (see also [@Weidmann0]) the operator is essentially self-adjoint on $$\label{do} \mathcal{D}_0(h)=\left\{ \psi\in \mathcal{D}_{\rm max}(h)\, |\, \psi \,\mbox{has compact support in}\, {\mathbb{R}}\right\},$$ where $$\begin{aligned} \label{lara} \mathcal{D}_\mathrm{max} (h) = \left\{ \psi \in L^2({\mathbb{R}}, {\mathbb{C}}^2)\, |\, \psi \, \mbox{abs. cont.,} \, h\psi \in L^2({\mathbb{R}}, {\mathbb{C}}^2) \right\}.\end{aligned}$$ In fact by Lemma \[good-core\] in Appendix \[s.a.\] we know that $h$ is essentially self-adjoint on $C_0^\infty ({\mathbb{R}},{\mathbb{C}}^2)$. We denote its self-adjoint extension by $h$ again. The second type of operators is defined on the half-line $$\begin{aligned} \label{amelie} h_k = \sigma_1 (-{\mathrm{i\,}}\partial_x) + \sigma_2\left(\tfrac{k}{x} -A\right) + V \quad \mbox{on} \quad L^2((0, \infty), {\mathbb{C}}^2), \end{aligned}$$ with $A,V \in L^2_{\rm{loc}}([0,\infty), {\mathbb{R}})$ and $k \in ({\mathbb{Z}}+ \tfrac{1}{2} )\cup \{0\}$. The maximal domain of $h_k$ is given by $$\begin{aligned} \label{lara2} \mathcal{D}_\mathrm{max} (h_k) = \{\psi \in L^2({\mathbb{R}}^+, {\mathbb{C}}^2)\, |\, \psi \ \mathrm{abs. cont.}, \ h_k\psi \in L^2({\mathbb{R}}^+, {\mathbb{C}}^2))\}.\end{aligned}$$ For these half-line operators we distinguish two cases: When $|k|{\geqslant}\tfrac{1}{2}$ the operator $h_k$ is in the limit point case at $+\infty$ [@Weidmann2 Korollar 15.21] and in the limit point case at $0$ (see Proposition \[lpcin0\] in Appendix \[s.a.\]). Thus $h_k$ is essentially self-adjoint on $\mathcal{D}_0(h_k)$ (defined as in ). We denote its self-adjoint extension by $h_k$ again. Using Lemma \[good-core\] in Appendix \[s.a.\] it actually holds that $C_0^\infty((0,\infty),{\mathbb{C}}^2)$ is also an operator core for $h_k$. \[ws1\] We note that by [@Weidmann2 Satz 15.6] the domains of self-adjointness of $h$ and $h_k$, $|k|{\geqslant}1/2$, coincide with their maximal domains. In the case when $k=0$ the operator is in the limit point case at $+\infty$ and in the limit circle case at $0$. According to the theory of Sturm-Liouville operators (see [@Weidmann2 Satz 15.12] or [@Weidmann0]) $h_0$ has a one-parameter family of self-adjoint realisations with corresponding domains $$\label{domain1} \mathcal{D}^\alpha(h_0) = \big\{ \mathcal{D}_\mathrm{max} (h_0)\, |\, \lim_{x \to 0}\psi_1(x)\cos\alpha -\psi_2(x)\sin \alpha =0\big \},\quad \alpha \in [0, 2\pi).$$ In the sequel we work with the self-adjoint realisation of $h_0$ on $\mathcal{D}^0(h_0)\equiv \mathcal{D}(h_0)$ and denote, as before, the resulting operator by the same symbol. \[ws2\] Let $\chi\in C^\infty((0,\infty),[0,1])$ be a smooth function supported away from zero with bounded first derivative. By the definition of the domains of self-adjointness (see Remark \[ws1\]) and and the fact that $k/x $ is a bounded function on the support of $\chi$ we have that $$\begin{aligned} \label{chid} \chi \mathcal{D}(h_k)\subset\mathcal{D}(h_0), \end{aligned}$$ whenever the two operators have the same potentials $V$ and $A$. \[basicHS\] Consider $h, h_0$ with $V\in L^2_{\rm loc}$ and $A=0$. Then for bounded intervals $I \subset {\mathbb{R}}$, $I_0\subset (0, \infty)$ the operators $\chi_I (h-{\mathrm{i\,}})^{-1}$, $\chi_{I_0} (h_0-{\mathrm{i\,}})^{-1}$ are Hilbert-Schmidt with Hilbert-Schmidt norms $$\label{HSbound1} \left\| \chi_I \frac{1}{(h-{\mathrm{i\,}})} \right\|_{\rm HS} {\leqslant}\frac{1}{\sqrt{2}} \,|I|^{1/2}\,,$$ $$\label{HSbound2} \left\| \chi_{I_0} \frac{1}{(h_0-{\mathrm{i\,}})} \right\|_{\rm HS} {\leqslant}|I_0|^{1/2}\,.$$ Intertwining the resolvents by the unitary transformation $$\begin{aligned} \label{theu} \big[U\psi] (x) = \exp \left({\mathrm{i\,}}\sigma_1\int_0^xV(s)\mathrm{d}s\right)\psi(x)\end{aligned}$$ on $L^2({\mathbb{R}}, {\mathbb{C}}^2)$, respectively on $L^2((0, \infty), {\mathbb{C}}^2)$, the proof reduces to the case $V=0$. Then, the statement on $\chi_I (h-{\mathrm{i\,}})^{-1}$ is a direct consequence of the Kato-Seiler-Simon inequality (see [@SeilerSimon] or [@Simon_TI Thm. 4.1]). For the claim on the operator $\chi_{I_0} (h_0-{\mathrm{i\,}})^{-1}$ on $L^2((0, \infty), {\mathbb{C}}^2)$ we use directly the resolvent kernel (computed in the Appendix \[expres\]) given by $$\label{resolventkernel} \frac{1}{\sigma_1(-{\mathrm{i\,}}\partial_x) -{\mathrm{i\,}}}(x_1,x_2) = \begin{cases} ie^{-x_1} \begin{pmatrix} \sinh x_2 & \cosh x_2 \\ \sinh x_2 & \cosh x_2 \end{pmatrix} & \mathrm{for} \ x_1 > x_2 {\geqslant}0 \\[0.5cm] ie^{-x_2} \begin{pmatrix} \sinh x_1 & -\sinh x_1 \\ -\cosh x_1 & \cosh x_1 \end{pmatrix} & \mathrm{for} \ x_2 > x_1 {\geqslant}0. \end{cases}$$ Since $\chi_{I_0}(x_1) (\sigma_1(-{\mathrm{i\,}}\partial_x)-{\mathrm{i\,}})^{-1}(x_1,x_2)$ is square-integrable, we obtain that $\chi_{I_0} (h_0-{\mathrm{i\,}})^{-1}$ is a Hilbert-Schmidt operator. In fact this is true for any function $f\in L^2((0,\infty))$; the norm may be computed as $$\label{hswithf} \begin{split} \left\| f \frac{1}{(h_0-{\mathrm{i\,}})} \right\|_{\rm{HS}}^2 &= \int_0^\infty \int_0^\infty |f(x_1)|^2 \left\|\frac{1}{\sigma_1(-{\mathrm{i\,}}\partial_x)-{\mathrm{i\,}}}(x_1,x_2)\right\|^2_{M_2({\mathbb{C}})} \mathrm{d}x_1\mathrm{d}x_2 \\ &= \int_0^\infty |f(x_1)|^2 \int_0^\infty e^{-2|x_1-x_2|} \, \mathrm{d}x_2\mathrm{d}x_1 {\leqslant}\|f\|_2^2. \end{split}$$ Lorentz transformations and resolvent identities {#loro} ================================================ It is known from the classical theory of electrodynamics that Lorentz boosts enables one to transform magnetic fields into electric ones and vice-versa. We are allowed to use this principle here, since the time dependent Dirac equation is invariant under Lorentz transformations (see Chapter 3 and Section 4.2 of [@Thaller]). In this section we use Lorentz boosts to transform the Hamiltonian to another operator whose magnetic vector potential vanishes at infinity. In our case the speed of the boost is given by the ratio of $A$ and $V$ and will, therefore, depend on the space variable. Recall that in $2+1$ space-time dimension a Lorentz boost in direction ${\bf n}\in {\mathbb{R}}^2$ with speed $\beta<1$ (in general, smaller than the speed of light) is represented by the operator $L_{\Lambda}=e^{{\bf n}\cdot{\boldsymbol \sigma}\theta/2}$, where $\beta=\tanh{\theta}$. Let us point out a transformation property of the $\sigma$-matrices: Let $a=0$ or $a = -\infty$ and $\theta \in C^1((a, \infty), {\mathbb{R}})$. Observe that $$\begin{aligned} e^{-\sigma_2\theta /2 } \sigma_1 e^{\sigma_2\theta/2 } & = e^{-\sigma_2\theta } \sigma_1 = (\cosh \theta - \sigma_2 \sinh \theta) \sigma_1,\end{aligned}$$ therefore, $$\begin{aligned} \label{lorentzfree} \begin{split} e^{-\sigma_2\theta/2 } \sigma_1 (-{\mathrm{i\,}}\partial_1) e^{\sigma_2\theta/2 } & = e^{-\sigma_2\theta/2 } \sigma_1 e^{\sigma_2\theta/2 } e^{-\sigma_2\theta/2 } (-{\mathrm{i\,}}\partial_1) e^{\sigma_2\theta/2 } \\ & = (\cosh \theta - \sigma_2 \sinh \theta) \sigma_1 \big(-{\mathrm{i\,}}\partial_1 -{\mathrm{i\,}}\sigma_2 \tfrac{\theta'}{2}\big) \\ & = \cosh \theta (1 - \sigma_2 \tanh \theta) \sigma_1 \big(-{\mathrm{i\,}}\partial_1 -{\mathrm{i\,}}\sigma_2 \tfrac{\theta'}{2}\big) \end{split}\end{aligned}$$ on the subspace ${C_0^\infty((a, \infty), {\mathbb{C}}^2)}$. Recall that for potentials $V,A$ satisfying Hypothesis (H1) (for $a=-\infty$) or (H2) (for $a=0$) the ratio fulfills $\|A_2/V_2\|_\infty<1$. This enable us to define the following objects: Let $\theta(x) = \tanh^{-1}(\beta(x))$, where $\beta:= A_2/V=A_2/V_2$ and $\gamma:=\cosh \theta$ (then clearly, $\gamma^{-1} =\sqrt{(1-\beta^2)}$). For the Dirac operator $h$ on $L^2({\mathbb{R}},{\mathbb{C}}^2)$ (as given in ), with potentials satisfying (H1), we compute $$\begin{aligned} \label{maintransformA} \begin{split} e^{-\sigma_2\theta/2 } h e^{\sigma_2\theta/2 } & = \gamma (1 - \sigma_2 \beta) \sigma_1 \big(-{\mathrm{i\,}}\partial_1 -{\mathrm{i\,}}\sigma_2 \tfrac{\theta'}{2}\big) + \big(1-\sigma_2 \tfrac{A_2}{V}\big) V - \sigma_2 A_1\\ &= M \Big[\sigma_1(-{\mathrm{i\,}}\partial_1) +V/\gamma -\gamma(1 + \sigma_2\beta)\sigma_2 A_1 + \sigma_3\tfrac{\theta'}{2} \Big] \end{split}\end{aligned}$$ on $C_0^\infty({\mathbb{R}},{\mathbb{C}}^2)$, where $M:= \gamma(1-\sigma_2\beta)$ is a bounded multiplication operator with bounded inverse. Analogously, for the operator $h_k$ (defined in ), with potentials satisfying (H2), we have $$\begin{aligned} \label{maintransformB} \begin{split} e^{-\sigma_2\theta/2 } &h_k e^{\sigma_2\theta/2 } = M \Big[\sigma_1(-{\mathrm{i\,}}\partial_1) +V/\gamma +\gamma(1 + \sigma_2\beta)\sigma_2 \big(\tfrac{k}{x} -A_1\big) + \sigma_3\tfrac{\theta'}{2} \Big] \end{split}\end{aligned}$$ on $C_0^\infty((0,\infty),{\mathbb{C}}^2)$. Note that, abusing notation, we use the symbol $M$ in both cases, however, the former acts on $L^2({\mathbb{R}}^2,{\mathbb{C}}^2)$ and the latter on $L^2((0,\infty),{\mathbb{C}}^2)$. Summarising, we have the following identities on $C_0^\infty({\mathbb{R}},{\mathbb{C}}^2)$ and $C_0^\infty((0,\infty),{\mathbb{C}}^2)$, $$\label{them} e^{-\sigma_2\theta/2 } h e^{\sigma_2\theta/2 }=M\tilde{h}\quad\mbox{and}\quad e^{-\sigma_2\theta/2 } h_k e^{\sigma_2\theta/2 }=M\tilde{h}_k,$$ respectively. Where the operator $$\tilde h := \sigma_1(-{\mathrm{i\,}}\partial_1) -\gamma(1 + \sigma_2\beta)\sigma_2 A_1 +V/\gamma + \sigma_3 \tfrac{\theta'}{2} \quad \mathrm{on} \ L^2({\mathbb{R}},{\mathbb{C}}^2),$$ is essentially self-adjoint on $C_0^\infty({\mathbb{R}}, {\mathbb{C}}^2)$. Similarly, $$\tilde h_k := \sigma_1(-{\mathrm{i\,}}\partial_1) +\gamma(1 + \sigma_2\beta)\sigma_2 \big(\tfrac{k}{x} -A_1\big) +V/\gamma + \sigma_3 \tfrac{\theta'}{2} \quad \mathrm{on} \ L^2((0,\infty),{\mathbb{C}}^2),$$ is, for $k \in {\mathbb{Z}}+ \tfrac{1}{2}$, in the limit point case at $0$ and thus essentially self-adjoint on $C_0^\infty((0,\infty),{\mathbb{C}}^2)$. The latter holds since $A_2=0$ in a vicinity of zero (and hence so is $\beta$). Therefore, $M\tilde h $ and $M\tilde h_k $ are closed operators on the domains $\mathcal{D} (\tilde h)$ and $\mathcal{D} (\tilde h_k)$, respectively. \[resolventestimate\] Consider the operator $h$ and $h_k$ with potentials satisfying Hypotheses [(H1)]{} and [(H2)]{}, respectively. Then $\mathcal{D} (\tilde h) = \mathcal{D} (M\tilde h) =e^{-\sigma_2\theta/2 }\mathcal{D} (h)$, respectively $\mathcal{D} (\tilde h_k) = \mathcal{D} (M\tilde h_k) =e^{- \sigma_2\theta/2}\mathcal{D} (h_k)$ for $k \in {\mathbb{Z}}+ \tfrac{1}{2}$. In addition, the resolvent sets fulfill $ \varrho( h)= \varrho(M\tilde h)$ and for any $z\in \varrho(h)$ we have $$\begin{aligned} &(M\tilde h-z)^{-1}=e^{- \sigma_2\theta/2} (h-z)^{-1}e^{ \sigma_2\theta/2},\\&\big\| (M\tilde h-z)^{-1}\big\| {\leqslant}\big\|( h-z)^{-1}\big\| \,\big\| e^{|\theta|} \big\|_{\infty}.\end{aligned}$$ Similarly, for $k \in {\mathbb{Z}}+ \tfrac{1}{2}$, $ \varrho( h_k)= \varrho(M\tilde h_k)$ and for any $z\in \varrho(h_k)$ holds $$\begin{aligned} &(M\tilde h_k-z)^{-1}=e^{- \sigma_2\theta/2} (h_k-z)^{-1}e^{ \sigma_2\theta/2},\\&\big\| (M\tilde h_k-z)^{-1}\big\| {\leqslant}\big\|( h_k-z)^{-1}\big\|\,\big\| e^{|\theta|} \big\|_{\infty}.\end{aligned}$$ We give the proof only for the operator $h$, since for $h_k$ one can proceed analogously. The equality $\mathcal{D}(\tilde h)=\mathcal{D}(M\tilde h)$ is a direct consequence of the bounded invertibility of $M$ (with inverse $M^{-1} = \gamma(1+\sigma_2\beta)$). Note that the relations are also valid for $C^1$-functions with compact support, which form an invariant space under $e^{\pm\theta/2 \sigma_2}$ transformations. In addition, since $C^1_0({\mathbb{R}},{\mathbb{C}}^2)$ is contained in $\mathcal{D} (\tilde h)$ and in $\mathcal{D} ( h)$, it is also an operator core for $\tilde h$ and $h$. Using we easily get that, for any $f \in C^1_0({\mathbb{R}},{\mathbb{C}}^2)$, $$\begin{aligned} \label{eq:1} \big\|e^{-\sigma_2\theta/2}f \big\|_{M\tilde h} & {\leqslant}\|e^{-\sigma_2\theta/2}\|_\infty \|f\|_{h},\\\label{eq:2} \big\|e^{\sigma_2\theta/2}f \big\|_{ h} &{\leqslant}\|e^{\sigma_2\theta/2}\|_\infty \|f\|_{M \tilde h}. \end{aligned}$$ Let $\varphi\in \mathcal{D}(h)$ and $(\varphi_n)_{n\in {\mathbb{N}}}\subset C^1_0({\mathbb{R}},{\mathbb{C}}^2)$ be a sequence that converges to $\varphi$ in the $h$-graph norm. Due to the sequence $(e^{-\sigma_2\theta/2}\varphi_n)_{n\in{\mathbb{N}}}$ is Cauchy in the $M\tilde h$-graph norm. Hence $$\lim_{n\to\infty}e^{-\sigma_2\theta/2}\varphi_n=e^{-\sigma_2\theta/2}\varphi\in \mathcal{D}(M \tilde h).$$ Thus we get that $e^{-\sigma_2\theta/2}\mathcal{D}(h)\subset\mathcal{D}(M \tilde h)$. The opposite inclusion can be shown along the same lines using the inequality . In order to derive the resolvent bound observe that $\mathcal{D}(M\tilde h)= e^{-\sigma_2\theta/2 }\mathcal{D}(h)$ implies the operator identity, for any $z\in {\mathbb{C}}$, $$\begin{aligned} e^{-\sigma_2\theta/2 } (h-z) e^{\sigma_2\theta/2 } = (M \tilde h-z) \quad \mathrm{on} \quad \mathcal{D}(M \tilde h).\end{aligned}$$ Let $z\in \varrho(h)$. Since $h-z: \mathcal{D}(h) \to L^2({\mathbb{R}},{\mathbb{C}}^2)$ is bijective with bounded inverse we conclude $M\tilde h-z : \mathcal{D}(M\tilde h) \to L^2({\mathbb{R}},{\mathbb{C}}^2)$ is also bijective with bounded inverse. In addition, $$\begin{aligned} \big\| (M\tilde h-z)^{-1} \big\| {\leqslant}\big\| e^{\theta/2 \sigma_2} \big\| \big\|(h-z)^{-1} \big\| \big\| e^{-\theta/2 \sigma_2} \big\|.\end{aligned}$$ We close this section by illustrating how to deduce resolvent identities of the type presented in Lemma \[resolventestimate\] when $V<A$ at $\infty$. To this end we consider the operator $h$ as given in with $A,V$ fulfilling $A, V \in L^p_{\rm loc} ({\mathbb{R}}, {\mathbb{R}})$ with $p {\geqslant}2 $ such that $A = A_1 + A_2$, $V= V_1+V_2$, where $A_1,V_1$ have compact support and $A_2, V_2 \in C^1({\mathbb{R}}, {\mathbb{R}})$ fulfill 1. $A_2$ is supported away from $0$ and $ {\rm supp}(V_2) \subset {\rm supp}(A_2)$, 2. $\|V_2/A_2\|_\infty <1$, 3. the derivative $(V_2/A_2)'$ is bounded on ${\mathbb{R}}$. Due to this assumptions we can choose $\beta = V_2/A_2=V_2/A$ and set $\theta = \tanh^{-1} \beta$, $\gamma^{-2} =1-\beta^2$ as above. Then, as in , we obtain $$\begin{aligned} \begin{split} e^{-\sigma_2\theta/2 } h e^{\sigma_2\theta/2 } & = \gamma (1 - \sigma_2 \beta) \sigma_1 \big(-{\mathrm{i\,}}\partial_1 -{\mathrm{i\,}}\sigma_2 \tfrac{\theta'}{2}\big) - \big(1-\sigma_2 \tfrac{V_2}{A}\big)\sigma_2A +V_1\\ &= M \Big[\sigma_1(-{\mathrm{i\,}}\partial_1) -\sigma_2 A/\gamma +\gamma(1 + \sigma_2\beta)V_1 + \sigma_3\tfrac{\theta'}{2} \Big] \end{split}\end{aligned}$$ on $C_0^\infty({\mathbb{R}},{\mathbb{C}}^2)$, using again the notation $M =\gamma(1-\sigma_2\beta)$. Beside of the residual terms $\gamma(1 + \sigma_2\beta)V_1$, $\sigma_3\theta'/2$, the operator $$\begin{aligned} \hat h := \sigma_1(-{\mathrm{i\,}}\partial_1) -\sigma_2 A/\gamma +\gamma(1 + \sigma_2\beta)V_1 + \sigma_3\tfrac{\theta'}{2} \quad \mathrm{on} \ L^2((0,\infty),{\mathbb{C}}^2)\end{aligned}$$ has a magnetic vector potential $A/\gamma$. As in the case $A_2< V_2$, we conclude Consider the operator $h$ with potentials $A,V$ satisfying [(H1$^\prime$)]{}. Then $\mathcal{D} (\hat h) = \mathcal{D} (M\hat h) =e^{-\sigma_2\theta/2 }\mathcal{D} (h)$ and the resolvent sets fulfill $ \varrho( h)= \varrho(M\hat h)$. For any $z\in \varrho(h)$ holds $$\begin{aligned} &(M\hat h-z)^{-1}=e^{- \sigma_2\theta/2} (h-z)^{-1}e^{ \sigma_2\theta/2}\quad \mbox{and}\\&\big\| (M\hat{ h}-z)^{-1}\big\| {\leqslant}\big\|(h-{\mathrm{i\,}})^{-1}\big\|\big\| e^{|\theta|} \big\|_{\infty}.\end{aligned}$$ The same statement is valid for $h_k$ with corresponding conditons and operators $\hat h_k$, $k \in {\mathbb{Z}}+\tfrac{1}{2}$. Hilbert-Schmidt bounds {#proofhilbert} ====================== In this section we prove Theorems \[mainlemma\] and \[hshk\]. We have verified these results already for the Dirac operator with purely electric potentials in Proposition \[basicHS\]. To treat the general case we use the connection between $h$ ($h_k$) and $\tilde{h}$ ($\tilde{h}_k$) established in the previous section. For the next Lemmas recall the definitions of the self-adjoint operators $h$ and $h_k$ and of $\tilde{h}$ and $\tilde{h}_k$ given in Sections \[basic\] and \[loro\], respectively. \[relativ\] Assume that $A,V$ satisfy Hypothesis [(H1)]{}. Then, there is a bounded operator $S\equiv S(A,V)$ such that $$\begin{aligned} \big (\tilde{h}-{\mathrm{i\,}}\big)^{-1} =U_2^*(-{\mathrm{i\,}}\sigma_1\partial_x-{\mathrm{i\,}})^{-1} S, \end{aligned}$$ where $[U_2 \psi] (x) = \exp \Big( {{\mathrm{i\,}}\sigma_1 \int_0^x (V_2/\gamma)(s) \mathrm{d}s}\Big)\psi(x)$. Recall that $$\begin{aligned} \tilde h=-{\mathrm{i\,}}\sigma_1\partial_x -\gamma(1 + \sigma_2\beta)\sigma_2 A_1 +V_1/\gamma +V_2/\gamma+ \sigma_3 \tfrac{\theta'}{2},\end{aligned}$$ where $\theta =\tanh^{-1}(A_2/V)$ has a uniformly bounded derivative. For shorthand notation we set $W:=-\gamma(1 + \sigma_2\beta)\sigma_2 A_1 +V_1/\gamma$. By the second resolvent identity we obtain $$\begin{aligned} \big (\tilde{h}-{\mathrm{i\,}}\big)^{-1}& = \left(-{\mathrm{i\,}}\sigma_1\partial_x+W+V_2/\gamma-{\mathrm{i\,}}\right)^{-1}\Big[1- \sigma_3 \tfrac{\theta'}{2} \, \big(\tilde{h}-{\mathrm{i\,}}\big)^{-1} \Big]\\ & = U_2^*\left(-{\mathrm{i\,}}\sigma_1\partial_x+\widetilde{W}-{\mathrm{i\,}}\right)^{-1}U_2\Big[1- \sigma_3 \tfrac{\theta'}{2} \, \big(\tilde{h}-{\mathrm{i\,}}\big)^{-1} \Big],\end{aligned}$$ where $\widetilde{W}=U_2 W U_2^*$. By the assumptions $|\widetilde{W}| \in L^p$ for some $p{\geqslant}2$, which implies that $\widetilde{W}$ is relatively compact with respect to $-{\mathrm{i\,}}\sigma_1\partial_x$ (use Kato-Seiler-Simon inequality). In particular, $\widetilde{W}(-{\mathrm{i\,}}\sigma_1\partial_x+\widetilde{W}-{\mathrm{i\,}})^{-1}$ is bounded. Finally, by the second resolvent identity we have $$\begin{aligned} \left(-{\mathrm{i\,}}\sigma_1\partial_x+\widetilde{W}-{\mathrm{i\,}}\right)^{-1}= \big(-{\mathrm{i\,}}\sigma_1\partial_x-{\mathrm{i\,}}\big)^{-1} \Big[1- \widetilde{W}\left(-{\mathrm{i\,}}\sigma_1\partial_x+\widetilde{W}-{\mathrm{i\,}}\right)^{-1}\Big],\end{aligned}$$ from which follows the claim. Throughout the rest of this section we write $F:=\sigma_2\theta/2=\sigma_2 A_2/(2V)$ (see Section \[loro\]). By a simple perturbational argument it suffices to prove the statement for $\xi=0$, i.e. for $h(0)=h$. Using Equation and Lemma \[resolventestimate\] we compute $$\begin{aligned} \mathbbm{1}_{I}(h-{\mathrm{i\,}})^{-1}&= \mathbbm{1}_{I}e^F (M\tilde h-{\mathrm{i\,}})^{-1} e^{-F}\\ &= \mathbbm{1}_{I}e^F (\tilde h-{\mathrm{i\,}})^{-1}\Big[(\tilde h-{\mathrm{i\,}}) (M\tilde h-{\mathrm{i\,}})^{-1}\Big] e^{-F},\end{aligned}$$ where the operator in $[...]$ is bounded by Lemma \[resolventestimate\] and the Closed Graph Theorem. Thus, we find some constant $c$ such that $$\begin{aligned} \left\|\mathbbm{1}_{I}(h-{\mathrm{i\,}})^{-1}\right\|_{\rm{HS}} {\leqslant}c \big\|\mathbbm{1}_{I}(\tilde h-{\mathrm{i\,}})^{-1}\big\|_{\rm{HS}}. \end{aligned}$$ The claim is now a direct consequence of Lemmas \[relativ\] and Proposition \[basicHS\]. Observe that $$\begin{aligned} \mathbbm{1}_{I}(h_k-{\mathrm{i\,}})^{-1} =e^F\mathbbm{1}_I(\tilde{h}_k-{\mathrm{i\,}})^{-1} \Big[(\tilde{h}_k-{\mathrm{i\,}}) (M\tilde{h}_k-{\mathrm{i\,}})^{-1}e^{-F} \Big],\end{aligned}$$ which implies, by the Closed Graph Theorem and Lemma \[mainlemma\], that $$\begin{aligned} \label{i1} \left\| \mathbbm{1}_{I}(h_k-{\mathrm{i\,}})^{-1} \right\|_{\rm{HS}} {\leqslant}c \big\| \mathbbm{1}_{I}(\tilde{h}_k-{\mathrm{i\,}})^{-1} \big\|_{\rm{HS}}, \end{aligned}$$ for some constant $c>0$. Here $$\tilde h_k = \sigma_1(-{\mathrm{i\,}}\partial_x) +\gamma(1 + \sigma_2\beta)\sigma_2 \big(\tfrac{k}{x} -A_1\big) +V_1/\gamma +V_2/\gamma+ \sigma_3 \tfrac{\theta'}{2}.$$ In order to make the argument more transparent we write $$\begin{aligned} \tilde h_k = \sigma_1(-{\mathrm{i\,}}\partial_x) +\sigma_2\tfrac{k}{x} +W +V_2/\gamma,\end{aligned}$$ where $W=W_1+W_2$ and $$\begin{aligned} W_1:=V_1/\gamma -\gamma(1 + \sigma_2\beta)\sigma_2 A_1,\end{aligned}$$ which has compact support and is obviously in $L^p({\mathbb{R}}^+,{\mathbb{C}}^{2\times 2})$ for some $p>2$, and $$\begin{aligned} W_2:=\gamma(1 + \sigma_2\beta)\sigma_2 \tfrac{k}{x}-\sigma_2\tfrac{k}{x} +\sigma_3 \tfrac{\theta'}{2}=\Big((\gamma-1)+\gamma \beta \sigma_2\Big) \sigma_2\tfrac{k}{x} +\sigma_3 \tfrac{\theta'}{2}.\end{aligned}$$ Due to the support properties of $A_2$ (recall that $\beta=A_2/V$ and $\gamma=(1-\beta^2)^{-1/2}$), the function $W_2$ is supported away from zero. In addition, observe that $W_2$ is uniformly bounded and hence $$\begin{aligned} \label{i2a} \big\| \mathbbm{1}_{I}(\tilde{h}_k-{\mathrm{i\,}})^{-1} \big\|_{\rm{HS}}{\leqslant}c \big\| \mathbbm{1}_{I}(\tilde{h}_k-W_2-{\mathrm{i\,}})^{-1} \big\|_{\rm{HS}}.\end{aligned}$$ According to Corollary \[corol2\] (from Section \[last\]) $W_1$ is an infinitesimally small perturbation with respect to $\tilde h_k -W= \sigma_1(-{\mathrm{i\,}}\partial_x) +\sigma_2\tfrac{k}{x} +V_2/\gamma$ and therefore $$\begin{aligned} \label{i2} \big\| \mathbbm{1}_{I}(\tilde{h}_k-W_2-{\mathrm{i\,}})^{-1} \big\|_{\rm{HS}}{\leqslant}c \big\| \mathbbm{1}_{I}(\tilde{h}_k-W-{\mathrm{i\,}})^{-1} \big\|_{\rm{HS}}.\end{aligned}$$ Let us define the self-adjoint operator $$h_0:=\sigma_1(-{\mathrm{i\,}}\partial_x) +V_2/\gamma$$ with domain, $\mathcal{D}(h_0)$, given by for $\alpha=0$. We can compare the resolvents of $\tilde{h}_k-W$ and $h_0$ as follows: Define $\chi\in C^\infty((0,\infty),[0,1])$ such that $\chi=0$ on $(0,\tfrac{1}{2})$ and $\chi=1$ on $[1,\infty)$. We compute $$\begin{aligned} \mathbbm{1}_{I}\big(\tilde{h}_k-W-{\mathrm{i\,}}\big)^{-1} &= \mathbbm{1}_{I}\chi \big(\tilde{h}_k-W-{\mathrm{i\,}}\big)^{-1}\\ &= \mathbbm{1}_{I} (h_0-{\mathrm{i\,}})^{-1}\Big[ (h_0-{\mathrm{i\,}})\chi \big(\tilde{h}_k-W-{\mathrm{i\,}}\big)^{-1}\Big].\end{aligned}$$ Observe that the operator in $[...]$ is bounded by Remark \[ws2\] and the Closed Graph Theorem (and its norm will depend on $|k|$). Thus there is a $c_{|k|}$ such that $$\begin{aligned} \label{i3} \left\| \mathbbm{1}_{I}(\tilde{h}_k-W-{\mathrm{i\,}})^{-1} \right\|_{\rm{HS}}{\leqslant}c_{|k|} \left\|\mathbbm{1}_{I} (h_0-{\mathrm{i\,}})^{-1}\right\|_{\rm{HS}}\,.\end{aligned}$$ This implies the result by Proposition \[basicHS\]. Proof of Theorem \[lastmainthm2\] {#last} ================================= The main object of this section is the proof of Theorem \[lastmainthm2\] for the Dirac operators $h_k$. As we already mentioned $h_k$ needs special care due to the $k/x$-singularity. Let us explain this a little further: Recall that for $A=V=0$ $$\begin{aligned} \label{amelie2} h_k = \sigma_1 (-{\mathrm{i\,}}\partial_x) + \sigma_2\tfrac{k}{x} \quad \mbox{on} \quad L^2((0, \infty), {\mathbb{C}}^2). \end{aligned}$$ Observe that, for $\varphi \in C^\infty_0((0,\infty),{\mathbb{C}}^2)$, $$\label{hhhh} \begin{split} \|\varphi\|_{h_k}^2 &= \|\varphi'\|^2 +\|\varphi\|^2 - {\bigg\langle \varphi,\sigma_3\frac{k}{x^2}\varphi \bigg\rangle} + \left\| \frac{k}{x} \varphi\right\|^2 \\ &{\geqslant}\|\varphi'\|^2 + \|\varphi\|^2 + (k^2 -|k|)\left\| \frac{1}{x} \varphi\right\|^2. \end{split}$$ Using this and the standard Hardy inequality on the half-line $$\begin{aligned} \label{eq:6} \int_0^\infty |\varphi'(x)|^2 dx {\geqslant}\frac{1}{4} \int_0^\infty \frac{|\varphi'(x)|^2}{x^2} dx\end{aligned}$$ one can show, for the case $|k|>1/2$, that $L^p$-perturbations can be controlled by the $h_k$-graph norm, whenever $p{\geqslant}2$. For the important case $|k|=1/2$, this argument does not work since the Hardy inequality becomes critical. Instead, a version of the Hardy-Sobolev-Maz’ya inequality on the half-line, proven recently in [@FrankLoss2012], allows us to control $L^p$-perturbations, but only for $p> 2$. These observations, made precise in Theorem \[perturbhk\] and Corollary \[corol2\], enable us to show that the operator $\mathbbm{1}_{(0,R)}(h_k-{\mathrm{i\,}})^{-1}$ is compact, provided the potentials $A,V$ are locally in $L^p$ for $p> 2$ (Corollary \[corol1\]). The latter is an important ingredient for the proof of Theorem \[lastmainthm2\] at the end of this section. \[hardy\] In view of and it is clear that $1/x$ is a perturbation with respect to $\big(-{\mathrm{i\,}}\sigma_1\partial_x + \sigma_2\tfrac{k}{x}\big)$, provided the strict inequality $|k|>1/2$ holds. \[perturbhk\] For $|k| {\geqslant}\tfrac{1}{2}$ consider $h_k$ with $A=V=0$. Then, any multiplication operator $M \in L^p((0, \infty), {\mathbb{C}}^{2\times 2})$ with $p>2$ is infinitesimally $h_k$-bounded. In addition, any multiplication operator $M \in L^2((0, \infty), {\mathbb{C}}^{2\times 2})$ is infinitesimally $h_k$-bounded for $|k| >\tfrac{1}{2}$. Let $\varphi \in C^\infty_0((0,\infty),{\mathbb{C}}^2)$ and $|k|{\geqslant}\tfrac{1}{2}$, then $$\begin{aligned} \nonumber \|\varphi\|_{h_k}^2 &= \|\varphi'\|^2 +\|\varphi\|^2 - {\bigg\langle \varphi,\sigma_3\frac{k}{x^2}\varphi \bigg\rangle} + \left\| \frac{k}{x} \varphi\right\|^2 \\\label{jakelin} &{\geqslant}\|\varphi'\|^2 + \|\varphi\|^2 + (k^2 -|k|)\left\| \frac{1}{x} \varphi\right\|^2 \\\nonumber & {\geqslant}\|\varphi'\|^2 +\|\varphi\|^2 - e^{-4(|k|-\frac{1}{2})^{2}} \frac{1}{4}\left\| \frac{1}{x} \varphi\right\|^2. \end{aligned}$$ Using the one-dimensional Sobolev inequality $\|\varphi\|_\infty^2{\leqslant}\kappa \|\varphi\|^2+\kappa^{-1}\|\varphi'\|^2$ valid for $\kappa>1$, we get that $$\begin{aligned} \label{joseflikeslatinas} \|\varphi\|_{h_k}^2&{\geqslant}\big(1- \mu(k) \big)\kappa \|\varphi\|_\infty^2+ (1-\kappa^{2})\|\varphi\|^2 + \mu(k)\left( \|\varphi'\|^2 -\frac{1}{4}\left\| \frac{1}{x} \varphi\right\|^2 \right),\end{aligned}$$ where $\mu(k) :=e^{-4(|k|-\frac{1}{2})^{2}} \in (0,1]$. (Note that the first term on the right hand side of equals zero when $k=1/2$.) By the Hardy-Sobolev-Maz’ya inequality on the half-line (see [@FrankLoss2012 Thm. 1.2]) we obtain, for $q \in (2, \infty)$ and $\theta =\frac{1}{2}(1-2q^{-1})$, a constant $c_\theta$ (depending only on $\theta$) such that $$\begin{aligned} c_\theta\|\varphi\|^2_q &{\leqslant}\left(\|\varphi'\|^2 - \frac{1}{4}\left\| \frac{1}{x} \varphi\right\|^2\right)^\theta \big(\|\varphi\|^2\big)^{1-\theta}\\ &{\leqslant}\epsilon \theta\left(\|\varphi'\|^2 - \frac{1}{4}\left\| \frac{1}{x} \varphi\right\|^2\right) + (1-\theta)\epsilon^{-\frac{\theta}{1-\theta}}\|\varphi\|^2. \end{aligned}$$ In the last step we use Young’s inequality with $\epsilon\in (0,1)$. Combining this with we conclude that $$\begin{aligned} \label{edgardolikesperuvianfood} \begin{split} \mu(k)c_\theta\|\varphi\|^2_q + (1-\mu(k)) \kappa&\epsilon \theta \|\varphi\|_\infty^2 {\leqslant}\epsilon \theta\|\varphi\|_{h_k}^2 + c(\epsilon,\kappa,\theta)\|\varphi\|^2. \end{split}\end{aligned}$$ For $M \in L^p((0, \infty), {\mathbb{C}}^{2\times 2})$ with $p>2$ we choose $\theta = p^{-1}$ (hence $p^{-1}+q^{-1}=1/2$), then yields $$\begin{aligned} \label{carrete1} \|M\varphi\|^2 {\leqslant}\|M\|_p^2\|\varphi\|_q^2 {\leqslant}\|M\|^2_p (\mu(k) c_\theta)^{-1}\left( \epsilon\theta\|\varphi\|_{h_k}^2 + c(\epsilon,\kappa,\theta)\|\varphi\|^2 \right)\end{aligned}$$ for any $\epsilon \in (0, 1]$. If $M$ is a $L^2$-function and $|k| > \tfrac{1}{2}$ we use again (dropping the first term) to obtain $$\begin{aligned} \label{carrete2} \|M\varphi\|^2 {\leqslant}\|M\|_2^2\|\varphi\|_\infty^2 {\leqslant}\frac{\kappa^{-1}}{(1-\mu(k))}\|M\|_2^2 \|\varphi\|_{h_k}^2 +\tilde{c}(\epsilon,\kappa,\theta)\|\varphi\|^2.\end{aligned}$$ Since $h_k$ is essentially self-adjoint on $C^\infty_0((0,\infty),{\mathbb{C}}^2)$ inequalities and imply the claim. \[corol2\] For $|k| {\geqslant}\tfrac{1}{2}$ consider $h_k$ with $A,V \in L_{\rm loc}^p((0, \infty), {\mathbb{R}})$ for some $p>2$. Then any multiplication operator $M \in L^{s}((0, \infty), {\mathbb{C}}^{2\times 2})$, $s>2$, with compact support is infinitesimally $h_k$-bounded. Let $\chi\in C^\infty({\mathbb{R}}^+,[0,1])$ be a smooth cut-off function which equals $1$ on the support of $M$ and vanishes for large $x$. Then, for any $\varphi \in C_0^\infty({\mathbb{R}}^+,{\mathbb{C}}^2)$ and $\epsilon\in (0,1)$ we find, by Theorem \[perturbhk\], a constant $c_\epsilon$ such that $$\begin{aligned} \label{rose} \|M\varphi\|=\|M\chi\varphi\|{\leqslant}\epsilon \big\|\big (-{\mathrm{i\,}}\sigma_1\partial_x + \sigma_2\tfrac{k}{x}\big)\chi \varphi \big\|+ c_\epsilon \|\varphi\|.\end{aligned}$$ Let us write $W:=V-\sigma_2 A \in L_{\rm loc}^p((0, \infty), {\mathbb{R}})$. Using again Theorem \[perturbhk\] we find a constant $c>0$ with $$\begin{aligned} \big\|\big (-{\mathrm{i\,}}\sigma_1\partial_x + \sigma_2\tfrac{k}{x}\big)\chi \varphi \big\|&{\leqslant}\|h_k\chi\varphi\|+\|W\chi\varphi\|\\ &{\leqslant}\|h_k\chi\varphi\|+\tfrac{1}{2} \big\|\big (-{\mathrm{i\,}}\sigma_1\partial_x + \sigma_2\tfrac{k}{x}\big)\chi \varphi \big\|+c\|\varphi\|\\ &{\leqslant}\|h_k\varphi\|+\tfrac{1}{2} \big\|\big (-{\mathrm{i\,}}\sigma_1\partial_x + \sigma_2\tfrac{k}{x}\big)\chi \varphi \big\|+(c+\|\chi'\|_\infty)\|\varphi\|.\end{aligned}$$ We get the desired result by combining this with . \[corol1\] For $|k| {\geqslant}\tfrac{1}{2}$ consider $h_k$ with $A,V \in L_{\rm loc}^p((0, \infty), {\mathbb{R}})$ for some $p>2$. Then, $h_k$ is a locally compact operator, i.e. for any $R >0$ the operator $\mathbbm{1}_{(0,R)}(h_k-{\mathrm{i\,}})^{-1}$ is compact. For $R >0$ let $\chi \in C^\infty([0,\infty),[0,1])$ be a smooth cutoff-function with $\chi (x) =1$ for $x{\leqslant}R$ and $\chi (x)=0$ for $x{\geqslant}R+1$. We compare $h_k$ with the reference operator $$\begin{aligned} h_{\rm ref} = \sigma_1(-{\mathrm{i\,}}\partial_x) +\sigma_2\frac{1}{x} \quad \mbox{on} \quad L^2((0, \infty), {\mathbb{C}}^2),\end{aligned}$$ which is known to be locally compact (see e.g. [@KMS1995]). We compute the resolvent difference $$\begin{aligned} \chi ^2 \frac{1}{h_{\rm ref}-{\mathrm{i\,}}} - \frac{1}{h_k-{\mathrm{i\,}}}\chi ^2 =&\ \frac{1}{h_k-{\mathrm{i\,}}}\left( (h_k-{\mathrm{i\,}}) \chi ^2 - \chi ^2 (h_{\rm ref}-{\mathrm{i\,}})\right) \frac{1}{h_{\rm ref}-{\mathrm{i\,}}} \\ =&\ \frac{1}{h_k-{\mathrm{i\,}}}\left( (V-\sigma_2 A) \chi ^2 -2{\mathrm{i\,}}\sigma_1 \chi \chi ' \right) \frac{1}{h_{\rm ref}-{\mathrm{i\,}}} \\ & + (k-1)\frac{1}{h_k-{\mathrm{i\,}}}x^{-1/4} \chi ^2\sigma_2 x^{-3/4}\frac{1}{h_{\rm ref}-{\mathrm{i\,}}}. \end{aligned}$$ Using that $h_{\rm ref}$ is locally compact and that $(V-\sigma_2 A) \chi $, $x^{-1/4}\chi $, and $2{\mathrm{i\,}}\sigma_1 \chi'$ are relatively $h_k$-bounded (see Corollary \[corol2\]), it suffices to show that $$\begin{aligned} \chi x^{-3/4}\frac{1}{h_{\rm ref}-{\mathrm{i\,}}} \end{aligned}$$ is a compact operator. To this end we first recall that $x^{-2}$ is bounded with respect to $h_{\rm ref}^2$ in the sense of quadratic forms (see Remark \[hardy\]). Since exponentiating to the power $3/4$ is operator monotonic we conclude that $x^{-3/4}|h_{\rm ref}-{\mathrm{i\,}}|^{-{3/4}}$ is bounded. Therefore, by the relation $$\begin{aligned} \chi x^{-3/4}\frac{1}{h_{\rm ref}-{\mathrm{i\,}}} = \ & x^{-3/4}\left|\frac{1}{h_{\rm ref}-{\mathrm{i\,}}}\right|^{3/4} {\rm sgn} \left(\frac{1}{h_{\rm ref}-{\mathrm{i\,}}} \right) \left|\frac{1}{h_{\rm ref}-{\mathrm{i\,}}}\right|^{1/4}\chi \ +\\ &x^{-3/4}\frac{1}{h_{\rm ref}-{\mathrm{i\,}}}\big(-{\mathrm{i\,}}\sigma_1\chi '\big) \frac{1}{h_{\rm ref}-{\mathrm{i\,}}}\end{aligned}$$ it suffices to show that $\chi |h_{\rm ref}-{\mathrm{i\,}}|^{-1/4}$ is compact. This, however, follows from the identity $$\begin{aligned} \left|\frac{1}{h_{\rm ref}-{\mathrm{i\,}}}\right|^{1/4}\chi &= \left(\frac{1}{h_{\rm ref}^2+1}\right)^{1/8}\chi \\ &= B(\tfrac{7}{8},\tfrac{1}{8})^{-1}\int_0^\infty \frac{1}{h_{\rm ref}^2+s+1} \chi \, \frac{1}{s^{1/8}} \,{\rm d}s\, \end{aligned}$$ (here $B(x,y)$ denotes the beta function), since a Riemannian integral of compact operators is also compact. \[rm1\] If we assume further that $|k|>1/2$ then, in view of Theorem \[perturbhk\], the statements of Corollaries \[corol2\] and \[corol1\] are also valid for $L^2$-perturbations. In this proof we slightly modify the argument of [@Last1996 Theorem 6.2] for the operator defined on the half-line. Let $\mathcal{B}({\mathbb{R}})$ denote the Borel $\sigma$-algebra on ${\mathbb{R}}$. Given any $\psi \in L^2({\mathbb{R}}^+, {\mathbb{C}}^2)$ we write its associated spectral measure (with respect to $h_k$) as $$\begin{aligned} \mu_{\psi} : \mathcal{B}({\mathbb{R}}) \to [0, \infty), \qquad \Omega \mapsto {\big\langle \psi,\mathbbm{1}_\Omega(h_k)\psi \big\rangle}. \end{aligned}$$ Since $\mu_\psi$ is absolutely continuous with respect to the Lebesgue measure it can be decomposed as a sum of mutually singular measures $\mu_\psi=\mu_{\psi,1}+\mu_{\psi,2}$, where $\mu_{\psi,2}({\mathbb{R}})<\|\psi\|^2/4$ and $\mu_{\psi,1}$ is a uniformly Lipschitz continuous measure, i.e. there is a constant $C>0$ such that for any interval with Lebesgue measure $|I|<1$, $\mu_{\psi,1}(I)<C|I|$ (this can be verified decomposing the Radom-Nykodym derivative, $f_\psi$, associated to $\mu_{\psi}$ as $f_\psi=f_\psi\mathbbm{1}_{\{f_\psi<\alpha\}} +f_\psi\mathbbm{1}_{\{f_\psi>\alpha\}} $ for $\alpha>0$ sufficiently large; for a more general statement involving uniform $\alpha$-Hölder continuity see [@Last1996 Theorem 4.2]). For $j\in\{1,2\}$ define $\psi_j:=\mathbbm{1}_{S_j}(h_k)\psi$ where $S_j\subset {\mathbb{R}}$ is the support of the measure $\mu_{\psi,j}$. Then, for any $\Omega \in \mathcal{B}({\mathbb{R}}) $, we have $$\begin{aligned} \mu_{\psi_j}(\Omega)&={\big\langle \psi_j,\mathbbm{1}_\Omega(h_k) \psi_j \big\rangle} ={\big\langle \psi,\mathbbm{1}_{\Omega\cap S_j}(h_k) \psi \big\rangle}\\ &=\mu_{\psi}(\Omega\cap S_j)=\mu_{\psi,1}(\Omega\cap S_j) +\mu_{\psi,2}(\Omega\cap S_j)= \mu_{\psi, j}(\Omega),\end{aligned}$$ where in the last equality we use that the measures $\mu_{\psi,j}$ are disjointly supported. Thus, we get that $\mu_\psi=\mu_{\psi_1}+ \mu_{\psi_2}$. Note that $\psi_1\not=0$ since $$\|\psi_1\|^2=\mu_{\psi_1}({\mathbb{R}})=\|\psi\|^2- \mu_{\psi_2}({\mathbb{R}}){\geqslant}3\|\psi\|^2 /4.$$ For any $R>1$ we have $$\begin{aligned} \label{eq:5} \| x^{p/2} e^{-{\mathrm{i\,}}h_k t}\psi \|^2 {\geqslant}\| R^{p/2} \mathbbm{1}_{ (R, \infty )}e^{-{\mathrm{i\,}}h_k t}\psi \|^2 {\geqslant}R^{p} \big( \|\psi\|^2 - \| \mathbbm{1}_{ (0,R )} e^{-{\mathrm{i\,}}h_k t}\psi \|^2 \big).\end{aligned}$$ We observe that since $\psi_1$ is orthogonal to $\psi_2$ the triangular inequality yields $$\label{eq:4} \begin{split} \|\mathbbm{1}_{ (0,R )} e^{-{\mathrm{i\,}}h_k t}\psi \|^2&{\leqslant}2\|\mathbbm{1}_{ (0,R )} e^{-{\mathrm{i\,}}h_k t}\psi_1\|^2+2\|\psi_2\|^2\\&{\leqslant}2\|\mathbbm{1}_{ (0,R )} e^{-{\mathrm{i\,}}h_k t}\psi_1\|^2+\tfrac{1}{2}\|\psi\|^2. \end{split}$$ In order to use Theorem \[hshk\] we replace the cut-off function above by one supported away from zero. Note that by Corollary \[corol1\] and the RAGE theorem we find a $T_0>0$ such that for all $T>T_0$ one has that $$\begin{aligned} \langle\| \mathbbm{1}_{ (0,R )} e^{-{\mathrm{i\,}}h_k t}\psi_1 \|^2\rangle_T&=\langle\| \mathbbm{1}_{ (0,1)} e^{-{\mathrm{i\,}}h_k t}\psi_1 \|^2\rangle_T+\langle\| \mathbbm{1}_{ (1,R)} e^{-{\mathrm{i\,}}h_k t}\psi_1 \|^2\rangle_T\\ &{\leqslant}\tfrac{1}{8}\|\psi\|^2+\langle\| \mathbbm{1}_{ (1,R)} e^{-{\mathrm{i\,}}h_k t}\psi_1 \|^2\rangle_T.\end{aligned}$$ Combining the latter bound with and we readily obtain, for $T>T_0,$ $$\begin{aligned} \label{eq:7} \begin{split} \big\langle \| x^{p/2} e^{-{\mathrm{i\,}}h_k t}\psi \|^2 \big\rangle_T {\geqslant}R^{p} \big( \tfrac{1}{4}\|\psi\|^2 - 2 \big\langle\| \mathbbm{1}_{ (1,R )} e^{-{\mathrm{i\,}}h_k t}\psi_1 \|^2 \big\rangle_T \big). \end{split}\end{aligned}$$ Next we recall (see [@Last1996 Theorem 3.2]) that given a self-adjoint operator $H$ and a Hilbert-Schmidt operator $A$ one finds a constant $c_\varphi$ such that $$\begin{aligned} \label{Last3.2} \langle \|Ae^{-{\mathrm{i\,}}t H}\varphi\|\rangle_T{\leqslant}c_\varphi \|A\|^2_{\rm HS}T^{-1}\end{aligned}$$ provided the $H$-spectral measure associated to $\varphi$ is Lipschitz continuous. Applying this and Theorem \[hshk\] we obtain $$\begin{aligned} \begin{split} \label{eq:31} \langle\| \mathbbm{1}_{ (1,R)} e^{-{\mathrm{i\,}}h_k t}\psi_1 \|^2\rangle_T&= \langle\| \mathbbm{1}_{ (1,R)} \mathbbm{1}_\triangle(h_k) e^{-{\mathrm{i\,}}h_k t}\psi_1 \|^2\rangle_T\\ &{\leqslant}c_{\psi_1}T^{-1}\| \mathbbm{1}_{ (1,R)}(h_k-{\mathrm{i\,}})^{-1} (h_k-{\mathrm{i\,}})\mathbbm{1}_\triangle(h_k) \|_{\rm HS}^2 \\ &{\leqslant}c_{\psi_1} c_{\triangle} C_k R T^{-1} \equiv \tfrac{1}{2} \widehat C_k(\psi_1, \Delta, k) R T^{-1}, \end{split}\end{aligned}$$ where $c_{\triangle} =\|(h_k-{\mathrm{i\,}})\mathbbm{1}_\triangle\|^2$. Hence, using the latter bound in we get, for $T>T_0$, that $$\begin{aligned} \big\langle \| x^{p/2} e^{-{\mathrm{i\,}}h_k t}\psi \|^2 \big\rangle_T & {\geqslant}R^{p} \big( \tfrac{1}{4}\|\psi\|^2 - \widehat C_k(\psi_1, \Delta, k) R T^{-1} \big)\\ &=\frac{1}{8^{p+1}\widehat C_k{(\psi_1, \Delta, k)}^p}\|\psi\|^{2p+2}\, T^{p},\end{aligned}$$ where in the last equality we have chosen $$\begin{aligned} R\equiv R(T)=\frac{\|\psi\|^2}{8\widehat C_k(\psi_1, \Delta, k) } \ T.\end{aligned}$$ Finally note that the inequality is trivially fulfilled for finite $T\in [0,T_0]$ by just choosing the constant $ C_k(\psi_1, \Delta, k) $ suitably. [**Acknowledgments.**]{} The authors want to thank Jean-Marie Barbaroux, Jean-Claude Cuenin and Karl-Michael Schmidt for useful discussions and remarks. J.M. also likes to thank the [*Faculdad de Física de la Pontificia Universidad Católica de Chile*]{} for the hospitality during his research stay. J.M. has been supported by SFB-TR12 “Symmetries and Universality in Mesoscopic Systems" of the DFG. E.S. has been supported by Fondecyt (Chile) project 1141008 and Iniciativa Científica Milenio (Chile) through the Millenium Nucleus RC–120002 “Física Matemática” . Remarks on Self-adjointness of Dirac operators {#s.a.} ============================================== In this section we state and prove some facts concerning self-adjointness of the one-dimensional Dirac operators discussed in Section \[basic\]. These facts are well known, however, they are not easy to find in the standard literature. \[lpcin0\] Let $|k| {\geqslant}\tfrac{1}{2}$, then $h_k$ is in the limit point case at $0$. It suffices to show that there is a solution to the eigenvalue problem $$\begin{aligned} \label{evp} h_k\varphi=\lambda\varphi\end{aligned}$$ which is not square-integrable at $0$. According to [@EasthamSchmidt2008 Theorem1] (see also [@Titchmarsh1961]), for $k{\geqslant}\tfrac{1}{2}$, there is a unique solution to with the asymptotic behaviour $$\begin{aligned} u(r)=(o(1), 1+o(1))^{\rm T}x^{k}\quad\mbox{as}\quad x\to 0.\end{aligned}$$ (Note that [@EasthamSchmidt2008 Theorem 1] is only stated for the case $A=0$, however, the same argument applies provided $A$ is integrable at zero.) Let $w$ be a linear independent solution of such that the Wronski determinant $W(u,w):=u_1w_2-u_2w_1\equiv 1$. Assume that $\liminf_{x\to 0} |w(x)|x^{k}=0$, then clearly $\liminf_{x\to 0} W(u,w)(x)=0$ which is a contradiction. Hence $w$ can not be square-integrable at $0$. An analogous argument holds also when $k{\leqslant}-1/2$. \[good-core\] Let $k\in {\mathbb{Z}}+\tfrac{1}{2}$, then $C_0^\infty((0,\infty),{\mathbb{C}}^2)$ is dense in $\mathcal{D}_0(h_k)$ with respect to the $h_k$-graph norm. The analogous statement holds for $C_0^\infty({\mathbb{R}},{\mathbb{C}}^2)\subset \mathcal{D}_0(h)$ in the $h$-graph norm. We give the details of the proof only for the operator defined on the whole real line. First note that clearly $C_0^\infty({\mathbb{R}},{\mathbb{C}}^2)$ is a subset of $\mathcal{D}_0(h)$. Let $\psi \in \mathcal{D}_0(h)$ and let $K$ be a compact set which contains the support of $\psi$. Since $\psi\in C_0({\mathbb{R}},{\mathbb{C}}^2)$ and $(V-\sigma_2 A)\in L^2_{\rm loc}$ we have that $$-{\mathrm{i\,}}\sigma_1\psi'=h\psi-(V-\sigma_2 A)\psi \in L^2({\mathbb{R}},{\mathbb{C}}^2),$$ which implies that $\psi\in H^1({\mathbb{R}},{\mathbb{C}}^2)$. Let $(\psi_n)_{n\in{\mathbb{N}}}\subset C_0^\infty$ be a sequence of mollifiers of $\psi$ whose support is also contained in $K$. We estimate $$\begin{aligned} \|\psi-\psi\|_h^2&=\|\psi-\psi_n\|_2^2+\|h(\psi-\psi_n)\|_2^2{\leqslant}\|\psi-\psi_n\|_{H^1}^2 +\|(V-\sigma_2 A)(\psi-\psi_n)\|_2\\ &{\leqslant}\|\psi-\psi_n\|_{H^1}^2+\|\psi-\psi_n\|_\infty^2\,\|(V-\sigma_2 A)\mathbbm{1}_{K}\|_2^2.\end{aligned}$$ By the $H^1$-convergence of mollifiers we know that $\|\psi-\psi_n\|_{H^1}^2\to 0$. Moreover, the Sobolev inequality in dimension one implies that $\|\psi-\psi_n\|_\infty^2\to 0$. Hence, $\psi_n$ converges to $\psi$ in the graph norm, as claimed. The argument for the operator $h_k$ is completely analogous. Just note that $k/x \in L^2_{\rm loc}((0,\infty))$. Computation of a resolvent kernel {#expres} ================================= In order to compute a resolvent kernel of the operator $\sigma_1(-{\mathrm{i\,}}\partial_x)$ on $L^2((0, \infty), {\mathbb{C}}^2)$ with the boundary condition $\psi_1(0) =0$, we use the unitary matrix $\hat U = \tfrac{1}{\sqrt{2}}(\mathbbm{1} +{\mathrm{i\,}}\sigma_3)$ to transform the problem to the operator $$\sigma_2({\mathrm{i\,}}\partial_x) = \frac{1}{2} (\mathbbm{1} + {\mathrm{i\,}}\sigma_3) \sigma_1(-{\mathrm{i\,}}\partial_x) (\mathbbm{1} - {\mathrm{i\,}}\sigma_3) = \hat U \sigma_1(-{\mathrm{i\,}}\partial_x) \hat U^*$$ on $L^2((0, \infty), {\mathbb{C}}^2)$ with the same boundary conditions. The corresponding relation for the resolvent is $$\label{resolventapp} \frac{1}{\sigma_1(-{\mathrm{i\,}}\partial_x) -{\mathrm{i\,}}} = \hat U^* \frac{1}{\sigma_2({\mathrm{i\,}}\partial_x) -{\mathrm{i\,}}} \, \hat U.$$ We note that, by [@Weidmann2 Section 15.5], the kernel of the resolvent can be given in terms of a fundamental system $u_1(z,\,\cdot \,), u_2(z,\,\cdot \,)$ of the ODE $(\sigma_1(-{\mathrm{i\,}}\partial_x) -z)u =0$. For $z \in {\mathbb{C}}\setminus {\mathbb{R}}$ this is given as follows $$\frac{1}{\sigma_2({\mathrm{i\,}}\partial_x) -z} (x_1,x_2) = \begin{cases} \sum_{j,k =1}^2 m_{j,k}^+(z) \overline{u_j(\overline z, x_1)}u_k ^{\rm T} (z,x_2) & \mathrm{if} \; x_1 > x_2>0, \\[0.1cm] \sum_{j,k =1}^2 m_{j,k}^-(z) \overline{u_j(\overline z, x_1)}u_k ^{\rm T}(z,x_2) & \mathrm{if} \; x_2 > x_1 >0, \end{cases}$$ where and $m^+(z), m^-(z)$ are $2\times 2$ matrices whose coefficients are given in terms of certain complex numbers $m_a(z), m_b(z)$ that are chosen such that $m_a(z) u_1(z,\,\cdot \,)+ u_2(z,\,\cdot \,)$ fulfill the boundary condition at $0$ and $m_b(z) u_1(z,\,\cdot \,)+ u_2(z,\,\cdot \,)$ is square-integrable at $\infty$ (see [@Weidmann2 Section 15.5]). Hence, using the fundamental system $$u_1(z,x) = \begin{pmatrix} \cos zx \\ \sin zx \end{pmatrix}, \quad \quad \quad u_2(z,x) = \begin{pmatrix} -\sin zx \\ \cos zx \end{pmatrix},$$ we obtain that $m_a({\mathrm{i\,}}) =0, m_b({\mathrm{i\,}}) = {\mathrm{i\,}}$. Therefore, $$m^+({\mathrm{i\,}}) = \begin{pmatrix} 0 & -1 \\ 0 & {\mathrm{i\,}}\end{pmatrix}, \quad \quad \quad m^-({\mathrm{i\,}}) = \begin{pmatrix} 0 & 0 \\ -1 & {\mathrm{i\,}}\end{pmatrix}.$$ In addition, using that $$u_1(i,x) = \overline{u_1(-{\mathrm{i\,}},x)} = \begin{pmatrix} \cosh x \\ {\mathrm{i\,}}\sinh x \end{pmatrix}, \quad \quad \quad u_2(i,x) = \overline{u_2(-{\mathrm{i\,}},x)} = \begin{pmatrix} -{\mathrm{i\,}}\sinh x \\ \cosh x \end{pmatrix},$$ we get the explicit resolvent kernel $$\frac{1}{\sigma_2({\mathrm{i\,}}\partial_x) -{\mathrm{i\,}}} (x_1,x_2) = \begin{cases} e^{-x_1} \begin{pmatrix} {\mathrm{i\,}}\sinh x_2 & -\cosh x_2 \\ \sinh x_2 & {\mathrm{i\,}}\cosh x_2 \end{pmatrix} & \mathrm{if} \ x_1 > x_2>0, \\[0.5cm] e^{-x_2} \begin{pmatrix} {\mathrm{i\,}}\sinh x_1 & \sinh x_1 \\ -\cosh x_1 & {\mathrm{i\,}}\cosh x_1 \end{pmatrix} & \mathrm{if} \ x_2 > x_1>0. \end{cases}$$ Finally, by we get . Proof of Corollaries \[appl1\] and \[appl2\] {#proofappl} ============================================ We observe that in the case of a translation symmetry in $x_2$-direction, we can write $$\begin{aligned} \langle\| |x_1|^{p/2}e^{-{\mathrm{i\,}}tH } \psi\|^2\rangle_T = \int_{-\infty}^\infty \big \langle \big\| |x_1|^{p/2} e^{-{\mathrm{i\,}}th(\xi) } \widehat\psi (\, \cdot \, , \xi) \big \|^2\big\rangle_T \, {\rm d}\xi\end{aligned}$$ By assumption we find $\xi_0>0$ large enough such that $$\begin{aligned} M:= \big\{\xi \in [-\xi_0,\xi_0] \,|\, \widehat\psi (\, \cdot \, , \xi) \neq 0, \ \widehat\psi (\, \cdot \, , \xi) \in P_{ac}(h(\xi)) L^2({\mathbb{R}}, {\mathbb{C}}^2)\big\}\end{aligned}$$ has non-zero Lesbegue measure. Using that $$\begin{aligned} \mathbbm{1}_\Delta(H) = \int_{\mathbb{R}}^\oplus \mathbbm{1}_\Delta (h(\xi)) {\rm d} \xi\end{aligned}$$ (see [@Reed_simon_4 Theorem XIII.85]), we conclude $\widehat\psi (\, \cdot \, , \xi) \in \mathbbm{1}_\Delta (h(\xi)) L^2({\mathbb{R}}, {\mathbb{C}}^2)$ for a.e. $\xi\in {\mathbb{R}}$ if $\psi \in \mathbbm{1}_\Delta (H) L^2({\mathbb{R}}^2, {\mathbb{C}}^2)$. Hence, Theorem \[lastmainthm1\] implies $$\begin{aligned} \langle\| |x_1|^{p/2}e^{-{\mathrm{i\,}}t H } \psi\|^2\rangle_T {\geqslant}\int_M\big \langle \big\| |x_1|^{p/2} e^{-{\mathrm{i\,}}th(\xi) } \widehat\psi (\, \cdot \, , \xi) \big \|^2\big\rangle_T \, {\rm d}\xi {\geqslant}T^p \int_M C_\xi (\psi, \Delta ,p) {\rm d}\xi \,,\end{aligned}$$ which give us the desired bound since $\int_M C_\xi (\psi, \Delta ,p) {\rm d}\xi =: C (\psi, \Delta ,p) > 0$. Now let us consider the case, when $H$ is spherically symmetric and fulfills the assumptions of Corollary \[appl1\]. We write $$\begin{aligned} H \cong \bigoplus_{k \in \, {\mathbb{Z}}+\frac{1}{2}} h_k,\end{aligned}$$ and use for $$\begin{aligned} \label{kunigunde} \psi =\sum_{j \in {\mathbb{Z}}} \psi_j \in \bigoplus_{j \in {\mathbb{Z}}}P_{ac}(h_j)L^2({\mathbb{R}}^+,{\mathbb{C}}^2) \end{aligned}$$ non-zero the estimate $$\begin{aligned} \| |{{\bf x}}|^p e^{-{\mathrm{i\,}}tH}\psi \|^2 = \sum_{j \in {\mathbb{Z}}} \| r^p e^{-{\mathrm{i\,}}t h_j}\psi_j \|^2 {\geqslant}\sum_{j =-l}^{l} \| r^p e^{-{\mathrm{i\,}}t h_j}\psi_j \|^2,\end{aligned}$$ where $l \in {\mathbb{N}}$ is chosen to be so large that $\sum_{j =-l}^{l} \| \psi_j \|^2 {\geqslant}\tfrac{1}{2}\|\psi\|$. Observing that $\mathbbm{1}_\Delta(H) = \bigoplus_{j \in {\mathbb{Z}}} \mathbbm{1}_\Delta(h_j)$, we deduce from Theorem \[lastmainthm2\] and that $$\begin{aligned} \| |{{\bf x}}|^p e^{-{\mathrm{i\,}}tH}\psi \|^2 {\geqslant}\sum_{j =-l}^{l} C_j(\psi_j, \Delta ,p)\, T^p = C(\psi, \Delta ,p) \, T^p\end{aligned}$$ (where we set $C_j(\psi_j, \Delta ,p)=0$ if $\psi_j=0$), with $C(\psi, \Delta ,p) >0$. [10]{} A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim. The electronic properties of graphene. , 81:109–162, 2009. J.-M. Combes. Connections between quantum dynamics and spectral properties of time-evolution operators. In [*Differential equations with applications to mathematical physics*]{}, volume 192 of [*Math. Sci. Engrg.*]{}, pages 59–68. Academic Press, Boston, MA, 1993. M. S. P. Eastham and K. M. Schmidt. Asymptotics of the spectral density for radial [D]{}irac operators with divergent potentials. , 44(1):107–129, 2008. C. L. Fefferman and M. I. Weinstein. Honeycomb lattice potentials and [D]{}irac points. , 25(4):1169–1220, 2012. C. L. Fefferman and M. I. Weinstein. Wave packets in honeycomb structures and two-dimensional [D]{}irac equations. , 326(1):251–286, 2014. R. L. Frank and M. Loss. Hardy-[S]{}obolev-[M]{}az’ya inequalities for arbitrary domains. , 97(1):39–54, 2012. I. Guarneri. Spectral properties of quantum diffusion on discrete lattices. , 10(2):95–100, 1989. B. Huard, J. A. Sulpizio, N. Stander, K. Todd, B. Yang, and D. Goldhaber-Gordon. Transport measurements across a tunable potential barrier in graphene. , 98:236803, 2007. O. Klein. Die [R]{}eflexion von [E]{}lektronen an einem [P]{}otentialsprung nach der relativistischen [D]{}ynamik von [D]{}irac. , 53(3-4):157–165, 1929. Y. Last. Quantum dynamics and decompositions of singular continuous spectra. , 142(2):406–445, 1996. J. Mehringer and E. Stockmeyer. Confinement-deconfinement transitions for two-dimensional [D]{}irac particles. , 266(4):2225–2250, 2014. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I.V. Grigorieva, and A.A. Firsov. Electric field effect in atomically thin carbon films. , 306(5696):666–669, 2004. M. Reed and B. Simon. . Academic Press, New York, 1978. K. M. Schmidt. Dense point spectrum and absolutely continuous spectrum in spherically symmetric [D]{}irac operators. , 7(4):459–475, 1995. K. M. Schmidt and O. Yamada. Spherically symmetric [D]{}irac operators with variable mass and potentials infinite at infinity. , 34(3):211–227, 1998. E. Seiler and B. Simon. Bounds in the [Y]{}ukawa2 quantum field theory: upper bound on the pressure, [H]{}amiltonian bound and linear lower bound. , 45(2):99–114, 1975. B. Simon. , volume 35 of [*London Mathematical Society Lecture Note Series*]{}. Cambridge University Press, Cambridge-New York, 1979. B. Simon. Schrödinger semigroups. , 7(3):447–526, 1982. N. Stander, B. Huard, and D. Goldhaber-Gordon. Evidence for [K]{}lein tunneling in graphene $p\mathrm{\text{-}}n$ junctions. , 102:026807, 2009. B. Thaller. . Texts and Monographs in Physics. Springer-Verlag, Berlin, 1992. E. C. Titchmarsh. On the nature of the spectrum in problems of relativistic quantum mechanics. , 12:227–240, 1961. J. Weidmann. , volume 1258 of [*Lecture Notes in Mathematics*]{}. Springer-Verlag, Berlin, 1987. J. Weidmann. . Mathematische Leitfäden \[Mathematical Textbooks\]. B. G. Teubner, Stuttgart, 2003. Anwendungen \[Applications\]. A. F. Young and P. Kim. Quantum interference and [K]{}lein tunnelling in graphene heterojunctions. , 5(3):222–226, 2009.

--- abstract: 'The Bethe-Salpeter equation for ground state of two fermions exchanging a gauge boson presents divergences in the momentum transverse, even in the ladder aproximation projected in light-front. Gauge theories with light-front gauge also present the difficulty associated to the instantaneous term of the propagator of a system composed by fermions bosons-exchange interaction. We used a prescription that allowed an apropriate description of the singularity in the propagator of the gauge boson in the light-front.' author: - | B.M.Pimentel$^{a}$, J.H.O.Sales$^{a}$ and Tobias Frederico$^{b}$\ $^{a}$Instituto de Física Teórica-UNESP, 01405-900 São Paulo, Brazil.\ $^{b}$Instituto Tecnológico de Aeronaútica, CTA, 12228-900\ São José dos Campos, Brazil. title: '**Gauge field divergences in the light-front**' --- Light-Front Dynamics: Definition ================================ Beginning from Dirac’s idea [@dirac] of representing the dynamics of the quantum system at ligth-front times $x^{+}=t+z$, we derive the Green’s function from the covariant propagator that evolutes the system from one light-front hyper-surface to another one. The light-front Green’s function is the probability amplitude for an initial state at $x^{+}=0$ do evolvey to a final state in the Fock-state at some $x^{+}$, where the evolution operator is defined by the light-front Hamiltonian [@Kogut] The Scalar Field Propagator =========================== The Feynman propagator for the scalar field is $$S(x^{\mu })=\int \frac{d^{4}k}{\left( 2\pi \right) ^{4}}\frac{ie^{-ik^{\mu }x_{\mu }}}{k^{2}-m^{2}+i\varepsilon }. \label{1}$$ and in terms of light-front variables [@jhs2002], we have $$S(x^{+})=\frac{1}{2}\int \frac{dk^{-}dk^{+}dk^{\perp }}{\left( 2\pi \right) }\frac{ie^{\frac{-i}{2}k^{-}x^{+}}}{k^{+}\left( k^{-}-\frac{k_{\perp }^{2}+m^{2}-i\varepsilon }{k^{+}}\right) }. \label{2}$$ The Fourier transform of the single boson state propagator to the in the light-front time is giver by: $$\widetilde{S}(k^{-})=\int dk^{+}dk^{\perp }\frac{i}{k^{+}\left( k^{-}-\frac{k_{\perp }^{2}+m^{2}-i\varepsilon }{k^{+}}\right) }. \label{3}$$ Fermion Field ============= Let $S_{\text{F}}$ denote fermion field propagator in covariant theory $$S_{\text{F}}(x^{\mu })=\int \frac{d^{4}k}{\left( 2\pi \right) ^{4}}\frac{i(\rlap\slash k_{\text{on}}+m)}{k^{2}-m^{2}+i\varepsilon }e^{-ik^{\mu }x_{\mu }}, \label{4}$$ where $\rlap\slash k_{\text{on}}=\frac{1}{2}\gamma ^{+}\frac{(k^{\perp })^{2}+m^{2}}{k^{+2}}+\frac{1}{2}\gamma ^{-}k^{+}-\gamma ^{\perp }k^{\perp }$. Using light-front variables in Eq.(\[4\]), we have $$S_{\text{F}}(x^{+})=\frac{i}{2}\int \frac{dk^{-}dk^{+}dk^{\perp }}{\left( 2\pi \right) }\left[ \frac{\rlap\slash k_{on}+m}{k^{+}\left( k^{-}-k_{on}^{-}+\frac{i\varepsilon }{k^{+}}\right) }+\frac{\gamma ^{+}}{2k^{+}}\right] e^{\frac{-i}{2}k^{-}x^{+}}. \label{5}$$ We note that for the fermion field, light-front propagator differs from the Feynmam propagator by an instantaneous propagator. Gauge Boson Propagator ====================== Let $S^{\mu \nu }$gauge propagator, $$S^{\mu \nu }(x^{\mu })=\int \frac{d^{4}k}{\left( 2\pi \right) ^{4}}\frac{ie^{-ik^{\mu }x_{\mu }}}{k^{2}+i\varepsilon }\left[ \frac{-nkg^{\mu \nu }+n^{\mu }k^{\nu }+n^{\nu }k^{\mu }}{nk}\right] , \label{6}$$ where we choose the light-front gauge $A^{+}=0$, $n^{\mu }=(1,0,0,-1)$ and the metric tensor is given from [@Kogut]$.$ The light-front components (\[6\]) can be as written $S^{+-}=S^{-+}=S^{++}=S^{+\perp }=0$ and $$S^{--}=4\frac{ik^{-}}{k^{+}(k^{2}+i\varepsilon )},\text{ }S^{-\perp }=S^{\perp -}=2\frac{ik^{\perp }}{k^{+}(k^{2}+i\varepsilon )},\text{ }S^{\perp \perp }=-1\frac{i}{k^{2}+i\varepsilon } \label{7a}$$ Interaction in First Order ========================== We consider the fermion-antifermion system in the light-front with one-gauge boson exchange ($A^{+}=0$), for which the interaction Lagrangian density is given by $$\mathcal{L}_{I}=g\overline{\Psi }_{1}\gamma _{\mu }A^{\mu }\Psi _{1}+g\overline{\Psi }_{2}\gamma _{\nu }A^{\nu }\Psi _{2}. \label{8}$$ The fermion corresponds to the field $\Psi $ with rest masses $m$ and the exchanged gauge boson to the field $A^{\mu }$ with mass $\mu =0.$ The coupling constant is $g.$ The perturbative correction to the two-body propagator which comes from the  exchange of one intermediate virtual boson, is $$\begin{aligned} \Delta S_{g^{2}}(x^{+}) &=&\left( ig\right) ^{2}\int d\overline{x}_{1}^{+}d\overline{x}_{2}^{+}S_{k^{\prime }}(x^{+}-\overline{x}_{1}^{+})(\gamma _{\mu })S_{k}(\overline{x}_{1}^{+}) \label{9} \\ &&S^{\mu \nu }(\overline{x}_{2}^{+}-\overline{x}_{1}^{+})S_{p}(x^{+}-\overline{x}_{2})(\gamma _{\nu })S_{p^{\prime }}(\overline{x}_{2}^{+}). \notag\end{aligned}$$ The intermediate boson propagates between the time interval $\overline{x}_{2}^{+}-\overline{x}_{1}^{+}.$ The labels in the particle propagators $k$ and $p$ indicates initial and $k^{\prime }$ and $p^{\prime }$ final states. Performing the Fourier transform from $x^{+}$ to $P^{-}$ and for the total kinematical momentum $P^{+}$, which we choose positive, and $P^{\perp }$. The double integration in $k^{-}$ is performed analytically in Eq.(\[10a\]), $$\begin{aligned} \Delta S_{g^{2}}(P^{-}) &=&\frac{-\left( ig\right) ^{2}i}{(4\pi )^{2}}\int \frac{dk^{-}dk^{\prime ^{-}}}{k^{+}k^{\prime ^{+}}(P^{+}-k^{\prime +})(P^{+}-k^{+})} \\ &&\left\{ \frac{\rlap\slash k_{on}^{\prime }+m}{\left( k^{\prime -}-k_{on}^{\prime -}+\frac{i\varepsilon }{k^{\prime +}}\right) }\right. \begin{array}{c} \gamma _{-} \end{array} \frac{\rlap\slash k_{on}+m}{\left( k^{-}-k_{on}^{-}+\frac{i\varepsilon }{k^{+}}\right) } \\ &&\frac{4\left( k^{-}-k^{\prime -}\right) }{(q^{+})^{2}\left( k^{-}-k^{\prime -}-q_{on}^{-}+\frac{i\varepsilon }{q^{+}}\right) }\frac{\rlap\slash p_{on}^{\prime }+m}{\left( p^{\prime -}-p_{on}^{\prime -}+\frac{i\varepsilon }{p^{\prime +}}\right) } \\ && \begin{array}{c} \gamma _{-} \end{array} \frac{\rlap\slash p_{on}+m}{\left( p^{-}-p_{on}^{-}+\frac{i\varepsilon }{p^{+}}\right) }+\end{aligned}$$ $$\begin{aligned} &&+\frac{\rlap\slash k_{on}^{\prime }+m}{\left( k^{\prime -}-k_{on}^{\prime -}+\frac{i\varepsilon }{k^{\prime +}}\right) } \notag \\ && \begin{array}{c} \gamma _{-} \end{array} \frac{\rlap\slash k_{on}+m}{\left( k^{-}-k_{on}^{-}+\frac{i\varepsilon }{k^{+}}\right) }\frac{2\left( k^{\perp }-k^{\prime \perp }\right) }{(q^{+})^{2}\left( k^{-}-k^{\prime -}-q_{on}^{-}+\frac{i\varepsilon }{q^{+}}\right) } \notag \\ &&\frac{\rlap\slash p_{on}^{\prime }+m}{\left( p^{\prime -}-p_{on}^{\prime -}+\frac{i\varepsilon }{p^{\prime +}}\right) } \begin{array}{c} \gamma _{\perp } \end{array} \frac{\rlap\slash p_{on}+m}{\left( p^{-}-p_{on}^{-}+\frac{i\varepsilon }{p^{+}}\right) }+\left[ \gamma _{\perp }\rightarrow \gamma _{-}\right] + \notag \\ &&\frac{\rlap\slash k_{on}^{\prime }+m}{\left( k^{\prime -}-k_{on}^{\prime -}+\frac{i\varepsilon }{p^{\prime +}}\right) } \begin{array}{c} \gamma _{\perp } \end{array} \frac{\rlap\slash k_{on}+m}{\left( k^{-}-k_{on}^{-}+\frac{i\varepsilon }{p^{+}}\right) } \notag \\ &&\frac{(-1)}{(q^{+})^{2}\left( k^{-}-k^{\prime -}-q_{on}^{-}+\frac{i\varepsilon }{q^{+}}\right) } \label{10a} \\ &&\frac{\rlap\slash p_{on}^{\prime }+m}{\left( p^{\prime -}-p_{on}^{\prime -}+\frac{i\varepsilon }{p^{\prime +}}\right) } \begin{array}{c} \gamma _{\perp } \end{array} \left. \frac{\rlap\slash p_{on}+m}{\left( p^{-}-p_{on}^{-}+\frac{i\varepsilon }{p^{+}}\right) }\right\} , \notag\end{aligned}$$ Conclusion ========== From equation (\[10a\]), we verified the existence of singularity in the components $(--)$ and $(-\perp )$ in the coordinate $q^{+}=k^{+}-k^{\prime +}.$ We hoped to remove those singularity using the technique of displacement $\left( \delta ^{+}\right) $ of the pole of the phase space in $q^{+}$ [@jhs97]. T.F and B.M. Pimentel thank to CNPq for partial support. J.H.O. Sales is supported by FAPESP/Brazil pos-doctoral fellowship. We acknowledge discussions with J.F.Libonati and J.Messias. [9]{} P. A. M. Dirac, Rev. Mod. Phys. **21**, 392 (1949). A.Harindranath, *Light-Front Quantization and Non-Perturbative QCD*, editors J.P.Vary and F.Wölz, International Institute of Theoretical and Applied Physics (1997). S.D.Glasek and K.G.Wilson Phys.Rev.**D49** (1994) 4214. J.H.O. Sales, T.Frederico, B.V. Carlson and P.U.Sauer, Phys. Rev. **C63**:064003(2001). J.H.O. Sales, T.Frederico, B.V. Carlson and P.U.Sauer, Phys. Rev. **C61**:044003(2000). J.P.B.C.de Melo, J.H.O.Sales, T.Frederico and P.U.Sauer, Nucl. Phys.**A631** (1998) 574c.

--- abstract: 'Cathodoluminescence spectroscopy (CL) allows characterizing light emission in bulk and nanostructured materials and is a key tool in fields ranging from materials science to nanophotonics. Previously, CL measurements focused on the spectral content and angular distribution of emission, while the polarization was not fully determined. Here we demonstrate a technique to access the full polarization state of the cathodoluminescence emission, that is the Stokes parameters as a function of the emission angle. Using this technique, we measure the emission of metallic bullseye nanostructures and show that the handedness of the structure as well as nanoscale changes in excitation position induce large changes in polarization ellipticity and helicity. Furthermore, by exploiting the ability of polarimetry to distinguish polarized from unpolarized light, we quantify the contributions of different types of coherent and incoherent radiation to the emission of a gold surface, silicon and gallium arsenide bulk semiconductors. This technique paves the way for in-depth analysis of the emission mechanisms of nanostructured devices as well as macroscopic media.' author: - 'Clara I. Osorio' - Toon Coenen - Benjamin Brenny - Albert Polman - 'A. Femius Koenderink' bibliography: - 'references\_polarimetry.bib' title: 'Angle-resolved cathodoluminescence imaging polarimetry' --- Introduction ============ Among many recent developments in microscopy, optical electron-beam spectroscopy techniques such as cathodoluminescence imaging (CL) have emerged as powerful probes to characterize materials and nanophotonic structures and devices. In CL, one collects light emitted in response to a beam of energetic electrons ($0.1-30$ keV), for example in a scanning electron microscope (SEM). The time-varying evanescent electric field around the electron-beam interacts with polarizable matter creating coherent emission, such as transition radiation (TR) [@Adamo_PRL12; @Bashevoy_OE07]. The spot size of the focused electron beam and the extent of the evanescent field about the electron trajectory define the interaction resolution to be below $\sim20$ nm, while the interaction time ($<1$ fs) determines the broadband character of the excitation. Aside from coherent emission, incoherent emission can also be generated both by the primary beam and by slower secondary electrons, which excite electronic transitions in matter [@Abajo_RMP07; @Yacobi]. The relative importance of the coherent and incoherent contributions provides information about the material composition and electronic structure. Spectral analysis of the cathodoluminescence as a function of the electron beam position allows the local characterization of the structure and defects of semiconductors [@Edwards_SemicondSciTech2011; @Sauer_PRL2000; @Ton-That_PRB2012], the functioning of nanophotonic devices [@Fontcuberta_PRB2009], and to map the optical resonances of plasmonic and metamaterial structures [@Zhu_PRL10]. Recently developed techniques for detection of CL enable the identification of the band structure and Bloch modes of photonic crystals [@Yamamoto_OE09; @Yamamoto_OE11; @Adamo_PRL12; @Ma_JPC14; @Sapienza_NM12], the dispersion of surface plasmons [@losstw; @Bashevoy_OE07], and the directivity and Purcell enhancement of plasmonic nano-antennas [@coenen_NL11; @yamamoto_NL11]. Besides frequency and linear momentum, the vectorial nature of light provides a third degree of freedom rich in information about the physics of light generation and scattering, encoded in the polarization of emitted light. In materials characterization, for instance, polarization gives direct access to the local orientation of emission centers and anisotropies in the host material. In nanophotonics, polarization plays a fundamental role (together with directionality) in determining the interaction between emitters and nanostructures. Furthermore, it is increasingly recognized that mapping and controlling the polarization of light is key to harnessing the wide range of opportunities offered by metamaterials and metasurfaces. Recent breakthroughs in chirality-enhanced antennas [@Gorodetski_PRL13], photonic topological insulators [@lu14], and the photonic equivalent of the spin-Hall effect [@onoda04; @yin13; @li13; @connor14], indicate the emerging importance of mapping the full polarization properties of nanophotonic structures. Polarization measurements of CL emission, however, have been limited to fully polarized emission and in particular to linearly polarized signals [@coenen_OE12; @Coenen_NC14]. In this letter we introduce a novel technique to access full polarization information in cathodoluminescence spectroscopy. Based on a polarization analysis method previously demonstrated in optical microscopes [@fallet_MEMS11; @arteaga_OE14; @kruk_ACSP14; @Osorio_SR15], we integrate a rotating-plate polarimeter in the detection path of the angle-resolved CL setup. Using the Mueller matrix formalism for the light collection system, we determine the Stokes parameters for CL emission, that is, all parameters required to completely describe the polarization state of the light, which can be polarized, partially polarized or totally unpolarized. We demonstrate the great potential of this new measurement technique by analyzing the angle-resolved polarization state of directional plasmonic bullseye and spiral antennas. Furthermore, and exploiting the unique capabilities of CL excitation, we measured the emission from metals and semiconductors. For these materials, we can separate coherent and incoherent emission mechanisms, with further applications in nanoscale materials science. CL Polarimetry ============== {width="70.00000%"} In our measurements, the $30$ keV electron beam from a scanning electron microscope (SEM) excites the sample. An aluminum paraboloid mirror collects and redirects the resulting CL emission out of the SEM. The outcoming beam is focused onto a fiber-coupled spectrometer or projected onto a 2D CCD array [@coenen_NL11; @coenen_APL11; @Sapienza_NM12], as shown in Fig. \[Fig1\](a). The wave-vector distribution of the CL emission can be retrieved from the CCD image, as every transverse point in the beam corresponds to a unique emission angle, in a procedure analogous to other Fourier imaging techniques [@Lieb_JosaB04; @Kosako_NP10; @curto10; @Aouani_NL11; @Sersic_NJP11; @Belacel_NL13]. Measuring polarization for all emission angles of CL presents several challenges. First, it requires determining the relative phase difference between field components, a task not achievable with only linear polarizers as in Ref. [@coenen_OE12]. Second, the paraboloid mirror performs a non-trivial transformation on the signal as it propagates from the sample to the detector plane. The shape of the mirror introduces a rotation of the vector components of light due to the coordinate transformation and, consequently, a change in the main polarization axes. In addition, the angle and polarization-dependent Fresnel coefficients of the mirror modify the polarization of the light upon reflection  [@Bruce_OPT06; @Bruce_04]. As a function of the angle of incidence, the mirror partially polarizes unpolarized light and transforms linearly to elliptically polarized light. To address these challenges, we included a rotating-plate polarimeter in the beam path of our CL system, composed of a quarter wave plate (QWP) and a linear polarizer [@Berry_ApplOp77; @Born_Wolf; @Chipman]. Figure \[Fig1\](a) shows the polarizing elements in a schematic of the setup. Depending on their orientation, these two elements act either as a linear polarizer or as a right or left handed circular polarizer. As shown in Fig. \[Fig1\](b), we measure the intensities $I_j$ transmitted by six different settings of the polarimeter (horizontal, vertical, $45^{\circ}$, $135^{\circ}$, right and left handed circular) in order to determine the Stokes parameters of the light: $$\begin{aligned} \label{stokes_eq} S_0 &=&I_{H}+I_{V} \nonumber\\ S_1 &=& I_{H}-I_{V}\nonumber\\ S_2 &=& I_{45}-I_{135}\nonumber\\ S_3 &=&I_{RHC}-I_{LHC}.\end{aligned}$$ These four parameters are the most general representation of polarization and can be used to retrieve any polarization-related quantity [@Born_Wolf]. The raw polarization-filtered CCD images are projected onto \[$\theta,\varphi$\]-space as indicated in Fig. \[Fig1\](b) using a ray-tracing analysis of the mirror, after which the Stokes parameters in the detection plane are determined. To transform these to Stokes parameters in the sample plane, we determine the Mueller matrix of the light collection system that accounts for the effects of the mirror on the polarization. In addition to the geometrical transformation, the analysis takes into account the Fresnel coefficients of the mirror for $s$- and $p$- polarized light. Due to the 3D shape of the mirror, each element of the Mueller matrix is a function of the emission angle, i.e., there is a Mueller matrix for each emission angle. The supplementary information describes in more detail how the Mueller matrix was calculated and how we benchmark these calculations using fully polarized transition radiation (see Fig. S2). {width="\textwidth"} The Stokes parameters in the sample plane allow determining *any* figure of merit for polarization. Given that both incoherent and coherent radiation may be generated in CL, the degree of polarization ($DOP$), and the degrees of linear ($DOLP$) and circular polarization ($DOCP$) will be especially relevant. Defined as the ratio of polarized, linearly or circularly polarized light to total intensity, they are given by $DOP=\sqrt{S_1^2+S_2^2+S_3^2}/S_0$, $DOLP=\sqrt{S_1^2+S_2^2}/S_0$ and $DOCP=S_3/S_0$. Equivalently, the ratio of unpolarized light to total intensity is given by $1-DOP$ so a $DOP$ smaller than $1$ corresponds to partially polarized light. CL Polarimetry on plasmonic structures ====================================== Bullseye antennas ----------------- To demonstrate the full potential of angle-resolved CL polarimetry we investigate the emission of a plasmonic bullseye structure with a pitch $d = 600$ nm, milled into a single-crystal gold substrate. Bullseyes are well-known for their ability to strongly direct light scattered by nanoscale apertures [@Lezec_Sc02], generated by fluorescence [@Jun_NC11; @Aouani_NL11] or thermal emission [@Norris_OE10]. Figure \[Fig2\](a) shows a scanning electron micrograph of the structure indicating the excitation position. The electron beam launches a circular surface plasmon polariton (SPP) wave which radiates outwards and scatters coherently from the grooves of the bullseye. The scattered fields interfere to give rise to directional emission. In our measurements, the emission is spectrally filtered by a $40$ nm bandwidth bandpass filter centered at $\lambda_{0} = 750$ nm (see Fig. S1 in the supplement for full spatial and spectral mapping). Figures \[Fig2\](b-d) represent the main steps of our polarimetric analysis for CL. Figure \[Fig2\](b) shows the angular intensity patterns measured for the six settings of the polarimeter (indicated by the arrows) after a coordinate transformation of the raw intensity data. Figure \[Fig2\](c) shows the Stokes parameters in the detection plane calculated using Eq. \[stokes\_eq\] from the patterns in Fig. \[Fig2\](b). The leftmost panel corresponds to the total intensity distribution, $S_0$. The bullseye emits in a narrow doughnut pattern without any azimuthal variations, consistent with the azimuthal symmetry of both excitation position and bullseye structure. The polar angle at which most of the CL is emitted, $\theta = 15^{\circ}$, corresponds to the grating equation $\theta = \sin^{-1}(k_{SPP}-m2\pi/d)/k_{0}$. In this spectral regime, the grating order $m=1$ is the only relevant order, $k_{0}=2\pi/\lambda_{0}$ and $k_{SPP}$ is the SPP wave-vector, calculated using the optical constants for gold from spectroscopic ellipsometry. The other panels in Fig. \[Fig2\](c) show the Stokes parameters $S_{1}$, $S_{2}$, and $S_{3}$ in the detection plane ($yz$-plane in Fig. \[Fig1\]) normalized to $S_{0}$, such that it is possible to see polarization features outside the areas of very bright emission. Next, we transform the data collected by the detector to the polarization state of the emitted light in the sample plane, by multiplying the Stokes parameters at the detection plane with the mirror’s inverse Mueller matrix. Among the quantities that the Stokes parameters allow retrieving, here we will focus on the electric field components. Figure \[Fig2\](d) shows the reconstructed spherical field vector amplitudes $|E_\phi|$ and $|E_\theta|$ that constitute the natural $s$- and $p$- polarization basis relevant to map the far-field generated by a localized radiating object. The figure shows that the $|E_{\theta}|$ distribution is strong and azimuthally symmetric while $|E_{\varphi}|$ is close to zero. Therefore, the measured emission of the bullseye is a narrow doughnut beam with a fully linear, radial polarization. Radial polarization is expected for the bulls-eye radiation as SPPs scatter out while maintaining their $p$-polarized character at the grooves. The polarization can alternatively be cast into Cartesian components. Figure \[Fig2\](d) shows the double-lobe patterns of $|E_{x}|$ and $|E_{y}|$, which are rotated $90^{\circ}$ relative to each other. The $|E_{z}|$ component is azimuthally symmetric and shows several emission rings. The outer rings correspond to transition radiation (TR) from the excitation position, which is modulated to yield a fringe pattern due to interference with SPPs scattered off the bullseye grooves [@kuttge_PRB09]. Since the electric field must be transverse to the propagation direction, the $|E_{z}|$ component vanishes at near-normal angles and therefore the main SPP emission beam from the bullseye (the narrow ring) appears relatively weak in $|E_z|$. While the emission in the sample plane is completely linearly polarized, a nonzero circular polarized signal is measured in the detection plane ($S_3$ in Fig. \[Fig2\](c)), which indicates the effect of the mirror and the importance of using the Mueller matrix analysis to correct for it. {width="80.00000%"} Non-symmetric geometries ------------------------ CL polarimetry is an unique tool to explore the relation between the symmetry of a system and its polarization response. While the symmetry of a bullseye structure excited right at the center ensures the clear TM polarization emission shown in Ref. \[Fig2\](d), this is not longer the case when launching an off-center circular SPP wave on the structure. Figure \[Fig3\] shows measurements for electron beam excitation in the center, halfway between the center and the edge, and at the edge of the central bullseye plateau, as indicated in the SEM micrographs on top of the figure. Figure \[Fig3\](a) shows $|E_{\theta}|$ and $|E_{\varphi}|$ for the three excitation positions. For off-center excitation, the zenithal field distribution $|E_{\theta}|$ is no longer symmetric, being stronger towards the left than towards the right of the image. This type of asymmetric beaming has also been observed in angular intensity measurements on asymmetric gratings [@Jun_NC11], spirals [@Rui_SR13], and asymmetrically excited (patch) antennas [@Mohtashami_ACSP14; @Coenen_NC14]. Besides the asymmetry, the off-center excitation also leads to a non-zero azimuthal field contribution, $|E_{\varphi}|$, which is similar in strength to the zenithal field contribution. The excitation position and the center of the bullseye defines a mirror symmetry that expresses itself as a nodal line for $|E_{\varphi}|$ at $\varphi = 90^{\circ}$ and $\varphi = 270^{\circ}$. At far off-center excitation, the azimuthal and zenithal field distributions are very rich in structure and for certain angular ranges the emission becomes elliptically or circularly polarized. The effect of off-center excitation is most clearly seen in Fig. \[Fig3\](b), which shows the degree of linear ($DOLP$) and circular ($DOCP$) polarization. Owing to the mirror symmetry of sample and excitation, the $DOCP$ remains close to zero along the axis of the electron beam displacement. Yet, away from this axis the emission becomes elliptical with opposite handedness on either side of the axis as dictated by mirror symmetry. For edge excitation, the complementary multi-lobe $|E_{\theta}|$ and $|E_{\varphi}|$ patterns lead to a rich behavior, where the emission changes from fully linear to almost fully circular polarization several times. Rather than breaking symmetry by changing the excitation position, it is also possible to study scattering and emission by intrinsic asymmetry and handedness of structures such as Archimedean spirals. Spirals enjoy a growing interest since it was shown that they can enhance the extraordinary transmission of single nanoapertures for particular helicities [@Drezet_OE08], and transfer polarization and orbital angular momentum to scattered photons [@Rui_SR13; @Rui_OE12]. This can result in a polarization-dependent directional beaming [@Rui_OL11] and demonstrates strong photon spin-orbit coupling effects [@Gorodetski_PRL13]. We fabricated Archimedean spiral gratings with clockwise (CW) and anti-clockwise (ACW) orientation, as shown in Fig. \[Fig4\](a,b), and used CL polarimetry to study the effect of spiral asymmetry and handedness on the far-field polarization, again taking a pitch $d$ = 600 nm and $\lambda_{0}$ = 750 nm. We excite the spirals in their origin as indicated in Fig. \[Fig4\]. Figures \[Fig4\](c,d) show the Cartesian components of the far-field emission of the spirals, which better reflect the handedness than the spherical fields. Since the groove pitch is the same for spirals and bullseyes, the angular spread of these patterns is similar, however they no longer have a minimum at the normal and the s-like shape in $|E_{x}|$ and $|E_{y}|$ clearly reflects the handedness of the spiral. {width="0.8\columnwidth"} In contrast with bullseyes excited at their center, the spirals can induce ellipticity in the polarization of the light even when excited in their origin, as shown in Fig. \[Fig4\](e,f). This is particularly evident in the region of higher intensity in the vicinity of the normal, where the $DOCP$ is close to $\pm1$. Thereby, the spirals are highly directional sources of circularly polarized light. Mirrored spirals simply exhibit mirrored patterns (where the $y$-axis defines the mirror symmetry), conserving intensities and field strengths, while the sign of the $DOCP$ changes. This result indicates that swapping spiral handedness not only flips the helicity of the output field but, in addition, it mirrors the distribution of intensity over angle. For spirals with smaller pitch we find similar but even stronger effects of handedness, aided by the fact that their radiation pattern is more strongly off-normal (see Fig. \[FigS3\](g-l)). In that case the $|E_{z}|$ distribution is also clearly chiral. The data shown in this section proves that polarimetry analysis of CL, in combination with precise electron beam positioning, provides direct insight into the complex emission behavior of nanophotonic structures. Measuring directionality and polarization of the emission from emitters coupled to single nanostructures is of paramount importance when designing and testing the performance of structures like optical antennas, plasmonic resonators, and metasurfaces. CL polarimetry applied to incoherent emitters ============================================= {width="40.00000%"} In addition to characterizing fully coherent radiation, CL polarimetry allows us to determine whether the measured radiation contains an unpolarized contribution such as in the case of incoherent luminescence from bulk or nanostructured materials. This is shown in Fig. \[Fig4\], where we compare azimuthally averaged zenithal cross cuts of the polarized ($S_{0}\times DOP$) and unpolarized ($S_{0}\times(1-DOP)$) emission intensities for single-crystal Au, Si and GaAs and compare them to calculations. The emission from Au at $\lambda_0 = 850$ nm in Fig. \[Fig4\](a) is expected for coherent TR (see also Fig. S2 in the supplement) and hence fully polarized. The data indeed shows excellent agreement with a calculated TR emission distribution. In the case of GaAs in Fig. \[Fig4\](b), the emission is dominated by very bright incoherent radiative band-to-band recombination measured at $\lambda_{0} = 850$ nm. This luminescence is fully isotropic and unpolarized *inside* the material, but large differences between $s$- and $p$- Fresnel transmission coefficients for the semiconductor-vacuum interface partially polarize the emission as seen in the data. Figure \[Fig4\](b) shows that unpolarized light is indeed dominant. The weak polarized emission has a very different angular emission distribution, that agrees well with Fresnel calculations (see Fig. S3 in the supplement for more information). Lastly, Si is a material that displays weak luminescence that it is comparable to TR [@Brenny_JAP14]. Indeed, the polarized intensity for Si at $\lambda_{0} = 650$ nm shown in Fig. \[Fig4\](c) constitutes $\sim 31.7\%$ of the total emission, which is much more significant than for GaAs, although unpolarized emission remains the dominating contribution. These examples show that angle-resolved polarimetry measurements provide quantitative and precise information about the origin of emission of different materials. This technique enables the separation of polarized and unpolarized emission, and therefore it can be used to determine the different mechanisms that simultaneously contribute to cathodoluminescence (see Fig. S2 in the supplement for a quantitative analysis of TR). Moreover, for the polarized part of the emission we can map the electric field components and their relative phase. Since it does not require any prior knowledge of the sample (unlike the method described in Ref. [@Brenny_JAP14]), this method is very general and can be applied to any (nanostructured) material. Conclusion ========== We have demonstrated ‘angle-resolved cathodoluminescence imaging polarimetry’ as a new microscopy tool to map the vectorial electromagnetic scattering properties of nanostructured and bulk materials. We determine the complete polarization state of emitted light as a function of angle from six CL intensity measurements in the detection plane, in combination with a mathematical transformation that corrects for the polarizing effect of the CL mirror. Due to the high resolution of the electron beam excitation, the wave-vector resolved polarization properties of locally excited plasmonic nano-antennas can be extracted with a spatial resolution for the excitation of $20$ nm. The angle-resolved polarization measurements of the emission of bullseye and spiral nanoantennas demonstrate how structural symmetry and handedness translate into the helicity of emitted light. These results show that angle-resolved cathodoluminescence polarimetry can be extremely valuable for the development of metallic and dielectric antennas for spin-resolved and chiral spectroscopy as well as for the study of photon spin Hall effects. Besides its relevance for nanophotonics, we demonstrate that our technique opens new perspectives for materials science not accessible with optical microscopes. Measuring the Stokes parameters generally enables the separation of incoherent and coherent CL generation, as we demonstrated for direct and indirect semiconductor materials. Our measurements on relatively simple samples of Au, GaAs and Si show the potential of the technique for the analysis of bulk materials which could be useful for many material inspection tasks. For optoeletronics, the nanoscale characterization of emission polarization from inorganic LEDs stacks, nanowires and quantum dots stands out in particular. The technique also introduces the possibility of locally studying material anisotropy, birefringence and optical activity. Methods ======= Sample fabrication ------------------ We fabricated bullseye and spiral structures by patterning a single-crystal Czochralski-grown Au $\langle100\rangle$ pellet which was mechanically polished to obtain a sub-$10$ nm RMS roughness. The patterning was done by using a $30$ keV Ga$^{+}$ ion beam in a FEI Helios NanoLab dual beam system at $9.7$ pA beam current and a dwell time of $10$ s per pixel. In the bullseye design the central plateau has a diameter of 1.2 m ($2$ times the pitch, $600$ nm) and the duty cycle of the circular grating consisting of $8$ grooves is $50\%$. The spiral design is based on an Archimedean spiral where the first half period of the spiral is omitted. For the spiral we show data both for $600$ and $440$ nm pitches. Both for the spirals and the bullseyes the groove depth was $\sim 110$ nm. The measurements on silicon were performed on a polished p-type (boron doping level $10^{15}-10^{16}$ cm$^{-3}$) single-crystal $\langle 100 \rangle$ wafer. The measurements on GaAs were done on a polished single-crystal $\langle 100 \rangle$ wafer. Measurements ------------ The measurements were performed in a FEI XL-30 SFEG ($30$ keV electron beam, $30$ nA current) equipped with a home-built CL system [@coenen_NL11; @coenen_APL11; @Sapienza_NM12]. To obtain the polarization state of the emission, we perform a series of six measurement of the angular CL pattern using a 2D back-illuminated CCD array. Each measurement was taken in a different setting of the polarimeter, defined by a specific combination of QWP and polarizer angles. Handedness of circularly polarized light was defined from the point of the view of the source, following the IEEE standard. In this case, right-handed circularly polarized light rotates anti-clockwise and left-handed circularly polarized light rotates clockwise. We use the known transition radiation pattern from an Au surface to calibrate the optical detection system. A $40$ nm band pass color filter spectrally selected the measured emission. For the bullseye and spiral measurements we used $30$ s integration time, which is a good compromise between a small spatial drift of the electron beam and a good signal-to-noise in CL. For the TR emission from single-crystal gold and the measurements on silicon we used $120$ s integration time since TR emission and luminescence are position independent and the measurement is not affected by spatial drift of the electron beam. For the measurements on GaAs we used a much lower current ($0.9$ nA) and integration time ($1$ s) due to the very bright band gap luminescence. For every setting of the polarimeter, we collected a dark reference measurement where we blank the electron beam (with the same integration time as the CL measurement), which was subtracted from the data in the post-processing stage. Possible sources of errors on the measurements include e-beam drift (in the case of position dependent samples), bleaching/contamination during measurements leading to a reduction in CL signal, fluctuations in current and mirror alignment.\ [**Supplementary information**]{} The Supporting Information provides further information about the calculation of the Mueller matrix, the calibration of set up, the spectral response of the antennas and the data analysis for the bulk material measurements.\ [**Notes**]{} A.P. is co-founder and co-owner of Delmic BV, a startup company that develops a commercial product based on the ARCIS cathodoluminescence system that was used in this work.\ [**Acknowledgements**]{} The authors thank Abbas Mohtashami for his help with the fabrication and Henk-Jan Boluijt for the cartoon in Fig. \[Fig1\](a). This work is part of the research program of the Stichting voor Fundamenteel Onderzoek der Materie (FOM), which is financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO). This work is part of NanoNextNL, a nanotechnology program funded by the Dutch ministry of economic affairs. It is also supported by the European Research Council (ERC). Supporting Information ====================== Spectral measurements on bullseye --------------------------------- To study the spectral response of the bullseye from Fig. 2 (main text), we raster scan the electron beam over the central part of the structure in 15 nm steps, and collect a CL spectrum for every pixel using a visible/NIR fiber-coupled Czerny-Turner spectrometer [@coenen_NL11]. The scan includes the central plateau, the first groove, and part of the first ridge. The spectra are corrected for the system response using the TR emission of the unstructured gold substrate [@kuttge_PRB09]. We then radially average the spectrum to obtain a map showing emitted intensity as a function of wavelength and radius, taking into account the proper Jacobian so we can directly compare the intensities (shown in Fig. \[FigS1\](a)). Figure \[FigS1\](b) shows individual spectra for specific radial distances as indicated in (a) by the white dashed lines. We observe several features in the scan. On the plateau, the emission is quite broadband except for a strong dip in intensity around $\lambda_{0}$ = $650$ nm. The CL intensity increases towards the edge of the plateau, then drops in the groove, and becomes bright on the first ridge again. The observed spectral features can be attributed to a mix of resonant and diffractive effects. {width="80.00000%"} Because the groove is wide enough, it supports a zero-th order standing wave mode on the bottom of the groove around $\lambda_{0}$ = 580 nm, which is cut-off for longer wavelengths leading to a low intensity in the red [@Brucoli_PRB11; @Schoen_NL13]. The dip at $\lambda_{0} = 650$ nm can be attributed to an interesting experimental artifact related to the diffraction of SPPs. Consistent with the grating equation mentioned in the main text, the bullseye emits very close to the normal at this wavelength. Because the bullseye structure is highly directional, a major part of this diffracted beam is lost through the $600$ m hole in the mirror right above the sample. In fact, one could use this dip to find the wavelength and spatial position at which such a structure attains maximum directionality in the normal direction. In this case, the effect is strongest for the central positions in the bullseye because the beam is exactly normal to the sample as is visible in Fig. 3(a) for $\lambda_{0}$ = $750$ nm. Even though the dip is strongest in the center of the plateau, it is visible for every radial excitation position in this map (even within the groove), indicating that the extended bullseye geometry always causes a significant fraction of CL emission to be in the normal direction. Calculation of the Mueller matrices ----------------------------------- The Mueller matrix of an optical element accounts for the effect of the element on the polarization state of an incident field [@Born_Wolf]. The Mueller matrices of a linear polarizer and a QWP are well-known for example. We use the Mueller formalism to relate the Stokes vector in the detection plane to the Stokes vector describing the sample emission polarization. The resulting Mueller matrix of the mirror contains both geometric and polarizing effects of the mirror in the emission polarization. To retrieve the Mueller matrix of our light collection system, we calculate how the electric field components $E_{\theta}$ and $E_{\varphi}$ in the emission plane transform to $E_{y}$ and $E_{z}$ in the detection plane [@Bruce_OPT06; @Bruce_04]. To that end we calculate how fully isotropic $p$-polarized and $s$-polarized emission is projected onto the detection plane by the parabolic mirror. We use the geometrical methods described in section 3 of Ref. [@coenen_OE12] to account for the parabolic reflector and we use the full complex Fresnel reflection coefficients to accurately describe the light reflection on the mirror. These coefficients were calculated using tabulated optical constants [@Palik] for the central frequency of the collection bandwidth. As the reflection angle at the mirror is different for every wave-vector emanating from the sample, each element of the Mueller matrix is a function of the emission angle [@Bruce_OPT06; @Bruce_04]. Instead of calculating the Mueller matrix, one could also envision experimentally retrieving it. This requires a precisely controlled radial and azimuthal polarization source as standard. While transition radiation could serve as a fully radial source, a fully azimuthal source is not readily available. The Mueller matrices can be used in two directions. Either one can invert measured data from the detector plane to sample coordinates, or in the opposite direction, one can predict how a given source will appear on the detector plane. For our analysis of TR emission, in Fig. \[FigS2\] we apply Mueller matrices to theoretical TR emission to predict the measured data for each setting of the polarimeter. In this case, we combine the mirror Mueller matrix with the Mueller matrices of a linear polarizer and a QWP, which are a function of the selected analyzer angles $\alpha$ and $\beta$ [@Chipman]. We note that for fully polarized sources this is analogous to the approach in Ref. [@coenen_OE12] where the Jones matrix of the polarizing element operates on the Jones electric field vector. Polarimetry of transition radiation emission -------------------------------------------- Transition radiation (TR) emission occurs whenever an electron traverses an interface between two dielectric media. The electron locally polarizes the material close to the interface, giving rise to a well-defined broadband vertically-oriented point-dipole-like source [@revjav; @coenen_APL11; @coenen_OE12]. This makes it a useful source to test our CL polarimetry technique. Here we perform polarimetry measurements on an unstructured part of the single-crystal Au substrate from which only TR emission is expected, using the same technique used for the bullseye. Figure \[FigS2\](a) shows the TR data at $\lambda_{0}$ = $850$ nm for every polarimeter setting. The top left panel includes the angular coordinate system for reference. {width="70.00000%"} In addition to the TR measurements, we calculated the emission pattern for a z-oriented dipole on top of an Au substrate and its polarization components. The dipolar far-field for $\lambda_{0} = 850$ nm was calculated from the asymptotic far-field expressions [@nanooptics] using ellipsometry data for the optical constants of the gold substrate. We then calculate the expected filtered pattern using the appropriate Mueller matrices for the paraboloid mirror and the polarimeter components. Figure \[FigS2\](b) shows the result of this calculation including the angular acceptance of the mirror, in order to allow a good comparison with the data. The excellent agreement of both angular distributions and relative intensities between data and calculation indicates that TR emission is indeed dipolar and that the mirror correction works well. The fringes in the data are an experimental artifact, probably due to interference between multiple reflections of the optical elements on the detector. We can use the measurements from Fig. \[FigS2\](a) to determine the Stokes parameters in the detection plane of the CCD and then use the Mueller matrix formalism to correct for the effects of the mirror. This allows us to retrieve the Stokes parameters in the sample plane from which we can determine the different field amplitudes. Figure \[FigS2\](c) shows the Stokes parameters in both the detection and sample planes, where $S_{1}$, $S_{2}$ and $S_{3}$ have been normalized by $S_{0}$ so that we can clearly observe the polarization distortions due to the mirror. In the detection plane the Stokes parameters display complex patterns that are very similar to those shown for the bullseye in Fig. 2(c), since both cases are dominated by purely radial polarization. In the sample plane the behavior of the Stokes parameters is much simpler, $S_{0}$ has barely changed and $S_{1}$ is close to 1 while $S_{2}$ and $S_{3}$ are very small. Again, TR is expected to be fully radially polarized, so there should be no diagonal ($S_{2}$) or circular ($S_{3}$) components. The striking difference between the Stokes parameters in the two planes underscores the importance of the Mueller matrix correction to provide accurate results. Fig \[FigS2\](d) shows the spherical and Cartesian field amplitudes that have been retrieved from the Stokes parameters in sample space. The fields nicely reveal the expected radially polarized nature of the emission. The amplitudes have been plotted using a single color scale to allow a quantitative comparison between the different components. Both the relative amplitudes and amplitude distributions match very well with the calculated dipolar fields shown in (e). These results demonstrate that cathodoluminescence polarimetry can reliably be used as a quantitative tool for deducing the far-field polarization distribution of a nanoscale emitter. Silicon and GaAs polarimetry ---------------------------- Calculating the contributions of TR, polarized and unpolarized luminescence to the total emission from Si or GaAs requires determining their angular profiles. An essential part of this process are the transmission coefficients $T_{p}$ and $T_{s}$ at the sample-vacuum interface, shown in Fig. 5(a) for the case of Si at $\lambda_{0}$ = 650 nm. The large contrast between the two coefficients at angles above $\sim$ 20$^{\circ}$ leads to more $p$-polarized light exiting the Si than $s$-polarized light. Especially near the Brewster angle, the $p$-component of the field is transmitted significantly better than the $s$-component, with a contrast between intensity transmission $T_{p}$ and $T_{s}$ exceeding 4 (see Fig. 5(a)). This difference in transmission coefficients results in a CL emission profile slightly different than the Lambertian cos($\theta$) profile (blue curve in Fig. \[FigS4\](b)). The polarizing effect of the interface does not noticeably affect the emission pattern of the total luminescence, but the unpolarized luminescence (consisting of equal amounts of $s$- and $p$-polarized light) is markedly narrower. Accordingly, the polarized part of the luminescence is stronger at higher angles and, interestingly, it follows a very similar profile to that of TR for Si at $\lambda_{0}$ = 650 nm, which has been calculated using formulas derived in Ref. [@revjav]. The same formulas are used to calculate the gray line describing TR from Au at $\lambda_{0}$ = 850 nm in Fig. 5(a). {width="0.9\columnwidth"} Once the theoretical profiles for the different emission processes are calculated, it is possible to determine their relative contributions to the total emission. The fraction of polarized and unpolarized luminescence is fully specified by the Fresnel equations. This gives enough information to compare calculations to data from GaAs with very good agreement, as that is fully dominated by luminescence. The case of Si is more complex as TR also plays a role, so the polarized emission is comprised of coherent TR as well as polarized luminescence. We determine the ratio of TR and luminescence by fitting the total intensity ($S_{0}$) to a linear combination of the two processes. Fresnel calculations again predict the polarized and unpolarized contributions to the luminescence so that we can combine all three components. For both the calculations and the experiments we scale the angular distributions by the overall (integrated) emission intensity, and find good (absolute) agreement, as was shown in Fig. 5(b). The polarized signal constitutes $\sim 32\%$ of the total emission. More explicitly, we find that TR contributes $\sim 21\%$ of the total CL intensity, so polarized luminescence contributes $\sim 11\%$ and unpolarized luminescence $\sim 68\%$. After separating the polarized and unpolarized components we can retrieve the different electric field components. Both the TR and the polarized luminescence should be $p$-polarized, which we verify from the experimental data using the Mueller analysis to determine the radial and azimuthal field amplitudes. This is shown for Si in Fig. \[FigS4\](c), where we indeed observe that almost all of the amplitude is in the $|E_{\theta}|$ component, i.e. for $p$- polarization. This demonstrates that we can separate the unpolarized and polarized emission even from mostly incoherently radiating semiconductors and still retrieve the correct electric fields for the polarized portion of the emission.

--- abstract: 'Explicit current-dependent expressions for anisotropic longitudinal and transverse nonlinear magnetoresistivities are represented and analyzed on the basis of a Fokker-Planck approach for two-dimensional single-vortex dynamics in a washboard pinning potential in the presence of point-like disorder. Graphical analysis of the resistive responses is presented both in the current-angle coordinates and in the rotating current scheme. The model describes nonlinear anisotropy effects caused by the competition of point-like (isotropic) and anisotropic pinning. Nonlinear guiding effects are discussed and the critical current anisotropy is analyzed. Gradually increasing the magnitude of isotropic pinning force this theory predicts a gradual decrease of the anisotropy of the magnetoresistivities. The physics of transition from the new scaling relations for anisotropic Hall resistance in the absence of point-like pins to the well-known scaling relations for the point-like disorder is elucidated. This is discussed in terms of a gradual isotropizaton of the guided vortex motion, which is responsible for the existence in a washboard pinning potential of new (with respect to magnetic field reversal) Hall voltages. It is shown that whereas the Hall conductivity is not changed by pinning, the Hall resistivity can change its sign in some current-angle range due to presence of the competition between *i*- and *a*-pins.' address: - | Institute of Theoretical Physics, National Science Center-Kharkov Institute of Physics and Technology, 61108, Kharkov, Ukraine;\ Kharkov National University, Physical Department, 61077, Kharkov, Ukraine - 'Kharkov National University, Physical Department, 61077, Kharkov, Ukraine' author: - 'Valerij A. Shklovskij' - 'Oleksandr V. Dobrovolskiy' title: 'Influence of Point-like Disorder on the Guiding of Vortices and the Hall Effect in a Washboard Planar Pinning Potential' --- INTRODUCTION ============ The importance of flux-line pinning in preserving the superconductivity in a magnetic field has been generally recognized since the discovery of type-II superconductivity. But till now the mechanism of flux-line pinning and creep in superconductors (and particularly in the high-*$T_c$* superconductors (HTSC’s)) is still a matter of controversy and great current interest, especially in the cases of strong competition between different types of pins. One of the open issues in the field is the influence of *isotropic* point-like disorder on the vortex dynamics in the *anisotropic* washboard planar pinning potential (PPP) for the case of arbitrary orientation of the transport current with respect to the PPP “channels” where the *guiding of vortices* can be realized. The importance of this issue may be substantiated by ubiquitous presence of point-like pins in those high- and low-*$T_c$* superconductors which were used so far for resistive measurements of the guided vortex motion$^{1-9}$. The first attempt to discuss the influence of isotropic point-like disorder on the guiding of vortices was made by Niessen and Weijsenfeld$^1$ still in 1969. They studied guided motion *in the flux flow regime* by measuring transverse voltages of cold-rolled sheets of a Nb-Ta alloy for different magnetic fields *H*, transport current densities *J*, temperatures *T*, and different angles $\alpha$ between the rolling and current direction. The (*H,J,T*,$\alpha$)-dependences of the cotangent of the angle $\beta$ between the average vortex velocity $\langle \textbf{v}\rangle$ and the vector **J** direction were presented. For the discussion, a simple theoretical model was suggested, based on the assumption that vortex pinning and guiding can be described in terms of an isotropic pinning force ${\bf F}_p^i$ plus a pinning force ${\bf F}_p^a$ with a fixed direction which was perpendicular to the rolling direction. The experimentally observed dependence of the transverse and longitudinal voltages on the magnetic field *in the flux flow regime* as a function of the angle $\alpha$ was in agreement with this model. Unfortunately, in spite of the correct description of a geometry of the motive forces of a problem (see below Fig. 1) it was impossible within the flux flow approach$^1$ to calculate theoretically the *nonlinear* (*J, T*, $\alpha$)-dependences of the average pinning forces $\langle{\bf F}_p^i\rangle$ and $\langle{\bf F}_p^a\rangle$ which determine the experimentally observed cot$\beta(J,T,\alpha)$ dependences. The *nonlinear guiding* problem was exactly solved at first only for the washboard PPP (i.e. for ${\bf F}_p^i=0 $) within the framework of the two-dimensional single-vortex stochastic model of anisotropic pinning based on the Fokker-Planck equation with a concrete form of the pinning potential$^{10,11}$. Two main reasons stimulated these theoretical studies. First, in some HTCS’s twins can easily be formed during the crystal growth$^{2-5,8}$. Second, in layered HTCS’s the system of interlayers between parallel *ab*-planes can be considered as a set of unidirectional planar defects which provoke the intrinsic pinning of vortices$^{12}$. Rather simple formulas were derived$^{11}$ for the experimentally observable *nonlinear* even$(+)$ and odd$(-)$ (with respect to the magnetic field reversal) longitudinal and transverse magnetoresistivities $\rho_{\|,\perp}^\pm(j,\theta,\alpha,\varepsilon)$ as functions of the dimensionless transport current density $j,$ dimensionless temperature $\theta,$ and relative volume fraction $0<\varepsilon<1$ occupied by the parallel twin planes directed at an angle $\alpha$ with respect to the current direction. The $\rho_{\|,\perp}^\pm$-formulas were presented as linear combinations of the even and odd parts of the function $\nu(j,\theta,\alpha,\varepsilon)$ which can be considered as the probability of overcoming the potential barrier of the twins$^{11}$; this made it possible to give a simple physical treatment of the nonlinear regimes of vortex motion (see below item II.C). Besides the appearance of a relatively large even transverse $\rho_\perp^+$ resistivity, generated by the guiding of vortices along the channels of the washboard PPP, explicit expressions for *two new nonlinear anisotropic Hall resistivities* $\rho_{||}^-$ *and* $\rho_\perp^-$ were derived and analyzed. The physical origin of these *odd* contributions caused by the subtle interplay between even effect of vortex guiding and the odd Hall effect. Both new resistivities were going to zero in the linear regimes of the vortex motion (i.e. in the thermoactivated flux flow (TAFF) and the ohmic flux flow (FF) regimes) and had a bump-like current or temperature dependence in the vicinity of highly nonlinear resistive transition from the TAFF to the FF. As the new odd resistivities arose due to the Hall effect, their characteristic scale was proportional to the small Hall constant as for ordinary odd Hall effect investigated earlier$^{10}$. It was shown$^{11}$ that appearance of these new odd $\rho_{|| , \bot }^-$ contributions leads to the new specific angle-dependent “scaling” relations for the PPP which demonstrate the so-called anomalous Hall behavior in the type-II superconductors. Here we should to emphasize that the anomalous behavior of the Hall effect in many high-temperature and in some conventional superconductors in the mixed state remains one of the challenging issues in the vortex dynamics$^{5,12,16}$. The problem at issues includes several remarkable experimental facts: a) the Hall effect sign reversal in the vortex state with respect to the normal state at temperatures near $\emph{T}_c$ and for moderate magnetic fields; b) the Hall resistivity “scaling” relation $\rho_\perp\sim\rho_\| ^ \beta$ exists with $1\leq\beta\leq2$, where $\rho_\perp$ is the Hall resistivity and $\rho_\parallel$ is the longitudinal resistivity; c) the influence of pinning on the “Hall anomaly” and scaling relation. Assuming that the “bare” Hall coefficient $\alpha_H$ is constant, two different scaling laws have been derived earlier theoretically for different pinning potentials$^{11,17}$. Vinokur et al. have shown$^{17}$ that a scaling law $\rho_\perp=\delta\rho_\parallel^2$ (where $\delta=n\alpha_H c^2/B\Phi_0$ is the Hall conductivity, $n=\pm1$, $c$ is the speed of light, $B$ is the magnetic field and $\Phi_0$ is the magnetic flux quantum) is the general feature of any isotropic vortex dynamics with an average pinning force directed along the average vortex velocity vector. Later it was shown$^{11}$ that for purely anisotropic *a*-pins that create a washboard planar pinning potential, the form of corresponding “scaling” relation is highly anisotropic due to the reason that pinning force for *a*-pins is directed perpendicular to the pinning planes. If $\alpha$ is the angle between parallel pinning planes and direction of the current density vector $\mathbf{j}$, then for $\alpha=0$ the scaling law has the form $\rho_\bot=-n(\alpha_H/\eta)\rho_\|$ ($\eta$ is the vortex viscosity) which was interpreted previously$^{11}$ as a scaling law with $\beta=1$, whereas for $\alpha=\pi/2$ the scaling relation is more complex$^{11}$. The $\rho_{\perp}^-$, as it is shown in this paper, can be presented as a sum of the three contributions with the different signs. The graphical analysis in Sec. III of this paper represents a some range of the $(\alpha,j)$-values where the theory predicts a nonlinear change of the $\rho_{\perp}^-$ sign. Let us consider another specific feature of the purely anisotropic guiding model$^{10,11}$. From the mathematical viewpoint, the nonlinear anisotropic problem, as solved in Ref. 11, reduces to the Fokker-Planck equation of the one-dimensional vortex dynamics$^{13}$ because the vortex motion is unpinned in the direction which is parallel to the PPP channels. As a consequence, a critical current $j_c$ exists only for the direction which is strictly perpendicular to the PPP channels ($\alpha=0$); $j_c(\alpha)=0$ for any other direction ($0<\alpha\le \pi/2$). However, the measurements of the magnetoresistivity show$^{1-8}$ that $j_c(\alpha)>0$ for all $\alpha$ (although $j_c(\alpha)$ may be anisotropic). So, in spite of some merits of a model with a washboard PPP, which was the first exactly solvable stochastic nonlinear model of anisotropic pinning, it cannot describe the $j_c$-anisotropy of the experimentally measured samples. Due to this reason later it was suggested$^{14,20}$ another simple model, which demonstrates this $j_c$-anisotropy for all $\alpha$ on the basis of the bianisotropic pinning potential formed by the sum of two washboard PPP’s in two mutually perpendicular directions. In contradistinction to the nonlinear model with uniaxial PPP$^{11}$, this bianisotropic nonlinear model predicts a $j_c(\alpha)$-anisotropy and relates it to the guiding anisotropy, describing the appearance of two step-like and two bump-like singularities in the $\rho_{\|,\perp}^+$ and $\rho_{\|,\perp}^-$ (Hall) resistive responses, respectively. Although several proposals to realize experimentally this bianisotropic model were discussed so far$^{14}$, the corresponding experiments, however, are still absent. At the same time, the experimental study of vortex dynamics in the PPP is always accompanied with a presence of a certain level of point-like disorder. So, as far as the analysis of existing experimental data is concerned, none of the present theoretical studies in the limiting cases of purely anisotropic or isotropic pinning are sufficient. The more general approach is needed. The objective of this paper is to present results of a theory for the calculation of the nonlinear magnetoresistivity tensor at arbitrary value of competition between point-like and anisotropic planar disorder for the case of in-plane geometry of experiment. This approach will give us the experimentally important theoretical model which demonstrates the $j_c$-anisotropy for all $\alpha$ and predicts a nonlinear change of the $\rho_\perp^-$ sign at some set of parameters (without change of the Hall *conductivity*) due a competition of the washboard PPP and a point-like disorder. The organization of the article is as follows. In Sec. II we derive main results of the $i+a$ pinning problem and consider two main limiting cases of purely $a$- or $i$-pinning. In Sec. III we represent the graphical analysis of different types of nonlinear responses, in particular, the $(j,\alpha)$ graphs of the $\rho_{\parallel,\perp}^\pm$ magnetoresistivities and the resistive response in a rotating current scheme. In Sec. IV we conclude with a general discussion of our results. Main relations ============== Formulation of the problem. --------------------------- The Langevin equation for a vortex moving with velocity $\mathbf{v}$ in a magnetic field $\mathbf{B}=\mathbf{n}B$ ($B\equiv|\mathbf{B}|$, $\mathbf{n}=n\mathbf{z}$, $\mathbf{z}$ is the unit vector in the $z$-direction and $n=\pm 1$) has the form $$\label{F1} \eta_{0}\mathbf{v}+n\alpha_{H}\mathbf{v}\times\mathbf{z}=\mathbf{F}_{L}+\mathbf{F}_{p}^{a}+\mathbf{F}_{p}^{i}+\mathbf{F}_{th},$$ where $\mathbf{F}_{L}=n(\Phi_{0}/c)\mathbf{j}\times\mathbf{z}$ is the Lorentz force ($\Phi_{0}$ is the magnetic flux quantum, $c$ is the speed of light), $\mathbf{F}_{p}^{a}=-\nabla U_{p}(x)$ is the anisotropic pinning force ($U_{p}(x)$ is the washboard planar pinning potential), $\mathbf{F}_{p}^{i}$ is the isotropic pinning force, induced by uncorrelated point-like disorder , $\mathbf{F}_{th}$ is the thermal fluctuation force, $\eta_{0}$ is the vortex viscosity, and $\alpha_{H}$ is the Hall constant. ![System of coordinates $xy$ (with the unit vectors $\mathbf{x}$ and $\mathbf{y}$) associated with the PPP planes and the system of coordinates $x'y'$ associated with the direction of the current density vector $\mathbf{j}$; $\alpha$ is the angle between the channels of the PPP and $\mathbf{j}$, $\beta$ is the angle between the average velocity vector of the vortices $\textbf{v}$ and the vector $\mathbf{j}$; $\mathbf{F}_{L}$ is the Lorentz force; $<\mathbf{F}_{p}^{i}>$ and $<\mathbf{F}_{p}^{a}>$ are the average isotropic and anisotropic pinning forces, respectively, $\mathbf{F}_{I}$ is the average effective motive force for a vortex. Here for simplicity we assume $\epsilon=0$.[]{data-label="fig1"}](fig1.eps) For purely isotropic pinning (i.e. for $\mathbf{F}_{p}^{a}=0$)[ Eq.]{} (\[F1\]) was earlier solved$^{17}$ for $\mathbf{F}_{th}=0$, using the fact that $$\label{F2} \mathbf{F}_{p}^{i}=-\eta_{i}(\upsilon)\mathbf{v},$$ where $ \eta_{i}(\upsilon)$ is velocity-dependent viscosity and $\upsilon\equiv|\mathbf{v}|$. Below we will show (see Eq. (8) and item D of Sec. II), that the solution, obtained in Ref. 17, can be presented in terms of the probability function of overcoming the effective current- and temperature-dependent potential barrier of isotropic pinning $\nu_{i}(F_{I})$, which is simply related to $\eta_{i}(\upsilon)$. In the absence of point-like disorder (i.e. for $\mathbf{F}_{p}^{i}=0$) [Eq.]{} (\[F1\]) was reduced to the Fokker-Planck equation, which was solved$^{10,11}$, assuming that the fluctuational force $\mathbf{F}_{th}(t)$ is represented by a Gaussian white noise, whose stochastic properties are assigned by the relations $$\label{F3} \langle F_{th,i}(t)\rangle=0, \, \langle F_{th,i}(t) F_{th,j}(t') \rangle=2T\eta_{0}\delta_{ij}\delta(t-t'),$$ where $T$ is the temperature in energy units. In what follows we derive the solution of [Eq.]{} (\[F1\]), using for $\mathbf{F}_{p}^{i}$ the assumption (\[F2\]), which reduces [Eq.]{} (\[F1\]) to the equation $$\label{F4} \eta\mathbf{v}+n\alpha_{H}\mathbf{v}\times\mathbf{z}=\mathbf{F}_{L}+\mathbf{F}_{p}^{a}+\mathbf{F}_{th},$$ where $\eta=\eta(\upsilon)\equiv\eta_{0}+\eta_{i}(\upsilon)$. Using the result of Ref. 11, the selfconsistent solution of the [Eq.]{} (\[F4\]) can be represented as $$\label{F5} \begin{array}{l} \eta(\upsilon)\langle v_{x} \rangle=F_{a}\nu_{a}(F_{a})/(1+\tilde{\epsilon}^{2}),\\ \\ \eta(\upsilon)\langle v_{y} \rangle=F_{Ly}+n\tilde{\epsilon}F_{a}\nu_{a}(F_{a})/(1+\tilde{\epsilon}^{2}), \end{array}$$ where $\nu_{a}(F_{a})$ is the probability of overcoming the PPP under the influence the effective moving force $F_{a}\equiv F_{Lx}-n\tilde{\epsilon}F_{Ly}$, $F_{Lx}$ and $F_{Ly}$ are the Lorentz force components acting along the vector $\mathbf{x}$ and $\mathbf{y}$, respectively, $\tilde{\epsilon}\equiv\epsilon Z(\upsilon)$, $\epsilon\equiv\alpha_{H}/\eta_{0}$ and $Z(\upsilon)\equiv\eta_{0}/\eta(\upsilon)$ with an obvious condition $0\leq Z(\upsilon)\leq1$. [Eqs.]{} (\[F5\]) can be rewritten as $$\label{F6} \eta(\upsilon)\langle \mathbf{v} \rangle=\mathbf{F}_{I},$$ where $F_{Ix}$ and $F_{Iy}$ are corresponding right-hand parts of [Eqs.]{} (\[F5\]). From [Eq.]{} (\[F6\]) we have $$\label{F7} \eta(\upsilon)\upsilon=F_{I},$$ where $F_{I}\equiv(F_{Ix}^2+F_{Iy}^2)^{1/2}$ and we omitted for simplicity the symbol of averaging for $\mathbf{v}$. Then from [Eq]{}. (\[F7\]) follows that $\upsilon=\upsilon(F_{I})$ and thus it is possible to represent $\eta_{i}(\upsilon)$ and $Z(\upsilon)$ in terms of $F_{I}$: $\eta_{i}(\upsilon)=\eta_{i}[\upsilon=\upsilon(F_{I})]\equiv\tilde{\eta}_{i}(F_{I})$ and $$\label{F8} Z(\upsilon)=Z[\upsilon=\upsilon(F_{I})]\equiv\nu_{i}(F_{I}).$$ Here $\nu_{i}(F_{I})$ has a physical meaning of the probability to overcome the effective potential barrier of isotropic pinning under the influence of effective ($\upsilon$-dependent through the $\tilde{\epsilon}$-dependence) force $F_{I}$. Then in terms of the $\nu_{i}(F_{I})$ [Eq.]{} (\[F6\]) takes the selfconsistent form $$\label{F9} \eta_{0}\mathbf{v}=\nu_{i}(F_{I})\mathbf{F}_{I},$$ which can be highly simplified for a small dimensionless Hall constant $(\epsilon\ll 1)$. Really, in this limit $\tilde{\epsilon}=\epsilon\nu_{i}(F_{i})$, where $F_{i}\equiv F_{I} (\epsilon=0)$, and the right-hand part of the [Eq.]{} (\[F6\]) becomes $\upsilon$-independent, i.e. is represented only in terms of the known quantities. Just in this limit all subsequent results of the paper will be discussed. The nonlinear resistivity and conductivity tensors -------------------------------------------------- The average electric field induced by the moving vortex system is given by $$\label{F10} \mathbf{E}=(1/c)\mathbf{B}\times\mathbf{v}=n(B/c)(-\upsilon_{y}\mathbf{x}+\upsilon_{x}\mathbf{y}),$$ where $\mathbf{x}$ and $\mathbf{y}$ are the unit vectors in $x$- and $y$-direction, respectively. From formulas (\[F9\]) and (\[F10\]) we find the dimensionless magnetoresistivity tensor $\hat{\rho}$ (having components measured in units of the flux-flow resistivity $\rho_{f}\equiv\Phi_{0}B/\eta_{0}c^{2}$) for the nonlinear law $\mathbf{E}=\hat{\rho}(j)\mathbf{j}$ $$\label{F11} \begin{array}{l}\hat{\rho}= \left(\begin {array}{cc} \rho_{xx}& \rho_{xy}\\ \rho_{yx}& \rho_{yy} \end {array}\right )=\\ \qquad{}\\ \qquad{} \left(\begin {array}{cc} \nu_{i}(F_{I})&-n\epsilon\nu_{i}^2(F_{i})\nu_{a}(F_{Lx})\\ n\epsilon\nu_{i}^2(F_{i})\nu_{a}(F_{Lx})&\nu_{i}(F_{I})\nu_{a}(F_{a})\end {array}\right ). \end{array}$$ The conductivity tensor $\hat{\sigma}$ (the components of which are measured in units of $1/\rho_{f}$), which is the inverse of the tensor $\hat{\rho}$, has the form $$\label{F12} \begin{array}{l}\hat{\sigma}=\left(\begin {array}{cc} \sigma_{xx}& \sigma_{xy}\\ \sigma_{yx}& \sigma_{yy} \end {array}\right )=\\ \qquad{}\\ \qquad{}\left(\begin {array}{cc} [\nu_{i}(F_{I})]^{-1} & n\epsilon \\ -n\epsilon & [\nu_{i}(F_{I})\nu_{a}(F_{a})]^{-1} \end {array}\right ). \end{array}$$ From [Eqs.]{} (\[F11\]) and (\[F12\]) we see that the off-diagonal components of the $\hat{\rho}$ and $\hat{\sigma}$ tensors satisfy the Onsager relation ($\rho_{xy}=-\rho_{yx}$ in the general nonlinear case and $\sigma_{xy}=-\sigma_{yx}$). All the components of the $\hat{\rho}$-tensor and the diagonal components of the $\hat{\sigma}$-tensor are functions of the current density $j$ through the external force value $F_{L}$, the temperature $T$, the angle $\alpha$, and the dimensionless Hall parameter $\epsilon$. For the following (see item E.2 of Sec. II) it is important, however, to stress that the off-diagonal components of the $\hat{\sigma}$ (i.e. the dimensional Hall conductivity terms $\delta=n\epsilon/\rho_f$) are not influenced by a presence of the $i$- and $a$-pins$^{16}$. The experimentally measurable resistive responses refer to a coordinate system tied to the current (see Fig. 1). The longitudinal and transverse (with respect to the current direction) components of the electric field, $E_{\parallel}$ and $E_{\perp}$, are related to $ E_{x}$ and $E_{y}$ by the simple expressions $$\label{F13} \begin{array}{ll} E_{\parallel}=E_{x}\sin\alpha+E_{y} \cos \alpha,\\ \\ E_{\perp}=-E_{x}\cos\alpha+E_{y}\sin\alpha.\\ \end{array}$$ Then according to [Eqs.]{} (\[F13\]), the expressions for the experimentally observable longitudinal and transverse (with respect to the $\mathbf{j}$-direction ) magnetoresistivities $\rho_{\parallel}\equiv E_{\parallel}/j $ and $\rho_{\perp}\equiv E_{\perp}/j$ have the form: $$\label{F14} \begin {array}{ll} \rho_{\parallel}=\rho_{xx}\sin^{2}\alpha +\rho_{yy}\cos^{2}\alpha ,\\ \\ \rho_{\perp}=\rho_{yx}+(\rho_{yy}-\rho_{xx})\sin\alpha\cos\alpha.\\ \end{array}$$ Note, however, that the magnitudes of the $\rho_{\parallel,\perp}$, given by [Eqs.]{} (\[F14\]), are, in general, depend on the direction of the external magnetic field $\mathbf{B}$ along $z$ axis due to the $n\epsilon$-dependence of the $F_{I} $ and $F_{a}$ forces in arguments of the $\nu_{i}$ and $\nu_{a}$ functions, respectively. In order to consider only $n$-independent magnitudes of the $\rho_{\parallel}$- and $\rho_{\perp}$-resistivities we should introduce the even(+) and the odd($-$) with respect to magnetic field reversal $(\rho^{\pm}\equiv (\rho(n)\pm\rho(-n))/2)$ longitudinal and transverse dimensional magnetoresistivities, which in view of Eqs. (\[F14\]) have the form: $$\label{F15} \begin {array}{ll} \rho_{\parallel}^{+}=\rho_{f}[\sin^{2}\alpha + \nu_{a}(F_{Lx})\cos^{2}\alpha ]\nu_{i}(F_{i}),\\ \\ \rho_{\parallel}^{-}=\rho_{f}[[\sin^{2}\alpha+\nu_{a}(F_{Lx})\cos^{2}\alpha]\nu_{i}^{-}(F_{I})+\\ \\ \qquad{} \nu_{i}(F_{i})\nu_{a}^{-}(F_{a})\cos^{2}\alpha].\\ \end{array}$$ $$\label{F16} \begin {array}{ll} \rho_{\perp}^{+}=-\rho_{f}\nu_{i}(F_{i})[1-\nu_{a}(F_{Lx})]\sin\alpha\cos\alpha,\\ \\ \rho_{\perp}^{-}=\rho_{f}[n\epsilon\nu_{a}(F_{Lx})\nu_{i}^{2}(F_{i})+\\ \\ \qquad\{\nu_{a}^{-}(F_{a})\nu_{i}(F_{i})-\nu_{i}^{-}(F_{I})[1-\nu_{a}(F_{Lx})]\}\times\\ \\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\sin\alpha\cos\alpha].\\ \end{array}$$ Here $\nu^{-}$ are the odd ($\nu^{-}\equiv\nu^{-}(n)=(\nu(n)-\nu(-n))/2$) components of the functions $\nu_{i}(F_{I})$ and $\nu_{a}(F_{a})$, and for $\nu^{-}_{a}(F_{a})$ we have the expansion in terms of $\epsilon<<1$: $$\label{F17} \nu_{a}^{-}\simeq -n\epsilon\nu_{i}(F_{i})F_{Ly}[d\nu_{a}(F_{Lx})/dF_{Lx}].$$ [Eqs.]{} (\[F15\])-(\[F16\]) are accurate to the first order in $\epsilon<<1$ and contain a lot of new physical information, which will be analyzed below (see item E of Sec. II). However, before this analysis it is instructive to discuss in short the main physically important features of two main limiting cases of purely anisotropic $a$-pinning and isotropic $i$-pinning, which follow from [Eqs.]{} (\[F15\])-(\[F16\]), when $\nu_{i}=1$ or $ \nu_{a}=1$, respectively. Anisotropic a-pinning. ---------------------- Setting $\nu_{i}=1$ we obtain rather simple formulas, which were derived firstly$^{11}$ for the experimentally observable nonlinear even and odd longitudinal and transverse anisotropic magnetoresistivities $\rho_{\parallel,\perp}^{\pm}(j,\theta,\alpha,\varepsilon_{a})$ as functions of the transport current density $j$, dimensionless temperature $\theta$ and relative volume fraction $0\leq\varepsilon_{a}\leq1$, occupied by the parallel twin planes, directed at an angle $\alpha$ with respect to the current direction: $$\label{F18} \rho_{\parallel a}^{+}=\rho_{f}[\nu_{a}^{+}\cos^{2}\alpha+\sin^{2}\alpha],\quad \rho_{\perp a}^{+}=\rho_{f}(\nu_{a}^{+}-1)\sin\alpha\cos\alpha,$$ $$\label{F19} \rho_{\parallel a}^{-}=\rho_{f}\nu_{a}^{-}\cos^{2}\alpha,\quad\rho^{-}_{\perp a}=\rho_{f}[n\epsilon\nu_{a}^{+}+\nu_{a}^{-}\sin\alpha\cos\alpha].$$ Here $\nu_{a}=\nu_{a}(F)$ is considered as the probability of overcoming the potential barrier of the washboard PPP in the $x$-direction under the influence of the effective force $F\equiv F_{Lx}-n\epsilon F_{Ly}$$^{11}$. This $\nu_a$-function describes an essentially nonlinear transition from the linear low-temperature thermoactivated flux flow (TAFF) regime of vortex motion to the ohmic flux flow (FF) regime. It is a step-like function of $j$ or $\theta$ for a small fixed temperature or current density respectively (see Figs. 4, 5 in Ref. 11). It follows from Eqs. (18)-(19) that for $\alpha\not=0, \pi/2$ the observed resistive response contains not only the ordinary longitudinal $\rho_{||a}^+(\alpha)$ and transverse $\rho_{\perp a}^-(\alpha)$ magnetoresistivities, but also two new components induced by the pinning anisotropy: an [*even transverse*]{} $\rho_{\perp a}^+(\alpha)$ and an [*odd longitudinal*]{} component $\rho_{||a}^-(\alpha)$. The physical origin of the $\rho_{\perp a}^+(\alpha)$ (which is independent of $\epsilon$) is related in an obvious way with the guided vortex motion along the "channels” of the washboard pinning potential in the TAFF regime. On the other hand, the component $\rho_{||a}^-(\alpha)$ is proportional to the odd component $\nu_a^-$, which is zero at $\epsilon=0$ and has a maximum in the region of the nonlinear transition from the TAFF to the FF regime at $\epsilon\not=0$ (see Figs. 6, 7 in Ref. 11) The $(j,\theta)$-dependence of the odd transverse (Hall) resistivity $\rho_{\perp a}^-(j,\theta)$ has contributions both, from the even $\nu_a^+\approx\nu_a$ and from the odd $\nu_a^-$ components of the $\nu_a(j,\theta)$-function. Their relative magnitudes are determined by the angle $\alpha$ and the effective Hall constant $\epsilon$. Note, that as the odd longitudinal $\rho_{\parallel a}^-$ and odd transverse $\rho_{\perp a}^-$ magnetoresistivities arise by virtue of the Hall effect, their characteristic scale is proportional to $\epsilon<<1$ (see Eqs. (19)). The appearance of these new odd Hall contributions follows from emergence of a certain equivalence of $xy$-directions for the case, where a guiding of the vortex along the channels of the washboard anisotropic pinning potential is realized$^{18}$ at $\alpha\neq0,\pi/2$ and leads to the new specific angle-dependent “scaling” relations for the Hall conductivity$^{11}$ for the case $\epsilon\tan\alpha<<1$ $$\label{F20}n\epsilon=(\rho_{\perp a}^{-}-\rho_{\parallel a}^{-}\tan\alpha)\cos^{2}\alpha/(\rho_{\parallel a}^{+}-\rho_{f}\sin^{2}\alpha).$$ Here the dimensionless Hall constant $\epsilon<<1$ is uniquely related to three experimentally observable nonlinear resistivities $\rho_{\parallel a}^{+}, \rho_{\parallel a}^{-}, \rho_{\perp a}^{-}$, and the “scaling” relation (\[F20\]) depends on the angle $\alpha$. This relation differs substantially from the power-law scaling relations, obtained in the isotropic case$^{17}$ (see below). In the particular case $\alpha=0$ we regain the results$^{10}$, specifically $\epsilon=\rho_{\perp a}^{-}/\rho_{\parallel a}^{+}$ (in Ref. 10 $\epsilon=\rho_{\perp}/\rho_{\parallel}$), i.e. a linear relationship between $\rho_{\perp a}^{-}$ and $\rho_{\parallel a}^{+}$. Eq. (20) may be represented in another form $$\rho_{\perp a}^-(\alpha)={\delta}\nu_a(\alpha){\rho_f}^2 -\rho_{||a}^-(\alpha)\tan\alpha$$ which is more suitable for considering scaling relations in longitudinal ($\alpha=\pi/2$) and transverse ($\alpha=0$) LT-geometries of experiment$^{11}$. In these geometries second term in the right hand side of Eq. (21) is zero and we obtain that $$\rho_{\perp a}^-= \tilde{\delta}({\rho_{||a}^+})^2,$$ $$\begin{array}{ll} \tilde{\delta}(\alpha=\pi/2)\equiv\tilde{\delta_L}={\delta}\nu_a(0,\theta);\\ \\\tilde{\delta}(\alpha=0)\equiv\tilde{\delta_T}={\delta}/\nu_a(j,\theta). \end{array}$$ From Eqs. (22)-(23) follows that $\tilde{\delta}$ may be interpreted as an *effective* Hall conductivity in LT-geometries which is suppressed for $\alpha=\pi/2$ ($\tilde{\delta_L}<{\delta}$) and enhanced for $\alpha=0$ ($\tilde{\delta_T}>{\delta}$) in comparison with a bare Hall conductivity ${\delta}$. The physical reason for this influence of $\nu_a$-function on the $\tilde{\delta}$ behavior in LT-geometries is simple. Namely, it appears as a result of the fact that in the case of anisotropic pinning the driving force $F\equiv F_{Lx}-n\epsilon F_{Ly}$, which determines the probability of overcoming the potential barrier ( and therewith also determines the magnitude of the component of the vortex velocity perpendicular to the channels of the PPP), is the sum of two forces. The first of these is the transverse component of the Lorentz force, $F_{Lx}$= $F_{L}\cos\alpha$, and the other is the *transverse* component of the Hall force $ F_{H}=n\epsilon F_{Ly}$ which is proportional to the longitudinal (relative to the PPP planes) component of the velocity of guided vortex motion. This second force $ F_{H}$, which changes its sign (relative to the sign of $F_{L}$ ) upon reversal of the sign of the external magnetic field, is the reason for appearance of new, Hall-like in their origin, $\nu^-$-terms in the formulas for the resistive responses in Eqs. (19). Returning to the physics of suppression and enhancement of the $\tilde{\delta}$ in LT-geometries we should keep in mind that only longitudinal component of the vortex velocity (with respect to the current direction) $v_l$ is responsible for the appearance of the transverse Hall voltage. Thus, in L-geometry $v_l$ and $\tilde{\delta}$ are suppressed by PPP-barriers, whereas in T-geometry $v_l$ is not influenced by them and $\tilde{\delta}$ looks like enhanced quantity. On the contrary, the behavior of the transverse component of the vortex velocity $v_t$, which determines the longitudinal voltage, in LT-geometries is opposite. Isotropic i-pinning. -------------------- For this case we put $ \nu_{a}=1$ and from [Eqs.]{} (\[F15\])-(\[F16\]) follows that $$\label{F21} \rho_{\parallel}^{+}=\rho_{\parallel i }=\rho_{f}\nu_{i}(F_{L}),\quad \rho_{\perp}^{-}=\rho_{\perp i}=\rho_{f}n\epsilon\nu_{i}^{2}(F_{L}),$$ where $F_{L}=F_{i}(\nu_{a}=1)=\mid\mathbf{F}_{L}\mid$. From [Eqs.]{} (\[F21\]) the well-known scaling relation $ \rho_{\parallel i}\sim(\rho_{\perp i})^{2}$, derived firstly in Ref. 17, follows. Note that $\rho_{\perp i}^{+}=\rho_{\parallel i}^{-}=0$ in this case, i.e. nonlinear resistive response is isotropic. Competition between a- and i-pinning. ------------------------------------- Equations (\[F15\])-(\[F16\]) for the magnetoresistivities $\rho_{\parallel,\perp}^{\pm}$ at arbitrary value of competition between point-like and anisotropic planar disorder for the in-plane geometry of experiment can be represented in a more suitable form, if we take into account [Eqs.]{} (\[F18\])-(\[F19\]) and (\[F21\]): $$\label{F22}\rho_{\parallel}^{+}=\nu_{i}(F_{i})\cdot\rho_{\parallel a}^{+},\quad \rho_{\perp}^{+}=\nu_{i}(F_{i})\cdot\rho_{\perp a}^{+},$$ $$\label{F23}\rho_{\parallel}^{-}=\nu_{i}^{-}\rho_{\parallel a}^{+}+\nu_{i}(F_{i})\cdot\rho_{\parallel a}^{-},$$ $$\label{F24} \rho_{\perp}^{-}=\rho_{f}n\epsilon\nu_{a}\nu_{i}^{2}+\rho_{f}\{\nu_{a}^{-}\nu_{i}-\nu^{-}_{i}[1-\nu_{a}]\}\sin2\alpha/2.$$ Here $\nu_{i}(F_{i})$ is the probability function $\nu_{i}$ of anisotropic argument $F_{i}=[F^{2}_{Lx}\nu^{2}_{a}(F_{Lx})+F_{Ly}^{2}]^{1/2}$, the magnetoresistivity $\rho_{\parallel,\perp a}^{\pm}$ and the $\nu_{a}\equiv\nu_{a}(F_{Lx})$-functions in [Eqs.]{} (\[F22\])-(\[F24\]) are the same as those in item C of Sec. II; $\nu_{i}^{-}=\nu_{i}^{-}[F_{I}(n)]$ and $F_{I}(n)=[F_{Ly}^{2}+ F_{Lx}^{2}\nu_{a}^{2}(F_{a})+2n\epsilon\nu_{i}(F_{i})F_{Lx}F_{Ly}\nu_{a}(1-\nu_{a})]^{1/2}$. It is easy to check, that previous results of items C and D of Sec. II follow from [Eqs.]{} (\[F22\])-(\[F24\]) in the limits of purely anisotropic (i.e. for $\nu_{i}=1$, $\nu_{i}^{-}=0$) and isotropic (i.e. for $\nu_{a}=1$, $\nu_{a}^{-}=0$) pins. In this subsection it must be suffice to discuss in short the main physically important features of these equations. First of all, the magnetoresistivities $\rho_{\parallel,\perp}^{\pm}$ can be found, if the $\nu_{a}$- and $\nu_{i}$- functions are known. Moreover, the converse statement is also valid: it is possible to reconstruct these functions from ($j$, $\theta$, $B$)-dependent resistive measurements, using only [Eqs.]{} (\[F22\]), where the Hall terms are ignored. [Eqs.]{} (\[F23\]) and (\[F24\]), which arise due to the Hall effect, have a rather complicated structure, which reflects a more pronounced competition between isotropic and anisotropic disorder in the Hall-mediated resistive responses. Let us outline the main new physical results, following from [Eqs.]{} (\[F22\])-(\[F24\]). ### Point-like disorder and vortex guiding. For the discussion of the influence of point-like pins on the guiding of vortices in the anisotropic pinning potential it is sufficient to analyze [Eqs.]{} (\[F22\]). Whereas for the purely anisotropic pinning ($\nu_{i}=1$) a critical current density $j_{c}$ exists only for direction, which is strictly perpendicular to the PPP ($\alpha=0$) and $j_{c}(\alpha)=0$ for any other direction ($0<\alpha\leq\pi/2$) due to the guiding of vortices along the channels of a washboard potential, in [Eqs.]{} (\[F22\]) the factor $\nu_{i}(F_{i})$ ensures that an anisotropic critical current density $j_{c}(\alpha,\theta)$ exists for arbitrary angles $\alpha$. It is interesting, however, to note, that the angular dependence of the ratio $\rho_{\perp}/\rho_{\parallel}$, which determines the angle $\beta$ between $\mathbf{j}$ and $\mathbf{v}$ for $a$-pins in Ref. 11, according to the relation $$\label{F25}\cot\beta=-\frac{\rho_{\perp a}^{+}}{\rho_{\parallel a}}=\frac{1-\nu_{a}}{\tan\alpha+\nu_{a}\cot\alpha}$$ is not influenced by the isotropic disorder, because factor $\nu_{i}(F_{i})$ in [Eqs.]{} (\[F22\]) vanishes from [Eq.]{} (\[F25\]). Physically it means, that character of anisotropy in the case of competition between $i$- and $a$-pinning is determined only by $<\mathbf{F}_{p}^{a}>=[\nu_{a}(F_{Lx},\theta)-1]\mathbf{F}_{Lx}$, (see Fig. 1), i.e. by the average pinning force of the PPP. Isotropic pins influence only the magnitude of the average $\mathbf{v}$-vector, because $<\mathbf{F}_{p}^{i}> \parallel \mathbf{v}\parallel\mathbf{F}_{I}$. So, the polar resistivity diagram $\rho(\alpha)$, which can be measured experimentally $^{5}$, is influenced by point-like pins, because from [Eqs.]{} (\[F11\]) follows, that $$\begin {array}{ll} \label{F26}\rho(\alpha)=\rho_{f}[\rho_{xx}^{2}\sin^{2}\alpha+\rho_{yy}^{2}\cos^{2}\alpha]^{1/2}=\\ \\ \qquad{} \rho_{f}\nu_{i}(F_{i})(\sin^{2}\alpha+\nu_{a}^{2}\cos^{2}\alpha)^{1/2}.\\ \end{array}$$ ### New Hall voltages and scaling relations. As it follows from [Eqs.]{} (\[F23\])-(\[F24\]), the odd longitudinal $\rho_\|^-$ and transverse $\rho_\perp^-$ magnetoresistivities contain terms with the $\nu_i^-$-function. They possess a highly anisotropic current- and temperature-dependent bump-like behavior. They tend to zero in the linear regime of vortex motion. For $\alpha=0,\pi/2$ these new terms disappear, because $\nu_i^-=\nu_a^-=0$ at these limits. As it was in the case of purely a-pinning (see item C of Sec. II), the appearance of these new odd Hall contributions follows from the emergence of a certain equivalence of $xy$-directions due to a guiding of vortices along the channels of the washboard pinning potential for the case with $\alpha\not=0,\pi/2$. Note also, that $\rho_\|^-$ includes two terms with similar signs, whereas in $\rho_\perp^-$ there are terms with opposite signs. The latter can give rise to the well-known sign change in the $(j,\theta,H)$-dependence of the Hall resistivity below $\mathrm T_c$$^{12}$. From [Eqs.]{} (\[F22\])-(\[F24\]) new anisotropic “scaling” relations for the dimensionless Hall constant $\epsilon$ can be derived. For this purpose we exclude $\nu_{i}^{-}$ from [Eqs.]{} (\[F22\])-(\[F24\]), for $\nu_{a}^{-}$ use [Eq.]{} (\[F17\]); and after some algebra in the limit $\epsilon\cdot\tan\alpha<<1$ we have: $$\label{F27}n\epsilon=\frac{2\rho_{\perp}^{-}\cdot\rho_{\parallel}^{+}+\rho_{f}\sin2\alpha(1-\nu_{a})\nu_{i}\rho_{\parallel}^{-}}{[2\nu_{a}\rho_{\parallel}^{+}-\sin2\alpha\cdot\rho_{f}\nu_{i}F_{Ly}\nu_{a}^{'}]\rho_{f}\nu_{i}^{2}}.$$ It is easy to check that from [Eq.]{} (\[F27\]) follows scaling relations $\delta=n\epsilon/\rho_{f}=\rho_{\perp i}/(\rho_{\parallel i})^2$ (for $i$-pins at $\nu_a=1$) and [Eq.]{} (\[F20\]) (for $a$-pins at $\nu_i=1$) As it follows from Eqs. (25) and (27), just the same “scaling” relations as given by Eqs. (22) and (23) for *a*-pins, exist also for (*i*+*a*)-pins (with a replacement of corresponding *a*-resistivities in Eq. (8) by $\rho_f\rho_{\perp}^{-}$ and $\rho_f\rho_{\parallel}^{+}$). Physically it follows from the fact that point-like disorder does not change the angular dependence of the ratio $\rho_{\perp a}^{+}/\rho_{\parallel a}^{+}$, which determines the angle $\beta$ between $\mathbf{j}$ and average velocity vector $\langle\mathbf{v}\rangle$ for $a$-pins$^{11}$, and influences only the magnitude of $\langle\mathbf{v}\rangle$$^{16}$. Grafical analysis of nonlinear regimes. ======================================= Pinning potential and $\nu$-function behavior. ---------------------------------------------- In order to analyze different types of nonlinear anisotropic $(j, \theta, \alpha)$-dependent magnetoresistivity responses, given by formulas (\[F22\])-(\[F24\]), we should bear in mind that these responses, as is seen from formula (\[F11\]), are completely determined by the $(j,\theta)$-behavior of the functions $\nu_{a}(F_{a})$ and $\nu_{i}(F_{I})$, having a sense of the probabilities to overcome the effective potential barriers of the $a$- and $i$-pins, respectively. A simple analytical model for the calculation of the $(j, \theta)$-dependent $\nu$-functions was given earlier$^{11,13,20}$. We will use for both $\nu_{i}$ and $\nu_{a}$ functions the one-dimensional periodic pinning potential $U_{p}(x)$ (see Fig. 2), which has a simple analytical form$^{11,20}$: $$\label{F28} U_{p}(x)= \left\{ \begin{array}{crr} \, - F_{p}x, \qquad\qquad 0 \leqslant x \leqslant b,\\ \\ F_{p}(x-2b), \qquad b \leqslant x \leqslant 2b,\\ \\ 0, \qquad\qquad\qquad 2b \leqslant x \leqslant h , \\ \end{array} \right.$$ where $F_{p}$ is the pinning force ($F_{p}=U_{0}/b$, where $U_{0}>0$ is the depth of the potential well and $2b$ is the width of the well). This form of $U_{p}(x)$ allows to define as the properties of a given pinning center (by the parameters $U_{0}$ and $b$), as well as the density of such centers (by the parameter $\varepsilon=2b/h$, where $h$ is the period of the $U_{p}(x)$). ![Model pinning potential $U_{p}(x)$: $h$ is the period of the potential, $2b$ is the width of the potential well, $U_{0}$ is the depth of the potential well, $\varepsilon=2b/h$ characterizes the concentration of the pinning planes.[]{data-label="fig2"}](fig2.eps) Calculation of the $\nu(j, \theta)$ function on the basis of the pinning potential, given by [Eq.]{} (\[F28\]), was done $^{11}$ and can be represented here in the form$^{11}$ $$\label{F29} \nu(f, \theta,\varepsilon)=\frac{2f(f^{2}-1)^{2}}{2f(f^{2}-1)(f^{2}-1+\varepsilon)-\varepsilon\theta G},$$ where $G=\{(3f^{2}+1)\cosh(f/\theta\varepsilon)+(f^{2}-1)\cosh[(f(1-\\ \\ \qquad2\varepsilon))/(\theta\varepsilon)]-2f(f-1)\cosh[f(1-\varepsilon)/\theta\varepsilon-(1/\theta)]-\\ \\ \qquad\quad2f(f+1)\cosh[(f(1-\varepsilon)/\theta\varepsilon+1/\theta]\}/\sinh(f/\theta\varepsilon). $\ Here and below we have for the time being dropped the indices $a$ and $i$ from the physical quantities pertaining to pinning potentials $U_{pa}$ and $U_{pi}$ and formula (\[F29\]) describes equally the pinning on both potentials. For convenience of qualitative analysis of the formulas following dimensionless parameters were used: $f=Fb/U_{0}$ is the effective motive force, which specifies its ratio to the pinning force $F_{p}=U_{0}/b$, $\theta=T/U_{0}$ is the temperature. The effect of the external force $F$ acting on the vortices consists in a lowering of the potential barrier for vortices localized at pinning centers and, hence, an increase in their probability of escape from them. Increasing the temperature also leads to an increase in the probability to escape of the vortices from the pinning centers through an increase in the energy of thermal fluctuations of the vortices. Thus the pinning potential of a pinning center, which for $F, T\rightarrow0$ leads to localization of the vortices, can be suppressed by both an external force and by temperature. A detailed quantitative and qualitative analysis of the behavior of $\nu(f,\theta,\epsilon)$ as a function of all the parameters and its asymptotic behavior as a function of each are described$^{11}$. Here we will pay particular attention only to the typical curves of $\nu(f,\theta,\epsilon)$ as a function of the parameters $f$ and $\theta$, which describe the nonlinear dynamics of the vortex system as a function of the external force acting on the vortices in the direction perpendicular to the pinning centers and as a function of temperature (see Figs. $4$ and $5$ in Ref. 11). As we see from those figures, the form of the $\nu(f)$ and $\nu(\theta)$ curves is determined by the values of the fixed parameters $\theta$ and $f$. The monotonically increasing function $\nu(f)$ reflects the nonlinear transition of the vortex motion from the TAFF to the FF regime with the increasing external force at low temperatures ($T \ll U_{0}$), while at high temperatures ($T \gg U_{0})$ the FF regime is realized in the entire range of variation of the external force (even at small forces) because of the effect of thermal fluctuation on the vortices. The monotonically increasing function $\nu(\theta)$ reflects the nonlinear transition from a dynamical state corresponding to the value of the external force at zero temperature to the FF saturation regime. The width of the transition from the TAFF to the FF regime on the $\nu(f)$ and $\nu(\theta)$ curves depends on substantially different on the increasing of the parameters $\theta$ and $f$, respectively. Namely, with increasing $\theta$ the function $\nu(f)$ shifts leftward and becomes less steep (see Fig. 4 in Ref. 11). That is, the higher the temperature, the smoother the transition from the TAFF to the FF regime and the lower the values of the external force, at which it occurs. With increasing $f$ the $\nu(\theta)$ curve also shifts leftward, it becomes steeper (see Fig. 5 in Ref. 11). Consequently, the greater the suppression of the potential barrier of the pinning center by the external force, the sharper the transition from the TAFF to the FF regime and the lower the temperature at which it occurs. These graphs will be needed later on when we will discuss the physical interpretation of the observed guiding-depended resistive responses. We also note that the dependence of the probability function $\nu(\varepsilon)$ on the concentration of pinning centers decreases monotonically from the value $\nu(0)=1$, which corresponds to the absence of pinning centers, and that it becomes steeper with decreasing fixed parameters $f$ and $\theta$, owing to the growth of the probability density for finding the vortices at the pinning centers with decreasing temperature and external force. Dimensionless form of the $\rho_{\parallel,\perp}^{\pm}$-responses. ------------------------------------------------------------------- Let us turn to the dimensionless parameters by which one can in general case take into account the difference of the potentials $U_{a}$ and $U_{i}-$ specifically, the difference of their periods $h_{a}$, $h_{i}$, the potential well depths $U_{0a}$, $U_{0i}$ and the width $b_{a}$, $b_{i}$. We introduce some new parameters: $\varepsilon=(\varepsilon_{a}\varepsilon_{i})^{1/2}$ is the average concentration of pinning centers, $U_{0}=(U_{0a}U_{0i})^{1/2}$ is the average depth of potential well, $\kappa=(\varepsilon_{i}/\varepsilon_{a})^{1/2}=(h_{a}b_{i}/h_{i}b_{a})^{1/2}$, and $p=(U_{0a}/U_{0i})^{1/2}$ , where the parameters $\kappa$ and $p$ are measures of the corresponding anisotropies. The temperature will be characterized by new parameters: $\theta_{a}=pT/U_{0}=T/U_{0a}$ and $\theta_{i}=(1/p)T/U_{0}=T/U_{0i}$, which are the ratio of the energy of thermal fluctuations of the vortices to the average potential well depth $U_{0a}$ and $U_{0i}$, respectively. The current density will be measured in units of $j_{c}=cU_{0}/\Phi_{0}h$, where $h=(h_{a}h_{i})^{1/2}$. Then the dimensionless parameters $f_{a}$ and $f_{i}$, which specify the ratio of the external forces $F_{a}$ and $F_{i}$ to the pinning forces $F_{pa}=U_{0a}/b_a$ and $F_{pi}=U_{0i}/b_i$ ($\nu_{a}$ and $\nu_{i}$ are the even functions of their arguments), we denote as $f_{a}=F_{a}/F_{pa}$ and $f_{i}=F_{i}/F_{pi}$. The values of the external force $F$, at which the heights of the potential barriers $U_{0a}$ and $U_{0i}$ vanish at $T=0$ correspond (at $\alpha=0$ and $\alpha=\pi/2$) to the critical current densities $j_{ca}=qj_{c}$ and $j_{ci}=j_{c}/q$ respectively, where $q=p/\kappa$. In general case of nonzero temperature and $0<\alpha<\pi/2$ it is possible to consider the angle-dependent crossover current densities $j_{ca}(\alpha)$ and $j_{ci}(\alpha)$ (see below) which correspond to change in the vortex dynamics from the TAFF regime to a nonlinear regime. The condition, that determines the temperature region, in which the concept of critical current densities is physically meaningful is $0 \leqslant T \ll U_{0}$, because for $T \gtrsim U_{0}$ the transition from the TAFF to the nonlinear regime is smeared, and the concept of critical current loses its physical meaning. It is possible now to rewrite [Eqs.]{} (\[F15\])-(\[F16\]) in the dimensionless form in order to represent them as functions of $j$, $\theta$, $\alpha$ at given values of parameters $\varepsilon$, $\epsilon$, $q$, $k$. $$\label{F30}\rho_{\parallel}^{+}=\nu_{i}(f_{i})[\sin^2\alpha+\nu_{a}(f_{a})\cos^2\alpha)],$$ $$\label{F31}\rho_{\perp}^{+}=-\nu_{i}(f_{i})[1-\nu_{a}(f_{a})]\sin2\alpha/2,$$ $$\label{F32}\rho_{\parallel}^{-}=\nu_{i}^{-}(\tilde{f_{i}})[\sin^2\alpha+\nu_{a}(f_{a})\cos^2\alpha)]+\nu_{a}^{-}(\tilde{f_{a}})\nu_{i}(f_{i})\cos^{2}\alpha,$$ \ $$\label{F33} \begin{array}{l}\rho_{\perp}^{-}=n\epsilon\nu_{a}(f_{a})\nu_{i}^{2}(f_{i})+ \\ \\ +\{\nu_{a}^{-}(\tilde{f_{a}})\nu_{i}(f_{i})- \nu^{-}_{i}(\tilde{f_{i}})[1-\nu_{a}(f_{a})]\}\sin2\alpha/2, \end{array}$$\ $$\label{F34} \textrm{where} \qquad f_{a}=jq^{-1}\cos\alpha,\quad{}\\$$ $$\label{F35} f_{i}=jq(\sin^2\alpha+\nu_{a}^2\cos^2\alpha)^{1/2},$$\ and $\tilde{f_{a}}=jq^{-1}[\epsilon\nu_{i}(f_{i})\sin\alpha+n\cos\alpha]$, $$\begin{array}{ll}\tilde{f_{i}}=jq\{\sin^2\alpha+\nu_{a}^2(\tilde{f_{a}})\cos^2\alpha)-n\epsilon\nu_{i}(f_{i})\nu_{a}(f_{a})\times \\ \\\qquad\qquad\qquad\qquad\qquad\qquad\times [1-\nu_{a}(f_{a})]\sin2\alpha\}^{1/2}. \end{array}$$ Here\ \ $\nu_{a}(f_{a})=\nu_{a}(f_{a}, \theta_{a}, \varepsilon_{a}/\kappa)$,$\nu_{a}(\tilde{f_{a}})=\nu_{a}(\tilde{f_{a}}, \theta_{a}, \varepsilon_{a}/\kappa)$,\ \ $\nu_{i}(f_{i})=\nu_{i}(f_{i}, \theta_{i}, \varepsilon_{i}\kappa)$, $\nu_{i}(\tilde{f_{i}})=\nu_{i}(\tilde{f_{i}}, \theta_{i}, \varepsilon_{i}\kappa)$,\ $$\begin{array}{crr} \textrm{and} \qquad &\nu^{\pm}_{i}(\widetilde{f}_{i})=\{\nu_{i}[\widetilde{f}_{i}(n)]\pm\nu_{i}[\widetilde{f}_{i}(-n)]\}/2,&\\ \\ \quad &\nu^{\pm}_{a}(\widetilde{f}_{a})=\{\nu_{a}[\widetilde{f}_{a}(n)]\pm\nu_{a}[\widetilde{f}_{a}(-n)]\}/2.& \end{array}$$ In [Eqs.]{} (33)-(38) we also denoted $\nu_{i}(f_{i})\equiv\nu^{+}_{i}(\widetilde{f_{i}})$ and $\nu_{a}(f_{a})\equiv\nu^{+}_{a}(\widetilde{f_{a}})$ for simplicity. Before following graphical analysis of the $\rho_{\parallel,\perp}^{\pm}$ dependences given by [Eqs.]{} (\[F30\])-(\[F33\]), we should point out the magnitude of some parameters which will be used for presentation of the graphs. It is important to remind here that the parameter $q$ determines the value of anisotropy between $\nu_{i}$ and $\nu_{a}$ critical current densities, whereas the parameter $k$ describes the anisotropy magnitude of the width of nonlinear transition from the TAFF to the FF regime for $\nu_{i}$ and $\nu_{a}$ function. More definitely, if $q>1$, then $j_{ca}=qj_{c}>j_{ci}=j_{c}/q$ and influence of the $i$-pins on the vortex dynamics decreases with $q$-increasing. For $q<1$ the situation is opposite and anisotropy effects may be fully suppressed with $q$-decreasing. So, for the observation of pronounced competition between $i$- and $a$-pins $q\approx1$ should be taken. The temperature dependences of the $\rho_{\parallel}^{+}(\alpha)$ at small current densities under conditions of the presence both isotropic and anisotropic pinning potential were studied experimentally$^{8}$. Arrhenius analysis of these dependences within the frames of suggested here theoretical approach have shown that for the samples$^{8}$ the $U_{0a}=4031$K, $U_{0i}=1568$K, $b_{a}=400$ nm, $b_{i}=2000$ nm at $T\approx8$K. Then for these samples $q\approx1.6$, $\kappa\approx0.5$, $\theta\approx0.003$. It was also pointed out$^{8}$ that the best fitting of the experimental and theoretical curves was established for $b_{i}/b_{a}=15$, from which follows $\kappa\approx0.25$. So for all graphs below we used $q=1.6$, $\kappa=0.25$, $\theta=0.003$, $\epsilon=0.01$ and if it is not pointed out specially, $\varepsilon_{a}=1$ and $\varepsilon_{i}=0.1$. Note also that for the even longitudinal resistivity $\rho^{+}_{\parallel}$ and the even transverse resistivity $\rho^{+}_{\perp}$ for a small Hall effect, terms proportional to $\epsilon\ll 1$ are absent (see [Eqs.]{} (\[F30\])-(\[F31\])) and only contributions describing the competition between isotropic pinning and nonlinear guiding effect on the PPP in terms of the even $\nu_{i}$ and $\nu_{a}$ functions are presented. Graphical analysis of current-angular dependences. -------------------------------------------------- ### $(j,\alpha)$-presentation of $\nu_{a}$ and $\nu_{i}$. In order to discuss graphical $(j,\alpha)$-behavior of the resistive responses we will use $\nu_{a}$ and $\nu_{i}$ functions of their arguments $f_{a}$ and $f_{i}$, respectively, in the form given by [Eq.]{} (\[F29\]). Then these functions are, as a corresponding $\nu$-function$^{11}$, the step-functions in $j$ (at fixed $\theta$) or in $\theta$ (at fixed $j$). For every of the $\nu$-functions it is useful to determine the “crossover current densities” $j_{ci}(\alpha)$ and $j_{ca}(\alpha)$, as those which correspond to the middle point of a sharp step-like nonlinear transition from the TAFF to the FF regime. As it follows from [Eqs.]{} (\[F34\])-(\[F35\]), we can present $f_{a}$ and $f_{i}$ as $f_{a}=j/j_{ca}(\alpha)$ with $j_{ca}(\alpha)=q/\cos\alpha$, and $f_{i}=j/j_{ci}(\alpha)$ with $j_{ci}(\alpha)\approx 1/q\cos\alpha$ for $\alpha\ll\pi/4$ and $j_{ci}(\alpha)\approx 1/\nu_{a}q\cos\alpha$ for $\tan^{2}\alpha\ll\nu_{a}^{2}(j,\alpha)$; $j_{ci}(\alpha)\approx 1/q\sin\alpha$ for $\alpha>\pi/4$ because [Eq.]{} (\[F35\]) can be presented in two equivalent forms, namely $f_{i}=jq\cos\alpha\sqrt{\tan\alpha^{2}+\nu_{a}^{2}}= jq\sin\alpha\sqrt{1+(\nu_{a}/\tan\alpha)^{2}}$. ![The current-angle dependence of the anisotropic probability function $\nu_{a}(j,\alpha)$. In all following graphs the parameters $q=1.6$, $\kappa=0.25$, $\theta=0.003$, $\epsilon=0.01$, $\varepsilon_{a}=1$, and $\varepsilon_{i}=0.1$ (unless otherwise stated).[]{data-label="fig3"}](fig3.eps) ![The current-angle dependence of the average effective motive force for a vortex $f_{i}(j,\alpha)$.[]{data-label="fig4"}](fig4.eps) ![The current-angle dependence of the isotropic probability function $\nu_{i}(j,\alpha)$.[]{data-label="fig5"}](fig5.eps) The behavior of $\nu_{a}(j,\alpha)$ function (see Fig. 3) is rather evident from the $f_{a}(j,\alpha)$ behavior. Namely, for all $\alpha\neq\pi/2$ (i.e. for $f_{a}\neq0$)) the $\nu_{a}$ with a current increasing consistently follows next stages: a) slow increasing at $0<j\lesssim j_{ca}$ in the TAFF regime, where $\nu_{a}\ll1$, b) sharp step-like increasing with a width of the order of $j_{ca}$ which corresponds to nonlinear transition from the TAFF to the FF regime, c) second stage of slow increasing for $j\gtrsim2j_{ca}$ which corresponds to the FF regime (see also item C of Sec. II). It follows from the expression for $j_{ca}(\alpha,q)$ that an increasing of $\alpha$ and (or) $q$ leads to a broadening of the step of the order of $j_{ca}$ and its shift to the larger current densities $j\approx j_{ca}$. The anisotropy of $f_{i}(\alpha)$ (see [Eq.]{} (\[F35\]) and Fig. 4) can be divided into two types: simple (“external”) which depends on $\cos^2\alpha$, and more complex (“internal”), given by $\nu_{a}(\alpha)$. The first (external) anisotropy stems from the “tensorial” $\alpha$-dependence which exists also in the linear (TAFF and FF) regimes of the flux motion. The second (internal) is through the $\alpha$-dependence of $\nu_{a}$, which in the region of transition from the TAFF to the FF regime is substantially nonlinear ([Eq.]{} (\[F29\]) and Fig. 3). The appearance of nonzero $\sin^2\alpha$ term in $f_{i}$ for $\alpha\neq0$ physically describes the guiding of vortices along the channels of the PPP in the presence of $i$-pins for the current densities $j\lesssim~j_{ci}(\alpha)$. The influence of $\nu_{a}$-anisotropy on $\nu_{i}$ is different for different values of the angle $\alpha$ (see Fig. 5). For $\alpha>\pi/4$ the anisotropy of $\nu_{a}(\alpha)$ does not influence the value of $f_{i}(\alpha)$ because $(\nu_{a}/\tan\alpha)^{2}\ll 1$ in the expression for $j_{ci}(\alpha)$. On the contrary, for $\alpha\ll\pi/4$ the influence of $a$-pins on $\nu_{i}(\alpha)$ is most effective for that range of current density, where $\nu_{a}^{2}>\tan^{2}\alpha$, due to the inequality $\tan^{2}\alpha\ll1$. Thus, the $\nu_{i}$ and $\nu_{a}$ as functions of the angle $\alpha$ at $j=const$ behave themselves oppositely (see Figs. 3, 5): $\nu_{i}$ increases monotonically with $\alpha$-increasing, whereas $\nu_{a}$ - monotonically decreases. For $j\gtrsim~j_{ca}(\alpha)$ and at small angles which meet the condition $\tan^{2}\alpha\ll 1$, the behavior of the $\nu_{i}$ and $\nu_{a}$ qualitatively similar in $\alpha$ and opposite in $q$. In case where $\tan^{2}\alpha>1$, the $\nu_{i}$ and $\nu_{a}$ behavior is qualitatively different and stems from the ($\alpha,q$)-dependences of the corresponding crossover current densities. In contradistinction to $\nu_{a}$, the transition of $\nu_{i}$ from the TAFF to the FF depends weakly from $\alpha$ and $q$; it moves to the lower current densities with $q$-increasing for $\alpha>\pi/4$ and moves to the higher ones for $\alpha\ll\pi/4$. In general, the $\nu_{a}$ behavior is more anisotropic than $\nu_{i}$ behavior. The $\nu_{i}$ anisotropy appears only in the TAFF regime, whereas $\nu_{a}$ anisotropy exists as in the TAFF, as well in the FF regime. And this anisotropy is greater in the current density as the angle $\alpha$ is greater. The $\nu_{i}$ and $\nu_{a}$ transition width at $\alpha=const$ is defined by $\varepsilon_{i}$ and $\varepsilon_{a}$ parameters, respectively, and it increases for $\varepsilon_{i}\rightarrow 1$ and $\varepsilon_{a}\rightarrow 1$. ### $(j,\alpha)$-presentation of even magnetoresistivities. Now we are in a position to discuss the results of the presentation of [Eq.]{} (\[F22\]) in the form of graphs. First we note that according to [Eqs.]{} (\[F22\]), the even resistive responses can be represented as the products of corresponding isotropic and anisotropic $\nu$-functions. For this reason the graphical analysis of the $\rho_{\parallel}^{+}(j,\alpha)$ and $\rho_{\perp}^{+}(j,\alpha)$, after the above-mentioned consideration of the $\nu_{i}(j,\alpha)$ (see Fig. 5), can be reduced to the construction and analysis of the $\rho_{\parallel a}^{+}(j,\alpha)$ and $\rho_{\perp a}^{+}(j,\alpha)$ graphs. ![The current-angle dependence of the dimensionless even longitudinal anisotropic magnetoresistivity $\rho_{\parallel a}^{+}(j,\alpha)$ for the value of the parameter $\varepsilon_{a}=1$.[]{data-label="fig6"}](fig6.eps) ![The current-angle dependence of the dimensionless even longitudinal magnetoresistivity $\rho_{\parallel}^{+}(j,\alpha)$ for the value of the parameter $\varepsilon_{a}=1$.[]{data-label="fig7"}](fig7.eps) ![The current-angle dependence of the dimensionless even longitudinal magnetoresistivity $\rho_{\parallel}^{+}(j,\alpha)$ for the value of the parameter $\varepsilon_{a}=0.01$.[]{data-label="fig8"}](fig8.eps) Let us begin with a discussion of $\rho_{\parallel a }^{+}$ behavior (see [Eq.]{} (\[F18\]) and Fig. 6). For all $\alpha\neq 0$ due to the term $\sin^2\alpha$ in [Eq.]{} (\[F18\]) a critical current density $j_{c}$ exists only for direction, which is strictly perpendicular to the PPP ($\alpha=0$) (as it was shown in item E.1 of Sec. II) and $j_{c}(\alpha)=0$ for any other direction ($0<\alpha\leq\pi/2$) due to the guiding of vortices along the channels of a washboard potential (see also Fig. 8 in Ref. 11). In the FF-regime the isotropization of the $\rho_{\parallel a}^{+}$ arises due to the vortex slipping over the PPP channels. Thus at small angles $\alpha$ the $\nu_{a}$ function strongly influences the $\rho_{\parallel a }^{+}$, whereas for $\alpha\rightarrow \pi/2$ this influence is not so effective due to the external anisotropy, which is proportional to the $\sin^2\alpha$ term. Returning now to the consideration of the $\rho_{\parallel}^{+}(j,\alpha)$ graph we refer to the [Eq.]{} (\[F30\]). It is necessary to pay special attention to the TAFF behavior of these curves at small currents and temperatures, which follows from the full pinning of vortices by point-like pins. This behavior is completely different (for $\alpha\neq0$) from the non-TAFF behavior of the corresponding graphs for the case of purely anisotropic pinning (see Fig. 8 in Ref. 11), which is provocated by the guiding of vortices along the channels of the PPP. At high current densities and (or) temperatures appears the FF regime, because the vortex motion transverse to the $a$-pins becomes substantial and longitudinal resistivity practically becomes isotropic. In these limiting cases the $\rho^{+}_{\parallel}(j)$ magnitudes are equal to unity (Fig. 7). For the angles $0<\alpha<\pi/2$ the $\rho_{\parallel}^{+}(j)$ behavior follows substantially the properties of one multiplier. The qualitative behavior of these multipliers, depending on the $j$ and $\alpha$ magnitude is very different as determined by different behavior of their crossover current densities $j_{ci}$ and $j_{ca}$. The priority of a sharp rise of the appearance $\nu_{i}$ or $\nu_{a}$ functions depends on the competition between the crossover current densities $j_{ci}$ and $j_{ca}$, respectively. That is why it may appear a “step” on some of the $\rho_{\parallel}^{+}(j)$ curves (for $q>1$ and $\alpha\neq 0, \pi/2$) when the next sequence of the vortex motion regimes is realized: a) full $i$-pinning in the TAFF regime ($0<j\lesssim~j_{ci}$); b) nonlinear transition from the TAFF to the FF regime for $i$-pins ($j\gtrsim2j_{ci}$), c) practically linear the FF regime as a consequence of the guiding of vortices along the channels of the washboard PPP (on the $\rho_{\parallel}^{+}(j,\alpha)$ surface one can see the horizontal sections at $j\approx j_{ca}$, see Figs. 7, 8); d) nonlinear transition to the FF regime of vortex motion transverse to the $a$-pins for $j\gtrsim j_{ca}$ and, at last, e) a free FF motion for $j\gg j_{ca}$. With decreasing of the $q$ the a)-e) corresponding regions along the current density axis $j$ can overlap each other and a common nonlinear transition appears instead of b)-d) regions. For the limiting cases $\alpha=0,~\pi/2$, a guiding of vortices is absent and the $\rho_{\parallel}^{+}(j)$ LT-behavior is simply related to the $\nu_{i}$ and $\nu_{a}$ behavior. If parameter $\varepsilon_{a}$ is decreasing, then the width of the transition of $\nu_{a}$ from the TAFF to the FF is also decreasing. Such enhancement of the $\nu_{a}$ steepness leads to appearance of the minimum in $\alpha$ for the $\rho_{\parallel}^{+}(j,\alpha)$ graph (see Fig. 8). ![The current-angle dependence of the dimensionless even transverse anisotropic magnetoresistivity $\rho_{\perp a}^{+}(j,\alpha)$[]{data-label="fig9"}](fig9.eps) ![The current-angle dependence of the dimensionless even transverse magnetoresistivity $\rho_{\perp}^{+}(j,\alpha)$. Pay attention to the inverted direction of the axes in comparison with Fig.9.[]{data-label="fig10"}](fig10.eps) Now we pass to a discussion of the $\rho_{\perp a}^{+}(j,\alpha)$ and $\rho_{\perp}^{+}(j,\alpha)$ graphs. As it follows from [Eq.]{} (\[F18\]), the $\rho_{\perp a}^{+}<0$ and has a minimum in $\alpha$ for all $j=const$. The $\rho_{\perp a}^{+}$ reaches its maximal magnitude for $\alpha\approx \pi/4$ due to the factor $\sin\alpha\cos\alpha$ and realization of guiding in the TAFF regime for $j\lesssim~j_{ca}$ (see. Fig. 9a in Ref. 11 and Fig.9). Therefore, the most favorable angle for its observation is near $\alpha=\pi/4$. In considered case the origin of this minimum has the same reason as a low ($j$, $\alpha$)-behavior of the $\rho^{+}_{\parallel}(j,\alpha)$ curves in Fig. 7, namely it stems from existence of the TAFF regime for the point-like pins at small $j$-values. As is seen in Fig. 9, the position and the magnitude of this $\rho^{+}_{\perp a}$-minimum strongly depends on the $\alpha$-value. It is very much pronounced for $q>1$ and strongly suppressed for $q<1$ by influence of the $i$-pins. With increasing of the current density $j\gtrsim~j_{ca}$ a position of the minimum in $\alpha$ is shifting due to the competition of two multipliers in the $\rho_{\perp a}^{+}(j,\alpha)$ expression [Eq.]{} (\[F18\])): $\sin2\alpha$ is decreasing for $\alpha\rightarrow\pi/2$, whereas $(1-\nu_{a}(j,\alpha))$ is increasing with $\alpha$-increasing for $j=const$, and decreasing for $j$-increasing for $\alpha=const$ due to the transition to the FF regime. For all $\alpha$ and current densities $j\gtrsim2j_{ca}$ the $\nu_{a}\approx1$, and for this reason $\rho_{\perp a}^{+}\rightarrow0$. The $q$-influence is defined by $j_{ca}(\alpha)$ and determines the region of appearance of a small value of the $\rho_{\perp a}^{+}$ for the current densities $j\approx j_{ca}$. Since the $\rho_{\perp}^{+}$, according to [Eq.]{} (\[F31\]), is the product of the $\rho_{\perp a}^{+}$ and $\nu_{i}(f_{i})$, so this graph (see Fig. 10) can be reduced to the product of the graphs in Fig. 5 and Fig. 9. The transition from the TAFF to the FF regime is highly anisotropic in $\alpha$; this causes a shift of the maximal $\rho_{\perp}^{+}(j,\alpha)$ magnitude in the direction of a small angle $\alpha\ll\pi/4$ for the $j=const$. That is why in view of $i$-pinning presence the $\rho_{\perp}^{+}(j,\alpha)$, as distinct from $\rho_{\perp a}^{+}(j,\alpha)$, has the minimum both in $\alpha$ and in $j$. This statement follows from the fact that influence of $i$-pinning leads to the $\rho_{\perp}^{+}\rightarrow0$ for $0<j\lesssim~j_{ci}(\alpha)$ due to the $\nu_{i}\ll1$. For the current densities $j\gtrsim~j_{ca}(\alpha)$ the $\rho_{\perp}^{+}(j,\alpha)$ behavior is determined exclusively by the above-mentioned $\rho_{\perp a}^{+}(j,\alpha)$ behavior. ### $(j,\alpha)$-presentation of odd magnetoresistivities. Before following discussion of the odd resistive responses we should remind the reader about the bump-like behavior of the current and temperature dependence of the $\nu^{-}$ functions (see Figs. 6 and 7 in Ref. 11 ), because $\nu^{-}_{i}$ and $\nu^{-}_{a}$ functions, as it follows from [Eqs.]{} (\[F32\])-(\[F33\]), give an important contribution to the odd responses. The $\nu^{-}(j)$ and $\nu^{-}(\theta)$ curves for the case of $\epsilon\ll1$ in fact are proportional to the derivatives of the corresponding $\nu^{+}(j)$ and $\nu^{+}(\theta)$ curves, which have a step-like behavior as a function of their arguments (see Ref. 11 for the detailed discussion of this point and [Eq.]{} (\[F17\]) in this paper). As the $\rho^{-}_{\parallel}$ and $\rho^{-}_{\perp}$ resistivities given by [Eqs.]{} (\[F23\])-(\[F24\]) arise by virtue of the Hall effect, their characteristic scale is proportional to $\epsilon\ll1$, as for [Eqs.]{} (\[F19\]) for purely anisotropic pins. ![The current-angle dependence of the function $\nu_{a}^{-}(j,\alpha)$.[]{data-label="fig11"}](fig11.eps) ![The current-angle dependence of the function $\nu_{i}^{-}(j,\alpha)$.[]{data-label="fig12"}](fig12.eps) The position of the characteristic peak in the $\nu^{-}_{i}$ and $\nu^{-}_{a}$ functions is different for $q\neq1$, because parameter $q$ determines the anisotropy of the critical current densities for $i$- and $a$- pins. So, if $q$ is not very close to the unity, the position of the $i$- and $a$- peaks cannot coincide, and in this case the current and temperature odd resistive dependences $\rho^{-}_{\parallel,\perp}$ can have a bimodal behavior. For the $\rho^{-}_{\parallel}$ curves such dependences will correspond to existence of the resistive “steps” on the $\rho^{+}_{\parallel}$ curves (see Fig. 7), because for $\epsilon\ll1$ we can consider the $\rho^{-}_{\parallel}$ dependences as derivatives of the $\rho^{+}_{\parallel}$ curves. From this viewpoint it is easy to understand the previous assertion in item E.2 of Sec. II that $\rho^{-}_{\parallel}$ includes two terms (every proportional to the $\nu^{-}_{i}$ and $\nu^{-}_{a}$, respectively) with similar signs. Now we will discuss the $\nu_{a}^{-}$ and $\nu_{i}^{-}$ as a function of ($j,\alpha$) and the parameter $q$ in detail. Really, due to the smallness of the Hall constant, the $\nu_{a}^{-}$ and $\nu_{i}^{-}$ tend to zero in the regions of the linear TAFF and FF regimes of the $\nu_{a}$ and $\nu_{i}$ function, respectively. The $\nu_{a}^{-}$ and $\nu_{i}^{-}$ functions have a sharp peak (see Fig. 11, 12) in the region of sharp change of the $\nu_{a}$ and $\nu_{i}$ increasing (for $j\approx j_{ca}$ or $j\approx j_{ci}$, respectively). With $\alpha$- and $q$-increasing the width and the height of the $\nu_{a}^{-}$ maximum also increases with simultaneous shift of the maximum to the higher current densities due to the relation $j_{ca}(\alpha)\approx q/\cos\alpha$. The $\nu_{i}^{-}$ peak is located in the angle range $0<\alpha\lesssim\pi/4$, which corresponds to a change of the angular dependence of the crossover current density $j_{ci}(\alpha)$ from the angles $\alpha\gtrsim\pi/4$ to the angles $\alpha\ll\pi/4$ (see. Fig. 12). The $\nu_{i}^{-}$ maximum shifts to a smaller current densities with $q$-increasing due to the $j_{ci}(\alpha)\approx1/\nu_{a}q\cos\alpha$,. The magnitudes of the $\nu_{a}^{-}$ and $\nu_{i}^{-}$ are compete by an order of magnitude for $0<\alpha\lesssim\pi/4$ and all $q$-values which satisfy a condition $j_{ca}\approx j_{ci}$. ![The current-angle dependence of the odd longitudinal magnetoresistivity $\rho_{\parallel}^{-}(j,\alpha)$.[]{data-label="fig13"}](fig13.eps) ![The current-angle dependence of the odd transverse magnetoresistivity $\rho_{\perp}^{-}(j,\alpha)$. The characteristic minimum (which is shown by the arrow) is in the region $0<\alpha<\pi/4$ and $j\approx j_{ci}(\alpha)$. The minimum is shown as two neighboring minimums due to the step-like behavior of the calculation. Pay attention to the inverted direction of the axes in comparison with Fig.13.[]{data-label="fig14"}](fig14.eps) Now let us discuss a graphical presentation of the [Eq.]{} (\[F32\]), which can be represented as $\rho_{\parallel}^{-}=B_{1}+B_{2}$, where $B_{1}=\nu_{i}^{-}\rho_{\parallel a}^{+}$, and $B_{2}=\rho_{f}\nu_{a}^{-}\nu_{i}\cos^{2}\alpha$. Taking into account that every factor in the $B_{1}$ and $B_{2}$ is positive (see Figs. 5, 6, 11, 12), we can conclude that $\rho_{\parallel}^{-}\geq0$ for all values of the $j,\alpha,q$. Proceeding to the analysis of the $B_{1}$ and $B_{2}$ $(j,\alpha,q)$-behavior in details we consider first those limiting cases in which $a$- or $i$- pinning is dominant i.e. $\nu_{i}\approx1$ or $\nu_{a}\approx1$, respectively. If $a$-pinning is dominant (i.e. for $q\gg1$), then $\nu_{i}^{-}\rightarrow0$, and [Eq.]{}(\[F32\]) has the form $\rho_{\parallel}^{-}\approx\rho_{\parallel a}^{-}=\nu_{a}^{-}\cos^{2}\alpha$. For the opposite case (i.e. for $q<1$), conversely, $\nu_{a}^{-}\rightarrow0$, and $\rho_{\parallel}^{-}\approx\nu_{i}^{-}\rho_{\parallel a}^{+}$. The $\rho_{\parallel}^{-}(j,\alpha)$ graph presentation is especially simple because it may be depicted with the aid of Figs. 5, 11, 12. In the general case, i.e. for $q\approx1$, we should consider the $B_{1}$ and $B_{2}$ separately because dominant type of pinning is absent. The $B_{1}$ is proportional both $\nu_{i}^{-}$, which is nonzero for $0<\alpha<\pi/4$ and $j(\alpha)\approx j_{ci}(\alpha,q)$ (see Fig. 12), and the factor $\rho_{\parallel a}^{+}$ with a graph, shown in Fig. 6. As a result, the $B_{1}$ has a sharp maximum for the $0<\alpha<\pi/4$ and $j(\alpha)\approx j_{ci}(\alpha,q)$. The second term $B_{2}$ is proportional both the factor $\nu_{a}^{-}$ and the factor $\nu_{i}\cos^2\alpha$. The contribution of the first factor is maximal for $\alpha\approx\pi/2$ and current densities $j(\alpha)\approx j_{ca}(\alpha,q)$, whereas the $\nu_{i}\cos^2\alpha$ contribution is maximal for $0<\alpha\lesssim\pi/4$ and $j(\alpha)\approx j_{ci}(\alpha,q)$. Therefore these factors compete so that the resulting maximum of $B_{2}$ shifts from $\alpha\approx\pi/2$ to the $\pi/4\lesssim\alpha<\pi/2$. It is relevant to note that the condition $j_{ci}(\alpha)<j_{ca}(\alpha)$ for $q\approx1$ and $\pi/4\lesssim\alpha<\pi/2$ is always fulfilled. That is why the maximal contribution of the $B_{2}$ is realized for $j(\alpha)\approx j_{ca}(\alpha,q)$ because in this region of the current densities the $\nu_{i}\rightarrow1$ for $\pi/4\lesssim\alpha<\pi/2$. Therefore the $\rho_{\parallel}^{-}$ behavior is determined mainly by the $B_{2}$ behavior, and the $B_{1}$ contribution is essential for $0<\alpha\lesssim\pi/4$ and $j\approx j_{ci}(\alpha)$. The $\rho_{\perp}^{-}(j,\alpha)$ dependence is the most complicated. For the sake of simplicity the analysis we represent the $\rho_{\perp}^{-}$ as a sum $\rho_{\perp}^{-}=\rho_{f}[A_{1}+(A_{2}+A_{3})\sin2\alpha]$, where $A_{1}=n\epsilon\nu_{a}\nu_{i}^{2}$, $A_{2}=\nu_{a}^{-}\nu_{i}/2$, $A_{3}=-\nu^{-}_{i}(1-\nu_{a})/2$. First we consider the limiting cases of purely isotropic or anisotropic pinning ($\nu_{a}\rightarrow1$ or $\nu_{i}\rightarrow1$, respectively). For $i$-pinning we have $\rho_{\perp}^{-}=\rho_{f}n\epsilon\nu_{i}^{2}$, from which follows (Vinokur et al.$^{17}$) a scaling relation $\rho_{\perp}\sim\rho_{\parallel}^2$. For the case of purely anisotropic pinning $\rho_{\perp}^{-}=\rho_{f}\{n\epsilon\nu_{a}+(\nu_{a}^{-}\sin2\alpha)/2\}$, and the scaling relation is $\rho_{\perp}\sim\rho_{\parallel}$ (see also Ref. 16). Now we consider every term in the $\rho_{\perp}^{-}(j,\alpha)$ in detail. The $A_{1}$ contribution can be reduced in fact to the multiplication of the graph in Fig. 3 by the graph in Fig. 5 squared; the result is essentially nonzero for $j\gtrsim j_{ca}(\alpha,q)$. The $A_{2}$ contribution was described above (see the $B_{2}$ term in the $\rho_{\parallel}^{-}$ without taking into account the $\cos^2\alpha$ anisotropy). Note also that both terms ($A_{1}$ and $A_{2}$) are positive for $n\epsilon>0$. The $A_{3}$ behavior is of great interest because the $A_{3}<0$ for $n\epsilon>0$. Let us consider the cases $q>1$ and $q<1$, which correspond to the $a$-, or $i$-pinning domination, respectively. Then, for $\alpha<\pi/4$: a\) for $q<1$ we have $j_{ci}(\alpha)>j_{ca}(\alpha)$ and the sharp maximum of the $\nu^{-}_{i}$ is suppressed by the factor $(1-\nu_{a})\rightarrow0$. As a result, the $A_{3}$ contribution can be ignored. b\) for $q\geq1$ the opposite inequality follows, i.e. $j_{ci}(\alpha)<j_{ca}(\alpha)$. Then for $j\approx j_{ci}(\alpha)$ the $A_{3}$ term is dominant because $\nu_{a}\ll1$ and $\nu_{i}^{-}\rightarrow n\epsilon$ in this $(j,\alpha)$-region (see Fig. 3 and Fig. 5). As a result, the $\rho_{\perp}^{-}(j,\alpha,q)$ change the sign for $j\approx j_{ci}(\alpha)$ and $0<\alpha\lesssim\pi/4$. Since the scale of the $\nu_{i}^{-}\ll\nu_{i}$, the amplitude of the minimum is small in comparison with the $\rho_{\perp}^{-}$ magnitude. Thus, a competition of the $a$- and $i$-pinning leads to the qualitatively important conclusion that the $\rho_{\perp}^{-}$ can change its sign at a certain range of $(\alpha,j,q)$-values, namely for $j\approx j_{ci}(\alpha,j,q)$, $0<\alpha\lesssim\pi/4$, and $q>1$. Resistive response in a rotating current scheme. ------------------------------------------------ ### Polar diagram. An experimental study of the vortex dynamics in $\rm { YBa_{2}Cu_{3}O_{7-\delta}}$ crystals with unidirectional twin planes was recently done using a modified rotating current scheme$^{4,5}$. In that scheme it was possible to pass current in an arbitrary direction in the $ab$ plane of the sample by means of four pairs of contacts placed in the plane of the sample. Two pairs of contacts were placed as in the conventional four-contact scheme, and the other two pairs were rotated by $90^{\circ}$ with respect to the first (see the illustration in Fig. $1$ of Ref. 4). By using two current sources connected to outer pair of contacts, one can continuously vary the direction of the current transport in the sample. By simultaneously measuring the voltage in the two directions, one can determine directly the direction and magnitude of the average velocity vector of the vortices in the sample as a function of the direction and magnitude of the transport current density vector. This made it possible to obtain the angular dependence of the resistive response on the direction of the current with respect to the pinning planes on the same sample. The experimental data$^{4,5}$ attest to the anisotropy of the vortex dynamics in a certain temperature interval which depends on the value of the magnetic field. A rotating current scheme was used$^4$ to measure the polar diagrams of the total magnetoresistivity $\rho(\alpha)$, where $\rho=(\rho^{2}_{x}+\rho^{2}_{y})^{1/2}$ is the absolute value of the magnetoresistivity, $\rho_{x}$ and $\rho_{y}$ are the $x$ and $y$ components of the magnetoresistivity in an $xy$ coordinate system, and $\alpha$ is the angle between the current direction and the $oy$ axis (parallel to the channels of the $a$-pinning centers). In the case of a linear anisotropic response the polar diagram of the resistivity is an ellipse, as can easily be explained. In the case of a nonlinear resistive response the polar diagram of the resistivity is no longer an ellipse and has no simple interpretation. In this subsection we carry out a theoretical analysis of the polar diagrams of the magnetoresistivity $\rho$ in the general nonlinear case in the framework of a stochastic model of $a+i$ pinning. This type of angular dependence $\rho(\alpha)$ is informative and convenient for theoretical analysis. For a sample with specific internal characteristics of the pinning (such as $q$, $\varepsilon_{a}$, $\varepsilon_{i}$, and $\kappa$) at a given temperature and current density the function $\rho(\alpha)$ is contained by the resistive response of the system in entire region of angles $\alpha$ and makes it possible to compare the resistive response for any direction of the current with respect to the direction of the planar pinning centers. In addition, in view of the symmetric character of the $\rho(\alpha)$ curves, their measurements makes it possible to establish the spatial orientation of the system of the planar pinning centers with respect to the boundaries of the sample if this information is not known beforehand. ![Series of graphs of the function $\rho(\alpha)$ for a sequence of the parameter $j$: 0.63 (1), 0.65 (2), 0.75 (3), 1.00 (4), 1.50 (5), 1.92 (6), 2.00 (7), 2.50 (8), 4.00 (9), 20.0 (10) for $\varepsilon_{a}=1$.[]{data-label="fig15"}](fig15.eps) ![Series of graphs of the function $\rho(\alpha)$ for a sequence of the parameter $j$: 0.63 (1), 0.65 (2), 0.75 (3), 1.00 (4), 1.50 (5), 1.92 (6), 2.00 (7), 2.50 (8), 4.00 (9), 20.0 (10) for $\varepsilon_{a}=0.1$.[]{data-label="fig16"}](fig16.eps) Now for analysis of the $\rho(\alpha)$ curves we imagine that vector **j** is rotated continuously from an angle $\alpha=\pi/2$ to $\alpha=0$. The characteristic form of the $\rho(\alpha)$ curves will obviously be determined by the sequence of dynamical regimes through which the vortex system passes as the current density vector is rotated. By virtue of the symmetry of the problem, the $\rho(\alpha)$ curves can be obtained in all regions of angles $\alpha$ from the parts in the first quadrant. We recall that in respect to the two systems of pinning centers it is possible to have the linear TAFF and FF regimes of vortex dynamics and regimes of nonlinear transition between them. The regions of nonlinear transitions are determined by the corresponding values of the crossover current densities $j_{ci}(\alpha,q)$ and $j_{ca}(\alpha,q)$. Now let us consider the typical $\rho(\alpha)$ dependences which are presented in Fig. 15 and 16 for a sequence of a current density magnitude. We remind that the polar diagram graphs represented below are constructed, as the previous graphs in Figs. 3-7, 9-14 for the next values of parameters: $q=1.6$ (i.e. for the case with dominant $a$-pins), $\kappa=0.25$, $\theta=0.003$, $\epsilon=0.01$, $\varepsilon_{i}=0.1$, $\varepsilon_{a}=1$ (Fig. 15), and $\varepsilon_{a}=0.1$ (Fig. 16). Note that $\rho(\alpha)$ is the product of two multipliers: one is the $\nu_{i}(f_{i})$ dependence, which was earlier studied in Fig. 4 of item C.1 of Sec. III, and other is the $\sqrt{\sin^2\alpha+\nu_{a}^2\cos^2\alpha}$ factor, which qualitative behavior is close to the $\rho_{\parallel a}^{+}(j,\alpha)$ dependence (see Fig. 6 in item C.2 of Sec. III). Let us analyze the $\rho(\alpha)$ behavior for the series of values of the current density $j$. When the angle $\alpha$ changes from $0$ to $\pi/2$ the function $\rho(\alpha)$ grows monotonically from $\rho(0)=\nu_{a}(j/q)\nu_{i}(jq\nu_{a}(j/q))$ to $\rho(\pi/2)=\nu_{i}(jq)$. In Fig. 15 curves 1-6 of the function $\rho(\alpha)$ have the shape of the 8-figure drawn along $ox$-axis (strongly elongated for the curves 1, 2). This anisotropy can be determined by the relation of the magnitudes of the half-axis at the direction $\alpha=\pi/2$ to the transverse half-axis for any fixed magnitude of the current density. The curves 1-6 of the $\rho=\rho(\alpha)$ graph has the 8-form elongated along $ox$-axis. It is caused by the step-like behavior of the $\nu_{i}$-function, corresponding for the curves 1, 2 to the crossover from the TAFF to the FF regime. That is why the magnitude of the $\rho(\pi/2)$ for the curve 2 is rather greater than for the first one. With $\alpha$-increasing the $\nu_{i}$-function is in the TAFF-region (see Fig.5), which provocates the $\rho(\alpha)\ll1$ in the case where the condition $j<j_{ci}(\alpha)$ is satisfied. Therefore, with $j$-increasing the magnitude of the angle $\alpha$, which separates the TAFF and the FF regions of the $\nu_{i}$-function at a fixed value of the current density, decreases to the $\alpha\ll1$. As the $\nu_{i}(j,\alpha=\pi/2)$ is in the FF region (i.e. $j\gtrsim~j_{ci}(\alpha=\pi/2)$), so the anisotropy of the 8-curve decreases for curves 3-6. The $\rho=\rho(\alpha)$ behavior of the curves 5-6 is more isotropic in the region $\alpha\ll\pi/4$ than behavior of the curves 1-4. If the condition $j>j_{ca}(0)$ is satisfied, an appearance of the nonzero resistance in corresponding region follows. Its magnitude is smaller than $\rho(\pi/2)$ for the curves 7, 8, 9 and practically is equal to the $\rho(\pi/2)$ for the curve 10. Note, that for the $\alpha=0,\pi$ and $j_{ca}<j\lesssim3j_{ca}/2$ one can see the minimum, which decreases with $j$-increasing and disappears in the case where the condition $j\gtrsim3j_{ca}/2$ is satisfied. So, for large magnitudes of the current densities the $\rho(\alpha)$ behavior becomes more isotropic. It is necessary to pay attention for the $\rho=\rho(\alpha)$ behavior in the case where $\varepsilon_{a}=0,1$ (see Fig. 16) for the same series of the magnitudes of the current densities. The behavior of the curves 6, 7, 8, 9 differs from the above-mentioned case, but the behavior of the curves 1-5, 10 retains the same. This fact is caused by the influence of the parameter $\varepsilon_{a}$ on the $\nu_{i}$ behavior only in the area of its sharp step-like behavior at the $j\simeq j_{ca}(\alpha)$. Note, that the $\nu_{i}$ contribution is dominant in the region $\alpha\ll1$ as well as the above-mentioned anisotropy of the $\rho_{\perp a}^{+}(j,\alpha)$ (see Fig. 5 and Fig. 7). As decreasing of the $\varepsilon_{a}$ causes the more narrow crossover from the TAFF- to the FF-regime, the $\nu_{i}(\alpha)$ has a minimum at fixed magnitude of the current density. The magnitude of this minimum decreases with the $j$-increasing and the minimum shifts from the $\alpha\simeq\pi/4$ to the $\alpha\simeq\pi/2$. The influence of the parameter $q$ acts on the crossover current densities $j_{ci}$ and $j_{ca}$ only quantitatively, but does not change an evolution of the curves 1-10 qualitatively. ### $\Theta_{E}(\alpha)$-dependence. Let us examine theoretically in our model a new type of the experimental dependence, recently studied$^4$ for $\Theta_{E}(\alpha)$, where $\Theta_{E}$ is the angle between $\mathbf{j}$-vector and the electric field vector $\mathbf{E}$ measured at fixed values of the current density and temperature. Taking into account that in the *xy* coordinate system the magnetoresistivity components are $\rho_{x}=\rho_{xx}\sin\alpha=\nu_{i}(F_i)\sin\alpha$, $\rho_{y}=\rho_{yy}\cos\alpha=\nu_{i}(F_i)\nu_{a}(F_a)\cos\alpha$, we obtain the following simple relation: $\tan\Theta_{E}(\alpha)=\rho_{x}/\rho_{y}=\tan\alpha/\nu_a(F_a)$, or $$\label{F36}\Theta_{E}(\alpha)=\arctan(\tan\alpha/\nu_a(F_a)).$$ Note, that the $\nu_{i}$ term, describing the $i$-pinning, is absent in [Eq.]{}(\[F36\]). Then it follows from the latter that the $\nu_{a}(j,\alpha,\theta)=\tan\alpha/\tan\Theta_{E}(j,\alpha,\theta)$, i.e. the $\nu_{a}(j,\alpha,\theta)$ function can be found from the experimental dependence $\Theta_{E}(\alpha)$. Unfortunately, the dependence $\Theta_{E}(\alpha)$ for the series of the temperature values was experimentally found$^{4}$ so far only for the FF-regime (see. Fig. 2 in Ref. 4). The $\Theta_{E}(j,\alpha)$ dependence is presented in Fig. 17. It shows all changes in the $\Theta_{E}(j,\alpha)$ behavior also for the TAFF-regime. ![Series of graphs of the function $\Theta_{E}(\alpha)$ for a sequence of the parameter $j$: 1 (1), 1.7 (2), 2.2 (3), 3.5 (4), 20 (5) for $T=8K$.[]{data-label="fig17"}](fig17.eps) Let us analyze the [Eq.]{} (\[F36\]) in detail. The $\Theta_{E}(j,\alpha)$ is the odd function of the angle $\alpha$, and its magnitude increases monotonically with the $\alpha$-increasing for all values of the $j$ due to the monotonical decreasing of the $\nu_{a}(j,\alpha)$ function (see. Fig. 17). It follows from [Eq.]{} (\[F36\]) that the period of the function $\Theta_{E}(\alpha)$ is equal to $\pi$. One more important limiting case is realized for $\nu_{a}\approx1$,which corresponds to the limit of isotropic pinning. Depending on the inequality between the $j$ magnitude and the crossover current density $j_{ca}(\alpha)\approx q/\cos\alpha$, one can separate two regions where the $\Theta_{E}(j,\alpha)$ behavior is qualitatively different. If $\emph{A}$ is the argument of the arctangent function in [Eq.]{} (\[F36\]), then in that region $j,\alpha,q$, where the inequality $j\gtrsim~j_{ca}(\alpha)$ is true (the FF regime for $\nu_{a}(j,\alpha)$, see also Fig. 3), the magnitude of the $\Theta_{E}\approx\emph{A}$ as $\emph{A}\ll1$. And for the case $j\lesssim~j_{ca}$ (the TAFF regime of the $\nu_{a}(j,\alpha)$) the value $\Theta_{E}\approx\pi/2-\emph{A}^{-1}$, as $\emph{A}\gg1$. Note, that the parameter $\varepsilon_{a}$ influences the $\Theta_{E}(j,\alpha)$ by changing the character of the step-like crossover of the $\nu_{a}(j,\alpha)$ (the smaller the $\varepsilon_{a}$, the sharper the crossover). The value of the parameter $q$, as well as above-mentioned, determines the magnitude of the $j_{ca}(\alpha)$ (and, therefore the position of the boundaries in $j$ of the regions of quite different $\Theta_{E}(\alpha)$ behavior) at fixed $\alpha$. ### Critical current density anisotropy. Under the critical current density we mean the current density, which corresponds to the electric field strength on the sample $E=1 \mu V/cm$. Let us determine the $j_{c}(\alpha)$ behavior graphically by crossing the $E_{\parallel}^{+}=j\rho_{\parallel}^{+}(j)$ graph and the plain $E=E_{c}$ in the polar coordinates. For all angles $\alpha$ the point of crossing for these graphs determines the critical current density magnitude for the defined direction, and the crossing line of the graphs presents the dependence $j_{c}(\alpha)$. Let us remind the reader that as in above-mentioned sections, in the nonlinear law $E_{\parallel}^{+}=j\rho_{\parallel}^{+}(j)$ we measure $j$ and $\rho$ in the values of the $j_{0}=cU_{0}/\Phi_{0}dh$ and $\rho_{f}=\rho_{n}B/B_{c2}$, respectively. That is why the $E$ magnitude we have to measure in the $E_{0}=j_{0}\rho_{f}$. As well as in item B of Sec. III we use the data from Ref. 8, where for the niobium samples $\rho_{n}\approx5,5\cdot10^{-6}$ Ohm$\cdot$cm, $B\approx150$ Gs, $B_{c2}\approx$ 17 kGs, $\rho_{f}\approx5\cdot10^{-8}$Ohm$\cdot$cm, $U_{0}=2500 K$, and $d=2.5\cdot10^{-6} cm$. Therefore, $E_{0}\approx6\cdot10^{-4}V/cm$, and for $E_{c}=1 \mu V/cm$ we have to cross the dimensionless $\rho_{\parallel}^{+}(j)\cdot j$ graph by the plain $E\approx0.002$. ![Series of graphs of the function $j_{c}(\alpha)$ for the parameter pairs: $E_c=0.002$, $q$=1.6 (1); $E_c=0.002$, $q$=3 (2); $E_c=2$, $q$=1.6 (3).[]{data-label="fig18"}](fig18.eps) Now we will discuss the $j_{c}(\alpha)$ as a function of $\alpha,q,E_c$, and $\varepsilon$ in detail. The $j_{c}(\alpha)$-anisotropy can be determined by the relation of the magnitudes of the half-axis at the direction $\alpha=0$ to the transverse half-axis for any fixed magnitude of the parameters $q$, $E_c$. The $j_{c}(\alpha)$ decreases monotonically from $j_{c}(0)$ with $\alpha$-increasing and has a minimum for $\alpha=\pi/2$. It is caused by the fact that, as it was shown in item C.1 of Sec. II, the $a$-pinning (with high values of the $j_{ca}$ for $q>1$) does not influence the $i$-pinning for $\alpha=\pi/2$. Therefore, the inequality for the crossover current densities $j_{ci}(\alpha)<j_{ca}(\alpha)$ for $q>1$ leads to the corresponding inequality for the critical current densities $j_{c}(0)<j_{c}(\pi/2)$. The $q$ influences the $j_{c}(\alpha)$ behavior (as in item D.1 of Sec. II) only quantitatively: with $q$-increasing the ratio $j_{c}(0)/j_{c}(\pi/2)$ grows and visa versa. It is caused by the $j_{c}(0)$-increasing and $j_{c}(\pi/2)$-decreasing due to the $\alpha$-behavior of the corresponding crossover current densities $j_{ci}(\alpha)$ and $j_{ca}(\alpha)$. The smaller the $\varepsilon_a$, the sharper the crossover between the $j_{c}(\alpha)$ regions of slowly and quickly decreasing as a function of the $\alpha$. With $E_c$-increasing the nonlinear law $E_{\parallel}^{+}=\rho_{\parallel}^{+}(j)j$ is satisfied for the larger values of the current density. That is why with $\alpha$-increasing from $0$ to $\alpha^{\ast}$ values (for which the condition $\tan^{2}\alpha^{\ast}\ll\nu_{a}^{2}(j,\alpha^{\ast})$ is satisfied) the $\nu_{a}$-function is in the FF regime and $j_{c}(\alpha)$ decreases slowly. When the condition $\alpha>\alpha^{\ast}$ is true the $\nu_{a}$-function has a step-like crossover from the FF to the TAFF regime and $j_{c}(\alpha)$ decreases quickly. So, the $\alpha^{\ast}$ behavior as a function of the parameters $q$ and $E_c$ is qualitatively different: it increases with $E_c$-increasing and decreases with $q$-increasing. On the increase of the $E_c$ by the several orders of magnitude the $j_{c}(\alpha)$ curve degenerates into a circumference due to the isotropization of the $j_{ci}(\alpha)$ and $j_{ca}(\alpha)$ behavior for the high $j$-values. Otherwise, with $E_c$-decreasing the $j_{c}(\alpha)$ curve degenerates into a narrow loop, because the $j_{ci}(\alpha)$ and $j_{ca}(\alpha)$ behavior for a small $j$ is very anisotropic. CONCLUSION. =========== In the present work we have theoretically examined the strongly nonlinear anisotropic two-dimensional single-vortex dynamics of a superconductor with coexistence of the anisotropic washboard PPP and isotropic pinning potential as function of the transport current density $j$ and the angle $\alpha$ between the direction of the current and PPP planes at a fixed temperature $\theta$. The experimental realization of the model studied here can be based on both naturally occurring$^{2-5}$ and artificially created$^{6-8}$ systems with $i+a$ pinning structures. The proposed model has made it possible for the first time (as far as we know) to give a consistent description of the nonlinear anisotropic current- and temperature-induced depinning of vortices for an arbitrary direction relative to the anisotropy of the washboard PPP. In the framework of this model one can successfully analyze theoretically certain observed resistive responses which are used for studying anisotropic pinning in a number of new experimental techniques$^{4}$ (the polar diagram of $\rho(\alpha)$, the $\Theta_{E}(\alpha)$ curve described by formula [Eq.]{} (\[F36\])) as well as new Hall responses specific for the $i+a$ pinning problem. A quantitative description of the anisotropic nonlinear resistive properties of the problem under study is done in the framework of the stochastic model on the basis of the Fokker-Planck approach. The main nonlinear components of the problem are the anisotropic $\nu_{a}(F_{a})$ and isotropic $\nu_{i}(F_{i})$ probability functions for the vortices to overcome the potential barriers of $a$- and $i$-pinning centers under the action of anisotropic motive forces $F_{a}$ and $F_{i}$, respectively. The latter include both the “external” parameters $j,\alpha,\theta$ and the “internal” parameters $q,\varepsilon_i,\varepsilon_a$ which describe the intensity and anisotropy of the pinning. As can be seen from [Eqs.]{} (\[F30\])-(\[F33\]), the magnetoresistivities $\rho_{\parallel,\perp}^\pm(j,\alpha,\theta)$ are, in general, nonlinear combinations of the experimentally measured $\nu_{i}$ and $\nu_{a}$ functions ($\nu_{i}$ can be measured independently from the $\rho_{\parallel,\perp}^+(\alpha=\pi/2)$, see [Eq.]{} (\[F30\]) and $\nu_{a}$ - from the $\Theta_{E}(\alpha)$, see [Eq.]{} (\[F36\])). Therefore, the nonlinear (in $j$) resistive behavior of the vortex system can be caused by factors of both an anisotropic and isotropic pinning origin. It is important to underline that whereas the structure of the $\nu_{a}(F_{a})$ and $F_{a}$ is the same as for purely $a$-pinning problem, the structure of the $\nu_{i}(F_{i})$ and $F_{i}$ is strongly different from the structure of the purely $i$-pinning problem due to the fact that $F_{i}$, as motive force of the ($i+a$)-problem, is nonlinear and anisotropic (see [Eqs.]{} (\[F34\])-(\[F35\])) and Figs. 3, 4, 5). Two main new features appear due to the introduction of the isotropic $i$-pins into the initially anisotropic $a$-pinning problem. First, unlike the stochastic model of uniaxial anisotropic pinning studied previously$^{10,11}$, where the critical current density $j_c$ is indeed equal to zero for all directions (excepting $\alpha=0$) due to the guiding of vortices, in the given $i+a$ model the anisotropic critical current density $j_c(\alpha)$ exists for all directions because $i$-pins “quench” the guiding of vortices in the limit $(j,\theta)\rightarrow0$. Second, the Hall resistivity response functions $\rho_{\perp}^{-}(j,\alpha)$ can have a change of sign in a certain range of $(j,\alpha,q)$ (at fixed dimensionless Hall constant $\epsilon=\alpha_H/\eta$ and the dimensional Hall conductivity $\delta=n\epsilon/\rho_f)$, whereas the sign of the $\rho_{\parallel}^{-}(j,\alpha)$ does not change. It should be noted that recently$^{8}$ the nonlinear (in $\theta$) anisotropic longitudinal and transverse resistances of Nb films deposited on facetted sapphire substrates were measured at different angles $\alpha$ between $\mathbf{j}$ and facet ridges in a broad range of temperature and relatively small magnetic field $\mathbf{H}$. The experimental data were in good agreement with the theoretical model described here. The measured $\rho_\parallel^+(\theta,\alpha)$ dependences can be fitted using the probability functions $\nu_a$ and $\nu_i$ in the form proposed here (see [Eq.]{} (\[F29\])) with the anisotropic and isotropic pinning potential given by [Eq.]{} (\[F28\]). The periods and depths of the potential wells were estimated from the experimental data$^8$ and were used here (see Sec.III) for the theoretical analysis of different types of nonlinear anisotropic $(j,\alpha)$-dependent magnetoresistivity responses, given by [Eqs.]{} (\[F30\])-(\[F33\]), in the form of graphs (see Figs. 3-18). Whether these theoretical results can explain a new portion of the $(j,\alpha)$-dependent $i+a$ resistivity data measured at $\theta=const$ (in particular, for the samples investigated earlier$^8$ at small current densities) remains to be seen. [20]{} A.K. Niessen and C.H. Weijsenfeld, J. Appl. Phys. **40**, 384 (1969). A.A. Prodan, V.A. Shklovskij, V.V. Chabanenko et al., Physica C **302**, 271 (1998). V.V. Chabanenko, A.A. Prodan, V.A. Shklovskij et al., Physica C **314**, 133 (1999). H.Pastoriza, S.Candia, and G.Nieva, Phys. Rev. Lett. **83**, 1026 (1999). G. D’Anna, V. Berseth, L. Forro, A. Erb, E. Walker, Phys. Rev. B [**61**]{}, 4215 (2000). M. Huth, K. A. Ritley, J. Oster et al., Adv. Funct. Mater. **12**, 333-341 (2002). O.K.Soroka, M.Huth, V.A. Shklovskij et al., Physica C **388-389**, 773,(2003). O.K.Soroka,“Vortex Dynamics in Superconductors in the Presence of Anisotropic Pinning” Ph. D. Thesis, J. Gutenberg University, Mainz, 2005. O.K.Soroka, V.A.Shklovskij, M.Huth et al., to be published. Y. Mawatari, Phys. Rev. B **56**, 3433 (1997). V.A. Shklovskij, A.A. Soroka, A.K. Soroka, Zh Eksp. Teor. Fiz. **116**, 2103 (1999) \[JETP 89, 1138 (1999)\]. G. Blatter, M.V. Feigel’man, V.B. Geshkenbein et al., Rev. Mod. Phys. [**66**]{}, 1125 (1994). O.V. Usatenko and V.A. Shklovskij, J. Phys. A 27, 5043 (1994). V.A. Shklovskij, Phys. Rev. B [**65**]{}, 092508 (2002). V.A. Shklovskij, J. Low Temp. Phys. [**130**]{}, 407 (2003). V.A. Shklovskij, J. Low Temp. Phys. [**139**]{}, 289 (2005). V.M. Vinokur, V.B. Geshkenbein, M.V. Feigel’man, and G. Blatter, Phys. Rev. Lett. [**71**]{}, 1242 (1993). V.A. Shklovskij. 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--- abstract: | We study the finite sample behavior of Lasso-based inference methods such as post double Lasso and debiased Lasso. Empirically and theoretically, we show that these methods can exhibit substantial omitted variable biases (OVBs) due to Lasso not selecting relevant controls. This phenomenon can be systematic in finite samples and occur even when the coefficients are very sparse and the sample size is large and larger than the number of controls. Therefore, relying on the existing asymptotic inference theory can be problematic in empirical applications. We compare the Lasso-based inference methods to modern high-dimensional OLS-based methods and provide practical guidance. **Keywords:** Lasso, post double Lasso, debiased Lasso, OLS, omitted variable bias, limited variability, finite sample analysis author: - 'Kaspar Wüthrich[^1] Ying Zhu[^2]\' bibliography: - 'bibliography.bib' date: 'First draft on arXiv: March 20, 2019. This draft: .' title: 'Omitted variable bias of Lasso-based inference methods: A finite sample analysis[^3] ' --- Introduction ============ Researchers are often interested in making statistical inferences on a single parameter (for example, the effect of a treatment or a policy), while controlling for confounding factors. In more and more economic applications, the number of potential control variables ($p$) is becoming large relative to the sample size ($n$), either due to the inherent richness of the data, the desire of researchers to specify flexible functional forms, or both. In such problems, a natural approach is to use the least absolute shrinkage and selection operator (Lasso), introduced by @tibsharini1996regression, to select the relevant controls (i.e., those with nonzero coefficients) and then run OLS with the selected controls. However, this approach has been criticized because, unless the magnitude of the coefficients associated with the relevant controls is very small, it requires these coefficients to be well separated from zero to ensure that Lasso selects them. This critique has motivated the development of post double Lasso [@belloni2014inference] and debiased Lasso [@javanmard2014confidence; @vandergeer2014asymptotically; @Zhang_Zhang]. The breakthrough in this literature is that it does not require the aforementioned separation condition, and the Lasso not selecting relevant controls yields negligible asymptotic biases under certain conditions on $n$, $p$, and the degree of sparsity. Since their introduction, post double Lasso and debiased Lasso have quickly become the most popular inference methods for problems with many control variables. Given the rapidly growing (asymptotic) theoretical and applied literature on these methods, it is crucial to take a step back and examine the performance of these new procedures in empirically relevant settings as well as to better understand their merits and limitations relative to other alternatives. In particular, there is a misconception that the post double Lasso and debiased Lasso are immune to under-selection of the Lasso because they do not require the above-mentioned separation condition. Empirically and theoretically, this paper shows that, in finite samples, under-selection can result in substantial OVBs of these methods and yield invalid inferences. We also compare the post double Lasso and debiased Lasso to modern high-dimensional OLS-based inference procedures. Let us consider the linear model $$\begin{aligned} Y_{i} & = & D_{i}\alpha^{*}+X_{i}\beta^{*}+\eta_{i},\label{eq:main-y}\\ D_{i} & = & X_{i}\gamma^{*}+v_{i}.\label{eq:main-d}\end{aligned}$$ Here $Y_{i}$ is the outcome, $D_{i}$ is the scalar treatment variable of interest, and $X_{i}$ is a $(1\times p)$-dimensional vector of additional control variables. To focus on the impact of under-selection and facilitate the exposition, we assume that (\[eq:main-y\]) and (\[eq:main-d\]) share the same set of $k$ non-zero coefficients. In this paper, we study the performance of post double Lasso and the debiased Lasso for estimating and making inferences (e.g., constructing confidence intervals) on the treatment effect $\alpha^{\ast}$. We present extensive simulation evidence demonstrating that post double Lasso and debiased Lasso can exhibit substantial OVBs relative to the standard deviations due to the Lasso not selecting all the relevant controls. Our simulation results can be summarized as follows. (i) Large OVBs are persistent across a range of empirically relevant settings and can occur even when $n$ is large and larger than $p$, and $k$ is small (e.g., when $n=10000$, $p=4000$, $k=5$). (ii) For the same $(n,p,k)$, noise variances, and magnitude of coefficients, there can be no OVBs at all, small OVBs, or substantial OVBs, depending on the variance of the relevant controls. (iii) When the controls exhibit limited variability, the performance of Lasso-based inference methods can be very sensitive to the choice of the regularization parameters; under sufficient variability, post double Lasso is less sensitive. (iv) There is no simple recommendation for how to choose the regularization parameters.[^4] (v) The OVBs can lead to invalid inferences and under-coverage of confidence intervals. In addition to the simulations, we conduct Monte Carlo studies based on two empirical applications: The analysis of the effect of 401(k) plans on savings by @belloni2017program and the study of the racial test score gap by @fryerlevitt2013. We draw samples of different size from the large original datasets and compare the subsample estimates to the estimates based on the original data. This exercise mimics random sampling from a large super-population. In both applications, we find substantial biases even when $n$ is considerably larger than $p$, and document that the magnitude of the biases varies substantially depending on the regularization choice. The existing (asymptotic) theory provides little insight about the OVBs of the Lasso-based inference methods documented in our simulation studies. In terms of formal results, it only implies an upper bound of $\texttt{constant}\cdot\frac{k\log p}{n}$ for the bias. Here the (positive) $\texttt{constant}$ does not depend on $(n,p,k)$ and bears little meaning in the existing theory which simply assumes $\frac{k\log p}{\sqrt{n}}\rightarrow0$ among other sufficient conditions.[^5] The asymptotic upper bound $\texttt{constant}\cdot\frac{k\log p}{n}$ is only informative about the *least favorable* case and does not explain the following practically relevant questions: (i) When do OVBs arise? (ii) Why can the OVBs be drastically different despite $(n,p,k)$, noise variances, and absolute values of coefficients being the same? (iii) What is the magnitude of OVBs in the *most favorable* cases? (iv) How severe can the OVBs be in finite samples where $\frac{k\log p}{n}$ is not small enough? To explain (i) and (ii), we provide theoretical conditions under which the OVBs occur systematically and establish a novel result on the under-selection of the Lasso. To explain (iii) and (iv), we derive new informative lower and upper bounds on the OVBs of post double Lasso and the debiased Lasso proposed by @vandergeer2014asymptotically. Our analyses are non-asymptotic and allow us to study the OVBs for fixed $(n,p,k)$, but are also informative when $\frac{k\log p}{n}\rightarrow0$ or $\frac{k\log p}{n}\rightarrow\infty$. Our theoretical results reveal that, in finite samples, the OVBs are not just simple linear functions of $\frac{k\log p}{n}$ but depend on $n$, $p$, and $k$ in a more complex way. In one of our results, we derive explicit universal constants, allowing us to compute precise lower bounds and perform “comparative statics” given features of the underlying empirical problems. This lower bound analysis of the OVBs is the first of its kind in the literature. In contrast to upper bound analyses, it is informative about the most favorable cases and thus the finite sample limitations of Lasso-based inference methods. Our results suggest that the OVBs can be substantial relative to the standard deviation derived from the existing theory. As a consequence, the confidence intervals proposed in the literature can exhibit under-coverage. In the main part of the paper, we focus on post double Lasso and present results for the debiased Lasso in the appendix. Post double Lasso consists of two Lasso selection steps: A Lasso regression of $Y_{i}$ on $X_{i}$ and a Lasso regression of $D_{i}$ on $X_{i}$. In the third step, the estimator of $\alpha^{\ast}$, $\tilde{\alpha}$, is the OLS regression of $Y_{i}$ on $D_{i}$ and the union of controls selected in the two Lasso steps. For the setup of (\[eq:main-y\])–(\[eq:main-d\]), post double has a clear advantage over a post (single) Lasso OLS. As @belloni2014inference [p.614] put it: “Intuitively, this procedure \[post double Lasso\] works well since we are more likely to recover key controls by considering selection of controls from both equations instead of just considering selection of controls from the single equation”. However, OVBs can still arise whenever the relevant controls are selected in neither Lasso steps. To formally study the OVBs, one has to first understand when such “double under-selection” is likely to occur in finite samples. This task is difficult because it requires necessary results on the Lasso’s inclusion to show that double under-selection can occur with high probability. To our knowledge, no formal results exist that can explain the phenomenon “double under-selection with high probability”, and this is likely why the literature provides no characterization of the finite sample OVBs. In this paper, under some classical assumptions, we prove that if the ratios of the absolute values of the non-zero coefficients and the variances of the controls are no greater than half the regularization parameters[^6] (or, in other words, if the products of the absolute values of the non-zero coefficients and the standard deviations of the corresponding controls are small relative to the noise’s standard deviation), Lasso fails to select these controls in both steps with high probability. Our new result on the under-selection of the Lasso is the key ingredient for deriving a lower bound formula for the OVB of the post double Lasso estimator $\tilde{\alpha}$. The OVB lower bound is characterized by the complicated interplay between the probability of double under-selection and the omitted coefficients, and is not a simple linear function of $\frac{k\log p}{n}$ in general. It can be substantial compared to the standard deviation obtained from the asymptotic distribution in @belloni2014inference[^7], even in settings where $k$ is very small, $n$ is large and larger than $p$. As $\frac{k\log p}{n}$ gets large, depending on the configuration of $\left(\beta_{K}^{*},\,\gamma_{K}^{*},\,\alpha^{*}\right)$, the OVB lower bounds scale as $\left|\alpha^{*}\right|$ or as $\frac{\sigma_{\eta}}{\sigma_{v}}$, where $\sigma_{\eta}^{2}$ is the variance of the $\eta_{i}$s in (\[eq:main-y\]) and $\sigma_{v}^{2}$ is the variance of the $v_{i}$s in (\[eq:main-d\]). This raises the question of how severe the OVBs can be when $\frac{k\log p}{n}\rightarrow\infty$ and both Lasso steps are inconsistent in terms of the prediction errors (and $\ell_{2}$-errors). To answer this question, we would like a meaningful upper bound (other than just $\infty$) on the OVBs, but there is no such result in the existing literature. Interestingly enough, under this undesirable regime, we show that the upper bounds on OVBs scale as $\left|\alpha^{*}\right|$ or $\frac{\sigma_{\eta}}{\sigma_{v}}$ with high probability. The occurrence of double under-selection and the magnitude of OVBs both depend on the absolute values of the non-zero coefficients and the standard deviations of the corresponding controls in both Lasso steps through their products. Therefore, under-selection cannot be avoided or mitigated by rescaling the covariates and the OVB lower bound remains the same after rescaling. Any normalization of $X_{i}$ simply leads to rescaled coefficients and vice versa, while their products stay the same. This result suggests an equivalence between “small” (nonzero) coefficient problems and problems with “limited” variability in the relevant controls. By rescaling the controls, the former can always be recast as the latter and vice versa. Given our theoretical results, all else equal, limited variability in the relevant controls makes it more likely for the Lasso to omit them. Limited variability is ubiquitous in applied economic research and there are many instances where it occurs by design. First, it naturally arises from small cells (i.e., when there are only a few observations in some of the cells defined by covariate values). Small cells are prevalent in specifications that include many two-way interactions and are saturated in at least a subset of controls.[^8] When the controls are discrete, limited overlap a major concern in research designs relying on unconfoundedness-type identification assumptions can be viewed as a small cell problem [e.g., @rothe2017robust]. Moreover, categorical controls, when incorporated through a set of indicator variables, give rise to small cells if some of the categories are sparsely populated. Second, when researchers perform subsample analyses, there are often controls that exhibit limited variability within subsamples. Third, in times series and “large $T$” panel data applications, persistence in the controls over time can lead to limited variability. Finally, many empirical settings feature high-dimensional fixed effects that suffer from limited variability. Some authors propose to penalize the fixed effects [e.g., @kock2019uniform] with the Lasso regularization, while others do not [e.g., @belloni2016cluster]. The results in this paper suggest that penalizing fixed effects with the Lasso regularization can be problematic. Our theoretical analyses of the OVBs of the post double Lasso have important implications for inference based on the procedure proposed in @belloni2014inference. On the one hand, @belloni2014inference show that the assumptions $\frac{k\log p}{\sqrt{n}}\rightarrow0$, $\left|\alpha^{*}\right|$ being bounded from above, and $\left(\sigma_{\eta},\,\sigma_{v}\right)$ being bounded away from zero and above (among other regularity conditions) are *sufficient* for establishing the asymptotic normality and unbiasedness of their post double Lasso procedure, regardless of whether under-selection is present or not. On the other hand, we show empirically relevant settings where double under-selection occurs with high probability, and also show that in such settings, small $\frac{k\log p}{\sqrt{n}}$ and $\frac{\left|\alpha^{*}\right|\sigma_{v}}{\sigma_{\eta}}$ are *almost necessary* for a good performance of the post double Lasso. Unfortunately in practice, because $k$ and $\alpha^{*}$ are fundamentally unknown, it is impossible to know whether or not $\frac{k\log p}{\sqrt{n}}$ and $\frac{\left|\alpha^{*}\right|\sigma_{v}}{\sigma_{\eta}}$ are small enough. Moreover, the requirement of small $\frac{k\log p}{\sqrt{n}}$ is actually quite demanding in high dimensional settings. For example, even with $n=10000>p=4000$ and $k=5$, we have $\frac{k\log p}{\sqrt{n}}\approx0.41$. In view of our theoretical results, simulations, and empirical evidence, a natural question is how to make statistical inference in a reliable manner, especially when limited variability is present. The case where $p$ is comparable to but still smaller than $n$ deserves special attention because of its prevalence in applied economic research.[^9] In this case, recently developed high-dimensional OLS-based inference methods [e.g., @cattaneo2018inference; @jochmans2018heteroscedasticity; @kline2018leaveout; @dadamo2018cluster] constitute a natural alternative to Lasso-based procedures. These modern OLS-based methods are based on (unbiased) OLS estimators and variance estimators that are robust to the inclusion of many controls.[^10] Based on extensive simulations, we find that OLS with standard errors proposed by @cattaneo2018inference is unbiased (as expected) and demonstrates excellent size accuracy, irrespective of the degree of variability in the controls. Another advantage of OLS-based methods is that, unlike the Lasso-based inference methods, they do not rely on any sparsity assumptions. This is important because sparsity assumptions may not be satisfied in applications and, as we show in this paper, the OVBs of the Lasso-based inference procedures can be substantial even when $k$ is very small, $n$ is very large and larger than $p$. However, OLS yields somewhat wider confidence intervals than the Lasso-based inference methods, which suggests that there is a trade-off between coverage accuracy and the width of the confidence intervals. Our analyses suggest several recommendations concerning the use of Lasso-based inference methods in empirical studies. First, one should always perform robustness checks with respect to the choice of the regularization parameters. Our simulation and theoretical results suggest the following heuristic: If the estimates of $\alpha^{*}$ are robust to increasing the theoretically recommended regularization parameters in the two Lasso steps, post double Lasso could be a reliable and efficient method. Second, our findings highlight the usefulness of augmenting the final OLS regression in post double Lasso with control variables motivated by economic theory and prior knowledge, as suggested by @belloni2014inference. Third, when $p$ is smaller than $n$, modern high-dimensional OLS-based inference methods constitute a viable alternative to Lasso-based inference methods. Our paper highlights the intrinsic limitations of basing inference procedures on variable selection in empirically relevant settings, and opens up interesting avenues for future research. For instance, we expect under-selection to have important implications for the finite sample behavior of many other linear and nonlinear inference procedures based on the Lasso [e.g., @belloni2012sparse; @belloni2014inference; @farrell2015robust; @belloni2017program; @chernozhukov2018double]. Furthermore, our paper motivates systematic analyses of the practical usefulness of Lasso-based inference procedures and other modern high-dimensional methods. For example, @angrist2019machine investigate the usefulness of post double Lasso for regression-based sensitivity analyses, and present simulation evidence on the finite sample behavior of Lasso-based instrumental variables (IV) methods. Finally, we emphasize that our results do not contradict the existing asymptotic theory on post double Lasso and the debiased Lasso, but rather offer a better understanding of when these methods perform well in finite samples and when they do not. This paper should not be viewed as a criticism of these methods and there is no doubt that they provide significant theoretical improvements over earlier procedures (e.g., post single Lasso OLS). Instead, our findings should be viewed as a warning that relying on the existing asymptotic inference theory for post double Lasso and debiased Lasso can be problematic in empirical applications and these methods can still suffer from the consequence of under-selection in finite samples. Lasso and post double Lasso {#sec:lasso_post_double_lassp} =========================== The Lasso {#sec:lasso} --------- Consider the following linear regression model $$Y_{i}=X_{i}\theta^{*}+\varepsilon_{i},\qquad i=1,\dots,n,\label{eq:1}$$ where $\left\{ Y_{i}\right\} _{i=1}^{n}=Y$ is an $n$-dimensional response vector, $\left\{ X_{i}\right\} _{i=1}^{n}=X$ is an $n\times p$ matrix of covariates with $X_{i}$ denoting the $i$th row of $X$, $\left\{ \varepsilon_{i}\right\} _{i=1}^{n}=\varepsilon$ is a zero-mean error vector, and $\theta^{*}$ is a $p$-dimensional vector of unknown coefficients. The Lasso estimator of $\theta^{*}$ is given by $$\hat{\theta}\in\arg\min_{\theta\in\mathbb{R}^{p}}\frac{1}{2n}\sum_{i=1}^{n}\left(Y_{i}-X_{i}\theta\right)^{2}+\lambda\sum_{j=1}^{p}\left|\theta_{j}\right|,\label{eq:las}$$ where $\lambda$ is the regularization parameter. For example, if $\varepsilon\sim\mathcal{N}\left(0_{n},\,\sigma^{2}I_{n}\right)$ and $X$ is a fixed design matrix with normalized columns (i.e., $\frac{1}{n}\sum_{i=1}^{n}X_{ij}^{2}=b$ for all $j=1,\dots,p$), @bickel2009simultaneous set $\lambda=2\sigma\sqrt{\frac{2b\left(1+\tau\right)\log p}{n}}$ (where $\tau>0$) to establish upper bounds on $\sqrt{\sum_{j=1}^{p}(\hat{\theta}_{j}-\theta_{j}^{*})^{2}}$ with high probability guarantee. @wainwright2009sharp sets $\lambda$ proportional to $\frac{\sigma}{\phi}\sqrt{\frac{b\log p}{n}}$, where $\phi\in(0,\,1]$ is a measure of correlation between the covariates with nonzero coefficients and those with zero coefficients, to establish perfect selection. Besides the classical choices in @bickel2009simultaneous and @wainwright2009sharp mentioned above, other choices of $\lambda$ are available in the literature. For instance, @belloni2013least develop a data-dependent approach and @belloni2012sparse and @belloni2016cluster propose choices that accommodate heteroscedastic and clustered errors. In the case of nearly orthogonal $X$ (which is typically required to ensure a good performance of the Lasso), these choices of $\lambda$ have a similar magnitude as those in @bickel2009simultaneous and @wainwright2009sharp. Finally, a very popular practical approach for choosing $\lambda$ is cross-validation. However, only a few theoretical results exist on the properties of Lasso when $\lambda$ is chosen using cross-validation; see, for example, @homrighausen2013thelasso [@homrighausen2014leaveoneout] and @chetverikov2017cv. Post double Lasso ----------------- The model implies the following reduced form model for $Y_{i}$: $$\begin{aligned} Y_{i} & = & X_{i}\pi^{*}+u_{i},\label{eq:reduce-1}\end{aligned}$$ where $\pi^{*}=\gamma^{*}\alpha^{*}+\beta^{*}$ and $u_{i}=\eta_{i}+\alpha^{*}v_{i}$. The post double Lasso, introduced by @belloni2014inference, essentially exploits the Frisch-Waugh theorem, where the regressions of $Y$ on $X$ and $D$ on $X$ are implemented with the Lasso: $$\begin{aligned} \hat{\pi} & \in & \textrm{arg}\min_{\pi\in\mathbb{R}^{p}}\frac{1}{2n}\sum_{i=1}^{n}\left(Y_{i}-X_{i}\pi\right)^{2}+\lambda_{1}\sum_{j=1}^{p}\left|\pi_{j}\right|,\label{eq:las-1}\\ \hat{\gamma} & \in & \textrm{arg}\min_{\gamma\in\mathbb{R}^{p}}\frac{1}{2n}\sum_{i=1}^{n}\left(D_{i}-X_{i}\gamma\right)^{2}+\lambda_{2}\sum_{j=1}^{p}\left|\gamma_{j}\right|.\label{eq:las-2}\end{aligned}$$ The final estimator $\tilde{\alpha}$ of $\alpha^{*}$ is then obtained from an OLS regression of $Y$ on $D$ and the union of selected controls $$\left(\tilde{\alpha},\,\tilde{\beta}\right)\in\textrm{arg}\min_{\alpha\in\mathbb{R},\beta\in\mathbb{R}^{p}}\frac{1}{2n}\sum_{i=1}^{n}\left(Y_{i}-D_{i}\alpha-X_{i}\beta\right)^{2}\quad\textrm{s.t. }\beta_{j}=0\;\forall j\notin\left\{ \hat{I}_{1}\cup\hat{I}_{2}\right\} ,\label{eq:double}$$ where $\hat{I}_{1}=\textrm{supp}\left(\hat{\pi}\right)=\left\{ j:\,\hat{\pi}_{j}\neq0\right\} $ and $\hat{I}_{2}=\textrm{supp}\left(\hat{\gamma}\right)=\left\{ j:\,\hat{\gamma}_{j}\neq0\right\} $. Evidence on the OVB of post double Lasso {#sec:evidence} ======================================== This section presents evidence on the under-selection of the Lasso and the OVB of post double Lasso. Section \[sec:numerical\_evidence\] analyzes a simple numerical example, Section \[sec:simulation\_evidence\] presents simulation evidence for different choices of the regularization parameters, and Section \[sec:empirical\_evidence\] revisits two empirical applications. In our simulations, we vary the variance of the control variables while fixing their coefficients. In particular, we consider settings where the control variables exhibit limited variability. It is important to note that all the results remain unchanged when we rescale the controls and transform the limited variability problem into a small coefficients problem; see Section \[sec:inference\] for a further discussion. Numerical example {#sec:numerical_evidence} ----------------- To study the under-selection of the Lasso, we simulate data according to the linear model , where $X_{i}\sim \mathcal{N}\left(0,\sigma_x^2I_{p}\right)$ is independent of $\varepsilon_{i}\sim\mathcal{N}(0,1)$ and $\left\{ X_{i},\varepsilon_{i}\right\} _{i=1}^{n}$ consists of i.i.d. entries. We set $n=500$, $p=200$, and consider a sparse setting where $\theta^{*}=(\underbrace{1,\dots,1}_{k},0,\dots,0)^{T}$ and $k=5$. We employ the classical recommendation for the regularization parameter by @bickel2009simultaneous.[^11] Figure \[fig:performance\_illu\] displays the average number of selected (relevant and irrelevant) covariates as a function of the degree of variability, $\sigma_x$. The variability of the covariates significantly affects the selection performance of the Lasso. The average number of selected covariates is monotonically increasing in $\sigma_x$, ranging from approximately zero when $\sigma_x=0.05$ up to five when $\sigma_x=0.5$. \[fig:performance\_illu\] Next, we investigate the implications of the under-selection of Lasso for post double Lasso. We simulate data according to the structural model –, where $X_{i}\sim \mathcal{N}\left(0,\sigma_x^2I_{p}\right)$, $\eta_{i}\sim\mathcal{N}(0,1)$, and $v_{i}\sim \mathcal{N}(0,1)$ are independent of each other and $\left\{ X_{i},\eta_{i},v_{i}\right\} _{i=1}^{n}$ consists of i.i.d. entries. Our object of interest is $\alpha^{*}$. We set $n=500$, $p=200$, $\alpha^{*}=0$, and consider a sparse setting where $\beta^{*}=\gamma^{*}=(\underbrace{1,\dots,1}_{k},0,\dots,0)^{T}$ and $k=5$. In both Lasso steps, we employ the recommendation for the regularization parameter by @bickel2009simultaneous. Figure \[fig:fsd\_dml\] displays the finite sample distribution of post double Lasso for different values of $\sigma_x$. For comparison, we plot the distribution of the “oracle estimator” of $\alpha^\ast$, a regression of $Y_i-X_i\pi^\ast$ on $D_i-X_i\gamma^\ast$. \[fig:fsd\_dml\] For $\sigma_x=0.05$, the post double Lasso estimator exhibits a small bias. Increasing $\sigma_x$ to $0.15$ shifts the distribution to the right, resulting in a large bias relative to the standard deviation. For $\sigma_x=0.3$, post double Lasso is biased and exhibits a larger standard deviation. Finally, for $\sigma_x=0.5$, the post double Lasso estimator is approximately unbiased and its distribution is centered at the true value $\alpha^{*}=0$. We emphasize that these striking differences arise despite $(n,p,k,\sigma_v,\sigma_\eta)$ and the coefficients being the same in these four designs. The bias of post double Lasso is caused by the OVB arising from the two Lasso steps not selecting all the relevant covariates. Figure \[fig:hist\_sel\_dml\] displays the number of selected relevant covariates (i.e., the cardinality of $\hat{I}_{1}\cup\hat{I}_{2}$ in ). \[fig:hist\_sel\_dml\] With very high probability, none of the relevant controls get selected when $\sigma_x\le 0.15$. The selection performance improves as $\sigma_x$ increases, until, with high probability, all relevant regressors get selected when $\sigma_x=0.5$.[^12] We further note that the shape of the finite sample distribution depends on $\sigma_x$. This distribution is a mixture of the distributions of OLS conditional on the two Lasso steps selecting different combinations of covariates.[^13] For $\sigma_x=0.05$, $\sigma_x=0.15$, and $\sigma_x=0.5$, the finite sample distributions are well-approximated by normal distributions (albeit centered at different values). The reason is that, when $\sigma_x=0.05$ or $\sigma_x=0.15$, none of the relevant controls get selected with high probability, and, when $\sigma_x=0.5$, the two Lasso steps almost always select all the relevant controls. In between these two cases, when $\sigma_x=0.3$, the finite sample distribution is a mixture of distributions with different means (depending on how many controls get selected) and has a larger standard deviation. The event where none of the relevant covariates are selected is of particular interest. Figure \[fig:cond\_bias\] displays the probability of this event as a function of $\sigma_x$ as well as the OVB conditional on this event, which is computed as $$\frac{1}{\sum_{r=1}^{R}S_{r}}\sum_{r=1}^{R}\left(\tilde{\alpha}_{r}-\alpha^\ast\right)\cdot S_{r}.$$ In the above formula, $R$ is the total number of simulation repetitions, $S_{r}$ is an indicator which is equal to one if nothing gets selected and zero otherwise, and $\tilde{\alpha}_{r}$ is the estimate of $\alpha^\ast$ in the $r$[th]{} repetition. As the variability in the controls increases from $\sigma_x=0.05$ to $\sigma_x=0.4$, the probability that nothing gets selected is decreasing from one to zero. The OVB increases until $\sigma_x=0.35$ and is not defined for $\sigma_x>0.35$ because the empirical probability that nothing gets selected is zero in this case. \[fig:cond\_bias\] Importantly, the issue documented here is not a “small sample” phenomenon, but persists even in large sample settings. To illustrate, Figures \[fig:fsd\_large\] and \[fig:hist\_sel\_large\] display the finite sample distributions and the number of selected relevant covariates when $(n,p,k)=(10000,4000,5)$. Even in large samples, the finite sample distribution may not be centered at the true value and the bias can be large relative to the standard deviation because, with high probability, none of the relevant controls are selected.[^14] Compared to the results for $(n,p,k)=(500,200,5)$, under-selection and large biases occur at lower values of $\sigma_x$, and all the relevant controls get selected when $\sigma_x=0.15$. \[fig:fsd\_large\] \[fig:hist\_sel\_large\] Simulation evidence {#sec:simulation_evidence} ------------------- Section \[sec:numerical\_evidence\] considers a simple numerical example and the classical regularization choice by @bickel2009simultaneous based on the true $\sigma^2$. Here we present simulation evidence for three popular feasible choices of the regularization parameter: The heteroscedasticity-robust proposal in @belloni2012sparse ($\lambda_{\text{BCCH}}$)[^15], the regularization parameter with the minimum cross-validated error ($\lambda_{\text{min}}$), and the regularization parameter with the minimum cross-validation error plus one standard deviation ($\lambda_{\text{1se}}$). We start by investigating the selection performance. We simulate data based on the DGP of Section \[sec:numerical\_evidence\], where $(n,p,k)=(500,200,5)$. To illustrate the impact of changing the sample size, we also show results for $n=1000$ and $n=200$. The results are based on 1,000 simulation repetitions. Figure \[fig:sel\] displays the average number of selected covariates as a function of $\sigma_x$. Lasso with $\lambda_{\text{BCCH}}$ selects the lowest number of covariates. Choosing $\lambda_{\text{1se}}$ leads to a somewhat higher number of selected covariates and results in moderate over-selection for larger values of $\sigma_x$. Lasso with $\lambda_{\text{min}}$ selects the highest number of covariates and exhibits substantial over-selection. Figure \[fig:sel\_rel\] shows the corresponding numbers of selected relevant covariates. We note that, when $\sigma_x=0.1$, the Lasso does not select all the relevant covariates even with $\lambda_{\text{min}}$ which can result in substantial over-selection. For all regularization choices, the selection performance improves as the sample size increases. \[fig:sel\] \[fig:sel\_rel\] To investigate the impact of under-selection on post double Lasso, we simulate data according to the DGP of Section \[sec:numerical\_evidence\], where $(n,p,k)=(500,200,5)$, and also consider $n=1000$ and $n=200$. The results are based on 1,000 simulation repetitions. Appendix \[app:additional\_simulations\] presents additional simulation evidence, where we vary $k$, the distribution of $X_i$, the error terms $(\eta_i,v_i)$, the true value $\alpha^\ast$, and also consider a heteroscedastic DGP. We employ the same regularization choice in both Lasso steps. Figure \[fig:dml\_bias\_std\] presents evidence on the bias of post double Lasso. To make the results easier to interpret, we report the ratio of the bias to the empirical standard deviation. \[fig:dml\_bias\_std\] Under limited variability, post double Lasso with $\lambda_{\text{BCCH}}$ can exhibit biases that are up to two times larger than the standard deviation when $n=200$ and still comparable to the standard deviation when $n=1000$. The relationship between $\sigma_x$ and the ratio of bias to standard deviation is non-monotonic: It is increasing for small $\sigma_x$ and decreasing for larger $\sigma_x$. The bias is somewhat smaller than the standard deviation when $\lambda=\lambda_{\text{1se}}$. Setting $\lambda=\lambda_{\text{min}}$ yields the smallest ratio of bias to standard deviation. For all choices of the regularization parameters, the ratio of bias to standard deviation is decreasing in the sample size. Finally, we note that when $\sigma_x$ is large enough such that there is no under-selection, post double Lasso performs well and is approximately unbiased for all regularization parameters. The additional simulation evidence reported in Appendix \[app:additional\_simulations\] confirms these results and further shows that $\alpha^\ast$ is an important determinant of the performance of post double Lasso because of its direct effect on the magnitude of the coefficients and the error variance in the reduced form equation . Moreover, we show that, while choosing $\lambda=\lambda_{\text{min}}$ works well when $\alpha^\ast=0$, this choice can yield poor performances when $\alpha^\ast\ne 0$ (c.f. Figure \[fig:dml\_bias\_std\_app\], DGP A5).[^16] Thus, there is no simple recommendation for how to choose the regularization parameters in practice. The substantive performance differences between the three regularization choices suggest that post double Lasso is sensitive to the regularization parameters when $\sigma_x$ is small (but not quite zero). To further investigate this issue, we compare the results for $\lambda_{\text{BCCH}}$, $0.5 \lambda_{\text{BCCH}}$, and $1.5 \lambda_{\text{BCCH}}$. Figures \[fig:sel\_sensitivity\] and \[fig:sel\_rel\_sensitivity\] display the average numbers of selected and selected relevant covariates. The differences in the selection performance are substantial. Lasso with $0.5 \lambda_{\text{BCCH}}$ over-selects for all $n$ and $\sigma_x$, while Lasso with $1.5 \lambda_{\text{BCCH}}$ under-selects unless $\sigma_x$ and $n$ are large. The differences get smaller as the sample size increases. Figure \[fig:dml\_bias\_std\_sens\] displays the ratio of bias and standard deviation for post double Lasso. Choosing $0.5 \lambda_{\text{BCCH}}$ yields small biases relative to the standard deviations for all $\sigma_x$. By contrast, choosing $1.5 \lambda_{\text{BCCH}}$ yields biases that can be more than four times larger than the standard deviations when $n=200$, and still be substantial when $n=1000$. For larger values of $\sigma_x$, post double Lasso is less sensitive to the choice of the regularization parameters. In Section \[sec:inference\], based on our theoretical results, we discuss how to interpret and use robustness checks with respect to the regularization parameters in empirical applications. \[fig:sel\_sensitivity\] \[fig:sel\_rel\_sensitivity\] \[fig:dml\_bias\_std\_sens\] In sum, our simulation evidence shows (i) that under-selection can lead to large biases relative to the standard deviation, (ii) that, under limited (but not quite zero) variability, the performance of post double Lasso is very sensitive to the choice of the regularization parameters, and (iii) that there is no simple recommendation for how to choose the regularization parameters in practice. Empirical evidence {#sec:empirical_evidence} ------------------ ### The effect of 401k plans on savings {#sec:401k} Here we revisit the analysis of the causal effect of 401(k) plans on individual savings. We use the same dataset as in @CH2004 and @belloni2017program, which we refer to for more details and descriptive statistics. The data contain information about $n=9915$ observations from a sample of households from the 1991 Survey of Income and Program Participation (SIPP). We focus on the effect of eligibility for 401(k) plans ($D$) on total wealth ($Y$).[^17] We consider two different specifications of the control variables ($X$). 1. **Two-way interactions specification.** We use the same set of controls as in @Benjamin2003 and @CH2004: Seven dummies for income categories, five dummies for age categories, family size, four dummies for education categories, as well as indicators of marital status, two-earner status, defined benefit pension status, individual retirement account (IRA) participation status, and homeownership. Following common empirical practice, we augment this baseline specification with all two-way interactions. After removing collinear columns there are $p=167$ control variables. 2. **Quadratic spline & interactions specification.** This specification is due to @belloni2017program. It contains indicators of marital status, two-earner status, defined benefit pension status, IRA participation status, and homeownership status, second-order polynomials in family size and education, a third-order polynomial in age, a quadratic spline in income with six breakpoints, as well as interactions of all the non-income variables with each term in the income spline. After removing collinear columns there are $p=272$ control variables. Table \[tab:results\_401k\] presents post double Lasso estimates based on the whole sample with $\lambda_{\text{BCCH}}$, $0.5\lambda_{\text{BCCH}}$ and $1.5\lambda_{\text{BCCH}}$. For comparison, we also report OLS estimates with and without covariates. For both specifications, the results are qualitatively similar across the different regularization choices and similar to OLS with all controls. This is not surprising given our simulation evidence, since $n$ is much larger than $p$. Nevertheless, there are some non-negligible quantitative differences between the point estimates. A comparison to OLS without control variables shows that omitting controls can yield substantial OVBs in this application. [1.25]{}[1]{} [lcc]{}\[tab:results\_401k\] & & [\ ]{} [\ ]{} Method & Point estimate & Robust std. error [\ ]{} Post double Lasso ($\lambda_{\text{BCCH}}$) & 6624.47 & 2069.73\ Post double Lasso ($0.5\lambda_{\text{BCCH}}$) & 6432.36 & 2073.35\ Post double Lasso (1.5$\lambda_{\text{BCCH}}$) & 7474.51 & 2053.00\ OLS with all covariates & 6751.91 & 2067.86\ OLS without covariates & 35669.52 & 2412.02\ [\ ]{} Method & Point estimate & Robust std. error [\ ]{} Post double Lasso ($\lambda_{\text{BCCH}}$) & 4646.58 & 2014.63\ Post double Lasso ($0.5\lambda_{\text{BCCH}}$) & 5648.14 & 1988.81\ Post double Lasso (1.5$\lambda_{\text{BCCH}}$) & 4472.32 & 2027.72\ OLS with all covariates & 5988.41 & 2033.02\ OLS without covariates & 35669.52 & 2412.02\ [1.25]{}[1.4]{} To investigate the impact of under-selection, we perform the following exercise. We draw random subsamples of size $n_s\in \{400,800,1600\}$ with replacement from the original dataset. This exercise mimics random sampling from a large super-population. Based on each subsample, we estimate $\alpha^\ast$ using post double Lasso with $\lambda_{\text{BCCH}}$, $0.5\lambda_{\text{BCCH}}$ and $1.5\lambda_{\text{BCCH}}$ and compute the bias as the difference between the average subsample estimate and the point estimate based on the original data with the same regularization parameters (cf. Table \[tab:results\_401k\]). The results are based on 2,000 simulation repetitions. Figures \[fig:app\_401k\_1\] and \[fig:app\_401k\_2\] display the bias and the ratio of bias to standard deviation for both specifications. We find that post double Lasso can exhibit large finite sample biases. The biases under the quadratic spline & interactions specification tend to be smaller (in absolute value) than the biases under the two-way interactions specification. Interestingly, the ratio of bias to standard deviation may not be monotonically decreasing in $n_s$ (in absolute value) due to the standard deviation decaying faster than the bias. Finally, we find that post double Lasso can be very sensitive to the choice of the regularization parameters. \[fig:app\_401k\_1\] \[fig:app\_401k\_2\] ### Racial differences in the mental ability of children In this section, we revisit @fryerlevitt2013’s analysis of the racial differences in the mental ability of young children. @fryerlevitt2013 use data from two sources: The Early Childhood Longitudinal Study Birth Cohort (ECLS-B) and the US Collaborative Perinatal Project (CPP). Here we focus on the CCP data, which contains information on women who gave birth in 12 medical centers from 1959 to 1965. We restrict the sample to black and white children and focus on the black-white test score gap. Our final sample includes $n=30002$ observations. We choose the standardized test score in the Stanford-Binet and Wechsler Intelligence Test at the age of seven as our outcome variable ($Y$).[^18] The variable of interest ($D$) is an indicator for black. Control variables ($X$) include rich information on socio-demographic characteristics, the home environment, and the prenatal environment. We refer to Table 1B in @fryerlevitt2013 for descriptive statistics. We use the same specification as in @fryerlevitt2013, excluding interviewer fixed effect.[^19] After removing collinear terms there are $p=78$ controls. Table \[tab:results\_testscores\] shows the results for post double Lasso with $\lambda_{\text{BCCH}}$, $0.5\lambda_{\text{BCCH}}$ and $1.5\lambda_{\text{BCCH}}$, as well as OLS with and without covariates based on the whole sample. As expected, since $n=30002$ is much larger than $p=78$, all methods except for OLS without covariates yield similar results. [1.25]{}[1]{} \[tab:results\_testscores\] ------------------------------------------------ ---------------- ------------------- Method Point estimate Robust std. error Post double Lasso ($\lambda_{\text{BCCH}}$) -0.6770 0.0114 Post double Lasso ($0.5\lambda_{\text{BCCH}}$) -0.6762 0.0114 Post double Lasso (1.5$\lambda_{\text{BCCH}}$) -0.6778 0.0114 OLS with all covariates -0.6694 0.0115 OLS without covariates -0.8538 0.0105 : Results based on original data [1.25]{}[1.4]{} To investigate the impact of under-selection, we draw random subsamples of size $n_s\in \{400,800,1600\}$ with replacement from the original dataset. In each sample, we estimate $\alpha^\ast$ using post double Lasso with $\lambda_{\text{BCCH}}$, $0.5\lambda_{\text{BCCH}}$ and $1.5\lambda_{\text{BCCH}}$ and compute the bias as the difference between the average estimate based on the subsamples and the estimate based on the original data with the same regularization parameters. The results are based on 2,000 simulation repetitions. Figure \[fig:app\_test\] displays the bias and the ratio of bias to standard deviation. While the bias is decreasing in $n_s$, it can be substantial and larger than the standard deviation when $n_s$ is small. Moreover, the performance of post double Lasso is very sensitive to the choice of the regularization parameters. For $\lambda=0.5\lambda_{\text{BCCH}}$, post double Lasso is approximately unbiased for all $n_s$, whereas, for $\lambda=1.5\lambda_{\text{BCCH}}$, the bias is comparable to the standard deviation even when $n_s=1600$. \[fig:app\_test\] Theoretical analysis {#sec:inference} ==================== This section provides a theoretical explanation for the findings in Section \[sec:evidence\]. We first establish a new necessary result for the Lasso’s inclusion and then derive lower and upper bounds on the OVBs of post double Lasso. These results hold for fixed $\left(n,\,p,\,k\right)$, and are also informative when $\frac{k\log p}{n}\rightarrow0$ or $\frac{k\log p}{n}\rightarrow\infty$. Throughout this section, we assume the regime where $p$ is comparable to or even much larger than $n$; that is, $p\asymp n$ or $p\gg n$. For the convenience of the reader, here we collect the notation to be used in the theoretical analyses. Let $1_{m}$ denote the $m-$dimensional (column) vector of “1”s and $0_{m}$ is defined similarly. The $\ell_{1}-$norm of a vector $v\in\mathbb{R}^{m}$ is denoted by $\left|v\right|_{1}:=\sum_{i=1}^{m}\left|v_{i}\right|$. The $\ell_{\infty}$ matrix norm (maximum absolute row sum) of a matrix $A$ is denoted by $\left\Vert A\right\Vert _{\infty}:=\max_{i}\sum_{j}\left|a_{ij}\right|$. For a vector $v\in\mathbb{R}^{m}$ and a set of indices $T\subseteq\left\{ 1,\dots,m\right\} $, let $v_{T}$ denote the sub-vector (with indices in $T$) of $v$. For a matrix $A\in\mathbb{R}^{n\times m}$, let $A_{T}$ denote the submatrix consisting of the columns with indices in $T$. For a vector $v\in\mathbb{R}^{m}$, let $\textrm{sgn}(v):=\left\{ \textrm{sgn}(v_{j})\right\} _{j=1,...,m}$ denote the sign vector such that $\textrm{sgn}(v_{j})=1$ if $v_{j}>0$, $\textrm{sgn}(v_{j})=-1$ if $v_{j}<0$, and $\textrm{sgn}(v_{j})=0$ if $v_{j}=0$. We denote $\max\left\{ a,\,b\right\} $ by $a\vee b$ and $\min\left\{ a,\,b\right\} $ by $a\wedge b$. Model setup ----------- We consider the structural model , which can be written in matrix notation as $$\begin{aligned} Y & = & D\alpha^{*}+X\beta^{*}+\eta,\label{eq:20}\\ D & = & X\gamma^{*}+v.\label{eq:21-1}\end{aligned}$$ Following standard practice, we work with centered data, i.e., $\bar{D}=\frac{1}{n}\sum_{i=1}^{n}D_{i}=0$, $\bar{X}=\left\{ \frac{1}{n}\sum_{i=1}^{n}X_{ij}\right\} _{j=1}^{p}=0_{p}$, and $\bar{Y}=\frac{1}{n}\sum_{i=1}^{n}Y_{i}=0$. In matrix notation, the reduced form becomes $$\begin{aligned} Y & = & X\pi^{*}+u,\label{eq:reduce}\end{aligned}$$ where $\pi^{*}=\gamma^{*}\alpha^{*}+\beta^{*}$ and $u=\eta+\alpha^{*}v$. We make the following assumptions about model (\[eq:20\])(\[eq:21-1\]). \[ass:inference\_normality\] The error terms $\eta$ and $v$ consist of independent entries drawn from $\mathcal{N}\left(0,\,\sigma_{\eta}^{2}\right)$ and $\mathcal{N}\left(0,\,\sigma_{v}^{2}\right)$, respectively, where $\eta$ and $v$ are independent of each other. \[ass:inference\_sparsity\_incoherence\] The following are satisfied: (i) $\beta^{*}$ and $\gamma^{*}$ are exactly sparse with $k\left(\le\min\left\{ n,\,p\right\} \right)$ non-zero coefficients and $K=\left\{ j:\,\beta_{j}^{*}\neq0\right\} =\left\{ j:\,\gamma_{j}^{*}\neq0\right\} \neq\emptyset$; (ii) $$\left\Vert \left(X_{K^{c}}^{T}X_{K}\right)\left(X_{K}^{T}X_{K}\right)^{-1}\right\Vert _{\infty}=1-\phi\label{eq:14}$$ for some $\phi\in(0,\,1]$, where $K^{c}$ is the complement of $K$; (iii) $X_{K}^{T}X_{K}$ is a diagonal matrix with the diagonal entries $X_{j}^{T}X_{j}=s\neq0$ for all $j\in K$, and $X_{j}^{T}X_{j}\leq s$ for all $j\in K^{c}$. Known as the incoherence condition due to @wainwright2009sharp, part (ii) in Assumption \[ass:inference\_sparsity\_incoherence\] is needed for the exclusion of the irrelevant controls. Note that if the columns in $X_{K^{c}}$ are orthogonal to the columns in $X_{K}$ (but within $X_{K^{c}}$, the columns need not be orthogonal to each other), then $\phi=1$. Obviously a special case of this is when the entire $X$ consists of mutually orthogonal columns (which is possible if $n\geq p$). To provide some intuition for (\[eq:14\]), let us consider the simple case where $k=1$ and $K=\left\{ 1\right\} $, $X$ is centered (such that $\left\{ \frac{1}{n}\sum_{i=1}^{n}X_{ij}\right\} _{j=1}^{p}=0_{p}$), and the columns in $X_{-1}=\left[X_{2},\,X_{3},\dots,\,X_{p}\right]$ are normalized such that the standard deviations of $X_{1}$ and $X_{j}$ (for any $j\in\left\{ 2,3,\dots,p\right\} $) are identical. Then, $1-\phi$ is simply the maximum of the absolute (sample) correlations between $X_{1}$ and each of the $X_{j}$s with $j\in\left\{ 2,3,\dots,p\right\} $. Assumptions \[ass:inference\_normality\] and \[ass:inference\_sparsity\_incoherence\] are classical; in particular, parts (ii) and (iii) of Assumption \[ass:inference\_sparsity\_incoherence\] are often viewed favorable to the performance of Lasso. Our goal here is to demonstrate that, even in these simple yet classical settings, the finite sample OVBs of post double Lasso can be substantial relative to the standard deviation provided in the existing literature. Stronger necessary results on the Lasso’s inclusion {#sec:nonasymptotic} --------------------------------------------------- \[sec:main\_results\] Post double Lasso exhibits OVBs whenever the relevant controls are selected in neither nor . To the best of our knowledge, there are no formal results strong enough to show that, with high probability, Lasso can fail to select the relevant controls in both steps. Therefore, we first establish a new necessary result for the (single) Lasso’s inclusion in Lemma \[prop:fixed\_design\]. Throughout Section \[sec:inference\], we focus on fixed designs (of $X$) to highlight the essence of the problem; see Appendix \[sec:Random-design\] for an extension to random designs. \[prop:fixed\_design\] In model , suppose the $\varepsilon_{i}$s are independent over $i=1,\dots,n$ and $\varepsilon_{i}\sim\mathcal{N}\left(0,\sigma^{2}\right)$, where $\sigma\in\left(0,\,\infty\right)$;[^20] $\theta^{*}$ is exactly sparse with at most $k\left(\le\min\left\{ n,\,p\right\} \right)$ non-zero coefficients and $K=\left\{ j:\,\theta_{j}^{*}\neq0\right\} \neq\emptyset$. Let Assumption \[ass:inference\_sparsity\_incoherence\](ii)-(iii) hold. We solve the Lasso (\[eq:las\]) with $\lambda\geq\frac{2\sigma}{\phi}\sqrt{\frac{s}{n}}\sqrt{\frac{2\left(1+\tau\right)\log p}{n}}$ (where $\tau>0$). Let $E_{1}$ denote the event that $\textrm{sgn}\left(\hat{\theta}_{j}\right)=-\textrm{sgn}\left(\theta_{j}^{*}\right)$ for at least one $j\in K$, and $E_{2}$ denote the event that $\textrm{sgn}\left(\hat{\theta}_{l}\right)=\textrm{sgn}\left(\theta_{l}^{*}\right)$ for at least one $l\in K$ with $$\left|\theta_{l}^{*}\right|\leq\frac{\lambda n}{2s}.\label{eq:min}$$ Then, we have $$\mathbb{P}\left(E_{1}\cap\mathcal{\mathcal{E}}\right)=\mathbb{P}\left(E_{2}\cap\mathcal{\mathcal{E}}\right)=0\label{eq:zero}$$ where $\mathcal{E}$ is defined in (\[eq:event\]) of Appendix \[appendix a1\] and $\mathbb{P}\left(\mathcal{E}\right)\geq1-\frac{1}{p^{\tau}}$. If (\[eq:min\]) holds for all $l\in K$, we have $$\mathbb{P}\left(\hat{\theta}=0_{p}\right)\geq1-\frac{1}{p^{\tau}}.\label{eq:nec}$$ Lemma \[prop:fixed\_design\] shows that for large enough $p$, Lasso fails to select any of the relevant covariates with high probability if (\[eq:min\]) holds for all $l\in K$. If such conditions hold with respect to both and , then Lemma \[prop:fixed\_design\] implies that the relevant controls are selected in neither nor with high probability, i.e., at least $1-\frac{2}{p^{\tau}}$ (cf. Panels (a) and (b) of Figure \[fig:hist\_sel\_dml\] and Panel (a) of Figure \[fig:hist\_sel\_large\]). Suppose that $$\lambda=\frac{2\sigma}{\phi}\sqrt{\frac{s}{n}}\sqrt{\frac{2\left(1+\tau\right)\log p}{n}}.\label{eq:choice}$$ Then becomes $$\left|\theta_{l}^{*}\right|\sqrt{\frac{s}{n}}\leq\phi^{-1}\sigma\sqrt{\frac{2\left(1+\tau\right)\log p}{n}},\,\,\,\,l\in K.$$ The product “$\left|\theta_{l}^{*}\right|\sqrt{\frac{s}{n}}$” suggests that normalizing $X_{j}$ to make $\frac{1}{n}\sum_{i=1}^{n}X_{ij}^{2}=1$ for all $j=1,\dots,p$ does not change the conclusions in Lemma \[prop:fixed\_design\]. Such normalization simply leads to rescaled coefficients and estimates (by a factor of $\sqrt{\frac{s}{n}}$). In particular, the choice of $\lambda\geq\frac{2\sigma}{\phi}\sqrt{\frac{s}{n}}\sqrt{\frac{2\left(1+\tau\right)\log p}{n}}$ in Lemma \[prop:fixed\_design\] becomes $\lambda=\lambda_{norm}\geq\frac{2\sigma}{\phi}\sqrt{\frac{2\left(1+\tau\right)\log p}{n}}$; also, $\left|\theta_{j}^{*}\right|\leq\frac{\lambda n}{2s}$ (where $\lambda\geq\frac{2\sigma}{\phi}\sqrt{\frac{s}{n}}\sqrt{\frac{2\left(1+\tau\right)\log p}{n}}$ without normalization) is replaced by $\sqrt{\frac{s}{n}}\left|\theta_{j}^{*}\right|\leq\frac{\lambda_{norm}}{2}$ (where $\lambda_{norm}\geq\frac{2\sigma}{\phi}\sqrt{\frac{2\left(1+\tau\right)\log p}{n}}$ with normalization). For $n=10000$, $p=4000$, $\sigma=1$, $\phi=0.95$, and $\tau=0.5$ in (\[eq:choice\]), Lemma \[prop:fixed\_design\] says that, with probability at least $0.97$, none of the relevant covariates are selected if $\max_{l\in K}\left|\theta_{l}^{*}\right|\leq0.05$ for $\sqrt{\frac{s}{n}}=1$ and $\max_{l\in K}\left|\theta_{l}^{*}\right|\leq0.5$ for $\sqrt{\frac{s}{n}}=0.1$. Therefore, everything else equal, limited variability in the relevant controls makes it more likely for the Lasso to omit them. Limited variability in the irrelevant controls $X_{K^{c}}$ does not matter for the selection performance of the Lasso because, for $j\in K^{c}$, $\left|\theta_{j}^{*}\right|\sqrt{\frac{1}{n}X_{j}^{T}X_{j}}=0$ regardless. Note that (\[eq:zero\]) implies $\mathbb{P}\left(\hat{\theta}_{l}\neq0\right)\leq\frac{1}{p^{\tau}}$ for any $l\in K$ subject to (\[eq:min\]). In comparison, @wainwright2009sharp shows that whenever $\theta_{l}^{*}\in\left(\lambda\frac{n}{s}\textrm{sgn}\left(\theta_{l}^{*}\right),\,0\right)$ or $\theta_{l}^{*}\in\left(0,\,\lambda\frac{n}{s}\textrm{sgn}\left(\theta_{l}^{*}\right)\right)$ for some $l\in K$, $$\mathbb{P}\left[\textrm{sgn}\left(\hat{\theta}_{K}\right)=\textrm{sgn}\left(\theta_{K}^{*}\right)\right]\leq\frac{1}{2}.\label{eq:wainwright}$$ Constant bounds in the form of (\[eq:wainwright\]) cannot explain that, with high probability, Lasso fails to select the relevant covariates in both and when $p$ is sufficiently large. \[more choices\]Under the assumptions in Lemma \[prop:fixed\_design\], the choices of regularization parameters $\lambda_{1}$ and $\lambda_{2}$ coincide with those in @bickel2009simultaneous when $\phi=1$; e.g., the columns in $X_{K^{c}}$ are orthogonal to the columns in $X_{K}$ (but within $X_{K^{c}}$, the columns need not be orthogonal to each other). If $\phi\approx 1$, similar results as those in Lemma \[prop:fixed\_design\] as well as in Sections \[sec:bias\_post\_double\_lasso\], \[sec:post\_double\_upper\_bounds\], and \[sec:bias\_debiased\_lasso\] hold for any choices of regularization parameters derived from the principle that $\lambda$ should be no smaller than $2\max_{j=1,\dots,p}\left|\frac{X_{j}^{T}\varepsilon}{n}\right|$ with high probability. These choices constitute what has been used in the vast majority of literature [e.g., @bickel2009simultaneous; @wainwright2009sharp; @belloni2012sparse; @belloni2013least; @belloni2014inference]. Choosing regularization parameters according to the principle discussed in Remark \[more choices\] prevents the inclusion of overly many irrelevant controls. This property is needed to ensure a good performance of post double Lasso, as discussed in Section \[sec:simulation\_evidence\] and Footnote \[footnote:over-selection\]. Lower bounds on the OVBs {#sec:bias_post_double_lasso} ------------------------ Proposition \[prop:bias\_post\_double\_formula\] derives a lower bound formula for the OVB of post double Lasso. We focus on the case where $\alpha^{*}=0$ because the conditions required to derive the explicit formula are difficult to interpret when $\alpha^{*}\ne0$. The reason is that the error in the reduced form equation involves $\alpha^{*}$, such that the choice of $\lambda_{1}$ in depends on the unknown $\alpha^{*}$. On the other hand, it is possible to provide easy-to-interpret scaling results (without explicit constants) for cases where $\alpha^{*}\ne0$, as we will show in Propositions \[prop:bias\_post\_double\_selection\] and \[prop:bias\_post\_double\_selection-1\]. \[prop:bias\_post\_double\_formula\] Let Assumptions \[ass:inference\_normality\] and \[ass:inference\_sparsity\_incoherence\] hold. Suppose $\lambda_{1}=2\phi^{-1}\sigma_{\eta}\sqrt{\frac{s}{n}}\sqrt{\frac{2\left(1+\tau\right)\log p}{n}}$, $\lambda_{2}=2\phi^{-1}\sigma_{v}\sqrt{\frac{s}{n}}\sqrt{\frac{2\left(1+\tau\right)\log p}{n}}$; for all $j\in K$ and $\left|a\right|,\left|b\right|\in(0,\,1]$, $$\text{both}\quad\beta_{j}^{*}\sqrt{\frac{s}{n}}=a\phi^{-1}\sigma_{\eta}\sqrt{\frac{2\left(1+\tau\right)\log p}{n}}\quad\text{ and}\quad\gamma_{j}^{*}\sqrt{\frac{s}{n}}=b\phi^{-1}\sigma_{v}\sqrt{\frac{2\left(1+\tau\right)\log p}{n}}.\label{eq:16}$$ In terms of $\tilde{\alpha}$ obtained from (\[eq:double\]), we have $$\left|\mathbb{E}\left(\tilde{\alpha}-\alpha^{*}\vert\mathcal{M}\right)\right|\geq\underset{:=\underline{\text{OVB}}}{\underbrace{\max_{r\in(0,1]}T_{1}\left(r\right)T_{2}\left(r\right)}}$$ where $$\begin{aligned} T_{1}\left(r\right) & = & \frac{\left(1+\tau\right)\left|ab\right|\phi^{-2}\sigma_{\eta}\frac{k\log p}{n}}{4\left(1+\tau\right)\phi^{-2}b^{2}\sigma_{v}\frac{k\log p}{n}+\left(1+r\right)\sigma_{v}},\\ T_{2}\left(r\right) & = & 1-k\exp\left(\frac{-b^{2}\left(1+\tau\right)\log p}{4\phi^{2}}\right)-\frac{1}{p^{\tau}}-\exp\left(\frac{-nr^{2}}{8}\right), \end{aligned}$$ for any $r\in(0,\,1]$, and $\mathcal{M}$ is an event with $\mathbb{P}\left(\mathcal{M}\right)\geq1-k\exp\left(\frac{-b^{2}\left(1+\tau\right)\log p}{4\phi^{2}}\right)-\frac{2}{p^{\tau}}$.[^21] \[rem:absenceOVB\] Let Assumptions \[ass:inference\_normality\] and \[ass:inference\_sparsity\_incoherence\] hold. As in Proposition \[prop:bias\_post\_double\_formula\], let $\lambda_{1}=2\phi^{-1}\sigma_{\eta}\sqrt{\frac{s}{n}}\sqrt{\frac{2\left(1+\tau\right)\log p}{n}}$ and $\lambda_{2}=2\phi^{-1}\sigma_{v}\sqrt{\frac{s}{n}}\sqrt{\frac{2\left(1+\tau\right)\log p}{n}}$. For all $j\in K,\;a>3,\,b>3$, if $$\text{either}\quad\left|\beta_{j}^{*}\right|\sqrt{\frac{s}{n}}=a\phi^{-1}\sigma_{\eta}\sqrt{\frac{2\left(1+\tau\right)\log p}{n}}\quad\text{or}\quad\left|\gamma_{j}^{*}\right|\sqrt{\frac{s}{n}}=b\phi^{-1}\sigma_{v}\sqrt{\frac{2\left(1+\tau\right)\log p}{n}},\label{eq:15}$$ then $\mathbb{P}\left[\textrm{supp}\left(\hat{\pi}\right)=\textrm{supp}\left(\pi^{*}\right)\right]\geq1-\frac{1}{p^{\tau}}$ or $\mathbb{P}\left[\textrm{supp}\left(\hat{\gamma}\right)=\textrm{supp}\left(\gamma^{*}\right)\right]\geq1-\frac{1}{p^{\tau}}$ by standard arguments. As a result, $\mathbb{P}\left(\left\{ \hat{I}_{1}\cup\hat{I}_{2}\right\} =K\right)\geq1-\frac{1}{p^{\tau}}$, where $\hat{I}_{1}$ and $\hat{I}_{2}$ are defined in (\[eq:double\]); i.e., the final OLS step (\[eq:double\]) includes all the relevant controls with high probability. By similar argument as in Section \[sec:appendix\_post\_double\_lasso\], on the high probability event $\left\{ \left\{ \hat{I}_{1}\cup\hat{I}_{2}\right\} =K\right\} $, the OVB of $\tilde{\alpha}$ is zero. Panels (d) of Figures \[fig:fsd\_dml\]\[fig:hist\_sel\_dml\] and Panels (b) of Figures \[fig:fsd\_large\]\[fig:hist\_sel\_large\] illustrate this phenomenon. Proposition \[prop:bias\_post\_double\_formula\] and Remark \[rem:absenceOVB\] show that the OVBs can be drastically different when we fix $\left(n,\,p,\,k,\,\sigma_{v},\,\sigma_{\eta},\,\alpha^{*}\right)$ in (\[eq:20\])(\[eq:21-1\]), but vary the products of the absolute values of the non-zero coefficients and the standard deviations of the corresponding controls: - Under (\[eq:16\]), $\mathbb{P}\left(\hat{I}_{1}=\hat{I}_{2}=\emptyset\right)\geq1-\frac{2}{p^{\tau}}$ and the OVBs are bounded from below by $\underline{OVB}$, which decreases as $\left|a\right|$ decreases. By fixing $\left(n,p,\phi,\sigma_{\eta},\sigma_{v},\tau\right)$, one can easily see from (\[eq:16\]) that the lower bound $\underline{OVB}$ depends on $\left|\beta_{j}^{*}\right|\sqrt{\frac{s}{n}}$ through $\left|a\right|$ and $\left|\gamma_{j}^{*}\right|\sqrt{\frac{s}{n}}$ through $\left|b\right|$, for $j\in K$. Therefore, under-selection cannot be avoided or mitigated by rescaling the covariates and $\underline{OVB}$ remains the same after rescaling. This fact suggests that the problem of “limited‘" variability can be recast as a “small‘" coefficient problem and vice versa. - Under (\[eq:15\]), $\mathbb{P}\left(\left\{ \hat{I}_{1}\cup\hat{I}_{2}\right\} =K\right)\geq1-\frac{1}{p^{\tau}}$ and the OVBs are zero. Under Assumptions \[ass:inference\_normality\]–\[ass:inference\_sparsity\_incoherence\] and if $\sigma_{v}$ is bounded away from zero and $\sigma_{\eta}$ is bounded from above, by contrast, the existing theory would imply that the biases of post double Lasso are bounded from above by $\texttt{constant}\cdot\frac{k\log p}{n}$, irrespective of whether Lasso fails to select the relevant controls or not, and how small $\left|a\right|$ and $\left|b\right|$ are. The (positive) $\texttt{constant}$ here does not depend on $\left(n,\,p,\,k,\,\frac{s}{n},\,\beta_{K}^{*},\,\gamma_{K}^{*},\,\alpha^{*}\right)$, and bears little meaning in the asymptotic framework which simply assumes $\frac{k\log p}{\sqrt{n}}\rightarrow0$ among other sufficient conditions. \[The existing theoretical framework makes it difficult to derive an informative $\texttt{constant}$, and to our knowledge, the literature provides no such derivation.\] The asymptotic upper bound $\texttt{constant}\cdot\frac{k\log p}{n}$ corresponds to the least favorable case and, thus, is uninformative about the most favorable cases that could vary in $\left(\frac{s}{n},\,\beta_{K}^{*},\,\gamma_{K}^{*},\,\alpha^{*}\right)$. By contrast, our lower bound analyses are informative about the most favorable cases, which are crucial for understanding the finite sample limitations of post double Lasso. For sufficiently large $p$, the configuration of $\left|\beta_{K}^{*}\right|$, $\left|\gamma_{K}^{*}\right|$, and $\frac{s}{n}$ in Proposition \[prop:bias\_post\_double\_formula\] leads to large $\mathbb{P}\left(\hat{I}_{1}=\hat{I}_{2}=\emptyset\right)$; meanwhile, the lower bound $\underline{OVB}$ is characterized by the interplay between $\mathbb{P}\left(\hat{I}_{1}=\hat{I}_{2}=\emptyset\right)$ (related to $T_{2}$) and the omitted coefficients (related to $T_{1}$). To gauge the magnitude of the OVB and explain why the confidence intervals proposed in the literature can exhibit under-coverage, it is instructive to compare $\underline{OVB}$ with $\ensuremath{\sigma_{\tilde{\alpha}}=\frac{1}{\sqrt{n}}\frac{\sigma_{\eta}}{\sigma_{v}}}$, the standard deviation (of $\tilde{\alpha}$) obtained from the asymptotic distribution in @belloni2014inference.[^22] Let us consider the following examples: $a=b=1$, $\sigma_{\eta}=\sigma_{v}=1$, $n=10000$, $p=4000$, $\tau=0.5$, and $\phi=0.95$; if $\left|\beta_{j}^{*}\right|\sqrt{\frac{s}{n}}\approx0.05$ and $\left|\gamma_{j}^{*}\right|\sqrt{\frac{s}{n}}\approx0.05$ for all $j\in K$, then $\frac{\underline{OVB}}{\sigma_{\tilde{\alpha}}}\approx0.52$ for $k=5$ and $\frac{\underline{OVB}}{\sigma_{\tilde{\alpha}}}\approx0.12$ for $k=1$. It is important to bear in mind that, the calculations of $\frac{\underline{OVB}}{\sigma_{\tilde{\alpha}}}$ are based on our theoretical *lower* bounds (for the OVBs) corresponding to the most favorable cases. The result $\frac{\underline{OVB}}{\sigma_{\tilde{\alpha}}}\approx0.52$ equals to the ratio of $\texttt{constant}\cdot\frac{k\log p}{n}$ (the typical asymptotic *upper* bound for the bias) to $\frac{1}{\sqrt{n}}$ (the typical scaling of the standard deviation $\sigma_{\tilde{\alpha}}$), with $\texttt{constant}=1.25$; for the second example with $k=1$, $\frac{\underline{OVB}}{\sigma_{\tilde{\alpha}}}\approx1.47\frac{k\log p}{\sqrt{n}}$. This comparison suggests that these *most favorable* cases are essentially the *least favorable* case. Now, let us recall the first example where $k=5$; changing $a=1$ to $a=0.1$ there yields $\frac{\underline{OVB}}{\sigma_{\tilde{\alpha}}}\approx0.12\frac{k\log p}{\sqrt{n}}$ when $\left|\beta_{j}^{*}\right|\sqrt{\frac{s}{n}}\approx0.005$, while changing $a=1$ to $a=3.1$ yields no OVBs when $\left|\beta_{j}^{*}\right|\sqrt{\frac{s}{n}}\approx0.16$, according to Remark \[rem:absenceOVB\]. These examples suggest that the most favorable cases may differ substantially from the least favorable case. In view of Proposition \[prop:bias\_post\_double\_formula\], $\underline{OVB}$ is not a simple linear function of $\frac{k\log p}{n}$ in general, but rather depends on $\left(n,p,k\right)$ and other factors in a more complex way. As $\frac{k\log p}{n}\rightarrow\infty$, we will show in Propositions \[prop:bias\_post\_double\_selection\] and \[prop:bias\_post\_double\_selection-1\] that, the OVB lower bounds scale as $\left|\alpha^{*}\right|$ or as $\frac{\sigma_{\eta}}{\sigma_{v}}$, depending on the configuration of $\left(\beta_{K}^{*},\,\gamma_{K}^{*},\,\alpha^{*}\right)$; as a consequence, $\frac{\underline{OVB}}{\sigma_{\tilde{\alpha}}}$ scales as $\frac{\sqrt{n}\left|\alpha^{*}\right|\sigma_{v}}{\sigma_{\eta}}$ or as $\sqrt{n}$. While the scaling $\sqrt{n}$ as $\frac{k\log p}{n}\rightarrow\infty$ can be easily seen from Proposition \[prop:bias\_post\_double\_formula\] where $\alpha^{*}=0$, Propositions \[prop:bias\_post\_double\_selection\] and \[prop:bias\_post\_double\_selection-1\] also consider cases where $\alpha^{*}\neq0$. When double under-selection occurs with high probability and $\frac{k\log p}{\sqrt{n}}$ is not small enough, according to the results above, $\tilde{\alpha}$ will perform poorly in general, except in one (albeit) extreme case. By Lemma \[prop:fixed\_design\] and (\[eq:16\]), $\left|\hat{\beta}-\beta^{*}\right|_{1}=\left|a\right|\frac{k\lambda_{1}n}{2s}$ and $\left|\hat{\gamma}-\gamma^{*}\right|_{1}=\left|b\right|\frac{k\lambda_{2}n}{2s}$ with probability at least $1-\frac{2}{p^{\tau}}$. If $\sigma_{v}$ is bounded away from zero, $\sigma_{\eta}$ is bounded from above, and $$\left|a\right|=\left|b\right|=o\left(1\right),\label{eq: small_OVB}$$ by similar argument in @belloni2014inference, we can show that $\sqrt{n}\left(\tilde{\alpha}-\alpha^{*}\right)$ is approximately normal and unbiased, *even if $\frac{k\log p}{\sqrt{n}}$ scales as a constant*. *If $\frac{k\log p}{n}$ scales as a constant*, then replacing (\[eq: small\_OVB\]) with $$\left|a\right|=\left|b\right|=o\left(\frac{1}{\sqrt{n}}\right),\label{eq: smaller_OVB}$$ yields the same conclusion. Holding other factors constant, the magnitude of OVBs decreases as $\left|\beta_{K}^{*}\right|$ and $\left|\gamma_{K}^{*}\right|$ decrease (i.e., as $\left|a\right|$ and $\left|b\right|$ decrease). As $\left|a\right|$ and $\left|b\right|$ become very small, the relevant controls become essentially irrelevant. The next results, Propositions \[prop:bias\_post\_double\_selection\] and \[prop:bias\_post\_double\_selection-1\], provide the scaling of OVB lower bounds under two different setups. These results consider cases where $\alpha^{*}\neq0$. Depending on the setups, some of the cases behave similarly to the case where $\alpha^{*}=0$ and some behave differently. For functions $f(n)$ and $g(n)$, we write $f(n)\succsim g(n)$ to mean that $f(n)\geq cg(n)$ for a universal constant $c\in(0,\,\infty)$ and similarly, $f(n)\precsim g(n)$ to mean that $f(n)\leq c^{'}g(n)$ for a universal constant $c^{'}\in(0,\,\infty)$; $f(n)\asymp g(n)$ when $f(n)\succsim g(n)$ and $f(n)\precsim g(n)$ hold simultaneously. As a general rule, $c$ constants denote positive universal constants that are independent of $n$, $p$, $k$, $\sigma_{\eta}$, $\sigma_{v}$, $s$, $\alpha^{*}$ and may change from place to place. These propositions are useful for understanding how the OVB lower bounds behave roughly as a function of $\left(n,\,p,\,k,\,\sigma_{\eta},\,\sigma_{v},\,\left|\alpha^{*}\right|\right)$, by abstracting from Proposition \[prop:bias\_post\_double\_formula\] all the finite sample subtleties (such as various universal constants) and assuming $\left|a\right|,\,\left|b\right|\asymp1$ (while $\left|a\right|,\,\left|b\right|\le1$) and $\phi^{-1}\precsim1$ (hence, $\phi^{-1}\asymp1$). Propositions \[prop:bias\_post\_double\_selection\] and \[prop:bias\_post\_double\_selection-1\] suggest that the OVB lower bounds do not simply scale as $\frac{k\log p}{n}$ even if $\left|a\right|,\,\left|b\right|\asymp1$ and $\phi^{-1}\precsim1$: When $\frac{k\log p}{n}\rightarrow0$, they scale as $\frac{\sigma_{\eta}}{\sigma_{v}}\frac{k\log p}{n}$ (in Proposition \[prop:bias\_post\_double\_selection\]) and as $\left|\alpha^{*}\right|\frac{k\log p}{n}$ (in Proposition \[prop:bias\_post\_double\_selection-1\]); when $\frac{k\log p}{n}\rightarrow\infty$, they scale as $\frac{\sigma_{\eta}}{\sigma_{v}}$ (in Proposition \[prop:bias\_post\_double\_selection\]) and as $\left|\alpha^{*}\right|$ (in Proposition \[prop:bias\_post\_double\_selection-1\]), instead of $\infty$. \[prop:bias\_post\_double\_selection\] Let Assumptions \[ass:inference\_normality\] and \[ass:inference\_sparsity\_incoherence\] hold. Suppose $\phi^{-1}\precsim1$ in (\[eq:14\]); the regularization parameters in (\[eq:las-1\]) and (\[eq:las-2\]) are chosen in a similar fashion as in Lemma \[prop:fixed\_design\] such that $\lambda_{1}\asymp\phi^{-1}\sigma_{\eta}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}$[^23] and $\lambda_{2}\asymp\phi^{-1}\sigma_{v}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}$; for all $j\in K$, $\left|\beta_{j}^{*}\right|\leq\frac{\lambda_{1}n}{2s}$ and $\left|\gamma_{j}^{*}\right|\leq\frac{\lambda_{2}n}{2s}$, but $\left|\beta_{j}^{*}\right|\asymp\sigma_{\eta}\sqrt{\frac{n}{s}}\sqrt{\frac{\log p}{n}}$ and $\left|\gamma_{j}^{*}\right|\asymp\sigma_{v}\sqrt{\frac{n}{s}}\sqrt{\frac{\log p}{n}}$. Let us consider $\tilde{\alpha}$ obtained from (\[eq:double\]). \(i) If $\alpha^{*}=0$, then there exist positive universal constants $c^{\dagger},c_{1},c_{2},c_{3},c^{*},c_{0}^{*}$ such that $$\left|\mathbb{E}\left(\tilde{\alpha}-\alpha^{*}\vert\mathcal{M}\right)\right|\geq c^{\dagger}\frac{\sigma_{\eta}}{\sigma_{v}}\left(\frac{k\log p}{n}\wedge1\right)\left[1-c_{1}k\exp\left(-c_{2}\log p\right)-\exp\left(-c_{3}n\right)\right],\label{eq:22}$$ where $\mathcal{M}$ is an event with $\mathbb{P}\left(\mathcal{M}\right)\geq1-c^{*}k\exp\left(-c_{0}^{*}\log p\right)$. \(ii) If $\alpha^{*}\gamma_{j}^{*}\in(0,\,-\beta_{j}^{*}]$, $\beta_{j}^{*}<0$ for $j\in K$ (or, $\alpha^{*}\gamma_{j}^{*}\in[-\beta_{j}^{*},\,0)$, $\beta_{j}^{*}>0$ for $j\in K$), then for some positive universal constants $c^{\dagger},c_{1},c_{2},c_{3},c^{*},c_{0}^{*}$, (\[eq:22\]) holds with $\mathbb{P}\left(\mathcal{M}\right)\geq1-c^{*}k\exp\left(-c_{0}^{*}\log p\right)$. To motivate the next proposition, note that as long as $\hat{I}_{1}=K$ or $\hat{I}_{2}=K$, the final OLS step (\[eq:double\]) corresponds to the oracle estimator. Under sufficient variability in $X_{K}$, when $\left|\pi_{K}^{*}\right|$ are small enough (so $\hat{I}_{1}=\emptyset$ with high probability), but $\left|\gamma_{K}^{*}\right|$ in (\[eq:21-1\]) are large enough (so $\hat{I}_{2}=K$ with high probability), post double Lasso coincides with the oracle estimator while OLS post (the single) Lasso (\[eq:las-1\]) omits the relevant controls with high probability. As we will show below, if $\left|\gamma_{K}^{*}\right|$ are also small enough, then like the OLS post (\[eq:las-1\]), post double Lasso can also yield substantial OVBs relative to the standard deviation. \[prop:bias\_post\_double\_selection-1\] Let Assumptions \[ass:inference\_normality\] and \[ass:inference\_sparsity\_incoherence\] hold. Suppose $\phi^{-1}\precsim1$ in (\[eq:14\]); the regularization parameters in (\[eq:las-1\]) and (\[eq:las-2\]) are chosen in a similar fashion as in Lemma \[prop:fixed\_design\] such that $\lambda_{1}\asymp\frac{\left(\sigma_{\eta}+\left|\alpha^{*}\right|\sigma_{v}\right)}{\phi}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}$ and $\lambda_{2}\asymp\phi^{-1}\sigma_{v}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}$; for all $j\in K$, $\left|\gamma_{j}^{*}\right|\leq\frac{\lambda_{2}n}{2s}$ but $\left|\gamma_{j}^{*}\right|\asymp\sigma_{v}\sqrt{\frac{n}{s}}\sqrt{\frac{\log p}{n}}$. Let us consider $\tilde{\alpha}$ obtained from (\[eq:double\]). \(i) If $\pi_{j}^{*}=0$ for all $j\in K$, then there exist positive universal constants $c^{\dagger},c_{1},c_{2},c_{3},c^{*},c_{0}^{*}$ such that $$\left|\mathbb{E}\left(\tilde{\alpha}-\alpha^{*}\vert\mathcal{M}\right)\right|\geq c^{\dagger}\left|\alpha^{*}\right|\left(\frac{k\log p}{n}\wedge1\right)\left[1-c_{1}k\exp\left(-c_{2}\log p\right)-\exp\left(-c_{3}n\right)\right],\label{eq:23}$$ where $\mathbb{P}\left(\mathcal{M}\right)\geq1-c^{*}k\exp\left(-c_{0}^{*}\log p\right)$. \(ii) For all $j\in K$, suppose $\left|\pi_{j}^{*}\right|\asymp\left(\sigma_{\eta}+\left|\alpha^{*}\right|\sigma_{v}\right)\sqrt{\frac{n}{s}}\sqrt{\frac{\log p}{n}}$, and we have either (1) $\alpha^{*}<0$, $\beta_{j}^{*}>0$, $\gamma_{j}^{*}>0$, $0<\pi_{j}^{*}\leq\frac{\lambda_{1}n}{2s}$, or (2) $\alpha^{*}>0$, $\beta_{j}^{*}<0$, $\gamma_{j}^{*}>0$, $-\frac{\lambda_{1}n}{2s}<\pi_{j}^{*}<0$. Then there exist positive universal constants $c^{\dagger},c_{1},c_{2},c_{3},c^{*},c_{0}^{*}$ such that (\[eq:23\]) holds with $\mathbb{P}\left(\mathcal{M}\right)\geq1-c^{*}k\exp\left(-c_{0}^{*}\log p\right)$. The different scaling in Propositions \[prop:bias\_post\_double\_selection\] and \[prop:bias\_post\_double\_selection-1\] comes from the fact that, the former constrains the magnitude of $\pi_{K}^{*}$ through constraining the magnitude of $\left(\beta_{K}^{*},\,\gamma_{K}^{*},\,\alpha^{*}\right)$, and the latter allows large magnitude of $\left(\beta_{K}^{*},\,\alpha^{*}\right)$ while constraining the magnitude of $\left(\pi_{K}^{*},\,\gamma_{K}^{*}\right)$. Proposition \[prop:bias\_post\_double\_selection-1\] suggests that, even if $\frac{k\log p}{\sqrt{n}}$ is small, the OVB lower bounds can still be substantial if $\left|\alpha^{*}\right|$ is large enough. To see this, suppose $\left|\alpha^{*}\right|\asymp\frac{n}{k\log p}$ and $\frac{k\log p}{n}=o\left(\frac{1}{\sqrt{n}}\right)$ (so that $\frac{k\log p}{\sqrt{n}}=o\left(1\right)$). Then the OVB lower bounds in Proposition \[prop:bias\_post\_double\_selection-1\] scale as a constant bounded away from zero. Even if $\frac{\sigma_{\eta}}{\sigma_{v}}\asymp1$, the ratios of these lower bounds to $\ensuremath{\sigma_{\tilde{\alpha}}=\frac{1}{\sqrt{n}}\frac{\sigma_{\eta}}{\sigma_{v}}}$ scale as $\frac{\left|\alpha^{*}\right|\sigma_{v}}{\sigma_{\eta}}\frac{k\log p}{\sqrt{n}}\asymp\sqrt{n}$. By contrast, in the setup of Proposition \[prop:bias\_post\_double\_selection\], the ratios of its lower bounds to $\ensuremath{\sigma_{\tilde{\alpha}}}$ scale as $\frac{k\log p}{\sqrt{n}}=o\left(1\right)$ (irrespective of $\frac{\sigma_{\eta}}{\sigma_{v}}$), whenever $\frac{k\log p}{n}=o\left(\frac{1}{\sqrt{n}}\right)$. Propositions \[prop:bias\_post\_double\_formula\]\[prop:bias\_post\_double\_selection-1\] on the OVB lower bounds have important implications for inference based on the post double Lasso procedure proposed in @belloni2014inference. On the one hand, the existing theoretical results on post double Lasso such as @belloni2014inference prove that the assumptions $\frac{k\log p}{\sqrt{n}}=o\left(1\right)$, $\left|\alpha^{*}\right|\precsim 1$, $\left|\sigma_{v}\right|\asymp 1$, and $\left|\sigma_{\eta}\right|\asymp 1$ (among other regularity conditions) are *sufficient* for establishing asymptotic normality and unbiasedness, regardless of whether under-selection is present or not. On the other hand, our Propositions \[prop:bias\_post\_double\_formula\]\[prop:bias\_post\_double\_selection-1\] show settings where double under-selection occurs with high probability, and suggest that in such settings, small $\frac{k\log p}{\sqrt{n}}$ and $\frac{\left|\alpha^{*}\right|\sigma_{v}}{\sigma_{\eta}}$ are *almost necessary* for a reliable performance of the post double Lasso, under any regularization choice derived from the principle mentioned in Remark \[more choices\]. \[rem:when\_post\_double\_work\] In view of our theoretical results and simulation evidence, if increasing $\lambda_{1}$ and $\lambda_{2}$ from the theoretically recommended choices (such as the one in @belloni2014inference) yield similar $\tilde{\alpha}$s, then the underlying model could be in the regime where either under-selection in both Lasso steps is unlikely, or the OVBs are simply negligible. Outside this regime, the magnitude and performance of post double Lasso can be quite sensitive to the increase of $\lambda_{1}$ and $\lambda_{2}$. The rationale behind this heuristic lies in that the final step of post double Lasso, (\[eq:double\]), is simply an OLS regression of $Y$ on $D$ and the union of selected controls from (\[eq:las-1\])(\[eq:las-2\]). One might ask by how much $\lambda_{1}$ and $\lambda_{2}$ should be increased for the robustness checks. For the regularization choice proposed in @belloni2014inference, our simulations suggest that an increase by $50\%$ works well in practice. As an example, Figure \[fig:dml\_bias\_std\_sens\] shows that the ratio of bias to standard deviation for the post double Lasso is less sensitive to increasing the regularization parameters when $\sigma_{x}\ge0.5$ because the selection performance of Lasso is more robust when $\sigma_{x}\ge 0.5$ (cf. Figures \[fig:sel\_sensitivity\] and \[fig:sel\_rel\_sensitivity\]). [\[]{}In Figures \[fig:sel\_sensitivity\], \[fig:sel\_rel\_sensitivity\], and \[fig:dml\_bias\_std\_sens\], we fix $\beta_{K}^{*}$ and $\gamma_{K}^{*}$ while varying $\sigma_{x}$, which is equivalent to varying the magnitude of $\beta_{K}^{*}$ and $\gamma_{K}^{*}$ while fixing $\sigma_{x}$, as discussed previously. In other words, we can simply replace the label $\sigma_{x}$ for the horizontal axis by the magnitude of $\beta_{K}^{*}$ and $\gamma_{K}^{*}$.[\]]{} Upper bounds on the OVBs {#sec:post_double_upper_bounds} ------------------------ Proposition \[prop:bias\_post\_double\_selection\] suggests that, the lower bounds on the OVBs scale as $\frac{\sigma_{\eta}}{\sigma_{v}}$ as $\frac{k\log p}{n}\rightarrow\infty$. Proposition \[prop:bias\_post\_double\_selection-1\] suggests that, the lower bounds on the OVBs scale as $\left|\alpha^{*}\right|$ as $\frac{k\log p}{n}\rightarrow\infty$. This prompts the question of how large the OVBs can be under the regime where $\frac{k\log p}{n}\rightarrow\infty$. To answer this question, we would like a meaningful upper bound (other than just $\infty$) on the OVBs, but there is no such result in the existing literature. Interestingly enough, we show that the upper bounds on the OVBs scale as $\frac{\sigma_{\eta}}{\sigma_{v}}$ or $\left|\alpha^{*}\right|$ with high probability, even if $\frac{k\log p}{n}\rightarrow\infty$ and the Lasso is inconsistent in the sense $\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(X_{i}\hat{\pi}-X_{i}\pi^{*}\right)^{2}}\rightarrow\infty$, $\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(X_{i}\hat{\gamma}-X_{i}\gamma^{*}\right)^{2}}\rightarrow\infty$ with high probability. To illustrate how the OVB upper bounds behave roughly as a function of $\left(n,\,p,\,k,\,\sigma_{\eta},\,\sigma_{v},\,\left|\alpha^{*}\right|\right)$, we again abstract from all the finite sample subtleties (such as various universal constants) and assume $\left|a\right|,\,\left|b\right|\asymp1$ (while $\left|a\right|,\,\left|b\right|\le1$) and $\phi^{-1}\precsim1$. \[prop:bias\_post\_double\_selection-upper\]Let Assumptions \[ass:inference\_normality\] and \[ass:inference\_sparsity\_incoherence\] hold. Suppose $\phi^{-1}\precsim1$ in (\[eq:14\]); the regularization parameters in (\[eq:las-1\]) and (\[eq:las-2\]) are chosen in a similar fashion as in Proposition \[prop:bias\_post\_double\_selection\] such that $\lambda_{1}\asymp\phi^{-1}\sigma_{\eta}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}$ and $\lambda_{2}\asymp\phi^{-1}\sigma_{v}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}$; for all $j\in K$, $\gamma_{j}^{*}=\gamma^{*}$, $\left|\beta_{j}^{*}\right|\leq\frac{\lambda_{1}n}{2s}$ and $\left|\gamma^{*}\right|\leq\frac{\lambda_{2}n}{2s}$, but $\left|\beta_{j}^{*}\right|\asymp\sigma_{\eta}\sqrt{\frac{n}{s}}\sqrt{\frac{\log p}{n}}$ and $\left|\gamma^{*}\right|\asymp\sigma_{v}\sqrt{\frac{n}{s}}\sqrt{\frac{\log p}{n}}$. Let us consider $\tilde{\alpha}$ obtained from (\[eq:double\]). Then for either $\alpha^{*}=0$, or $\alpha^{*}\neq0$ subject to the conditions in part (ii) of Proposition \[prop:bias\_post\_double\_selection\], there exist positive universal constants $c_{1},c_{2},c_{3},c_{4},c^{*},c_{0}^{*}$ such that $$\mathbb{P}\left(\left|\tilde{\alpha}-\alpha^{*}\right|\leq\overline{OVB}\vert\hat{I}_{1}=\hat{I}_{2}=\emptyset\right)\geq1-c_{1}k\exp\left(-c_{2}\log p\right)-c_{3}\exp\left(-c_{4}nr^{2}\right)$$ where $\mathbb{P}\left(\hat{I}_{1}=\hat{I}_{2}=\emptyset\right)\geq1-c^{*}\exp\left(-c_{0}^{*}\log p\right)$ and $$\overline{OVB}\asymp\max\left\{ \frac{\sigma_{\eta}}{\sigma_{v}}\left(\frac{k\log p}{n}\wedge1\right),\,\frac{\sigma_{v}\sigma_{\eta}\left(r\vee\frac{k\log p}{n}\right)}{\left(\frac{k\log p}{n}\vee1\right)\sigma_{v}^{2}}\right\}$$ for any $r\in(0,\,1]$. \[prop:bias\_post\_double\_selection-upper-1\]Let Assumptions \[ass:inference\_normality\] and \[ass:inference\_sparsity\_incoherence\] hold. Suppose $\phi^{-1}\precsim1$ in (\[eq:14\]); the regularization parameters in (\[eq:las-1\]) and (\[eq:las-2\]) are chosen in a similar fashion as in Proposition \[prop:bias\_post\_double\_selection-1\] such that $\lambda_{1}\asymp\frac{\sigma_{\eta}+\left|\alpha^{*}\right|\sigma_{v}}{\phi}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}$ and $\lambda_{2}\asymp\phi^{-1}\sigma_{v}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}$; for all $j\in K$, $\gamma_{j}^{*}=\gamma^{*}$, $\left|\gamma^{*}\right|\leq\frac{\lambda_{2}n}{2s}$, but $\left|\gamma^{*}\right|\asymp\sigma_{v}\sqrt{\frac{n}{s}}\sqrt{\frac{\log p}{n}}$. Let us consider $\tilde{\alpha}$ obtained from (\[eq:double\]). \(i) If $\pi_{j}^{*}=0$ for all $j\in K$, then there exist positive universal constants $c_{1},c_{2},c_{3},c_{4},c^{*},c_{0}^{*}$ such that $$\mathbb{P}\left(\left|\tilde{\alpha}-\alpha^{*}\right|\leq\overline{OVB}\vert\hat{I}_{1}=\hat{I}_{2}=\emptyset\right)\geq1-c_{1}k\exp\left(-c_{2}\log p\right)-c_{3}\exp\left(-c_{4}nr^{2}\right) \label{eq:OVB_upper}$$ where $\mathbb{P}\left(\hat{I}_{1}=\hat{I}_{2}=\emptyset\right)\geq1-c^{*}\exp\left(-c_{0}^{*}\log p\right)$ and $$\overline{OVB}\asymp\max\left\{ \left|\alpha^{*}\right|\left(\frac{k\log p}{n}\wedge1\right),\,\frac{\sigma_{v}\sigma_{\eta}\left(r\vee\frac{k\log p}{n}\right)}{\left(\frac{k\log p}{n}\vee1\right)\sigma_{v}^{2}}\right\}$$ for any $r\in(0,\,1]$. \(ii) If $0<\left|\pi_{j}^{*}\right|\leq\frac{\lambda_{1}n}{2s}$ but $\left|\pi_{j}^{*}\right|\asymp\left(\sigma_{\eta}+\left|\alpha^{*}\right|\sigma_{v}\right)\sqrt{\frac{n}{s}}\sqrt{\frac{\log p}{n}}$ for all $j\in K$, then there exist positive universal constants $c_{1},c_{2},c_{3},c_{4},c^{*},c_{0}^{*}$ such that (\[eq:OVB\_upper\]) holds with $\mathbb{P}\left(\hat{I}_{1}=\hat{I}_{2}=\emptyset\right)\geq1-c^{*}\exp\left(-c_{0}^{*}\log p\right)$ and $$\overline{OVB}\asymp\max\left\{ \left(\left|\alpha^{*}\right|\vee\frac{\sigma_{\eta}}{\sigma_{v}}\right)\left(\frac{k\log p}{n}\wedge1\right),\,\frac{\sigma_{v}\sigma_{\eta}\left(r\vee\frac{k\log p}{n}\right)}{\left(\frac{k\log p}{n}\vee1\right)\sigma_{v}^{2}}\right\}$$ for any $r\in(0,\,1]$. Suppose $\sigma_{v}\asymp1$ and $\frac{c^{'}k}{p^{c^{''}}}$ is small for some positive universal constants $c^{'}$ and $c^{''}$. As $\frac{k\log p}{n}\rightarrow\infty$ (where both Lasso steps are inconsistent in the sense that $\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(X_{i}\hat{\pi}-X_{i}\pi^{*}\right)^{2}}\rightarrow\infty$ and $\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(X_{i}\hat{\gamma}-X_{i}\gamma^{*}\right)^{2}}\rightarrow\infty$ with high probability), Proposition \[prop:bias\_post\_double\_selection-upper\] implies that $\overline{OVB}\asymp\frac{\sigma_{\eta}}{\sigma_{v}}$ and Proposition \[prop:bias\_post\_double\_selection-upper-1\] implies that $\overline{OVB}\asymp\left(\left|\alpha^{*}\right|\vee\frac{\sigma_{\eta}}{\sigma_{v}}\right)$, and $\mathbb{P}\left(\left|\tilde{\alpha}-\alpha^{*}\right|\leq\overline{OVB}\vert\hat{I}_{1}=\hat{I}_{2}=\emptyset\right)$ is large. Implications for inference and comparison to high-dimensional OLS-based methods =============================================================================== As shown in Section \[sec:inference\], the OVBs of post double Lasso have important consequences for making inference. In terms of simulation evidence, Figure \[fig:dml\_cov\] displays the coverage rates of 90% confidence intervals based on the DGP in Section \[sec:simulation\_evidence\] and shows that the OVB of post double Lasso can cause substantial under-coverage even when $n=1000$. \[fig:dml\_cov\] These results and those in Section \[sec:inference\] prompt the question of how to make inference in a reliable manner when under-selection and OVBs is a concern, such as, when limited variability is present. In many economic applications, $p$ is comparable to but still smaller than $n$. In such moderately high dimensional settings, OLS-based inference procedures provide a natural alternative to Lasso-based methods. Under classical conditions, OLS is the best linear unbiased estimator. Moreover, under normality and homoscedasticity, OLS admits exact finite sample inference for any fixed $(n,\,p)$ as long as $\frac{p+1}{n}\leq1$ (recalling that the number of regression coefficients is $p+1$ in ).[^24] Furthermore, unlike the Lasso-based inference methods, OLS does not rely on any sparsity assumptions. This is important because sparsity assumptions may not be satisfied in applications and, as we show in this paper, the OVBs of Lasso-based inference procedures can be substantial even when $k$ is very small, $n$ is large and larger than $p$. While OLS is unbiased, constructing standard errors is challenging when $p$ is large. For instance, @cattaneo2018inference show that the usual versions of Eicker-White robust standard errors are inconsistent under asymptotics where the number of controls grows as fast as the sample size. This result motivates a recent literature to develop high-dimensional OLS-based inference procedures that are valid in settings with many controls [e.g., @cattaneo2018inference; @jochmans2018heteroscedasticity; @kline2018leaveout]. Figures \[fig:dml\_bias\_std\_ols\]–\[fig:dml\_cov\_ols\] compare the finite sample performance of post double Lasso and OLS with the heteroscedasticity robust HCK standard errors proposed by @cattaneo2018inference. We do not report results for $n=200$ since OLS cannot be applied in this case. OLS is unbiased (as expected) and exhibits close-to-exact empirical coverage rates irrespective of the degree of variability in the controls. The additional simulation evidence in Appendix \[app:additional\_simulations\] confirms the excellent performance of OLS with HCK standard errors. Figure \[fig:dml\_ci\_ols\] displays the average length of 90% confidence intervals and shows that OLS yields somewhat wider confidence intervals than post double Lasso. \[fig:dml\_bias\_std\_ols\] \[fig:dml\_cov\_ols\] \[fig:dml\_ci\_ols\] In sum, our simulation results suggest that modern OLS-based inference methods that accommodate many controls constitute a viable alternative to Lasso-based inference methods. These methods are unbiased and demonstrate an excellent size accuracy, irrespective of the degree of variability in the relevant controls. However, there is a trade-off because OLS yields somewhat wider confidence intervals than post double Lasso. Recommendations for empirical practice {#sec:recommendation} ====================================== Here we summarize the practical implications of our results and provide guidance for empirical researchers. First, the simulation evidence in Section \[sec:evidence\] and Appendix \[app:additional\_simulations\] and the theoretical results in Section \[sec:bias\_post\_double\_lasso\] (cf. Remark \[rem:when\_post\_double\_work\]) suggest the following heuristic: If the estimates of $\alpha^{*}$ are robust to increasing the theoretically recommended regularization parameters in the two Lasso steps, post double Lasso could be a reliable and efficient method. Therefore, we recommend to always check whether empirical results are robust to increasing the regularization parameters. Based on our simulations, a simple rule of thumb is to increase by $50\%$ the regularization parameters proposed in @belloni2014inference. Robustness checks are standard in other contexts (e.g., bandwidth choices in regression discontinuity designs) and our results highlight the importance of such checks in the context of Lasso-based inference methods. Second, we show that even if the relevant controls are associated with large coefficients, the Lasso may not select such controls due to limited variability. Therefore, following @belloni2014inference, we recommend to always augment the union of selected controls with an “amelioration” set of controls motivated by economic theory and prior knowledge to mitigate the OVBs. Third, our simulations show that in moderately high-dimensional settings where $p$ is comparable to but smaller than $n$, recently developed OLS-based inference methods that are robust to the inclusion of many controls exhibit better size properties and not much wider confidence intervals than Lasso-based inference methods. This suggests that high-dimensional OLS-based procedures constitute a viable alternative to Lasso-based inference methods. Conclusion ========== Given the rapidly increasing popularity of Lasso and Lasso-based inference methods in empirical economic research, it is crucial to better understand the merits and limitations of these new tools, and how they compare to other alternatives such as the high-dimensional OLS-based procedures. This paper presents empirical and theoretical evidence on the finite sample behavior of post double Lasso and the debiased Lasso (in the appendix). Specifically, we analyze the finite sample OVBs arising from the Lasso not selecting all the relevant control variables. Our results have important practical implications, and we provide guidance for empirical researchers. We focus on the implications of under-selection for post double Lasso and the debiased Lasso in linear regression models. However, our results on the under-selection of the Lasso also have important implications for other inference methods that rely on Lasso as a first-step estimator. Towards this end, an interesting avenue for future research would be to investigate the impact of under-selection on the performance of the non-linear Lasso-based approaches proposed by @belloni2014inference, @farrell2015robust, @belloni2017program, and @chernozhukov2018double. In moderately high-dimensional settings where $p$ is smaller than but comparable to $n$, it would also be interesting to compare the non-linear treatment effects estimators in @belloni2014inference to the robust finite sample methods proposed by @rothe2017robust. Finally, this paper motivates further examinations of the practical usefulness of Lasso-based inference procedures and other modern high-dimensional methods. For example, in the context of the study of elite college effects by @dale2002estimating, @angrist2019machine find that post double Lasso can be useful for regression-based sensitivity analyses when there is an abundance of potential control variables. They further present interesting simulation evidence on the finite sample behavior of Lasso-based IV methods [e.g., @belloni2012sparse]. [Appendix to “Omitted variable bias of Lasso-based inference methods: A finite sample analysis” (for online publication)]{} Notation {#notation .unnumbered} ======== Here we collect additional notation that is not provided in the main text. The $\ell_{q}-$norm of a vector $v\in\mathbb{R}^{m}$ is denoted by $\left|v\right|_{q}$, $1\leq q\leq\infty$ where $\left|v\right|_{q}:=\left(\sum_{i=1}^{m}\left|v_{i}\right|^{q}\right)^{1/q}$ when $1\leq q<\infty$ and $\left|v\right|_{q}:=\max_{i=1,\dots,m}\left|v_{i}\right|$ when $q=\infty$. For a matrix $A\in\mathbb{R}^{n\times m}$, the $\ell_{2}-$operator norm of $A$ is defined as $\left\Vert A\right\Vert _{2}:=\sup_{v\in S^{m-1}}\left|Av\right|_{2}$, where $S^{m-1}=\left\{ v\in\mathbb{R}^{m}:\,\left|v\right|_{2}=1\right\} $. For a square matrix $A\in\mathbb{R}^{m\times m}$, let $\lambda_{\min}(A)$ and $\lambda_{\max}(A)$ denote its minimum eigenvalue and maximum eigenvalue, respectively. Proofs for the main results =========================== Lemma \[prop:fixed\_design\] {#appendix a1} ---------------------------- ### Preliminary {#preliminary .unnumbered} We will exploit the following Gaussian tail bound: $$\mathbb{P}\left(\mathcal{Z}\geq t\right)\leq\frac{1}{2}\exp\left(\frac{-t^{2}}{2\sigma^{2}}\right)$$ for all $t\geq0$, where $\mathcal{Z}\sim\mathcal{N}\left(0,\,\sigma^{2}\right)$. Note that the constant “$\frac{1}{2}$” cannot be improved uniformly. Given $\lambda\geq\frac{2\sigma}{\phi}\sqrt{\frac{s}{n}}\sqrt{\frac{2\left(1+\tau\right)\log p}{n}}$ where $\tau>0$ and the tail bound $$\mathbb{P}\left(\left|\frac{X^{T}\varepsilon}{n}\right|_{\infty}\geq t\right)\leq\exp\left(\frac{-nt^{2}}{2\sigma^{2}s/n}+\log p\right)\leq\frac{1}{p^{\tau}}$$ for $t=\frac{\sigma}{\phi}\sqrt{\frac{s}{n}}\sqrt{\frac{2\left(1+\tau\right)\log p}{n}}$, we have $$\lambda\geq2\left|\frac{X^{T}\varepsilon}{n}\right|_{\infty}\label{eq:lambda}$$ with probability at least $1-\frac{1}{p^{\tau}}$. Let the event $$\mathcal{E}=\left\{ \left|\frac{X^{T}\varepsilon}{n}\right|_{\infty}\leq\frac{\sigma}{\phi}\sqrt{\frac{s}{n}}\sqrt{\frac{2\left(1+\tau\right)\log p}{n}}\right\} .\label{eq:event}$$ Note that $\mathbb{P}\left(\mathcal{E}\right)\geq1-\frac{1}{p^{\tau}}$. Lemma \[prop:fixed\_design\] relies on the following intermediate results. *(i) On the event $\mathcal{E}$, (\[eq:las\]) has a unique optimal solution $\hat{\theta}$ such that $\hat{\theta}_{j}=0$ for $j\notin K$.* *(ii) If $\mathbb{P}\left(\left\{ \hat{\theta}_{j}\neq0,\,j\in K\right\} \cap\mathcal{E}\right)>0$, conditioning on $\left\{ \hat{\theta}_{j}\neq0,\,j\in K\right\} \cap\mathcal{E}$, we must have $$\left|\hat{\theta}_{j}-\theta_{j}^{*}\right|\geq\frac{\lambda n}{2s}.\label{eq:37-1}$$* Claim (i) above follows from the argument in @wainwright_2019. To show claim (ii), we develop our own proof. The proof for claim (i) above is based on a construction called Primal-Dual Witness (PDW) method developed by @wainwright2009sharp. The procedure is described as follows. 1. Set $\hat{\theta}_{K^{c}}=0_{p-k}$. 2. Obtain $(\hat{\theta}_{K},\,\hat{\delta}_{K})$ by solving $$\hat{\theta}_{K}\in\arg\min_{\theta_{K}\in\mathbb{R}^{k}}\left\{ \underset{:=g(\theta_{K})}{\underbrace{\frac{1}{2n}\left|Y-X_{K}\theta_{K}\right|_{2}^{2}}}+\lambda\left|\theta_{K}\right|_{1}\right\} ,\label{eq:sub}$$ and choosing $\hat{\delta}_{K}\in\partial\left|\theta_{K}\right|_{1}$ such that $\nabla g(\theta_{K})\vert_{\theta_{K}=\hat{\theta}_{K}}+\lambda\hat{\delta}_{K}=0$.[^25] 3. Obtain $\hat{\delta}_{K^{c}}$ by solving $$\frac{1}{n}X^{T}(X\hat{\theta}-Y)+\lambda\hat{\delta}=0,\label{eq:42}$$ and check whether or not $\left|\hat{\delta}_{K^{c}}\right|_{\infty}<1$ (the *strict dual feasibility* condition) holds. Lemma 7.23 from Chapter 7 of @wainwright_2019 shows that, if the PDW construction succeeds, then $\hat{\theta}=(\hat{\theta}_{K},\,0_{p-k})$ is the unique optimal solution of program (\[eq:las\]). To show that the PDW construction succeeds on the event $\mathcal{E}$, it suffices to show that $\left|\hat{\delta}_{K^{c}}\right|_{\infty}<1$. The details can be found in Chapter 7.5 of @wainwright_2019. In particular, under the choice of $\lambda$ stated in Lemma \[prop:fixed\_design\], we obtain that $\left|\hat{\delta}_{K^{c}}\right|_{\infty}<1$ and hence the PDW construction succeeds conditioning on $\mathcal{E}$ where $\mathbb{P}\left(\mathcal{E}\right)\geq1-\frac{1}{p^{\tau}}$. In summary, conditioning on $\mathcal{E}$, under the choice of $\lambda$ stated in Lemma \[prop:fixed\_design\], program (\[eq:las\]) has a unique optimal solution $\hat{\theta}$ such that $\hat{\theta}_{j}=0$ for $j\notin K$. We now show (\[eq:37-1\]). By construction, $\hat{\theta}=(\hat{\theta}_{K},\,0_{p-k})$, $\hat{\delta}_{K}$, and $\hat{\delta}_{K^{c}}$ satisfy (\[eq:42\]) and therefore we obtain $$\begin{aligned} \frac{1}{n}X_{K}^{T}X_{K}\left(\hat{\theta}_{K}-\theta_{K}^{*}\right)-\frac{1}{n}X_{K}^{T}\varepsilon+\lambda\hat{\delta}_{K} & = & 0_{k},\label{eq:7-1}\\ \frac{1}{n}X_{K^{c}}^{T}X_{K}\left(\hat{\theta}_{K}-\theta_{K}^{*}\right)-\frac{1}{n}X_{K^{c}}^{T}\varepsilon+\lambda\hat{\delta}_{K^{c}} & = & 0_{p-k}.\label{eq:8}\end{aligned}$$ Solving the equations above yields $$\hat{\theta}_{K}-\theta_{K}^{*}=\left(\frac{X_{K}^{T}X_{K}}{n}\right)^{-1}\frac{X_{K}^{T}\varepsilon}{n}-\lambda\left(\frac{X_{K}^{T}X_{K}}{n}\right)^{-1}\hat{\delta}_{K}.\label{eq:49}$$ In what follows, we will condition on $\left\{ \hat{\theta}_{j}\neq0,\,j\in K\right\} \cap\mathcal{E}$ and make use of (\[eq:lambda\])-(\[eq:event\]). Let $\Delta=\frac{X_{K}^{T}\varepsilon}{n}-\lambda\hat{\delta}_{K}$. Note that $$\left|\hat{\theta}_{K}-\theta_{K}^{*}\right|\geq\left|\left(\frac{X_{K}^{T}X_{K}}{n}\right)^{-1}\right|\left|\left|\lambda\hat{\delta}_{K}\right|-\left|\frac{X_{K}^{T}\varepsilon}{n}\right|\right|,\label{eq:7}$$ where the inequality uses the fact that $\left(\frac{X_{K}^{T}X_{K}}{n}\right)^{-1}$ is diagonal. In Step 2 of the PDW procedure, $\hat{\delta}_{K}$ is chosen such that $\left|\hat{\delta}_{j}\right|=1$ for any $j\in K$ with $\hat{\theta}_{j}\neq0$; we therefore obtain $$\left|\hat{\theta}_{j}-\theta_{j}^{*}\right|\geq\frac{n}{s}\left|\left|\lambda\right|-\left|\frac{X_{j}^{T}\varepsilon}{n}\right|\right|\geq\frac{\lambda n}{2s}$$ where the second inequality follows from (\[eq:lambda\]). ### Main proof {#main-proof .unnumbered} In what follows, we let $$\begin{aligned} E_{1} & = & \left\{ \textrm{sgn}\left(\hat{\theta}_{j}\right)=-\textrm{sgn}\left(\theta_{j}^{*}\right),\,\textrm{for some }j\in K\right\} ,\\ E_{2} & = & \left\{ \textrm{sgn}\left(\hat{\theta}_{j}\right)=\textrm{sgn}\left(\theta_{j}^{*}\right),\,\textrm{for some }j\in K\textrm{ such that (\ref{eq:min}) holds}\right\} ,\\ E_{3} & = & \left\{ \textrm{sgn}\left(\hat{\theta}_{j}\right)=\textrm{sgn}\left(\theta_{j}^{*}\right),\,\textrm{for some }j\in K\right\} .\end{aligned}$$ To show (\[eq:nec\]) in (iv), recall we have established that conditioning on $\mathcal{E}$, (\[eq:las\]) has a unique optimal solution $\hat{\theta}$ such that $\hat{\theta}_{j}=0$ for $j\notin K$. Therefore, conditioning on $\mathcal{E}$, the KKT condition for (\[eq:las\]) implies $$\frac{s}{n}\left(\theta_{j}^{*}-\hat{\theta}_{j}\right)=\lambda\textrm{sgn}\left(\hat{\theta}_{j}\right)-\frac{X_{j}^{T}\varepsilon}{n}\label{eq:kkt}$$ for $j\in K$ such that $\hat{\theta}_{j}\neq0$. We first show that $\mathbb{P}\left(E_{1}\cap\mathcal{E}\right)=0$. Suppose $\mathbb{P}\left(E_{1}\cap\mathcal{E}\right)>0$. We may then condition on the event $E_{1}\cap\mathcal{E}$. Case (i): $\theta_{j}^{*}>0$ and $\hat{\theta}_{j}<0$. Then, the LHS of (\[eq:kkt\]), $\frac{s}{n}\left(\theta_{j}^{*}-\hat{\theta}_{j}\right)>0$; consequently, the RHS, $\lambda\textrm{sgn}\left(\hat{\theta}_{j}\right)-\frac{X_{j}^{T}\varepsilon}{n}=-\lambda-\frac{X_{j}^{T}\varepsilon}{n}>0$. However, given the choice of $\lambda$, conditioning on $\mathcal{E}$, $\lambda\geq2\left|\frac{X^{T}\varepsilon}{n}\right|_{\infty}$ and consequently, $-\lambda-\frac{X_{j}^{T}\varepsilon}{n}\leq-\frac{\lambda}{2}<0$. This leads to a contradiction. Case (ii): $\theta_{j}^{*}<0$ and $\hat{\theta}_{j}>0$. Then, the LHS of (\[eq:kkt\]), $\frac{s}{n}\left(\theta_{j}^{*}-\hat{\theta}_{j}\right)<0$; consequently, the RHS, $\lambda\textrm{sgn}\left(\hat{\theta}_{j}\right)-\frac{X_{j}^{T}\varepsilon}{n}=\lambda-\frac{X_{j}^{T}\varepsilon}{n}<0$. However, given the choice of $\lambda$, conditioning on $\mathcal{E}$, $\lambda\geq2\left|\frac{X^{T}\varepsilon}{n}\right|_{\infty}$ and consequently, $\lambda-\frac{X_{j}^{T}\varepsilon}{n}\geq\frac{\lambda}{2}>0$. This leads to a contradiction. It remains to show that $\mathbb{P}\left(E_{2}\cap\mathcal{E}\right)=0$. We first establish a useful fact under the assumption that $\mathbb{P}\left(E_{3}\cap\mathcal{E}\right)>0$. Let us condition on the event $E_{3}\cap\mathcal{E}$. If $\theta_{j}^{*}>0$, we have $\frac{s}{n}\left(\theta_{j}^{*}-\hat{\theta}_{j}\right)=\lambda-\frac{X_{j}^{T}\varepsilon}{n}\geq\frac{\lambda}{2}>0$ (i.e., $\theta_{j}^{*}\geq\hat{\theta}_{j}$); similarly, if $\theta_{j}^{*}<0$, then we have $\frac{s}{n}\left(\theta_{j}^{*}-\hat{\theta}_{j}\right)=-\lambda-\frac{X_{j}^{T}\varepsilon}{n}\leq-\frac{\lambda}{2}<0$ (i.e., $\theta_{j}^{*}\leq\hat{\theta}_{j}$). Putting the pieces together implies that, for $j\in K$ such that $\textrm{sgn}\left(\hat{\theta}_{j}\right)=\textrm{sgn}\left(\theta_{j}^{*}\right)$, $$\left|\theta_{j}^{*}-\hat{\theta}_{j}\right|=\left|\theta_{j}^{*}\right|-\left|\hat{\theta}_{j}\right|.\label{eq:equal}$$ We now show that $\mathbb{P}\left(E_{2}\cap\mathcal{E}\right)=0$. Suppose $\mathbb{P}\left(E_{2}\cap\mathcal{E}\right)>0$. We may then condition on the event that $E_{2}\cap\mathcal{E}$. Because of (\[eq:min\]) and (\[eq:equal\]), we have $\left|\theta_{j}^{*}-\hat{\theta}_{j}\right|<\frac{\lambda n}{2s}$. On the other hand, (\[eq:37-1\]) implies that $\left|\theta_{j}^{*}-\hat{\theta}_{j}\right|\geq\frac{\lambda n}{2s}$. We have arrived at a contradiction. Consequently, we must have $\mathbb{P}\left(E_{2}\cap\mathcal{E}\right)=0$. In summary, we have shown that $\mathbb{P}\left(E_{1}\cap\mathcal{E}\right)=0$ and $\mathbb{P}\left(E_{2}\cap\mathcal{E}\right)=0$. Claim (i) in “Preliminary” implies that $\mathbb{P}\left(E_{4}\vert\mathcal{E}\right)=0$ where $E_{4}$ denotes the event that $\hat{\theta}_{j}\neq0$ for some $j\notin K$. Therefore, on $\mathcal{E}$, none of the events $E_{1}$, $E_{2}$ and $E_{4}$ can happen. This fact implies that, if (\[eq:min\]) is satisfied for all $l\in K$, we must have $$\mathbb{P}\left(\hat{\theta}=0_{p}\right)\geq1-\mathbb{P}\left(\mathcal{E}^{c}\right)\geq1-\frac{1}{p^{\tau}}.$$ Proposition \[prop:bias\_post\_double\_formula\] {#sec:appendix_post_double_lasso} ------------------------------------------------ We first show the case where $ab>0$. Let the events $$\begin{aligned} \mathcal{E}_{t_{1}} & = & \left\{ \left|\frac{X_{K}^{T}v}{n}\right|_{\infty}\leq t_{1},\,t_{1}>0\right\} ,\label{eq:50}\\ \mathcal{E}_{t_{2}}^{'} & = & \left\{ \frac{1}{n}\sum_{i=1}^{n}v_{i}^{2}\leq\sigma_{v}^{2}+t_{2},\:t_{2}\in(0,\,\sigma_{v}^{2}]\right\} .\nonumber \end{aligned}$$ By tail bounds for Gaussian and Chi-Square variables, we have $$\begin{aligned} \mathbb{P}\left(\mathcal{E}_{t_{1}}\right) & \geq & 1-k\exp\left(\frac{-nt_{1}^{2}}{2\frac{s}{n}\sigma_{v}^{2}}\right),\label{eq:32}\\ \mathbb{P}\left(\mathcal{E}_{t_{2}}^{'}\right) & \geq & 1-\exp\left(\frac{-nt_{2}^{2}}{8\sigma_{v}^{4}}\right).\nonumber \end{aligned}$$ In the following proof, we exploit the bound $$\begin{aligned} \mathbb{P}\left(\mathcal{E}_{t_{2}}^{'}\vert\hat{I}_{2}=\emptyset,\,\mathcal{E}_{t_{1}}\right) & \geq & \mathbb{P}\left(\mathcal{E}_{t_{2}}^{'}\cap\mathcal{E}_{t_{1}}\cap\left\{ \hat{I}_{2}=\emptyset\right\} \right)\nonumber \\ & \geq & \mathbb{P}\left(\mathcal{E}_{t_{1}}\right)+\mathbb{P}\left(\mathcal{E}_{t_{2}}^{'}\right)+\mathbb{P}\left(\hat{I}_{2}=\emptyset\right)-2\nonumber \\ & \geq & 1-\frac{1}{p^{\tau}}-k\exp\left(\frac{-nt_{1}^{2}}{2\frac{s}{n}\sigma_{v}^{2}}\right)-\exp\left(\frac{-nt_{2}^{2}}{8\sigma_{v}^{4}}\right)\label{eq:58}\end{aligned}$$ where the third inequality follows from Lemma \[prop:fixed\_design\], which implies $\hat{I}_{2}=\emptyset$ with probability at least $1-\frac{1}{p^{\tau}}$. Note that $\mathbb{P}\left(\mathcal{E}_{t_{1}}\cap\left\{ \hat{I}_{2}=\emptyset\right\} \right)\geq\mathbb{P}\left(\mathcal{E}_{t_{1}}\right)+\mathbb{P}\left(\hat{I}_{2}=\emptyset\right)-1\geq1-\frac{1}{p^{\tau}}-k\exp\left(\frac{-nt_{1}^{2}}{2\frac{s}{n}\sigma_{v}^{2}}\right)$, which is a “high probability” guarantee for sufficiently large $p$ and $t_{1}$. Thus, working with $\mathbb{P}\left(\mathcal{E}_{t_{2}}^{'}\vert\hat{I}_{2}=\emptyset,\,\mathcal{E}_{t_{1}}\right)$ is sensible under an appropriate choice of $t_{1}$ (as we will see below). We first bound $\frac{\frac{1}{n}D^{T}X_{K}}{\frac{1}{n}D^{T}D}\beta_{K}^{*}$. Note that $$\begin{aligned} \frac{\frac{1}{n}D^{T}X_{K}}{\frac{1}{n}D^{T}D}\beta_{K}^{*} & = & \left(\frac{D^{T}D}{n}\right)^{-1}\left[\frac{1}{n}\left(X_{K}\gamma_{K}^{*}+v\right)^{T}X_{K}\beta_{K}^{*}\right]\\ & = & \left(\frac{D^{T}D}{n}\right)^{-1}\left[\frac{1}{n}\gamma_{K}^{*T}X_{K}^{T}X_{K}\beta_{K}^{*}+\frac{1}{n}v^{T}X_{K}\beta_{K}^{*}\right]\\ & = & \frac{\frac{s}{n}\gamma_{K}^{*T}\beta_{K}^{*}+\frac{1}{n}v^{T}X_{K}\beta_{K}^{*}}{\frac{1}{n}\left(X_{K}\gamma_{K}^{*}+v\right)^{T}\left(X_{K}\gamma_{K}^{*}+v\right)},\end{aligned}$$ and that $\frac{s}{n}\gamma_{K}^{*T}\beta_{K}^{*}=2\left(1+\tau\right)ab\phi^{-2}\sigma_{\eta}\sigma_{v}\frac{k\log p}{n}$. Moreover, applying (\[eq:58\]) with $t_{1}=\left|b\right|\frac{\lambda_{2}}{4}$ and $t_{2}=r\sigma_{v}^{2}$ (where $r\in(0,\,1]$) yields $$\begin{aligned} \frac{s}{n}\gamma_{K}^{*T}\beta_{K}^{*}+\frac{1}{n}v^{T}X_{K}\beta_{K}^{*} & \geq & \frac{s}{n}\gamma_{K}^{*T}\beta_{K}^{*}-\left|\frac{1}{n}v^{T}X_{K}\right|_{\infty}\left|\beta_{K}^{*}\right|_{1}\nonumber \\ & \geq & \left(1+\tau\right)ab\phi^{-2}\sigma_{\eta}\sigma_{v}\frac{k\log p}{n}\label{eq:pos}\end{aligned}$$ as well as $$\begin{aligned} \frac{1}{n}\left(X_{K}\gamma_{K}^{*}+v\right)^{T}\left(X_{K}\gamma_{K}^{*}+v\right) & \leq & \frac{s}{n}\gamma_{K}^{*T}\gamma_{K}^{*}+\left|\frac{2}{n}v^{T}X_{K}\right|_{\infty}\left|\gamma_{K}^{*}\right|_{1}+\frac{1}{n}v^{T}v\nonumber \\ & \leq & 4\left(1+\tau\right)\phi^{-2}b^{2}\sigma_{v}^{2}\frac{k\log p}{n}+\sigma_{v}^{2}+r\sigma_{v}^{2}\label{eq:den}\end{aligned}$$ with probability at least $$1-k\exp\left(\frac{-b^{2}\left(1+\tau\right)\log p}{4\phi^{2}}\right)-\frac{1}{p^{\tau}}-\exp\left(\frac{-nr^{2}}{8}\right):=T_{2}\left(r\right).$$ Conditioning on $\mathcal{E}_{t_{1}}\cap\left\{ \hat{I}_{2}=\emptyset\right\} $ with $t_{1}=t^{*}=\left|b\right|\frac{\lambda_{2}}{4}$, putting the pieces together yields $$\frac{D^{T}X_{K}}{D^{T}D}\beta_{K}^{*}\geq\frac{\left(1+\tau\right)ab\phi^{-2}\sigma_{\eta}\frac{k\log p}{n}}{4\left(1+\tau\right)\phi^{-2}b^{2}\sigma_{v}\frac{k\log p}{n}+\sigma_{v}+r\sigma_{v}}:=T_{1}\left(r\right),\label{eq:47}$$ with probability at least $T_{2}\left(r\right)$. That is, $$\mathbb{P}\left(\frac{D^{T}X_{K}}{D^{T}D}\beta_{K}^{*}\geq T_{1}\left(r\right)\vert\hat{I}_{2}=\emptyset,\,\mathcal{E}_{t^{*}}\right)\geq T_{2}\left(r\right).$$ When $\alpha^{*}=0$ in (\[eq:20\]), the reduced form coefficients $\pi^{*}$ in (\[eq:reduce\]) coincide with $\beta^{*}$ and $u$ coincides with $\eta$. Given the conditions on $X,$ $\eta$, $v$, $\beta_{K}^{*}$ and $\gamma_{K}^{*}$, we can then apply (\[eq:nec\]) in Lemma \[prop:fixed\_design\] and the fact $\mathbb{P}\left(\hat{I}_{1}=\hat{I}_{2}=\emptyset\right)\geq\mathbb{P}\left(\hat{I}_{1}=\emptyset\right)+\mathbb{P}\left(\hat{I}_{2}=\emptyset\right)-1$ to show that $E=\left\{ \hat{I}_{1}=\hat{I}_{2}=\emptyset\right\} $ occurs with probability at least $1-\frac{2}{p^{\tau}}$. Note that with the choice $t_{1}=t^{*}=\left|b\right|\frac{\lambda_{2}}{4}$, $\mathbb{P}\left(E\cap\mathcal{E}_{t^{*}}\right)\geq\mathbb{P}\left(E\right)+\mathbb{P}\left(\mathcal{E}_{t^{*}}\right)-1\geq1-k\exp\left(\frac{-b^{2}\left(1+\tau\right)\log p}{4\phi^{2}}\right)-\frac{2}{p^{\tau}}$, which is a “high probability” guarantee given sufficiently large $p$.[^26] Therefore, it is sensible to work with $\mathbb{E}\left(\tilde{\alpha}-\alpha^{*}\vert\mathcal{M}\right)$ where $$\mathcal{M}=E\cap\mathcal{E}_{t^{*}}.\label{eq:M_event}$$ Given $E$, (\[eq:double\]) becomes $$\tilde{\alpha}\in\textrm{arg}\min_{\alpha\in\mathbb{R}}\frac{1}{2n}\left|Y-D\alpha\right|_{2}^{2},\qquad\textrm{while }\tilde{\beta}=0_{p}.\label{eq:double-1}$$ As a result, we obtain $\mathbb{E}\left(\tilde{\alpha}-\alpha^{*}\vert\mathcal{M}\right)=\mathbb{E}\left(\frac{\frac{1}{n}D^{T}X_{K}}{\frac{1}{n}D^{T}D}\beta_{K}^{*}\vert\mathcal{M}\right)+\mathbb{E}\left(\frac{\frac{1}{n}D^{T}\eta}{\frac{1}{n}D^{T}D}\vert\mathcal{M}\right)$ and $$\begin{aligned} \mathbb{E}\left(\frac{\frac{1}{n}D^{T}\eta}{\frac{1}{n}D^{T}D}\vert\mathcal{M}\right) & = & \frac{1}{\mathbb{P}\left(\mathcal{M}\right)}\mathbb{E}\left[\frac{\frac{1}{n}D^{T}\eta}{\frac{1}{n}D^{T}D}1_{\mathcal{M}}\left(D,\eta\right)\right]\nonumber \\ & = & \frac{1}{\mathbb{P}\left(\mathcal{M}\right)}\mathbb{E}_{D}\left\{ \mathbb{E}_{\eta}\left[\frac{\frac{1}{n}D^{T}\eta}{\frac{1}{n}D^{T}D}1_{\mathcal{M}}\left(D,\eta\right)\vert D\right]\right\} \nonumber \\ & = & \frac{1}{\mathbb{P}\left(\mathcal{M}\right)}\mathbb{E}_{D}\left\{ \frac{\frac{1}{n}\sum_{i=1}^{n}D_{i}\mathbb{E}_{\eta}\left[\eta_{i}1_{\mathcal{M}}\left(D,\eta\right)\vert D\right]}{\frac{1}{n}D^{T}D}\right\} \nonumber \\ & = & 0\label{eq:60}\end{aligned}$$ where $1_{\mathcal{M}}\left(D,\eta\right)=1\left\{ \left(v,\eta\right):\,\hat{I}_{1}=\hat{I}_{2}=\emptyset,\,\left|\frac{X_{K}^{T}v}{n}\right|_{\infty}\leq t^{*}\right\} $ (recall $X$ is a fixed design); the last line follows from $\frac{1}{n}\sum_{i=1}^{n}D_{i}=0$, the distributional identicalness of $\left(\eta_{i}\right)_{i=1}^{n}$ and that $\mathbb{E}_{\eta}\left[\eta_{i}1_{\mathcal{M}}\left(D,\eta\right)\vert D\right]$ is a constant over $i$s. It remains to bound $\mathbb{E}\left(\frac{\frac{1}{n}D^{T}X_{K}}{\frac{1}{n}D^{T}D}\beta_{K}^{*}\vert\mathcal{M}\right)=\mathbb{E}\left(\frac{\frac{1}{n}D^{T}X_{K}}{\frac{1}{n}D^{T}D}\beta_{K}^{*}\vert\hat{I}_{2}=\emptyset,\,\mathcal{E}_{t^{*}}\right)$. Note that conditioning on $\mathcal{E}_{t^{*}}$, $\frac{\frac{1}{n}D^{T}X_{K}}{\frac{1}{n}D^{T}D}\beta_{K}^{*}$ is positive by (\[eq:pos\]). Applying a Markov inequality yields $$\mathbb{E}\left(\frac{\frac{1}{n}D^{T}X_{K}}{\frac{1}{n}D^{T}D}\beta_{K}^{*}\vert\hat{I}_{2}=\emptyset,\,\mathcal{E}_{t^{*}}\right)\geq T_{1}\left(r\right)\mathbb{P}\left(\frac{D^{T}X_{K}}{D^{T}D}\beta_{K}^{*}\geq T_{1}\left(r\right)\vert\hat{I}_{2}=\emptyset,\,\mathcal{E}_{t^{*}}\right)\geq T_{1}\left(r\right)T_{2}\left(r\right).$$ Combining the result above with (\[eq:60\]) and maximizing over $r\in(0,\,1]$ gives the claim. We now show the case where $ab<0$. The argument is almost similar. In particular, we use $$\begin{aligned} \frac{s}{n}\gamma_{K}^{*T}\beta_{K}^{*}+\frac{1}{n}v^{T}X_{K}\beta_{K}^{*} & \leq & \frac{s}{n}\gamma_{K}^{*T}\beta_{K}^{*}+\left|\frac{1}{n}v^{T}X_{K}\right|_{\infty}\left|\beta_{K}^{*}\right|_{1}\\ & \leq & -\left(1+\tau\right)\left|ab\right|\phi^{-2}\sigma_{\eta}\sigma_{v}\frac{k\log p}{n}<0\end{aligned}$$ and replace (\[eq:pos\]) with $$-\frac{s}{n}\gamma_{K}^{*T}\beta_{K}^{*}-\frac{1}{n}v^{T}X_{K}\beta_{K}^{*}\geq\left(1+\tau\right)\left|ab\right|\phi^{-2}\sigma_{\eta}\sigma_{v}\frac{k\log p}{n}.$$ Note that $$\mathbb{E}\left(\alpha^{*}-\tilde{\alpha}\vert\mathcal{M}\right)=\mathbb{E}\left(\frac{-\frac{1}{n}D^{T}X_{K}}{\frac{1}{n}D^{T}D}\beta_{K}^{*}\vert\mathcal{M}\right)-\mathbb{E}\left(\frac{\frac{1}{n}D^{T}\eta}{\frac{1}{n}D^{T}D}\vert\mathcal{M}\right).$$ So the rest of the proof follows from the argument for the case where $ab>0$. Proposition \[prop:bias\_post\_double\_selection\] {#appendix:bias_post_double_lower_scaling} -------------------------------------------------- Part (i) of Proposition \[prop:bias\_post\_double\_selection\] follows immediately from the proof for Proposition \[prop:bias\_post\_double\_formula\]. It remains to establish part (ii) where $\alpha^{*}\neq0$, $\alpha^{*}\gamma_{j}^{*}\in(0,\,-\beta_{j}^{*}]$, $\beta_{j}^{*}<0$ for all $j\in K$ (or, $\alpha^{*}\gamma_{j}^{*}\in[-\beta_{j}^{*},\,0)$, $\beta_{j}^{*}>0$ for all $j\in K$). Because of these conditions, we have $$\left|\pi_{j}^{*}\right|=\left|\beta_{j}^{*}+\alpha^{*}\gamma_{j}^{*}\right|<\left|\beta_{j}^{*}\right|\quad\forall j\in K.$$ Note that $\left|\alpha^{*}\right|\leq\max_{j\in K}\frac{\left|\beta_{j}^{\ast}\right|}{\left|\gamma_{j}^{\ast}\right|}\asymp\frac{\sigma_{\eta}}{\sigma_{v}}$ and $$\begin{aligned} \left|\frac{X^{T}u}{n}\right|_{\infty} & = & \left|\frac{X^{T}\left(\eta+\alpha^{*}v\right)}{n}\right|_{\infty}\\ & \leq & \left|\frac{X^{T}\eta}{n}\right|_{\infty}+\left|\frac{\alpha^{*}X^{T}v}{n}\right|_{\infty}\\ & \precsim & \sigma_{\eta}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}+\frac{\sigma_{\eta}}{\sigma_{v}}\sigma_{v}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}\\ & \precsim & \phi^{-1}\sigma_{\eta}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}\end{aligned}$$ with probability at least $1-c_{1}^{'}\exp\left(-c_{2}^{'}\log p\right)$. The fact above justifies the choice of $\lambda_{1}$ stated in \[prop:bias\_post\_double\_selection\]. We can then apply (\[eq:nec\]) in Lemma \[prop:fixed\_design\] to show that $\hat{I}_{1}=\emptyset$ with probability at least $1-c_{5}\exp\left(-c_{6}\log p\right)$. Furthermore, under the conditions on $X$ and $\gamma_{K}^{*}$, (\[eq:nec\]) in Lemma \[prop:fixed\_design\] implies that $\hat{I}_{2}=\emptyset$ with probability at least $1-c_{0}\exp\left(-c_{0}^{'}\log p\right)$. Therefore, we have $$\mathbb{P}\left(\hat{I}_{1}=\hat{I}_{2}=\emptyset\right)\geq\mathbb{P}\left(\hat{I}_{1}=\emptyset\right)+\mathbb{P}\left(\hat{I}_{2}=\emptyset\right)-1\geq1-c_{1}^{''}\exp\left(-c_{2}^{''}\log p\right).$$ Given $u=\eta+\alpha^{*}v$, when $\alpha^{*}\neq0$, the event $\left\{ \hat{I}_{1}=\emptyset\right\} $ is not independent of $D$, so $\mathbb{E}\left(\frac{\frac{1}{n}D^{T}X_{K}}{\frac{1}{n}D^{T}D}\beta_{K}^{*}\vert E,\,\mathcal{E}_{t^{*}}\right)\neq\mathbb{E}\left(\frac{\frac{1}{n}D^{T}X_{K}}{\frac{1}{n}D^{T}D}\beta_{K}^{*}\vert\hat{I}_{2}=\emptyset,\,\mathcal{E}_{t^{*}}\right)$ (recalling $E=\left\{ \hat{I}_{1}=\hat{I}_{2}=\emptyset\right\} $). Instead of (\[eq:58\]), we apply $$\begin{aligned} \mathbb{P}\left(\mathcal{E}_{t_{2}}^{'}\vert E,\,\mathcal{E}_{t_{1}}\right) & \geq & \mathbb{P}\left(\mathcal{E}_{t_{2}}^{'}\cap\mathcal{E}_{t_{1}}\cap E\right)\\ & \geq & \mathbb{P}\left(\mathcal{E}_{t_{1}}\right)+\mathbb{P}\left(\mathcal{E}_{t_{2}}^{'}\right)+\mathbb{P}\left(E\right)-2\\ & \geq & 1-c_{1}^{''}\exp\left(-c_{2}^{''}\log p\right)-k\exp\left(\frac{-nt_{1}^{2}}{2\frac{s}{n}\sigma_{v}^{2}}\right)\\ & & -\exp\left(\frac{-nt_{2}^{2}}{8\sigma_{v}^{4}}\right),\qquad\textrm{for any }t_{2}\in(0,\,\sigma_{v}^{2}].\end{aligned}$$ The rest of the proof follows from the argument for Proposition \[prop:bias\_post\_double\_formula\] and the bounds above. Proposition \[prop:bias\_post\_double\_selection-1\] ---------------------------------------------------- Note that we have $$\begin{aligned} \left|\frac{X^{T}u}{n}\right|_{\infty} & = & \left|\frac{X^{T}\left(\eta+\alpha^{*}v\right)}{n}\right|_{\infty}\\ & \leq & \left|\frac{X^{T}\eta}{n}\right|_{\infty}+\left|\frac{\alpha^{*}X^{T}v}{n}\right|_{\infty}\\ & \precsim & \sigma_{\eta}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}+\left|\alpha^{*}\right|\sigma_{v}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}},\end{aligned}$$ which justifies the choice of $\lambda_{1}$ stated in Proposition \[prop:bias\_post\_double\_selection-1\]. For part (i), recall that $\pi_{K}^{*}=0_{k}$. By (i) of the intermediate results in “Preliminary” of Section \[appendix a1\], $\mathbb{P}\left(\hat{\pi}=0_{p}\right)=\mathbb{P}\left(\hat{I}_{1}=\emptyset\right)\geq1-c\exp\left(-c^{'}\log p\right)$. Substituting $\beta_{K}^{*}=-\alpha^{*}\gamma_{K}^{*}$ in $\frac{\frac{1}{n}D^{T}X_{K}}{\frac{1}{n}D^{T}D}\beta_{K}^{*}$ and following the rest of proof for Proposition \[prop:bias\_post\_double\_selection\] yields the claim. For part (ii), for all $j\in K$, note that $\beta_{j}^{*}>-\alpha^{*}\gamma_{j}^{*}>0$ in case (1) and $\beta_{j}^{*}<-\alpha^{*}\gamma_{j}^{*}<0$ in case (2). Under the conditions on $X$ and $\pi_{K}^{*}$, (\[eq:nec\]) in Lemma \[prop:fixed\_design\] implies that $\mathbb{P}\left(\hat{\pi}=0_{p}\right)=\mathbb{P}\left(\hat{I}_{1}=\emptyset\right)\geq1-c\exp\left(-c^{'}\log p\right)$. Substituting $\beta_{K}^{*}>-\alpha^{*}\gamma_{K}^{*}>0$ for case (1) and $\beta_{j}^{*}<-\alpha^{*}\gamma_{j}^{*}<0$ for case (2) in the derivation of the bounds for $\frac{1}{n}D^{T}X_{K}\beta_{K}^{*}$ and following the rest of proof for Proposition \[prop:bias\_post\_double\_selection\] yields the claim. Proposition \[prop:bias\_post\_double\_selection-upper\] \[appendix:Proposition-OVB-upper-proof\] ------------------------------------------------------------------------------------------------- Given $\left\{ \hat{I}_{1}=\hat{I}_{2}=\emptyset\right\} $, note that $\left|\tilde{\alpha}-\alpha^{*}\right|\leq\left|\frac{\frac{1}{n}D^{T}X_{K}}{\frac{1}{n}D^{T}D}\beta_{K}^{*}\right|+\left|\frac{\frac{1}{n}D^{T}\eta}{\frac{1}{n}D^{T}D}\right|$. We make use of the following bound on Chi-Square variables: $$\mathbb{P}\left[\frac{1}{n}\sum_{i=1}^{n}v_{i}^{2}-\mathbb{E}\left(\frac{1}{n}\sum_{i=1}^{n}v_{i}^{2}\right)\leq-\sigma_{v}^{2}r^{'}\right]\leq\exp\left(\frac{-nr^{'2}}{16}\right)\label{eq:41}$$ for all $r^{'}\geq0$. On the event $\left\{ \hat{I}_{1}=\hat{I}_{2}=\emptyset\right\} $, choosing $t_{1}=t^{*}=\frac{\left|\gamma^{*}\right|s}{4n}$ in (\[eq:50\]) and $r^{'}=\frac{1}{2}$ in (\[eq:41\]) yields $$\begin{aligned} \left|\frac{\frac{1}{n}D^{T}X_{K}}{\frac{1}{n}D^{T}D}\beta_{K}^{*}\right| & \leq & \frac{\frac{s}{n}\left|\gamma_{K}^{*T}\beta_{K}^{*}\right|+\left|\beta_{K}^{*}\right|_{1}t^{*}}{\frac{s}{n}\gamma_{K}^{*T}\gamma_{K}^{*}-2\left|\gamma_{K}^{*}\right|_{1}t^{*}+\frac{1}{2}\sigma_{v}^{2}}\\ & \leq & \frac{c_{3}\sigma_{\eta}\sigma_{v}\frac{k\log p}{n}}{c_{1}\frac{k\log p}{n}\sigma_{v}^{2}+c_{2}\sigma_{v}^{2}}\\ & \leq & c_{4}\frac{\sigma_{\eta}}{\sigma_{v}}\left(\frac{k\log p}{n}\wedge1\right)\end{aligned}$$ with probability at least $1-c_{5}k\exp\left(-c_{6}\log p\right)-\exp\left(\frac{-n}{64}\right)$. We can also show that $$\begin{aligned} & & \mathbb{P}\left(\left|\frac{1}{n}D^{T}\eta\right|\leq t\vert\hat{I}_{1}=\hat{I}_{2}=\emptyset\right)\\ & \geq & \mathbb{P}\left(\left\{ \left(\eta,v\right):\,\left|\frac{1}{n}D^{T}\eta\right|\leq t\right\} \cap\left\{ \hat{I}_{1}=\hat{I}_{2}=\emptyset\right\} \right)\\ & \geq & \mathbb{P}\left(\left|\frac{1}{n}D^{T}\eta\right|\leq t\right)+\mathbb{P}\left(\hat{I}_{1}=\hat{I}_{2}=\emptyset\right)-1\\ & \geq & 1-c_{5}\exp\left(-c_{6}\log p\right)-\mathbb{P}\left(\left|\frac{1}{n}D^{T}\eta\right|>t\right).\end{aligned}$$ Note that $\left|\frac{1}{n}D^{T}\eta\right|\leq k\left|\gamma^{*}\right|\left|\frac{1}{n}X^{T}\eta\right|_{\infty}+\left|\frac{1}{n}v^{T}\eta\right|$ where $$\begin{aligned} \mathbb{P}\left(\left|\frac{1}{n}v^{T}\eta\right|\precsim\sigma_{v}\sigma_{\eta}r\right) & \geq & 1-2\exp\left(-c_{7}nr^{2}\right)\quad\textrm{for any }r\in(0,\,1],\\ \mathbb{P}\left(\left|\frac{1}{n}X^{T}\eta\right|_{\infty}\precsim\sigma_{\eta}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}\right) & \geq & 1-2\exp\left(-c_{8}\log p\right).\end{aligned}$$ The inequalities above yield $$\begin{aligned} & \mathbb{P}\left\{ \left|\frac{1}{n}D^{T}\eta\right|\precsim\left[\left(\sigma_{v}\sigma_{\eta}r\right)\vee\underset{\asymp\sigma_{v}\sigma_{\eta}\frac{k\log p}{n}}{\underbrace{\left(k\left|\gamma^{*}\right|\sigma_{\eta}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}\right)}}\right]\vert\hat{I}_{1}=\hat{I}_{2}=\emptyset\right\} \\ \geq & 1-c_{5}\exp\left(-c_{6}\log p\right)-2\exp\left(-c_{7}nr^{2}\right)-2\exp\left(-c_{8}\log p\right).\end{aligned}$$ We have already shown that, conditioning on $\left\{ \hat{I}_{1}=\hat{I}_{2}=\emptyset\right\} $, $\frac{1}{n}D^{T}D\succsim\left(\frac{k\log p}{n}\vee1\right)\sigma_{v}^{2}$ with probability at least $1-c_{5}k\exp\left(-c_{6}\log p\right)-\exp\left(\frac{-n}{64}\right)$. As a consequence, $$\begin{aligned} & & \mathbb{P}\left\{ \left|\frac{\frac{1}{n}D^{T}\eta}{\frac{1}{n}D^{T}D}\right|\precsim\frac{\sigma_{v}\sigma_{\eta}\left(r\vee\frac{k\log p}{n}\right)}{\left(\frac{k\log p}{n}\vee1\right)\sigma_{v}^{2}}\vert\hat{I}_{1}=\hat{I}_{2}=\emptyset\right\} \\ & \geq & \mathbb{P}\left\{ \left|\frac{1}{n}D^{T}\eta\right|\precsim\sigma_{v}\sigma_{\eta}\left(r\vee\frac{k\log p}{n}\right)\,\,and\,\,\frac{1}{n}D^{T}D\succsim\left(\frac{k\log p}{n}\vee1\right)\sigma_{v}^{2}\vert\hat{I}_{1}=\hat{I}_{2}=\emptyset\right\} \\ & \geq & \mathbb{P}\left\{ \left|\frac{1}{n}D^{T}\eta\right|\precsim\sigma_{v}\sigma_{\eta}\left(r\vee\frac{k\log p}{n}\right)\vert\hat{I}_{1}=\hat{I}_{2}=\emptyset\right\} \\ & & +\mathbb{P}\left\{ \frac{1}{n}D^{T}D\succsim\left(\frac{k\log p}{n}\vee1\right)\sigma_{v}^{2}\vert\hat{I}_{1}=\hat{I}_{2}=\emptyset\right\} -1\\ & \geq & 1-c_{9}k\exp\left(-c_{10}\log p\right)-c_{12}\exp\left(-c_{11}nr^{2}\right).\end{aligned}$$ Putting the pieces above together yields $$\mathbb{P}\left(\left|\tilde{\alpha}-\alpha^{*}\right|\leq\overline{OVB}\vert\hat{I}_{1}=\hat{I}_{2}=\emptyset\right)\geq1-c_{1}^{'}k\exp\left(-c_{2}^{'}\log p\right)-c_{4}^{'}\exp\left(-c_{3}^{'}nr^{2}\right)$$ where $\overline{OVB}\asymp\max\left\{ \frac{\sigma_{\eta}}{\sigma_{v}}\left(\frac{k\log p}{n}\wedge1\right),\,\frac{\sigma_{v}\sigma_{\eta}\left(r\vee\frac{k\log p}{n}\right)}{\left(\frac{k\log p}{n}\vee1\right)\sigma_{v}^{2}}\right\} $. Proposition \[prop:bias\_post\_double\_selection-upper-1\] {#appendix:Proposition-OVB-upper-proof-1} ----------------------------------------------------------- For part (i), substituting $\beta_{K}^{*}=-\alpha^{*}\gamma_{K}^{*}$ in $\frac{\frac{1}{n}D^{T}X_{K}}{\frac{1}{n}D^{T}D}\beta_{K}^{*}$ and following the rest of proof for Proposition \[prop:bias\_post\_double\_selection-upper\] yields the claim. For part (ii), note that $\left|\beta_{j}^{*}\right|\leq\left|\pi_{j}^{*}\right|+\left|\alpha^{*}\gamma_{j}^{*}\right|\precsim\left(\sigma_{\eta}+\left|\alpha^{*}\right|\sigma_{v}\right)\sqrt{\frac{n}{s}}\sqrt{\frac{\log p}{n}}$ for all $j\in K$. Substituting $\left|\beta_{j}^{*}\right|\precsim\left(\sigma_{\eta}+\left|\alpha^{*}\right|\sigma_{v}\right)\sqrt{\frac{n}{s}}\sqrt{\frac{\log p}{n}}$ in the derivation of the upper bound for $\left|\frac{\frac{1}{n}D^{T}X_{K}}{\frac{1}{n}D^{T}D}\beta_{K}^{*}\right|$ and following the rest of proof for Proposition \[prop:bias\_post\_double\_selection-upper\] yields the claim. Debiased Lasso =============== In this section, we present theoretical and simulation results on the OVB of the debiased Lasso proposed by @vandergeer2014asymptotically. Theoretical results {#sec:bias_debiased_lasso} ------------------- The idea of debiased Lasso is to start with an initial Lasso estimate $\hat{\theta}=\left(\hat{\alpha},\,\hat{\beta}\right)$ of $\theta^{*}=\left(\alpha^{*},\,\beta^{*}\right)$ in equation (\[eq:main-y\]), where $$\left(\hat{\alpha},\,\hat{\beta}\right)\in\textrm{arg}\min_{\alpha\in\mathbb{R},\beta\in\mathbb{R}^{p}}\frac{1}{2n}\left|Y-D\alpha-X\beta\right|_{2}^{2}+\lambda_{1}\left(\left|\alpha\right|+\left|\beta\right|_{1}\right).\label{eq:21-2}$$ Given the initial Lasso estimator $\hat{\alpha}$, the debiased Lasso adds a correction term to $\hat{\alpha}$ to reduce the bias introduced by regularization. In particular, the debiased Lasso takes the form $$\tilde{\alpha}=\hat{\alpha}+\frac{\hat{\Omega}_{1}}{n}\sum_{i=1}^{n}Z_{i}^{T}\left(Y_{i}-Z_{i}\hat{\theta}\right),\label{eq:5-3}$$ where $Z_{i}=\left(D_{i},\,X_{i}\right)$ and $\hat{\Omega}_{1}$ is the first row of $\hat{\Omega}$, which is an approximate inverse of $\frac{1}{n}Z^{T}Z$, $Z=\left\{ Z_{i}\right\} _{i=1}^{n}$. Several different strategies have been proposed for constructing the approximate inverse $\hat{\Omega}$; see, for example, @javanmard2014confidence, @vandergeer2014asymptotically, and @Zhang_Zhang. We will focus on the widely used method proposed by @vandergeer2014asymptotically, which sets $$\begin{aligned} \hat{\Omega}_{1} & := & \hat{\tau}_{1}^{-2}\left(\begin{array}{cccc} 1 & -\hat{\gamma}_{1} & \cdots & -\hat{\gamma}_{p}\end{array}\right),\\ \hat{\tau}_{1}^{2} & := & \frac{1}{n}\left|D-X\hat{\gamma}\right|_{2}^{2}+\lambda_{2}\left|\hat{\gamma}\right|_{1},\end{aligned}$$ where $\hat{\gamma}$ is defined in (\[eq:las-2\]). \[prop:bias\_debiased\_lasso\] Let Assumption \[ass:inference\_normality\] and part (i) and (iii) of Assumption \[ass:inference\_sparsity\_incoherence\] hold. Suppose: with probability at least $1-\kappa$, $\left\Vert \left(Z_{-K}^{T}X_{K}\right)\left(X_{K}^{T}X_{K}\right)^{-1}\right\Vert _{\infty}\leq1-\frac{\phi}{2}$ for some $\phi\in(0,\,1]$ such that $\phi^{-1}\precsim1$, where $Z_{-K}$ denotes the columns in $Z=\left(D,\,X\right)$ excluding $X_{K}$; the regularization parameters in (\[eq:las-2\]) and (\[eq:21-2\]) are chosen in a similar fashion as in Lemma \[prop:fixed\_design\] such that $\lambda_{1}\asymp\phi^{-1}\left(\sqrt{\frac{s}{n}}\vee\sigma_{v}\right)\sigma_{\eta}\sqrt{\frac{\log p}{n}}$ and $\lambda_{2}\asymp\phi^{-1}\sigma_{v}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}$; for all $j\in K$, $\left|\beta_{j}^{*}\right|\leq\frac{\lambda_{1}n}{2s}$ and $\left|\gamma_{j}^{*}\right|\leq\frac{\lambda_{2}n}{2s}$, but $\left|\beta_{j}^{*}\right|\asymp\left[\sqrt{\frac{n}{s}}\vee\frac{n\sigma_{v}}{s}\right]\sigma_{\eta}\sqrt{\frac{\log p}{n}}$ and $\left|\gamma_{j}^{*}\right|\asymp\sigma_{v}\sqrt{\frac{n}{s}}\sqrt{\frac{\log p}{n}}$. Let us consider $\tilde{\alpha}$ obtained from (\[eq:5-3\]). If $\alpha^{*}=0$, then there exist positive universal constants $c^{\dagger},c_{1},c_{2},c_{3},c^{*},c_{0}^{*}$ such that $$\left|\mathbb{E}\left(\tilde{\alpha}-\alpha^{*}\vert\mathcal{M}^{'}\right)\right|\geq c^{\dagger}\frac{\sigma_{\eta}}{\sigma_{v}}\left(\frac{k\log p}{n}\wedge1\right)\left[1-2\kappa-c_{1}k\exp\left(-c_{2}\log p\right)-\exp\left(-c_{3}n\right)\right],$$ where $\mathcal{M}^{'}$ is an event with $\mathbb{P}\left(\mathcal{M}^{'}\right)\geq1-2\kappa-c^{*}k\exp\left(-c_{0}^{*}\log p\right)$. \[prop:bias\_debiased\_lasso-upper\] Let Assumption \[ass:inference\_normality\] and part (i) and (iii) of Assumption \[ass:inference\_sparsity\_incoherence\] hold. Suppose: with probability at least $1-\kappa$, $\left\Vert \left(Z_{-K}^{T}X_{K}\right)\left(X_{K}^{T}X_{K}\right)^{-1}\right\Vert _{\infty}\leq1-\frac{\phi}{2}$ for some $\phi\in(0,\,1]$ such that $\phi^{-1}\precsim1$, where $Z_{-K}$ denotes the columns in $Z=\left(D,\,X\right)$ excluding $X_{K}$; the regularization parameters in (\[eq:las-2\]) and (\[eq:21-2\]) are chosen in a similar fashion as in Proposition \[prop:bias\_debiased\_lasso\] such that $\lambda_{1}\asymp\phi^{-1}\left(\sqrt{\frac{s}{n}}\vee\sigma_{v}\right)\sigma_{\eta}\sqrt{\frac{\log p}{n}}$ and $\lambda_{2}\asymp\phi^{-1}\sigma_{v}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}$; for all $j\in K$, $\gamma_{j}^{*}=\gamma^{*}$, $\left|\beta_{j}^{*}\right|\leq\frac{\lambda_{1}n}{2s}$ and $\left|\gamma^{*}\right|\leq\frac{\lambda_{2}n}{2s}$, but $\left|\beta_{j}^{*}\right|\asymp\left[\sqrt{\frac{n}{s}}\vee\frac{n\sigma_{v}}{s}\right]\sigma_{\eta}\sqrt{\frac{\log p}{n}}$ and $\left|\gamma^{*}\right|\asymp\sigma_{v}\sqrt{\frac{n}{s}}\sqrt{\frac{\log p}{n}}$. Let us consider $\tilde{\alpha}$ obtained from (\[eq:5-3\]). If $\alpha^{*}=0$, then there exist positive universal constants $c_{1},c_{2},c_{3},c_{4},c^{*},c_{0}^{*}$ such that $$\mathbb{P}\left(\left|\tilde{\alpha}-\alpha^{*}\right|\leq\overline{OVB}\vert \hat{\theta}=0_{p+1},\,\hat{\gamma}=0_{p} \right)\geq1-c_{1}k\exp\left(-c_{2}\log p\right)-c_{3}\exp\left(-c_{4}nr^{2}\right)$$ where $\mathbb{P}\left( \hat{\theta}=0_{p+1},\,\hat{\gamma}=0_{p} \right)\geq1-2\kappa-c^{*}k\exp\left(-c_{0}^{*}\log p\right)$ and $$\overline{OVB}\asymp\max\left\{ \frac{\sigma_{\eta}}{\sigma_{v}}\left(\frac{k\log p}{n}\wedge1\right),\,\frac{\sigma_{v}\sigma_{\eta}\left(r\vee\frac{k\log p}{n}\right)}{\left(\frac{k\log p}{n}\vee1\right)\sigma_{v}^{2}}\right\}$$ for any $r\in(0,\,1]$. One can show that a population version of the mutual incoherence condition, $\left\Vert \left[\mathbb{E}\left(Z_{-K}^{T}\right)X_{K}\right]\left(X_{K}^{T}X_{K}\right)^{-1}\right\Vert _{\infty}=1-\phi$, implies $\left\Vert \left(Z_{-K}^{T}X_{K}\right)\left(X_{K}^{T}X_{K}\right)^{-1}\right\Vert _{\infty}\leq1-\frac{\phi}{2}$ with high probability (that is, $\kappa$ is small and vanishes polynomially in $p$). For example, we can apply (\[eq:77\]) in Lemma \[lem:A3\] with slight notational changes. The event $\mathcal{M}^{'}$ in Proposition \[prop:bias\_debiased\_lasso\] is the intersection of $\left\{ \hat{\theta}=0_{p+1},\,\hat{\gamma}=0_{p}\right\}$ and an additional event, both of which occur with high probabilities. The additional event is needed in our analyses for technical reasons. See Appendix \[sec:appendix\_debiased\_lasso\] for details. Proof for Propositions \[prop:bias\_debiased\_lasso\] and \[prop:bias\_debiased\_lasso-upper\] {#sec:appendix_debiased_lasso} ---------------------------------------------------------------------------------------------- Under the conditions in Proposition \[prop:bias\_debiased\_lasso\], (\[eq:nec\]) in Lemma \[prop:fixed\_design\] implies that $\hat{\gamma}=0_{p}$ with probability at least $1-c_{0}\exp\left(-c_{0}^{'}\log p\right)$. Conditioning on this event, $\hat{\Omega}_{1}=\left(\frac{1}{n}D^{T}D\right)^{-1}e_{1}$ where $e_{1}=\left(\begin{array}{cccc} 1 & 0 & \cdots & 0\end{array}\right)$. If $\alpha^{*}=0$, under the conditions in Proposition \[prop:bias\_debiased\_lasso\], we show that $\hat{\theta}=0_{p+1}$ with probability at least $1-2\kappa-c_{5}\exp\left(-c_{6}\log p\right)$. To achieve this goal, we slightly modify the argument for (\[eq:nec\]) in Lemma \[prop:fixed\_design\] by replacing (\[eq:event\]) with $\mathcal{E}=\mathcal{E}_{1}\cap\mathcal{E}_{2}$, where $$\begin{aligned} \mathcal{E}_{1} & = & \left\{ \left|\frac{Z^{T}\eta}{n}\right|_{\infty}\precsim\phi^{-1}\left(\sqrt{\frac{s}{n}}\vee\sigma_{v}\right)\sigma_{\eta}\sqrt{\frac{\log p}{n}}\right\} ,\\ \mathcal{E}_{2} & = & \left\{ \left\Vert \left(Z_{-K}^{T}X_{K}\right)\left(X_{K}^{T}X_{K}\right)^{-1}\right\Vert _{\infty}\leq1-\frac{\phi}{2}\right\} ,\end{aligned}$$ and $Z_{-K}$ denotes the columns in $Z$ excluding $X_{K}$. Note that by (\[eq:A37-1\]), $\mathbb{P}\left(\mathcal{E}_{1}\right)\geq1-c_{1}^{'}\exp\left(-c_{2}^{'}\log p\right)$ and therefore, $\mathbb{P}\left(\mathcal{E}\right)\geq1-\kappa-c_{1}^{'}\exp\left(-c_{2}^{'}\log p\right)$. We then follow the argument used in the proof for Lemma \[prop:fixed\_design\] to show $\mathbb{P}\left(E_{1}\cap\mathcal{E}\right)=0$ and $\mathbb{P}\left(E_{2}\cap\mathcal{E}\right)=0$, where $$\begin{aligned} E_{1} & = & \left\{ \textrm{sgn}\left(\hat{\beta}_{j}\right)=-\textrm{sgn}\left(\beta_{j}^{*}\right),\,\textrm{for some }j\in K\right\} ,\\ E_{2} & = & \left\{ \textrm{sgn}\left(\hat{\beta}_{j}\right)=\textrm{sgn}\left(\beta_{j}^{*}\right),\,\textrm{for some }j\in K\right\} .\end{aligned}$$ Moreover, conditioning on $\mathcal{E}$, $\hat{\alpha}=0$ and $\hat{\beta}_{K^{c}}=0_{p-k}$. Putting these facts together yield the claim that $\hat{\theta}=0_{p+1}$ with probability at least $1-2\kappa-c_{5}\exp\left(-c_{6}\log p\right)$. Letting $E=\left\{ \hat{\theta}=0_{p+1},\,\hat{\gamma}=0_{p}\right\} $ with $\mathbb{P}\left(E\right)\geq1-2\kappa-c_{1}\exp\left(-c_{2}\log p\right)$ and recalling the event $\mathcal{E}_{t^{*}}$ in the proof for Proposition \[prop:bias\_post\_double\_formula\], we can then show $$\begin{aligned} \mathbb{E}\left(\tilde{\alpha}-\alpha^{*}\vert\mathcal{M}^{'}\right) & = & \frac{1}{n}\mathbb{E}\left(\hat{\Omega}_{1}Z^{T}\eta\vert\mathcal{M}^{'}\right)+\mathbb{E}\left[\frac{D^{T}X_{K}}{D^{T}D}\left(\beta_{K}^{*}-\hat{\beta}_{K}\right)\vert\mathcal{M}^{'}\right]\\ & = & \mathbb{E}\left(\frac{\frac{1}{n}D^{T}\eta}{\frac{1}{n}D^{T}D}\vert\mathcal{M}^{'}\right)+\mathbb{E}\left(\frac{D^{T}X_{K}}{D^{T}D}\beta_{K}^{*}\vert\mathcal{M}^{'}\right)\\ & = & \mathbb{E}\left(\frac{D^{T}X_{K}}{D^{T}D}\beta_{K}^{*}\vert\mathcal{M}^{'}\right)\end{aligned}$$ where $\mathcal{M}^{'}=E\cap\mathcal{E}_{t^{*}}$ such that $\mathbb{P}\left(\mathcal{M}^{'}\right)\geq1-2\kappa-c_{3}^{*}k\exp\left(-c_{4}^{*}\log p\right)$ and the last line follows from the argument used to show (\[eq:60\]). The rest of argument is similar to what is used in showing Proposition \[prop:bias\_post\_double\_formula\]. However, because (\[eq:21-2\]) involves $D$, $\mathbb{E}\left(\frac{\frac{1}{n}D^{T}X_{K}}{\frac{1}{n}D^{T}D}\beta_{K}^{*}\vert E,\,\mathcal{E}_{t^{*}}\right)\neq\mathbb{E}\left(\frac{\frac{1}{n}D^{T}X_{K}}{\frac{1}{n}D^{T}D}\beta_{K}^{*}\vert\hat{\gamma}=0_{p},\,\mathcal{E}_{t^{*}}\right)$. Instead of (\[eq:50\]) and (\[eq:58\]), we apply $$\begin{aligned} \mathbb{P}\left(\mathcal{E}_{t_{2}}^{'}\vert E,\,\mathcal{E}_{t_{1}}\right) & \geq & \mathbb{P}\left(\mathcal{E}_{t_{2}}^{'}\cap\mathcal{E}_{t_{1}}\cap E\right)\\ & \geq & \mathbb{P}\left(\mathcal{E}_{t_{1}}\right)+\mathbb{P}\left(\mathcal{E}_{t_{2}}^{'}\right)+\mathbb{P}\left(E\right)-2\\ & \geq & 1-2\kappa-c_{1}\exp\left(-c_{2}\log p\right)-k\exp\left(\frac{-nt_{1}^{2}}{2\frac{s}{n}\sigma_{v}^{2}}\right)\\ & & -\exp\left(\frac{-nt_{2}^{2}}{8\sigma_{v}^{4}}\right),\qquad\textrm{for any }t_{2}\in(0,\,\sigma_{v}^{2}].\end{aligned}$$ Consequently, we have the claim in Proposition \[prop:bias\_debiased\_lasso\]. Following the argument used to show Proposition \[prop:bias\_post\_double\_selection-upper\], we also have the claim in Proposition \[prop:bias\_debiased\_lasso-upper\]. Simulations evidence -------------------- Here we evaluate the performance of the debiased Lasso proposed by @vandergeer2014asymptotically based on the simulation setting of Section \[sec:simulation\_evidence\]. We use cross-validation to choose the regularization parameters as it is the most commonly-used method in this literature. Figures \[fig:db\_bias\_std\] and \[fig:db\_cov\] present the results. Debiased Lasso exhibits substantial biases (relative to the standard deviation) and under-coverage for all values of $\sigma_x$ and its performance is very sensitive to the regularization choice. The performance tends to get worse as $\sigma_x$ increases because the bias decays slower than the standard deviation. A comparison to the results in Section \[sec:simulation\_evidence\] shows that post double Lasso performs better than debiased Lasso.[^27] \[fig:db\_bias\_std\] \[fig:db\_cov\] Additional simulations for post double Lasso {#app:additional_simulations} ============================================ In the main text, we consider a setting with $k=5$, normally distributed control variables, normally distributed homoscedastic errors terms, and $\alpha^\ast=0$. Here we provide additional simulation evidence based on a more general model:[^28] $$\begin{aligned} Y_{i} & = & D_{i}\alpha^{*}+X_{i}\beta^{*}+\sigma_{y}(D_{i},X_{i})\eta_{i},\label{eq:dgp_illustration_inference1_general}\\ D_{i} & = & X_{i}\gamma^{*}+\sigma_{d}(X_{i})v_{i},\label{eq:dgp_illustration_inference2_general}\end{aligned}$$ where $\eta_{i}$ and $v_{i}$ are independent of each other and $\left\{ X_{i},\eta_{i},v_{i}\right\} _{i=1}^{n}$ consists of i.i.d. entries. The object of interest is $\alpha^{*}$. We set $n=500$, $p=200$, and consider a sparse setting where $\beta^{*}=\gamma^\ast=(\underbrace{1,\dots,1}_{k},0,\dots,0)^{T}$. We consider six DGPs that differ with respect to $k$, the distributions of $X_i$, $\eta_{i}$, and $v_{i}$, the specifications of $\sigma_{y}(D_{i},X_{i})$ and $\sigma_{d}(X_{i})$, as well as $\alpha^\ast$. For DGP A1, we do not report the results for $\sigma_x< 0.2$ due to numerical issues with the computation of standard errors. The results are based on 1,000 simulation repetitions. $X_i$ $k$ $\sigma_y(D_i,X_i)$ $\sigma_d(X_i)$ $\eta_i$ $v_i$ $\alpha^\ast$ -------- ------------------------------------------------------------------- ----- ------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------- ----------------------------- ------------------------------ --------------- -- -- DGP A1 Indep. $\text{Bern}\left(\frac{1-\sqrt{1-4\sigma_x^2}}{2}\right)$ 5 1 1 $ \mathcal{N}(0,1)$ $ \mathcal{N}(0,1)$ 0 DGP A2 $\mathcal{N}(0,\sigma_x^2I_p)$ 5 1 1 $\frac{t(5)}{\sqrt{(5/3)}}$ $ \frac{t(5)}{\sqrt{(5/3)}}$ 0 DGP A3 $\mathcal{N}(0,\sigma_x^2I_p)$ 5 $\sqrt{\frac{(1+D_{i}\alpha^{*}+X_{i}\beta^{*})^{2}}{\frac{1}{n}\sum_{i}(1+D_{i}\alpha^{*}+X_{i}\beta^{*})^{2}}}$ $\sqrt{\frac{(1+X_{i}\gamma^{*})^{2}}{\frac{1}{n}\sum_{i}(1+X_{i}\gamma^{*})^{2}}}$ $ \mathcal{N}(0,1)$ $ \mathcal{N}(0,1)$ 0 DGP A4 $\mathcal{N}(0,\sigma_x^2I_p)$ 10 1 1 $ \mathcal{N}(0,1)$ $ \mathcal{N}(0,1)$ 0 DGP A5 $\mathcal{N}(0,\sigma_x^2I_p)$ 5 1 1 $ \mathcal{N}(0,1)$ $ \mathcal{N}(0,1)$ 1 DGP A6 $\mathcal{N}(0,\sigma_x^2I_p)$ 5 1 1 $ \mathcal{N}(0,1)$ $ \mathcal{N}(0,1)$ -1 Figures \[fig:dml\_bias\_std\_app\]–\[fig:dml\_ci\_app\] present the results. The two most important determinants of the performance of post double Lasso are $k$ and $\alpha^{*}$. To see why $\alpha^\ast$ is important, recall that the reduced form parameter and the error term in the first step of post double Lasso (i.e., program (\[eq:las-1\])) are $\pi^{*}=\alpha^{*}\gamma^{*}+\beta^{*}$ and $u_{i}=\eta_{i}+\alpha^{*}v_{i}$. This implies that the magnitude of $\pi^{*}$ as well as the variance of $u_i$ depend on $\alpha^{*}$. Consequently, the selection performance of Lasso in the first step is directly affected by $\alpha^{*}$. In the extreme case where $\alpha^{*}$ is such that $\pi^{*}=0_{p}$, Lasso does not select any controls with high probability if the regularization parameter is chosen according to the standard recommendations. The simulation results further show that there is no practical recommendation for choosing the regularization parameters. While $\lambda_{\text{min}}$ leads to the best performance when $\alpha^\ast=0$, this choice can yield poor performances when $\alpha^\ast> 0$. Finally, across all DGPs, OLS outperforms post double Lasso in terms of bias and coverage accuracy, but yields somewhat wider confidence intervals. \[fig:dml\_bias\_std\_app\] \[fig:dml\_cov\_app\] \[fig:dml\_ci\_app\] Theoretical results for random designs \[sec:Random-design\] ============================================================ Results ------- In this section, we provide some results in Lemma \[prop:random\_design\] for the Lasso with a random design $X$. The necessary result on the Lasso’s inclusion established in Lemma \[prop:random\_design\] can be adopted in a similar fashion as in Propositions \[prop:bias\_post\_double\_selection\]-\[prop:bias\_post\_double\_selection-upper-1\] and \[prop:bias\_debiased\_lasso\]-\[prop:bias\_debiased\_lasso-upper\] to establish the OVBs. We make the following assumption about (\[eq:1\]). \[ass:random\_design\] Each row of $X$ is sampled independently; for all $i=1,\dots,n$ and $j=1,\dots,p$, $\sup_{r\geq1}r^{-\frac{1}{2}}\left(\mathbb{E}\left|X_{ij}\right|^{r}\right)^{\frac{1}{r}}\leq\alpha<\infty$; for any unit vector $a\in\mathbb{R}^{k}$ and $i=1,\dots,n$, $\sup_{r\geq1}r^{-\frac{1}{2}}\left(\mathbb{E}\left|a^{T}X_{i,K}^{T}\right|^{r}\right)^{\frac{1}{r}}\leq\tilde{\alpha}<\infty$, where $X_{i,K}$ is the $i$th row of $X_{K}$ and $K=\left\{ j:\,\theta_{j}^{*}\neq0\right\} $. Moreover, the error terms $\varepsilon_{1},\dots,\varepsilon_{n}$ are independent such that $\sup_{r\geq1}r^{-\frac{1}{2}}\left(\mathbb{E}\left|\varepsilon_{i}\right|^{r}\right)^{\frac{1}{r}}\leq\sigma<\infty$ and $\mathbb{E}\left(X_{i}\varepsilon_{i}\right)=0_{p}$ for all $i=1,\dots,n$. Assumption \[ass:random\_design\] is known as the sub-Gaussian tail condition defined in @vershynin2012. Examples of sub-Gaussian variables include Gaussian mixtures and distributions with bounded support. The first and last part of Assumption \[ass:random\_design\] imply that $X_{ij},j=1,\dots,p,$ and $\varepsilon_{i}$ are sub-Gaussian variables and is used in deriving the lower bounds on the regularization parameters. The second part of Assumption \[ass:random\_design\] is only used to establish some eigenvalue condition on $\frac{X_{K}^{T}X_{K}}{n}$. \[ass:sparsity\_incoherence\_random\_design\] The following conditions are satisfied: (i) $\theta^{*}$ is exactly sparse with at most $k$ non-zero coefficients and $K\neq\emptyset$; (ii) $$\left\Vert \left[\mathbb{E}\left(X_{K^{c}}^{T}X_{K}\right)\right]\left[\mathbb{E}\left(X_{K}^{T}X_{K}\right)\right]^{-1}\right\Vert _{\infty}=1-\phi\label{eq:14-1}$$ for some $\phi\in(0,\,1]$ such that $\phi^{-1}\precsim1$; (iii) $\mathbb{E}\left(X_{ij}\right)=0$ for all $j\in K$ and $\mathbb{E}\left(X_{j}^{T}X_{j}\right)\leq s$ for all $j=1,\dots,p$; (iv) $$\begin{aligned} \max\left\{ \frac{\phi}{12(1-\phi)k^{\frac{3}{2}}},\,\frac{\phi}{6k^{\frac{3}{2}}},\,\frac{\phi}{k}\right\} \sqrt{\frac{\log p}{n}} & \leq & \alpha^{2}\quad\textrm{if }\phi\in\left(0,\,1\right),\label{eq:7-2}\\ \max\left\{ \frac{1}{6k^{\frac{3}{2}}},\,\frac{1}{k}\right\} \sqrt{\frac{\log p}{n}} & \leq & \alpha^{2}\quad\textrm{if }\phi=1,\label{eq:7-4}\\ \max\left\{ 2\tilde{\alpha}^{2},\,12\alpha^{2},\,1\right\} \sqrt{\frac{\log p}{n}} & \leq & \lambda_{\min}\left(\mathbb{E}\left[\frac{1}{n}X_{K}^{T}X_{K}\right]\right).\label{eq:7-3} \end{aligned}$$ Part (iv) of Assumption \[ass:sparsity\_incoherence\_random\_design\] is imposed to ensure that $$\begin{aligned} \left\Vert \left(\frac{1}{n}X_{K}^{T}X_{K}\right)^{-1}-\left[\mathbb{E}\left(\frac{1}{n}X_{K}^{T}X_{K}\right)\right]^{-1}\right\Vert _{\infty} & \precsim & \frac{1}{\lambda_{\min}\left(\mathbb{E}\left[\frac{1}{n}X_{K}^{T}X_{K}\right]\right)},\\ \left\Vert \frac{1}{n}X_{K^{c}}^{T}X_{K}\left(\frac{1}{n}X_{K}^{T}X_{K}\right)^{-1}\right\Vert _{\infty} & \leq & 1-\frac{\phi}{2},\end{aligned}$$ with high probability. To gain some intuition for , let us further assume $k\asymp1$, $X_{i}$ is normally distributed for all $i=1,\dots,n$, and $\mathbb{E}\left(X_{K}^{T}X_{K}\right)$ is a diagonal matrix with the diagonal entries $\mathbb{E}\left(X_{j}^{T}X_{j}\right)=s\neq0$. As a result, $\tilde{\alpha}=\alpha\asymp\sqrt{\frac{s}{n}}$ by the definition of a sub-Gaussian variable (e.g., @vershynin2012) and essentially require $\sqrt{\frac{\log p}{n}}\precsim\frac{s}{n}$. Given $$\mathbb{P}\left(\left|\frac{X^{T}\varepsilon}{n}\right|_{\infty}\geq t\right)\leq2\exp\left(\frac{-nt^{2}}{c_{0}\sigma^{2}\alpha^{2}}+\log p\right).\label{eq:38}$$ and $\lambda\geq\frac{c\alpha\sigma\left(2-\frac{\phi}{2}\right)}{\phi}\sqrt{\frac{\log p}{n}}$ for some sufficiently large universal constant $c>0$, we have $$\lambda\geq2\left|\frac{X^{T}\varepsilon}{n}\right|_{\infty}\label{eq:lambda-1}$$ with probability at least $1-c^{'}\exp\left(-c^{''}\log p\right)$. Define the following events $$\begin{aligned} \mathcal{E}_{1} & = & \left\{ \left|\frac{X^{T}\varepsilon}{n}\right|_{\infty}\precsim\frac{\alpha\sigma\left(2-\frac{\phi}{2}\right)}{\phi}\sqrt{\frac{\log p}{n}}\right\} ,\\ \mathcal{E}_{2} & = & \left\{ \lambda_{\max}(\hat{\Sigma}_{KK})\leq\frac{3}{2}\lambda_{\max}(\Sigma_{KK})\right\} ,\\ \mathcal{E}_{3} & = & \left\{ \left\Vert \left(\frac{1}{n}X_{K}^{T}X_{K}\right)^{-1}-\left[\mathbb{E}\left(\frac{1}{n}X_{K}^{T}X_{K}\right)\right]^{-1}\right\Vert _{\infty}\precsim\frac{1}{\lambda_{\min}\left(\mathbb{E}\left[\frac{1}{n}X_{K}^{T}X_{K}\right]\right)}\right\} ,\\ \mathcal{E}_{4} & = & \left\{ \left\Vert \frac{1}{n}X_{K^{c}}^{T}X_{K}\left(\frac{1}{n}X_{K}^{T}X_{K}\right)^{-1}\right\Vert _{\infty}\leq1-\frac{\phi}{2}\right\} .\end{aligned}$$ By (\[eq:38\]), $\mathbb{P}\left(\mathcal{E}_{1}\right)\geq1-c^{'}\exp\left(-c^{''}\log p\right)$; by (\[eq:S7\]), $\mathbb{P}\left(\mathcal{E}_{2}\right)\geq1-c_{1}^{'}\exp\left(-c_{1}^{''}\log p\right)$; by (\[eq:S4\]), $\mathbb{P}\left(\mathcal{E}_{3}\right)\geq1-c_{2}^{'}\exp\left(-c_{2}^{''}\left(\frac{\log p}{k^{3}}\right)\right)$; by (\[eq:77\]), $\mathbb{P}\left(\mathcal{E}_{4}\right)\geq1-c_{3}^{''}\exp\left(-b\left(\frac{\log p}{k^{3}}\right)\right)$, where $b$ is some positive constant that only depends on $\phi$ and $\alpha$. \[prop:random\_design\] Let Assumptions \[ass:random\_design\] and \[ass:sparsity\_incoherence\_random\_design\] hold. We solve the Lasso (\[eq:las\]) with $\lambda\geq\frac{c\alpha\sigma\left(2-\frac{\phi}{2}\right)}{\phi}\sqrt{\frac{\log p}{n}}$ for some sufficiently large universal constant $c>0$. Suppose $\mathbb{E}\left[X_{K}^{T}X_{K}\right]$ is a positive definite matrix. \(i) Then, conditioning on $\mathcal{E}_{1}\cap\mathcal{E}_{4}$ (which holds with probability at least $1-c_{1}\exp\left(-b\frac{\log p}{k^{3}}\right)$), (\[eq:las\]) has a unique optimal solution $\hat{\theta}$ such that $\hat{\theta}_{j}=0$ for $j\notin K$. \(ii) With probability at least $1-c_{1}\exp\left(-b\frac{\log p}{k^{3}}\right)$, $$\left|\hat{\theta}_{K}-\theta_{K}^{*}\right|_{2}\leq\frac{3\lambda\sqrt{k}}{\lambda_{\min}\left(\mathbb{E}\left[\frac{1}{n}X_{K}^{T}X_{K}\right]\right)}\label{eq:5gen-1}$$ where $\theta_{K}=\left\{ \theta_{j}\right\} _{j\in K}$ and $b$ is some positive constant that only depends on $\phi$ and $\alpha$; if $\mathbb{P}\left(\left\{ \textrm{supp}(\hat{\theta})=K\right\} \cap\mathcal{E}_{1}\cap\mathcal{E}_{2}\right)>0$, conditioning on $\left\{ \textrm{supp}(\hat{\theta})=K\right\} \cap\mathcal{E}_{1}\cap\mathcal{E}_{2}$, we must have $$\left|\hat{\theta}_{K}-\theta_{K}^{*}\right|_{2}\geq\frac{\lambda\sqrt{k}}{3\lambda_{\max}\left(\mathbb{E}\left[\frac{1}{n}X_{K}^{T}X_{K}\right]\right)}\geq\frac{\lambda\sqrt{k}}{3\sum_{j\in K}\left(\mathbb{E}\left[\frac{1}{n}X_{j}^{T}X_{j}\right]\right)}.\label{eq:4gen-1}$$ \(iii) If $\mathbb{E}\left(X_{K}^{T}X_{K}\right)$ is a diagonal matrix with the diagonal entries $\mathbb{E}\left(X_{j}^{T}X_{j}\right)=s\neq0$, then $$\left|\hat{\theta}_{j}-\theta_{j}^{*}\right|\leq\frac{7\lambda n}{4s}\qquad\forall j\in K\label{eq:5-1}$$ with probability at least $1-c_{1}\exp\left(-b\frac{\log p}{k^{3}}\right)$; if $\mathbb{P}\left(\left\{ \hat{\theta}_{j}\neq0,\,j\in K\right\} \cap\mathcal{E}_{1}\cap\mathcal{E}_{3}\cap\mathcal{E}_{4}\right)>0$, conditioning on $\left\{ \hat{\theta}_{j}\neq0,\,j\in K\right\} \cap\mathcal{E}_{1}\cap\mathcal{E}_{3}\cap\mathcal{E}_{4}$, we must have $$\left|\hat{\theta}_{j}-\theta_{j}^{*}\right|\geq\frac{\lambda n}{4s}\geq c_{0}\frac{\sigma}{\phi}\sqrt{\frac{n}{s}}\sqrt{\frac{\log p}{n}}.\label{eq:4-1}$$ \(iv) Suppose $K=\left\{ 1\right\} $ and $\mathbb{E}\left(X_{1}^{T}X_{1}\right)=s\neq0$. If $$\left|\theta_{1}^{*}\right|\leq\frac{\lambda n}{4s},\label{eq:min-1}$$ then we must have $$\mathbb{P}\left(\hat{\theta}=0_{p}\right)\geq1-c\exp\left(-b\log p\right).\label{eq:nec1}$$ The part $\frac{\lambda n}{4s}\geq c_{0}\frac{\sigma}{\phi}\sqrt{\frac{n}{s}}\sqrt{\frac{\log p}{n}}$ in bound (\[eq:4-1\]) follows from the fact that $\alpha\succsim\sqrt{\frac{s}{n}}=\sqrt{\mathbb{E}\left(\frac{1}{n}\sum X_{ij}^{2}\right)}$ where $j\in K$. Main proof for Lemma \[prop:random\_design\] -------------------------------------------- In what follows, we let $\Sigma_{KK}:=\mathbb{E}\left[\frac{1}{n}X_{K}^{T}X_{K}\right]$, $\hat{\Sigma}_{KK}:=\frac{1}{n}X_{K}^{T}X_{K}$, and $\lambda_{\min}\left(\Sigma\right)$ denote the minimum eigenvalue of the matrix $\Sigma$. The proof for Proposition \[prop:random\_design\](i) follows similar argument as before but requires a few extra steps. In applying Lemma 7.23 from Chapter 7.5 of @wainwright_2019 to establish the uniqueness of $\hat{\theta}$ upon the success of PDW construction, it suffices to show that $\lambda_{\min}(\hat{\Sigma}_{KK})\geq\frac{1}{2}\lambda_{\min}(\Sigma_{KK})$ and this fact is verified in (\[eq:S7-1\]) in the appendix. As a consequence, the subproblem (\[eq:sub\]) is strictly convex and has a unique minimizer. The details that show the PDW construction succeeds conditioning on $\mathcal{E}_{1}\cap\mathcal{E}_{4}$ (which holds with probability at least $1-c_{1}\exp\left(-b\frac{\log p}{k^{3}}\right)$) can be found in Lemma \[lem:A4\] (where $b$ is some positive constant that only depends on $\phi$ and $\alpha$). To show (\[eq:5gen-1\]), note that our choice of $\lambda$ and $\left|\hat{\delta}_{K}\right|\leq1$ yield $$\left|\Delta\right|\leq\left|\lambda\hat{\delta}_{K}\right|+\left|\frac{X_{K}^{T}\varepsilon}{n}\right|\leq\frac{3\lambda}{2}1_{k},$$ which implies that $\left|\Delta\right|_{2}\leq\frac{3\lambda}{2}\sqrt{k}$. Moreover, we can show $$\begin{aligned} \left|\hat{\theta}_{K}-\theta_{K}^{*}\right|_{2} & = & \frac{\left|\left(\frac{X_{K}^{T}X_{K}}{n}\right)^{-1}\Delta\right|_{2}}{\left|\Delta\right|_{2}}\left|\Delta\right|_{2}\label{eq:30-1}\\ & \leq & \frac{1}{\lambda_{\min}\left(\frac{1}{n}X_{K}^{T}X_{K}\right)}\frac{3\lambda}{2}\sqrt{k}.\nonumber \end{aligned}$$ Applying (\[eq:S7-1\]) and the bound $\left|\Delta\right|_{2}\leq\frac{3\lambda}{2}\sqrt{k}$ yields the claim. In showing (\[eq:4gen-1\]) in (ii) and (\[eq:4-1\]) in (iii), we will condition on $\left\{ \textrm{supp}(\hat{\theta})=K\right\} \cap\mathcal{E}_{1}\cap\mathcal{E}_{2}$ and $\left\{ \hat{\theta}_{j}\neq0,\,j\in K\right\} \cap\mathcal{E}_{1}\cap\mathcal{E}_{3}\cap\mathcal{E}_{4}$, respectively. To show (\[eq:4gen-1\]), note that in Step 2 of the PDW procedure, $\hat{\delta}_{K}$ is chosen such that $\left|\hat{\delta}_{j}\right|=1$ for any $j\in K$ whenever $\textrm{supp}(\hat{\theta})=K$. Given the choice of $\lambda$, we are ensured to have $$\left|\Delta\right|\geq\left|\left|\lambda\hat{\delta}_{K}\right|-\left|\frac{X_{K}^{T}\varepsilon}{n}\right|\right|\geq\frac{\lambda}{2}1_{k},$$ which implies that $\left|\Delta\right|_{2}\geq\frac{\lambda}{2}\sqrt{k}$. Moreover, we can show $$\left|\hat{\theta}_{K}-\theta_{K}^{*}\right|_{2}=\frac{\left|\left(\frac{X_{K}^{T}X_{K}}{n}\right)^{-1}\Delta\right|_{2}}{\left|\Delta\right|_{2}}\left|\Delta\right|_{2}\geq\frac{1}{\lambda_{\max}\left(\frac{1}{n}X_{K}^{T}X_{K}\right)}\frac{\lambda}{2}\sqrt{k}.\label{eq:30}$$ It remains to bound $\lambda_{\max}\left(\hat{\Sigma}_{KK}\right)$. We first write $$\begin{aligned} \lambda_{\max}(\Sigma_{KK}) & = & \max_{||h^{'}||_{2}=1}\mu^{'T}\Sigma_{KK}\mu^{'}\\ & = & \max_{||h^{'}||_{2}=1}\left[\mu^{'T}\hat{\Sigma}_{KK}\mu^{'}+\mu^{'T}(\Sigma_{KK}-\hat{\Sigma}_{KK})\mu^{'}\right]\\ & \geq & \mu^{T}\hat{\Sigma}_{KK}\mu+\mu^{T}(\Sigma_{KK}-\hat{\Sigma}_{KK})\mu\end{aligned}$$ where $\mu\in\mathbb{R}^{k}$ is a unit-norm maximal eigenvector of $\hat{\Sigma}_{KK}$. Applying Lemma \[lem:A1\](b) with $t=\tilde{\alpha}^{2}\sqrt{\frac{\log p}{n}}$ yields $$\mu^{T}\left(\Sigma_{KK}-\hat{\Sigma}_{KK}\right)\mu\geq-\tilde{\alpha}^{2}\sqrt{\frac{\log p}{n}}$$ with probability at least $1-c_{1}\exp\left(-c_{2}\log p\right)$, provided that $\sqrt{\frac{\log p}{n}}\leq1$; therefore, $\lambda_{\max}(\Sigma_{KK})\geq\lambda_{\max}(\hat{\Sigma}_{KK})-\tilde{\alpha}^{2}\sqrt{\frac{\log p}{n}}$. Because $\tilde{\alpha}^{2}\sqrt{\frac{\log p}{n}}\leq\frac{\lambda_{\max}(\Sigma_{KK})}{2}$ (implied by (\[eq:7-3\])), we have $$\lambda_{\max}(\hat{\Sigma}_{KK})\leq\frac{3}{2}\lambda_{\max}(\Sigma_{KK})\label{eq:S7}$$ with probability at least $1-c_{1}\exp\left(-c_{2}\log p\right)$. As a consequence, $$\left|\hat{\theta}_{K}-\theta_{K}^{*}\right|_{2}\geq\frac{1}{\lambda_{\max}\left(\frac{1}{n}X_{K}^{T}X_{K}\right)}\frac{\lambda}{2}\sqrt{k}\geq\frac{1}{\lambda_{\max}(\Sigma_{KK})}\frac{\lambda}{3}\sqrt{k}.$$ The second inequality in (\[eq:4gen-1\]) simply follows from the fact $\lambda_{\max}\left(\mathbb{E}\left[\frac{1}{n}X_{K}^{T}X_{K}\right]\right)\leq\sum_{j\in K}\left(\mathbb{E}\left[\frac{1}{n}X_{j}^{T}X_{j}\right]\right)$. To show (\[eq:5-1\]), note that $$\begin{aligned} \left|\hat{\theta}_{K}-\theta_{K}^{*}\right|_{\infty} & \leq & \left|\hat{\Sigma}_{KK}^{-1}\frac{X_{K}^{T}\varepsilon}{n}\right|_{\infty}+\lambda\left\Vert \hat{\Sigma}_{KK}^{-1}\right\Vert _{\infty}\nonumber \\ & \leq & \left\Vert \hat{\Sigma}_{KK}^{-1}\right\Vert _{\infty}\left|\frac{X_{K}^{T}\varepsilon}{n}\right|_{\infty}+\lambda\left\Vert \hat{\Sigma}_{KK}^{-1}\right\Vert _{\infty}\nonumber \\ & \leq & \frac{3\lambda}{2}\left\Vert \hat{\Sigma}_{KK}^{-1}\right\Vert _{\infty}.\label{eq:10-1}\end{aligned}$$ We then apply (\[eq:S4\]) of Lemma \[lem:A2\] in the appendix, and the fact $\left\Vert \hat{\Sigma}_{KK}^{-1}\right\Vert _{\infty}-\left\Vert \Sigma_{KK}^{-1}\right\Vert _{\infty}\leq\left\Vert \hat{\Sigma}_{KK}^{-1}-\Sigma_{KK}^{-1}\right\Vert _{\infty}$ (so that $\left\Vert \hat{\Sigma}_{KK}^{-1}\right\Vert _{\infty}\leq\frac{7n}{6s}$); putting everything yields the claim. To show (\[eq:4-1\]), we again carry over the argument in the proof for Lemma \[prop:fixed\_design\]. Letting $M=\hat{\Sigma}_{KK}^{-1}-\Sigma_{KK}^{-1}$, we have $$\begin{aligned} \left|\hat{\theta}_{K}-\theta_{K}^{*}\right| & = & \left|\left(\Sigma_{KK}^{-1}+M\right)\left[\left(\frac{X_{K}^{T}\varepsilon}{n}\right)-\lambda\hat{\delta}_{K}\right]\right|\\ & \geq & \left|\Sigma_{KK}^{-1}\left[\left(\frac{X_{K}^{T}\varepsilon}{n}\right)+\lambda\hat{\delta}_{K}\right]\right|-\left|M\left[\left(\frac{X_{K}^{T}\varepsilon}{n}\right)-\lambda\hat{\delta}_{K}\right]\right|\\ & \geq & \left|\Sigma_{KK}^{-1}\right|\left|\left|\lambda\hat{\delta}_{K}\right|-\left|\frac{X_{K}^{T}\varepsilon}{n}\right|\right|-\left\Vert M\right\Vert _{\infty}\left|\left(\frac{X_{K}^{T}\varepsilon}{n}\right)-\lambda\hat{\delta}_{K}\right|_{\infty}1_{k},\end{aligned}$$ where the third line uses the fact that $\Sigma_{KK}^{-1}$ is diagonal. Note that as before, the choice of $\lambda$ stated in Lemma \[prop:random\_design\] and the fact $\Sigma_{KK}^{-1}=\frac{n}{s}I_{k}$ yield $$\begin{aligned} \left|\hat{\theta}_{j}-\theta_{j}^{*}\right| & \geq & \frac{\lambda n}{2s}-\left\Vert M\right\Vert _{\infty}\left|\left(\frac{X_{K}^{T}\varepsilon}{n}\right)-\lambda\hat{\delta}_{K}\right|_{\infty}\\ & \geq & \frac{\lambda n}{2s}-\frac{3}{2}\lambda\left\Vert M\right\Vert _{\infty}.\end{aligned}$$ By (\[eq:S4\]) of Lemma \[lem:A2\] in the appendix, with probability at least $1-c_{1}\exp\left(-b\frac{\log p}{k^{3}}\right)$, $\left\Vert M\right\Vert _{\infty}\leq\frac{1}{6}\lambda_{\min}^{-1}(\Sigma_{KK})=\frac{n}{6s}$. As a result, we have (\[eq:4-1\]). The part $\frac{\lambda n}{4s}\geq c_{0}\frac{\sigma}{\phi}\sqrt{\frac{n}{s}}\sqrt{\frac{\log p}{n}}$ in bound (\[eq:4-1\]) follows from the fact that $\alpha\succsim\sqrt{\frac{s}{n}}=\sqrt{\mathbb{E}\left(\frac{1}{n}\sum X_{ij}^{2}\right)}$ where $j\in K$. To establish (\[eq:nec1\]), we adopt argument similar to what is used in showing (\[eq:nec\]) by applying the KKT condition $$\left(\frac{1}{n}X_{1}^{T}X_{1}\right)\left(\theta_{1}^{*}-\hat{\theta}_{1}\right)=\lambda\textrm{sgn}\left(\hat{\theta}_{1}\right)-\frac{X_{1}^{T}\varepsilon}{n}$$ and defining $\mathcal{E}=\mathcal{E}_{1}\cap\mathcal{E}_{4}$. Additional technical lemmas and proofs -------------------------------------- In this section, we show that the PDW construction succeeds with high probability in Lemma \[lem:A4\], which is proved using results from Lemmas \[lem:A1\]\[lem:A3\]. The derivations for Lemmas \[lem:A2\] and \[lem:A3\] modify the argument in @wainwright2009sharp and @ravikumar2010 to make it suitable for our purposes. In what follows, we let $\Sigma_{K^{c}K}:=\mathbb{E}\left[\frac{1}{n}X_{K^{c}}^{T}X_{K}\right]$ and $\hat{\Sigma}_{K^{c}K}:=\frac{1}{n}X_{K^{c}}^{T}X_{K}$. Similarly, let $\Sigma_{KK}:=\mathbb{E}\left[\frac{1}{n}X_{K}^{T}X_{K}\right]$ and $\hat{\Sigma}_{KK}:=\frac{1}{n}X_{K}^{T}X_{K}$. \[lem:A1\] (a) Let $\left(W_{i}\right)_{i=1}^{n}$ and $\left(W_{i}^{'}\right)_{i=1}^{n}$ consist of independent components, respectively. Suppose there exist parameters $\alpha$ and $\alpha^{'}$ such that $$\begin{aligned} \sup_{r\geq1}r^{-\frac{1}{2}}\left(\mathbb{E}\left|W_{i}\right|^{r}\right)^{\frac{1}{r}} & \leq & \alpha,\\ \sup_{r\geq1}r^{-\frac{1}{2}}\left(\mathbb{E}\left|W_{i}^{'}\right|^{r}\right)^{\frac{1}{r}} & \leq & \alpha^{'}, \end{aligned}$$ for all $i=1,\dots,n$. Then $$\mathbb{P}\left[\left|\frac{1}{n}\sum_{i=1}^{n}\left(W_{i}W_{i}^{'}\right)-\mathbb{E}\left[\frac{1}{n}\sum_{i=1}^{n}\left(W_{i}W_{i}^{'}\right)\right]\right|\geq t\right]\leq2\exp\left(-cn\left(\frac{t^{2}}{\alpha^{2}\alpha^{'2}}\wedge\frac{t}{\alpha\alpha^{'}}\right)\right).\label{eq:A37-1}$$ (b) For any unit vector $v\in\mathbb{R}^{d}$, suppose there exists a parameter $\tilde{\alpha}$ such that $$\sup_{r\geq1}r^{-\frac{1}{2}}\left(\mathbb{E}\left|a^{T}Z_{i}^{T}\right|^{r}\right)^{\frac{1}{r}}\leq\tilde{\alpha},$$ where $Z_{i}$ is the $i$th row of $Z\in\mathbb{R}^{n\times d}$, then we have $$\mathbb{P}(\left|\left|Zv\right|_{2}^{2}-\mathbb{E}\left(\left|Zv\right|_{2}^{2}\right)\right|\geq nt)\leq2\exp\left(-c^{'}n\left(\frac{t^{2}}{\tilde{\alpha}^{4}}\wedge\frac{t}{\tilde{\alpha}^{2}}\right)\right).$$ Lemma \[lem:A1\] is based on Lemma 5.14 and Corollary 5.17 in @vershynin2012. \[lem:A2\] Suppose Assumption \[ass:random\_design\] holds. For any $t>0$ and some constant $c>0$, we have $$\mathbb{P}\left\{ \left\Vert \hat{\Sigma}_{K^{c}K}-\Sigma_{K^{c}K}\right\Vert _{\infty}\geq t\right\} \leq2(p-k)k\exp\left(-cn\left(\frac{t^{2}}{k^{2}\alpha^{4}}\wedge\frac{t}{k\alpha^{2}}\right)\right),\label{eq:S2}$$ $$\mathbb{P}\left\{ \left\Vert \hat{\Sigma}_{KK}-\Sigma_{KK}\right\Vert _{\infty}\geq t\right\} \leq2k^{2}\exp\left(-cn\left(\frac{t^{2}}{k^{2}\alpha^{4}}\wedge\frac{t}{k\alpha^{2}}\right)\right).\label{eq:S3}$$ Furthermore, if $k\geq1$, $\frac{\log p}{n}\leq1$, $\tilde{\alpha}^{2}\sqrt{\frac{\log p}{n}}\leq\frac{\lambda_{\min}(\Sigma_{KK})}{2}$, and $\alpha^{2}\sqrt{\frac{\log p}{n}}\leq\frac{\lambda_{\min}(\Sigma_{KK})}{12}$, we have $$\begin{aligned} \mathbb{P}\left\{ \left\Vert \hat{\Sigma}_{KK}^{-1}\right\Vert _{2}\leq\frac{2}{\lambda_{\min}(\Sigma_{KK})}\right\} & \geq & 1-c_{1}^{'}\exp\left(-c_{2}^{'}\log p\right),\label{eq:S8}\\ \mathbb{P}\left\{ \left\Vert \hat{\Sigma}_{KK}^{-1}-\Sigma_{KK}^{-1}\right\Vert _{\infty}\leq\frac{1}{6\lambda_{\min}(\Sigma_{KK})}\right\} & \geq & 1-c_{1}\exp\left(-c_{2}\left(\frac{\log p}{k^{3}}\right)\right).\label{eq:S4} \end{aligned}$$ Let $u_{j^{'}j}$ denote the element $(j^{'},\,j)$ of the matrix difference $\hat{\Sigma}_{K^{c}K}-\Sigma_{K^{c}K}$. The definition of the $l_{\infty}$matrix norm implies that $$\begin{aligned} \mathbb{P}\left\{ \left\Vert \hat{\Sigma}_{K^{c}K}-\Sigma_{K^{c}K}\right\Vert _{\infty}\geq t\right\} & = & \mathbb{P}\left\{ \max_{j^{'}\in K^{c}}\sum_{j\in K}|u_{j^{'}j}|\geq t\right\} \\ & \leq & (p-k)\mathbb{P}\left\{ \sum_{j\in K}|u_{j^{'}j}|\geq t\right\} \\ & \leq & (p-k)\mathbb{P}\left\{ \exists j\in K\,\vert\,|u_{j^{'}j}|\geq\frac{t}{k}\right\} \\ & \leq & (p-k)k\mathbb{P}\left\{ |u_{j^{'}j}|\geq\frac{t}{k}\right\} \\ & \leq & (p-k)k\cdot2\exp\left(-cn\left(\frac{t^{2}}{k^{2}\alpha^{4}}\wedge\frac{t}{k\alpha^{2}}\right)\right), \end{aligned}$$ where the last inequality follows Lemma \[lem:A1\](a). Bound (\[eq:S3\]) can be derived in a similar fashion except that the pre-factor $(p-k)$ is replaced by $k$. To prove (\[eq:S4\]), note that $$\begin{aligned} \left\Vert \hat{\Sigma}_{KK}^{-1}-\Sigma_{KK}^{-1}\right\Vert _{\infty} & = & \left\Vert \Sigma_{KK}^{-1}\left[\Sigma_{KK}-\hat{\Sigma}_{KK}\right]\hat{\Sigma}_{KK}^{-1}\right\Vert _{\infty}\nonumber \\ & \leq & \sqrt{k}\left\Vert \Sigma_{KK}^{-1}\left[\Sigma_{KK}-\hat{\Sigma}_{KK}\right]\hat{\Sigma}_{KK}^{-1}\right\Vert _{2}\nonumber \\ & \leq & \sqrt{k}\left\Vert \Sigma_{KK}^{-1}\right\Vert _{2}\left\Vert \Sigma_{KK}-\hat{\Sigma}_{KK}\right\Vert _{2}\left\Vert \hat{\Sigma}_{KK}^{-1}\right\Vert _{2}\nonumber \\ & \leq & \frac{\sqrt{k}}{\lambda_{\min}(\Sigma_{KK})}\left\Vert \Sigma_{KK}-\hat{\Sigma}_{KK}\right\Vert _{2}\left\Vert \hat{\Sigma}_{KK}^{-1}\right\Vert _{2}.\label{eq:S5} \end{aligned}$$ To bound $\left\Vert \Sigma_{KK}-\hat{\Sigma}_{KK}\right\Vert _{2}$ in (\[eq:S5\]), we apply (\[eq:S3\]) with $t=\frac{\alpha^{2}}{\sqrt{k}}\sqrt{\frac{\log p}{n}}$ and obtain $$\left\Vert \hat{\Sigma}_{KK}-\Sigma_{KK}\right\Vert _{2}\leq\frac{\alpha^{2}}{\sqrt{k}}\sqrt{\frac{\log p}{n}},$$ with probability at least $1-c_{1}\exp\left(-c_{2}\frac{\log p}{k^{3}}\right)$, provided that $k^{-3}\frac{\log p}{n}\leq1$. To bound **$\left\Vert \hat{\Sigma}_{KK}^{-1}\right\Vert _{2}$** in (\[eq:S5\]), let us write $$\begin{aligned} \lambda_{\min}(\Sigma_{KK}) & = & \min_{||\mu^{'}||_{2}=1}\mu^{'T}\Sigma_{KK}\mu^{'}\nonumber \\ & = & \min_{||\mu^{'}||_{2}=1}\left[\mu^{'T}\hat{\Sigma}_{KK}\mu^{'}+\mu^{'T}(\Sigma_{KK}-\hat{\Sigma}_{KK})\mu^{'}\right]\nonumber \\ & \leq & \mu^{T}\hat{\Sigma}_{KK}\mu+\mu^{T}(\Sigma_{KK}-\hat{\Sigma}_{KK})\mu\label{eq:S6} \end{aligned}$$ where $\mu\in\mathbb{R}^{k}$ is a unit-norm minimal eigenvector of $\hat{\Sigma}_{KK}$. We then apply Lemma \[lem:A1\](b) with $t=\tilde{\alpha}^{2}\sqrt{\frac{\log p}{n}}$ to show $$\left|\mu^{T}\left(\Sigma_{KK}-\hat{\Sigma}_{KK}\right)\mu\right|\leq\tilde{\alpha}^{2}\sqrt{\frac{\log p}{n}}$$ with probability at least $1-c_{1}^{'}\exp\left(-c_{2}^{'}\log p\right)$, provided that $\sqrt{\frac{\log p}{n}}\leq1$. Therefore, $\lambda_{\min}(\Sigma_{KK})\leq\lambda_{\min}(\hat{\Sigma}_{KK})+\tilde{\alpha}^{2}\sqrt{\frac{\log p}{n}}$. As long as $\tilde{\alpha}^{2}\sqrt{\frac{\log p}{n}}\leq\frac{\lambda_{\min}(\Sigma_{KK})}{2}$, we have $$\lambda_{\min}(\hat{\Sigma}_{KK})\geq\frac{1}{2}\lambda_{\min}(\Sigma_{KK}),\label{eq:S7-1}$$ and consequently (\[eq:S8\]), $$\left\Vert \hat{\Sigma}_{KK}^{-1}\right\Vert _{2}\leq\frac{2}{\lambda_{\min}(\Sigma_{KK})}$$ with probability at least $1-c_{1}^{'}\exp\left(-c_{2}^{'}\log p\right)$. Putting the pieces together, as long as $\frac{\alpha^{2}}{\lambda_{\min}(\Sigma_{KK})}\sqrt{\frac{\log p}{n}}\leq\frac{1}{12}$, $$\left\Vert \hat{\Sigma}_{KK}^{-1}-\Sigma_{KK}^{-1}\right\Vert _{\infty}\leq\frac{\sqrt{k}}{\lambda_{\min}(\Sigma_{KK})}\frac{\alpha^{2}}{\sqrt{k}}\sqrt{\frac{\log p}{n}}\frac{2}{\lambda_{\min}(\Sigma_{KK})}\leq\frac{1}{6\lambda_{\min}(\Sigma_{KK})}\label{eq:19}$$ with probability at least $1-c_{1}\exp\left(-c_{2}\frac{\log p}{k^{3}}\right)$. \[lem:A3\] Let Assumption \[ass:random\_design\] hold. Suppose $$\left\Vert \mathbb{E}\left[X_{K^{c}}^{T}X_{K}\right]\left[\mathbb{E}(X_{K}^{T}X_{K})\right]^{-1}\right\Vert _{\infty}=1-\phi\label{eq:S1}$$ for some $\phi\in(0,\,1]$. If $k\geq1$ and $$\begin{aligned} \max\left\{ \frac{\phi}{12(1-\phi)k^{\frac{3}{2}}},\,\frac{\phi}{6k^{\frac{3}{2}}},\,\frac{\phi}{k}\right\} \sqrt{\frac{\log p}{n}} & \leq & \alpha^{2}\quad\textrm{if }\phi\in\left(0,\,1\right),\label{eq:7-2-1}\\ \max\left\{ \frac{1}{6k^{\frac{3}{2}}},\,\frac{1}{k}\right\} \sqrt{\frac{\log p}{n}} & \leq & \alpha^{2}\quad\textrm{if }\phi=1,\label{eq:7-4-1}\\ \max\left\{ 2\tilde{\alpha}^{2},\,12\alpha^{2},\,1\right\} \sqrt{\frac{\log p}{n}} & \leq & \lambda_{\min}(\Sigma_{KK}),\label{eq:7-3-1} \end{aligned}$$ then for some positive constant $b$ that only depends on $\phi$ and $\alpha$, we have $$\mathbb{P}\left[\left\Vert \frac{1}{n}X_{K^{c}}^{T}X_{K}\left(\frac{1}{n}X_{K}^{T}X_{K}\right)^{-1}\right\Vert _{\infty}\geq1-\frac{\phi}{2}\right]\leq c^{'}\exp\left(-b\left(\frac{\log p}{k^{3}}\right)\right).\label{eq:77}$$ Using the decomposition in @ravikumar2010, we have **$$\hat{\Sigma}_{K^{c}K}\hat{\Sigma}_{KK}^{-1}-\Sigma_{K^{c}K}\Sigma_{KK}^{-1}=R_{1}+R_{2}+R_{3},$$** where **** $$\begin{aligned} R_{1} & = & \Sigma_{K^{c}K}\left[\hat{\Sigma}_{KK}^{-1}-\Sigma_{KK}^{-1}\right],\\ R_{2} & = & \left[\hat{\Sigma}_{K^{c}K}-\Sigma_{K^{c}K}\right]\Sigma_{KK}^{-1},\\ R_{3} & = & \left[\hat{\Sigma}_{K^{c}K}-\Sigma_{K^{c}K}\right]\left[\hat{\Sigma}_{KK}^{-1}-\Sigma_{KK}^{-1}\right]. \end{aligned}$$ By (\[eq:S1\]), we have $\left\Vert \Sigma_{K^{c}K}\Sigma_{KK}^{-1}\right\Vert _{\infty}=1-\phi$. It suffices to show $\left\Vert R_{i}\right\Vert _{\infty}\leq\frac{\phi}{6}$ **** for $i=1,...,3$. For $R_{1}$, note that $$R_{1}=-\Sigma_{K^{c}K}\Sigma_{KK}^{-1}[\hat{\Sigma}_{KK}-\Sigma_{KK}]\hat{\Sigma}_{KK}^{-1}.$$ Applying the facts $\left\Vert AB\right\Vert _{\infty}\leq\left\Vert A\right\Vert _{\infty}\left\Vert B\right\Vert _{\infty}$ and $\left\Vert A\right\Vert _{\infty}\leq\sqrt{a}\left\Vert A\right\Vert _{2}$ for any symmetric matrix $A\in\mathbb{R}^{a\times a}$, we can bound $R_{1}$ in the following fashion: $$\begin{aligned} \left\Vert R_{1}\right\Vert _{\infty} & \leq & \left\Vert \Sigma_{K^{c}K}\Sigma_{KK}^{-1}\right\Vert _{\infty}\left\Vert \hat{\Sigma}_{KK}-\Sigma_{KK}\right\Vert _{\infty}\left\Vert \hat{\Sigma}_{KK}^{-1}\right\Vert _{\infty}\\ & \leq & (1-\phi)\left\Vert \hat{\Sigma}_{KK}-\Sigma_{KK}\right\Vert _{\infty}\sqrt{k}\left\Vert \hat{\Sigma}_{KK}^{-1}\right\Vert _{2}, \end{aligned}$$ where the last inequality uses (\[eq:S1\]). If $\phi=1$, then $\left\Vert R_{1}\right\Vert _{\infty}=0$ so we may assume $\phi<1$ in the following. Bound (\[eq:S8\]) from the proof for Lemma \[lem:A2\] yields $$\left\Vert \hat{\Sigma}_{KK}^{-1}\right\Vert _{2}\leq\frac{2}{\lambda_{\min}(\Sigma_{KK})}$$ with probability at least $1-c_{1}\exp\left(-c_{2}\log p\right)$. Now, we apply bound (\[eq:S3\]) from Lemma \[lem:A2\] with $t=\frac{\phi}{12(1-\phi)}\sqrt{\frac{\log p}{kn}}$ and obtain $$\mathbb{P}\left[\left\Vert \hat{\Sigma}_{KK}-\Sigma_{KK}\right\Vert _{\infty}\geq\frac{\phi}{12(1-\phi)}\sqrt{\frac{\log p}{kn}}\right]\leq2\exp\left(-c\left(\frac{\phi^{2}\log p}{\alpha^{4}(1-\phi)^{2}k^{3}}\right)\right),$$ provided $\frac{\phi}{12(1-\phi)\alpha^{2}k}\sqrt{\frac{\log p}{kn}}\leq1$. Then, if $\sqrt{\frac{\log p}{n}}\leq\lambda_{\min}(\Sigma_{KK})$, we are guaranteed that $$\mathbb{P}\left[\left\Vert R_{1}\right\Vert _{\infty}\geq\frac{\phi}{6}\right]\leq2\exp\left(-c\left(\frac{\phi^{2}\log p}{\alpha^{4}(1-\phi)^{2}k^{3}}\right)\right)+c_{1}\exp\left(-c_{2}\log p\right).$$ For $R_{2}$, note that $$\begin{aligned} \left\Vert R_{2}\right\Vert _{\infty} & \leq & \sqrt{k}\left\Vert \Sigma_{KK}^{-1}\right\Vert _{2}\left\Vert \hat{\Sigma}_{K^{c}K}-\Sigma_{K^{c}K}\right\Vert _{\infty}\\ & \leq & \frac{\sqrt{k}}{\lambda_{\min}(\Sigma_{KK})}\left\Vert \hat{\Sigma}_{K^{c}K}-\Sigma_{K^{c}K}\right\Vert _{\infty}. \end{aligned}$$ If $\frac{\phi}{6\alpha^{2}k}\sqrt{\frac{\log p}{kn}}\leq1$ and $\sqrt{\frac{\log p}{n}}\leq\lambda_{\min}(\Sigma_{KK})$, applying bound (\[eq:S2\]) from Lemma \[lem:A2\] with $t=\frac{\phi}{6}\sqrt{\frac{\log p}{kn}}$ yields $$\mathbb{P}\left[\left\Vert R_{2}\right\Vert _{\infty}\geq\frac{\phi}{6}\right]\leq2\exp\left(-c\left(\frac{\phi^{2}\log p}{\alpha^{4}k^{3}}\right)\right).$$ For $R_{3}$, applying (\[eq:S2\]) with $t=\phi\sqrt{\frac{\log p}{n}}$ to bound $\left\Vert \hat{\Sigma}_{K^{c}K}-\Sigma_{K^{c}K}\right\Vert _{\infty}$ and (\[eq:S4\]) to bound $\left\Vert \hat{\Sigma}_{KK}^{-1}-\Sigma_{KK}^{-1}\right\Vert _{\infty}$ yields $$\mathbb{P}\left[\left\Vert R_{3}\right\Vert _{\infty}\geq\frac{\phi}{6}\right]\leq c^{'}\left[\exp\left(-c\left(\frac{\phi^{2}\log p}{\alpha^{4}k^{3}}\right)\right)+\exp\left(-c\left(\frac{\log p}{k^{3}}\right)\right)\right],$$ provided that $\frac{\phi}{\alpha^{2}k}\sqrt{\frac{\log p}{n}}\leq1$ and $\sqrt{\frac{\log p}{n}}\leq\lambda_{\min}(\Sigma_{KK})$. Putting everything together, we conclude that $$\mathbb{P}\left[\left\Vert \hat{\Sigma}_{K^{c}K}\hat{\Sigma}_{KK}^{-1}\right\Vert _{\infty}\geq1-\frac{\phi}{2}\right]\leq c^{'}\exp\left(-b\left(\frac{\log p}{k^{3}}\right)\right)$$ for some positive constant $b$ that only depends on $\phi$ and $\alpha$. \[lem:A4\] Let the assumptions in Lemmas \[lem:A2\] and \[lem:A3\] hold. Suppose $\theta^{*}$ is exactly sparse with at most $k$ non-zero coefficients and $K=\left\{ j:\,\theta_{j}^{*}\neq0\right\} \neq\emptyset$. If we choose $\lambda\geq\frac{c\alpha\sigma\left(2-\frac{\phi}{2}\right)}{\phi}\sqrt{\frac{\log p}{n}}$ for some sufficiently large universal constant $c>0$, $\left|\hat{\delta}_{K^{c}}\right|_{\infty}\leq1-\frac{\phi}{4}$ with probability at least $1-c_{1}\exp\left(-b\frac{\log p}{k^{3}}\right)$, where $b$ is some positive constant that only depends on $\phi$ and $\alpha$. By construction, the subvectors $\hat{\theta}_{K}$, $\hat{\delta}_{K}$, and $\hat{\delta}_{K^{c}}$ satisfy the zero-subgradient condition in the PDW construction. With the fact that $\hat{\theta}_{K^{c}}=\theta_{K^{c}}^{*}=0_{p-k}$, we have $$\begin{aligned} \hat{\Sigma}_{KK}\left(\hat{\theta}_{K}-\theta_{K}^{*}\right)-\frac{1}{n}X_{K}^{T}\varepsilon+\lambda\hat{\delta}_{K} & = & 0_{k},\\ \hat{\Sigma}_{K^{c}K}\left(\hat{\theta}_{K}-\theta_{K}^{*}\right)-\frac{1}{n}X_{K^{c}}^{T}\varepsilon+\lambda\hat{\delta}_{K^{c}} & = & 0_{p-k}. \end{aligned}$$ The equations above yields $$\begin{aligned} \hat{\delta}_{K^{c}} & = & -\frac{1}{\lambda}\hat{\Sigma}_{K^{c}K}\left(\hat{\theta}_{K}-\theta_{K}^{*}\right)+X_{K^{c}}^{T}\frac{\varepsilon}{n\lambda},\\ \hat{\theta}_{K}-\theta_{K}^{*} & = & \hat{\Sigma}_{KK}^{-1}\frac{X_{K}^{T}\varepsilon}{n}-\lambda\hat{\Sigma}_{KK}^{-1}\hat{\delta}_{K}, \end{aligned}$$ which yields $$\hat{\delta}_{K^{c}}=\left(\hat{\Sigma}_{K^{c}K}\hat{\Sigma}_{KK}^{-1}\right)\hat{\delta}_{K}+\left(X_{K^{c}}^{T}\frac{\varepsilon}{n\lambda}\right)-\left(\hat{\Sigma}_{K^{c}K}\hat{\Sigma}_{KK}^{-1}\right)X_{K}^{T}\frac{\varepsilon}{n\lambda}.$$ Using elementary inequalities and the fact that $\left|\hat{\delta}_{K}\right|_{\infty}\leq1$, we obtain $$\left|\hat{\delta}_{K^{c}}\right|_{\infty}\leq\left\Vert \hat{\Sigma}_{K^{c}K}\hat{\Sigma}_{KK}^{-1}\right\Vert _{\infty}+\left|X_{K^{c}}^{T}\frac{\varepsilon}{n\lambda}\right|_{\infty}+\left\Vert \hat{\Sigma}_{K^{c}K}\hat{\Sigma}_{KK}^{-1}\right\Vert _{\infty}\left|X_{K}^{T}\frac{\varepsilon}{n\lambda}\right|_{\infty}.$$ By Lemma \[lem:A3\], $\left\Vert \hat{\Sigma}_{K^{c}K}\hat{\Sigma}_{KK}^{-1}\right\Vert _{\infty}\leq1-\frac{\phi}{2}$ with probability at least $1-c^{'}\exp\left(-\frac{b\log p}{k^{3}}\right)$; as a result, $$\begin{aligned} \left|\hat{\delta}_{K^{c}}\right|_{\infty} & \leq & 1-\frac{\phi}{2}+\left|X_{K^{c}}^{T}\frac{\varepsilon}{n\lambda}\right|_{\infty}+\left\Vert \hat{\Sigma}_{K^{c}K}\hat{\Sigma}_{KK}^{-1}\right\Vert _{\infty}\left|X_{K}^{T}\frac{\varepsilon}{n\lambda}\right|_{\infty}\\ & \leq & 1-\frac{\phi}{2}+\left(2-\frac{\phi}{2}\right)\left|\hat{X}^{T}\frac{\varepsilon}{n\lambda}\right|_{\infty}. \end{aligned}$$ It remains to show that $\left(2-\frac{\phi}{2}\right)\left|X^{T}\frac{\varepsilon}{n\lambda}\right|_{\infty}\leq\frac{\phi}{4}$ with high probability. This result holds if $\lambda\geq\frac{4\left(2-\frac{\phi}{2}\right)}{\phi}\left|X^{T}\frac{\varepsilon}{n}\right|_{\infty}$. In particular, Lemma \[lem:A1\](a) and a union bound imply that $$\mathbb{P}\left(\left|\frac{X^{T}\varepsilon}{n}\right|_{\infty}\geq t\right)\leq2\exp\left(\frac{-nt^{2}}{c_{0}\sigma^{2}\alpha^{2}}+\log p\right).$$ Thus, under the choice of $\lambda$ in Lemma \[lem:A4\], we have $\left|\hat{\delta}_{K^{c}}\right|_{\infty}\leq1-\frac{\phi}{4}$ with probability at least $1-c_{1}\exp\left(-b\frac{\log p}{k^{3}}\right)$. [^1]: Department of Economics, University of California, San Diego, 9500 Gilman Dr. La Jolla, CA 92093. Email: kwuthrich@ucsd.edu [^2]: Department of Economics, University of California, San Diego, 9500 Gilman Dr. La Jolla, CA 92093. Email: yiz012@ucsd.edu. [^3]: Alphabetical ordering; both authors contributed equally to this work. We would like to thank Stéphane Bonhomme, Gordon Dahl, Graham Elliott, Michael Jansson, Ulrich Müller, Andres Santos, Azeem Shaikh, Aman Ullah, Jeffrey Wooldridge, and seminar participants for their comments. We are especially grateful to Yixiao Sun for providing extensive feedback on an earlier draft. This paper was previously circulated as “Behavior of Lasso and Lasso-based inference under limited variability” and “Omitted variable bias of Lasso-based inference methods under limited variability: A finite sample analysis”. Zhu acknowledges a start-up fund from the Department of Economics at UCSD and the Department of Statistics and the Department of Computer Science at Purdue University, West Lafayette. [^4]: A natural idea to avoid OVBs due to the Lasso not selecting relevant controls is to choose a regularization parameter smaller than the recommended ones [e.g., @belloni2014inference]. However, as we show in simulations, this idea does not work in general and can lead to substantial biases. [^5]: The existing theoretical framework for the Lasso-based inference methods makes it difficult to derive an informative $\text{constant}$. To our knowledge, the literature provides no such derivations. [^6]: Note that the existing Lasso theory requires the regularization parameter to exceed a certain threshold, which depends on the standard deviations of the noise and the covariates. [^7]: Such a comparison is useful as the standard deviation in @belloni2014inference is the basis for their recommended inference procedures. We would also like to compare the OVBs to the conditional or unconditional finite sample standard deviation. However, this would require exact formulas of various selection probabilities (more than just bounds on the probabilities), which are impossible to derive for the Lasso in general. [^8]: Many applications of Lasso-based inference procedures feature such specifications [e.g., @belloni2014high; @belloni2014inference; @chen2015can; @decker2016health; @fremstad2017does; @knaus2018heterogenous; @jones2018what; @schmitz2017informal]. [^9]: In our theoretical results, however, $p$ is allowed to exceed $n$. [^10]: Note that traditional robust standard errors are inconsistent in settings with many controls [@cattaneo2018inference]. [^11]: Specifically, we set $\lambda=2\sigma\sqrt{\frac{2b(1+\tau)\log p}{n}}$, assuming that $\sigma$ is known. In practice, we first normalize $X_{i}$ such that $b=1$, run Lasso using $\lambda=2\sigma\sqrt{\frac{2(1+\tau)\log p}{n}}$, and then rescale the coefficients. [^12]: For all four values of $\sigma_x$, none of the controls with zero coefficients get selected with high probability. [^13]: Note that exact formulas of these mixture probabilities cannot be derived. [^14]: For both values of $\sigma_x$, none of the controls with zero coefficients get selected with high probability. [^15]: This approach is based on the following modified Lasso program: $$\hat{\theta}\in\arg\min_{\theta\in\mathbb{R}^{p}}\frac{1}{n}\sum_{i=1}^n\left(Y_i-X_i\theta \right)^2+\frac{\lambda}{n}\sum_{j=1}^p|\hat{l}_j\theta_j | \label{eq:general_lasso}$$ where $(\hat{l}_1,\dots,\hat{l}_p)$ are penalty loadings obtained using the iterative post Lasso-based algorithm developed in @belloni2012sparse. Our implementation is based on the Matlab code provided on the authors’ webpage: <https://voices.uchicago.edu/christianhansen/code-and-data/>. We set $\lambda=2c\sqrt{n}\Phi^{-1}(1-\varsigma/(2p))$, where $c=1.1$ and $\varsigma=0.1/\log n$ as recommended by @belloni2014inference [@belloni2017program]. Under homoscedasticity and with nearly orthogonal $X$, this regularization choice has a similar magnitude as the one in @bickel2009simultaneous.\[footnote\_bcch\] [^16]: \[footnote:over-selection\] A similar phenomenon arises with $0.5\lambda_{\text{BCCH}}$. This choice works well when $\alpha^\ast=0$, but yields biases when $\alpha^\ast\ne 0$. We found that, under our DGPs, this is related to the fact that when $\alpha^\ast\ne 0$, and differ in terms of the underlying coefficients and the noise level, which leads to differences in the (over-)selection behavior of the Lasso. [^17]: Here focus on the intention to treat effect of 401(k) eligibility on assets as in @Poterbaetal1994 [@Poterbaetal1995; @Poterbaetal1998] and @Benjamin2003. Some studies use 401(k) eligibility as an instrument for 401(k) participation [e.g., @CH2004; @belloni2017program]. [^18]: @fryerlevitt2013 also analyze test scores at the age of four years as well as eight months. [^19]: With interviewer fixed effects and based on the sample of all children, we were able to exactly replicate the results in Table 3 of @fryerlevitt2013. [^20]: The normality of $\varepsilon_{i}$ can be relaxed without changing the essence of our results. [^21]: Here (and similarly in the rest of propositions), we implicitly assume $p$ is sufficiently large such that $1-k\exp\left(\frac{-b^{2}\left(1+\tau\right)\log p}{4\phi^{2}}\right)-\frac{2}{p^{\tau}}>0$. Indeed, probabilities in such a form are often referred to as the “high probability” guarantees in the literature of (nonasymptotic) high dimensional statistics concerning large $p$ and small enough $k$. The event $\mathcal{M}$ is the intersection of $\left\{ \hat{I}_{1}=\hat{I}_{2}=\emptyset\right\}$ and an additional event; $\left\{ \hat{I}_{1}=\hat{I}_{2}=\emptyset\right\}$ occurs with probability at least $1-\frac{2}{p^{\tau}}$ and the additional event occurs with probability at least $1-k\exp\left(\frac{-b^{2}\left(1+\tau\right)\log p}{4\phi^{2}}\right)$. The additional event is needed in our analyses for technical reasons. See (\[eq:M\_event\]) of Appendix \[sec:appendix\_post\_double\_lasso\] for the definition of $\mathcal{M}$. [^22]: We thank Ulrich Müller for suggesting this comparison. [^23]: In general, $\lambda_{1}\asymp\phi^{-1}\left(\sigma_{\eta}+\left|\alpha^{*}\right|\sigma_{v}\right)\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}$ and the iterative algorithm for choosing $\lambda_{1}$ in @belloni2014inference described in Footnote \[footnote\_bcch\] achieves this scaling. Under the conditions on $\left(\beta_{K}^{*},\,\gamma_{K}^{*},\,\alpha^{*}\right)$ in Proposition \[prop:bias\_post\_double\_selection\], this scaling is equivalent to $\phi^{-1}\sigma_{\eta}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}$. [^24]: In fact, when the parameter space is unrestricted, OLS-based inference exhibits desirable optimality properties [e.g., @armstrong2016 Section 4.1]. [^25]: For a convex function $f:\,\mathbb{R}^{p}\mapsto\mathbb{R}$, $\delta\in\mathbb{R}^{p}$ is a subgradient at $\theta$, namely $\delta\in\partial f(\theta)$, if $f(\theta+\triangle)\geq f(\theta)+\left\langle \delta,\,\triangle\right\rangle $ for all $\triangle\in\mathbb{R}^{p}$. [^26]: Because $v$ and $\eta$ are independent of each other, the bound $\mathbb{P}\left(E\cap\mathcal{E}_{t^{*}}\right)\geq1-k\exp\left(\frac{-b^{2}\left(1+\tau\right)\log p}{4\phi^{2}}\right)-\frac{2}{p^{\tau}}$ can be further sharpened to $\mathbb{P}\left(E\cap\mathcal{E}_{t^{*}}\right)\geq\left(1-\frac{1}{p^{\tau}}\right)^{2}-k\exp\left(\frac{-b^{2}\left(1+\tau\right)\log p}{4\phi^{2}}\right)$. [^27]: We found that one of the reasons for the relatively poor performance of debiased Lasso is that $D$ is highly correlated with the relevant controls. Unreported simulation results show that debiased Lasso exhibits a better performance when $(D,X)$ exhibit a Toeplitz dependence structure as in the simulations reported by @vandergeer2014asymptotically. [^28]: This model and the multiplicative specification of heteroscedasticity are as in the simulations of @belloni2014inference.

--- abstract: 'Due to its interaction with the virtual electron-positron field in vacuum, the photon exhibits a nonzero anomalous magnetic moment whenever it has a nonzero transverse momentum component to an external constant magnetic field. At low and high frequencies this anomalous magnetic moment behaves as paramagnetic, and at energies near the first threshold of pair creation it has a maximum value greater than twice the electron anomalous magnetic moment. These results might be interesting in an astrophysical and cosmological context.' author: - 'S. Villalba-Chávez$^{\dag\ddag}$ and H. Pérez-Rojas$^{\ddag}$' date: 'July, $21 \ \ 2006$' title: 'Has the Photon an Anomalous Magnetic Moment?' --- It was shown by Schwinger[@Schwinger] in 1951 that electrons get an anomalous magnetic moment $\mu^\prime=\alpha/2\pi\mu_B$ (being $\mu_B=e\hbar/2m_0c$ the Bohr magneton) due to radiative corrections in quantum electrodynamics (QED), that is, due to the interaction of the electron with the background virtual photons and electron-positron pairs. We want to show that also, due to the interaction with the virtual quanta of vacuum, an anomalous photon magnetic moment arises. It is obtained from the expression for the photon self-energy in a magnetic field, calculated by Shabad[@shabad1; @shabad2] in an external constant magnetic field $\Pi_{\mu\nu}(x,x^{\prime\prime}\vert A^{ext})$ by starting from the electron-positron Green function in the Furry picture, and by using the Schwinger proper time method. The expression obtained was used by Shabad [@shabad2] to investigate the photon dispersion equation in vacuum in presence of an external magnetic field. It was found a strong deviation from the light cone curve near the energy thresholds for pair creation, which suggests that the photon propagation behavior in the external classical magnetic field is strongly influenced by the virtual electron-positron pairs of vacuum near these thresholds, showing a behavior similar to that of a massive particle. These phenomena become especially significant near the critical field $B_c=m_0^2/e \sim 4,41 \cdot 10^{13}$ Gauss, where $m_0, e$ are respectively the electron mass and charge. The photon magnetic moment might have astrophysical and cosmological consequences. For instance, photons passing by a strongly magnetized star, would experience an additional shift to the usual gravitational one produced by the star mass. In presence of an external field the current vector is non vanishing $j(x)_{\mu}=ie Tr\gamma_{\mu} G(x,x|A^{ext}) \neq 0$, where $G(x,x^{\prime}|A^{ext})$ is the electron-positron Green’s function in the external field. By calling the total electromagnetic field by $A^t_{\mu}= A^{ext}_{\mu}+A_{\mu}$, the QED Schwinger-Dyson equation for the photon field $A_\mu(x)$, propagating in the external field $A_\mu(x)^{ext}$ is $$\left[\square \eta_{\mu\nu}-\partial_\mu\partial_\nu\right] A^\nu(x)+\int\Pi_{\mu\nu}(x,x^\prime\vert A^{ext}) A^\nu(x^\prime) d^4x^\prime=0,\label{sdpmBF}$$ where $\mu,\nu=1,2,3,4$. The expression (\[sdpmBF\]) is actually the set of Maxwell equations in a neutral polarized vacuum, where the second term corresponds to the approximation of the four-current linear in $A_{\mu}$, where the coefficient is the polarization operator $\delta j_\mu (x)/\delta A^t_\nu (x^{\prime\prime})|_{A^t=A^{ext}}=\Pi_{\mu\nu}(x,x^{\prime\prime}\vert A_\mu^{ext})$. The external (constant and homogeneous) classical magnetic field is described by $A_\mu^{ext}(x)=1/2F_{\mu\nu}^{ext}x^\nu$, where the electromagnetic field tensor $F_{\mu \nu}^{ext}=\partial_\mu A_\nu^{ext}-\partial_\nu A_\mu^{ext}=B (\delta_{\mu 1}\delta_{\nu 2}-\delta_{\mu 2}\delta_{\nu 1})$ and $F^*_{\mu \nu}=\frac{i}{2}\epsilon_{\mu \nu \rho \kappa}F^{\rho \kappa}$ is its dual pseudotensor. To understand what follows it is necessary to recall some basic results developed in refs. [@shabad1],[@shabad2]. The presence of the constant magnetic field creates, in addition to the photon momentum four-vector $C^{4}_\mu=k_\mu$, three other orthogonal four-vectors which we write as four-dimensional transverse $k_\mu C^{i\mu}=0$ for $i=1,2,3$. These are $C^{1}_\mu= k^2 F^2_{\mu \lambda}k^\lambda-k_\mu (kF^2 k)$, $C^{2}_\mu=F^{*}_{\mu \lambda}k^\lambda$, $C^{3}_\mu=F_{\mu \lambda}k^\lambda$ ($C^{1,2,3}_{\mu}k^{\mu}=0$). We have $C^{4}_\mu C^{4\nu}=k_\mu k^\nu=0$ on the light cone. One gets from these four-vectors three basic independent scalars $k^2$, $kF^2k$, $kF^{*2}k$, which in addition to the field invariant ${\cal F}=\frac{1}{4}F_{\mu \rho}F^{\rho \mu}=\frac{1}{2}B^2$, are a set of four basic scalars of our problem. In momentum space it can be written the eigenvalue equation [@shabad1] $$\Pi_{\mu\nu}(k,k^{\prime\prime}\vert A_\mu^{ext})=\sum_i \pi^{(i)}_{n,n^\prime} a^{(i) \nu }a^{(i)}_\mu/(a^{(i)\nu }a^{(i)}_\nu ) \label{2}$$ In correspondence to each eigenvalue $\pi^{(i)}_{n,n^\prime}$ $i=1,2,3$ there is an eigenvector $a^{(i)\nu }$. The set $a^{(i)\nu }$ is obtained by simply normalizing the set of four vectors $C^{i}_\mu$. ($C^{4}_\mu=k_\mu$ leads to a vanishing eigenvalue due to the four-dimensional transversality property $\Pi_{\mu\nu}(k,k^{\prime\prime}\vert A_\mu^{ext})k_\mu=0$). The solution of the equation of motion (\[sdpmBF\]) can be written as a superposition of eigenwaves given by $$A_\mu(k)=\sum_{j=1}^4 \delta(k^2-\pi_j)a_\mu^j(k) \label{3}$$ By considering $a^{(i)}_\mu (x)$ as the electromagnetic four vector describing the eigenmodes, it is easy to obtain the corresponding electric and magnetic fields of each mode ${\bf e}^{(i)}= \frac{\partial }{\partial x_0}\vec{a}^{(i)}-\frac{\partial }{\partial {\bf x}}a^{(i)}_0$, ${\bf h}^{(i)}=\nabla\times\vec{a}^{(i)}$ (see [@shabad2]). From now on we specialize in a frame in which $x_3||B$. Then $kF^2k/2\mathcal{F}=-k_{\perp}^2$ and we name $z_1= k^2 + kF^2k/2\mathcal{F}=k_{\parallel}^2-\omega^2$. The previous results (see [@shabad2]) indicate the existence of three dispersion equations with the following structure $$k^2=\pi^{(i)}\left(z_1,k_{\perp}^2,eB\right).\ \\ \ i=1,2,3 \label{egg}$$ The eigenvalues $\pi^{(i)}$ contain only even functions of the external field through the scalars $kF^2k$, $kF^{*2}k$, and $e \sqrt{2 \cal F}=eB$ and can be expressed as a functional expansion in series of even powers of the product $e A_\mu^{ext}$ [@Fradkin]. One can solve (\[egg\]) for $z_1$ in terms of $k_{\perp}^2$. It results $$\omega^2=\vert\textbf{k}\vert^2+f_i\left(k_{\perp}^2,B\right) \label{eg2}$$ The term $ f_i$ contains the interaction of the photon with the virtual $e^{\pm}$ pairs in the external field in terms of the variables $k_{\perp}^2,B$. As it is shown in [@shabad2], it makes the photon dispersion equation to have a drastic departure from the light cone curve near the energy thresholds for free pair creation, We are thus in conditions to define an anomalous magnetic moment for the photon as $\mu_\gamma=-\partial \omega/\partial B$. Then $\mu_\gamma$ is a function of $B$. For weak fields ($B \ll B_c$), and frequencies small enough (see below), the function $f_i$ can be written as linear in $B$, the resulting dispersion law being then $$\omega=\vert\textbf{k}\vert- \mu_{\gamma} B \label{de0}$$ The first term corresponds to the light cone equation, which is modified by the second, which contains the contribution of the photon magnetic moment. The gauge invariance property $\pi^{(i)}(0,0)=0$ implies that the function $f_i(k_{\perp}^2,B)$ vanishes when $k_{\perp}^2=0$ [@proceeding]. This means that, due to gauge invariance, when the propagation is parallel to $\bf{B}$, $\mu_{\gamma}$ vanishes. Thus, in every mode of propagation $\mu_\gamma=0$ if $k_\perp=0$. Therefore the photon magnetic moment depends essentially on the perpendicular momentum component and this determines the optical properties of the quantum vacuum in presence of $B$. As a result, the problem of the propagation of light in empty space, in presence of an external magnetic field is similar to the problem of the dispersion of light in an anisotropic medium, where the role of the medium is played by the polarized vacuum in the external magnetic field. An anisotropy is created by the preferred direction in space along $\textbf{B}$. Therefore, the refraction index $n^{(i)}=\vert\textbf{k}\vert/\omega_i$ in mode $i$ is given in the case in which the approximate expression (\[de0\]) is valid as $$n^{(i)}=1+\frac{\mu_\gamma B }{\vert\textbf{k}\vert} \label{in}$$ For parallel propagation, $k_\perp=0$, for any mode it is obviously $n_i=1$. In [@shabad1; @shabad2] (see also [@proceeding]) it was shown that $\Pi_{\mu \nu}$ has singularities starting the value $z_1=-4m_0^2$, which is the first threshold for pair creation, corresponding to Landau quantum numbers $n=n^{\prime}=0$. Other pair creation thresholds are given by $k_{\perp}^{\prime}=m_0^2[(1+2 n B/B_c)^{1/2}+(1+2n^\prime B/B_c)^{1/2}]^2$, with the electron and positron in excited Landau levels $n,n^{\prime}\neq 0$). In what follows we will work in the transparency region, that is, out from the region for absorption due to the pair creation i.e., $\omega^2-k_\parallel^2\leq k_{\perp}^{\prime 2}$ (*i.e.* within the kinematic domain, where $\pi_{1,2,3}$ are real. We will be interested in two limits, i.e., when its energy is near the first pair creation threshold energy (and the magnetic field $B \sim B_c$), and when it is much smaller than it, $4m_0^2\gg \omega^2 $ and small fields $(B\ll B_c)$, in the one loop approximation. Below the first threshold the eigenvalues corresponding to the first and third modes do not contribute, whereas the second mode it is shown in [@prd], by using the formalism developed iny [@shabad1; @shabad2] that the eigenvalue near this threshold is in that limit $$\pi_2=-\frac{2\mu^{\prime}B}{m_0}\left[z_1 \exp\left(-\frac{k_{\perp}^2}{2eB}\right)\right].\label{pi22}\\$$ Here $\mu^\prime=(\alpha/2\pi)\mu_B$ is the anomalous magnetic moment of the electron. The exponential factor in (\[pi22\]) plays a very important role. If $2 e B\ll k_{\perp}^2 $, it would make the exponential factor negligible small and in the limit $B\to 0$, it vanishes (as well as $\mu_\gamma$ below). In the opposite case, if $k_{\perp}^2 \ll 2eB$, the exponential is of order unity. By considering the last assumption and the case of transversal propagation ($k_{\parallel}=0$), the dispersion equation for the second mode has the solution $$\omega^2= k_\parallel^2+k_\perp^2\left(1+\frac{2\mu^{\prime}B}{m_0}\right)^{-1} \label{de1}$$ from which it results that the photon energy can be expressed approximately as a linear function of the external field B, as indicated in (\[de0\]), where $$\mu_\gamma^{(2)}=\frac{\mu^\prime k_\perp^2}{m_0\omega} \label{de11}$$ Notice that, as pointed out above, in the limit $B=0$, if $k_\perp^2 \neq 0$, then $\mu_\gamma^{(2)}$ vanishes. By considering transversal propagation and $\omega \simeq k_{\perp}$ one can write $$\mu_\gamma^{(2)}=\frac{\mu^{\prime}\vert\textbf{k}_{\perp}\vert}{ m_0}. \label{lemm}$$ As $\mu_\gamma^{(2)}>0$, the magnetic moment is paramagnetic, which is to be expected since vacuum in a magnetic field behaves as paramagnetic [@Elizabeth]. For photon energies $\omega \sim 10^{-6}m_0$ and $B \sim 10^{4}$G, we have $\mu_{\gamma}\sim 10^{-6}\mu^{\prime}$. In a more exact approximation we must take into account the contribution from higher Landau quantum numbers. We will be interested now on the photon magnetic moment in the region near the thresholds, and for fields $B \lesssim B_c$. The eigenvalues of the modes can be written approximately [@Hugo2] as $$\pi_{n,n^{\prime}}^{(i)}\approx-2\pi\phi_{n,n^{\prime}}^{(i)}/\vert\Lambda\vert \label{eg5}$$ with $\vert\Lambda\vert=((k_\perp^{\prime 2 }-k_\perp^{\prime \prime 2})(k_\perp^{\prime 2}-\omega^2+k_\parallel^2))^{1/2}$ with $k_\perp^{\prime\prime 2}=m_0^2[(1+2nB/B_c)^{1/2}-(1+2n^{\prime}B/B_c)^{1/2}]^2$, is the squared threshold energy for excitation between Landau levels $n,n^{\prime}$ of an electron or positron. The functions $\phi_{n,n^{\prime}}^{(i)}$ are expressed in terms of Laguerre functions of the variable $k_\perp^2/2e B$. In the vicinity of the first resonance $n=n^{\prime}=0$ and considering $k_\perp\neq0$ and $k_\parallel\neq0$, according to [@shabad1; @shabad2] the physical eigenwaves are described by the second and third modes, but only the second mode has a singular behavior near the threshold and the function $\phi^{(2)}_{n n'}$ has the structure $$\phi_{0,0}^{(2)}\simeq-\frac{2\alpha e B m_0^2}{\pi}\textrm{exp}\left(-\frac{k_\perp^2}{2e B}\right)$$ In this case $k_{\perp}^{\prime\prime 2}=0$ and $k_{\perp}^{\prime 2}=4m_0^2$ is the threshold energy. By using the approximation given by (\[eg5\]) the dispersion equation (\[egg\]) is turned into a cubic equation in the variable $z_1$ that can be solved by applying the Cardano formula. We will refer in the following to (\[eg2\]) as the real solution of this equation. We should define the functions $ m_n=(k_\perp^{\prime}+k_\perp^{\prime\prime })/2$, $ m_{n^{\prime}}=(k_\perp^{\prime }-k_\perp^{\prime\prime })/2$ and $ \Lambda^{*}=4m_nm_{n^\prime} (k_\perp^{\prime 2}-k_\perp^2)$ to simplify the form of the solutions (\[eg2\]) of the equation ([\[egg\]]{}). The functions $f_{i}$ are dependent on $k_{\perp}^{2},k_{\perp}^{\prime 2},k_{\perp}^{\prime\prime 2}, B$, and are $$f_i^{(1)}=\frac{1}{3}\left[2k_\perp^{2}+k_\perp^{\prime 2}+\frac{\Lambda^{* 2}}{(k_\perp^{\prime\prime 2}-k_\perp^{\prime 2})\mathcal{G}^{1/3}}+ \frac{\mathcal{G}^{1/3}}{k_\perp^{\prime\prime 2}-k_{\perp}^{\prime 2}}\right] \label{fi}$$ where $\mathcal{G} =6 \pi\sqrt{3}D-\Lambda^{*3}+54 \pi^2 \phi_{n,n^{\prime}}^{(i) 2}(k_\perp^{\prime\prime 2}-k_\perp^{\prime 2})^2$ with $$D=\sqrt{-(k_\perp^{\prime 2}-k_\perp^{\prime\prime 2})^2\Lambda^{*3}\phi_{n,n^{\prime}}^{(i) 2}\left[1-\frac{27 \pi^2 \phi_{n,n^{\prime}}^{(i)2}(k_\perp^{\prime\prime2}-k_\perp^{\prime 2})^2}{\Lambda^{*3}}\right]}$$ Besides (\[fi\]), there are two other solutions of the above- mentioned cubic equation resulting from the substitution of (\[eg5\]) in (\[egg\]). These are complex solutions and are located in the second sheet of the complex plane of the variable $z_1=\omega^2-k_\parallel^2$ but they are not interesting to us in the present context. Now the magnetic moment of the photon can be calculated by taking the implicit derivative $\partial\omega/\partial B$ in the dispersion equation. From (\[egg\]) and (\[eg5\]) it is obtained that $$\mu_\gamma^{(i)}=\frac{\pi}{\omega(\vert \Lambda\vert^3-4\pi\phi_{n,n^\prime}^{(i)}m_n m_{n^\prime})}\left[\phi_{n,n^{\prime}}^{(i)}\left(A\frac{\partial m_n}{\partial B}+Q\frac{\partial m_{n^\prime}}{\partial B}\right)-\Lambda^2\frac{\partial \phi_{n,n^{\prime}}^{(i)}}{\partial B}\right] \label{mm2}$$ with $ A= 4m_{n^\prime}[z_1+(m_n+m_{n^\prime})(3m_n+m_{n^\prime})] $ and $ Q= 4m_{n}[z_1+(m_n+m_{n^\prime})(m_n+3m_{n^\prime})] $. In the vicinity of the first threshold $k_\perp^{\prime\prime 2}=0$, $k_\perp^{\prime 2}=4m_0^2$ and $\partial m_n/\partial B=0$ when $n=0$, therefore for the second mode the photon magnetic moment is given by $$\mu_\gamma^{(2)}=\frac{\alpha m_0^3\left(4m_0^2+z_1\right)\exp\left(-\frac{k_{\perp}^2}{2eB}\right)}{\omega B_c\left[(4m_0^2+z_1)^{3/2}+\alpha m_0^3 \frac{B}{B_c} \exp\left(-\frac{k_{\perp}^2}{2eB}\right)\right]}\left(1+\frac{k_{\perp}^2}{2eB}\right).\label{FRR1}$$ ![\[fig:Phvk\] Photon magnetic moment curve drawn with regard to perpendicular momentum squared, for the second mode $k_\perp^{\prime 2}=4m_0^2$ with $n=n^{\prime}=0$.](prl1.eps){width="3in"} It is easy to show that this function has a maximum near the threshold. If we consider $\omega$ near $2m_0$, the function $\mu_{\gamma}^{(2)}=f(X)$, where $X=\sqrt{4m_0^2-\omega^{2}}$ has a maximum for $X= {\pi\phi_{00}^{(2)}/m_0}^{1/3}$, which is very close to the threshold. Thus, near the first threshold and in the second mode of propagation the expression (\[mm2\]) has a maximum value when $k_\perp^2 \simeq k_\perp^{\prime 2}$. Therefore in a vicinity of the first pair creation threshold the magnetic moment of the photon has a resonance peak which is positive, indicating a paramagnetic behavior, and its value is given by $$\mu_{\gamma}^{(2)}=\frac{m_0^2(B+2B_c)}{3m_\gamma B^2}\left[2\alpha\frac{B}{B_c}\exp\left(-\frac{2B_c}{B}\right)\right]^{2/3} \label{mumax}$$ Obviously, (\[mumax\]) would vanish also for $B\to 0$. The maximum of (\[mumax\]) is given numerically by $$\mu_\gamma^{(2)}\approx 3\mu^\prime \left(\frac{1}{2\alpha}\right)^{1/3}\approx 12.85\mu^\prime$$ Thus, the maximum value achieved by the photon magnetic moment under the assumed conditions is larger than twice the anomalous magnetic moment of the electron. In (\[mumax\]) we introduced the quantity $m_{\gamma}$ which has meaning near the thresholds, and which could be named as the “dynamical mass” of the photon in presence of a strong magnetic field, which is defined by the equation $$m_{\gamma}^{(2)}=\omega (k_\perp^{\prime 2})=\sqrt{4m_0^2-m_0^2\left[2\alpha \frac{B}{B_c}\exp\left(-\frac{2B_c}{B}\right)\right]^{2/3}} \label{dm}$$ The “dynamical mass” accounts for the fact that the massless photon coexists with the massive pair near the thresholds, leading to a behavior very similar to that of a neutral massive vector particle bearing a magnetic moment. However, it does not violate gauge invariance since the condition $\Pi_{\mu \nu}(0,0,B) =0$ is preserved. The idea of a photon mass has been introduced previously, for instance in ref.[@Osipov], in a regime different from ours, in which $k_{\parallel}\gg 4m^2$. ![\[fig:mb1\] Photon magnetic moment behavior with regard to external magnetic field strength for the second mode.](prl2.eps){width="3in"} We conclude, thus, that for photons in a strong magnetic field a nonzero magnetic moment arises, which is paramagnetic, and has a maximum near the first threshold of pair creation. These results may have several interesting consequences. For instance, if we consider a photon beam of density $n_\gamma$, it carries a magnetization ${\cal M}=n_\gamma \mu_{\gamma}^{(2)}$ which contributes to increasing the field $B$ to $B^{\prime}= B + 4\pi{\cal M}$. Trough this mechanism, the radiation field might contribute to the increase of the external field. Both authors are indebted to A.E. Shabad for several comments and important remarks on the subject of this paper. H.P.R. thanks G. Altarelli, J. Ellis and P. Sikivie for comments, and to CERN, where part of this paper was written, for hospitality. [18]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , ****, (). , ****, (). , ****, (). , ****, (). ****, (). , , , , , . , ****, (). , , ****, ().

--- abstract: 'We present 3D magnetohydrodynamic (MHD) numerical simulations of the evolution of self–gravitating and weakly magnetized disks with an adiabatic equation of state. Such disks are subject to the development of both the magnetorotational and gravitational instabilities, which transport angular momentum outward. As in previous studies, our hydrodynamical simulations show the growth of strong $m=2$ spiral structure. This spiral disturbance drives matter toward the central object and disappears when the Toomre parameter $Q$ has increased well above unity. When a weak magnetic field is present as well, the magnetorotational instability grows and leads to turbulence. In that case, the strength of the gravitational stress tensor is lowered by a factor of about 2 compared to the hydrodynamical run and oscillates periodically, reaching very small values at its minimum. We attribute this behavior to the presence of a second spiral mode with higher pattern speed than the one which dominates in the hydrodynamical simulations. It is apparently excited by the high frequency motions associated with MHD turbulence. The nonlinear coupling between these two spiral modes gives rise to a stress tensor that oscillates with a frequency which is a combination of the frequencies of each of the modes. This interaction between MHD turbulence and gravitational instabilities therefore results in a smaller mass accretion rate onto the central object.' author: - 'Sébastien Fromang, Steven A. Balbus, Caroline Terquem and Jean–Pierre De Villiers' bibliography: - 'author.bib' title: ' Evolution of self–gravitating magnetized disks. II- Interaction between MHD turbulence and gravitational instabilities' --- Introduction ============ In systems such as the disks surrounding low mass protostars or active galactic nuclei, the simultaneous appearance of both gravitational and magnetic instabilities is expected. During the first stages of their evolution, for example, protoplanetary disks are expected to be rather massive because of strong infall from the parent molecular cloud. As the disk builds up in mass as a result of the collapse of an envelope, its surface mass density becomes large enough for gravitational instabilities to develop (e.g.,  ). These disks are also believed to be sufficiently ionized, at least over some extended regions, to be coupled to a magnetic field [@gammie96; @sanoetal00; @fromang02]. By modeling the outer parts of disks around quasi-stellar objects (QSOs) as steady, viscous, geometrically thin, and optically thick, @goodmanj03 has argued that they are self-gravitating. More precisely, he predicts self–gravitational instabilities to develop beyond about $10^{-2}$ parsecs from the central object. In addition, it has been suggested by that self-gravitating regions of disks around QSOs are likely to be coupled to a magnetic field. The stability of a thin, self-gravitating gas disk is controlled by the Toomre $Q$ parameter [@toomre64]: $$Q=\frac{c_s\kappa}{\pi G \Sigma} \, ,$$ where $c_s$ is the sound speed, $\kappa$ is the epicyclic frequency (see, e.g.,  ), $\Sigma$ is the disk surface mass density and $G$ is the gravitational constant. Gaseous disks are unstable against axisymmetric perturbations when $Q \le 1$, and against non-axisymmetric perturbations when $Q {\raisebox{-.8ex}{$\buildrel{\textstyle>}\over\sim$}}1$. Since analytical predictions of the nonlinear evolution of gravitational instabilities are difficult, there have been a large number of numerical simulations of gravitationally unstable disks. Despite the rather daunting technical problems of combining three-dimensional (3D) hydrodynamic calculations with rapid and accurate Poisson equation solvers, significant progress have been made. To do so, the energetics must be treated crudely, with the focus squarely on purely dynamical behavior. Using this strategy, the above $Q$ criterion for instability has been confirmed (and shown to still be approximately valid for disks of finite thickness), and the properties of the unstable modes have been studied as a function of the disk parameters . Several authors have investigated the saturation properties of the instability, and have shown that it is capable of transporting significant amount of mass and angular momentum in a few orbital times . The first calculations mostly used simple adiabatic equations of state (EOS). More recently, isothermal disks have also been studied [@pickett98; @pickett00a; @boss98; @mayer02]. Some new investigations also include a simplified treatment of the disk radiative cooling [@pickett03; @rice03; @boss02]. All these models were purely hydrodynamical, and neglected the effect of magnetic fields. However, it is known that stability of astrophysical disks is extremely sensitive to the presence of weak magnetic fields. In particular, the magnetorotational instability (MRI) completely disrupts laminar Keplerian flow when a subthermal magnetic field of any geometry is present. This was first understood by . Since then, it has been shown through many numerical simulations that the nonlinear outcome of the MRI is MHD turbulence, which, in common with gravitational instabilities, transports angular momentum outward (see , or @balbusaraa03 [-@balbusaraa03], for a review). Since disks around low–mass stars and around QSOs may be both magnetized and self–gravitating, the spiral structure gravitational transport described above must somehow develop in a medium in the throes of MHD turbulence. The question naturally arises as to how these two powerful instabilities interact with one another. What is the ultimate effect on the global properties of accretion disks, and in particular, on the critical transport properties of mass and angular momentum? To keep this initial investigation tractable, we must restrict ourselves here to an adiabatic EOS. But the dynamical behavior of “simple” adiabatic disks is still rich, and contains unanticipated findings. In a companion paper to this one (@fromangetal04a [-@fromangetal04a], hereafter paper I), we carried out 2D axisymmetric numerical simulations of the evolution of massive and magnetized disks. The results show that the MRI behaves in a self–gravitating environment as it does in zero mass disks. Turbulent transport of angular momentum causes the disk to evolve toward a two component structure: (1) an inner thin disk in Keplerian rotation fed by (2) an outer thick disk whose rotation profile deviates from Keplerian, strongly influenced by self-gravity. However, angular momentum transport by gravitational instabilities cannot develop in axisymmetric simulations, which leaves unanswered the question of the outcome of the interaction between both instabilities. This is the subject of the present paper. The plan of the paper is as follows: in section 2, we present our numerical methods. The initial state of our simulations will be described in section 3. We present our results in section 4 and, finally, give our conclusions in section 5. Numerical methods ================= Algorithms ---------- The calculations in this paper are based on the equations of ideal MHD: $$\begin{aligned} \frac{\partial \rho}{\partial t} + {{ \mbox{\boldmath{$\nabla$}} }}{{ \mbox{\boldmath{$\cdot$}} }}(\rho {\bf v}) = 0, \\ \rho \left( \frac{\partial {\bf v}}{\partial t} + {\bf v} {{ \mbox{\boldmath{$\cdot$}} }}{{ \mbox{\boldmath{$\nabla$}} }}{\bf v} \right) = - {{ \mbox{\boldmath{$\nabla$}} }}P - \rho {{ \mbox{\boldmath{$\nabla$}} }}\Phi + \frac{1}{4 \pi} ({{ \mbox{\boldmath{$\nabla$}} }}{{ \mbox{\boldmath{$\times$}} }}{\bf B}) {{ \mbox{\boldmath{$\times$}} }}{\bf B}, \\ \rho \left( \frac{\partial }{\partial t} + {\bf v} {{ \mbox{\boldmath{$\cdot$}} }}{{ \mbox{\boldmath{$\nabla$}} }}\right) \left( \frac{e}{\rho} \right) = -P {{ \mbox{\boldmath{$\nabla$}} }}{{ \mbox{\boldmath{$\cdot$}} }}{\bf v}, \\ \frac{\partial {\bf B}}{\partial t} = {{ \mbox{\boldmath{$\nabla$}} }}{{ \mbox{\boldmath{$\times$}} }}( {\bf v} {{ \mbox{\boldmath{$\times$}} }}{\bf B} ), \label{MHD equations}\end{aligned}$$ where $\rho$ is the mass density, $e$ is the energy density, $\bf{v}$ is the fluid velocity, $\bf{B}$ is the magnetic field, $P$ is the gas pressure and $\Phi=\Phi_s+\Phi_c$ is the total gravitational potential, which has contributions $\Phi_s$ from the disk self–gravity and $\Phi_c$ from a central mass. The Poisson equation determines the gravitational potential, $$\nabla^2 \Phi_s = 4 \pi G \rho,$$ and to close our system of equations, we adopt an adiabatic equation of state for a monoatomic gas: $$P = (\gamma -1)e, \quad \gamma = 5/3 . \label{EOS}$$ To solve these equations, we use the GLOBAL code . This uses standard cylindrical coordinates $(r, \phi, z)$ and time–explicit Eulerian finite differences. The magnetic field is evolved using the combined Method of Characteristics and Constrained Transport algorithm (MOC–CT), which preserves the divergence of the magnetic field to machine accuracy. Finally, we use outflow boundary conditions in the radial and vertical directions, and periodic boundary conditions in $\phi$. In its original form, GLOBAL did not include a Poisson solver, and the development of such a routine represents a major technical component of the results we report here. The calculation is done in two steps. The potential $\Phi_s$ is first computed at the grid boundary, using the spectral decomposition decribed below, and then calculated on the whole grid using a very rapid method. It is the first step, the boundary calculation, that is computationally expensive. In the expansion of $\Phi_s$, we have adopted the method of , which uses half–integer Legendre functions in the Green’s function. This method is better suited to cylindrical coordinates than the traditional expansion in spherical harmonics, which are of course tailored to spherical coordinates. Following , $\Phi_s$ may be written $$\Phi_s (r,\phi,z) = - \frac{G}{\pi \sqrt{r}} \int_{V} d\tau' \; \frac{\rho(r',\phi',z')}{\sqrt{r'}} \sum_{m=0}^{\infty} \epsilon_m Q_{m-1/2}(\chi) \cos m(\phi - \phi') \; . \label{gravpot calc}$$ Here, $d\tau' = r' dr' d\phi' dz'$ is the elementary volume element, and the integral is taken over the whole computational domain. $Q_{m-1/2}$ denotes the half–integer order Legendre function of the second type [@mathfunc]. The argument $\chi$ is a function of position: $$\chi = \frac{r^2+r'^{2}+(z-z')^2}{2rr'} \, .$$ The Legendre functions are computed once at the beginning of each simulation and stored in memory. At each time step, we calculate $\Phi_s$ using equation (\[gravpot calc\]), in which the sum over $m$ is truncated at some upper value $m_{max}$. We then calculate $\Phi_{s}$ everywhere on the grid, using a combination of a Fourier transform in $\phi$ and the 2D Successive Over Relaxation (SOR) Method [@hirsh88] in the $(r,z)$ plane. Although this is an efficient method, the calculation of the self–gravitating potential is still very demanding of computational resources. For the resolution $(N_r,N_{\phi},N_z)=(128,64,128)$ used in this paper, the time required by the Poisson solver still represents $\sim 40\%$ of the computation time for $m_{max}=8$. Diagnostics ----------- We introduce and define some key quantities that have been used to analyze the results of the simulations. We denote the ratio of the volume averaged thermal pressure to the volume averaged magnetic pressure as $\langle \beta \rangle$: $$\langle \beta \rangle = \frac{\langle P \rangle}{\langle B^2/8\pi \rangle} \, .$$ This parameter is used primarily as a measure of the initial magnetic field strength. In 3D numerical simulations of magnetized self–gravitating disks, angular momentum is transported by the sum of the Maxwell, Reynolds, and gravitational stress tensors. Following and @hawley00, we define the height and azimuthal averages (noted with an overbar) of each these respective stresses as: $$\begin{aligned} T^{Max}_{r\phi}(r,t) & = & - \frac{\overline{B_rB_{\phi}}}{4\pi} \, ,\\ T^{Ren}_{r\phi}(r,t) & = & \overline{\rho v_r v_{\phi}}-\frac{\overline{\rho v_r} \textrm{ } \overline{\rho v_{\phi}}}{\overline{\rho}} \, ,\\ T_{r\phi}^{grav}(r,t) & = & \frac{1}{4\pi G} \overline{ \frac{\partial \Phi_s}{\partial r} \frac{1}{r} \frac{\partial \Phi_s}{\partial \theta} } \, .\end{aligned}$$ As in paper I, volume averages of these quantities will be denoted as $\langle T^{Max}_{r\phi}\rangle(t)$, etc. Note that $T_{r\phi}^{grav}$ is associated with the gravitational torque resulting from non-axisymmetric disk structure. This quantity clear vanishes in an axisymmetric simulation ($m_{max}=0$). In this case, the standard $\alpha$ parameter can be defined as the sum of the Maxwell and Reynolds stress tensors normalized by the gas pressure: $$\alpha(r,t)=\frac{T^{Max}_{r\phi}(r,t)+ T^{Ren}_{r\phi}(r,t)}{\overline{P(r,t)}} \, . \label{eq:alpha}$$ Initial model ============= We start our simulations with a disk model which is as close as possible to hydrostatic equilibrium: $$- {{ \mbox{\boldmath{$\nabla$}} }}P - \rho {{ \mbox{\boldmath{$\nabla$}} }}\left( \Phi_s + \Phi_c \right) + \rho r \Omega ^2 {{ \mbox{\boldmath{$e$}} }_r} = {\bf 0} \, . \label{hydrostatic equilibrium}$$ Here $\Omega$ is the angular velocity and ${ \mbox{\boldmath{$e_r$}} }$ is the unit vector in the radial direction. The coordinate system has its origin on the disk center. The potential $\Phi_c$ is due to a central mass $M_c$. We chose $M_c= 2 M_d$, where $M_d$ is the disk mass. The initial disk model is gravitationally unstable. Because of the presence of the disk self–gravity, equation (\[hydrostatic equilibrium\]) has to be solved iteratively. We use the Self–Consistent Field (SCF) iterative method developed by @hachisu86. In this method, the radial profile of the angular velocity $\Omega$ or, equivalently, the specific angular momentum $j=r^2 \Omega$, is specified. Following @pickett96, we fix $j(r)$: $$j=j_{r0} \left(\frac{M_r+M_c}{M_d+M_c}\right)^{q}$$ where $j_{r0}$ and $q$ are constant, and $M_r$ is the disk mass within radius $r$. Setting $q=2$ gives a $j$ profile close to that used by @pickett96. We begin the iteration with an arbitrary mass density $\rho$, from which we can calculate $\Phi_s$. From $\rho$ and the above expression for $j$ we also calculate $\Omega$ (note that it still depends on the constant $j_{r0}$). The relation (\[hydrostatic equilibrium\]) is then integrated to give the value of the enthalpy $h$: $$h =\frac{5}{2}K \rho^{2/3} = C - \left( \Phi_s + \Phi_c \right) + \int r \Omega ^2 dr \, ,$$ where the constant $C$ and $j_{r0}$ are determined from the boundary conditions $\rho=0$ at $\left( r=R_{in}, z=0 \right)$ and at $\left( r=R_{out},z=0 \right)$. Here $R_{in}$ and $R_{out}$ are the radial boundaries of the disk. The new density field is then calculated from $h$ using the normalizing condition that $\rho_{max}=1$, which determines the polytropic constant $K$. Upon iterating this procedure, we converge to a model very close to equilibrium. The resulting disk model (with $R_{in}=0.25$ and $R_{out}=1$) has an $\Omega$ profile close to Keplerian and a density profile displayed in figure \[initial model\]. Note that the disk is rather thin, with an aspect ratio $H/r$ varying between $0.1$ and $0.2$. As noted above, the ratio $M_c/M_d$ is chosen in such a way that the Toomre $Q$ parameter is initially close to unity. The radial profile of $Q$ in the initial disk model is shown in figure \[initial toomre\]. The minimum value of $Q$ is approximately $1.1$, and $Q$ is close to unity over a large range of radii. We therefore expect strong non–axisymmetric gravitational instabilities to develop in this disk. Results ======= Table \[models\] lists the parameters of the different runs we present below. Column $1$ gives the label of the model. HD refers to a hydrodynamical run. Models T and P start with a purely toroidal and poloidal magnetic field, respectively. Column 2 gives the computational azimuthal domain and column 3 gives the highest fourier component of the gravitational potential. When $m_{max}=0$, i.e. when only the $m=0$ component in the Fourier expansion of $\Phi_s$ is included, gravitational instabilities cannot develop (recall that $Q>1$, so that the disk is stable against axisymmetric perturbations). Therefore, models T1 and P1 enable us to study the evolution of MHD turbulence and to compare it with previous work and with the 2D simulations of paper I. In models T2, T2$_{low}$, T2$^{*}$, T3 and P2, both gravitational and magnetic instabilities develop. In model T2$^{*}$, the non–axisymmetric part of $\Phi_s$ is included only after $6$ orbits, i.e. after MHD turbulence has established itself. Column 4 gives the ratio of the volume-averaged thermal and magnetic pressures and column 5 gives the resolution $(N_r,N_{\phi},N_z)$ of the run. Model $\phi$–range $m_{max}$ $\langle \beta \rangle$ Resolution ------------ -------------- ------------------- ------------------------- ---------------- -- HD $[0,\pi]$ $8$ $\infty$ $(128,64,64)$ T1 $[0,\pi /2]$ $0$ $8$ $(128,32,128)$ T2 $[0,\pi]$ $8$ $8$ $(128,64,128)$ T2$_{low}$ $[0,\pi]$ $8$ $8$ $(64,64,64)$ T2$^*$ $[0,\pi]$ $0$–$8^{\dagger}$ $8$ $(128,64,128)$ T3 $[0,\pi]$ $16$ $8$ $(128,64,128)$ P1 $[0,\pi /2]$ $0$ $300$ $(128,32,128)$ P2 $[0,\pi]$ $8$ $300$ $(128,64,128)$ : Model parameters. Column 2 gives the computational azimuthal domain, column 3 gives the highest Fourier component of the gravitational potential included in the calculation, column 4 gives the ratio of the volume averaged thermal and magnetic pressures and column 5 gives the resolution $(N_r,N_{\phi},N_z)$ of the run. Model HD is hydrodynamical. Models T and P start with a purely toroidal and poloidal magnetic field, respectively. When $m_{max}=0$, the disk self–gravitating potential is forced to stay axisymmetric. In model T2$^*$, $m_{max}=0$ at the beginning of the run and is set to 8 after a few orbits. []{data-label="models"}     [$^\dagger$ For this run, $m_{max}=0$ when $t \in [0,5.8]$, while $m_{max}=8$ for $t>5.8$.]{} In all the models, an adiabatic equation of state is used. The computational domain extends radially from 0.1 to 1.4, and vertically from $-0.2$ to $0.2$. In the azimuthal direction, the computational domain extends from 0 to either $\pi/2$ or $\pi$. The smaller range is used in the $m_{max}=0$ gravitationally stable cases. Indeed, @hawley00 and have shown that an azimuthal domain of $\pi/3$ is generally sufficient to describe the transport properties of MHD turbulence. When we allow for the development of gravitational instabilities, we restrict the azimuthal domain to the half disk $[0,\pi]$. This saves computational time, but of course allows only even modes to develop. The focus of the paper is not on the detailed spectrum of modes which appear in a given disk model, however, but on the interaction between MHD turbulence and the largest scale gravitational modes. This interaction should not be particularly sensitive to whether an integer number of modes exactly fits in the half disk. Time is measured in units of the orbital period at the initial outer edge $R_{out}=1$ of the disk model. Typical simulations are carried out for $8$ to $10$ orbits at this position. This corresponds to 60–80 orbits at the initial disk inner edge. The simulations are seeded by adding to the mass density at $r> 0.4$ random perturbations with a relative amplitude of $5 \times 10^{-3}$. We now describe in turn the hydrodynamical run, the simulations with only MHD turbulence, and the runs with both gravitational and magnetic instabilities. Control Hydrodynamical Run: Model HD ------------------------------------ The time evolution of the Fourier components of the density in the equatorial plane is shown in figure \[time\_fourier\_hydro\] for the modes $m=2, 4$ and $6$ (from top to bottom). The $m=2$ mode grows at the beginning of the simulation and saturates after $4$ orbits. Higher $m$ modes emerge after about $3$ orbits. Apart from the $m=4$ mode, which may also be linearly unstable, the $m>2$ modes appear to be non–linearly excited. The development of a $m=2$ spiral structure may be seen in figure \[image220 hydro\], which shows the logarithm of the density in the equatorial plane at $t=4.27$. (Note that the result of the simulation has been extended by symmetry to cover the range $[0,2\pi]$). This mode is clearly global. Its pattern speed is $\Omega_p=6.28$, which means that corotation (the radius where the gas angular velocity matches the pattern speed) is located at the initial outer edge of the disk. Such a mode is predicted to emerge by linear stability analyses of self-gravitating disks . The instability is due to the interaction between waves that propagate near the outer boundary and waves that reside inside the inner Lindblad resonance (where the pattern speed in the frame corotating with the planet matches the gas epicyclic frequency). This is located at $r\sim 0.6$ in our disk model. During the simulation, matter is driven toward the disk center by the gravitational torque associated with the spiral arms and, at $t \simeq 8$, $Q$ has become sufficiently high (${\raisebox{-.8ex}{$\buildrel{\textstyle>}\over\sim$}}2$) that the disk settles into a stable state. The results of this simulation are in agreement with theoretical expectations and with previous work, and show that the Poisson solver performs satisfactorily in the hydrodynamical regime. MHD Simulations in an Axisymmetric Gravitational Potential ---------------------------------------------------------- In the presence of a weak magnetic field, we expect our disk model to be unstable to the MRI, regardless of the field geometry. We first perform simulations in which only the MRI develops (models T1 and P1). This allows the properties of the ensuing MHD turbulence to be quantified and compared with previous work. To prevent the growth of non–axisymmetric gravitational instabilities, we retain only the $m=0$ component in the Fourier expansion of $\Phi_s$. The resolution is $(N_r,N_{\phi},N_z)=(128,32,128)$ and the azimuthal domain extends from 0 to $\pi/2$, which would be equivalent to a resolution of $128^3$ over a range of $2\pi$. ### Initial toroidal field: Model T1 We add to the equilibrium disk model described above a $\langle \beta \rangle=8$ toroidal magnetic field and run the simulation for about $10$ orbits (about $80$ orbits at the initial disk inner edge). Figure \[3d\_evol\_mhd\_axi\] shows the time evolution of the volume-averaged Maxwell and Reynolds stress tensors and the corresponding $\alpha$ parameter (see eq. \[\[eq:alpha\]\]). The Maxwell stress increases during the linear phase of the instability. It then saturates after 4 orbits, when the MRI breaks down into turbulence. The presence of turbulence is seen in figure \[density\_mhd\_axi\], which shows the density perturbation in the equatorial plane at $t=6.4$. Turbulent fluctuations are present over the full extent of the disk. It is clear from figure \[3d\_evol\_mhd\_axi\] that the Reynolds stress is significantly smaller than the Maxwell stress over the course of the simulation. This is in agreement with previous non self–gravitating global simulations of the MRI . The right panel of figure \[3d\_evol\_mhd\_axi\] shows the radial profile of $\alpha$ at the end of the simulation, i.e. at $t \simeq 10$. The typical value of $\alpha$ is a few times $10^{-2}$, similar to what was found in previous simulations starting with a toroidal field with a net flux . The Maxwell stress stays roughly constant during our simulation. This indicates that the resolution $(128,32,128)$ is large enough for the turbulence to be sustained over the duration of the run. We therefore adopt it in the following runs (which of necessity are limited in time by the fact that mass is accreted onto the central mass). ### Initial poloidal field: Model P1 {#sec:P1} To investigate the sensitivity of our results to the initial field geometry, we run the same calculation as in model T1 but with an initial poloidal magnetic field. We calculate the field from the (toroidal) component of the vector potential in the initial disk model: $$A_{\phi} \propto \rho \cos \left( 8 \pi \frac{r-R_{in}}{R_{out}-R_{in}} \right) \, . \label{Aphi}$$ This corresponds to 4 magnetic loops confined inside the disk. The first 2D simulations of a disk permeated by a weak vertical field showed the development and growth of “channel” solutions. In 3D, these solutions still exist but they quickly break down into turbulence, as predicted by the analysis of . Turbulence is more rapidly established when the field varies on a fairly small scale, which motivates the above choice of $A_\phi$. The radial and vertical components of the magnetic field are computed from $A_{\phi}$ and normalized such as to obtain the desired initial value of $\langle \beta \rangle$. Since the linear growth of the vertical field is much more rapid than that of the toroidal field (see below), we chose a much larger initial value of $\langle \beta \rangle=300$. The properties of the turbulence are similar to those found when the initial field is toroidal. Figure \[maxwell poloidal axi\] shows the time evolution of the Maxwell stress for both models T1 and P1. As expected, the linear instability is much more vigorous when a vertical field is present, because of the growth of the channel solutions. However, in both cases the stress saturates at a similar value and the level of turbulence is comparable. The evolution of the Maxwell stress in P1 is somewhat similar to what was obtained in paper I. The important difference is that in 3D, the stress saturates when turbulence is established and does not decay with time, as it does in 2D. Full MHD simulations -------------------- In this section we report the results of full 3D simulations including the development of both self-gravitational and magnetic instabilities. The resolution is $(N_r,N_{\phi},N_z)=(128,64,128)$ and the azimuthal domain extends from 0 to $\pi$. ### Initial Toroidal Field: Models T2, T2$^*$ and T3 In this sequence of models, we observe the simultaneous appearance of both MHD turbulence and the $m=2$ spiral arm familiar from the hydrodynamical calculation. To better understand how angular momentum is transported in the disk, we compare the time evolution of the different stresses with those obtained in the models described in the previous section. Figure \[grav toroidal\] shows the gravitational stress $\langle T_{r\phi}^{grav} \rangle$ as a function of time for both models T2 and HD. Somewhat surprisingly, the presence of both gravitational and MHD instabilities leads to an average $\langle T_{r\phi}^{grav} \rangle$ reduced by a factor $ \sim 2$ compared with the values obtained without a magnetic field. The magnetic torques do not lead to more vigorous gravitational instability. One possible explanation may be that turbulent motions tend to broaden the spiral arms by adding an extra fluctuating component to the thermal pressure, but there is more going on just this. Figure \[grav toroidal\] also shows that the gravitational stress varies nearly periodically with time and can reach very small values. This behavior is in fact associated with the near disappearance of the spiral arms, as can be seen in figure \[snapshot toroidal\]. These snapshots correspond to a maximum and a minimum of the gravitational stress, respectively. The spiral arms are sharp at $t=4.95$, whereas they lack definition at $t=5.09$. The arms form, disperse, and reform. This periodic variation is also seen in figure \[mdot toroidal\], which shows the mass accretion rate onto the central mass as a function of time. As expected, the accretion rate has a periodic component with the same frequency as that found in the gravitational stress. The period in both cases is $\sim 0.28$. In figure \[maxwell toroidal\], we compare the evolution of the Maxwell stress in models T2 and T2$^*$ (for which only the first $6$ orbits, during which $m_{max}=0$, are plotted). In contrast to the gravitational stress, the Maxwell stress is significantly larger when the disk is gravitationally unstable. This appears to be due to the systematic compression of the magnetic field lines along the spiral arms, as opposed, say, to an increase of the level of turbulent fluctuations. When the gravitational instability disappears after about $7$ orbits in model T2, for example, the Maxwell stress decreases to the same value as in model T2$^*$. As in the hydro model HD, the Toomre $Q$ parameter rises throughout the body of the disk over the course of the simulation (as mass is transported toward the inner region), until the gravitational instability ceases. But even by $t \simeq 8$, when there is no longer any gravitational transport, $Q$ is still larger in model HD than in model T2. This is because the gravitational instability is stronger in the hydrodynamical case, and the disk is depleted more rapidly. To check the sensitivity of these results to our choice of initial conditions, we conducted the following experiment. In model T2$^*$, the input parameters are the same as in model T2, but the non–axisymmetric part of $\Phi_s$ is included only after 6 orbits, i.e. only after MHD turbulence has been firmly established. Figure \[maxwell gravitational toroidal\] shows the time evolution of the volume averaged Maxwell and gravitational stress tensors for run T2$^*$. Until $t=7$, the development of MHD turbulence is the same as in model T1. However, in the time interval $t=7$–8, i.e. after gravitational instabilities have developed, $\langle T_{r\phi}^{Max} \rangle$ decreases to reach about one third of its value at $t=7$. The reason of this decline is not completely clear. One possibility may be that the compression of the (randomized) magnetic field in the spiral arms leads to more efficient reconnection of the field lines. Another possibility is that gravitational stresses feed off the density fluctuations generated by the MRI, thereby indirectly coupling the magnetic and gravitational energies. In any case, this behavior stands in contrast with was observed in run T2, where gravitational instabilities developed while the magnetic field was still ordered. In model T2$^*$, the gravitational stress tensor is roughly a factor of $2$ smaller than in model T2, but shows the same periodic variations. Our next comparison run, model T3, differs from model T2 only in the number $m_{max}$ of fourier coefficients in the expansion of $\Phi_s$. (T3 has $m_{max}=16$, T2 has $m_{max}= 8$.) Once again, very similar results emerge, and the choice of $m_{max}$ does not appear to be critical (cf. § \[modal analysis\]). To summarize: the evolution of a purely toroidal field in a gravitationally unstable disk leads to a reduction and strong periodicity in the gravitational stress (compared with a purely hydrodynamical model). Compared with gravitationally stable models, the Maxwell stress is larger or smaller depending repsectively on whether gravitational instabilities develop at the same time as MHD turbulence (magnetic field alignment in spiral arms) or after turbulence is established (reduction in magnetic stress as gravitational stress develops). The behavior of an initial poloidal field is considered next. ### Initial Poloidal Field: Model P2 Do our toroidal field findings extend to poloidal field behavior? To answer this question, we begin with an initial poloidal field, calculated as in section \[sec:P1\] above. Again, we start with $\langle \beta \rangle = 300$. Except for the initial field geometry, model P2 is the same as model T2. Figure \[grav poloidal\] shows the evolution of the gravitational stress tensor for both models P2 and HD (this is the equivalent of figure \[grav toroidal\]). As in the case of a toroidal field, a non–axisymmetric $m=2$ spiral grows and becomes nonlinear in model P2. The fact that gravitational instabilities develop earlier in model P2 than in models T2 and HD (see figures \[grav toroidal\] and \[grav poloidal\]) appears to be due to the fact that the strong linear magnetic instability associated with the poloidal field produces large perturbations of the density. Once again, we find that the gravitational stress is smaller than in model HD, and varies periodically with time with a period $\sim 0.38$ somewhat larger than for model P2. Figure \[maxwell poloidal\] shows the evolution of the Maxwell stress for models P1 and P2 (this is the equivalent of figure \[maxwell toroidal\]). Once again, the Maxwell stress is larger when the disk is gravitationally unstable and gravitational instabilities develop at the beginning of the run (model P2). Since the state of MHD turbulence in a saturated disk is the same whether an initial poloidal or toroidal field is used, we have not run a case “P2$^*$” with the non–axisymmetric part of $\Phi_s$ added later. Such a run is expected to be very similar to T2$^*$, since the initial turbulent states are similar. The history of run P2 and the above argument together suggest that toroidal and poloidal initial fields behave very similarly. ### Modal analysis {#modal analysis} The full MHD simulations described above suggest that a general feature of the evolution of gravitationally unstable turbulent disks is a periodic modulation of the gravitational stress. To understand the reason for this modulation, we now analyse in more detail the unstable modes that appear in models HD, T2 and P2. We are in particular interested in determining the power spectrum as a function of mode frequency $\sigma$. Following , we Fourier transform in time at each radial zone $r$ and at a fixed azimuth $\phi_0$ the function $\rho(r,\phi_0,t)$. To get the spectral time evolution, we carry out each Fourier transform over a series of 4 distinct time intervals. The number of time-steps used in each time interval gives a finite frequency resolution $d \sigma / 2 \pi=0.3$. The contours of constant power are then plotted as a function of frequency and radius for the various time intervals. Figure \[mode hydro\] shows the contours for model HD. The different panels correspond to different time intervals. In the first panel, i.e. at time $t \simeq 2.4$, we see the presence of a mode with frequency $\sigma/ 2\pi=2.5$ which extends over the whole disk. Its amplitude is small, peaking at about $3 \times 10^{-2}$. This mode is still present in the second panel, at time $t \simeq 3.4$, with a similar amplitude structure. However, a second low frequency mode with $\sigma/ 2\pi=1$ (i.e. with a corotation radius at the disk initial outer edge) is now apparent. Its amplitude is significant only in the disk outer parts, where it peaks at 0.2. This mode subsequently grows and completely dominates the high frequency ($\sigma / 2\pi=2.5$) mode in the third panel, i.e. at $t \simeq 4.4$. There its amplitude peaks at 0.7. This mode can of course be identified with the two–arm spiral seen in figure \[image220 hydro\]. At $t=4.4$ the mode is non–linear. Its amplitude does not increase further, as can be seen in the fourth and last panel, i.e. at $t \simeq 5.4$. At this later time, only the frequency of the mode has changed. It is now around 1.5. Both the finite frequency resolution $d \sigma$ and the increase of the central mass due to accretion may account for this shift in frequency. A third mode with a frequency twice that of the low frequency mode is seen in the last two panels. The relationship between the frequencies of these two modes and the fact that their radial structure is very similar suggests that they are harmonics of each other. \ Figure \[mode mhd\] shows the contour plots for model T2. In the first panel, at $t \simeq 2.4$, there is no dominant mode. Instead, there is a large number of high frequency (with mostly $\sigma/ 2 \pi=1.3$–3.8) perturbations. The amplitude of these fluctuations is a few times $10^{-2}$. They are associated with the growing MRI. In the second panel, at $t \simeq 3.4$, two modes emerge, but their amplitude is still rather low. However, these modes subsequently grow and are clearly seen with a larger amplitude in the third panel, at $t \simeq 4.4$. One of these modes has a frequency $\sigma / 2 \pi=1$ and is the same as that seen in the hydrodynamical simulations. Its corotation radius is located at the disk initial outer edge. Its amplitude peaks at a value of about 0.5 in the outer parts of the disk. The other mode has a frequency $\sigma/ 2 \pi=2.5$ and an amplitude $\sim 0.2$ constant over the whole disk. In particular, its amplitude in the disk inner parts is larger than that of the other mode. This mode is probably the same as that seen with a lower amplitude at early times in the hydrodynamical simulations (panels 1 and 2 of figure \[mode hydro\]). This suggests that this mode is a disk eigenmode which is excited in model T2 by the high frequency motions associated with the turbulence. We expect nonlinear coupling between these two modes to give rise to beat oscillations, i.e. oscillations with a frequency being a linear combination of the frequencies of the two modes. We have noted above that the gravitational stress tensor in model T2 oscillates with a period $\sim 0.28$. This is consistent with the frequency of the oscillations being $\sigma/2 \pi \simeq m( \sigma_{\rm HF} - \sigma_{\rm BF}) =3$, where $\sigma_{\rm HF}$ and $\sigma_{\rm BF}$ are the frequencies of the high and low–frequency modes, respectively. This suggests that the oscillations of the gravitational stress tensor result from a nonlinear coupling between these two modes. Figure \[mode poloidal\] shows the contour plots for model P2 in the time interval 2.72–3.88. the situation is similar to model T2. Here again two modes of comparable amplitude are present. One of this mode has a frequency $\sigma/2 \pi=1$ and can be identified with the mode which emerges in the hydrodynamical simulations. The other mode has a frequency $\sigma/ 2 \pi=2.1$. Given the finite frequency resolution $d \sigma/ 2 \pi$, this second mode may be the same as that identified in model T2. However, it may also be a different mode with a lower frequency. In any case, the situation is qualitatively the same here as in model T2. Since the two modes have comparable amplitude, they interact nonlinearly, which results in a periodic modulation of the gravitational stress. We now vary some of the parameters of model T2 in order to examine the sensitivity of these physical results to the numerical input parameters. Figure \[mode t2low\] shows the contour plots for model T2$_{low}$ in the time interval 2.72–3.88. The resolution for this run in the radial and vertical directions is half that of model T2. The similarity between this plot and the third panel of figure \[mode mhd\] demonstrates that our results do not depend strongly on the numerical resolution. Figure \[mode t3\] is the same as figure \[mode t2low\] but for model T3. Again, it is very similar to the third panel of figure \[mode mhd\]. Since, in model T3, only the parameter $m_{max}$ is different from model T2, figure \[mode t3\] suggests that the limited number of Fourier components included in the calculation of the self–gravitating potential does not qualitatively affect the main physical results presented in this paper. In conclusion, models P2, T2$_{low}$ and T3, taken together, suggest that the physical results described in this paper are insensitive to both the numerical setup of the simulations and the initial magnetic field topology. Discussion ========== In this paper, we have presented the first 3D numerical simulations of the evolution of self–gravitating and magnetized disks. We have investigated disks in which only gravitational or magnetic instabilities develop, and disks in which both types of instabilities occur. When no magnetic field is present, self-gravitating disks are unstable when the Toomre $Q$ parameter is close to unity. The spectrum of unstable modes in that case is dominated by a large scale two–arm spiral whose corotation radius is located near the disk outer edge. The instability is due to the interaction between waves which propagate near the disk outer boundary and inside the inner Lindblad resonance (ILR), respectively . When a magnetic field is present, two large scale modes grow in the disk. They both have $m=2$. One of this mode is the same as that seen in the hydrodynamical simulations. The second mode has a higher frequency. The ILR of the low–frequency mode, which is at $r \simeq 0.6$, is very close to the corotation radius of the high–frequency mode, which is at $r \simeq 0.5$. Such a pair of modes was seen in the hydrodynamical calculations of self–gravitating disks performed by . There, both modes were unstable because of the particular vortensity profile. In our simulations, in the absence of magnetic field, only the low–frequency mode is unstable. The high–frequency mode seems to be part of the spectrum of normal modes in that case, but it does not grow. The presence of MHD turbulence does not modify the spectrum of large scale (comparable to the disk radius) modes, as it acts on scales limited by the disk thickness. However, the fact that the high–frequency mode is unstable in the MHD simulations suggests that turbulence acts as a source for high–frequency oscillations. Nonlinear coupling between these two modes leads to an oscillation of the gravitational stress tensor. Note that such a coupling between two modes with coinciding resonances has been suggested to explain some of the features seen in numerical simulations of galactic disks by @taggeretal87. These authors argued that the proximity of the resonances made the coupling very efficient. The oscillation of the gravitational stress tensor is accompagnied by the periodic disappearance of the spiral arms in the disk. Also, the peak value of this stress is decreased by about half compared to the hydrodynamical simulations. The results reported here are robust and do not depend on the geometry of the magnetic field. They have important consequences for disks around AGN and protoplanetary disks. They first show that accretion of a self–gravitating disk onto the central star is slowed down when a magnetic field is present. They also show that the accretion is time–dependent, with a characteristic timescale for the variability being on the order of a fraction of the dynamical timescale at the outer edge of the region where the instabilities develop. As mentioned in the introduction, protoplanetary disks are probably self–gravitating in the early phases of their evolution. For a disk of about 100 AU, the work presented here suggests variability on a timescale $\sim 10^3$ years. The periodicity in the spatial distribution of knots in jets emanating from such objects is in the range 10–$10^3$ years [@reipurth00], and is usually thought of as being produced by a time–dependent accretion in the central parts of the disk. The simulations presented in this paper suggest that periodic modulations of the accretion rate might well be the result of the interplay between gravitational instabilities and MHD turbulence, a far from obvious source. Note that the first detection of near–IR variability in a sample of Class I protostars was performed recently by Park & Kenyon (2002). However, the poor time coverage of their data prevents a useful measure of the variability timescales to be extracted. Disks around AGN display time–dependent phenomena on a large range of timescales [@ulrichetal97]. The dynamical timescale for a disk orbiting a $10^8$ solar masses black hole at $10^{-2}$ parsecs is 9.3 years, and variations are observed on timescales up to years. This again is consistent with the processes described in this paper. Acknowledgments {#acknowledgments .unnumbered} =============== It is a pleasure to thank John Hawley, John Papaloizou and Michel Tagger for useful discussions. SF thanks the Department of Astronomy at UVa for hospitality during the course of this work. SF is supported by a scholarship from the French [*Ministère de l’Education Nationale et de la Recherche*]{}. SF and CT acknowledge partial support from the European Community through the Research Training Network “The Origin of Planetary Systems” under contract number HPRN-CT-2002-00308, from the [*Programme National de Planétologie*]{} and from the [*Programme National de Physique Stellaire*]{}. The simulations presented in this paper were performed at the Institut du Développement et des Resources en Informatique Scientifique.

--- 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. 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[^1]: Supported by Chinese National Natural Science Foundation Projects \# 11871483 and 11771391.

--- abstract: 'In this paper we introduce new methods to prove the finite cyclicity of some graphics through a triple nilpotent point of saddle or elliptic type surrounding a center. After applying a blow-up of the family, yielding a singular 3-dimensional foliation, this amounts to proving the finite cyclicity of a family of limit periodic sets of the foliation. The boundary limit periodic sets of these families were the most challenging, but the new methods are quite general for treating such graphics. We apply these techniques to prove the finite cyclicity of the graphic $(I_{14}^1)$, which is part of the program started in 1994 by Dumortier, Roussarie and Rousseau (and called DRR program) to show that there exists a uniform upper bound for the number of limit cycles of a planar quadratic vector field. We also prove the finite cyclicity of the boundary limit periodic sets in all graphics but one through a triple nilpotent point at infinity of saddle, elliptic or degenerate type (with a line of zeros) and surrounding a center, namely the graphics $(I_{6b}^1)$, $(H_{13}^3)$, and $(DI_{2b})$.' author: - | Robert Roussarie, Université de Bourgogne\ Christiane Rousseau, Université de Montréal[^1] title: Finite cyclicity of some center graphics through a nilpotent point inside quadratic systems --- Introduction ============ This paper is part of a long term program to prove the finiteness part of Hilbert’s 16th problem for quadratic vector fields, sometimes written $H(2) <\infty$, namely the existence of a uniform bound for the number of limit cycles of quadratic vector fields. The DRR program (see paper [@DRR94(1)]) reduces this problem to proving that 121 graphics (limit periodic sets) have finite cyclicity inside quadratic vector fields, and the long term program is to prove the finite cyclicity of all these graphics. This program has been an opportunity to develop new more sophisticated methods for analyzing the finiteness of the number of limit cycles bifurcating from graphics in generic families of $C^\infty$ vector fields, in analytic families of vector fields, and in finite-parameter families of polynomial vector fields. In this paper, we focus on some graphics in the latter case: graphics through a nilpotent point and surrounding a center inside quadratic systems. The general method is to use the Bautin trick, namely transforming a proof of finite cyclicity of a generic graphic into a proof of finite cyclicity of a graphic surrounding a center. This is possible in quadratic systems since the center conditions are well known: indeed all graphics through a nilpotent point and surrounding a center occur in the stratum of reversible systems. The systems of this stratum are symmetric with respect to an axis, and are also Darboux integrable with an invariant line and an invariant conic. In practice, the Bautin trick consists in dividing a displacement map $V$ in a center ideal, i.e. in writing it as a finite sum of generalized monomials times non vanishing functions of the form $$V(z)= \sum_{i=1}^n a_i m_i(1+h_i(z)),\label{type_V}$$ where each $a_i$ belongs to the center ideal in parameter space, $m_i$ is a generalized monomial in $z$ and $h_i(z)=o(1)$ behaves well under derivation. To compute the displacement map, we write it as a difference of compositions of regular transitions and Dulac maps near the singular points. The Dulac maps are calculated in $C^k$ normalizing coordinates for a family unfolding the vector field. In this paper, we develop some general additional methods, which allow to prove the finite cyclicity of the graphic $(I_{14}^1)$ (Figure \[graphics\](a)). In particular, for the unfolding of this graphic, it is very helpful to be able to claim that all regular transitions are the identity in the center case. This is possible if we exploit the fact that the centers occur when the system is symmetric, and if we choose cleverly the sections on which the different transition maps are defined. Also, in the center case, the Dulac maps have a simple form since the system is Darboux integrable. The methods can be summarized as follows. - We highlight that the change to $C^k$ normalizing coordinates in the neighborhood of the singular points on the blow-up locus can be done by an operator. This allows preserving the symmetry in the center case when changing to normalizing coordinates. - We introduce a uniform way of calculating the two types of Dulac maps when entering the blow-up through a much shorter proof than the one given in [@ZR]. - Although each Dulac map is not $C^k$, we can divide in the center ideal its difference to the corresponding Dulac map in the integrable case. - The method of the blow-up of the family allows reducing the proof of finite cyclicity of the graphic to the proof that a certain number of limit periodic sets have finite cyclicity. These limit periodic sets are defined in the blown-up space. The ones obtained in blowing up a nilpotent saddle are shown in Table \[tab.shhconvex\]. For all of them but one (the boundary limit periodic set), we can reduce the displacement map to a $1$-dimensional map, the number of zeros of which can be bounded by the Bautin trick and a derivation-division algorithm on a map of type . The boundary limit periodic set is more challenging, since we need to work with a 2-dimensional displacement map, the zeros of which we must study along the leaves of an invariant foliation coming from the blow-up. We introduce a generalized derivation operator, which allows performing a derivation-division algorithm on functions of the type $$V(r,\rho)= \sum_{i=1}^n a_im_i(1+h_i(r,\rho)),\label{type_V2}$$ where $h_i$ are ${\cal C}^k $-functions on monomials and $m_i$ are generalized monomials in $r$, $\rho$ (see definitions in Appendix II). During this process, we have to take into account that $r\rho=\mathrm{Cst}$. We have a partial result for every graphic, but one (namely $(H^3_{14})$), through a triple point at infinity: \[thMain1\] Let us consider the graphics $(I^1_{14})$, $(I^1_{6b})$, $(H^3_{13})$ and $(DI_{2b})$ through a triple point at infinity (see Figure 1). Then for any of them, the boundary periodic limit set obtained in the blowing up has a finite cyclicity. Theorem \[thMain1\] is not sufficient to prove that the given graphic has a finite cyclicity inside the family of quadratic vector fields. The reason is that, beside the boundary limit periodic set, other limit periodic sets (see for instance Table \[tab.shhconvex\] for $(I_{14}^1)$) are obtained in the blowing up and, as explained above, we have to prove that each of them has also a finite cyclicity. We present here a complete result for the first graphic: \[thMain2\] The graphic $(I^1_{14})$ has a finite cyclicity inside the family of quadratic vector fields. As for the finite cyclicity of the other graphics $(I_{6b}^1)$, $(H_{13}^3)$ and $(DI_{2b})$, we intend to address the problem in the next future. The finite cyclicity of $(H_{13}^3)$ should be straightforward with arguments identical to those used for $(I_{14}^1)$. It will be done simultaneously with the corresponding generic graphic $(H_{12}^3)$. Some of the limit periodic sets to be studied for $(I_{6b}^1)$ will involve four Dulac maps of second type. For these limit periodic sets, it is not possible to reduce the study of the cyclicity to a single equation. Hence, new methods will need to be adapted to treat the center case, when the periodic solutions correspond to a system of two equations in the four variables $r_1, \rho_1, r_2, \rho_2$, with $r_1\rho_1=\nu_1$ and $r_2\rho_2=\nu_2$. As for the graphic $(DI_{2b})$, some of the limit periodic sets to be studied involve four Dulac maps of second type, two of them through the semi-hyperbolic points $P_1$ and $P_2$ on the blown-up sphere. The techniques developed in this paper can be adapted for studying the boundary limit periodic sets of graphics of the DRR program through a nilpotent finite singular point. The only new difficulty in that case is to show that the three parameters of the leading terms in the displacement map do indeed generate the center ideal. We also hope to adapt them to study the boundary graphic of the hemicycle $(H^3_{14})$: there, the additional difficulty is the two semi-hyperbolic points along the equator. Proofs of Theorems \[thMain1\] and \[thMain2\] are given in Section 3 and Appendix II, where the detailed computations of cyclicity are found in Theorems \[thderdiv\], \[thpgeq2\] and \[thp1\]. Theorem \[thnormalformhyp\] in Appendix I, gives a statement about normal form for 3-dimensional hyperbolic saddle points in a way adapted to this paper. Theorem \[thtransgeneralhypsaddle\] of the same appendix gives a new proof for Dulac transitions near these saddle points, shorter than the one given in [@ZR]. Precise properties for the specific unfoldings deduced from the quadratic family are proved in Appendix III. These properties of some parameter functions are needed to obtain the results of finite cyclicity. Preliminaries ============= Normal form for the unfolding of a nilpotent triple point of saddle or elliptic type ------------------------------------------------------------------------------------ We consider graphics through one singular point, which is a triple nilpotent point of saddle or elliptic type. A germ of vector field in the neighborhood of such a point has the form $$\begin{aligned} \begin{split} \dot x&=y \\ \dot y&=\pm x^3+ bxy +\eta x^2y + yO(x^3)+ O(y^2).\end{split}\label{normal_form_DRS}\end{aligned}$$ The saddle case corresponds to the plus sign, and the elliptic case to the minus sign with $|b|\geq 2\sqrt{2}$. In the elliptic case, we limit ourselves here to the case $|b|>2\sqrt{2}$, which corresponds geometrically to a nilpotent point with hyperbolic points on the divisor of the quasi-homogeneous blow-up. The unfolding of such points has been studied by Dumortier, Roussarie and Sotomayor, [@DRS], including a normal form for the unfolding of the family. A different normal form has been used in [@ZR] for studying the finite cyclicity of generic graphics through such singular points, when we limit ourselves to $|b|>2\sqrt{2}$ in the elliptic case. This normal form is particularly suitable for applications in quadratic vector fields, where there is always an invariant line through a nilpotent point of multiplicity $3$. A germ of $C^\infty$ vector field in the neighborhood of a nilpotent point of multiplicity $3$ of saddle or elliptic type can be brought by an analytic change of coordinates to the form $$\begin{aligned} \begin{split} \dot x&=y+ ax^2, \\ \dot y&=y(x + \eta x^2 +o(x^2) + O(y)).\end{split}\label{normal_form_ZR}\end{aligned}$$ This requires an additional change of variable and scaling compared to what has been done in [@ZR]. The point is a nilpotent saddle when $a<0$ and a nilpotent elliptic point when $a>0$ (see Figure \[fig.topotype\]). The case $|b|=2\sqrt{2}$ corresponds to $a=\frac12$. For $a\neq\frac12$, a generic unfolding depending on a multi-parameter $\lambda= (\mu_1,\mu_2,\mu_3, \mu)$ has the form $$\begin{aligned} \begin{split} \dot x&=y+ a(\lambda)x^2 +\mu_2, \\ \dot y&=\mu_1+\mu_3 y + x^4h_1(x,\eps)+y(x + \eta x^2 +x^3 h_2(x,\lambda))+ y^2Q(x,y,\lambda),\end{split}\label{normal_form_family}\end{aligned}$$ where $h_1(x,\lambda)= O(|\lambda|)$. Moreover, $h_1, h_2, Q$ are $C^\infty$ functions, and $Q$ can be chosen of arbitrarily high order in $\lambda$. Finite cyclicity of a graphic ----------------------------- A *graphic* $\Gamma$ of a vector field $X_0$ , i.e. a union of trajectories and singular points, has *finite cyclicity* inside a family $X_\lambda$ if there exists $N\in\mathbb N$, $\eps>0$ and $\delta>0$ such that any vector field $X_\lambda$ with $|\lambda|<\delta$ has at most $N$ periodic solutions at a Hausdorff distance less than $\eps$ from $\Gamma$. If a graphic has a finite cyclicity, its *cyclicity* is the minimum of such numbers $N$. This means that when studying the finite cyclicity of a graphic $\Gamma$, we need to find a uniform bound for the number of periodic solutions that can appear from it, for all values of the multi-parameter in a small neighborhood $W$ of the origin. Typically we need to find a uniform bound for the number of fixed points of the Poincaré return map or, equivalently, for the number of zeros of some displacement map between two transversal sections to the graphic. With graphics containing a nilpotent singular point there is no way to make a uniform treatment for all $\lambda\in W$, and we typically cover $W$ by a finite number of sectors, on each of which we give a uniform bound. The method for doing this is the *blow-up of the family*, which was first introduced in [@R], and next applied to slow-fast systems in [@DR]. Blow-up of the family --------------------- We take the neighborhood of the origin in parameter-space of the form $\S^2\times [0,\nu_0)\times U$, where $U$ is a neighborhood of $0$ in $\mu$-space and we make the change of parameters $$(\mu_1,\mu_2,\mu_3)= (\nu^3\overline{\mu}_1,\nu^2\overline{\mu}_2,\nu\overline{\mu}_3),\label{change_parameters}$$ where $\ov{M}=(\overline{\mu}_1,\overline{\mu}_2,\overline{\mu}_3)\in\S^2$ and $\nu\in [0,\nu_0)$. Note that $\S^2$ is compact. Hence, to give an argument of finite cyclicity for the graphic $\Gamma$, it suffices to find a neighborhood of each $\ov{M}=(\overline{\mu}_1,\overline{\mu}_2,\overline{\mu}_3)\in\S^2$ inside $\S^2$, a corresponding $\nu_0>0$ and a corresponding $U$ on which we can give a bound for the number of limit cycles. In our study, we will consider special values $a_0$ of $a$. It is important to note that $a(\lambda)$ depends on $\lambda$, and hence that $a-a_0$ is in some sense a parameter in itself. The way to handle this program is to do a *blow-up of the family.* For this, we introduce the weighted blow-up of the singular point $(0,0,0)$ of the three-dimensional family of vector fields, obtained by adding the equation $\dot \nu=0$ to . The blow-up transformation is given by $$(x,y,\nu) = (r\ov{x}, r^2\ov{y}, r\rho),\label{blow-up_family}$$ with $r>0$ and $(\ov{x},\ov{y},\rho)\in \S^2$. After dividing by $r$ the transformed vector field, we get a family of $C^\infty$ vector fields $\ov{X}_{A}$, depending on the parameters $A=(a- a_0,\ov{M}, \mu)$. The foliation $\{\nu=r\rho=\mathrm{Cst}\}$ is invariant under the flow. The leaves $\{r\rho=\nu\}$, with $\nu>0$, are regular two-dimensional manifolds, while the critical locus $\{r\rho=0\}$ is stratified and contains the two strata (see Figure \[fig.strat\]): - $\S^1\times \R^+$ is the blow-up of $X_0$ (for $\lambda=0$); - $D_{\ov{\mu}}= \{\ov{x}^2+\ov{y}^2+\rho^2=1\mid \rho \geq0\}$, for any $\ov{\mu}\in \S^2$. Limit periodic sets in the blown-up family ------------------------------------------ The vector field $\ov{X}_A$ has singular points on $r=\rho=0$. For $a\neq \frac12$, there will be four distinct singular points (occuring in two pairs) corresponding to $\ov{y}=0$ (for $P_1$ and $P_2$) and $\ov{y}=\frac{1-2a}{2}$ (for $P_3$ and $P_4$): see Figure \[fig.strat\]. Their eigenvalues appear in Table \[eigenvalue\]. $r$ $\rho$ $ y $ ------- ------------ ----------- -------------- $P_1$ $ -a $ $\ \ a $ $-(1-2a)$ $P_2$ $\ \ a $ $ -a $ $\ \ (1-2a)$ $P_3$ $\ \ 1/2 $ $ -1/2 $ $-(1-2a)$ $P_4$ $ -1/2$ $\ \ 1/2$ $\ \ (1-2a)$ : The eigenvalues at $P_i$ ($i=1,2,3,4$) \[eigenvalue\] [*We will study the finite cyclicity of a graphic $\Gamma$ joining a pair of opposite points $P_i$ and $P_{i+1}$ in $\ov{X}$, with $i=1$ or $i=3$.* ]{} We consider a particular value $A_0=(a_0, \ov{M}_0, \mu_0)$. Here is the strategy for finding an upper bound for the number of limit cycles that appear for $A$ in a neighborhood of $A_0$. We determine the phase portrait of the family rescaling on $D_{\ov{\mu}}$: this allows determining *limit periodic sets* $\ov{\Gamma}$, which are formed by the union of $\Gamma$ with a finite number of trajectories and singular points on $D_{\ov{\mu}}$ joining $P_i$ and $P_{i+1}$, so that their orientation be compatible with that of $\Gamma$. The limit periodic sets to be studied appear in Table \[tab.shhconvex\] for the saddle case. They come from studying the phase portrait of the *family rescaling* $$\begin{aligned} \begin{split} \dot{\ov{x}}&= \ov{y}+a\ov{x}^2+\ov{\mu}_2,\\ \dot{\ov{y}}&=\ov{\mu}_1+\ov{\mu}_3\ov{y} +\ov{x}\,\ov{y},\end{split}\label{family_rescaling}\end{aligned}$$ obtained by putting $\rho=1$ and $r=0$. It then suffices to show that each limit periodic set has finite cyclicity, i.e. to show the existence of an upper bound for the number of periodic solutions of $\ov{X}_A$ for $A$ in a small neighborhood of $A_0$. {width="3.2cm"} {width="3.2cm"} {width="3.2cm"} --------------------------------- --------------------------------- ---------------------------------- Sxhh1 Sxhh2 Sxhh3 {width="3.2cm"} {width="3.2cm"} {width="3.2cm"} Sxhh4 Sxhh5 Sxhh6 {width="3.2cm"} {width="3.2cm"} Sxhh7 Sxhh8 {width="3.2cm"} {width="3.2cm"} Sxhh9 Sxhh10 : Convex limit periodic sets of hh-type for a graphic with a nilpotent saddle.[]{data-label="tab.shhconvex"} Proving the finite cyclicity of a limit periodic set ----------------------------------------------------- Typically, the kind of argument we will use for proving the finite cyclicity of a limit periodic set is the following: we look for the zeroes of a displacement map between two sections. The sections are 2-dimensional but, because of the invariant foliation, the problem can be reduced to a 1-dimensional problem and the conclusion follows by, either an iteration of Rolle’s theorem, or its generalization, namely a derivation-division argument. The technique can be adapted to non generic graphics occurring inside integrable systems: the proof in the generic case is transformed into a proof for the corresponding graphic, using some adequate division of the coefficients of the displacement map in the ideal of conditions for integrability. To compute the displacement map, we decompose the related transition maps between sections into compositions of Dulac maps in the neighborhood of the singular points and regular $C^k$ transitions elsewhere. Dulac maps ---------- The Dulac maps are the transition maps in the neighborhood of a singular point on $r=\rho=0$. They are computed when the system is in $C^k$ normal form. The normalizing theorem is Theorem \[thnormalformhyp\] of Appendix I. There, it is proved that the normal form is obtained by a normalizing operator ${\cal N}$, a crucial property for this paper. The theorem establishes the existence of a parameter-depending local change of coordinates of class ${\cal C}^k$ bringing the blow-up of in the neighborhood of one of the points $P_i$ into the normal form $\ov{ X}_A^N$ (up to $t\mapsto -t$) written in normal form coordinates $(\ov{Y},r,\rho)$ (provided that the eigenvalue in $r$ has a sign opposite to the two other eigenvalues). Using Table \[eigenvalue\], we take $\sigma=2(1-2a)$ near $\sigma_0=2(1-2a_0)$ for $P_3$ and $P_4$ when $a_0<\frac12$, and $\sigma=\frac{2a-1}{a} $ near $\sigma_0=\frac{2a_0-1}{a_0} $ for $P_1$ and $P_2$ when $a>\frac12$. The normal form $\ov{ X}_A^N$ is given by 1. If $\sigma_0\not\in \Q:$ $$\label{eq2p} \ov{X}^N_{A}: \begin{cases}{\dot r}=r, \\ {\dot \rho} =-\rho,\\ \dot{\ov{Y} }=-(\sigma+\varphi_{A}(\nu))\ov{Y}.\end{cases}$$ 2. If $\sigma_0=\frac{p}{q}\in \Q,$ with $(p,q)=1$ when $q\not =1:$ $$\label{eq3p} \ov{X}^N_{A}: \begin{cases}{\dot r}=r, \\ {\dot \rho} =-\rho,\\ {\dot{\ov{Y}}} =-\Big(\sigma+\varphi_{A}(\nu)\Big)\ov{Y}+\Phi_{A}(\nu,r^p\ov{Y}^q)\ov{Y}+\rho^p\eta_{A}(\nu) , \end{cases}$$ with $\eta_{A}\equiv 0$ when $\sigma_0\not\in \N$ ( $q\not =1$). The functions $\varphi_{A},\Phi_{A},\eta_{A}$ are polynomials of degree $ \leq K(k)$ increasing with $k,$ with smooth coefficients in $A$ and $\Phi_{A}(\nu,0)\equiv 0.$ We introduce the “compensator” function $\omega(\xi,\alpha)$, also denoted $\omega_\alpha(\xi)$, defined by $$\omega(\xi,\alpha)=\omega_\alpha(\xi)=\begin{cases}\frac{\xi^{-\alpha}-1}{\alpha}, & \alpha\not =0, \\ -\ln \xi, &\alpha=0.\end{cases} \label{compensator}$$ We propose in Appendix I a new computation of the Dulac maps previously studied in [@ZR]. There are two types of Dulac transitions. The first type of transition map goes from a section $\{r= r_0\}$ to a section $\{\rho = \rho_0\}$, or the other way around. This type of transition typically behaves as an affine map, which is a very strong contraction or dilatation. The study of the number of zeroes of a displacement involving only Dulac maps of the first type is reduced to the study of the number of zeroes of a 1-dimensional map. The second type of Dulac map is concerned with a transition map from a section $\{\ov{Y}=Y_0\}$ to, either a section $\{r=r_0\}$, or a section $\{\rho=\rho_0\}$. We take $\nu_0=r_0\rho_0.$ ### First type of Dulac map \[thtranstypeI\] We consider the Dulac map from the section $\{\rho= \rho_0\}$ to the section $\{r=r_0\}$, both parametrized by $(\ov{Y},\nu).$ Let $$\bar\sigma=\bar \sigma(\sigma,\nu)=\sigma+\varphi_{A}(\nu)$$ and $$\alpha=\alpha(\sigma,\nu)=\bar\sigma(\sigma,\nu)-\sigma_0.$$ The $\ov{Y}$-component of the transition map $D_{A}$ has the following expression: 1. If $\sigma_0\not\in \Q: $ $$\label{eq20d} D_{A}(\ov{Y},\nu)=\Big(\frac{\nu}{\nu_0}\Big)^{\bar\sigma} \ov{Y}.$$ 2. If $\sigma_0=\frac{p}{q}\in \Q$ with $(p,q)=1$ when $\sigma_0\not \in \N:$ $$\label{eq21d} D_{A}(\ov{Y},\nu)=\eta_{A}(\nu)\rho_0^p\Big(\frac{\nu}{\nu_0}\Big)^{\bar\sigma}\omega\Big(\frac{\nu}{\nu_0},\alpha\Big)+\Big(\frac{\nu}{\nu_0}\Big)^{\bar\sigma}\Big(\ov{Y}+\phi_{A}(\ov{Y},\nu)\Big),$$ with $\eta_{A}$ as in (\[eq3p\]). In particular, $\eta_{A}\equiv 0$ when $\sigma_0\not\in \N.$ The function family $\phi_{A}$ in (\[eq21d\]) is of order $O(\nu^{p+q\alpha}\omega^{q+1}\Big(\frac{\nu}{\nu_0},\alpha\Big)|\ln \nu|)$ and for any integer $l\geq 2,$ is of class ${\cal C}^{l-2}$ in $(\ov{Y},\nu^{1/l},\nu^{1/l}\omega\Big(\frac{\nu}{\nu_0},\alpha\Big), \nu,\mu,\sigma)$. ### Second type of Dulac map \[thtranstypeII\] We consider the Dulac map from the section $\{\ov{Y}=Y_0\},$ parametrized by $(r,\rho)$ to a section $\{r=r_0\}$ parameterized by $(\ov{Y},\nu)$. It has the form $(r,\rho)\mapsto(D_A(r,\rho), \nu)$, with its $\ov{Y}$-component, $(D_A(r,\rho)$, given by: 1. If $\sigma_0\not\in \Q: $ $$\label{eq18b} D_{A}(r,\rho)=\Big(\frac{r}{r_0}\Big)^{\bar\sigma} Y_0.$$ 2. If $\sigma_0=\frac{p}{q}\in \Q$ with $(p,q)=1$ when $\sigma_0\not \in \N:$ $$\label{eq19b} D_{A}(r,\rho)=\eta_{A}(\nu)\rho^p\Big(\frac{r}{r_0}\Big)^{\bar\sigma}\omega\Big(\frac{r}{r_0},\alpha\Big)+\Big(\frac{r}{r_0}\Big)^{\bar\sigma}\Big(Y_0+\phi_{A}(r,\rho)\Big),$$ with $\eta_{A}$ as in (\[eq3p\]) ($\eta_{A}\equiv 0$ when $\sigma_0\not\in \N).$ The function family $\phi_{A}$ in (\[eq19b\]) is of order $O(r^{p+q\alpha}\omega^{q+1}\Big(\frac{r}{r_0},\alpha\Big)|\ln r|)$ and, for any integer $l\geq 2,$ is of class ${\cal C}^{l-2}$ in $(r^{1/l},r^{1/l}\omega\Big(\frac{r}{r_0},\alpha\Big),\rho,\mu,\sigma)$. Applications to quadratic systems {#sect:quadratic} ================================= Quadratic systems with a nilpotent singular point at infinity -------------------------------------------------------------- \[thm.infty\] A quadratic system with a triple singularity point of saddle or elliptic type at infinity and a finite singular point of center type can be brought to the form $$\left\{\begin{array}{ll} \dot x&=-y+B_0x^2,\\ \dot y&=x+xy, \end{array}\right. \label{inf}$$ with $B_0>0$. For $B_0\neq1$, the full $5$-parameter unfolding inside quadratic systems is given with $B= B_0+\mu_0$ inside the family $$\left\{\begin{array}{ll} \dot x&=-y+Bx^2 +\mu_2y^2 + \left(\mu_4+ B\mu_5\right)x\\ \dot y&=x+xy+\mu_3y^2+(1-2B)\mu_5y. \end{array}\right. \label{infunfold}$$ For $B_0=1$, the full $5$-parameter unfolding inside quadratic systems is rather given with $B= 1+\mu_0$ inside the family $$\left\{\begin{array}{ll} \dot x&=-y+(1+\mu_0)x^2 +\mu_2y^2 + \mu_5x\\ \dot y&=x+(\mu_4+\mu_5)x^2+ xy+\mu_3y^2. \end{array}\right. \label{infunfold_B1}$$ The parameter $\mu_2$ (resp. $\mu_3$) corresponds to a nonzero multiple of the parameter $\mu_2$ (resp. $\mu_3$) in the blow-up of the family at the singular point. There is no parameter $\mu_1$ in this family since the connection along the equator is fixed. Moreover for we have: 1. $B_0>1$ for a nilpotent saddle; $B_0=\frac{3}{2}$ corresponds to $a=-\frac{1}{2}$ in ($b=0$ in ). 2. $B_0<1$ for an elliptic point; the elliptic point is of larger codimension, type 1 (the singular points in the blow-up coallesce by pairs) if $B_0=\frac{1}{2}$ (corresponding to $a=\frac{1}{2}$ in , i.e., $b=2\sqrt{2}$ in ). 3. The system has an invariant line $y=-1$ if $\mu_3-(1-2B)\mu_5=0$. 4. If $\mu_2=\mu_3=\mu_4=0$, the system has an invariant parabola $$y=\frac{2B-1}{2}x^2+(2B-1)\mu_5 x -\frac1{2B} + (2B-1)\mu_5^2. \label{Parabola.invariant}$$ The parabola $y=\frac12x^2-\frac12$ is invariant for system when $\mu_0=\mu_2=\mu_3=\mu_4=0$. 5. The integrability condition is $\mu_3=\mu_4=\mu_5=0$, for which we have the following graphics with return map - $B>1$: $(I_{14}^1)$, - $\frac12<B<1$: $(I_{6b}^1)$, - $0<B<\frac12$: $(H_{13}^3)$, - $B=0$: $(H_{14}^3)$, - $B=1$: $(DI_{2b})$. 6. The value of “$a$" in the corresponding normal form is $a=1-B$, and the parameters $\mu_2$ and $\mu_3$ correspond to $\mu_2$ and $\mu_3$ up to a nonzero constant. We can suppose that the nilpotent singular point at infinity is located on the y-axis, the other singular point at infinity on the x-axis, and the focus or center at the origin. Then the system can be brought to the form $$\left\{\begin{array}{ll} \dot x&=\delta_{10} x +\delta_{01} y +\delta_{20} x^2 +\delta_{11}xy,\\ \dot y&=\gamma_{10} x +\gamma_{01} y +\gamma_{11} xy +\gamma_{02} y^2. \end{array}\right. \label{inf.1}$$ Localizing the system at the singular point at infinity on y-axis by $v=\frac{x}{y}, \ \ w=\frac{1}{y}$, we have $$\left\{\begin{array}{ll} \dot v&=(\delta_{11}-\gamma_{02})v-\delta_{01} w+(\delta_{20}-\gamma_{11})v^2 +(\delta_{10}-\gamma_{01})vw-\gamma_{10}v^2w,\\ \dot w&=w(-\gamma_{02}-\gamma_{01} w -\gamma_{11}v-\gamma_{10} vw). \end{array}\right. \label{inf.2}$$ For the singular point $(0,0)$ of system to be nilpotent, we should have $\delta_{11}=\gamma_{02}=0$. The point is triple if $\gamma_{11}\neq 0$. We want the finite singular point to be a center, which corresponds in this case to the system being reversible with respect to a line. Because of our choice of singular points at infinity this line can only be the $y$-axis. Then $\delta_{10}=\gamma_{01}=0$. By a rescaling and still using the original coordinates $(x,y)$, we obtain the system . The change of coordinates $W=-w+(B_0-1)v^2$ brings the system into the equivalent form $$\left\{\begin{array}{ll} \dot V&=W\\ \dot W&=(B_0-1)V^3+(2B_0-3)VW+ o(V^3) + o(VW). \end{array}\right. \label{inf.4}$$ The classification of the nilpotent singularity at infinity follows. A general unfolding preserving the singular point at the origin (which is simple) is of the form (after scaling of $x$, $y$, and $t$) $$\left\{\begin{array}{ll} \dot x&=-y+Bx^2+ m_{10}x +m_{11}xy+m_{02}y^2\\ \dot y&=x+xy+ n_{01}y+n_{20}x^2+n_{02}y^2, \end{array}\right. \label{inf_unfold}$$ with $B$ close to $B_0$. We use a change of variable $(X,Y)= (x+ \zeta_1y, \zeta_2x+ y)$ for small $\zeta_1,\zeta_2$. The terms in $XY$ in the expression of $\dot X$ and the term in $X^2$ in the expression of $\dot Y$ vanish precisely when $$\begin{cases} (2B-1)\zeta_1-m_{11}(1+\zeta_1\zeta_2)+2\zeta_2m_{02}+ 2\zeta_1n_{02}(\zeta_1+\zeta_2)-\zeta_1^2\zeta_2=0,\\ (B-1)\zeta_2+(1+\zeta_2^2)n_{02}- \zeta_2^2n_{11}+\zeta_2^3m_{02}=0,\end{cases}$$ which can be solved for $(\zeta_1,\zeta_2)$ by the implicit function theorem except for $B_0=1$. When $B_0=1$, we replace the second equation by the vanishing of the term in $Y$ in in the expression of $\dot Y$, namely $$\zeta_1+\zeta_2- n_{01}+ m_{10}\zeta_1\zeta_2=0.$$ Again, we get a system that can be solved for $(\zeta_1,\zeta_2)$ by the implicit function theorem. Finite cyclicity of the boundary limit periodic sets of $(I_{14}^1)$, $(I_{6b})$ and $(DI_{2b})$ {#sect:boundary} ------------------------------------------------------------------------------------------------ In the whole paper, $*$ denotes a nonzero constant, which may depend on some parameters. \[thm:boundary\_graphic\] The boundary limit periodic sets of $(I_{14}^1)$, $(I_{6b})$ and $(DI_{2b})$ (see Figures \[graphics\] (a), (b) and (d) and \[boundary\_graphic\]) have finite cyclicity. ![The boundary graphic through $P_3$ and $P_4$ and the four sections $\Sigma_i$ and $\Pi_i$, $i=3,4$, in the normalizing coordinates.[]{data-label="boundary_graphic"}](boundary_graphic){width="5cm"} The finite cyclicity of the boundary limit periodic set is studied inside the family when $B_0\neq 1$, and we will discuss later the adjustment when $B_0=1$. [**Choice of parameters.**]{} We take as parameters $$M=( \ov{\mu}_3, \mu_4,\mu_5, \ov{\mu}_2,B_0-1)= (M_C, \ov{\mu}_2, B_0-1), \label{par_M}$$ with $(\ov{\mu}_2,\ov{\mu}_3)\in\S_1$ and $(B_0-1,\mu_4,\mu_5)$ in a small ball. The parameters $$M_C= (\ov{\mu}_3, \mu_4,\mu_5)\label{M_C}$$ unfold the integrable situation. We let $I_{C}$ be the ideal of germs of $C^k$-functions of the parameters generated by $\{\ov{\mu}_3, \mu_4,\mu_5\}$. 1. The symbol $O_P(M_C)$ refers to a function in the parameter $M$ belonging to the ideal $I_{C}$. 2. The symbol $O_G(M_C)$ refers to a function of $(X,M)$ which belongs to the ideal generated by $I_{C}$ inside the space of functions of $(X,M).$ Depending on the limit periodic set, we could have $X=\ov{x}_3$, where $\ov{x}_3$ is the normalizing coordinate near $P_3$, or $X=(r,\rho)$. [**The displacement map.**]{} It is better to consider the chart $\ov{y}=1$ in the blow-up. We take $C^k$ normalizing charts in the neighborhood of $P_3$ and $P_4$. As discussed above, these $C^k$ normalizing charts can be chosen symmetric one to the other under the center conditions. The normalizing coordinates are $(r,\rho,\ov{x}_i)$ near $P_i$. We consider sections $\Sigma_i= \{\ov{x}_i=X_0\}$ and $\Pi_i= \{r=r_0\}$ in the normalizing charts. The sections $\Sigma_i$ are parameterized by $(r,\rho)$, and the sections $\Pi_i$ by $(\ov{x}_i,\nu)$. Let $V= D_4\circ S - T\circ D_3$ be the displacement map from $\Sigma_3$ to $\Pi_4$: $T$ and $D_3$ follow the flow forward, while $S$ and $D_4$ follow the flow backwards. Let us first give the proof when $\sigma_i(0)\notin\Q$. The Dulac maps are defined from sections $\Sigma_i=\{\ov{x}_i=X_0\}$ to sections $\Pi_i=\{r=r_0\}$, with $X_0$ and $r_0$ fixed. Then the Dulac maps $D_i$ have the form $$D_i(r,\rho)= (C_i(M) r^{\ov{\sigma}_i}, r\rho).\label{D_i}$$ We can choose $X_0$ and $r_0$ so that $C_i(0)=1$, i.e. $X_0r_0^{-\sigma_0}=1$, and $C_3(M)= C_4(M)$ under the center conditions. The map $T$ has the form $$T(\ov{x}_3,\nu)= (H(\ov{x}_3,\nu),\nu).\label{map_T}$$ Because of the symmetry of the sections, then $H\equiv id$ under the center conditions. The planes $r=0$ and $\rho=0$ are invariant under the map $S$, which hence has the form $$S(r,\rho)= (r F(r,\rho), \rho F^{-1}(r,\rho)),\label{map_S}$$ with $F$ of class $C^k$, since $\nu=r\rho$ is invariant. Moreover, it is known from [@ZR] that $F(0,0)=1$ when the sections $\Sigma_i$ are symmetric. The displacement map then has the form $$\Delta (r,\rho)= \left(C_4(M) r^{\ov{\sigma}_4}F^{\ov{\sigma}_4}(r,\rho)- H\left(C_3(M)r^{\ov{\sigma}_3}\right),\nu\right).\label{map_V}$$ Let $V(r,\rho)$ be the first component of $\Delta$. Then periodic solutions correspond to zeroes of $V$. We now need to compute $F$ and $H$. [**Computation of $H$.**]{} The map $H$ is $C^k$ in $(\ov{x}_3, \nu)$. It has the form $$H(\ov{x}_3, \nu)= \ov{x}_3+ \eps_0(M) + \eps_1(M)\ov{x}_3 + O(\ov{x}_3^2)O_G(M_C),$$ with $\eps_0(M)=O_P(M_C), \eps_1(M)=O_P(M_C)$. For $\mu_2=\mu_3=\mu_4=0$, the system has the invariant parabola . The term $\mu_4 x$ in $\dot x$ is without contact, which yields that $$\eps_0(M)=*\mu_4(1+O(M))+O(\mu_3) + O(\mu_5)O(M)= *\mu_4(1+O(M))+O(\ov{\mu}_3\nu) + O(\mu_5)O(M),\label{eps_0}$$ where $*$ denotes a nonzero constant. Lemma \[proof\_eps\_0\] in Appendix II shows that the same is true for . Let us again take $\mu_2=\mu_3=\mu_4=0$. The divergence is then $(2B+1)x+ (1-B) \mu_5$. Proposition \[proof\_eps\_1\] in the Appendix II shows that $$\eps_1(M)= *\mu_5(1+O(M))+ O(\ov{\mu}_3\nu) + O(\mu_4). \label{eps_1}$$ [**The center ideal.**]{} The equations and imply that we can take $\{\eps_0,\eps_1,\ov{\mu}_3\}$ as generators of the center ideal $I_C$. [**Computation of $F$.**]{} The function $F$ has the form: $$F(r,\rho) = 1+*\ov{\mu}_3 \rho(1+ O(\rho)) + O(r)O_G(M_C).\label{eq_F}$$ Indeed, it is proved in Lemma \[coef\_rho\] in the Appendix that the second derivative of $\rho F(0,\rho)$ is a nonzero multiple of $\ov{\mu}_3$. Moreover, the blown-up vector field is integrable on $r=0$ for $\ov{\mu}_3=0$. [**Writing the displacement as a finite sum of terms.**]{} We need grouping all terms of the displacement map into a finite sum of the form . We will see that three terms are sufficient and show that $$V(r,\rho)= -\eps_0(M) (1+ h_0(r, \rho))- C_3(M) \eps_1(M)r^{\ov{\sigma}_3} (1+ h_1(r,\rho)) + *\ov{\mu}_3r^{\ov{\sigma}_3} \rho (1+ h_2(r,\rho)).\label{form_V}$$ We now explain how to group the different terms. The symbol $O(r^\delta)$ used in the sequel, is for an unspecified $\delta>0,$ which may vary from one formula to the other. Let us first consider the terms coming from $H\circ D_3$. Remember that $H$ is the identity when we have a center. Moreover, the map $H$ really takes place in the initial $(x,y)$-plane, where the center ideal is generated by $\{\eps_0,\eps_1,\mu_3\}$. Hence, the higher order terms of $H\circ D_3$ are of the form $$r^{2\ov{\sigma}_3} \left(\eps_0(M) k_0(r,\rho) + \eps_1(M)k_1(r,\rho) + \mu_3k_2(r,\rho)\right).$$ The first two terms contribute to $h_0(r,\rho)$ and $h_1(r,\rho),$ as contributions of order $O(r^\delta).$ As for the third term, we use the fact that $\mu_3=r\rho \ov{\mu}_3$. Hence it contributes to $h_2(r,\rho),$ also as a term of order $O(r^\delta).$ The term $C_3(M) r^{\ov{\sigma}_3}$ will be later grouped with the corresponding term $C_4(M) r^{\ov{\sigma}_4}$ coming from $D_4\circ S.$ Let us now consider the other terms coming from $D_4\circ S(r,\rho)= C_4(M) r^{\ov{\sigma}_4}F(r,\rho)^{\ov{\sigma}_4}$. Again we use that $F$ is the identity when there is a center, i.e. all its terms are divisible in the ideal $I_C$. One of them is the term $*\ov{\mu}_3 r^{\ov{\sigma}_4}\rho$ coming from the term $*\ov{\mu}_3 \rho$ of $F$. As mentioned above, all higher order terms $r^{\ov{\sigma}_4}o(\rho)$ have coefficients divisible by $\ov{\mu}_3$. Also, all terms in $r^{\ov{\sigma}_4}\rho O(r)$ can be distributed in $h_0$, $h_1$ and $h_2,$ as terms of order $O(r^\delta)$. Hence, we only need to consider the pure terms in $o(r^{\ov{\sigma}_4})$. It suffices to show that all such terms can be divided in $\{\eps_0,\eps_1\}$. This comes from the fact that the computation of the pure terms in $r$ can be done in the plane $\rho=0$, and that the system restricted to this plane does not contain any term in $\ov{\mu}_3$. Since $$\ov{\sigma}_4-\ov{\sigma}_3=\nu O_P(M_C) f(\nu) = r\rho O_P(M_C) f(\nu),\label{sigma_3_m_4}$$ with $f$ of class $C^k$, we can replace everywhere $\ov{\sigma}_4$ by $\ov{\sigma}_3,$ up to terms of order $O(r^\delta),$ distributed in $h_0,h_1$ and $h_2.$ We are left with the terms $C_3(M) r^{\ov{\sigma}_3} -C_4(M)r^{\ov{\sigma}_4}$. We write this as $$\begin{aligned} \begin{split} C_3(M) r^{\ov{\sigma}_3} -C_4(M)r^{\ov{\sigma}_4}&= (C_3(M) - C_4(M)) r^{\ov{\sigma}_3} + C_4(M)(r^{\ov{\sigma}_3} - r^{\ov{\sigma}_4})\\ &=(C_3(M) - C_4(M)) r^{\ov{\sigma}_3} + C_4(M)(\ov{\sigma}_3-\ov{\sigma}_4)r^{\ov{\sigma}_3} \omega(r, \ov{\sigma}_3-\ov{\sigma}_4).\end{split} \end{aligned}$$ The difference $C_3(M) - C_4(M)$ is $X_0r_0^{-\ov{\sigma}_3}(1- r_0^{\ov{\sigma}_3-\ov{\sigma}_4})$. Using , the two terms can be decomposed in sums of terms contributing to $h_0,h_1,h_2$, as terms of order $O(r^\delta).$ [**Finite cyclicity in the case $\sigma_0$ irrational.**]{} The displacement map $V$ in is a special case of a universal family $$a_0(1+ h_0(r,\rho)) + a_1r^{\ov{\sigma}_3} (1+ h_1(r,\rho)) +a_2 r^{\ov{\sigma}_3}\rho (1+ h_2(r,\rho)),\label{eq_V_irrational}$$ with $h_0,h_1$ of order $O(r^\delta)$ and $h_2$ is of order $O(\rho)+O(r^\delta).$ Using that these three functions are of order $o(1),$ we show in Theorem \[thderdiv\] below that this family has at most two small zeros along any curve $r\rho=\mathrm{Cst}$ for $r,\rho<\delta$ for some small $\delta$. This implies that, either $V$ has at most two small zeros, or $V$ is identically zero, in which case we have a center. [**Adjustment of the proof when $\sigma_0=\frac{p}{q}$ with $q>1$.**]{} The adjustments are minimal. Indeed, the formula of the Dulac map is more complicated: $$D_i(r,\rho)= (r^{\ov{\sigma}} (C_i(M) + \phi(r, \rho)), r\rho),\label{D_i_rational}$$ with $\phi(r,\rho)$ as in Theorem \[thtranstypeII\]. Hence $\phi(r, \rho)$ produces in $V$ new terms of order $O(r^\delta),$ distributed in $h_0,h_1,h_2.$ [**Adjustement of the proof when $\sigma_0=p$.**]{} Here the first component of $D_i(r,\rho)$ has an additional term of the form $$\kappa_i(r,\rho) = \eta_i(\nu)\rho^p r^{\ov{\sigma}_i}\omega\left(\frac{r}{r_0},\ov{\sigma}_i-p\right).$$ All higher order terms can be distributed in $h_0,h_1,h_2$ and we need only consider the term $\tilde E=\kappa_4\circ S-(1+\eps_1(M))\kappa_3=\left(\kappa_4\circ S-\kappa_4\right)+E$ with $E=\kappa_4(r,\rho)-(1+\eps_1(M))\kappa_3(r,\rho) $. 1. We consider first the term $\kappa_4\circ S-\kappa_4.$ Let $\beta=\ov{\sigma}_4-p$. We have that $$\kappa_4(rF)-\kappa_4(r)=\eta_4\nu^pr^\beta\underbrace{\Big[F^\beta\omega_\beta\Big(\frac{Fr}{r_0}\Big)-\omega_\beta\Big(\frac{r}{r_0}\Big)\Big]}_{G(r,\rho)}.$$ Let us consider $G(r,\rho)$: $$G(r,\rho)=F^\beta\Big(\omega_\beta\Big(\frac{Fr}{r_0}\Big)-\omega_\beta\Big(\frac{r}{r_0}\Big)\Big)+(F^\beta-1)\omega_\beta\Big(\frac{r}{r_0}\Big).$$ Since $$\omega_\beta\Big(\frac{Fr}{r_0}\Big)-\omega_\beta\Big(\frac{r}{r_0}\Big)=\Big(\frac{r}{r_0}\Big)^{-\beta}\frac{F^{-\beta}-1}{\beta},$$ we obtain that $$G(r,\rho)=-\frac{F^\beta-1}{\beta}\Big(\frac{r}{r_0}\Big)^{-\beta}+(F^\beta-1)\omega_\beta\Big(\frac{r}{r_0}\Big)=\frac{F^\beta-1}{\beta}\Big(-\Big(\frac{r}{r_0}\Big)^{-\beta}+\beta\omega_\beta\Big(\frac{r}{r_0}\Big)\Big),$$ i.e. $G(r,\rho)=-\frac{F^\beta-1}{\beta}$, and then $\kappa_4(rF)-\kappa_4(r)=-\eta_4\nu^pr^\beta\frac{F^\beta-1}{\beta}.$ As $F=1+*\bar\mu_3\rho (1+\rho \bar g(\rho))+rO_G(M_C),$ we have that $$\frac{F^\beta-1}{\beta}=*\bar\mu_3\rho (1+\rho \bar g(\rho))+rO_G(M_C),$$ and then that $$\kappa_4(rF)-\kappa_4(r)=-\eta_4\nu^pr^\beta(*\bar\mu_3\rho (1+\rho g(\rho))+rO_G(M_C)).$$ The term $rO_G(M_C))$ gives contributions of order $O(r^\delta)$ in $h_0,h_1,h_2.$ Next, the term $*\bar\mu_3\rho (1+\rho \bar g(\rho))$ gives the contribution $-*\eta_4\nu^{p-1}\rho (1+\rho\bar g(\rho))$ in $h_2.$ If $p\geq 2,$ this term is also of order $O(r\rho),$ and it is of order $O(\rho)$ if $p=1.$ 2. We consider now: $$\begin{aligned} \begin{split} E&=\rho^p\left[\left(\eta_4(\nu)-\eta_3(\nu)(1+\eps_1(M)\right)r^{\ov{\sigma}_3}\omega\left(\frac{r}{r_0},\ov{\sigma}_3-p\right)\right. \\ &\qquad+ \eta_4(\nu) \left(r^{\ov{\sigma}_4}-r^{\ov{\sigma}_3}\right)\omega\left(\frac{r}{r_0},\ov{\sigma}_3-p\right)\\ &\qquad \left.+ \eta_4(\nu) r^{\ov{\sigma}_4}\left(\omega\left(\frac{r}{r_0},\ov{\sigma}_3-p\right)- \omega\left(\frac{r}{r_0},\ov{\sigma}_4-p\right)\right)\right].\end{split} \end{aligned}$$ The second term in the bracket is of the form $$\eta_4(\nu) (\ov{\sigma}_3 - \ov{\sigma}_4)r^{\ov{\sigma}_3}\omega(r, \ov{\sigma}_3-\ov{\sigma}_4)\omega\left(\frac{r}{r_0},\ov{\sigma}_3-p\right).$$ Using , this term can be distributed in $h_0,h_1,h_2,$ [as terms of order $O(r^\delta).$]{} A similar argument holds for the third term. Indeed, we introduce a compensator $$\Omega(\xi, \alpha, \beta)= \Omega_{\alpha,\beta}(\xi)=\begin{cases} \frac{\omega(\xi,\alpha) - \omega(\xi,\beta)}{\alpha-\beta}, &\alpha\neq\beta, \\ \frac12(\ln \xi)^2, &\alpha=\beta,\end{cases}\label{Omega}$$ allowing to rewrite this term as $$\eta_4(\nu) r^{\ov{\sigma}_4}(\ov{\sigma}_3 - \ov{\sigma}_4)\Omega\left(\frac{r}{r_0}, \ov{\sigma}_3-p, \ov{\sigma}_4-p\right).$$ Again, using , this term can be distributed in $h_0,h_1,h_2,$ as terms of order $O(r^\delta)$. This allows writing the displacement map as a sum of four terms $$\begin{aligned} \begin{split} V(r,\rho)&= -\eps_0(M) (1+ h_0(r, \rho))- C_3(M) \eps_1(M)r^{\ov{\sigma}_3} (1+ h_1(r,\rho)) \\ &\qquad+ *\ov{\mu}_3r^{\ov{\sigma}_3} \rho (1+ h_2(r,\rho)) + K(M) r^{\ov{\sigma}_3}\rho^p \omega\left(\frac{r}{r_0},\ov{\sigma}_3-p\right),\label{form_Vp}\end{split}\end{aligned}$$ with $h_0,h_1$ of order $O(r^\delta).$ Moreover, $K(M)=\eta_4(\nu)-\eta_3(\nu)(1-\eps_1(M))= O_P(M_C)$. For $p\geq2$, we conclude that the cyclicity is at most $3$ by Theorem \[thpgeq2\]. For $p=1$, we will prove in Theorem \[thp1\] that the cyclicity is at most $2.$ To this end, we will use that $\eta_4(0)= - \eta_3(0)=\ov{\mu}_3$ and then that $K(M)=*\ov{\mu}_3+O(\nu)O_P(M_C),$ in order to rewrite $V$ as: $$\begin{aligned} \begin{split} V(r,\rho)&= -\eps_0(M) (1+ h_0(r, \rho))- C_3(M) \eps_1(M)r^{\ov{\sigma}_3} (1+ h_1(r,\rho)) \\ &+*\ov{\mu}_3r^{\ov{\sigma}_3} \rho (1+ h_2(r,\rho))+*\ov{\mu}_3r^{\ov{\sigma}_3}\rho \omega\left(\frac{r}{r_0},\ov{\sigma}_3-p\right)(1+ h_3(r,\rho)), \label{form_Vp3}\end{split}\end{aligned}$$ with $h_0,h_1$ and $h_3$ of order $O(r^\delta).$ Finite cyclicity of the boundary limit periodic sets of $(H_{13}^3)$ {#sect:boundaryH_13_3} -------------------------------------------------------------------- \[thm:boundary\_hemicycle\] The boundary limit periodic set of $(H_{13}^3)$ (see Figures \[graphics\](c) and \[boundary\_hemicycle\]) has finite cyclicity. ![The boundary graphic through $P_1$ and $P_2$ and the four sections $\Sigma_i$ and $\Pi_i$, $i=1,2$, in the normalizing coordinates.[]{data-label="boundary_hemicycle"}](hemicycle_boundary){width="7cm"} The proof is very similar to that of Theorem \[thm:boundary\_graphic\]. The graphic occurs in the family for $B<\frac12$, which corresponds to $\frac12<a<1$, but we prefer to use the following equivalent unfolding inside quadratic systems (only parameters’ names are changed so that they play similar role as in Theorem \[thm:boundary\_graphic\]) $$\left\{\begin{array}{ll} \dot x&=-y+Bx^2 +\mu_2y^2 + \mu_5 x\\ \dot y&=x+xy+\mu_3y^2+\mu_4y. \end{array}\right. \label{infunfold_bis}$$ The point $P_4$ (resp. $P_3$) is replaced by $P_1$ (resp. $P_2$). The quantity $\sigma_i$ is now given by $\sigma_i=\frac{2a-1}{a}$. The main difference with Theorem \[thm:boundary\_graphic\] is that the transition from $\Pi_2$ to $\Pi_1$ is replaced by the composition $T_r^{-1} \circ D_r^{-1} \circ T\circ D_\ell\circ T_\ell$. The transitions $T_\ell$ and $T_r$ are along the equator of the Poincaré sphere and hence preserve the connection (no translation terms). The saddle points $P_\ell$ and $P_r$ have inverse hyperbolicity ratios: $\tau_\ell= 1/\tau_r= \frac{1-B}{B}<1$. Hence, it is better to consider a displacement map $$V: \Sigma_2\rightarrow \Pi_r, \qquad V=T\circ D_\ell\circ T_\ell \circ D_2 - D_r\circ T_r\circ D_1\circ S.\label{displ_hemicycle}$$ The computation of $S$ is the same as before. [**Computation of $T_\ell$ and $T_r$.**]{} $T_r$ and $T_\ell$ are regular $C^k$-transitions with no translation terms. They can be computed in the coordinates $(v,w)= (-\frac{x}{y},\frac1{y})$. The transformed system in these coordinates is given in . The transitions take place along $w=0$. Along this line, $\mathrm{div}=(3-2B)v -2\mu_3$. Hence $T_r'(0)-T_\ell'(0)= O(\mu_3) = \nu O(\ov{\mu}_3)$. This property is preserved in the normalizing coordinates. [**Computation of $T$.**]{} The transition $T$ in studied in . The line $y=-1$ is invariant under $\mu_3=\mu_4$. Hence, the constant term is of the form $$T(0)=\eps_0(M)=*(\mu_4-\nu\ov{\mu}_3).\label{coef_const}$$ Under the condition $\eps_0=0$, we have $\mathrm{div}|_{y=-1}= (2B+1)x+\mu_5-\nu\ov{\mu}_3$. Hence, $$T'(0)= \eps_1(M)= *\mu_5+O(\mu_4) + O(\nu)O(\ov{\mu}_3).\label{coef_linear}$$ The equations and remain valid in the normalizing coordinates, and we call the corresponding coefficients $\tilde{\eps}_0$ and $\tilde{\eps}_1$. [**The Dulac maps $D_\ell$ and $D_r$.** ]{} We first localize the system using coordinates $(u,z)= (\frac{y}{x},\frac1{x})$. The normalizing coordinates are of the form $(\ov{u}_i,z)$, $i\in\{\ell,r\}$. Then, $$D_i(z)= \begin{cases} C_i(M) z^{\tau_\ell}, &\frac{1-B_0}{B_0} \notin \Q,\\ C_i(M) z^{\tau_\ell} (1 + \zeta(z, M)), &\frac {1-B_0}{B_0} \in \Q, \end{cases}$$ with $\zeta$, a ${\cal C}^k$-function on monomials (see Appendix II). [**The Dulac maps $D_1$ and $D_2$.** ]{} They are given in Theorem \[thtranstypeI\]. Since the connection along the equator is fixed, then the coefficient $\eta_i$ vanishes identically when $\sigma_0\in \N$. Hence, the displacement map $V(r,\rho)$ has the form $$V(r,\rho) = \tilde{\eps}_0(1+ h_0(r,\rho)) + *\tilde{\eps}_1r^{\ov{\sigma}_2+\tau_\ell}(1+ h_1(r,\rho)) - *\ov{\mu}_3 r^{\ov{\sigma}_2+\tau_\ell}\rho(1+ h_2(r,\rho)).$$ This equation contains no resonant monomials since $\ov{\sigma}_2+\tau_\ell= \frac{1-B-B^2}{B(1-B)}\neq1$ as soon as $B\neq\frac12$. We conclude that the cyclicity is at most two by Theorem \[thderdiv\]. Finite cyclicity of $(I_{14}^1)$ -------------------------------- We now prove Theorem \[thMain2\], i.e. that the graphic $(I_{14}^1)$ has finite cyclicity inside quadratic systems (see Figure \[graphics\](a)). [*Proof of Theorem \[thMain2\].*]{} Such a graphic occurs for system when $B_0>1$, and its deformation in quadratic systems is given in . As usual, we should normally consider all limit periodic sets of Table \[tab.shhconvex\]. It was shown in [@ZR] that a graphic through a nilpotent saddle point has finite cyclicity inside any $C^\infty$-unfolding under the generic conditions that the return map $P$ along the graphic has a derivative different from one and that the nilpotent saddle point has codimension 3. But the only limit periodic sets of Table \[tab.shhconvex\] for which we use the genericity hypotheses are the boundary limit periodic sets which have been treated in Theorem \[thm:boundary\_graphic\], and the intermediate and lower limit periodic sets of Sxhh1 and Sxhh5. For these limit periodic sets, we only have Dulac maps of the first type as in Theorem \[thtranstypeI\]. Hence, we can work with a 1-dimensional displacement map, which we take as $V: \Sigma_3\longrightarrow\Pi_4$, $V=D_4\circ S-T\circ D_3$ (see figure \[other\_graphic\]). As before the sections $\Sigma_i$ and $\Pi_i$ are parameterized by the normalizing coordinate $\ov{x}_i$ near $P_i$, which are chosen so that $S$ and $T$ are the identity in the center case. ![Intermediate and lower limit periodic sets of Sxhh1 and Sxhh5: the four sections $\Sigma_i$ and $\Pi_i$, $i=3,4$, in the normalizing coordinates near $P_3$ and $P_4$.[]{data-label="other_graphic"}](other_graphic){width="5cm"} The technique is to write $V$ in the form of a finite sum $$V(\ov{x}_3, \mu)=\tilde{\epsilon}_0+\nu^{\ov{\sigma}}\left( \sum_{i=1}^n \tilde{\eps}_ih_i(\ov{x}_3,\mu)\right), \label{finite_form}$$ for some $\ov{\sigma}>0$. The parameters are the same as in and . We write little details since they are very similar to [@RR]. [**The intermediate graphics.**]{} For these graphics, the map $V(\ov{x}_3, \mu)$ is $C^k$ in $\ov{x}_3$. Under the condition $\mu_2=\mu_3=0$ for a nilpotent saddle, has an invariant parabola for $\mu_4=0$, which is the only possible connection at a nilpotent saddle. Hence, $T$ has a constant term of the form $*\mu_4+ O(\mu_3) + \mu_5O(M)$. The constant term of the transition $S$ has the form $ O(\ov{\mu}_3)$ since $\ov{\mu}_2$ respects the symmetry, and hence does not contribute to the breaking of the connection. When $\sigma_0\notin \N$, this yields that the constant term $\tilde{\eps}_0$ in the displacement map has the form $\tilde{\eps}_0= *\mu_4 + O(\nu)O(\ov{\mu}_3) + \mu_5 O(M).$ When $\sigma_0=p\in\N$, there are additional terms $$\begin{aligned} \begin{split}& \eta_3\rho_0^p\left(\frac{\nu}{\nu_0}\right)^{\ov{\sigma}_3}\omega\left(\frac{\nu}{\nu_0},\alpha_3\right)- \eta_4\rho_0^p\left(\frac{\nu}{\nu_0}\right)^{\ov{\sigma}_4}\omega\left(\frac{\nu}{\nu_0},\alpha_4\right)\\ &\qquad=(\eta_3- \eta_4)\rho_0^p\left(\frac{\nu}{\nu_0}\right)^{\ov{\sigma}_3}\omega\left(\frac{\nu}{\nu_0},\alpha_3\right)\\ &\qquad\qquad+ \eta_4(\alpha_3-\alpha_4)\rho_0^p\left(\frac{\nu}{\nu_0}\right)^{\ov{\sigma}_3}\omega\left(\frac{\nu}{\nu_0},\alpha_3-\alpha_4\right) \omega\left(\frac{\nu}{\nu_0},\alpha_3\right) \\ &\qquad \qquad+ \eta_4(\alpha_3-\alpha_4)\rho_0^p\left(\frac{\nu}{\nu_0}\right)^{\ov{\sigma}_4}\Omega\left(\frac{\nu}{\nu_0},\alpha_3, \alpha_4\right). \end{split}\end{aligned}$$ In this expression $\eta_3-\eta_4=O_P(M_C)$ and $\alpha_3-\alpha_4= O_P(M_C)O(\nu)$. Hence, in all cases we have $$\tilde{\eps}_0= *\mu_4 + O(\nu)O(\ov{\mu}_3) + \mu_5 O(M)+ O(\nu)O_P(M_C).\label{eps_tilde_0}$$ The linear term has the form $\nu^{\ov{\sigma}_3} T'(0) - \nu^{\ov{\sigma}_4}S'(0))$. Moreover, $S'(0)|_{\rho=0}\equiv 1$ precisely when $\ov{\mu}_3=0$. Also, Lemma \[proof\_eps\_1\] shows that $T'(0)-1= *\mu_5+O(\mu_4)+O(\mu_3) $. Considering that $\ov{\sigma}_3-\ov{\sigma}_4= O(\nu)$, then $$\nu^{\ov{\sigma}_4}= \nu^{\ov{\sigma}_3}(1+ (\ov{\sigma}_3-\ov{\sigma}_4)\omega(\nu,\ov{\sigma}_3-\ov{\sigma}_4)= \nu^{\ov{\sigma}_3} (1+ O(\nu)).$$ This yields $$\tilde{\eps}_1= \nu^{\ov{\sigma}_3}\left(*\mu_5+ O(\mu_4)+O(\nu) O(\ov{\mu}_3)\right).\label{eps_tilde_1}$$ Now, because of the funneling effect, any nonlinearity on the side of $T$ has a high coefficient in $\nu$ which damps it. Hence, the only significant nonlinearities are on the side of $S$. We are sure that $S$ is nonlinear when $\ov{\mu}_3\neq 0$. This comes from the fact that the graphic belongs to a family of graphics. In the case of Sxhh1, this family ends in a lower graphic with a saddle point and its hyperbolicity ratio $\tau$ is different from $1$ precisely when $\ov{\mu}_3\neq0$, yielding that $S(\ov{x}_3)= C_0+ C_1\ov{x}_3^{\tau} + o(\ov{x}_3^\tau),$ with $C_1\not =0,$ for graphics near the saddle point, and hence that $S$ is nonlinear on the whole section $\Sigma_3$. Then, for any graphic occuring for a value $\ov{x}_{3,0}$, there exists $n$ such that $S^{(n)}(\ov{x}_3)= c_{n,3}\ov{\mu}_3\neq0$. Hence, $V^{(n)}(\ov{x}_{3,0})= \nu^{\ov{\sigma}_4}\left[c_{n,3}\ov{\mu}_3+ O(\nu)O_P(M_C)\right]= \tilde{\eps}_n$. Moreover, for all graphics except a few isolated ones we have that $n=2$. The same argument can be applied for Sxhh5 since the connection is fixed between the two saddles and the product of their hyperbolicity ratios is different from $1$ precisely when $\mu_3\neq0$. Hence, we have written $V$ under the form with $h_i(\ov{x}_3)=\ov{x}_3^i(1+O(\ov{x}_3))$. We conclude to finite cyclicity by means of Theorem \[thderdiv\]. [**The lower graphic of Sxhh1.** ]{} The study is very similar and divided in two cases. When $\ov{\mu}_3\neq0$, it was already shown in [@ZR] that the lower graphic of Sxhh1 has finite cyclicity. This comes from the fact that the hyperbolicity ratio $\tau$ at the saddle point is non equal to $1$ precisely when $\ov{\mu}_3\neq0$, in which case we conclude to finite cyclicity because of the nonlinearity of $S$. Hence the difficult case is the neighborhood of $\ov{\mu}_3=0$ since, for this value, $\tau_0=1$. In that case we reparameterize the section $\Sigma_3$ by means of $\tilde{x}_3= \ov{x}_3- c_0(M)$, so that $\tilde{x}_3=0$ corresponds to the unstable manifold of the saddle point on the blow-up sphere. Then, as before, we write $V$ as a sum of terms: $$V(\tilde{x}_3,M)=\tilde{\eps}_0h_0(\tilde{x}_3,M)+\ov{\mu}_3 \tilde{x}_3\omega(\tilde{x}_3,\tau-1)h_3(\tilde{x}_3,M) + \tilde{\eps}_1\tilde{x}_3h_1(\tilde{x}_3,M),$$ with $h_i(0,0)\neq0$. We conclude to finite cyclicity by means of Theorem \[thderdiv\]. [**The lower graphic of Sxhh5.** ]{} Such a graphic occurs for $\ov{\mu}_2>0$. Because the connection is fixed between the two saddles, the map $S$ can easily be computed and has the form $c_0+c_1\ov{x}_3^\tau + o(\ov{x}_3^\tau)$, where $\tau= 1-\frac{2\ov{\mu}_3}{\sqrt{-\frac{\ov{\mu}_2}{a}}+\ov{\mu}_3}$ is the product of the two hyperbolicity ratios. Again, we reparameterize the section $\Sigma_3$ by means of $\tilde{x}_3= \ov{x}_3- c_0(M)$, so that $\tilde{x}_3=0$ corresponds to the unstable manifold of the right saddle point on the blow-up sphere. This allows writing the map $V$ in the form $$\begin{cases} V(\tilde{x}_3)=\sum_{i=0}^{\max(\lfloor \tau\rfloor,1)} \tilde{\eps}_i \tilde{x}_3^ih_i(\tilde{x}_3,M) + \ov{\mu}_3 \tilde{x}_3^\tau h_\tau(\tilde{x}_3,M),&\tau_0\notin\N,\\ V(\tilde{x}_3)=\sum_{i=0}^{\tau_0} \tilde{\eps}_i \tilde{x}_3^ih_i(\tilde{x}_3,M) + \ov{\mu}_3 \tilde{x}_3^{\tau_0}\omega(\tilde{x}_3,\tau-\tau_0) h_\tau(\tilde{x}_3,M),&\tau_0\in\N,\end{cases}$$ with $h_i(0,0)\neq0$. We conclude to finite cyclicity by means of Theorem \[thderdiv\]. $\Box$ Appendix I — Hyperbolic fixed points ==================================== We will consider germs of smooth family of $3$-dimensional vector fields $X_{\mu,\sigma}$ at $(0)\in \R^{3},$ with coordinates $(u,v,y),$ which are quasi-linear of the form: $$\label{eq1} X_{\mu,\sigma}: \begin{cases}{\dot u}= u, \\ {\dot v} =-v,\\ {\dot y} = -\sigma y+F_\mu(u,v,y), \end{cases}$$ where $\sigma$ is a parameter in a neighborhood of $\sigma_0\in \R^+$, and $\mu$ a parameter in a neighborhood of $\mu_0$ in some Euclidean space. Moreover, $F_\mu= O(|(u,v,y)|^2)$ at the origin, for any value of the parameter $(\mu,\sigma).$ [*The system has the first integral: $\nu=uv.$*]{} **Normal form** --------------- It is possible to find local normal form coordinates for $X_{\mu,\sigma}$ by a coordinate change preserving the coordinates $u$ and $v.$ More precisely, we have the following normal form result: \[thnormalformhyp\] There exists a normalizing operator $\mathcal{N}$ defined on each pair $(X_{\mu,\sigma},k)$, where $X_{\mu,\sigma}$ is a family as above and $k\in \N^*$, such that , $$\mathcal{N}(X_{\mu,\sigma},k)= \left(\delta_k, K(k), \eps_k, \eta_k,G_{\mu,\sigma}\right),$$ where $$(u,v,y)\rightarrow (u,v,Y=G_{\mu,\sigma}(u,v,y)),$$ is a parameter-depending change of coordinates of class $C^k$ defined defined for $|\sigma-\sigma_0|\leq \delta$, $|\mu-\mu_0|<\eps_k,$ and $|(u,v,y)|<\eta_k$, such that $dG_{\mu,\sigma}(0,0,0)=\mathrm{Id},$ which brings $X_{\mu,\sigma}$ to the following polynomial normal form of degree $K(k)$: 1. If $\sigma_0\not\in \Q:$ $$\label{eq2} X^N_{\mu,\sigma}: \begin{cases}{\dot u}=u, \\ {\dot v} =-v,\\ {\dot Y} =-(\sigma+\varphi_{\mu,\sigma}(\nu))Y.\end{cases}$$ 2. If $\sigma_0=\frac{p}{q}\in \Q,$ with $(p,q)=1$ when $q\not =1:$ $$\label{eq3} X^N_{\mu,\sigma}: \begin{cases}{\dot u}=u, \\ {\dot v} =-v,\\ {\dot Y} =-\Big(\sigma+\varphi_{\mu,\sigma}(\nu)\Big)Y+\Phi_{\mu,\sigma}(\nu,u^pY^q)Y+v^p\eta_{\mu,\sigma}(\nu) , \end{cases}$$ with $\eta_{\mu,\sigma}\equiv 0$ when $\sigma_0\not\in \N$ ( $q\not =1$). The functions $\varphi_{\mu,\sigma},\Phi_{\mu,\sigma},\eta_{\mu,\sigma}$ are polynomials of degree $ \leq K(k)$, with $C^\infty$ coefficients in $(\mu,\sigma)$ and $\Phi_{\mu,\sigma}(\nu,0)\equiv 0.$ The proof is standard in the literature, and we only recall the main steps. The degree $K(k)$ can be determined algorithmically from the eigenvalues $\{1,-1,-\sigma_0\}$. The number $\delta_k$ is chosen sufficiently small so as not to introduce any new resonant terms of degree $\leq K(k)$ for some $\sigma\in [\sigma_0-\delta_k, \sigma_0+\delta_k]$. The first step is to bring the system to normal form up to degree $K(k)$ $$\label{eqR} X^p_{\mu,\sigma}: \begin{cases}{\dot u}=u, \\ {\dot v} =-v,\\ {\dot z} =P(\sigma,\mu, u,v, z) + R(\sigma,\mu, u,v, z).\end{cases}$$ where $P(\sigma,\mu, u,v, z)$ is a polynomial in $u,v,z$ of degree $K(k)$ containing only resonant terms, and $R(\sigma,\mu, u,v, z)= o(|(u,v,z)|^{K(k)}$. This can be done by means of a polynomial change of coordinate $$y= z+ \sum_{\substack{i+j+\ell=2\\ i-j+\sigma_0(\ell-1)\neq0}}^{K(k)} a_{ij\ell}r^i\rho^jz^\ell.$$ Because this change of coordinate is tangent to the identity and contains no resonant monomial, then it is uniquely determined. The second step is to kill the remainder $R$ in . For this purpose, we decompose $R$ as $R= R_1+R_2$, with $R_1= O(u^{\lfloor K(k)/2\rfloor})$ and $R_2=O(|(v,z)|^{\lfloor K(k)/2\rfloor})$. Each part is killed by the homotopy method. The details are exactly the same as in [@IY]. Again, this step is algorithmic. **Properties of compensators** ------------------------------ This section is devoted to properties of different fonctions useful for the expression of the results, and in particular the so-called compensators $\omega_\alpha(\xi)$ and $\Omega_{\alpha,\beta}(\xi)$ defined in and . First, we introduce the analytic function $$\kappa(\eta)= \begin{cases} \frac{e^\eta-1}{\eta},&\eta\not =0,\\ 1, &\eta=0.\end{cases}\label{def:kappa}$$ The following Lemma gives some useful properties of $\kappa$: \[lemkappa\] The function $\kappa$ is an entire analytic real function whose series is given by $\kappa(\eta)=\sum_0^{+\infty}\frac{\eta^n}{(n+1)!}$. It follows that $\frac{d\kappa}{d\eta}(\eta)<\kappa(\eta)<e^\eta$ for $\eta>0.$ Moreover, $\kappa(\eta)>0,$ $\frac{d\kappa}{d\eta}(\eta)>0$, and $\frac{d^2\kappa}{d\eta^2}(\eta)>0,$ for all $\eta\in \R.$ We have that $\kappa(\eta)=\frac{1}{\eta}(\sum_0^{+\infty}\frac{\eta^n}{n!}-1)=\sum_0^{+\infty}\frac{\eta^n}{(n+1)!}$ and then: $\frac{d\kappa}{d\eta}(\eta)=\sum_0^{+\infty}\frac{\eta^n}{n!(n+2)}.$ The inequalities $\frac{d\kappa}{d\eta}(\eta)<\kappa(\eta)<e^\eta$ for $\eta>0,$ follow trivially. Clearly, $\kappa(\eta)\not =0$ for all $\eta\in \R\setminus \{0\}$ and as $\kappa(0)=1,$ it follows that $\kappa(\eta)>0$ for all $\eta\in \R.$ Next, as $\frac{d\kappa}{d\eta}(\eta)=\frac{\eta e^\eta-e^\eta+1}{\eta^2},$ any root $\eta\not =0$ of $\frac{d\kappa}{d\eta}(\eta)=0$ verifies that $e^\eta=\frac{1}{1-\eta}.$ Comparing the series of these two functions, we see that $e^\eta<\frac{1}{1-\eta}$ for $\eta\in ]0,1[.$ The inequality $\frac{d\kappa}{d\eta}(\eta)>0$ is trivially verified when $\eta\geq 1.$ Finally, when $\eta<0,$ we put $\eta=-\delta,$ with $\delta\in \R^+$. The trivial inequality: $e^\delta>1+\delta,$ for $\delta\in \R^+$ implies that $e^\eta<\frac{1}{1-\eta}$ for $\eta<0.$ As $\frac{d\kappa}{d\eta}(0)=\frac{1}{2},$ we have that $\frac{d\kappa}{d\eta}(\eta)>0$ for all $\eta\in \R.$ To finish, since $\frac{d^2\kappa}{d\eta^2}(\eta)=\frac{(\eta^2-2\eta+2 )e^\eta-2}{\eta^3},$ any root $\eta\not =0$ of $\frac{d^2\kappa}{d\eta^2}(\eta)=0$ verifies that $e^\eta=\frac{1}{1-\eta+\frac{1}{2}\eta^2}$, or equivalently $e^{-\eta}=1-\eta+\frac{1}{2}\eta^2$. Let $g(\eta)=e^{-\eta}-1+\eta-\frac{1}{2}\eta^2$. Let us show that $g(\eta)\neq0$ for $\eta\neq0$. Indeed, $g'(\eta)= -e^{-\eta}+1-\eta<0$. The numerator of $\frac{d\kappa}{d\eta}(\eta)$ is $-e^{\eta}g'(\eta)$ and is positive for $\eta\neq0$. Hence, $g'(\eta)<0$ for $\eta\neq0$, and since $g(0)=0$, then $\eta g(\eta)<0$ for $\eta\neq0$. As $\frac{d^2\kappa}{d\eta^2}(0)=\frac{1}{3},$ we have that $\frac{d^2\kappa}{d\eta^2}(\eta)>0$ for all $\eta\in \R.$ The following lemma gives the relation of $\omega$ defined in with $\kappa$, and interesting properties which can be easily deduced using this relation: \[lemomega\] We have that $\omega(\xi,\alpha)=-\kappa(-\alpha\ln \xi)\ln \xi.$ The compensator $\omega $ verifies the following estimates 1. $ \omega(\xi,\alpha)\leq -\ln \xi$ if $\alpha\leq 0$ and $\omega(\xi,\alpha)\leq -\xi^{-\alpha}\ln \xi$ if $\alpha\geq 0,$ and then $$\label{eq23} \omega(\xi,\alpha)=O(\xi^{-|\alpha|}|\ln \xi|).$$ 2. $$\label{eq24} \omega(\xi,\alpha)\rightarrow +\infty \ \ \mathrm{when}\ \ (\xi,\alpha)\rightarrow (0,0).$$ Using properties of $\kappa$ given in Lemma \[lemkappa\], it follows that: 1. If $\alpha\geq 0,$ i.e $-\alpha \ln \xi \geq 0,$ then $\omega(\xi,\alpha)=-\kappa(-\alpha \ln \xi) \ln \xi$ is less than $-e^{-\alpha \ln \xi} \ln \xi =-\xi^{-\alpha}\ln \xi.$ 2. If $\alpha\leq 0,$ i.e $-\alpha \ln \xi \leq 0,$ then $\omega(\xi,\alpha)=-\kappa(-\alpha \ln \xi) \ln \xi \leq -\ln \xi$ (indeed, $\kappa$ is increasing, $\kappa(0)= 1,$ yielding $\kappa(\eta)\leq 1$ when $\eta\leq 0).$ The estimate (\[eq23\]) follows from these two inequalities. In order to prove (\[eq24\]), we take any $K>0.$ 1. If $-\alpha\ln \xi \geq -K,$ we have that $\kappa(-\alpha\ln \xi)\geq \kappa(-K),$ as $\kappa$ is increasing, and then $\omega(\xi,\alpha)\geq -\kappa(-K)\ln \xi.$ 2. If $-\alpha\ln \xi \leq -K$ (in particular $\alpha\leq 0$), we have that $$\omega(\xi,\alpha)=\frac{1-e^{-\alpha\ln \xi}}{|\alpha|}\geq \frac{1-e^{-K}}{|\alpha|},$$ from which (\[eq24\]) follows. In parallel with the compensator $\Omega$ introduced in , we introduce the symmetric function $${\cal K}(\eta,\delta)=\begin{cases} \frac{\kappa(\eta)-\kappa(\delta)}{\eta-\delta}, &\eta\not =\delta,\\\frac{d\kappa}{d\eta}(\eta),&\eta=\delta. \end{cases}\label{def:K}$$ This yields $$\Omega(\xi,\alpha,\beta)={\cal K}(-\alpha\ln \xi,-\beta\ln \xi)\ln^2\xi.$$ The useful properties of $\Omega(\xi,\alpha,\beta)$ are given by the following lemma: \[lemOmega\] $\Omega_{\alpha,\beta}(\xi)=O(\xi^{-\gamma}\ln^2 \xi), $ where $\gamma=\mathrm{max}\{|\alpha|,|\beta|\}$, and $\Omega_{\alpha,\beta}(\xi)\rightarrow +\infty, $ when $(\xi,\alpha,\beta)\rightarrow (0,0,0).$ To prove the two claims, we just have to use the Mean Value Theorem for the function ${\cal K}$: there exists $\theta\in [\eta,\delta],$ such that ${\cal K}(\eta,\delta)=\frac{d\kappa}{d\eta}(\theta).$ Let us begin by the first claim. Let us start with the case $\alpha\geq \beta$. Then ${\cal K}(-\alpha\ln \xi,-\beta\ln \xi)= \frac{d\kappa}{d\eta}(\theta),$ for some $\theta\in [-\beta\ln \xi,-\alpha\ln \xi].$ As $\frac{d\kappa}{d\eta}(\eta)$ is an increasing function (see Lemma \[lemkappa\]), we have that ${\cal K}(-\alpha\ln \xi,-\beta\ln \xi)\leq \frac{d\kappa}{d\eta}(-\alpha\ln \xi).$ If $\alpha\leq 0,$ we use that $\frac{d\kappa}{d\eta}(-\alpha\ln \xi)\leq \frac{d\kappa}{d\eta}(0)=\frac{1}{2}$ to obtain that $\Omega_{\alpha,\beta}(\xi)\leq \frac{1}{2}\ln^2\xi.$ If $\alpha\geq 0,$ again using Lemma \[lemkappa\], we have that $\frac{d\kappa}{d\eta}(-\alpha\ln \xi)\leq e^{-\alpha\ln \xi}=\xi^{-\alpha}$, and then that: $\Omega_{\alpha,\beta}(\xi)\leq \xi^{-\alpha}\ln^2\xi.$ We can summarize the two possibilities by writing that $\Omega_{\alpha,\beta}(\xi)\leq\xi^{-|\alpha|}\ln^2\xi,$ as soon as $\alpha\geq \beta$ and $\xi$ and $|\alpha|$ sufficiently small. Using the symmetry of $\Omega_{\alpha,\beta}(\xi)$ we can permute $\alpha $ and $\beta$ in the above argument to obtain finally that $\Omega_{\alpha,\beta}(\xi)=O(\xi^{-\gamma}\ln^2 \xi), $ where $\gamma=\mathrm{max}\{|\alpha|,|\beta|\}.$ We now prove the second claim. By symmetry on $\alpha$ and $\beta$ it suffices to prove the claim for $\alpha\geq \beta.$ As above, we can write that $\Omega_{\alpha,\beta}(\xi)=\frac{d\kappa}{d\eta}(\theta)\ln^2\xi,$ for some $\theta\in [-\beta\ln \xi,-\alpha\ln \xi].$ Now, we want to bound $\Omega_{\alpha,\beta}$ from below. Since $\frac{d\kappa}{d\eta}$ is increasing, $\Omega_{\alpha,\beta}(\xi)\geq \frac{d\kappa}{d\eta}(-\beta\ln\xi)\ln^2\xi.$ If $\beta\geq 0,$ we just use that $\frac{d\kappa}{d\eta}(-\beta\ln\xi)\geq \frac{d\kappa}{d\eta}(0)=\frac{1}{2},$ to obtain that $\Omega_{\alpha,\beta}(\xi)\geq \frac{1}{2}\ln^2\xi.$ If $\beta\leq 0,$ we have to compute $\frac{d\kappa}{d\eta}(-\beta\ln\xi)=\frac{d\kappa}{d\eta}(|\beta|\ln\xi)=\frac{d\kappa}{d\eta}(\ln\xi^{|\beta|}).$ As $\frac{d\kappa}{d\eta}(\eta)=\frac{(\eta-1) e^\eta+1}{\eta^2},$ we have that $\frac{d\kappa}{d\eta}(-\beta\ln\xi) =\frac{(|\beta|\ln \xi-1)\xi^{|\beta|} +1}{|\beta|^2\ln^2\xi} $ and then: $\Omega_{\alpha,\beta}(\xi)\geq \frac{(|\beta|\ln \xi-1)\xi^{|\beta|} +1}{|\beta|^2},$ yielding that $\Omega_{\alpha,\beta}(\xi)\rightarrow +\infty.$ This yields the conclusion. **Transition along the trajectories** -------------------------------------- More precisely, let $W$ be a neighborhood of the origin in $\R^3$, and $\Pi\subset \{u=u_0\},$ for $u_0>0 $, be a section. The neighborhood $W$ can be chosen sufficiently small so that the trajectory starting at any point in $W\cap \{u>0\}$ reaches $\Pi$ for a finite positive time (in particular, $W\cap \Pi=\emptyset).$ We consider the transition $T_{\mu,\sigma}$ from the points in $W\cap \{u>0\}$ to the section $\Pi.$ We will compute $T_{\mu,\sigma},$ in the ${\cal C}^k$-coordinates given by Theorem \[thnormalformhyp\]. In this system of coordinates the family is the smooth family of polynomial vector fields $X^N_{\mu,\sigma}$ (this means polynomial in $(u,v,y)$ with smooth coefficients in $(\mu,\sigma)).$ We take $\Pi=[-Y_0,Y_0]\times [0,v_0]\times \{u_0\}$ for some $Y_0>0,v_0>0.$ On $\Pi$, we replace the coordinate $v$ by $\nu=u_0v,$ with $\nu\in [0,\nu_0=u_0v_0].$ Then, we can write $T_{\mu,\sigma}(u,v,Y)=(\widetilde Y_{\mu,\sigma}(u,v,Y),\nu=uv).$ The expression of the $Y$-component $\widetilde Y_{\mu,\sigma}$ is given by the following Theorem: \[thtransgeneralhypsaddle\] Let $\bar\sigma=\bar \sigma(\sigma,\nu)=\sigma+\varphi_{\mu,\sigma}(\nu)$ and $\alpha=\alpha(\sigma,\nu)=\bar\sigma(\sigma,\nu)-\sigma_0$, where $\varphi_{\mu,\sigma}$ is the polynomial family introduced in Theorem \[thnormalformhyp\]. The $Y$-component of the transition map $T_{\mu,\sigma}$ has the following expression on $W\cap\{u>0\}$: 1. If $\sigma_0\not\in \Q: $ $$\label{eq4} \widetilde Y_{\mu,\sigma}(u,v,Y)=\Big(\frac{u}{u_0}\Big)^{\bar\sigma} Y.$$ 2. If $\sigma_0=\frac{p}{q}\in \Q$ with $(p,q)=1,$ when $\sigma_0\not \in \N:$ $$\label{eq5} \widetilde Y_{\mu,\sigma}(u,v,Y)=\eta_{\mu,\sigma}(\nu)v^p\Big(\frac{u}{u_0}\Big)^{\bar\sigma}\omega\Big(\frac{u}{u_0},\alpha\Big)+\Big(\frac{u}{u_0}\Big)^{\bar\sigma}\Big(Y+\phi_{\mu,\sigma}(Y,u,v)\Big),$$ where $\eta_{\mu,\sigma}$ is the same as in (\[eq3\]) (in particular, $\eta_{\mu,\sigma}\equiv 0$ when $\sigma_0\not\in \N).$ The function family $\phi_{\mu,\sigma}$ in (\[eq5\]) is of order $O\left(u^{p+q\alpha}\omega^{q+1}\Big(\frac{u}{u_0},\alpha\Big)|\ln u|\right)$ and, for any integer $l\geq 2,$ is of class ${\cal C}^{l-2}$ in $(Y,u^{1/l},u^{1/l}\omega\Big(\frac{u}{u_0},\alpha\Big),v,\mu,\sigma)$. [*Proof.*]{} The time to go from a point $(u,v,Y)\in W\cap\{u>0\}$ to the section $\Pi$ along the flow of $X^N_{\mu,\sigma}$ is equal to $-\ln\frac{u}{u_0}.$ Expression (\[eq4\]) follows trivially from the integration of the third line of the system (\[eq2\]). Then, from now on, we will assume that $\sigma_0\in \Q$ and we will study the integration of the system (\[eq3\]). The trajectory through the point $(u,v,Y)$ is equal to $(ue^t,ve^{-t},Y(t))$ where $Y(t)$ is solution of the $1$-dimensional non-autonomous differential equation: $$\label{eq6} {\dot Y}(t) =-\bar \sigma Y(t)+\Phi_{\mu,\sigma}(\nu,u^pe^{pt}Y(t)^q)Y(t)+e^{-pt}v^p\eta_{\mu,\sigma}(\nu),$$ with initial condition $Y(0)=Y.$ In order to eliminate the linear term in (\[eq6\]) we look for $Y(t)$ in the form $Y(t)=e^{-\bar\sigma t}Z(t).$ As $\dot Y(t)=e^{-\bar\sigma t}\dot Z(t)-\bar\sigma Y(t),$ and letting $\bar \sigma=\frac{p}{q}+\alpha$, we obtain the following differential equation for $Z(t):$ $$\label{eq7} \dot Z=\Phi_{\mu,\sigma}(\nu,e^{-q\alpha t}u^pZ^q)Z+ e^{\alpha t}v^p\eta_{\mu,\sigma}(\nu),$$ with initial condition $Z(0)=Y.$ Note that the term in $\eta_{\mu,\sigma}$ is only present when $q=1.$ The $1$-dimensional non-autonomous differential equation (\[eq7\]) is smooth in $(t,Z,\sigma,\nu,u,v,\mu)$ and can be integrated for any time $t\in [0,-\ln\frac{u}{u_0}].$ If $Z(t)$ is the solution of (\[eq7\]) with initial condition $Z(0)=Y,$ we will have that $$\label{eq8} \widetilde Y_{\mu,\sigma}(u,v,Y)=\Big(\frac{u}{u_0}\Big)^{\bar\sigma}Z\Big(-\ln\frac{u}{u_0}\Big).$$ The above expression has to be studied for $u>0$ (we extend $\widetilde Y$ along $\{u=0\}$ by $\widetilde Y_{\mu,\sigma}(0,v,Y)=0).$ We first study the integration of (\[eq7\]). To begin, it is easy to get rid of the term $e^{\alpha t}v^p\eta_{\mu,\sigma}(\nu)$ in (\[eq7\]). Let us consider the analytic function $$\Theta(t,\alpha)=\begin{cases}\frac{e^{\alpha t}-1}{\alpha}, & \alpha\not =0,\\ t,&\alpha=0.\end{cases}$$ which verifies $\dot \Theta=e^{\alpha t}.$ We have that $\Theta(t,\alpha)=t\kappa(\alpha t)$ and then $ \omega(\xi,\alpha)=\Theta(-\ln \xi,\alpha).$ Putting $Z(t)=v^p\eta_{\mu,\sigma}(\nu)\Theta(t,\alpha)+\bar Z(t),$ we see that $\bar Z(t)$ is the solution of the differential equation $$\label{eq9} \dot{\bar Z}=\Phi_{\mu,\sigma}\Big(\nu,u^pe^{-q\alpha t}(v^p\eta_{\mu,\sigma}(\nu)\Theta(t,\alpha)+\bar Z)^q\Big)(v^p\eta_{\mu,\sigma}(\nu)\Theta(t,\alpha)+\bar Z),$$ with initial condition $\bar Z(0)=Y.$ As $\Phi_{\mu,\sigma}(\nu,0)\equiv 0,$ we can write $\Phi_{\mu,\sigma}(\nu,\xi)=\xi H_{\mu,\sigma}(\nu,\xi),$ where $H_{\mu,\sigma}$ is a smooth function. Now, let us notice that $e^{\alpha t}=\dot \Theta= 1+\alpha\Theta.$ Moreover the map $t\rightarrow \Theta(t,\alpha)$ is invertible (for any $\alpha$). Then, we can change the time $t$ by the time $\Theta$ in the differential equation (\[eq9\]). We obtain the new equation $$\label{eq10} \frac{d\bar Z}{d\Theta}=u^p\bar H(\Theta,\bar Z,u,v,\nu,\alpha,\mu,\sigma)$$ with $$\label{eq11} \bar H=(1+\alpha \Theta)^{-(1+q)}(v^p\eta\Theta+\bar Z)^{q+1}H_{\mu,\sigma}\Big(\nu,u^p(1+\alpha \Theta)^{-q}(v^p\eta\Theta+\bar Z)^q\Big),$$ where $\eta=\eta_{\mu,\sigma}(\nu).$ Let $\Psi\left(\Theta,Y,u,v,\nu,\alpha,\mu,\sigma\right)$ be the solution of (\[eq10\]), with the “time” $\Theta.$ [*Up to now, $\Theta$ is seen as an independent variable; in particular it is independent from $\alpha$*]{}. For $t=-\ln\frac{u}{u_0},$ then $\Theta=\omega_\alpha(\frac{u}{u_0}),$ yielding $$\label{eq12} Z\Big(-\ln\frac{u}{u_0}\Big)=\Psi\left(\omega\Big(\frac{u}{u_0},\alpha\Big),Y,u,v,\nu,\alpha,\mu,\sigma\right)+v^p\eta_{\mu,\sigma} (\nu)\omega\Big(\frac{u}{u_0},\alpha\Big),$$ and then, the computation of $\widetilde Y_{\mu,\sigma}(u,v,Y)$ reduces to the computation of $\Psi\left(\omega\Big(\frac{u}{u_0},\alpha\Big),Y,u,v,\mu,\sigma\right).$ One difficulty in the study of $\Psi\left(\omega\Big(\frac{u}{u_0},\alpha\Big),Y,u,v,\nu,\alpha,\mu,\sigma\right)$ is that $\omega\Big(\frac{u}{u_0},\alpha\Big)\rightarrow+\infty$ if $u\rightarrow 0.$ To overcome this difficulty we will exploit the fact that the right hand side of (\[eq10\]) is divisible by $u^p.$ We first study the differential equation (\[eq10\]). We put $u=U^l$ and change the time $\Theta$ by the time $\tau=U\Theta$ (and not just by $u\Theta,$ as it could seem more natural). The equation (\[eq10\]) is replaced by the following equation $$\label{eq13} \frac{d\bar Z}{d\tau}=U^{pl-1}\bar H\Big(\frac{\tau}{U},\bar Z,U^p,v,\nu,\alpha,\mu,\sigma\Big),$$ where $\bar H$ is given by (\[eq11\]). Let $\bar G$ be the right hand side of (\[eq13\]). It is smooth for $U>0,$ but since it is function of $\alpha\frac{\tau}{U},$ it is not well-defined in a whole neighborhood of the point $\{(\tau,\bar Z,U,v,\nu,\alpha,\mu,\sigma)=(0,0,0,0,0,0,\mu_0,\sigma_0)\}.$ Fortunately, we only need to integrate (\[eq13\]) in a closed domain $\ov{\cal D}$: [**Definition of $\ov{\cal D}$.**]{} The domain $\ov{\cal D}$ is defined in the space $(\tau,U,\bar Z,v,\nu,\alpha,\mu,\sigma)$ defined by 1. $U\in [0,U_1],$ $|\alpha|\leq \alpha_0$ and $\tau\in [0,U\omega (\frac{U^l}{u_0},\alpha)]$, where $U_1,\alpha_0>0$ are chosen arbitrarily small (the time $\tau=U\omega( \frac{U^l}{u_0},\alpha)$ corresponds to the time $t=-\ln \frac{u}{u_0}=-l\ln \frac{U}{u_0})$, 2. $(\bar Z,v,\nu,\alpha,\mu,\sigma)\in \A,$ an arbitrarily small closed neighborhood of the value $(0,0,0,0,\mu_0,\sigma_0).$ We want to prove that $\bar G$ is of class ${\cal C}^{l-2}$ on $\ov{\cal{D}}.$ We will first prove a technical lemma about the partial derivatives of the function $\bar G.$ Let us denote by $\partial_m \bar G$ any partial derivative of $\bar G$ corresponding to a multi-index $m=(m_1,\ldots,m_s)$ associated to the variables $\tau,U,\bar Z,v,\nu,\alpha,\mu,\sigma$ and the coordinates of $\mu.$ Let $|m|=m_1+\cdots +m_s$ be the degree of $m.$ We will note by $\delta,$ a strictly positive number, which can be made arbitrarily small by appropriately choosing $U_1$ and $\A.$ We have the following: \[lembarD\] Let be $\sigma_0=\frac{p}{q}$ as above. Let $m$ be any multi-index such that $|m|\leq l-2.$ Then, for any $\delta>0$, there exists a domain $\ov{\cal{D}}$ as above, such that *on the restriction to the domain $\ov{\cal{D}}$* we have that $$\label{eq13_bis} \partial_m \bar G =O(U^{pl-|m|-1-\delta}).$$ Recall that $\bar G=U^{pl}\bar H,$ where $\bar H$ is given by (\[eq11\]) and $\Theta$ is replaced by $\frac{\tau}{U}.$ The proof is straightforward, but rather tedious, and we just give the main steps. First, let us notice that on $\ov{\cal{D}}$ we have that, for any $s\in \Z$: $$\label{eq16} \Big(1+\alpha \frac{\tau}{U}\Big)^s=(1+\alpha \Theta)^s=e^{s\alpha t}= O(U^{-|sl\alpha|}).$$ Also, using Lemma \[lemomega\], we have that: $$\frac{\tau}{U}=\Theta=\kappa(\alpha t)t \leq e^{|\alpha|t}t\leq l U^{-|l\alpha|}|\ln U|.$$ These estimations imply that $\Big(1+\alpha \frac{\tau}{U}\Big)^{-(q+1)}$ and $\frac{\tau}{U}$ have an order $O(U^{-\delta}).$ As $\bar H$ is bounded on $\bar {\cal D},$ we have that $\bar G=O(U^{pl-1-\delta}).$ This is the expected result for $m=0.$ Next, we use the expression of the partial derivatives of $\bar G,$ in terms of the functions $\Theta,$ $(1+\alpha\Theta)^{-q}$ or $(1+\alpha\Theta)^{-(q+1)}$ and the partial derivatives of $H_{\mu,\sigma},$ evaluated on $\ov{\cal D}$ (these partial derivatives are bounded on $\ov{\cal D}).$ We have for instance that: $$\frac{\partial }{\partial U}(1+\alpha\Theta)^{-q}=-ql\alpha (1+\alpha\Theta)^{-(q)}\frac{1}{U}=O(U^{-1-\delta}).$$ As $(1+\alpha\Theta)^{-q}=O(U^{-\delta}),$ we remark that the order in $U$ has discreased by one unit (modulo an order in $\delta).$ [*It is easy to see that this observation can be generalized for any partial derivative: the previous order in $U$ decreases by one unity for each first order partial derivation (modulo an order in $\delta).$*]{} Then, starting with $\bar G=O(U^{pl-1-\delta})$ for $m=0,$ the estimation (\[eq13\]) for any multi-index $m$ follows directly by recurence from this fall of order (let us notice that, in a symbolic way, we have: $``\delta+\delta=\delta").$ [**End of the proof of Theorem \[thtransgeneralhypsaddle\]** ]{} Lemma \[lembarD\] says that each partial derivative $\partial_m\bar G$ can be extended continuously on $\tau=U=0$ by giving it the value zero at these points. Then, as the function $\bar G$ is smooth on $\ov{\cal D}\setminus \{\tau=U=0\},$ the restriction of $\bar G$ to $\ov{\cal{D}}$ is a function of differentiability class ${\cal C}^{l-2},$ on the whole domain $\ov{\cal{D}},$ [*including the points on $\{\tau=U=0\},$*]{} when we give to each partial derivative of $\bar G$ or order less than $l-2$ the value $0$ at these points. Let $\B$ be a closed neighborhood of $(0,0,0)$ in the $(\tau,\alpha,U)$-plane, containing the closed set $$\{(\tau,\alpha,U)\ | \ \tau\in [0,-lU\ln\frac{U}{U_0}], \ |\alpha |\leq \alpha_0,\ U\in [0,U_1]\}$$ that we have introduced above in the definition of $\ov{\cal{D}}$. The closed domain $\ov{\cal{D}}$ is contained in the neighborhood $\A\times \B.$ Using the Whitney Theorem for the extention of differentiable functions (see \[M\] for instance), we can find a ${\cal C}^{l-2}$-function $\widetilde G$ on a $\A\times \B$ such that $\widetilde G|_{\ov{\cal{D}}}\equiv \bar G$ (here, this extention can also be easily constructed by hand, in an elementary way). For times $\tau \in [0,-lU\ln\frac{U}{U_0}]$ the flow $\Psi(\tau,\bar Z,U,v,\nu,\alpha,\mu,\sigma)$ of the differential equation (\[eq10\]): $\frac{d\bar Z}{d\tau}=\bar G$ coincides with the flow $\widetilde \Psi(\tau,\bar Z,U,v,\nu,\alpha,\mu,\sigma)$ of the differential equation $\frac{d\bar Z}{d\tau}=\widetilde G.$ This equation is of differentiability class ${\cal C}^{l-2}$ on $\A\times \B,$ as well as its flow $\widetilde \Psi.$ In particular, we have that $$\label{eq17} \bar Z\Big(-\ln\frac{u}{u_0}\Big)=\widetilde \Psi\left(U\omega\Big(\frac{U^l}{u_0},\alpha\Big),Y,U,v,\nu,\alpha,\mu,\sigma\right),$$ is a ${\cal C}^{l-2}$-function of $(Y,U,U\omega\Big(\frac{U^l}{u_0},\alpha\Big),v,\nu,\alpha,\mu,\sigma)$, i.e. is a ${\cal C}^{l-2}$-function in the variables $\left(Y,u^\frac{1}{l}, u^\frac{1}{l}\omega(\frac{u}{u_0},\alpha),v,\nu,\alpha,\mu,\sigma\right)$, [*a function which is defined on a neighborhood of the point $(0,0,0,0,0,\mu_0,\sigma_0).$*]{} We can replace $\alpha$ (outside $\omega$) by its expression in $(\sigma,\nu)$ and $\nu$ by $uv$ to obtain finally that $ \bar Z\Big(-\ln\frac{u}{u_0}\Big)$ is a ${\cal C}^{l-2}$-function of $\left(Y,u^\frac{1}{l},u^\frac{1}{l}\omega(\frac{u}{u_0},\alpha),v,\mu,\sigma\right)$. As $\bar Z (0)=Y,$ we can write $$\label{eq18} \bar Z\Big(-\ln\frac{u}{u_0}\Big)=Y+\phi_{\mu,\sigma}(Y,u,v),$$ where $$\label{eq19} \phi_{\mu,\sigma}=\widetilde \Psi\left(U\omega\Big(\frac{U^l}{u_0},\alpha\Big),Y,U,v,\nu,\alpha,\mu,\sigma\right)-Y$$ is a ${\cal C}^{l-2}$-function of $\left(Y,u^\frac{1}{l},u^\frac{1}{l}\omega(\frac{u}{u_0},\alpha),v,\mu,\sigma\right).$ Finally, collecting the different terms in (\[eq8\]), (\[eq12\]), (\[eq18\]) and (\[eq19\]), we obtain the expression (\[eq5\]) in Theorem \[thtransgeneralhypsaddle\], for the transition function $\widetilde Y_{\mu,\sigma}(u,v,Y).$ We can estimate $\phi_{\mu,\sigma}$ from the differential equation (\[eq9\]) for $\bar Z(t).$ If $G(t,\bar Z, u,v,\nu,\alpha,\sigma,\mu)$ is the right hand side of (\[eq9\]), we have that $G=O(u^pe^{-q\alpha t}\Theta^{q+1})$ on the domain $\ov{\cal D}$ defined above. As $t\leq -\ln\frac{u}{u_0} $ on $\ov{\cal D},$ then $\Theta(t,\alpha)\leq \omega\Big(\frac{u}{u_0},\alpha\Big)$, yielding $G=O(u^{p+q\alpha}\omega^{q+1}(\frac{u}{u_0},\alpha)).$ From this estimate of the order of $G$, it follows that $$\phi_{\mu,\sigma}=\bar Z\Big(-\ln \frac{u}{u_0}\Big)-Y=O(u^{p+q\alpha}\omega^{q+1}\Big(\frac{u}{u_0},\alpha\Big)|\ln u|),$$ which is the estimation in the statement of Theorem \[thtransgeneralhypsaddle\]. $\Box$ **Transitions between sections** -------------------------------- Theorem \[thtransgeneralhypsaddle\] gives the expression of the transition $T_{\mu,\sigma}=(\nu,\widetilde Y_{\mu,\sigma}),$ starting from any point $(u,v,Y)$ in the domain $W\cap\{u>0\}$ and landing on a section $\Pi\subset \{u=u_0\},$ for some $u_0>0$ (we can extend trivially $T_{\mu,\sigma}$ to the whole neighborhood $W$ by taking $\widetilde Y_{\mu,\sigma}(u,v,0)=0).$ We apply this to get Theorems \[thtranstypeI\] and  \[thtranstypeII\] after changing $(u,v)\mapsto(r,\rho)$. [**Discussion of Theorems \[thtranstypeI\] and \[thtranstypeII\].**]{} A previous version of Theorems \[thtranstypeI\] and \[thtranstypeII\] was given in Theorems 4.10 and 4.14 of [@ZR]. It is interesting to compare their proofs and formulations with the proofs and formulations in the present paper. 1. The proof in the present version is unified: Theorem \[thtransgeneralhypsaddle\] gives a formula for a global transition from any point in a $3$-dimensional neighborhood $W,$ formula which is easy to restrict on the two different types of section $\Sigma.$ Next, the proof of Theorem \[thtransgeneralhypsaddle\], even if it is based on the same normal form, is much shorter than the proofs of Theorems 4.10 and 4.14 given in [@ZR]. The reason seems to be that in [@ZR] the transition function $\widetilde Y$ and its partial derivatives are directly estimated by a variational method. In the present paper, we have replaced the $1$-dimensional non-autonomous differential system: $\dot{\bar Z}=\bar G,$ which is not defined in a neighborhood of the point $\{(\tau,\bar Z,U, v, \nu,\alpha,\mu,\sigma)=(0,0,0,0,0,0,\mu_0,\sigma_0)\},$ by a differential equation: $\dot{\bar Z}=\widetilde G,$ differentiable on a neighborhood of this point. As a consequence, we obtain almost without computation that the function $\phi_{\mu,\sigma}$ is differentiable (in terms of fractional power and a compensator of some variable). In fact, the heavy computations made in [@ZR] are replaced by an implicit use of the Cauchy Theorem for differential equations. 2. We can compare the statements in [@ZR] and in the present paper. We restrict the comparison to the only non-trivial case: $\sigma_0\in \Q$. The transition function called here $\widetilde Y_{\mu,\sigma}$ is given by the formula (4.11) of Theorem 4.10 of [@ZR]. We can observe that it is quite similar to the above formula (\[eq16\]), up to the changes of notations. The same remarks are valid for the transition of type II which is treated in Theorem 4.14 in [@ZR]. The only important difference is in the form and properties of the function $\phi_{\mu,\sigma},$ which is called $\phi$ or $\theta$ in [@ZR]. We will comment on this in the next items. 3. The function $\phi_{\mu,\sigma}$ in Theorem \[thtranstypeI\] is of order $O(\nu^{p+q\alpha}\omega^{q+1}\Big(\frac{\nu}{\nu_0},\alpha\Big)|\ln \nu|).$ This order has to be compared with the order given for the function $\phi$ in Theorem 4.10 of [@ZR] which is exactly the same order for $\alpha<0$, but equal to $O(\nu^{p}\omega^{q+1}\Big(\frac{\nu}{\nu_0},\alpha\Big)|\ln \nu|)$ for $\alpha>0.$ This minor difference is probably due to the difference in the method of proof. It is less easy to compare the order of $\phi_{\mu,\sigma}$ in Theorem \[thtranstypeII\] with the order of $\theta$ in Theorem 4.14 of [@ZR]. 4. In Theorem 4.10 of [@ZR], $\phi$ is a ${\cal C}^\infty$-function of $\omega\Big(\frac{\nu}{\nu_0},\alpha\Big)$ and other variables. Since $\omega\rightarrow +\infty$ for $\nu\rightarrow 0,$ this means that the domain of $\phi$ has to be unbounded. This implies that it is not possible to deduce directly the order of the partial derivatives of $\phi.$ This order is obtained by using variational methods and heavy computations. On the contrary, the formulation given in Theorems \[thtranstypeI\] and \[thtranstypeII\], permits a direct deduction of the order of any partial derivative of $\phi_{\mu,\sigma}.$ Let us show this on an example for a transition map of type I. Considering any $l\in \N$ and observing that $\phi_{\mu,\sigma}$ is of order $O(\nu^{p-\delta}),$ we can write $$\phi_{\mu,\sigma}=\nu^{p-\frac{1}{l}}\bar\phi_{\mu,\sigma},$$ where $\bar\phi_{\mu,\sigma}$ is a ${\cal C}^{l-p-3}$-function in $(Y,\nu^{1/l},\nu^{1/l}\omega\Big(\frac{\nu}{\nu_0},\alpha\Big),\mu,\sigma)$. As a consequence any partial derivative of $\phi_{\mu,\sigma}$ in terms of $Y,\mu,\sigma,$ of degree less than $l-p-3,$ is of order $O(\nu^{p-\frac{1}{l}}).$ Taking into account that we can take $l$ arbirarily large, this order in very similar to the order obtained in Theorem 4.10 of [@ZR]. **Appendix II—Counting the number of roots** ============================================ **Differentiable functions on monomials** ----------------------------------------- We come back to the notations of Section \[sect:quadratic\]: $r,\rho$ are variables defined in a compact neighborhood $\A$ of $(0,0)$ in the first quadrant $Q=\{r\geq 0, \rho\geq 0\}.$ [*We will always choose $\A$ to be a rectangle $[0,r_1]\times [0,\rho_1],$ in order to have connected curves $l_\nu=\{(r,\rho)\in \A\ | \ r\rho=0\}.$*]{} In the following definitions we will use also compensators $\omega_\gamma$ and $\Omega_{\gamma,\delta},$ depending on other parameters $\gamma,\delta.$ We will often use the shortened notation $\omega_\gamma,\Omega_{\gamma,\delta}$ for $\omega_\gamma\Big(\frac{r}{r_0}\Big),\Omega_{\gamma,\delta}\Big(\frac{r}{r_0}\Big).$ Moreover, changing $r$ to $\frac{r}{r_0}$, we can of course suppose that $r_0=1$. We consider a multi-parameter $\lambda$ in a compact neighborhood $\B$ of a value $\lambda_0$ in some euclidean space ${\cal E}.$ The neighborhood $\B$ will be chosen sufficiently small to have the desired properties. We also consider functions which are differentiable on real powers of $r,\rho$ and compensators in $r.$ We give a precise definition of this notion. 1. A *primary monomial (monomial in short),* is an expression $M=r^a,\ \rho^b,$ $\ r^a\omega_\gamma(r)^{c},\ r^a\Omega_{\gamma_1,\gamma_2}(r)^d$ or $\omega_\gamma(r)^{-e}$ where $a,b,c,d,e$ and $\gamma,\gamma_1,\gamma_2$ are smooth functions of $\lambda.$ Moreover $a,b,e$ are strictly positive and $\gamma(\lambda_0)=\gamma_1(\lambda_0)=\gamma_2(\lambda_0)=0$ (we can have $\gamma=\alpha$ or $\beta$ and $(\gamma_1,\gamma_2)=(\alpha,\beta)).$ For instance, $r^\frac{2}{3},\ \rho^\frac{1}{5}, \omega_\alpha^{-1}, r\Omega_{\alpha,\beta}$ are primary monomials but not $r^\alpha$ or $\omega_\alpha^\alpha.$ A monomial $M$ defines a $\lambda$-family of functions $M(r,\rho,\lambda)$ on $Q=\{r\geq 0,\ \rho\geq 0\},$ $M$ is smooth for $r>0$ and, by Lemmas \[lemomega\] and \[lemOmega\], it can be extended continuously along $\{r=0\});$ we have that $M(0,0,\lambda_0)=0$ (i.e. $M=o(1),$ in terms of some distance of $(r,\rho,\lambda)$ to $(0,0,\lambda_0)).$ 2. We say that a function $f(r,\rho,\lambda)$ on $\A\times\B$ is a *${\cal C}^k$-function on the monomials $M_1,\ldots ,M_l$* if there exists a ${\cal C}^k$-function $\tilde f(\xi_1,\ldots,\xi_l,\lambda)$ defined on $\widetilde \A\times \B,$ where $\widetilde \A$ is a neighborhood of $0\in \R^l$ such that $ f(r,\rho,\lambda)=\tilde f(M_1,\ldots ,M_l,\lambda).$ If the number of monomials and their type is not specified, we just say that $f$ is a [${\cal C}^k$-function on monomials.]{} Clearly, the space of [${\cal C}^k$-functions on monomials,]{} defined on $\A\times\B$ is a ring. The classical theorems of differential calculus (Taylor formula, division theorem and so on) can be extended to these functions by applying them to the function $\tilde f.$ Since the differentiability class $k$ is finite, there will be falls of differentiability class in these operations: Lemma \[lemderivfunct\] is one example. For this reason, we will consider functions $f$ with the property to be [*${\cal C}^k$-functions on monomials, for any $k\in \N$*]{} (but with a choice of monomials and a size of the neighborhood $\A\times\B$ that may depend on $k$). The functions $\psi_{\mu,\sigma}(Y,u,v),\psi_{\mu,\sigma}(Y,\nu)$ and $\psi_{\mu,\sigma}(u,v)$ introduced in the statements of Theorems \[thtransgeneralhypsaddle\] are, \[thtranstypeI\], and \[thtranstypeII\] are examples of ${\cal C}^k$-functions on monomials for any $k,$ which use only the single compensator $\omega_\alpha.$ The functions $h_i$ entering in the expression of the displacement map $V$ in Section 3 are using other compensators $\omega_\gamma$, and also $\Omega_{\alpha,\beta}.$ **Procedure of division-derivation for functions with $2$ variables** --------------------------------------------------------------------- In this section, $h(r,\rho,\lambda)=o(1)$ will mean that $h(0,0,\lambda_0)=0.$ We want to bound the number of roots of an equation $\{V(r,\rho,\lambda)=0\}$ along the curves $l_\nu=\{r\rho=\nu\ |\ (r,\rho)\in \A\},$ for $\nu>0$ and a neighborhood $\A\times\B$ sufficiently small. The function $V$ is expressed using ${\cal C}^k$-functions on monomials. To obtain this bound, we will apply Rolle’s Theorem, and to this end we will use recurrently the Lie-derivative $L_{\cal X}$ of $V$ by the vector field $${\cal X}=r\frac{\partial}{\partial r}-\rho\frac{\partial}{\partial \rho}.\label{vf}$$ Hence, we need some properties of $L_{\cal X}$ acting on ${\cal C}^k$-functions on monomials. It is easy to see that: $$\begin{cases}L_{\cal X}r^a=ar^a,\\ L_{\cal X}\rho^b=-b\rho^b,\\ L_{\cal X}\omega_\gamma=-(1+\gamma \omega_\gamma),\\ L_{\cal X}\Omega_{\gamma_1,\gamma_2}=-(\omega_{\gamma_1}+\gamma_2 \Omega_{\gamma_1,\gamma_2}).\end{cases}$$ From this, it follows that \[lemderivfunct\] If $f$ is a ${\cal C}^k$-function on monomials, then $L_{\cal X}f$ is a ${\cal C}^{k-1}$-function on monomials and $L_{\cal X}f=o(1).$ If $M$ is any monomial, $L_{\cal X}M$ is a linear combinaison of monomials. Then, $L_{\cal X}f=\sum_i\frac{\partial \tilde f}{\partial \xi_i}L_{\cal X}M_i,$ is a ${\cal C}^{k-1}$-function on monomials and, since each monomial is $o(1),$ this function $L_{\cal X}f$ is also $o(1)$. For the procedure of division-derivation we will need more general monomials than the admissible ones: 1. A *general monomial* is an expression $M=r^a\rho^b\prod_i \omega_i^{c_i}\prod_j\Omega_j^{d_j}$ where $i$ and $j$ belong to finite sets of indices. The coefficients $a,b,c_i,d_j,$ as well as the internal parameters of the compensators $\omega_i,\Omega_j,$ are smooth functions of $\lambda$ (without any restriction on sign). Let $a(\lambda_0)=a^0,b(\lambda_0)=b^0.$ 2. A general monomial is *resonant* if $a^0=b^0$ (in this case the polynomial part $r^{a^0}\rho^{b^0}$ of $M$ reduces to the first integral $\nu^{a^0}$). Seen as a function of $(r,\rho,\lambda),$ such a monomial is in general not defined for $r=0$ and $\rho=0.$ An interesting property is that if $M$ is a general monomial, then $M^{-1}$ is also a general monomial. For convenience, if $\omega_i=\omega(r,\gamma_i)$ we will use the contracting expressions: $\omega=(\omega_i)_i,\ \gamma=(\gamma_i)_i,\ c=(c_i) _i,\ \prod_i\omega_i^{c_i}=\omega^c, \sum_i \gamma_ic_i=\gamma c.$ A first easy result, which will be the principal tool in the proof of Theorem \[thderdiv\] below, is the following: \[lemderivmon\] We consider an expression $f=M(1+h)$ where $M=r^a\rho^b\omega^c$ is a general non-resonant monomial [without $\Omega$-factor]{} and $h$ is a ${\cal C}^k$-function on monomials, of order $o(1).$ Then, on a sufficiently small neighborhood $\B$, we can write: $$\label{eq25} L_{\cal X}f=(a-b+\gamma c)M(1+g),$$ with $g,$ a ${\cal C}^{k-1}$-function on monomials, of order $o(1).$ We have that $L_{\cal X}f=L_{\cal X}M(1+h)+ML_{\cal X}h.$ Using the formula of derivation for $\omega$, we obtain that $L_{\cal X}M=(a-b+\gamma c+c\omega^{-1})M.$ As $M$ is non-resonant, we have that $a^0-b^0\not =0$ and, if $\B$ is a sufficiently small neighborhood of $\lambda_0,$ we will also have that $a-b+\gamma c\not =0$ on $\B.$ Then, we obtain that: $$L_{\cal X}f=(a-b+\gamma c)\Big(1+\frac{c\omega^{-1}}{a-b+\gamma c}\Big)M(1+h)+ML_{\cal X}h.$$ We can write this expression as $L_{\cal X}f=(a-b+\gamma c)M(1+g),$ with $$g=h+\frac{c\omega^{-1}(1+h)+L_{\cal X}h}{a-b+\gamma c}.$$ It follows from Lemmas \[lemderivfunct\] and \[lemderivmon\] that $g$ is a ${\cal C}^{k-1}$-function on monomials, of order $o(1).$ We want to use the algorithm of division-derivation in order to prove the following result: \[thderdiv\] Let $V(r,\rho,\lambda)$ be a function on $\A\times\B\cap \{r>0,\ \rho>0\},$ of the form $$\label{eq26} V(r,\rho,\lambda)=\sum_{i=1}^l A_i(\lambda)M_i\Big(1+g_i(r,\rho,\lambda)\Big),$$ where: 1. the [leading monomials]{} $M_i=r^{a_i}\rho^{b_i}\omega^{c_i}$ are general monomials, without $\Omega$-factor ($\omega=(\omega_j)_j,$ $c_i=(c_i^j)_j$ with $j\in J,$ a finite set), 2. the functions $g_i$ are ${\cal C}^k$-functions on monomials, with $k\geq l,$ and of order $o(1),$ 3. the functions $A_i(\lambda)$ are continuous, 4. the monomials $M_jM_i^{-1}$ for $i\not =j$ are non-resonant, i.e. $$\label{eq27} a_j^0-a_i^0-b_j^0+b_i^0\not =0\ \ \mathrm{ for}\ \ i\not =j.$$ Then, if $\A\times\B$ is chosen sufficiently small, [i)]{} either the function $V$ has at most $l-1$ isolated roots counted with their multiplicity, on each curve $l_\nu=\{r\rho=\nu\}\subset \A,$ [ii)]{} or $V$ is identically zero. We suppose that $V$ is defined for $\lambda\in \B$ (some neighborhood of $\lambda_0)$ and we define the following closed subsets: $$\B_i=\{ \lambda\in \B \ | \ A_i(\lambda)\geq A_j(\lambda), \forall j=1,\ldots,l\}.$$ Of course we have $\B=\cup_i \B_i$, and it is sufficient to prove the result for any $\B_i$ (and $\B$ sufficiently small). Then let us pick any $i=1,\ldots,l.$ By reordering the indices, we can suppose that we have picked $i=l.$ The algorithm of division-derivation consists in the production of a sequence of functions: $V_0=V,V_1,\ldots,V_{l-1},$ such that each $V_j$ is a summation similar to $V$ but only on $l-j$ terms, and is defined on a smaller neighborhood $\A^j\times \B^j$ of $(0,0,\lambda_0).$ To define $V_1,$ we first divide $V$ by $M_1(1+g_1)$ [*(a division step).*]{} This is made on a neighborhood $\A^1\times \B^1\subset \A\times\B$ chosen such that $1+g_1(r,\rho,\lambda)\not =0$ for all $(r,\rho,\lambda)\in \A^1\times \B^1.$ On this neigborhood we consider the function: $$\frac{V}{M_1(1+g_1)}=A_1+\sum_{i=2}^k A_iM_iM_1^{-1}\Big(1+\tilde g_i\Big),$$ where the function $\tilde g_i,$ defined by $1+\tilde g_i=\frac{1+g_i}{1+g_1},$ is ${\cal C}^k$ on monomials and of order $o(1).$ Next we apply the operator $L_{\cal X}$ [*(a derivation step)*]{}. Since the monomials $M_iM_1^{-1}$ are non resonant for $i\not =1,$ we can apply Lemma \[lemderivmon\] to obtain the following function $V_1$ on $\A^1\times \B^1$: $$V_1=L_{\cal X}\Big[\frac{V}{M_1(1+g_1)}\Big]= \sum_{i=2}^l(a_i-a_1-b_i+b_1) A_iM_iM_1^{-1}\Big(1+ g_i^1(y,z)\Big),$$ with the function $ g_i^1$, ${\cal C}^k$ on monomials and of order $o(1).$ The effect of the derivation is to kill the first term $A_1$, thus reducing by one the number of terms in the summation. Except from this fact, the terms of the summation are completely similar to the ones in $V$, but with the functions $A_i$ replaced by $(a_i-a_1-b_i+b_1) A_i$, and the monomials $M_i$ replaced by the monomials $M_iM_1^{-1}.$ For the recurrence step of order $j+1=1,\ldots,k-1,$ we assume that we have a function: $$V_j= \sum_{i=j+1}^l\Big(\prod_{m=1}^{j}(a_i-b_i-a_m+b_m)\Big) A_i(\lambda)M_iM_j^{-1}\Big(1+ g_i^j\Big),$$ defined on some neighborhood $\A^{j}\times \B^{j}$ with functions $g_i^j,$ ${\cal C}^{k-j}$ on monomials and of order $o(1).$ As in the first step from $V$ to $V_1,$ we divide $V_j$ by $M_{j+1}M_j^{-1}\Big(1+ g_{j+1}^j\Big),$ which is possible on some neighborhood $\A^{j+1}\times \B^{j+1}\subset \A^{j}\times \B^j,$ and next apply the differential operator $L_{\cal X}$ to produce a function $$V_{j+1}= \sum_{i=j+2}^l\Big(\prod_{m=1}^{j+1}(a_i-b_i-a_m+b_m)\Big) A_i(\lambda)M_iM_{j+1}^{-1}\Big(1+ g_i^{j+1}\Big),$$ where the $ g_i^{j+1}$ are ${\cal C}^{k-j-1}$ on monomials and of order $o(1).$ Performing the $l-1$ steps of the recurrence, we end up with a function $$V_{l-1}= (a_l-b_l-a_1+b_1)\cdots (a_l-b_{l}-a_{l-1}+b_{l-1})A_l(\lambda)M_lM_{l-1}^{-1}\Big(1+ g_l^{l}\Big),$$ where $g_l^l$ is ${\cal C}^{k-l}$ on monomials and of order $o(1).$ As $g^l_l=o(1),$ and at least ${\cal C}^{0}$ on monomials, we can choose a last neighborhood $\A^{l}\times \B^{l}\subset \A^{l-1}\times \B^{l-1},$ such that the function $1+g_l^l$ is nowhere zero on it. We restrict now $\lambda\in W_l=\B^l\cap \B_l.$ On this set we have the following alternative: $A_l(\lambda)\not =0$ or $A_1(\lambda)=\cdots =A_l(\lambda)=0.$ In the last case, the function $V$ is identical to $0$ and has no isolated roots. Then we just have to look at values $\lambda$ where $A_l(\lambda)\not =0.$ For such a value of $\lambda,$ the function $V_{l-1}$ itself is nowhere zero on $\A^l\times W.$ Consider now any curve $l_\nu$ in $\A^l.$ Recall that the derivation $L_{\cal X}$ of a function $G$ corresponds to the derivation of $G$ along the flow of ${\cal X}$ and that $l_\nu$ is an orbit of this vector field. Then, as $V_{l-1}$ is equal to the derivation of $V_{l-2},$ up to a non-zero function, Rolle’s Theorem applied to $V_{l-2},$ implies that the restriction of this function to $l_\nu,$ has at most one root (let us notice that $\l_\nu$ is connected!). The same argument based on Rolle’s Theorem can be applied by recurrence to obtain for each $j\leq l,$ that the function $V_{l-j}$ has at most $j-1$ roots, counted with their multiplicity. Finally, the function $V$ has at most $l-1$ roots counted with their multiplicity on $\l_\nu\cap \A^l,$ for $\lambda\in W_l.$ We obtain the result by considering in the same way the different subsets $\B_i.$ 1. Even if $V$ is a summation on admissible monomials, it is clear that, in general, the division step may produce general monomials. This is the reason why we begin with general monomials in (\[eq26\]). 2. Using the first integral $r\rho=\nu,$ we can rewrite the leading monomial $M_i$ in the form $M_i=\nu^{b_i}r^{a_i-b_i}\omega^{c_i};$ We call $\bar M_i=r^{a_i-b_i}\omega^{c_i}$ a [reduced monomial.]{} The sum (\[eq26\]) may be written in reduced form, with $p_i=a_i-b_i$: $$\label{eq28} V(r,\rho,\lambda)=\sum_{i=1}^l \nu^{b_i}A_i(\lambda)r^{p_i}\omega^{c_i}\Big(1+g_i(r,\rho,\lambda)\Big),$$ 3. The non-resonance condition (\[eq27\]) in Theorem \[thderdiv\] is equivalent to the condition that the $p_i(\lambda_0)=p_i^0$ in (\[eq28\]) are two by two distinct. Up to a change of indices and a reordering, we can suppose in this case that $p_1^0<p_2^0\cdots <p_l^0$. Let us note that some of $p_i^0$ may be negative, and also that one of them may be equal to zero. **The results of finite cyclicity for the boundary limit periodic set** ----------------------------------------------------------------------- We now want to apply Theorem \[thderdiv\] to the displacement function $V$ in the text. We write $\bar\sigma_3=\sigma_0+\alpha.$ After putting this function in the reduced form (\[eq27\]), we have the following. 1. In the case $\sigma_0\not \in \N$, the function $V$ is given in and we have the sequence of monomials: $\{1,r^{\sigma_0+\alpha}, r^{\sigma_0-1+\alpha}\}$. This allows applying Theorem \[thderdiv\], yielding that the boundary limit periodic set is at most $2$. 2. In the case $\sigma_0=p\in \N,$ the function $V$ is given in or , and the sequence of monomials is: $\{1,r^{p+\alpha},r^{p-1+\alpha}, r^{\alpha}\omega_\alpha \}.$ We have two resonant leading monomials when $p\not =1$, and even $3$ when $p=1.$ Theorem \[thderdiv\] does not apply in none of these cases. Hence, we give a direct proof for $\sigma_0\in \N,$ using exactly the same procedure of derivation-division as in Theorem \[thderdiv\], but based on a more refined estimation than the formula (\[eq25\]) used to prove Theorem \[thderdiv\]. Recall that the parameter was called $M$ in this context. It will not be sufficient to consider the leading reduced monomials for $M=M_0$, and we will have to look more precisely at the form of certain remainders. We need the following result: \[lemdersingmon\] $$\label{eq29} L_{\cal X}\Big[r^\alpha \omega_\alpha\Big(1+O(r^\delta)\Big)\Big]=-r^{\alpha} \Big(1+O(r^\delta)\Big),$$ We have that $L_{\cal X}\Big[r^\alpha\omega_\alpha\Big(1+O(r^\delta)\Big)\Big]= L_{\cal X}\Big[r^\alpha\omega_\alpha\Big] (1+O(r^\delta))+r^\alpha\omega_\alpha O(r^\delta).$ Now, $L_{\cal X}\Big[r^\alpha\omega_\alpha\Big]=\alpha r^\alpha\omega_\alpha -r^\alpha r^{-\alpha}.$ As $r^{-\alpha}=1+\alpha\omega_\alpha,$ we have that $L_{\cal X}\Big[r^\alpha\omega_\alpha\Big]=-r^\alpha.$ Since $r^\alpha\omega_\alpha O(r^\delta)$ is of order $O(r^\delta)$ (for a smaller $\delta$), we obtain (\[eq29\]) by grouping the terms. The formula (\[eq29\]) is wrong in general if we replace the remainder by the more general remainder $o(1).$ Let us consider for instance the expression $f=r^\alpha\omega_\alpha(1+\rho).$ We have that $L_{\cal X}f=-r^{\alpha} (1+\rho)-r^\alpha\omega_\alpha\rho=-r^{\alpha} (1+\rho+ \omega_\alpha\rho).$ The term $\omega_\alpha\rho$ is not of order $o(1).$ Let $\A,\B$ be neighborhoods defined as above. First we have the following result when $\sigma_0\not =1$: \[thpgeq2\] Consider the case $\sigma_0=p\in \N$, with $p\not =1.$ Then the cyclicity of the boundary limit periodic set is at most $3$, namely for sufficiently small neighborhoods $\A$ and $\B$, the equation $V(r,\rho,M)=0$ has at most $3$ roots, counted with their multiplicities, on each curve $l_\nu\subset \A.$ Recall that the displacement map $V$ is given by $$\label{eq31} V(r,\rho)=*\varepsilon_0(1+h_0)+*\varepsilon_1 r^{p+\alpha}(1+h_1)+*\bar\mu_3 \nu r^{p-1+\alpha}(1+h_2)+*K(M)\nu^p r^{\alpha}\omega_\alpha.$$ The sequence of leading monomials in (\[eq31\]) does not verify the condition of non-resonance. To overcome this difficulty, we will use that there is no remainder in the last term, and that $h_0$ is of order $O(r^\delta).$ For $h_1$ and $h_2$, it will be sufficient to know that they are $o(1).$ As in the proof of Theorem \[thderdiv\], we define the partition $\B=\B_1\cup \B_2\cup \B_3 \cup \B_4$ in terms of the coefficients in (\[eq31\]). At each step we will have to restrict the size of $\B.$ We will not recall it. As the three last leading monomials in (\[eq31\]) are $o(1),$ the cyclicity is trivially $0$ when $M\in\B_1.$ We suppose now that $M\in \B_2 \cup \B_3 \cup \B_4.$ Using (\[eq29\]), we obtain: $$L_{\cal X}\frac{ V}{1+h_0}=*\varepsilon_1r^{p+\alpha} (1+g_1) +*\bar\mu_3\nu r^{p-1+\alpha}(1+g_2)+*K(M)\nu^pr^\alpha.$$ Now, the sequence of leading monomials $\{r^{p+\alpha},r^{p-1+\alpha},r^\alpha\}$ verifies the condition of non-resonance and we can apply Theorem \[thderdiv\] to $L_{\cal X}\frac{ V}{1+h_0}.$ Then, this function has at most $2$ roots, and the function $V$ itself has at most $3$ roots, when $M\in \B_2 \cup \B_3 \cup \B_4.$ Finally, we have \[thp1\] Consider the case $\sigma_0=1.$ Then the cyclicity of the boundary limit periodic set is at most $2$. We can start with the formula (\[eq31\]) which is valid for any $p\in \N.$ Moreover, for $p=1$ we have that $K(M)=\eta_4(\nu)-\eta_3(\nu)(1+\varepsilon_1)=*\bar\mu_3+O(\nu)O_P(M_C).$ This is a direct consequence of the fact that the linear part of the system at the points $P_3$ and $P_4$ is given, up to a constant, by $\dot r=r,\ \dot \rho =-\rho,\ \dot{\bar y}=-\sigma(\bar y+\bar\mu_3 \rho).$ Then, we can split the last term in (\[eq31\]) as the sum $*\bar\mu_3\nu r^{\alpha}\omega_\alpha+\nu r^{\alpha}\omega_\alpha O(\nu)O_P(M_C).$ The second term gives contributions of order $O(r^\delta)$ in $h_0,h_1$ and $h_2$, and produces a remainder $h_3$ of order $O(r^\delta)$ for the last leading monomial $r^{\alpha}\omega_\alpha.$ Then, for $p=1,$ the displacement map $V$ takes the form: $$\label{eq32} V(r,\rho)=*\varepsilon_0(1+h_0)+*\varepsilon_1 r^{1+\alpha}(1+h_1)+*\bar\mu_3 \nu r^{\alpha}(1+h_2)+*\bar\mu_3\nu r^{\alpha}\omega_\alpha(1+h_3)$$ The sequence of leading monomials in (\[eq32\]) does not verify the condition of non-resonance. To overcome this difficulty, we will use that $h_0$ and $h_3$ are of order $O(r^\delta).$ It will be sufficient to know that $h_1$ and $h_2$ are $o(1).$ As in the proof of Theorem \[thpgeq2\], the cyclicity is $0$ if $|\varepsilon_0|\geq \mathrm{ max}\{|\varepsilon_1|,|\bar\mu_3|\}.$ Otherwise, let us consider $L_{\cal X}\frac{ V}{1+h_0}.$ Using (\[eq29\]), we have that $$L_{\cal X}\frac{ V}{1+h_0}=*\varepsilon_1r^{1+\alpha} (1+g_1) +*\bar\mu_3\nu \Big[\alpha r^{\alpha}(1+h_2)+r^\alpha L_{\cal X}h_2\Big]+*\bar\mu_3\nu r^\alpha(1+g_3),$$ with $ g_3$ of order $O(r^\delta).$ Grouping the different terms, we obtain $$L_{\cal X}\frac{ V}{1+h_0}=r^{1+\alpha}\Big[*\varepsilon_1 (1+g_1)+*\bar\mu_3\rho (1+*\alpha+g_4)\Big],$$ where $g_4=*\alpha h_2+L_{\cal X}h_2+g_3$ is of order $o(1).$ Now, the sequence of leading monomials $\{1,\rho \}$ verifies the condition of non-resonance and we can apply Theorem \[thderdiv\] to $r^{-1-\alpha}L_{\cal X}\frac{ V}{1+h_0}.$ This function has at most $1$ root, yielding that $V$ itself has at most $2$ roots, if $|\varepsilon_0|\leq\mathrm{max}\{|\varepsilon_1|,|\bar\mu_3|\}.$ Appendix III ============ \[proof\_eps\_0\] The parameter function $\eps_0$ in the expression of the displacement map $V$ has the form for system . Since the system has an invariant parabola for $\mu_0=\mu_2=\mu_3=\mu_4=0$, it suffices to make the calculation for $\mu_0=\mu_2=\mu_3=\mu_5=0$. The system is integrable when $\mu_4=0$, with integrating factor $(1+y)^3$. Hence, it suffices to show that the following Melnikov integral is a nonzero multiple of $\mu_4$. Indeed, $$\int_{y=\frac12x^2-\frac12} \mu_4 \frac{x^2}{(1+y)^3}\,dx= \int_{-\infty}^\infty 8\mu_4 \frac{x^2}{(1+x^2)^3}\,dx= *\mu_4.$$ \[proof\_eps\_1\] The parameter function $\eps_1$ in the expression of the displacement map $V$ has the form for both systems and . It has been proved in [@DER96] (see for instance Theorem 3.5) that it suffices to show that $\int \mathrm{div} \,dt =*\mu_5$ along the invariant parabola when all parameters but $\mu_5$ vanish. Two different calculations are needed for the cases and . In the first case, the invariant parabola is given by . Then, $$\begin{aligned} \begin{split} \int \mathrm{div} \,dt&=\lim_{X_0\to \infty}\int_{-X_0}^{X_0} \frac{(2B+1)x + (1-B) \mu_5}{-y+ Bx^2+ B\mu_5x}\, dx\\ &=\lim_{X_0\to \infty}\left((2B+1) \ln\frac{1+B(X_0+(B-1)\mu_5)^2+o(\mu_5)}{1+B(X_0-(B-1)\mu_5)^2+o(\mu_5)} \right.\\ &\qquad \left.+2B^{3/2}(B-1) \mu_5 \left(\arctan\left(\sqrt{B}(X_0+O(\mu_5))\right)- \arctan\left(\sqrt{B}(-X_0+O(\mu_5))\right) \right)\right)\\ &= 2B^{3/2}(B-1)\pi \mu_5 + o(\mu_5).\end{split} \end{aligned}$$ The second case of is easier since the invariant parabola $y=\frac12 x^2+ \frac12$ is independent of $\mu_5$. Then $$\int_{y=\frac12 x^2+ \frac12} \mathrm{div} \,dt= \int_{-\infty}^\infty 2\mu_5 \frac{dx}{x^2+1} = 2\pi \mu_5.$$ \[coef\_rho\] The second derivative of the map $S=\rho F(0,\rho)$, where $F$ is defined in is a nonzero multiple of $\ov{\mu}_3$. We first localize the system at the nilpotent point at infinity using the coordinates $(v,w)= (-\frac{x}{y}, \frac1{y})$: after mutiplication by $w$, this yields $$\begin{aligned} \begin{split} \dot v&=w+ (1-B)v^2-\mu_2- \mu_3v+vw((3B-1)\mu_5+\mu_4) + v^2w,\\ \dot w&=vw - \mu_3w - (1-2B)\mu_5w^2+vw^2. \end{split} \label{coord_vw} \end{aligned}$$ A similar localization can be done for . We now let the blow-up $(v,w)= (r\ov{x}, r^2)$ for $w>0$, and we consider the restriction of the blow-up system to the $(\rho,\ov{x})$-plane for $r=0$, (after multiplication by $2$) $$\begin{aligned} \begin{split} \dot \rho&=-\rho(\ov{x}-\ov{\mu}_3\rho)= P(\rho, \ov{x}),\\ \dot{\ov{x}}&=2+(1-2B)\ov{x}^2-2 \ov{\mu}_2\rho^2-\ov{\mu}_3\ov{x}\rho= Q(\rho,\ov{x}). \end{split} \end{aligned}$$ Note that this system is the same for and . The singular points occur at $\ov{x}=\pm \beta$ with $\beta=\sqrt{\frac2{2B-1}}$. We localize at $P_3$ using $x_3=\beta-\ov{x}$ and at $P_4$ using $x_4=\beta+\ov{x}$. Hence, the system at $P_4$ is obtained from that at $P_3$ through $(x_3,\beta)\mapsto (-x_4,-\beta)$. The map is between two sections $\{\ov{x}_i= X_0\}$ in the normal form coordinates $\ov{x}_i$ near $P_i$ and we take $X_0$ small. The section $\{\ov{x}_4= X_0\}$ (resp $\{\ov{x}_3= X_0\}$) has equation $\ov{x}= f_4(\rho)=- x_0 +O(\rho)$ (resp. $\ov{x}= f_3(\rho)= x_0 +O(\rho)$). A formula for the second derivative was given in [@ZR] (Proposition 5.2), namely $$\begin{aligned} \begin{split} S''(0)&= S'(0)\left[2\left(f_4'(0)S'(0)\left(\frac{P_\rho'}{Q}\right)(0,f_4(0))- f_3'(0)\left(\frac{P_\rho'}{Q}\right)(0,f_3(0))\right)\right.\\ &\qquad+ \left. \int_{f_3(0)}^{f_4(0)} \left(\frac{P_{\rho\rho}''}{Q}(0,\ov{x})-2\frac{P_\rho'Q_\rho'}{Q^2}(0,\ov{x})\right)\exp\left(\int_{f_3(0)}^{\ov{x}} \left(\frac{P_\rho'}{Q}\right)(0,x) dx\right) d\ov{x}\right].\end{split}\label{derivee_seconde}\end{aligned}$$ Here, $S'(0)=1$. We call the three terms in the bracket $2I_1$, $2I_2$ and $I_3$. Let us first consider $I_3$. $$I_3= 4\ov{\mu}_3(2+(1-2B)x_0^2)^{\frac1{2(1-2B)}}\int_{x_0}^{-x_0} (1-B\ov{x}^2)(2+(1-2B)\ov{x}^2)^{\frac{8B-5}{2(1-2B)}} d\ov{x}.\label{formula_I3}$$ There are two different cases for $f_j'(0)$ depending whether $B_0= \frac 34$ or not. [**The case $B_0=\frac34$.**]{} In this case, the singular point has equal eigenvalues and a Jordan normal form for nonzero $\ov{\mu}_3$. Hence, the change of coordinate to normal form is tangent to the identity and $f_3'(0),f_4'(0) =O(\ov{\mu}_3)O(X_0)$. Also the integral part of $I_3$ in is equal to $-2\left(\frac{3}{2}x_0-\ln \frac{2+x_0}{2-x_0}\right)\not =0$. The result follows in that case. [**The case $B_0\neq\frac34$.**]{} In this case, the change of coordinates to normal form is given by $\ov{x}= \beta-\left(\ov{x}_3-\frac{\ov{\mu}_3}{3-4B}\rho\right) +O(|(\rho,\ov{x}_3)|^2)$ for $P_3$ (resp. $\ov{x}=- \beta+\left(\ov{x}_4+\frac{\ov{\mu}_3}{3-4B}\rho\right) +O(|(\rho,\ov{x}_4)|^2)$ for $P_4$), yielding $f_i'(0)= \frac{\ov{\mu}_3}{3-4B}(1+ O(X_0))$; $$2I_1+2I_2=[[+]]\ov{\mu}_3\frac{4}{3-4B}\,\frac{x_0}{2+(1-2B)x_0^2}$$ As for the integral part in $I_3$, it is given by $$\frac23 2^{\frac{5-8B}{2(2B-1)}} x_0\left[-3\phantom{,}_2F_1\left(\frac12,\frac{5-8B}{2(1-2B)};\frac32; \frac{2B-1}{2}x_0^2\right) +Bx_0^2\phantom{,}_2F_1\left(\frac32,\frac{5-8B}{2(1-2B)};\frac52; \frac{2B-1}{2}x_0^2\right) \right], \label{eq_I3}$$ where $\phantom{,}_2F_1(a,b;c;z)$ is the Gauss hypergeometric function defined by $$\phantom{,}_2F_1(a,b;c;z)= \sum_{i=0}^\infty \frac{(a)_n(b)_n}{(c)_n} \,\frac{z^n}{n!},$$ with $$(a)_0=1, \qquad (a)_n=a(a+1)\dots (a+n-1).$$ The function $\phantom{,}_2F_1(a,b;c;z)$ is analytic in the whole plane, except for a singularity at $z=1$. Moreover, $\phantom{,}_2F_1(a,b;c;0)=1$ and $$\begin{aligned} \begin{split} &\phantom{,}_2F_1(a,b;c;z)= \frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}\phantom{,}_2F_1(a,b;a+b-c+1;1-z)\\ &\qquad+ (1-z)^{c-a-b}\frac{\Gamma(c)\Gamma(a+b-c)}{\Gamma(a)\Gamma(b)}\phantom{,}_2F_1(c-a,c-b;c-a-b+1;1-z) \end{split}\end{aligned}$$ for $z\in (-1,1)$. This yields that near $z=1$ $$\phantom{,}_2F_1(a,b;c;z)= \frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}+ \frac{\Gamma(c)\Gamma(a+b-c)}{\Gamma(a)\Gamma(b)}(1-z)^{c-a-b}.\label{hyperg_1}$$ In the two hypergeometric functions appearing in , the exponent of $(1-z)$ in is $$c-a-b= \frac{4B-3}{2(1-2B)}\begin{cases} <0, &B>\frac34,\\ >0,& B<\frac34.\end{cases}.$$ Hence, the first (resp. second) term in is dominant when $B<\frac34$ (resp. $B>\frac34$). We treat the two cases. [**The case $B<\frac34$.**]{} For $\frac{2B-1}{2}x_0^2$ close to $1$, the bracket part of is close to $$-3\frac{\Gamma(\frac32)\Gamma(\frac{4B-3}{2(1-2B)})}{\Gamma(1)\Gamma(\frac{B-1}{1-2B})} + Bx_0^2\frac{\Gamma(\frac52)\Gamma(\frac{4B-3}{2(1-2B)})}{\Gamma(1)\Gamma(\frac{B-1}{1-2B}+1)} =\frac{\Gamma(\frac32)\Gamma(\frac{4B-3}{2(1-2B)})}{\Gamma(1)\Gamma(\frac{B-1}{1-2B})} \left (-3+\frac{3(1-2B)}{2(B-1)} Bx_0^2\right),$$ since $\Gamma(x+1)=x\Gamma(x)$. We let $x_0^2= \frac2{2B-1}- \delta$, with $\delta>0$ small. Using that $\Gamma(\frac32)=\frac12\sqrt{\pi},$ the integral part of $I_3$ in is close to $$\begin{cases} -\frac{3\sqrt{\pi}}2\frac{\Gamma(\frac{4B-3}{2(1-2B)})}{\Gamma(\frac{B-1}{1-2B})}\frac{2B-1}{2(B-1)}(2-B\delta), &B_0\neq1\\ \frac{3\sqrt{\pi}}4 \frac{\Gamma(\frac{4B-3}{2(1-2B)})}{\Gamma(\frac{-B}{1-2B})} Bx_0^2+ O(B-B_0), &B_0=1.\end{cases}$$ The coefficient is nonzero for $\delta>0$ as soon as $B_0\neq1$ (resp. $B_0=1$) and $\frac{B-1}{1-2B} $ (resp. $-\frac{B}{1-2B}$) is not a negative integer, which is the case for $B>\frac12$. This shows that $I_3$ grows as $(2+(1-2B)x_0^2)^{\frac1{2(1-2B)}}$, while $2(I_1+I_2)$ grows as $(2+(1-2B)x_0^2)^{-1}$. Hence, \[\[$I_3$\]\] is dominant when $B<\frac34$, and $2(I_1+I_2)+I_3= *\ov{\mu}_3\neq0$ when $B<\frac34$. [**The case $B>\frac34$.**]{} \[\[For $\frac{2B-1}{2}x_0^2$ close to $1$, the bracket part of has two parts $J_3'$ and $J_3''$. $$J_3'= -\frac{3\sqrt{\pi}}2\frac{\Gamma(\frac{4B-3}{2(1-2B)})}{\Gamma(\frac{B-1}{1-2B})}\frac{2B-1}{2(B-1)}(2+O(\delta)).$$ $$\begin{aligned} \begin{split}J_3''&=\left(1-\frac{2B-1}{2} x_0^2\right)^{\frac{4B-3}{2(1-2B)}}\left(-3\frac{\Gamma(\frac32)\Gamma(\frac{3-4B}{2(1-2B)})}{\Gamma(\frac12)\Gamma(\frac{5-8B}{2(1-2B)})} + Bx_0^2\frac{\Gamma(\frac52)\Gamma(\frac{3-4B}{2(1-2B)})}{\Gamma(\frac32)\Gamma(\frac{5-8B}{2(1-2B)})}+O(\delta)\right)\\ &= \frac32\left(1-\frac{2B-1}{2} x_0^2\right)^{\frac{4B-3}{2(1-2B)}}\frac{\Gamma(\frac{3-4B}{2(1-2B)})}{\Gamma(\frac{5-8B}{2(1-2B)})}(Bx_0^2 -1+O(\delta))\\ & -\frac{3}{3-4B}\left(1-\frac{2B-1}{2} x_0^2\right)^{\frac{4B-3}{2(1-2B)}}(1+O(\delta)). \end{split}\end{aligned}$$ This yields the corresponding parts $I_3'$ and $I_3''$ for $I_3$, considering that $\delta=2+(1-2B)x_0^2$: $$\begin{cases} I_3'=*\ov{\mu}_3 J_3'\delta^{\frac1{2(1-2B)}},\\ I_3''= -\ov{\mu}_3\frac{4x_0}{3-4B}\delta^{-1}+ O(1).\end{cases}$$ Considering that $\frac1{2(1-2B)}\in (-1,0)$, Then $2(I_1+I_2)+I_3=*\ov{\mu}_3 \delta^{\frac1{2(1-2B)}} (1+O(\delta))\neq0$. \]\] [99]{} F. Dumortier, R. Roussarie, Duck cycles and centre manifolds, Memoirs of A.M.S., vol. 121, n$^\circ 577$ (1996) 1–100. F. Dumortier, R. Roussarie and C. Rousseau, Hilbert’s 16th problem for quadratic vector fields, J. Differential Equations [**110**]{} (1994), no. 1, 86–133. F. Dumortier, M. El Morsalani and C. Rousseau, Hilbert’s 16th problem for quadratic systems and cyclicity of elementary graphics, Nonlinearity [**9**]{} (1996), 1209–1261. F. Dumortier, R. Roussarie and S. Sotomayor, [*Generic 3-parameter families of vector fields in the plane, unfoldings of saddle, focus and elliptic singularities with nilpotent linear parts.*]{} Springer Lecture Notes in Mathematics 1480, 1–164 (1991). Y. Ilyashenko and S. Yakovenko, Finitely-smooth normal forms of local families of diffeomorphisms and vector fields, Russian Mathematical Surveys [**46**]{} (1991), 1–43. R. Roussarie, Desingularisation of unfoldings of cuspidal loops, in: “Geometry and analysis in nonlinear dynamics”. H. Broer, F. Takens, Eds. Pitman Research Notes in Math. Series, n$^\circ 222,$ Longman Scientific and Technical (1992) 41–55. R. Roussarie and C. Rousseau, Finite cyclicity of nilpotent graphics of pp-type surrounding a center, Bull. Belg. Math. Soc. Simon Stevin [**15**]{} (2008), 547–614. H. Zhu and C. Rousseau, Finite cyclicity of graphics with a nilpotent singularity of saddle or elliptic type, J. Differential Equations [**178**]{} (2002), 325–436. [^1]: This research was supported by NSERC in Canada.

--- abstract: 'Recently N. Nitsure showed that for a coherent sheaf ${{\mathcal F}}$ on a noetherian scheme the automorphism functor ${\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}}$ is representable if and only if ${{\mathcal F}}$ is locally free. Here we remove the noetherian hypothesis and show that the same result holds for the endomorphism functor ${\underline{{\mathrm{End}\,}}}_{{{\mathcal F}}}$ even if one asks for representability by an algebraic space.' author: - Niko Naumann title: 'Representability of ${\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}}$ and ${\underline{{\mathrm{End}\,}}}_{{{\mathcal F}}}$' --- [MSC2000: 14A25]{} Statement of results {#s1} ==================== {#s11} Let $X$ be a scheme and ${{\mathcal F}}$ a quasi-coherent ${{\mathcal O}}_X$-module of finite presentation. We are interested in the representability of the following two functors on the category of $X$-schemes: $$\begin{aligned} {\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}} (X') & := & {\mathrm{Aut}\,}_{{{\mathcal O}}_{X'}} (f^* {{\mathcal F}}) \\ {\underline{{\mathrm{End}\,}}}_{{{\mathcal F}}} (X') & := & {\mathrm{End}\,}_{{{\mathcal O}}_{X'}} (f^* {{\mathcal F}})\end{aligned}$$ for $f : X' \to X$ an $X$-scheme. The result is as follows: \[thm11\] Let $X$ be a scheme and ${{\mathcal F}}$ a quasi-coherent ${{\mathcal O}}_X$-module of finite presentation. Then the following are equivalent: 1. ${{\mathcal F}}$ is locally free. 2. ${\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}}$ is representable by a scheme. 3. ${\underline{{\mathrm{End}\,}}}_{{{\mathcal F}}}$ is representable by a scheme. If $X$ is locally noetherian, these conditions are also equivalent to the following: 1. ${\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}}$ is representable by an algebraic space. 2. ${\underline{{\mathrm{End}\,}}}_{{{\mathcal F}}}$ is representable by an algebraic space. {#s12} The equivalence of 1) and 2) in theorem \[thm11\] in case $X$ is noetherian is the main result of [@N]. Our proof follows the ideas of [*loc.cit.*]{} closely. The main steps are contained in the following two lemmas: \[t1\] Let $A$ be a local ring and $M$ a finitely presented $A$-module which is [*not*]{} free. Then there is a local homomorphism $A \to B$ such that $$M \otimes_A B \cong B^n \oplus (B/b)^m \; ,$$ for some $0 \neq b \in B , b^2 = 0$ and $m \ge 1,n \ge 0$.\ If $A$ is noetherian, $B$ can be chosen to be artin. We observe that in the last statement of the lemma the noetherian hypothesis is indispensable: let $(B , {{\mathfrak{m}}})$ be a local ring such that there is $0 \neq b \in \bigcap_{n \ge 1} {{\mathfrak{m}}}^n$. Clearly $(b^2) \subsetneq (b)$, so after dividing out $(b^2)$ one gets a ring $B$ as in the lemma but for any local homomorphism $f : B \to C$ with $C$ [*noetherian*]{} one clearly has $f (b) = 0$. \[t2\] Let $S$ be a scheme and $S_0 \subseteq S$ a closed subscheme defined by a nilpotent ideal sheaf. Assume $X$ is a flat $S$-scheme and $f : X \to Y$ is an $S$-morphism such that $f \times {\mathrm{id}}_{S_0}$ is an isomorphism. Then $f$ is an isomorphism. {#s13} In order to treat the representability of ${\underline{{\mathrm{End}\,}}}_{{{\mathcal F}}}$ we will use the following observation: \[t3\] Under the assumptions of 1.1 the obvious natural transformation of (set-valued) functors ${\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}} \to {\underline{{\mathrm{End}\,}}}_{{{\mathcal F}}}$ is relatively representable by an open immersion. For completeness we also include a proof of the next lemma which is essentially lemma 5 of [@N] and shows the relative representability of a “parabolic” sub-group functor:\ Let $X$ be a scheme and $$\label{eq:1} 0 \longrightarrow {{\mathcal F}}' \longrightarrow {{\mathcal F}}\longrightarrow {{\mathcal F}}'' \longrightarrow 0$$ a short exact sequence of quasi-coherent ${{\mathcal O}}_X$-modules with ${{\mathcal F}}'$ finitely presented and ${{\mathcal F}}''$ locally free. For any morphism $f : Y \to X$, the sequence $f^* ((\ref{eq:1}))$ is exact because ${{\mathcal F}}''$ is in particular ${{\mathcal O}}_X$-flat and it makes sense to consider $$P (Y) := \{ \alpha \in {\mathrm{Aut}\,}_{{{\mathcal O}}_Y} (f^* {{\mathcal F}}) {\, | \,}\alpha (f^* {{\mathcal F}}') \subseteq f^* {{\mathcal F}}' \} \subseteq {\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}} (Y) \; .$$ \[t4\] In the above situation, the natural transformation $P \hookrightarrow {\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}}$ is relatively representable by a closed immersion. For basic facts about (relative) representability we refer to [@BLR], 7.6. Proofs ====== {#s21} In this subsection we dispense with the easy implications of theorem \[thm11\], the assumptions and notations of which we now assume:\ As ${\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}}$ and ${\underline{{\mathrm{End}\,}}}_{{{\mathcal F}}}$ are clearly Zariski sheaves the problem of representing them is Zariski local on $X$, i.e. we can assume that $X$ is affine and ${{\mathcal F}}$ corresponds to a free module of finite rank. In this case, representability of both ${\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}}$ and ${\underline{{\mathrm{End}\,}}}_{{{\mathcal F}}}$ is obvious; we have proved the implications 1) $\Rightarrow$ 2) and 1) $\Rightarrow$ 2’). Finally, the implications 2) $\Rightarrow$ 3) and 2’) $\Rightarrow$ 3’) are trivial. {#s22} [**Proof of lemma \[t1\]:**]{} Let $(A , {{\mathfrak{m}}})$ be a local ring and $M$ a finitely presented $A$-module which is not free. We will find the required local homomorphism $A \to B$ as a suitable quotient of $A$:\ Let $$\label{eq:2} A^m \xrightarrow{\alpha} A^n \xrightarrow{\beta} M \rightarrow 0$$ be a minimal presentation of $M$, i.e. $n = \dim_k (M / {{\mathfrak{m}}}M)$ where $k := A / {{\mathfrak{m}}}$ is the residue field of $A$. Then $M$ is free if and only if $\alpha = 0$: clearly $\alpha = 0$ is sufficient for freeness of $M$ and conversely, if $M$ is free, it is necessarily so of rank $n$, hence $\beta$ is a surjective endomorphism of $A^n$ which must be an isomorphism by a standard application of Nakayama’s lemma, c.f. [@M], thm. 2.4., hence $\alpha = 0$.\ For any $J \subseteq {{\mathfrak{m}}}$, (\[eq:2\]) $\otimes_A A / J$ is a minimal presentation of the $A / J$-module $M / JM$. If we denote by $I \subseteq A$ the ideal generated by the coefficients of any matrix representation of $\alpha$ and note that the minimality of (\[eq:2\]) implies $I\subseteq {{\mathfrak{m}}}$ we find that $M / JM$ is $A / J$-free if and only if $\alpha\otimes id_{A/J}=0$ if and only if $I \subseteq J$. As $M$ is not $A$-free we have $I \neq 0$ and as $I$ is finitely generated we get ${{\mathfrak{m}}}I \subsetneq I$, again by Nakayama’s lemma. By Zorn’s lemma, using again that $I$ is finitely generated, there is an ideal $J$ with ${{\mathfrak{m}}}I \subseteq J \subsetneq I$ and which is maximal subject to these conditions (indeed, any ascending chain of such ideals admits its union as an upper bound because I is finitely generated). We claim that $B := A / J$ is as required:\ By the maximality of $J$ the ideal ${\overline{I}}:= I / J$ is non-zero principal: ${\overline{I}}= (b) , 0 \neq b \in B$ and we neccessarily have $b^2 = 0$: if not, we would have $b \in (b^2)$, i.e. $b = xb^2$ or $b (1-xb) = 0$ for some $x \in B$. As $b \in {\overline{{{\mathfrak{m}}}}}:= {{\mathfrak{m}}}/ J$, the maximal ideal of $B , 1-xb$ was a unit of $B$, so we would have $b = 0$. We now show that $M\otimes_{A} B$ has the desired structure: any coefficient $\alpha_{ij}$ of a matrix representation of $\alpha \otimes {\mathrm{id}}_B$ is of the form $\alpha_{ij} = bu_{ij} , u_{ij} \in B$. As by construction ${\overline{{{\mathfrak{m}}}}}b = 0$ we see that if $\alpha_{ij} \neq 0$, then $u_{ij} \in B^*$. We get a matrix equation $(\alpha_{ij}) = b (u_{ij})$ and $(u_{ij})$ can be chosen with $u_{ij} = 0$ or $u_{ij} \in B^*$, all $i,j$. Then the usual Gau[ß]{}-algorithm can be applied to $(u_{ij})$, showing that indeed $M \otimes_A B \cong B^n \oplus (B / b)^m$ for some $m,n\geq 0$. As, by construction, $M \otimes_A B$ is not $B$-free, we finally see that $m \ge 1$. If $A$ is noetherian we can start the construction of $B$ by first dividing out a suitable high power of ${{\mathfrak{m}}}$: Indeed, if $M / {{\mathfrak{m}}}^n M$ was free for all $n \ge 1$ we would have $I \subseteq \bigcap_{n \ge 1} {{\mathfrak{m}}}^n = (0)$. Then the ring $B$ we obtain in the above construction is noetherian local with ${\overline{{{\mathfrak{m}}}}}$ nilpotent, hence zero-dimensional, i.e. $B$ is artin local. [**Proof of lemma \[t2\]:**]{} We can assume that the ideal sheaf ${{\mathcal I}}$ of $S_0 \subseteq S$ satisfies ${{\mathcal I}}^2 = 0$. Our assertion is local on $S, X$ and $Y$ and thus reduces to the following:\ Given a ring $k$ and an ideal $I \subseteq k$ of square zero, if $f : A \to B$ is a morphism of $k$-algebras with $B$ $k$-flat and such that $f \otimes_k {\mathrm{id}}_{k / I}$ is an isomorphism, then $f$ is an isomorphism:\ 1) $f$ is surjective: any $b \in B$ can be written $$b = f (a) + \sum_j\alpha_jb_j'\; \mbox{ ,some} \; \alpha_j \in I , b'_j \in B , a \in A \; .$$ Applying this to the $b'_j$ we get (for some $\alpha_{ij} \in I , b''_{ij} \in B , a_j \in A)$: $$b = f (a) + \sum_j\alpha_j(f(a_j)+\sum_{ij}\alpha_{ij}b_{ij}'')=f(a+\sum_j\alpha_j a_j )\; .$$ 2) $f$ is injective: For $K := $ker$ (f)$ the $k$-flatness of $B$ implies $K / IK = 0$ and the same argument as in 1) shows that $K = 0$. [**Proof of 2) $\Rightarrow$ 1) in theorem \[thm11\]:**]{} Under the notations of \[s11\] we assume that ${\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}}$ is representable by a scheme and, by contradiction, that ${{\mathcal F}}$ is not locally free. Note that the assumption on representability is stable under base-change $Y\rightarrow X$. So, base-changing to a suitable local ring of $X$, we can assume $X = {\mathrm{Spec}\,}(A)$ with $A$ local and ${{\mathcal F}}$ corresponding to a finitely presented $A$-module $M$ which is not free. According to lemma \[t1\] we can assume $M \cong A^n \oplus (A / a)^m$ for some $0 \neq a \in A$ with $a^2 = 0$ and $m \ge 1$. Let $G \to S:= {\mathrm{Spec}\,}(A)$ be the group-scheme representing ${\underline{{\mathrm{Aut}\,}}}_M$ and put $S_0 := {\mathrm{Spec}\,}(A / a)$. The sub-functor $G' \hookrightarrow G$ of automorphisms preserving (base-changes of) the direct summand $(A / a)^m$ is represented by a closed sub-group scheme (still to be denoted $G'$) according to lemma \[t4\]. Let $P \subseteq {\mathrm{Gl}\,}_{n+m , S}$ denote the standard parabolic sub-group of automorphisms preserving the rank $m$ direct summand. $P$ is flat over $S$, as can be seen over ${\mathrm{Spec}\,}({{\mathbb{Z}}})$. There is a morphism of $S$-groups $f : P \to G'$ which on points is given by sending $\left( \begin{smallmatrix} \alpha & 0 \\ \beta & \gamma \end{smallmatrix} \right)$ to $\left( \begin{smallmatrix} \alpha & 0 \\ \pi \beta & \overline{\gamma} \end{smallmatrix} \right)$, where $\alpha \in {\mathrm{Aut}\,}_A (A^n) , \gamma \in {\mathrm{Aut}\,}_A (A^m) , \beta : A^n \to A^m$ and $\overline{\gamma} \in {\mathrm{Aut}\,}_{A/a} ((A / a)^m)$ denotes the reduction of $\gamma$ and $\pi : A^m \to (A / a)^m$ is the natural map. This “point-wise” description of $f$ is immediately checked to be functorial and a homomorphism and hence does indeed define a morphism of $S$-groups. Obviously, $f \times {\mathrm{id}}_{S_0}$ is an isomorphism, hence so is $f$ by lemma \[t2\]. This is however a contradiction, because $f (S) : P (S) \to G' (S)$ is not injective, as $f (S) ({\mathrm{id}}_{A^n} \oplus (1-a) {\mathrm{id}}_{A^m}) = 1$ and $a \neq 0 , m \ge 1$. {#s23} [**Proof of lemma \[t3\]:**]{} Given a scheme $X$, a quasi-coherent ${{\mathcal O}}_X$-module ${{\mathcal F}}$ of finite presentation and some $\varphi \in {\mathrm{End}\,}_{{{\mathcal O}}_X} ({{\mathcal F}})$ we have to show that there is an open sub-scheme $X_0 \subseteq X$ such for all $f : Y \to X , f^* (\varphi) \in {\mathrm{Aut}\,}_{{{\mathcal O}}_Y} (f^* {{\mathcal F}}) \subseteq {\mathrm{End}\,}_{{{\mathcal O}}_Y} (f^* {{\mathcal F}})$ if and only if $f$ factors through $X_0$. Consider ${{\cal G}}:= {\mathrm{coker}\,}(\varphi)$ and the exact sequence of ${{\mathcal O}}_X$-modules $$\label{eq:3} {{\mathcal F}}\xrightarrow{\varphi} {{\mathcal F}}\xrightarrow{} {{\cal G}}\xrightarrow{} 0 \; .$$ We claim that $f^* (\varphi)$ is an automorphism if and only if $f^* ({{\cal G}}) = 0$: as $f^* ((\ref{eq:3}))$ is exact, necessity is obvious. If, conversely, $f^* ({{\cal G}}) = 0$ then for any $y \in Y$ $f^* (\varphi)_y$ is a surjective endomorphism of the finitely generated ${{\mathcal O}}_{Y,y}$-module ${{\mathcal F}}_y$, hence is an isomorphism, hence so is $f^* (\varphi)$. So the sought for $X_0 \subseteq X$ is the complement of the support of ${{\cal G}}$ which is open, because ${{\cal G}}$ is finitely presented. [**Proof of lemma \[t4\]:**]{} Given a scheme $X$ and a short exact sequence $0 \to {{\mathcal F}}' \to {{\mathcal F}}\to {{\mathcal F}}'' \to 0$ of quasi-coherent ${{\mathcal O}}_X$-modules with ${{\mathcal F}}'$ finitely presented and ${{\mathcal F}}''$ locally free and some $\alpha \in {\mathrm{Aut}\,}_{{{\mathcal O}}_X} ({{\mathcal F}})$, we have to show the representability by a closed sub-scheme of $X$ of the following functor on $X$-schemes: $$F (Y \xrightarrow{f} X) := \left\{ \begin{array}{ccl} * & , & f^* (\alpha) (f^* {{\mathcal F}}') \subseteq f^* {{\mathcal F}}' \\ \emptyset &, & \mbox{otherwise} \; . \end{array} \right.$$ Clearly, $F$ is a Zariski sheaf, so the problem is local on $X$, i.e. we can assume that $X = {\mathrm{Spec}\,}(A)$ is affine, ${{\mathcal F}}''$ corresponds to some $A^n$, ${{\mathcal F}}'$ corresponds to some $A$-module $M$ for which there is a presentation $A^a \to A^b \to M \to 0$ and ${{\mathcal F}}$ corresponds to some $A$-module $N$. The exact sequence $0 \to {{\mathcal F}}' \to {{\mathcal F}}\to {{\mathcal F}}'' \to 0$ then becomes an exact sequence $0 \to M \xrightarrow{\iota} N \xrightarrow{\pi} A^n \to 0$ of $A$-modules and we are given some $\alpha \in {\mathrm{Aut}\,}_A (N)$. Consider $\nu := \pi \alpha \iota$. As all the above sequences are exact after [*any*]{} base-change, we have $F (Y \xrightarrow{f} X) \neq \emptyset \iff f^* (\nu) = 0$. We have a diagram (defining $\psi$): $$\xymatrix{ A^a \ar[r] & A^b \ar[r] \ar[dr]^{\psi} & M \ar[r] \ar[d]^{\nu} & 0 \\ & & A^n & }$$ which is exact after any base-change, hence $f^* (\nu) = 0 \iff f^* (\psi) = 0$, for any $f : Y \to X$. So the closed sub-scheme of $X$ we are looking for is the one defined by the ideal of $A$ generated by the coefficients of any matrix representation of $\psi$. [**Proof of 2’) $\Rightarrow$ 1) in theorem \[thm11\]:**]{} Under the notations of \[s11\] we assume that ${\underline{{\mathrm{End}\,}}}_{{{\mathcal F}}}$ is representable by a scheme. Then so is ${\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}}$ by lemma \[t3\], hence ${{\mathcal F}}$ is locally free by what has been shown in \[s22\]. {#s24} \ [**Proof of 3) $\Rightarrow$ 1) and 3’) $\Rightarrow$ 1) in theorem \[thm11\]:**]{} Under the notations of 1.1 we assume that $X$ is locally noetherian and that either 3) or 3’) holds as well as, by contradiction, that ${{\mathcal F}}$ is not locally free. By lemma \[t3\] we know in either case that ${\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}}$ is representable by an algebraic space. Using the last assertion of lemma \[t1\] we can assume $X = {\mathrm{Spec}\,}(A)$ with $A$ artin local. Then ${\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}}$ is representable by a scheme according to [@K], p. 25, 7) contradicting what we proved in \[s22\]. [**Acknowledgements.**]{} I would like to thank H. Frommer and M. Volkov for interesting discussions and G. Weckermann for excellent type-setting. [9999]{} S. Bosch, W. Lütkebohmert, M. Raynauld, Néron Models, Ergebnisse der Mathematik, 3. Folge, Band 21, Springer, Heidelberg 1990. D. Knutson, Algebraic Spaces, Springer LNM 203. H. Matsumura, Commutative ring theory, Cambridge studies in advanced mathematics 8, 1997. N. Nitsure, Representability of ${\mathrm{Gl}\,}_E$, arXiv:math.AG/0204047. [Mathematisches Institut der WWU Münster\ Einsteinstr. 62\ 48149 Münster\ Germany\ e-mail: naumannn@uni-muenster.de]{}

--- abstract: 'Unbiased data collection is essential to guaranteeing fairness in artificial intelligence models. Implicit bias, a form of behavioral conditioning that leads us to attribute predetermined characteristics to members of certain groups and informs the data collection process. This paper quantifies implicit bias in viewer ratings of TEDTalks, a diverse social platform assessing social and professional performance, in order to present the correlations of different kinds of bias across sensitive attributes. Although the viewer ratings of these videos should purely reflect the speaker’s competence and skill, our analysis of the ratings demonstrates the presence of overwhelming and predominant implicit bias with respect to race and gender. In our paper, we present strategies to detect and mitigate bias that are critical to removing unfairness in AI.' author: - 'Rupam Acharyya\*,Shouman Das\*,Ankani Chattoraj, Oishani Sengupta,Md Iftekar Tanveer' bibliography: - 'reference.bib' title: Detection and Mitigation of Bias in Ted Talk Ratings --- Introduction ============ Machine-learning techniques are being used to evaluate human skills in areas of social performance, such as automatically grading essays [@alikaniotis2016automatic; @taghipour2016neural], outcomes of video based job interviews [@chen2017automated; @Naim2016], hirability [@Nguyen2016], presentation performance [@Tanveer2015; @Chen2017a; @Tanveer2018] etc. These algorithms automatically quantify the relative skills and performances by assessing large quantities of human annotated data. Companies and organizations world-wide are increasingly using commercial products that utilize machine learning techniques to assess these areas of social interaction. However, the presence of implicit bias in society reflected in the annotators and a combination of several other unknown factors (e.g. demographics of the subjects in the datasets, demographics of the annotators) creates systematic imbalances in human datasets. Machine learning algorithms (neural networks in most cases) trained on such biased datasets automatically replicate the imbalance [@o2016weapons] naturally present in the data and result in producing *unfair* predictions. Examining the impact of implicit bias in social behavior requires extensive, diverse human data that is spontaneously generated and reveals the perception of success in social performance. In this paper, we analyze ratings of TED Talk videos to quantify the amount of social bias in viewer opinions. TED Talks present a platform where speakers are given a short time to present inspiring and socially transformative ideas in an innovating and engaging way. In its mission statement, the TED organization describes itself as a “global community, welcoming people from every discipline and culture” and makes an explicit commitment to “change attitudes, lives, and ultimately the world” [@tedtalk]. Since TED Talks offer a platform to speakers from diverse backgrounds trying to convince people of their professional skills and achievements, the platform lends itself to a discussion of several critical issues regarding fairness and implicit bias: How can we determine the fairness of viewer ratings of TED Talk videos? Can we detect implicit bias in the ratings dataset? Are ratings influenced by the race and gender of the speaker? Ideally, these ratings should depend on the perception of the speaker’s success and communicative performance; not on the speaker’s gender or ethnicity. For instance, our findings show that while a larger proportion of viewers rate white speakers in a confidently positive manner, speaker of other gender identities and ethnic backgrounds receive a greater number of mixed ratings and elicit wider differences of opinion. In addition, men and women are rated as positive or negative with more consistency, while speakers identifying with other gender identities are rated less consistently in either direction. With this assumption, we conducted computational analysis to detect and mitigate bias present in our data. We utilize a state of the art metric “*Disparate Impact*” as in [@feldman2015certifying] for measuring fairness, and three popular methods of bias correction— 1. pre-processing [@calmon2017optimized; @kamiran2012data], 2. in-processing [@calders2010three; @kamishima2011fairness], and 3. post-processing [@hardt2016equality] . We compared such predictions of the ratings with the actual ratings provided by the viewers of the TED talks and found that our model prediction performs better w.r.t. a standard fairness metric. Our experiments show that if the traditional machine learning models are trained on a dataset without any consideration of the data bias, the model will make decision in an unwanted way which could be highly unfair to an unprivileged group of the society. In short, major contributions of the paper are as follows, 1. We show that public speaking ratings can be biased depending on the race and gender of a speaker. We utilize the state-of-the-art fairness measuring metric to identify the biases present in the TED talk public speaking rating. 2. We propose a systematic procedure to detect unfairess in the TEDTalk public speaking ratings. This method can be adopted by any machine learning practitioner or data scientist at industry who are concerned with fairness in their datasets or machine learning models. Related Works ============= With the increased availability of huge amount of data, data-driven decision making has emerged as a fundamental practice to all sorts of industries. In recent years, data scientists and the machine learning community put conscious effort to detect and mitigate bias from data sets and respective models. Over the years, researchers have used multiple notions of fairness as the tools to get rid of bias in data that are outlined below: - *‘individual fairness’*, which means that similar individuals should be treated similarly [@dwork2012fairness] - *‘group fairness’*, which means that underprivileged groups should be treated same as privileged groups [@pedreschi2009measuring; @pedreshi2008discrimination]. - *‘fairness through awareness’*, which assumes that an algorithm is fair as long as its outcome or prediction is not dependent on the use of protected or sensitive attributes in decision making [@grgic2016case]. - *‘equality of opportunity’*, mainly used in classification task which assumes that the probability of making a decision should be equal for groups with same attributes [@hardt2016equality]. - *‘counterfactual fairness*, very close to equality of opportunity but the probability is calculated from the sample of counterfactuals [@russell2017worlds; @kusner2017counterfactual] which ensures that the predictor probability of a particular label should be same even if the protected attributes change to different values . The fairness measures mentioned above can be characterized as both, manipulation to data and implementation of a supervised classification algorithm. One can employ strategies of detecting unfairness in a machine learning algorithm, observe [@zliobaite2015survey] and removing them by, - Pre-processing: this strategy involves processing the data to detect any bias and mitigating unfairness before training any classifiers [@calmon2017optimized; @kamiran2012data]. - In-processing: this technique adds a regularizer term in the loss function of the classifier which gives a measurement of the unfairness of the classifier [@calders2010three; @kamishima2011fairness]. - Post-processing: this strategy manipulates predictor output which makes the classifier fair under the measurement of a specific metric [@hardt2016equality]. For our analysis, we follow this well established paradigm and use an open-source toolkit AIF360 [@aif360-oct-2018] to detect and mitigate bias present in the data set and classifiers at all three stages: the pre-processing, the in-processing and the post-processing step. Data Collection =============== We analysed the TedTalk data collected from the [ted.com](ted.com) website. We crawled the website and gathered information about TedTalk videos which have been published on the website for over a decade (2006-2017). These videos cover a wide range of topics, from contemporary political, social issues to modern technological advances. The speakers who delivered talks at the TedTalk platform are also from a diverse background; including but not limited to, scientists, education innovators, celebrities, environmentalists, philosophers, filmmakers etc. These videos are published on the [ted.com](ted.com) website and are watched by millions of people around the world who can give ratings to the speakers. The rating of each talk is a collection of fourteen labels such as beautiful, courageous, fascinating etc. In this study we try to find if there is any implicit bias in the rating of the talks with respect to the race and gender of the speaker. Some properties of the full dataset is given in table \[tab:datasize\]. Each viewer can assign three out of fourteen labels to a talk and we use the total count for each label of rating for our analysis. In figure \[fig:avg\_rating\], average number of ratings in each of the fourteen categories is shown as a bar plot. Our preliminary observation reveals some disparities among the rating labels e.g. the label ‘inspiring’ has significantly higher count than other labels. **Property** **Quantity** ---------------------------------- -------------- Total number of Talks 2,383 Average number of views per talk 1,765,071 Total length of all talks 564.63 Hours Average rating labels per talk 2,498.6 : TED talk Dataset Properties: Information about the TED talk videos that are used in our method of detecting unfairness[]{data-label="tab:datasize"} ![Average number of ratings per video in each of the fourteen categories in the TedTalk dataset. observe that the the positive ratings outnumbers the negative rating on average which inform us about the pattern of the viewer’s rating. Viewer’s are more inclined to give positive ratings. []{data-label="fig:avg_rating"}](figures/rating_barplot.pdf){width="1.\columnwidth"} Data Annotation and Normalization --------------------------------- Our raw data includes information about the topic of the talk, number of views, date of publications, the transcripts used by the speakers, count of rating labels given by the viewers etc. However, this raw data does not come with the protected features such as gender and race of the speakers. We identified gender and race as potential sensitive attributes from our preliminary analysis. Thereby, to annotate the data for these protected features, we used amazon mechanical turk. We assigned three turkers for each talk, and annotated the race and gender of the speaker. To estimate the reliability of the annotation, we performed the Krippendorff’s alpha reliability test [@krippendorff2018content] on the annotations. We found a value of 93% for the Krippendorff’s alpha on our annotations. If there is a disagreement among the annotations of different turkers, we take the majority vote whenever possible or, we manually investigate to do the annotations. For gender, we used three categories: male, female, others, and for race, we used four categories, White, African American or Black, Asian, others. In our analysis, we used the total number of views $(V)$, the transcripts of the talks $(T)$ and the rating $(Y)$ given by the viewers. For preprocessing we employ the following steps, - First we normalize the number of views using the min-max normalization technique. $$V = \frac{V-V_{\min}}{V_{\max}-V_{\min}}$$ - For handling the transcripts, we utilize a state-of-the-art text embedding method ‘doc2vec’ as described in  [@le2014distributed] - We also scale the rating $(Y)$ of each video. Note that $Y$ consists of fourteen numbers $(y_1,\cdots, y_{14})$ which represents the number of fourteen rating categories. We scale it as, $$Y_{\textrm{scaled}} = \frac{Y}{\sum_{i=1}^{14}y_i}$$ Finally, to train our classification model, we binarize our rating lables with a threshold of the median across all talks. This means that for a given talk the rating label beautiful can be 0 or 1, courageous label can be 0 or 1 and so on for all fourteen possible lables per talk. Here 0 indicates the label does not hold and 1 indicates that the label holds true for the respective talk. **Gender** **Count** **Race** **Count** ------------ ----------- ---------- ----------- Female 768 White 1901 Male 1596 Asian 210 Other 19 Black 169     Other 103 : Total count of protected attributes (race and gender) in the TedTalk dataset. One preliminary obeservation is white male outnumbers all other groups by a huge margin, also third gender speakers are underrepresented in the data set.[]{data-label="tab:protected_attrinutes"} Observation in Data =================== Before we train our machine learning models for predicting TedTalk rating, we performed several exploratory analysis on the data set. First, we counted the total number of different gender and races of the speakers, see Table \[tab:protected\_attrinutes\]. We notice that there is a significant imbalance in the gender and race count. The count of ‘white male’ speakers outnumbers all the other groups combined. Also, the number of speakers from the third gender community is very small which shows an noteworthy imbalance in the dataset.\ {width="1.\linewidth"} {width="1.\linewidth"} There is also a clear and visible difference in the distribution of the ratings for different groups corresponding to the protected attributes: race and gender. Figure \[fig:race\_smooth\_hist\] and figure \[fig:gender\_smooth\_hist\] represents smoothed histogram across different groups for several ratings where we observed substantial difference in the distribution. Interestingly we find one expected bias and one counter intuitive bias when considering ratings based on race. The expected bias is that talks of white speakers are rated to be beautiful and courageous with high confidence as compared to speakers of other races, the sharp blue distributions in top row of Figure \[fig:race\_smooth\_hist\] are indicative of that. The wider green and red distributions on the top row of Figure \[fig:race\_smooth\_hist\] confirm that the society is more confused and less confident while rating the talk of a non-white speaker as beautiful or courageous. On the other hand, counter intuitively, we find that viewers rate talks of speakers of other race as more fascinating than other races (sharpness of red distribution in \[fig:race\_smooth\_hist\] in bottom row, left panel). Though counter intuitive, such bias is also not acceptable and clearly not fair. This highlights the diversity and difficulty of the issues of fairness and bias, since bias may not always stand out to be against the “expected” unprivileged class. Our work hence highlights the need to be careful when accounting for fairness as in fair society all types of biases should be removed.\ Furthermore we also looked at the way viewers rate speakers based on gender. Even under this category we observed that male speakers are confidently rated to have given beautiful and courageous talks when compared to any other gender (sharpness, and less width of blue distributions in figure \[fig:gender\_smooth\_hist\] indicates that). Though the confidence and tendency of rating in favor of male speakers drops substantially under informative and inspiring category (comparable width of green blue and red distributions) but is still slightly more than that for the speakers that are female (bi-modality of the distribution is an indication of confused rating behaviour) and of other gender. Besides highlighting the prevalence of biases with respect to gender and race, these results point out the need consider both expected and unexpected biases in data.\ {width="1.\linewidth"} To dive deep into the nature of prevalent bias with respect to gender of a speaker we further divided the rating labels into positive and negative, where positive includes say, “beautiful", “courageous" etc. and negative includes say, “confusing", “unconvincing" etc. Now if we carefully notice \[fig:correlation\], we will see that there is lot more structure in ratings for male and female speakers and a lot more variability in the matrix for other speakers when positive and negative rating labels are considered. Note that, in our dataset a viewer can choose to rate with any of three possible labels. The chaos in the third matrix (rightmost) of figure \[fig:correlation\] indicates that, for speakers of other genders, viewers tend to choose a mix of positive and negative labels. However for male and female speakers viewers are more sure and choose either all positive or all negative labels depending on their liking of the talk. This indicates that in general there is extreme confusion among viewers about their decision of whether or not they like a talk given by a speaker of other gender. This can also be identified as an indicator for the viewer’s double minded intention when they rate those talks. Methodology =========== In this section we will describe the methods we used to design a bias free rating prediction. We explored all three methods as suggested in  [@d2017conscientious], 1) Pre-Processing, 2) In-Processing and 3) Post Processing. We utilized the aforementioned toolkit AIF360 to implement these methods. This is explained in Figure \[fig:pipeline\] which is adopted from [@aif360-oct-2018].For each of the three steps, we quantify the amount of fairness with respect to a well defined metric, in this case *‘Disparate Impact’*  [@biddle2006adverse]. In our TedTalk dataset, the fourteen rating categories are transformed to binary labels as described in Data section, i.e the category ’beautiful’ can be 0 or 1 depending on if it was rated true or not. Using these binary labels we calculate the disparate impact in the rating of a video. Disparate impact can be understood with an example, let’s say we consider the rating category ‘beautiful’. For female speakers, we can estimate how likely are they to get a beautiful rating. Disaprate impact compares such probabilities for all possible groups like male with female, female with other gender, male with other gender and so on. Disparate impact equals to 1 identifies as a fair case since it means both groups that are compared are equally likely to be rated beautiful. {width="1.\linewidth"} Pre Processing -------------- As a first step we modify the original data before training the rating classifier. We then feed the modified and bias free data to the classifier for training as in \[fig:pipeline\] top row. The main driving force for this step is the fact that raw data is inherently biased in almost all problems pertaining to social science. The pre-processing step attempts to make the data bias free, so that the classifier trained with this unbiased data makes fair predictions. When we look into our data in details, we see from Table \[tab:protected\_attrinutes\] that the there is a strong imbalance of bias in the data with respect to race and gender. We observe that our data has a stronger bias related to race than for gender. So with the goal to make fair predictions, we first preprocess the data and attempt to remove the dominant bias with respect to race.\ There are several ways to remove bias from the training data before feeding it to the classifier  [@calmon2017optimized; @feldman2015certifying; @kamiran2012data; @zemel2013learning]. We have chosen the disparate impact remover method  [@feldman2015certifying] as the metric we have used is disparate impact metric. We use the disparate impact remover from AIF360 for this purpose. {width="0.7\linewidth"} The result of the preprocesing step is shown in Figure \[fig:di\_remover\]. This figure is showing the disparate impact across all rating labels of the dataset. We computed the disparate impact metric for the original label of the dataset and observed that it is far from 1 for most of the rating categories as shown by the blue bars. This exhibits the amount of unfairness present in the original dataset. The huge existing bias in original data highlights the need to design a fair predictor of the rating. Hence, we trained a logistic regression model with both the original dataset and the pre-processed dataset. We observed that prediction on preprocessed data has disparate impact closer to 1 (Figure \[fig:di\_remover\]) than when it is trained with the orginal biased data. Our result hence shows that it is important to identify the cause of dominant bias in the data and remove it by preprocessing to gain substantial improvement in fair predictions. In Processing ------------- In the last section we showed that it is possible to build a fair model by removing the bias present in the original data before training a classification model. However it is not always possible to do preprocessing, because there may not be access to the original data or it might be hard to identify the cause of bias in the data or it may be time consuming to re-annotate data and so on. Under such scenarios the solution is to design a fair classifier that uses a fair algorithm instead. So this allows us to still train the model with a biased dataset as input but the predictions employed by the fair algorithm are fair. The algorithm and the classifier in this case identifies the presence of bias in the dataset used for training and adjusts appropriately (based on the amount of bias) while predicting corresponding labels. Examples of such in processing techniques for bias mitigation can be found in  [@art2018; @kamishima2012fairness; @zhang2018mitigating]. For example, in our case, we hope that even if there is some bias in our dataset in favor of male, our rating classifier will make sure it accounts for that and weighs females and speakers of other genders equally to get rid of such unwanted unfairness. This will then generate fair rating predictions.\ For the TED talk data we have used the “Prejudice Remover" to do in processing  [@kamishima2012fairness]. The intuition is that we attempt to make the dependence of rating on sensitive attributes as strong as the nonsensitive attributes. For example, assume that female speaker is unprivileged and male speaker is privileged, that is ratings are heavily biased by male speakers in the real data. In that case the goal of the classifier and the learning algorithm is to make sure that rating predictions depend equally on male and female speaker and is not strongly influenced by male speakers only instead even if that is the case in the biased data.\ The result of in-processing is shown in Figure \[fig:inproc\]. This figure is showing the disparate impact across all rating labels of the dataset. Similar to the pre-processing part we computed the disparate impact metric (in this case we compute the metric for gender bias) for the original label of the dataset and observed that it is far from 1 for most of the rating categories as shown by the blue bars,. This exhibits the amount of unfairness present in the original dataset w.r.t gender. We then trained a logistic regression model and a Prejudice Remover model [@kamishima2012fairness] with the original dataset. We observed that prediction of the Prejudice Remover model has disparate impact closer to 1 (Figure \[fig:di\_remover\]) than the prediction of the logistic regression model for most of the categories. {width="0.7\linewidth"} Post Processing --------------- We tried many popular methods  [@hardt2016equality; @pleiss2017fairness] available in literature to employ post processing in our data. However, none of the methods gained significant improvement in making fair predictions. This highlights two important issues: 1) Nature of bias strongly drives what technique works best in terms of making fair predictions, in our case gender and race bias cannot be removed by just post processing technique 2) Not all types of dataset can used to make fair prediction with a fixed processing technique, in our case, as shown, post processing is unsuitable for the TED talk data videos. Conclusion from analysis ------------------------ We have employed a principled technique for removing bias and making fair rating predictions for TED talk videos. We first identify the dominant cause of the bias (in our case it is racial bias) in the data and remove it by pre processing, we then remove the other non-dominant causes of bias (gender bias in our case) by in-processing and we show that post processing does not make any improvement in fair predictions for our data. This highlights that choice of processing to achieve fair predictions heavily depends of types of bias and the dataset being used. This establishes the need to explore all possible biases and all possible techniques to obtain best results in terms of fairness. Discussion ========== Ensuring fairness in a machine learning model is a complex, multi-faceted issue which depends not only on the challenges presented by the immediate task but also on the definition of fairness that is in use. Unfairness or bias may arise at any stage of a machine learning pipeline. Thus, while developing machine learning algorithms which can make fair predictions for a classification task is important, creating a data collection process that is free from bias is essential for the model’s ultimate success. For example, Holstein et al. [@holstein2019improving] show that commercial machine-learning product teams express the need for tools that can actively guide the data collection process to ensure fairness in the downstream machine learning task. In our study, we demonstrate a systematic method of detecting bias using the AIF360 toolkit and show that various bias mitigation techniques can be applied to our data set at various stages of a machine learning pipeline (i.e. pre-processing, in-processing, post-processing). This procedure can be followed by any machine learning practitioner or data scientist who has to deal with real world data to make machine learning models for decision making.\ We have applied a state of the art fairness measuring metric ‘Disparate impact’ to a new and diverse dataset ‘TedTalk’, revealing sharp differences in the perception of speakers based on their race and gender. Based on our findings, we demonstrate that in TED Talk ratings, viewers rate ‘white male’ speakers confidently, while all other speakers are rated in a weaker manner. On the other hand, viewers rating shows a great deal variability when rating any other groups combined.\ Our study shows that detecting and erasing bias in the data collection process is essential to resolving issues related to fairness in AI. While the existing literature about fairness in AI [@dwork2012fairness; @pedreschi2009measuring; @russell2017worlds] have focused almost exclusively on detecting bias in specific domains such as recidivism prediction, admission decision-making, face detection etc, the analysis of implicit bias and . In all these scenarios, researchers come up with a quantifiable measurement of fairness and train the model which ensures the model is fair. However, in a more diverse domain where the users and system interacts in a complex way (e.g. recommendation system, chatbots etc), there is a lot of work left to do to detect and erase unfairness. While we have evaluated the Ted Talk dataset with specific metrics that are appropriate to it, there is an urgent need to develop metrics that are adapted just as uniquely to the needs of each dataset and its possible implementations. Identifying different kinds of social bias that affects in applications directly engaging with social influence and opinion formation is absolutely necessary for creating fair and balanced artificial intelligence systems.

--- abstract: | The one-dimensional partially asymmetric simple exclusion process with open boundaries is considered. The stationary state, which is known to be constructed in a matrix product form, is studied by applying the theory of $q$-orthogonal polynomials. Using a formula of the $q$-Hermite polynomials, the average density profile is computed in the thermodynamic limit. The phase diagram for the correlation length, which was conjectured in [@me99](J. Phys. A [**32**]{} (1999) 7109), is confirmed. \[Keywords: asymmetric simple exclusion, exact solution, density profile, $q$-orthogonal polynomials\] author: - | Tomohiro SASAMOTO\ [*Department of Physics, Graduate School of Science,*]{}\ [*University of Tokyo,*]{}\ [*Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan*]{} title: 'Density Profile of the One-Dimensional Partially Asymmetric Simple Exclusion Process with Open Boundaries' --- Introduction ============ \[intro\] The one-dimensional asymmetric simple exclusion process (ASEP) [@L; @Sp] is a system of particles which hop preferentially in one direction on a one-dimensional lattice with hard-core exclusion interaction. The ASEP has been studied extensively since it is one of the few models which show rich non-equilibrium behaviors and is exactly solvable [@Derrida98]. Besides, the ASEP has applications to many interesting problems such as the hopping conductivity, growth processes and the traffic flows [@SZ]. In this article, we consider the stationary state of the ASEP with open boundary conditions. That is, the system is connected to particle resevoirs at boundaries. The case where particles can hop only in one direction, which we refer to as the “totally asymmetric” case in the sequel, was solved in [@DEHP; @SD]. The current and the density profile were calculated exactly in the thermodynamic limit. The phase diagram for the current and the correlation length were identified. The system exhibits phase transitions depending on the parameters at the boundaries. Recently the obtained phase diagram was discussed from the point of view of the domain wall dynamics [@KSKS]. The partially asymmetric case with the open boundary conditions was partially solved in [@me99]. The current was evaluated in the thermodynamic limit. The phase diagram for the current was identified. It turned out to be the same as the one obtained by mean-field approximation [@ER] or by employing a plausible assumption [@Sandow]. The phase diagram for the correlaiton length was also obtained by assuming that the correlation length is given by the logarithm of the ratio of the largest and the second largest eigenvalue of a certain matrix which plays a similar role as a transfer matrix does in equilibrium statistical mechanical models. It was shown that the phase diagram has a richer structure than that for the totally asymmetric case. The average density profile was, however, not calculated in [@me99]. In this sence the obtained phase diagram for the correlation length has remained a conjecture. The purpose of this paper is the confirmation of this phase diagram. By using the explicit formula for the Poisson kernel of the $q$-Hermite polynomials, the average density profile in the thermodynamic limit is calculated for the partially asymmetric case. It turns out that the phase diagram was correctly predicted in [@me99]. In this article, we only consider the case where hoppings of particles at the boundaries and those at the bulk part of the system are compatible. In other words, when we allow the particle input at the left boundary and the particle output at the right boundary, the hopping rate to the right is assumed to be larger than that to the left. When hoppings at the boundaries and those at bulk is imcompatible, the current becomes zero in the thermodynamic limit. The situation seems to be similar to the closed boundary condition where particles can not enter or go out of the system [@SS]. Off course, when we consider the finite chain, the current remains to be positive. We remark that the asymptotic current for this case was evaluated in [@BECE]. The paper is organized as follows. In the next section, the definition of the model is given in terms of the master equation. The so-called matrix product ansatz, which gives the stationary state in the form of matrix product, is also explained. Some properties of the $q$-Hermite polynomials and the relationship to the matrix product ansatz are explained in section \[q-H\]. The section \[density\] is the main section of this article. First, the one-point funciton is represented in the form of double integrals. Second, the average density profile in the thermodynamic limit are summerlized whereas the evaluation of the integrals are relegated to Appendices. The phase diagram for the correlation length is identified. The concluding remarks are given in the last section. Definition of Model and Matrix Product Ansatz ============================================= The one-dimensional asymmetric simple exclusion process (ASEP) is defined as follows. During the infinitesimal time interval $\d t$, each particle jumps to the right nearest neighboring site with probability $p_R \d t$ and to the left nearest neighboring site with probability $p_L \d t$. If the chosen site is already occupied, the particle does not move due to the exclusion rule. More than one particle can not be on the same site. Each site can be either empty or occupied. The case where particles can hop only in one direction, i.e., the case where either $p_L = 0$ or $p_R=0$ is called the “totally asymmetric” case. The $p_R=p_L$ case is called the “symmetric” case whereas the case where particles hop in both directions with different rates will be referred to as the “partially asymmetric” case. In addition, we allow the particle input at the left end of the chain with rate $\alpha$ and allow the particle output at the right end of the chain with rate $\beta$ (Fig. 1). Here the length of the chain is denoted by $L$. In this article, we restrict our attention to the partially asymmetric case since the totally asymmetric case and the symmetric case was already solved in [@DEHP; @SD] and in [@me96] respectively. The restrictions on the parameters are $0 < p_L < p_R$ and $\alpha,\beta>0$. More formally, the process is defined in terms of the master equation. Each configuration of the system is indicated by $\{\tau_1,\tau_2,\ldots,\tau_L\}$ where $\tau_j$ $(j=1,2,\ldots,L)$ denotes the particle number at site $j$. Namely $\tau_j=0$ if the site $j$ is empty whereas $\tau_j=1$ if the site $j$ is occupied. Let $P(\tau_1,\tau_2,\ldots,\tau_L;t)$ denote the probability that the system has the configulation $\{ \tau_1,\tau_2,\ldots,\tau_L\}$ at time $t$. Then the time evolution of the ASEP is described by the following master equation, $$\begin{aligned} &\quad \frac{\d}{\d t} P(\tau_1,\tau_2,\ldots,\tau_L;t) \notag \\ &= \alpha (2\tau_1-1) P(0,\tau_2,\ldots,\tau_L;t) \notag \\ &\quad + \sum_{j=1}^{L-1} (\tau_j-\tau_{j+1}) \left[ p_L P(\tau_1,\tau_2,\ldots,0,1,\ldots,\tau_L;t) \right. \notag \\ &\quad \left. - p_R P(\tau_1,\tau_2,\ldots,1,0,\ldots,\tau_L;t) \right] \notag \\ &\quad + \beta (1-2\tau_L) P(\tau_1,\tau_2,\ldots,\tau_{L-1},1;t). \label{mas-eq}\end{aligned}$$ For instance, the master equation for $L=2$ case reads $$\label{mas-eq-L2} \frac{\d}{\d t} \begin{bmatrix} P(00;t)\\ P(01;t)\\ P(10;t)\\ P(11;t) \end{bmatrix} = - \begin{bmatrix} \alpha & -\beta & 0 & 0\\ 0 & \alpha + p_L +\beta & -p_R & 0\\ -\alpha & -p_L & p_R & -\beta\\ 0 & -\alpha & 0 & \beta \end{bmatrix} \begin{bmatrix} P(00;t)\\ P(01;t)\\ P(10;t)\\ P(11;t) \end{bmatrix}.$$ One can confirm himself that the dynamics of the ASEP is correctly encoded in the master equation (\[mas-eq\]). When time $t$ goes to infinity, the system is expected to reach the stationary state. The probability distribution in the stationary state will be denoted as $P(\tau_1,\tau_2,\ldots,\tau_L)$. For instance, except for the normalization, the stationary state for $L=2$ case is the eigenvector of the $4\times 4$ matrix in the right hand side of (\[mas-eq-L2\]) with the eigenvalue zero. Explicitly, it reads $$\label{stationary-2} \begin{bmatrix} P(00)\\ P(01)\\ P(10)\\ P(11) \end{bmatrix} = \text{Const.} \begin{bmatrix} \frac{1}{\alpha^2} \\ \frac{1}{\alpha\beta}\\ \frac{1}{p_R} \left( \frac{p_L}{\alpha\beta} + \frac{1}{\alpha} +\frac{1}{\beta} \right) \\ \frac{1}{\beta^2} \end{bmatrix}.$$ In [@DEHP], it was shown that the probability distribution of the ASEP in the stationary state for general $L$ can be written in the form of the matrix product as $$\label{originalMPA} P(\tau_1,\tau_2,\ldots,\tau_L) = \frac{1}{Z_L} \langle W| \prod_{j=1}^{L}(\tau_j D + (1-\tau_j) E)| V\rangle,$$ where $D$ and $E$ are square matrices and $\langle W|$ and $|V\rangle$ are vectors satisfying following relations, $$\begin{gathered} \label{mat-cond1} p_R DE-p_L ED = \zeta (D+E), \\ \label{mat-cond2} \alpha \langle W| E = \zeta\langle W|, \quad \beta D |V\rangle = \zeta |V\rangle.\end{gathered}$$ Here $\zeta$ is an arbitrary number. If one defines the matrix $C$ by $$\label{eq:def-c} C=D+E,$$ the normalization $Z_L$ is given by $$Z_L = \langle W| C^L |V\rangle.$$ Here, for the case of $L=2$, we check that the state (\[originalMPA\]) indeed gives the stationary state of the process by using the algebraic relations (\[mat-cond1\]) and (\[mat-cond2\]). For $L=2$, (\[originalMPA\]) reads $$\begin{bmatrix} P(00) \\ P(01) \\ P(10) \\ P(11) \end{bmatrix} = \frac{1}{Z_2} \begin{bmatrix} \langle W| E^2 |V\rangle \\ \langle W| ED |V\rangle \\ \langle W| DE |V\rangle \\ \langle W| D^2 |V\rangle \end{bmatrix}.$$ Three components, $P(00),P(01),P(11)$ can be calculated by simply using (\[mat-cond2\]). On the other hand, one computes $P(10)$ first changing the order of matrices $D,E$ by (\[mat-cond1\]) and then using (\[mat-cond2\]). Hence we get $$\begin{bmatrix} P(00) \\ P(01) \\ P(10) \\ P(11) \end{bmatrix} = \frac{1}{Z_2} \begin{bmatrix} \frac{\zeta^2}{\alpha^2}\\ \frac{\zeta^2}{\alpha\beta}\\ \frac{\zeta^2}{p_R} \left( \frac{p_L}{\alpha\beta} + \frac{1}{\alpha} +\frac{1}{\beta} \right)\\ \frac{\zeta^2}{\beta^2} \end{bmatrix}.$$ One can compare this expression with (\[stationary-2\]) to see that this expression indeed gives the stationary state for $L=2$ case. One also sees that the arbitrary parameter $\zeta$ appears in the same way for all components, $P(00),P(01),P(10),P(11)$. Changing the parameter $\zeta$ only changes the normalization $Z_2$. This is true for general $L$ as well. In this article, the proof that the state (\[originalMPA\]) gives the stationary state for general $L$ is not given. See [@DEHP]. We express several physical quantities in the form of matrix products. The one-point function $\langle n_j \rangle_L$ is defined as the probability that the site $j$ is occupied. In other words, $\langle n_j \rangle_L$ is the average density at site $j$. The two-point function $\langle n_j n_k\rangle_L$ is defined as the probability that the sites $j$ and the site $k$ are both occupied. Higher correlation functions are defined similarly. In the matrix language, they are computed by $$\begin{aligned} \label{def-1pt} \langle n_j \rangle_L &= \langle W| C^{j-1} D C^{L-j} |V\rangle /Z_L , \\ \label{def-2pt} \langle n_j n_k\rangle_L &= \langle W| C^{j-1} D C^{k-j-1} D C^{L-k}|V\rangle /Z_L ,\end{aligned}$$ and so on. The current through the bond between site $j$ and site $j+1$ is defined by $ J_L^{(j)} = p_R \langle n_j (1-n_{j+1})\rangle - p_L \langle n_j (1-n_{j-1})\rangle. $ In the steady state, the current is independent of $j$ and hence is denoted by $J_L$. It is given by $$\label{def-current} J_L = \zeta \frac{\langle W|C^{L-1}|V\rangle} {\langle W|C^{L}|V\rangle} = \zeta \frac{Z_{L-1}} {Z_L}.$$ Once one finds a representation of these algebraic relations, by using the above formula, one can in principle calculate the physical quantities such as the particle current $J_L$, the one-point function $\langle n_j \rangle_L$, the two-point function $\langle n_j n_k\rangle_L$ and the higher correlation functions. We note that the process has an obvious particle-hole symmetry. When we look at holes instead of particles, they tend to hop to the left with rate $p_R$ and to the right with rate $p_L$ with hard-core exclusion. In addition, they are injected at right end with rate $\beta$ and they are removed at the left end with rate $\alpha$. In other words, the process is invariant under the changes, $$\begin{aligned} \notag \text{particle} &\leftrightarrow \text{hole} \\ \label{symmetry} \alpha &\leftrightarrow \beta \\ \notag \text{site number} \,\, j &\leftrightarrow \text{site number} \,\, L-j+1.\end{aligned}$$ Due to this symmetry, it is sufficient to obtain the density for the right half of the system. The density for the left half of the system is obtained by using the above symmetry as $$\label{ri-le} \langle n_j \rangle_L (\alpha,\beta) = 1-\langle n_{L-j+1} \rangle_L (\beta,\alpha),$$ where the dependence of $\langle n_j \rangle_L$ on the parameters $\alpha$ and $\beta$ are eplicitly indicated. Before closing the section, we present some simulation results (Fig. 2 and Fig. 3). Figure 2 shows the space-time diagrams for several choices of parameters. It is clear that the properties of the system crutially depend on the values of the boundary parameters. The differeces become more transparent when we consider the particle current or the the average density profile of the stationary state. In principle, the stationary state is achieved only in the infinite time limit. However, we see from Fig. 2 that the system practically goes into a stationary state after some transient time. Hence if we average the density over a long time after the transient time, it would be regarded as the average density profile of the stationary state practically. The results are shown in Fig. 3. When $\alpha$ is small and $\beta$ is large, the bulk density is low. It decays sharply near the right boundary. This is called the low density phase. Conversely, when $\alpha$ is large and $\beta$ is small, the bulk density is high. It decays sharply near the left boundary. This is called the high density phase. When $\alpha=\beta$ is small, the low density region and the high density region coexist (coexistence line). Finally, when both $\alpha$ and $\beta$ are large enough, the density takes the value $1/2$ at bulk and decays slowly near both the boundaries. This is called the maximal current phase. Our main tasks in the following are to obtain the average density profiles in Fig. 4 exactly. Representation of Algebra and $q$-Hermite Polynomials ===================================================== \[q-H\] First we introduce some notations for later convenience. We introduce the $q$-number, $$\{n\} = 1-q^n,$$ and the $q$-shifted factorial, $$\begin{aligned} (a;q)_n &= \label{q-shi-fac-1} (1-a)(1-aq)(1-aq^2)\cdots(1-aq^{n-1}), \\ (a;q)_0 &= 1. \label{q-shi-fac-2}\end{aligned}$$ We also define $$\label{q-prod-inf} (a;q)_{\infty} = \prod_{j=0}^{\infty}(1-aq^j),$$ for $|q|<1$. Since products of $q$-shifted factorials appear so often, we use the notations, $$\begin{aligned} (a_1,a_2,\cdots,a_k;q)_{\infty} &= (a_1;q)_{\infty}(a_2;q)_{\infty}\cdots (a_k;q)_{\infty}, \\ (a_1,a_2,\cdots,a_k;q)_{n} &= (a_1;q)_{n}(a_2;q)_{n}\cdots (a_k;q)_{n} .\end{aligned}$$ Next a representation of the algebraic relaiton (\[mat-cond1\]) and (\[mat-cond2\]) is given. If we define $$D = 1+d, \hspace{10mm} E = 1+e,$$ we see that these relations become $$\begin{gathered} \label{q-b} d e - q e d = 1-q, \\ \label{w-coh} \langle W| e = a \langle W|, \hspace{10mm} d |V\rangle = b |V\rangle,\end{gathered}$$ where we put $$\begin{gathered} q\ = p_L/p_R, \\ a = \frac{1-\tilde{\alpha}} {\tilde{\alpha}}, \quad b = \frac{1-\tilde{\beta}} {\tilde{\beta}},\end{gathered}$$ with $\tilde{\alpha}=\alpha/(p_R-p_L), \tilde{\beta}=\beta/(p_R-p_L)$. Since $0< p_R<p_L$, we have $0<q<1$. In this article, we take the following representation for the matrices $d, e$ and the vectors $\langle W|, |V\rangle$, $$\begin{gathered} \label{de-qb} d = \begin{bmatrix} 0 & \{1\}^{\frac{1}{2}} & 0 & 0 & \cdots \\ 0 & 0 & \{2\}^{\frac{1}{2}} & 0 & \\ 0 & 0 & 0 & \{3\}^{\frac{1}{2}} & \\ \vdots & & & \ddots &\ddots \\ \end{bmatrix}, \hspace{10mm} e = \begin{bmatrix} 0 & 0 & 0 & 0 & \cdots \\ \{1\}^{\frac{1}{2}} & 0 & 0 & 0 & \\ 0 & \{2\}^{\frac{1}{2}} & 0 & 0 & \\ 0 & 0 & \{3\}^{\frac{1}{2}} & 0 & \\ \vdots & & & \ddots & \ddots \end{bmatrix}, \\ \label{WV-qb} \langle W| = \kappa\,_c\langle a| = \kappa \left( 1, \frac{a}{\sqrt{(q;q)_1} }, \frac{a^2}{ \sqrt{(q;q)_2} },\ldots \right), \hspace{10mm} |V \rangle = \kappa \, |b\rangle_c = \kappa \left( \begin{matrix} 1\\ \displaystyle\frac{b}{ \sqrt{(q;q)_1}}\\ \displaystyle\frac{b^2}{ \sqrt{(q;q)_2} }\\ \vdots \end{matrix} \right).\end{gathered}$$ The constant $\kappa$ is takes as $\kappa^2=(ab;q)_{\infty}$ so that $\langle W|V\rangle=1$. It should be noticed that there exists another useful representaion of the algebraic relations (\[mat-cond1\]) and (\[mat-cond2\]). It was first given in [@DEHP] and was used to obtain the phase diagram of the correlation length in [@me99]. The advantage of the represetation (\[de-qb\]) and (\[WV-qb\]) is that the commutation relation of the matrices $d$ and $e$ turns out to be a simple diagonal matrix. We have $$\label{de-com} de-ed = (1-q) \begin{bmatrix} 1 & 0 & 0 & 0 & \cdots \\ 0 & q & 0 & 0 & \\ 0 & 0 & q^2 & 0 & \\ 0 & 0 & 0 & q^3 & \\ \vdots & & & & \ddots \end{bmatrix}.$$ This fact will play an important role for the calculation of the average density profile in the next section. Next we list some properties of the continuous $q$-Hermite polynomials . The proofs can be found for instance in [@AAR; @GR]. The continuous $q$-Hermite polynomials $\{ H_n(x|q) |\, n=0,1,2,\ldots\}$ are defined by the three term recurrence relation, $$\label{rec-qh} H_{n+1}(x;q) + (1-q^n)H_{n-1}(x;q) = 2 x H_n(x;q),$$ with the initial condition, $$\label{init-cond} H_{-1}(x;q) = 0, \hspace{10mm} H_0(x;q) = 1.$$ They are explicitly given by the formula, $$H_n(\cos\theta|q) = \sum_{k=0}^{n} \frac{(q;q)_n }{ (q;q)_k (q;q)_{n-k} } e^{i(n-2k)\theta}.$$ The orthogonality relation reads $$\label{ortho} \int_{0}^{\pi} H_n(\cos\theta|q) H_m(\cos\theta|q) (e^{2i\theta},e^{-2i\theta};q)_{\infty}\d \theta = 2\pi \frac{ (q;q)_n }{ (q;q)_{\infty} } \delta_{mn}.$$ The generating function is also known and is given by $$\label{gen} \sum_{n=0}^{\infty} \frac{ H_n(\cos\theta|q) }{ (q;q)_n } \lambda^n = \frac{1}{(\lambda e^{i\theta},\lambda e^{-i\theta};q)_{\infty} }$$ for $|\lambda| < 1$. To calculate the average density profile, we also need the so-called Poisson kernel, $$\label{Mehler} \sum_{n=0}^{\infty} \frac{ H_n(\cos\theta|q) H_n(\cos\varphi|q) r^n } {(q;q)_n} = \frac{ (r^2;q)_{\infty} } { (r e^{i(\theta+\varphi)}, r e^{-i(\theta+\varphi)}, r e^{i(\theta-\varphi)}, r e^{-i(\theta-\varphi)} ;q)_{\infty} }.$$ This formula is called $q$-Mehler formula in the mathematics literature. Here we notince that, if we introduce $$\label{p-Hermite} p_n(x) = H_n(x|q)/\sqrt{(q;q)_n},$$ the three-term recurrence relation is rewritten into the form, $$\label{p-eigen} \begin{bmatrix} 0 & \{1\}^{\frac{1}{2}} & 0 & 0 & \cdots \\ \{1\}^{\frac{1}{2}} & 0 & \{2\}^{\frac{1}{2}} & 0 & \\ 0 & \{2\}^{\frac{1}{2}} & 0 & \{3\}^{\frac{1}{2}} & \\ \vdots & & \ddots & \ddots &\ddots \\ \end{bmatrix} \begin{bmatrix} p_ 0(x)\\ p_ 1(x)\\ p_ 2(x)\\ \vdots \end{bmatrix} = 2x \begin{bmatrix} p_ 0(x)\\ p_ 1(x)\\ p_ 2(x)\\ \vdots \end{bmatrix}.$$ In other words, $|p(x)\rangle =\,^t(p_0(x), p_1(x), \ldots)$ is formally an eigenvector of the matrix $d+e$ with eigenvalue $2x$. This is the basic relationship between the representaion of the algebra (\[mat-cond1\]),(\[mat-cond2\]) and the theory of $q$-orthogonal polynomials. Finally, the completeness of the continuous $q$-Hermite polynomials reads $$\label{complete} 1 = \frac{(q;q)_{\infty}} {2\pi} \int_0^{\pi} \d \theta (e^{i\theta},e^{-i\theta};q)_{\infty} |p(\cos\theta)\rangle \langle p(\cos\theta) |.$$ Calculation of Density Profile ============================== \[density\] In this section, the average density profile is calculated by using the formula (\[Mehler\]) of the continuous $q$-Hermite polynomials. We first recall the asymptotic behaviors of the normalization $Z_L$ and the current in the thermodynamic limit $J=\lim_{L\rightarrow\infty} J_L$. The asymptotic expressions for $Z_L$ were given in [@me99] and are summarized as follows: - For phase $A$ (low-density phase; $a>1$ and $a>b$ ; $\tilde{\alpha} < \frac12$ and $\tilde{\alpha} < \tilde{\beta}$) $$Z_L \simeq \frac{(a^{-2};q)_{\infty}} {(b/a;q)_{\infty}} [(1+a)(1+a^{-1})]^L ,$$ - For phase $B$ (high-density phase; $b>1$ and $a<b$ ; $\tilde{\beta} < \frac12$ and $\tilde{\alpha} > \tilde{\beta}$) $$Z_L \simeq \frac{(b^{-2};q)_{\infty}} {(a/b;q)_{\infty}} [(1+b)(1+b^{-1})]^L ,$$ - For phase $C$ (maximal current phase; $0<a,b<1$ ; $\tilde{\beta} > \frac12$ and $\tilde{\alpha} > \frac12$) $$Z_L \simeq \frac{(ab;q)_{\infty} (q;q)_{\infty}^3 4^{L+1}} {\sqrt{\pi}(a,b;q)_{\infty}^2 L^{ \frac{3}{2} } } ,$$ - On the coexistense line ($a=b>1$ ; $\tilde{\alpha} = \tilde{\beta} < \frac12$) $$Z_L \simeq \frac{(a-a^{-1}) (a^{-2};q)_{\infty} L} {(q;q)_{\infty}} [(1+a)(1+a^{-1})]^{L-1}.$$ Using (\[def-current\]), the current in the thermodynamic limit is readily computed as - For phase A ($\tilde{\alpha} < \frac12$ and $\tilde{\beta} >\tilde{\alpha}$) $$\label{current-A} J=(p_R-p_L)\tilde{\alpha} (1-\tilde{\alpha}),$$ - For phase B ($\tilde{\beta} < \frac12$ and $\tilde{\alpha} >\tilde{\beta}$) $$\label{current-B} J=(p_R-p_L)\tilde{\beta} (1-\tilde{\beta}),$$ - For phase C ($\tilde{\alpha} > \frac12$ and $\tilde{\beta} >\frac12$) $$\label{current-C} J=\frac{p_R-p_L}{4}.$$ The phase diagram for the current is depicted in Fig. 4. Now we turn to consider the average density profile. As you can see from the figures in Fig. 4, the average density is almost constant at bulk part except on the coexistence line. Hence we are interested in the average bulk density and how the density decays near the boundaries. When the average density decays like $e^{-r/\xi}$ with $r$ distance from the boundary, we refer to $\xi$ as the correlation length in this paper. As for the average density near the boundares, it is sufficient to compute the density near the right boundary due to the symmetry (\[symmetry\]). The relation (\[ri-le\]) enables us to know the average density profile near the left boundary. On the other hand, the coexistence line should be treated separately. Since it is easier to calculate the density difference than the density itself, we rewrite the density at site $j$ as $$\label{density-decompose} \langle n_j \rangle_L = \sum_{k=j}^{L-1} (\langle n_k \rangle_L - \langle n_{k+1} \rangle_L) + \langle n_L \rangle_L .$$ At the right boundary, we have $$\begin{aligned} \label{d-right} \langle n_L \rangle_L &= \frac{1}{Z_L} \langle W| C^{L-1} D |V\rangle \notag\\ &= \frac{1}{\tilde{\beta}} \frac{Z_{L-1}}{Z_L} \notag\\ &\rightarrow \frac{J}{\beta}. \hspace{10mm} (L\rightarrow\infty)\end{aligned}$$ Using (\[current-A\])-(\[current-C\]), the density at the right boundary is easily calculated. Next we notice that $$\begin{aligned} \langle n_k \rangle_L - \langle n_{k+1} \rangle_L &= \frac{1}{Z_L} (\langle W| C^{k-1}DC^{L-k}|V\rangle - \langle W| C^{k}DC^{L-k-1}|V\rangle ) \notag\\ &= \frac{1}{Z_L} \langle W|C^{k-1}(DC-CD)C^{L-k-1}|V\rangle \notag\\ &= \frac{1}{Z_L} \langle W|C^{k-1}(DE-ED)C^{L-k-1}|V\rangle \notag\\ &= \frac{1}{Z_L} \langle W|C^{k-1}(de-ed)C^{L-k-1}|V\rangle.\end{aligned}$$ Now one can represent $\langle W|C^{k-1}(de-ed)C^{L-k-1}|V\rangle$ in the form of double integrals. The calculation proceeds as follows. First we notice that $$\begin{aligned} &\quad \langle W|C^{k-1}(de-ed)C^{L-k-1}|V\rangle \notag\\ &= \kappa^2 (q;q)_{\infty}^2 \int_0^{\pi} \frac{\d \theta}{2\pi} (e^{2i\theta},e^{-2i\theta};q)_{\infty} \int_0^{\pi} \frac{\d \varphi}{2\pi} (e^{2i\varphi},e^{-2i\varphi};q)_{\infty} \notag\\ &\quad \times \, _c \langle a| C^{k-1} |p(\cos\theta)\rangle \langle p(\cos\theta)| (de-ed) C^{L-k-1} |p(\cos\varphi)\rangle \langle p(\cos\varphi)| b\rangle_c \notag\\ &= (ab;q)_{\infty} (q;q)_{\infty}^2 \int_0^{\pi} \frac{\d \theta}{2\pi} \int_0^{\pi} \frac{\d \varphi}{2\pi} (e^{2i\theta},e^{-2i\theta},e^{2i\varphi},e^{-2i\varphi};q)_{\infty} [2(1+\cos\theta)]^{k-1}[2(1+\cos\varphi)]^{L-k-1} \notag\\ &\quad\times \, _c \langle a| p(\cos\theta)\rangle \langle p(\cos\theta)| (de-ed) |p(\cos\varphi)\rangle \langle p(\cos\varphi)| b\rangle_c . \end{aligned}$$ Here the formula (\[gen\]) gives $$\label{ap-pb} _c \langle a| p(\cos\theta)\rangle = \frac{1}{(ae^{i\theta},a^{-i\theta};q)_{\infty}}, \quad \langle p(\cos\varphi)| b\rangle_c = \frac{1}{(be^{i\theta},b^{-i\theta};q)_{\infty}}.$$ The remaining term $ \langle p(\cos\theta)| (de-ed) |p(\cos\varphi)\rangle $ can also be computed by using the fact (\[de-com\]) and the formula (\[Mehler\]). We see that $$\begin{aligned} \langle p(\cos\theta)| (de-ed) |p(\cos\varphi)\rangle &= (1-q) \sum_{n=0}^{\infty} p_n(\cos\theta) p_n(\cos\varphi) q^n \notag\\ &= (1-q) \sum_{n=0}^{\infty} \frac{ H_n(\cos\theta|q) H_n(\cos\varphi|q) q^n } {(q;q)_n} \notag\\ &= \frac{ (q;q)_{\infty} } { (q e^{i(\theta+\varphi)}, q e^{-i(\theta+\varphi)}, q e^{i(\theta-\varphi)}, q e^{-i(\theta-\varphi)} ;q)_{\infty} }\end{aligned}$$ Hence we have $$\begin{gathered} \langle W|C^{k-1}(de-ed)C^{L-k-1}|V\rangle \notag\\ = (ab;q)_{\infty}(q;q)_{\infty}^3 \int_0^{\pi} \frac{\d \theta}{2\pi} \int_0^{\pi} \frac{\d \varphi}{2\pi} \frac{ (e^{2i\theta},e^{-2i\theta},e^{2i\varphi},e^{-2i\varphi};q)_{\infty} [2(1+\cos\theta)]^{k-1}[2(1+\cos\varphi)]^{L-k-1} } {(ae^{i\theta}, ae^{-i\theta}, qe^{i(\theta+\varphi)}, qe^{-i(\theta+\varphi)}, qe^{i(\theta-\varphi)}, qe^{-i(\theta-\varphi)}, be^{i\varphi}, be^{-i\varphi};q)_{\infty} }\end{gathered}$$ for $0<a,b<1$. Summing with respect to $k$ from $j$ to $L-1$, we get $$\begin{gathered} \sum_{k=j}^{L-1} (\langle n_k \rangle_L - \langle n_{k+1} \rangle_L) \notag\\ = \frac{(ab;q)_{\infty}(q;q)_{\infty}^3}{Z_L} \int_0^{\pi} \frac{\d \theta}{2\pi} \int_0^{\pi} \frac{\d \varphi}{2\pi} \frac{ (e^{2i\theta},e^{-2i\theta},e^{2i\varphi},e^{-2i\varphi};q)_{\infty} } {(ae^{i\theta}, ae^{-i\theta}, qe^{i(\theta+\varphi)}, qe^{-i(\theta+\varphi)}, qe^{i(\theta-\varphi)}, qe^{-i(\theta-\varphi)}, be^{i\varphi}, be^{-i\varphi};q)_{\infty} } \notag\\ \times \frac{ [2(1+\cos\theta)]^{j-1}[2(1+\cos\varphi)]^{L-j} -[2(1+\cos\theta)]^{L-1} } { 2\cos\varphi - 2 \cos\theta }, \label{diff}\end{gathered}$$ when $0<a,b<1$. Hence, substitution of this expression into (\[density-decompose\]) leads to the expression of the average density profile for $0<a,b<1$. Finally, the analytic continuation of (\[diff\]) gives the expression of the density porfile for other values of the parameters as well. This is conveniently done by writing (\[diff\]) as a contour integral of the two complex variables $z_1=e^{i\theta}$ and $z_2=e^{i\varphi}$. That is, we rewrite (\[diff\]) as $$\label{I1-I2} \sum_{k=j}^{L-1} (\langle n_k \rangle_L - \langle n_{k+1} \rangle_L) = \frac{1}{Z_L}(I_1+I_2),$$ where $$\begin{aligned} I_1 &= \frac14 (ab;q)_{\infty}(q;q)_{\infty}^3 \int\frac{\d z_1}{2\pi i z_1} \int\frac{\d z_2}{2\pi i z_2} \notag\\ &\quad \frac{(z_1^2,z_1^{-2},z_2^2,z_2^{-2};q)_{\infty} [(1+z_1)(1+z_1^{-1})]^{L-1}} {(az_1,a z_1^{-1},qz_1 z_2, q z_1^{-1} z_2^{-1}, q z_1 z_2^{-1},q z_1^{-1} z_2^{-1}, bz_2,b z_2^{-1};q)_{\infty} (z_2 + z_2^{-1} -z_1-z_1^{-1})}, \notag\\ I_2 &= \frac14 (ab;q)_{\infty}(q;q)_{\infty}^3 \int\frac{\d z_1}{2\pi i z_1} \int\frac{\d z_2}{2\pi i z_2} \notag\\ &\quad \frac{(z_1^2,z_1^{-2},z_2^2,z_2^{-2};q)_{\infty} [(1+z_1)(1+z_1^{-1})]^{j-1} [(1+z_2)(1+z_2^{-1})]^{L-j} } {(az_1,a z_1^{-1},q z_1 z_2, q z_1^{-1} z_2^{-1}, q z_1 z_2^{-1},q z_1^{-1} z_2^{-1}, bz_2,b z_2^{-1};q)_{\infty} (z_2 + z_2^{-1} -z_1-z_1^{-1})}.\end{aligned}$$ When $0<a,b<1$, the contours of $z_1$ and $z_2$ are both unit circles. For other valures of the parameters, the contours are deformed as poles in the integrands move in and out of the unit circles. The evaluation of the integrals $I_1$ and $I_2$ in the thermodynamic limit for each phase is relegated to Appendices. Basically, the integral $I_1$ gives the density at bulk region whereas the integral $I_2$ gives the density near the right boundary. The results are summerized in the following. We thus obtain the phase diagram shown in Fig. 5. Notice that this phase diagram was corrrectly predicted in [@me99]. Therein the correlation length for each phase was assumed to be given by the logarithm of the ratio of the largest and the second largest eigenvalue of the matrix $C$. - Phase $C$ ($\tilde{\alpha} > 1/2$ and $\tilde{\beta}>1/2$; $0<a,b<1$) In this phase, the average density at bulk is $1/2$. The average density decays near the right bounary as $$\label{density-max} \langle n_j \rangle_L = \frac12 - \frac{1}{2\sqrt{\pi} l^{\frac12}}.$$ Here and in the following, we set $l=L-j+1$. The density decays algebraically and hence the correlation length is infinite. This average density profile is exactly the same as that for the totally asymmetric case. The density decay at the left boundary can be obtained from the symmetry relation (\[ri-le\]). - Phase $A_1$ ($ \tilde{\alpha} < \tilde{\beta} < \tilde{\alpha}/[(1-\tilde{\alpha})q+\tilde{\alpha}]$ and $\tilde{\beta} < 1/2$; $aq<b<a$ and $b>1$ ) The average density at bulk is $\tilde{\alpha}$. The density near the right boundary decays exponentially as $$\label{density-A1} \langle n_j \rangle_L = \tilde{\alpha} - \frac{(b^{-2}q,q;q)_{\infty}} {(a^{-1}b^{-1}q,ab^{-1}q;q)_{\infty}} \left[ \frac{\tilde{\alpha}(1-\tilde{\alpha}) } {\tilde{\beta}(1-\tilde{\beta})} \right]^l (1-2\tilde{\beta}),$$ with the correlation length $$\xi^{-1} = \ln \frac{\tilde{\beta}(1-\tilde{\beta})} {\tilde{\alpha}(1-\tilde{\alpha})}.$$ On the other hand, the density near the left boundary takes the constant value $\tilde{\alpha}$. This fact is obtained by conbining the average density profile near the right boundary for the high-density phase below and the symmetry (\[symmetry\]). - Phase $A_2$ ($q/(1+q) < \tilde{\alpha} < 1/2$ and $\tilde{\beta} > 1/2$; $1<a<q^{-1}$ and $b<1$) In the bulk region, the average density takes the constant value $\tilde{\alpha}$. On the other hand, the density profile near the right boundary decays exponentially as $$\label{density-A2} \langle n_j \rangle_L = \tilde{\alpha} - \frac{(a-b)(1-ab)(abq,a^{-1}bq;q)_{\infty} (q;q)_{\infty}^4} {(a-1)^2(b-1)^2(aq,a^{-1}q,bq;q)_{\infty}} \frac{[4\tilde{\alpha}(1-\tilde{\alpha})]^l} {\sqrt{\pi} l^{\frac32}},$$ with the correlation length $$\xi^{-1} = -\ln 4[\tilde{\alpha}(1-\tilde{\alpha})].$$ But the decay is not purely exponential but with algebraic corrections. At the left boundary, the density takes the constant value $\tilde{\alpha}$. - Phase $A_3$ ( $\tilde{\beta} > \tilde{\alpha}/[(1-\tilde{\alpha})q+\tilde{\alpha}]$ and $\tilde{\alpha} < q/(1+q)$; $a>q^{-1}$ and $b<aq$) The average density at bulk is $\tilde{\alpha}$. Near the right boundary, the density decays exponentially as $$\label{density-A3} \langle n_j \rangle_L = \tilde{\alpha} - \frac{(1-ab)(1-(aq)^{-1})} {(1-b(aq)^{-1})(1+aq)} \left[\frac{(1+aq)(1+(aq)^{-1})} {(1+a)(1+a^{-1})} \right]^l$$ with the correlation length $$\xi^{-1} = \ln \frac{q} {[\tilde{\alpha}+(1-\tilde{\alpha})q]^2}.$$ This density profile has no correspondense for the totally asymmetric case. At the left boundary, the density takes the constant value $\tilde{\alpha}$. - Phase $B_1$ ($ \tilde{\alpha}q/[(1-\tilde{\alpha})+q\tilde{\alpha}] < \tilde{\beta} < \tilde{\beta} $ and $\tilde{\alpha} < 1/2$; $bq<a<b$ and $a>1$) The average density at bulk is $1-\tilde{\beta}$. As can be seen from the calculation in Appendix, The density takes the constant value near the right boundary. $$\label{density-B} \langle n_j \rangle_L = 1-\tilde{\beta}$$ Since this phase is related to the phase $A_1$ through the symmetry (\[symmetry\]), the denisty decays exponentially near the left boundary with the correlation length $$\xi^{-1} = \ln \frac{\tilde{\alpha}(1-\tilde{\alpha})} {\tilde{\beta}(1-\tilde{\beta})}.$$ - Phase $B_2$ ($q/(1+q) < \tilde{\beta} < 1/2$ and $\tilde{\alpha} > 1/2$; $a<1$ and $1<b<q^{-1}$) This phase is symmetric to phase $A_2$ through the symmetry (\[symmetry\]). The average density at bulk and near the right boundary is $1-\tilde{\beta}$. Near the left boundary, the density decays exponentially with the correlation length $$\xi^{-1} = -\ln 4[\tilde{\beta}(1-\tilde{\beta})].$$ - Phase $B_3$ ($a<bq$ and $b>q^{-1}$) ($\tilde{\beta} < \tilde{\alpha}q/[1-\tilde{\alpha}+\tilde{\alpha}q]$ and $\tilde{\beta} < q/(1+q)$; $a<bq$ and $b>q^{-1}$) This phase is symmetric to phase $A_3$ through the symmetry (\[symmetry\]). The average density at bulk and near the right boundary is $1-\tilde{\beta}$. The density decays exponentially with the correlation length $$\xi^{-1} = \ln \frac{q} {[\tilde{\beta}+(1-\tilde{\beta})q]^2},$$ near the left boundary. - Coexistence line ($\tilde{\alpha}=\tilde{\beta}<1/2$; $a=b>1$) The average density shows linear profile at bulk, $$\label{density-coex} \langle n_j \rangle_L = \tilde{\alpha} + (1-2\tilde{\alpha})\frac{j}{L}.$$ This is essentially the same as the result for the totally asymmetric case. Before closing the section, we present a simulation result for the correlation length to show the differece between the phase $A_2$ and $A_3$. The simulation was done on $\tilde{\beta}=1$ line. For the totally asymmetric case, the system is in the $A_2$ phase on this line. The the correlation length is given by $\xi_{A_2}=1/\ln4[\alpha(1-\alpha)]$. On the other hand, for the partially asymmetric case, the system is in the $A_3$ phase when $\tilde{\alpha}<q/(1+q)$. The correlation length is given by $\xi_{A_3}=1/\ln\frac{q}{[\tilde{\beta}+(1-\tilde{\beta})q]^2}$. The differences between these two expressions become large especially when $q$ and $\alpha$ are small. Especially, as $\alpha \rightarrow 0$, $\xi_{A_2}$ goes to zero whereas $\xi_{A_3}$ goes to $-1/\ln q$. In Fig. 6 the simulation result for the correlation length is shown for $p_R=1.0,\,P_L=0.9,\,\beta=10$ which corresponds to $q=0.9,\,\tilde{\beta}=1$. It is clear that the the correlatin lenght approaches the finite value as $\alpha \rightarrow 0$ and is well described by the formula for $\xi_{A_3}$. Concluding Remarks ================== \[conc\] In this article, we have computed the average density profile of the partially asymmetric simple exclusion process with open boundaries. The calculation has been done for a wide rage of parameters satisfying $0<p_L<p_R$ and $\alpha>0,\beta>0$. The phase diagram for the correlation length has been obtained. It has turned out that the phase diagram was correctly predicted in the earlier paper [@me99]. In [@me99], the phase diagram was obtained by assuming that the correlation length is given by the logarithm of the ratio of the largest and the second largest eigenvalues of the matrix $C$. The discussions were only for the phases with the exponentially decaying profile. In this article, we have not only confirmed this fact but also obtained the asymptotic expressions of the average density profile for all phases. There are two key facts which allowed us to calculate the average density profile exactly in the thermodaynamic limit. One is that the commuation relation of the matrices $D,E$ becomes a simple diagonal matrix and the other is the formula (\[Mehler\]) of the $q$-Hermite polynomials. There seems to be many possible applications and generalizations of the analysis of this article. First, it is possible to generalize the analysis in this paper to the partially asymmetric exclusion process on a ring with a single defect particle [@Mallick; @Jafa]. The corresponding totally asymmetric case was already solved in [@Mallick]. Second, the case where $p_L>p_R$ is also interesting. Although the current was evaluated in [@BECE], more exact results are desirable. Third, it would be interesting to apply the simialr analysis to the multi-species models [@EFGM; @EKKM; @AHR98-1; @ADR; @MMR]. Compared to the ASEP, much less is known about these models. Several investigatios are now in progress [@RSS]. The results about these will be reported elsewhere. Acknowledgment {#acknowledgment .unnumbered} ============== The author would like to thank P. Deift, E. R. Speer and N. Rajewsky for fruitful discussions and comments. He also thanks the continuous encouragement of M. Wadati. The author is a Research Fellow of the Japan Society for the Promotion of Science. Evaluation of Integral $I_1$ ============================ In this appendix, the integral $I_1$, $$\begin{aligned} I_1 &= \frac14 (ab;q)_{\infty}(q;q)_{\infty}^3 \int\frac{\d z_1}{2\pi i z_1} \int\frac{\d z_2}{2\pi i z_2} \notag\\ &\quad \frac{(z_1^2,z_1^{-2},z_2^2,z_2^{-2};q)_{\infty} [(1+z_1)(1+z_1^{-1})]^{L-1}} {(az_1,a z_1^{-1},qz_1 z_2, q z_1^{-1} z_2^{-1}, q z_1 z_2^{-1},q z_1^{-1} z_2^{-1}, bz_2,b z_2^{-1};q)_{\infty} (z_2 + z_2^{-1} -z_1-z_1^{-1})}, \label{I1def}\end{aligned}$$ is evaluated. First, for the case where $a,b<1$, both of the contours of $z_1$ and $z_2$ are unit circles. We have $$\begin{gathered} I_1 = I_1^{(0)} = \int_0^{\pi} \frac{\d \theta}{2\pi} \int_0^{\pi} \frac{\d \varphi}{2\pi} \frac{ (e^{2i\theta},e^{-2i\theta},e^{2i\varphi},e^{-2i\varphi};q)_{\infty} } {(ae^{i\theta}, ae^{-i\theta}, qe^{i(\theta+\varphi)}, qe^{-i(\theta+\varphi)}, qe^{i(\theta-\varphi)}, qe^{-i(\theta-\varphi)}, be^{i\varphi}, be^{-i\varphi};q)_{\infty} } \notag\\ \times \frac{ [2(1+\cos\theta)]^{L-1} } { 2\cos\varphi - 2 \cos\theta }. \label{I10}\end{gathered}$$ Second, consider the case where $a$ becomes larger one but $b$ is still smaller than one. We assume $$\label{na} a>aq>aq^2>\cdots >aq^{n^{(a)}} >1> aq^{n^{(a)}+1}>\cdots .$$ Then the contour of $z_2$ is still a unit circle but the contour of $z_1$ has to be modified to include all poles at $z_1=aq^k$ ($k=0,1,\ldots,n^{(a)}$) and to exclude all poles at $z_1=(a q^k)^{-1}$ ($k=0,1,\ldots,n^{(a)}$). Seperating the contributions from poles at $z_1=aq^{k}$ and $z_1=(aq^k)^{-1}$, the integral $I_1$ can be rewritten as $$\begin{aligned} \label{eq:I1-1} I_1 &= I_1^{(0)} - \frac{(ab;q)_{\infty} (q;q)_{\infty}^2}{2 a} \sum_{k=0}^{n^{(a)}} \frac{(-)^k q^{k(k-1)/2} (a^2 q^{2k},a^{-2} q^{-2k};q)_{\infty} [\lambda_k^{(a)}]^{L-1} } {(q;q)_k (a^2 q^k;q)_{\infty}} \notag\\ &\quad\times \int_{C_0} \frac{\d z_2}{2\pi i z_2} \frac{(z_2^2,z_2^{-2};q)_{\infty}} {(aq^{k+1}z_2,aq^{k+1}z_2^{-1},a^{-1}q^{-k}z_2, a^{-1}q^{-k}z_2^{-1},b z_2,bz_2^{-1};q)_{\infty}} . \end{aligned}$$ Here the contour $C_0$ denotes the unit circle. The $\lambda_k^{(c)}$’s are defined by $$\lambda_k^{(c)} = (1+c q^k)(1+c^{-1}q^{-k}),$$ for $c=a,b$ and $k=0,1,2,\ldots$. The intgral can be evaluated explicitly by using the general formula, $$\label{int-aw} \int_C \frac{\d z}{2\pi i z} \frac{(z^2,z^{-2};q)_{\infty}} {(az,az^{-1},bz,bz^{-1},cz,cz^{-1},dz,dz^{-1};q)_{\infty}} = \frac{2(abcd;q)_{\infty}} {(q,ab,ac,ad,bc,bd,cd;q)_{\infty}}.$$ The contour $C$ is such that it includes all poles of the type $f q^{k}$ and excludes all poles of the type $f^{-1}q^{-k}$ with $f=a,b,c,d$ and $k=0,1,2,\ldots$. The parameters $a,b,c,d$ in this formula has nothing to do with the $a,b,c,d$ which appear in the rest of this article. This formula plays a crutial role in proving the orthogonaliry relation of the Askey-Wilson polynomials. The proof can be found in [@AW85]. Now we get $$\begin{aligned} \label{I1a} I_1 &= I_1^{(0)} + I_1^{(a)}, \notag\\ I_1^{(a)} &= -\frac{(ab;q)_{\infty}}{a} \sum_{k=0}^{n^{(a)}} \frac{(-)^k q^{k(k-1)/2} (a^2 q^{2k},a^{-2}q^{-2k};q)_{\infty}} {(q;q)_k (a^2 q^k,ab q^{k+1}, a^{-1}b q^{-k};q)_{\infty}} [\lambda_k^{(a)}]^{L-1} . \end{aligned}$$ When $b$ also becomes larger than one, there appear the terms which come from poles at $z_2=bq^k$ and $z_2=(bq^k)^{-1}$ in (\[I1def\]). When (\[na\]) and $$\label{nb} b>bq>bq^2>\cdots >bq^{n^{(b)}} >1> bq^{n^{(b)}+1}>\cdots$$ hold, we have $$\begin{aligned} \label{I1ab} I_1 &= I_1^{(0)} + I_1^{(a)} + I_1^{(b,0)} + I_1^{(b,b)} + I_1^{(b,a)}, \\ I_1^{(b,0)} &= \frac{(ab;q)_{\infty}(q;q)_{\infty}^2}{2 b} \sum_{k=0}^{n^{(b)}} \frac{(-)^k q^{k(k-1)/2} (b^2q^{2k},b^{-2}q^{-2k};q)_{\infty} } {(q;q)_{k}(b^2q^k;q)_{\infty}} \notag\\ &\quad\times \int_{C_0}\frac{\d z_1}{2\pi i z_1} \frac{(z_1^2,z_1^{-2};q)_{\infty} [(1+z_1)(1+z_1^{-1})]^{L-1}} {( az_1,az_1^{-1}, bq^{k+1}z_1,bq^{k+1}z_1^{-1}, b^{-1}q^{-k}z_1,b^{-1}q^{-k}z_1^{-1};q)_{\infty}} , \\ I_1^{(b,b)} &= -\sum_{k=0}^{n^{(b)}} \sum_{m=k+1}^{n^{(b)}} \frac{(-)^k q^{k(k-1)/2+(m-k)(m-k-1)/2} [\lambda_m^{(b)}]^{L-1} } {b (q;q)_k (q;q)_{m-k-1}} \notag\\ &\quad\times \frac{(q,ab,b^2q^{2k},b^{-2}q^{-2k},b^2q^{2m},b^{-2}q^{-2m};q)_{\infty}} {(q^{m-k},b^2 q^k,b^2q^{m+k+1},b^{-2}q^{-k-m}, abq^m,ab^{-1}q^{-m};q)_{\infty} } , \\ I_1^{(b,a)} &= \sum_{k=0}^{n^{(b)}} \sum_{m:bq^k > aq^m} \frac{(-)^{k+m} q^{k(k+1)/2+m(m+1)/2} [\lambda_m^{(a)}]^{L-1} } {b (q;q)_k (q;q)_{m}} \notag\\ &\quad\times \frac{(q,ab,b^2q^{2k},b^{-2}q^{-2k},a^2q^{2m},a^{-2}q^{-2m};q)_{\infty}} {(b^2 q^k,a^2 q^m,abq^{k+m+1},a^{-1}bq^{k+1-m}, ab^{-1}q^{m-k},a^{-1}b^{-1}q^{-k-m};q)_{\infty}} .\end{aligned}$$ Lastly, when $a<1$ and (\[nb\]) holds, we have $$\label{I1b} I_1 = I_1^{(0)} + I_1^{(b,0)} + I_1^{(b,b)}.$$ Now we turn to the calculation of the asymptotic expression of the integral $I_1$. - [The case $a<1$ and $b<1$]{} In this case, $I_1=I_1^{(0)}$. We evaluate the asymptotic behavior of $I_1^{(0)}$ by employing the steepest decent method. First we change the variable from $\theta,\varphi$ to $u,y$ as $$\begin{gathered} \label{change1} 1+\cos\theta = 2 e^{-u/L}, \\ \label{change2} 1+\cos\varphi = 2 y e^{-u/L},\end{gathered}$$ to obtain $$\begin{aligned} \label{I1-1} I_1 = -\frac{4^{L+2}}{L^{\frac23}} \int_{0}^{\infty} \d u u^{\frac12} e^{-u} \sqrt{ \frac{1-e^{-u/L}} {u/L } } e^{-u/L} \int_{0}^{e^{u/L}} \d y \frac{ y^{\frac12}\sqrt{1-y e^{-u/L}} } {1-y} \notag\\ \times \frac{ (e^{2i\theta},e^{-2i\theta},e^{2i\varphi},e^{-2i\varphi};q)_{\infty} } {(ae^{i\theta}, ae^{-i\theta}, qe^{i(\theta+\varphi)}, qe^{-i(\theta+\varphi)}, qe^{i(\theta-\varphi)}, qe^{-i(\theta-\varphi)}, be^{i\varphi}, be^{-i\varphi};q)_{\infty} }.\end{aligned}$$ Here $\theta$ and $\varphi$ are considerd as functions in $u$ and $y$ through (\[change1\]) and (\[change2\]) respectively. Now we can take the limit $L\rightarrow\infty$ in the integrand. Changing the variable $y$ back to $\varphi$ by $$1+\cos\varphi = 2 y,$$ we have $$I_1 \simeq -\frac{\sqrt{\pi} (q;q)_{\infty}^2 4^{L+1} } {2 (a;q)_{\infty}^2 L^{\frac32} } \int_{0}^{\pi} \d \varphi \frac{(e^{2i\varphi},e^{-2i\varphi};q)_{\infty}} {(qe^{i\varphi},qe^{i\varphi},qe^{-i\varphi},qe^{-i\varphi}, be^{i\varphi},be^{i\varphi};q)_{\infty}} .$$ Using the formula (\[int-aw\]), we get $$I_1 \simeq -\frac{\pi^{\frac32} (1-b) 4^{L+1} } {(a,b;q)_{\infty}^2 L^{\frac32}}.$$ - [The case $a<1$ and $b>1$]{} The integral $I_1$ is given by (\[I1b\]). The main contributions come from $I_1^{(0)}$ and $I_1^{(b,0)}$. Each contribution behaves as $4^L$. whilest the normalization $Z_L$ behaves as $[(1+b)(1+b^{-1})]^L$ for this case. Since $(1+b)(1+b^{-1})$ is larger than $4$, $I_1$ is negligible compared to $Z_L$. So we do not compute the explicit expression. - [The case $a>1$]{} The main contribution comes from the $k=0$ term in the summation of $I_1^{(a)}$. We have $$I_1 \simeq -\frac{a(1-ab)(a^{-2};q)_{\infty}[(1+a)(1+a^{-1})]^{L-1}} {(a^{-1}b;q)_{\infty}}.$$ Evaluation of Integral $I_2$ ============================ In this Appendix, the integral $I_2$, $$\begin{aligned} I_2 &= \frac14 (ab;q)_{\infty}(q;q)_{\infty}^3 \int\frac{\d z_1}{2\pi i z_1} \int\frac{\d z_2}{2\pi i z_2} \notag\\ &\quad \frac{(z_1^2,z_1^{-2},z_2^2,z_2^{-2};q)_{\infty} [(1+z_1)(1+z_1^{-1})]^{j-1} [(1+z_2)(1+z_2^{-1})]^{L-j} } {(az_1,a z_1^{-1},qz_1 z_2, a z_1^{-1} z_2^{-1}, q z_1 z_2^{-1},q z_1^{-1} z_2^{-1}, bz_2,b z_2^{-1};q)_{\infty} (z_2 + z_2^{-1} -z_1-z_1^{-1})}, \label{I2def}\end{aligned}$$ is evaluated. First, for the case where $a,b<1$, both of the contours of $z_1$ and $z_2$ are unit circles. We have $$\begin{gathered} I_2 = I_2^{(0)} = \int_0^{\pi} \frac{\d \theta}{2\pi} \int_0^{\pi} \frac{\d \varphi}{2\pi} \frac{ (e^{2i\theta},e^{-2i\theta},e^{2i\varphi},e^{-2i\varphi};q)_{\infty} } {(ae^{i\theta}, ae^{-i\theta}, qe^{i(\theta+\varphi)}, qe^{-i(\theta+\varphi)}, qe^{i(\theta-\varphi)}, qe^{-i(\theta-\varphi)}, be^{i\varphi}, be^{-i\varphi};q)_{\infty} } \notag\\ \times \frac{ [2(1+\cos\theta)]^{j-1} [2(1+\cos\varphi)]^{L-j} } { 2\cos\varphi - 2 \cos\theta } . \label{I20}\end{gathered}$$ Since we will calculate the average density profile near the right boundary, we set $l=L-j+1$. Similarly to the case of the integal $I_1$, when $a$ or $b$ or both of them become larger than one, there apper other contributions besides $I_2^{(0)}$. When (\[na\]) and (\[nb\]) hold, the result is $$\begin{aligned} I_2 &= I_2^{(0)} + I_2^{(a,0)} + I_2^{(a,a)} + I_2^{(a,b)} + I_2^{(b,0)} + I_2^{(b,b)} + I_2^{(b,a)}, \label{I2all} \\ I_2^{(a,0)} &= -\frac{(ab;q)_{\infty}(q;q)_{\infty}^2}{2 a} \sum_{k=0}^{n^(a)} \frac{(-)^k q^{k(k-1)/2} (a^2q^{2k},a^{-2}q^{-2k};q)_{\infty} [\lambda_k^{(a)}]^{L-l} } {(q;q)_{k}(a^2q^k;q)_{\infty}} , \notag\\ &\quad\times \int_{C_0}\frac{\d z_2}{2\pi i z_2} \frac{(z_2^2,z_2^{-2};q)_{\infty} [(1+z_2)(1+z_2^{-1})]^{l-1}} {(aq^{k+1}z_2,aq^{k+1}z_2^{-1}, a^{-1}q^{-k}z_2,a^{-1}q^{-k}z_2^{-1}, bz_2,bz_2^{-1};q)_{\infty}} , \\ I_2^{(a,a)} &= \sum_{k=0}^{n^{(a)}} \sum_{m=k+1}^{n^{(a)}} \frac{(-)^k q^{k(k-1)/2+(m-k)(m-k-1)/2} [\lambda_k^{(a)}]^{L-l} [\lambda_m^{(a)}]^{l-l} } {a (q;q)_k (q;q)_{m-k-1}} \notag\\ &\quad\times \frac{(q,ab,a^2q^{2k},a^{-2}q^{-2k},a^2q^{2m},a^{-2}q^{-2m};q)_{\infty}} {(q^{m-k},a^2 q^k,a^2q^{m+k+1},a^{-2}q^{-k-m}, abq^m,ba^{-1}q^{-m};q)_{\infty} } , \\ I_2^{(a,b)} &= -\sum_{k=0}^{n^{(a)}} \sum_{m:aq^k > bq^m} \frac{(-)^{k+m} q^{k(k+1)/2+m(m+1)/2} [\lambda_k^{(a)}]^{L-l} [\lambda_m^{(b)}]^{l-l} } {a (q;q)_k (q;q)_{m}} \notag\\ &\quad\times \frac{(q,ab,a^2q^{2k},a^{-2}q^{-2k},b^2q^{2m},b^{-2}q^{-2m};q)_{\infty}} {(a^2 q^k,b^2 q^m,abq^{k+m+1},ab^{-1}q^{k+1-m}, a^{-1}bq^{m-k},a^{-1}b^{-1}q^{-k-m};q)_{\infty}} , \\ I_2^{(b,0)} &= \frac{(ab;q)_{\infty}(q;q)_{\infty}^2}{2 b} \sum_{k=0}^{n^{(b)}} \frac{(-)^k q^{k(k-1)/2} (b^2q^{2k},b^{-2}q^{-2k};q)_{\infty} [\lambda_k^{(b)}]^{l-1} } {(q;q)_{k}(b^2q^k;q)_{\infty}} \notag\\ &\quad\times \int_{C_0}\frac{\d z_1}{2\pi i z_1} \frac{(z_1^2,z_1^{-2};q)_{\infty} [(1+z_1)(1+z_1^{-1})]^{L-l}} {( az_1,az_1^{-1}, bq^{k+1}z_1,bq^{k+1}z_1^{-1}, b^{-1}q^{-k}z_1,b^{-1}q^{-k}z_1^{-1};q)_{\infty}} , \\ I_2^{(b,b)} &= -\sum_{k=0}^{n^{(b)}} \sum_{m=k+1}^{n^{(b)}} \frac{(-)^k q^{k(k-1)/2+(m-k)(m-k-1)/2} [\lambda_k^{(b)}]^{l-1} [\lambda_m^{(b)}]^{L-l} } {b (q;q)_k (q;q)_{m-k-1}} \notag\\ &\quad\times \frac{(q,ab,b^2q^{2k},b^{-2}q^{-2k},b^2q^{2m},b^{-2}q^{-2m};q)_{\infty}} {(q^{m-k},b^2 q^k,b^2q^{m+k+1},b^{-2}q^{-k-m}, abq^m,ab^{-1}q^{-m};q)_{\infty} } , \\ I_2^{(b,a)} &= \sum_{k=0}^{n^{(b)}} \sum_{m:bq^k > aq^m} \frac{(-)^{k+m} q^{k(k+1)/2+m(m+1)/2} [\lambda_k^{(b)}]^{l-1} [\lambda_m^{(a)}]^{L-l} } {b (q;q)_k (q;q)_{m}} \notag\\ &\quad\times \frac{(q,ab,b^2q^{2k},b^{-2}q^{-2k},a^2q^{2m},a^{-2}q^{-2m};q)_{\infty}} {(b^2 q^k,a^2 q^m,abq^{k+m+1},a^{-1}bq^{k+1-m}, ab^{-1}q^{m-k},a^{-1}b^{-1}q^{-k-m};q)_{\infty}} .\end{aligned}$$ Now we consider the asymptotic expression of the integral $I_2$. We take the limit $L\rightarrow\infty$ at first and then take the limit $l\rightarrow\infty$. - [The case $a,b<1$]{} In this case, $I_2=I_2^{(0)}$. The evaluation for this case proceeds analogously to the evaluation of $I_1^{(0)}$. Changing the variables $\theta,\varphi$ to $u,y$ as in (\[change1\]) and (\[change2\]), we get $$\begin{aligned} \label{I2-1} I_2 = -\frac{4^{L+2}}{L^{\frac23}} \int_{0}^{\infty} \d u u^{\frac12} e^{-u} \sqrt{ \frac{1-e^{-u/L}} {u/L } } e^{-u/L} \int_{0}^{e^{u/L}} \d y \frac{ y^{l-\frac12}\sqrt{1-y e^{-u/L}} } {1-y} \notag\\ \times \frac{ (e^{2i\theta},e^{-2i\theta},e^{2i\varphi},e^{-2i\varphi};q)_{\infty} } {(ae^{i\theta}, ae^{-i\theta}, qe^{i(\theta+\varphi)}, qe^{-i(\theta+\varphi)}, qe^{i(\theta-\varphi)}, qe^{-i(\theta-\varphi)}, be^{i\varphi}, be^{-i\varphi};q)_{\infty} }.\end{aligned}$$ We take the limit $L\rightarrow\infty$ in this expression and change the varialbe $y$ back to $\varphi$ to get $$I_2 \simeq -\frac{\sqrt{\pi} (q;q)_{\infty}^2 4^{L+2} } {2 (a;q)_{\infty}^2 L^{\frac23}} \int_{0}^{\pi} \d \phi \left[ \frac12 (1+\cos\varphi) \right]^{l} \frac{(qe^{2i\varphi},qe^{-2i\varphi};q)_{\infty}} {(qe^{i\varphi},qe^{i\varphi},qe^{-i\varphi},qe^{-i\varphi}, be^{i\varphi},be^{i\varphi};q)_{\infty}}.$$ We can consider the limit $l\rightarrow\infty$ by using the steepest descent method. We have $$I_2 \simeq -\frac{ 2 \cdot 4^{L+1}\pi } {(a,b;q)_{\infty}^2 L^{\frac23} l^{\frac12}}.$$ - [The case $1<a<q^{-1}$ and $b<1$]{} In this case, the integral $I_2$ is given by $I_2 = I_2^{(0)}+I_2^{(a,0)}$. The main contribution comes from the $k=0$ term in $I_2^{(a,0)}$. $$\begin{aligned} \label{I2a0} I_2 &\simeq -(ab,a^{-2};q)_{\infty} (q;q)_{\infty}^2 [(1+a)(1+a^{-1})]^{L-l} \notag\\ &\quad\times \int_{C_0} \frac{\d z_2}{2\pi i z_2} \frac{(z_2^2,z_2^{-2};q)_{\infty} [(1+z_2)(1+z_2^{-1})]^{l-1}} {(a^{-1}z_2,a^{-1}z_2^{-1},aqz_2, aqz_2^{-1},bz_2,bz_2^{-1};q)_{\infty}}.\end{aligned}$$ Taking the limit $l\rightarrow\infty$ by using the steeptest descent method, we get $$I_2 \simeq -\frac{4^l (ab,a^{-2};q)_{\infty}(q;q)_{\infty}^2 [(1+a)(1+a^{-1})]^{L-l} } {a \sqrt{\pi} l^{\frac32} (a^{-1},qa,b;)_{\infty}^2}.$$ - [The case $a>q^{-1}$ and $aq>b$]{} In general, the integral for this case has the expression (\[I2all\]). The main contribution comes from the $k=0,m=1$ term in $I_2^{(a,a)}$. We have $$\label{I2aa} I_2 \simeq -\frac{(1-ab)(1-a^{-2}q^{-2})(a^{-2};q)_{\infty} [(1+a)(1+a^{-1})]^{L-l} [(1+aq)(1+a^{-1}q^{-1})]^{l-1}} {(a^{-1}bq^{-1};q)_{\infty}}.$$ - [The case $a>b>aq$ and $b>1$]{} The main contribution for this case comes from the $k=0,m=1$ term in $I_2^{(a,b)}$. We have $$\label{I2ab} I_2 \simeq -\frac{(1-ab)(q,a^{-2},b^{-2};q)_{\infty} [(1+a)(1+a^{-1})]^{L-l} [(1+b)(1+b^{-1})]^{l-1} } {a (a^{-1}b,a^{-1}b^{-1},ab^{-1}q;q)_{\infty}} .$$ - [The case $b>1$ and $b>a$]{} In this case, the normalization $Z_L$ behaves as $[(1+b)(1+b^{-1})]^L$. All contributions to $I_2$ can be neglected compared to $Z_L$. Hence we do not compute the asymptotic expression explicitly for this case. Figure Captions Fig. 1 : One-dimensional partially asymmetric simple exclusion process with open boundaries. Particles have hard-core exclusion interaction and tend to hop to the right (resp. left) nearest neiboring site with rate $p_R$ (resp. $p_L$). There are also particle injection (resp. ejection) at the left (resp. right) edge. Fig. 2 : Space-time diagram of the ASEP from Monte-Carlo simulations. The holizontal axis represents the site number $j$ whereas the vertical axis represents time. The existence of pariticle is represented as a black point. The lattice length is taken to be $L=200$. The bulk hopping rates are taken to be $p_R=1,\,p_L=0$. After some transient time, the system practically goes to a steady state. The steady state depends crutially on the values of the boundary parameters. Fig. 3 : Average density profile of the ASEP from Monte-Carlo simulations. The holizontal axis represents the site number $j$ whereas the vertical axis represents the average density. The lattice length is taken to be $L=200$. The bulk hopping rates are taken to be $p_R=1,\,p_L=0$ Fig. 4 : The phase diagram of the current. Regions $A,B$ and $C$ are called the low-density phase, the high-density phase and the maximal current phase respectively. Fig. 5 : The phase diagram of the correlation length in the $\tilde{\alpha}$-$\tilde{\beta}$ plane for the partially asymmetric case. The low-density phase (resp. high-density phase) is divided into three phases, $A_1,A_2$ and $A_3$ (resp. $B_1,B_2$ and $B_3$). Fig. 6 : The correlation length $\xi$ for the case $p_R=1,p_L=0.9,\tilde{\beta}=1$. The solid line is the theoretical prediction given by $\xi=1/\ln\frac{q}{[\tilde{\beta}+(1-\tilde{\beta})q]^2}$ whereas the black dots are the simulation data. [99]{} T. Sasamoto, J. Phys. A [**32**]{} (1999) 7109. T. M. Ligget, [*Interacting Particle Systems*]{} (Springer-Verlag, New York, 1985). H. Spohn, [*Large Scale Dynamics of Interacting Particles*]{} (Springer-Verlag, New York, 1991). B. Derrida, Phys. Rep. [**301**]{} (1998) 65. B. Schmittmann and R. K. P. Zia, Statistical mechanics of driven diffusive systems, in [*Phase Transitions and Critical Phenomena,*]{} Vol 17, C. Domb and J. Lebowitz eds. (Academic, London, 1994). B. Derrida, M. R. Evans, V. Hakim and V. Pasquier, J. Phys. A [**26**]{} (1993) 1493. G. Schütz and E. Domany, J. Stat. Phys. [**72**]{} (1993) 277. A. B. Kolomeisky, G. M. Schütz, E. B. Kolomeisky, J. P. Straley, J. Phys. A [**31**]{} (1998) 6911. F. H. L. Essler and V. Rittenberg, J. Phys. A [**29**]{} (1996) 3375. S. Sandow, Phys. Rev. E [**50**]{} (1994) 2660. S. Sandow and G. M. Schütz, Europhys. Lett. [**26**]{} (1994) 7 R. A. Blythe, M. R. Evans, F. Colaiori and F. H. L. Essler, cond-mat/9910242 T. Sasamoto, S. Mori and M. Wadati, J. Phys. Soc. Jpn, [**65**]{} (1996) 2000. G. E. Andrews, R. Askey and R. Roy, [*Special Functions*]{} (Cambridge, Cambridge, 1999). G. Gasper and M. Rahman, [*Basic Hypergeometric Series*]{} (Cambridge, Cambridge, 1990). K. Mallick, J. Phys. A [**29**]{} (1996) 5375. F. H. Jafarpour, cond-mat/9908327. M. R. Evans, D. P. Foster, C. Godèche, and D. Mukamel, Phys. Rev. Lett. [**74**]{} (1995) 208; J. Stat. Phys. [**230**]{} (1995) 69. M. R. Evans, Y. Kafri, H. M. Koduvely, and D. Mukamel, Phys. Rev. Lett. [**230**]{} (1998) 425. P. F. Arndt, T. Heinzel and V. Rittenberg, J. Phys. A [**31**]{} (1998) 833; J. Stat. Phys. [**90**]{} (1998) 783; J. Phys. A [**31**]{} (1998) L45; cond-mat/9809123. F. C. Alcaraz, S. Dasmahaptra and V. Rittenberg, J. Phys. A [**31**]{} (1998) 845. K. Mallick, S. Mallick and N. Rajewsky, cond-mat/9903248. N. Rajewsky, T. Sasamoto and E. R. Speer, in preparation. Askey R. A. and Wilson J. A., [*Memoirs Amer. Math. Soc.*]{} [**319**]{} (1985). (500,500) (20,305)[\[0.35\][]{}]{} (50,295)[(1,0)[50]{}]{} (70,280)[$j$]{} (15,380)[(0,1)[70]{}]{} (-3,410) (20,265)[($A$) low density phase]{} (32,253)[$p_R=1.0, \,p_L=0.0$]{} (37,240)[$\alpha=0.2,\,\beta=1.0$]{} (220,308)[\[0.35\][]{}]{} (250,295)[(1,0)[50]{}]{} (270,280)[$j$]{} (215,380)[(0,1)[70]{}]{} (197,410) (200,265)[($C$) maximal current phase]{} (232,253)[$p_R=1.0, \,p_L=0.0$]{} (237,240)[$\alpha=1.0,\,\beta=1.0$]{} (20,5)[\[0.35\][]{}]{} (50,-5)[(1,0)[50]{}]{} (70,-20)[$j$]{} (15,80)[(0,1)[70]{}]{} (-3,110) (40,-35)[coexistence line]{} (32,-47)[$p_R=1.0, \,p_L=0.0$]{} (37,-60)[$\alpha=0.2,\,\beta=0.2$]{} (220,0)[\[0.35\][]{}]{} (250,-5)[(1,0)[50]{}]{} (270,-20)[$j$]{} (215,80)[(0,1)[70]{}]{} (197,110) (220,-35)[($B$) high density phase]{} (232,-47)[$p_R=1.0, \,p_L=0.0$]{} (237,-60)[$\alpha=1.0,\,\beta=0.2$]{} (500,300) (0,190)[\[0.6\][]{}]{} (70,182)[(1,0)[50]{}]{} (90,170)[$j$]{} (-3,212)[(0,1)[70]{}]{} (-20,240) (30,155)[($A$) low density phase]{} (42,140)[$p_R=1.0, \,p_L=0.0$]{} (47,125)[$\alpha=0.2,\,\beta=0.25$]{} (200,190)[\[0.6\][]{}]{} (270,182)[(1,0)[50]{}]{} (290,170)[$j$]{} (220,155)[($C$) maximal current phase]{} (242,140)[$p_R=1.0, \,p_L=0.0$]{} (247,125)[$\alpha=1.0,\,\beta=1.0$]{} (0,0)[\[0.6\][]{}]{} (70,-5)[(1,0)[50]{}]{} (90,-20)[$j$]{} (50,-35)[coexistence line]{} (42,-50)[$p_R=1.0, \,p_L=0.0$]{} (47,-65)[$\alpha=0.2,\,\beta=0.2$]{} (200,0)[\[0.6\][]{}]{} (270,-5)[(1,0)[50]{}]{} (290,-20)[$j$]{} (230,-35)[($B$) high density phase]{} (242,-50)[$p_R=1.0, \,p_L=0.0$]{} (247,-65)[$\alpha=0.25,\,\beta=0.2$]{} (500,300) (0,100)[\[1.0\][]{}]{} (-20,270) (280,85)

--- abstract: 'We have explored a simple microscopic model to simulate a thermally activated rate process where the associated bath which comprises a set of relaxing modes is not in an equilibrium state. The model captures some of the essential features of non-Markovian Langevin dynamics with a fluctuating barrier. Making use of the Fokker-Planck description we calculate the barrier dynamics in the steady state and non-stationary regimes. The Kramers-Grote-Hynes reactive frequency has been computed in closed form in the steady state to illustrate the strong dependence of the dynamic coupling of the system with the relaxing modes. The influence of nonequilibrium excitation of the bath modes and its relaxation on the kinetics of activation of the system mode is demonstrated. We derive the dressed time-dependent Kramers rate in the nonstationary regime in closed analytical form which exhibits strong non-exponential relaxation kinetics of the reaction co-ordinate. The feature can be identified as a typical non-Markovian dynamical effect.' --- =0.0cm =-0.0cm =-1.0cm =21.0cm =15.5cm =0.2cm =0.5cm [**[Jyotipratim Ray Chaudhuri$^{\rm a}$, Gautam Gangopadhyay$^{\rm b}$,\ Deb Shankar Ray$^{\rm a}$]{}**]{} $^{\rm a}$[**[Indian Association for the Cultivation of Science]{}**]{}\ [**[Jadavpur, Calcutta 700 032, INDIA.]{}**]{} $^{\rm b}$[**[S. N. Bose National Centre for Basic Sciences]{}**]{}\ [**[JD Block, Sector III, Salt Lake City, Calcutta 700 091, INDIA.]{}**]{} **[I.Introduction]{}** More than half a century ago Kramers$^{1}$ considered the problem of activated rate processes by using a model Brownian particle trapped in a one dimensional well which is separated by a barrier of finite height from a deeper well. The particle was supposed to be immersed in a medium such that the medium exerts a frictional force on the particle but at the same time thermally activate it so that the particle may gain enough energy to cross the barrier. Over several decades the model has been the standard paradigm in many areas of physics and chemistry$^{2}$. The Kramers problem was to find the rate of escape from the well to the barrier. The motion of the particle is governed by the following phenomenological Langevin equation, $$\ddot{x}=-\frac{1}{m}\frac{\partial V(x)}{\partial x} - \gamma\dot{x} + \frac{1}{m} F(t) \hspace{0.2cm},$$ where $x$ is the coordinate of the particle of mass $m$ moving in a potential $V(x)$. $\gamma$ and $F(t)$ are the damping rate and the Gaussian stationary random force provided by the thermal bath respectively. The properties of noise can be summarized by the following two relations, $$\langle F(t)\rangle=0 \hspace{0.4cm}, \hspace{0.4cm} \langle F(0)F(t)\rangle=2 \gamma mKT \delta(t) \hspace{0.2cm}.$$ The Langevin equation (1) is equivalent to the Fokker-Planck equation for probability distribution $p=p(x,v,t)$ \[also known as Kramers equation\], $$\frac{\partial p}{\partial t}=\frac{1}{m}\frac{\partial V(x)}{\partial x} \frac{\partial p}{\partial v}-v\frac{\partial p}{\partial x} + \gamma \left[\frac{KT}{m} \frac{\partial^{2} p}{\partial v^{2}} +\frac{\partial} {\partial v}(vp) \right] \hspace{0.2cm}.$$ Kramers$^{1}$ obtained the steady state escape rate $k$ in the limiting cases of high and low damping rates in the following form, $$k=\left\{\begin{array}{lllll} \frac{\omega_{0}\omega_{b}}{2\pi\gamma}\exp[-\frac{E_{b}}{KT}] & & & \gamma\longrightarrow\infty \\ \gamma\frac{E_{b}}{KT}\exp[-\frac{E_{b}}{KT}] & & & \gamma\longrightarrow 0 \end{array}\right. \hspace{0.2cm},$$ where $\omega_{o}$ and $\omega_{b}$ are the frequencies associated with the curvature of the potential at the bottom of the well and at the barrier top, respectively. $E_{b}$ refers to the depth of the well. Kramers has also derived an expression for ‘intermediate’ value of $\gamma$ : $$\begin{aligned} k=\frac{\omega_{0}}{2\pi\omega_{b}}\left\{\left[ \left(\frac{\gamma}{2} \right)^{2}+\omega_{b}^{2}\right]^{\frac{1}{2}}-\frac{\gamma}{2}\right\} \exp(-E_{b}/KT)\hspace{0.2cm}.\end{aligned}$$ For non-Markovian random processes where one takes into account of the short internal time scales of the system compared to that of the thermal bath, the Langevin equation(1) gets replaced by its non-Markovian counterpart$^{3,4}$, sometimes called the generalized Langevin equation (GLE); $$\ddot{x}=-\frac{1}{m}\frac{\partial V(x)}{\partial x}-\int_{0}^{t}d\tau Z(t-\tau) \dot{x}(\tau) + \frac{1}{m}R(t) \hspace{0.2cm},$$ where $R(t)$ is Gaussian but non-Markovian such that $$\langle R(t) \rangle = 0 ,\hspace{1.0cm}\langle R(0)R(t) \rangle = Z(t)mKT \hspace{0.2cm}.$$ The memory function $Z(t)$ is expressed in terms of Fourier-Laplace components $$Z_{n}(\omega) = \int_{o}^{\infty} dt Z(t) e^{-in\omega t}$$ with $Z_{0}(\omega) = \gamma$ Based on equation (5) Adelman$^{5}$ obtained the generalized Fokker-Planck equation for a Brownian oscillator with a parabolic potential as given by ; $$\frac{\partial p}{\partial t} = -{\bar{\omega}}_{b}^{2} x \frac{\partial p} {\partial v} -v\frac{\partial p}{\partial x} + {\bar{\gamma}}\frac{\partial}{\partial v} (vp)+ {\bar{\gamma}} \frac{KT}{m}\frac{\partial^{2}p}{\partial v^{2}} + \frac{KT}{m}\left(\frac {{\bar{\omega}}_{b}^{2}}{\omega_{b}^{2}}-1\right)\frac{\partial^{2}p} {\partial v\partial x} \hspace{0.2cm},$$ where ${\bar{\gamma}}$ = ${\bar{\gamma}}(t)$ and ${{\bar{\omega}}_{b}^{2}} ={{\bar{\omega}}_{b}^{2}}(t)$ are now functions of time \[although bounded , they may not always provide long time limits\] which play a decisive role in the calculation of non-Markovian Kramers rate. Various workers have made use of generalized Langevin equation to treat the different aspects of the escape problem in the non-Markovian regime. For example, Grote and Hynes$^{4}$ considered the average motion of the particle in the vicinity of the barrier governed by GLE and found that on the average the particle is slowed down by friction and defining a reactive frequency $\lambda_{r}$ they showed that the average motion goes as $\exp(\pm\lambda_{r} t)$. The analysis of Hänggi and Mojtabai$^{6}$ on the other hand is based on the generalized Fokker-Planck equation of Adelman with a parabolic potential in the high friction limit. The generalized FP approach has also been adopted by Carmeli and Nitzan$^{7}$ to derive the expression for the steady-state escape rate in the high and low friction limit in the Markovian as well as non-Markovian regimes. A comprehensive overview has been given in Ref.(2). While the early post-Kramers development as summarized above is largely phenomenological, an interesting advancement in the theory of activated rate processes was made when the generalized Langevin equation was realized in terms of a microscopic model which comprises a system coupled linearly to a discrete set of harmonic oscillators. Using the properties of the bath and a normal mode analysis it was shown$^{8}$ that the reactive frequency $\lambda_{r}$ defined by Grote and Hynes$^{4}$ for the average motion across the barrier is actually a renormalised effective barrier frequency. The object of the present paper is twofold : First is to consider a simple variant of the system-heat bath model$^{9,10,11}$ to simulate the activated rate processes, where the associated bath is in a nonequilibrium state. The model incorporates some of the essential features of Langevin dynamics with a fluctuating barrier which had been heuristically and phenomenologically proposed earlier in several occasions.$^{10,13-17}$ While the majority of the treatments of the phenomenological fluctuating barrier rest on the reduction of the equations to overdamped limit$^{5,10,14}$, thus restricting the validity of the solutions in the large time limit, we take full account of the inertial terms in our calculation of barrier dynamics and probability distribution function both in the long time and in the short time nonstationary regimes. The Fokker-Planck description allows us to calculate Kramers-Grote-Hynes reactive frequency pertaining to these situations for non-Markovian dynamics in closed form. Second, since the theories of activated processes traditionally deal with stationary bath, the nonstationary activated processes has remained largely overlooked so far. We specifically address this issue and examine the influence of initial excitation and subsequent relaxation of a bath modes on the activation of the reaction co-ordinate. We show that relaxation of the nonequilibrium bath modes may result in strong non-exponential kinetics and a nonstationary Kramers rate. The physical situation that has been addressed is the following : We consider that at $t=0_-$, the time just before the system and the bath is subjected to an external excitation, the system is appropriately thermalized. At $t=0$, the excitation is switched on and the bath is thrown into a nonstationary state which behaves as a nonequilibrium reservoir. We follow the stochastic dynamics of the system mode after $t>0$. The important separation of the time scales of the fluctuations of the nonequilibrium bath and the thermal bath (to which it relaxes) is that the former effectively remains stationary on the fast correlation of the thermal noise. The outline of the paper is as follows; Following Ref. \[10\] we discuss in Sec.II a microscopic model to simulate an activated rate process where the system in question is not initially thermalized. Appropriate elimination of reservoir degrees of freedom leads to a nonlinear non-Markovian Langevin equation which governs the dynamics of a particle with a fluctuating barrier, stochasticity being contributed by both (additive) thermal noise and a slower (multiplicative) noisy relaxing nonequilibrium modes. The Fokker-Planck description is provided in Sec.III. The standard Markovian description and the generalized FP equation of Adelman’s form can be recovered in the appropriate limits. In Sec.IV we derive the expression for Kramers rate of barrier crossing in the non-Markovian but steady state regime and show that the Kramers-Grote-Hynes “reactive frequency” can be explicitly realized in this model in closed form. Sec.V is devoted to nonstationary aspect. We solve the time-dependent FP equation for nonstationary probability density and calculate the corresponding current. An expression for Kramers rate in the nonstationary regime in closed analytical form is derived. The paper is concluded in Sec.VI. **[II. The model and the Langevin equation]{}** We consider a model consisting of a system mode coupled to a set of relaxing modes considered as a semi-infinite dimensional system ({$q_k$}-subsystem) which effectively constitutes a nonequilibrium bath. This, in turn, is in contact with a thermally equilibrated reservoir. Both the reservoirs are composed of two sets of harmonic oscillators characterized by the frequency sets $\{\omega_{k}\}$ and $\{\Omega_{j}\}$ for the nonequilibrium and the equilibrium bath, respectively. The system-reservoir combination evolves under the total Hamiltonian $$\begin{aligned} H=\frac{p^{2}}{2m}+V(x)+\frac{1}{2}\sum_{j}(P_{j}^{2}+\Omega_{j}^{2}Q_{j}^{2}) +\frac{1}{2}\sum_{k}(p_{k}^{2}+\omega_{k}^{2}q_{k}^{2})\nonumber\\ -x\sum_{j}K_{j}Q_{j}-g(x)\sum_{k}q_{k}-\sum_{j,k}\alpha_{jk}q_{k}Q_{j} \hspace{0.2cm},\end{aligned}$$ the first two terms on the right hand side describe the system mode. The Hamiltonian for the thermal and nonequilibrium baths are described by the sets $\{Q_{j},P_{j}\}$ and $\{q_{j},p_{j}\}$ for coordinates and momenta, respectively. The coupling terms containing $K_{j}$ refers to the usual system-thermal bath linear coupling. The last two terms indicate the coupling of the nonequilibrium bath to the system and the thermal bath modes, respectively. Since in the present problem, H is considered to be classical and temperature, T high for the thermally activated problem we note that quantum effects do not play any significant role. Hamiltonian (9) is a simpler variant of that treated in Ref.\[10\]. For simplicity we take $m=1$ in (9) and for rest of the treatment. As shown in Ref. \[10\] the model (9) captures the essential features of fluctuating barrier dynamics. We recall the relevant aspect in the following discussions. Eliminating the equilibrium reservoir variables $\{Q_{j},P_{j}\}$ in an appropriate way $^{9,10}$ one may show that the nonequilibrium bath modes obey the following equations of motion, $${\ddot{q}}_{k}+\gamma{\dot{q}}_{k}+\omega_{k}^{2}q_{k}=g(x)+\eta_{k}(t) \hspace{0.2cm}.$$ This takes into account of the average dissipation ($\gamma$) of the nonequilibrium reservoir modes $q_{k}$ due to its coupling to thermal reservoir which induces fluctuations $\eta_{k}(t)$ characterized by $\langle \eta_{k}(t)\rangle=0$ and the usual fluctuation-dissipation theorem $\langle \eta_{k}(t)\eta_{k}(0)\rangle=2\gamma KT\delta(t)$. We mention here that moving from Eq.(9) to (10) generate cross terms of the form $\sum_j \gamma_{kj} q_j$, which are neglected for $j \neq k$. Proceeding similarly to eliminate the thermal reservoir variables from the equations of motion of the system mode one obtains $${\ddot{x}}+\gamma_{\rm eq}{\dot{x}}+V'(x)=\xi_{\rm eq}(t)+g'(x)\sum_{k}q_{k} \hspace{0.2cm},$$ where $\gamma_{\rm eq}$ refers to the dissipation coefficient of the system mode due to its direct coupling to the thermal bath providing fluctuations $\xi_{\rm eq}(t)$. Here we have $$\begin{aligned} \langle \xi_{\rm eq}(t)\rangle=0 \hspace{0.2cm}{\rm and} \hspace{0.2cm} \langle \xi_{\rm eq}(t)\xi_{\rm eq}(0)\rangle=2\gamma_{\rm eq}KT\delta(t) \hspace{0.2cm}.\end{aligned}$$ Now making use of the formal solutions of Eq.(10)$^{10}$ which takes into account of the relaxation of the nonequilibrium modes and integrating over the nonequilibrium modes with a Debye type frequency distribution of the form, $$\begin{aligned} \rho(\omega)=\left\{ \begin{array}{lllll} 3\omega^{2}/2\omega_{c}^{3} & & , & {\rm for} |\omega| \le \omega_{c}\\ 0 & & , & {\rm for}|\omega| > \omega_{c} \end{array} \right.\end{aligned}$$ where $\omega_{c}$ is the high frequency Debye cut-off, one finally arrives at the following Langevin equation of motion for the system mode, $${\ddot{x}}+\Gamma(x){\dot{x}}+\tilde{V}'(x)=\xi_{\rm eq}(t)+g'(x)\xi_{\rm neq} (t)\hspace{0.2cm}.$$ Here $\Gamma(x)$ is a system coordinate dependent dissipation constant composed of $\gamma_{\rm eq}$ and $\gamma_{\rm neq}$ as follows, $$\Gamma(x)=\gamma_{\rm eq}+\gamma_{\rm neq}[g'(x)]^{2}\hspace{0.2cm}.$$ $\xi_{\rm neq}$ refers to the fluctuations of the nonequilibrium bath modes which effectively cause a damping of the system mode by an amount $\gamma_{\rm neq}[g'(x)]^{2}$. Eq.(12) also includes the modification of the potential $V(x)$ in which the particle moves as $${\tilde{V}}(x)=V(x)-\frac{\omega_{c}}{\pi} \; \gamma_{\rm neq} \; g^{2}(x) \hspace{0.2cm}.$$ Eq.(12) thus describes the effective dynamics of a particle in a modified barrier, where the metastability of the well originates from the dynamic coupling $g(x)$ of the system mode with the nonequilibrium bath modes. It is necessary to stress here that $g(x)$, in general, is nonlinear. This nonlinearity has two immediate consequences. First, by virtue of the term ${\tilde{V}}'(x)$ in Eq.(12) it gives rise to a fluctuating barrier. Second, the term $g'(x) \xi_{\rm neq} (t)$ imparts a multiplicative noise term in Eq.(12) in addition to the usual additive noise term $\xi_{\rm eq} (t)$. We point out here that the problem of diffusion over a fluctuating barrier$^{13-17}$ of similar nature has been addressed earlier by a number of workers from the phenomenological point of view. For example, Stein et.al.$^{14}$ have calculated the decay of probability from the metastable state in the white noise limit and also for short finite correlation times for the fluctuating part of the potential. Riemann and Elston$^{15}$ have calculated an asymptotic rate formula when the particle is subjected to both dichotomous and thermal noise. The treatment followed in the aforesaid cases concerns overdamped situation and, in general, the validity is restricted to long time limit. In the present problem, however, we look at the stochastic process right from the moment the nonequilibrium excitation (followed by the relaxation) sets in. We are therefore forced to take into consideration of the inertial term in Eq.(12) on its usual footing. We now turn to the another aspect of the problem. In order to define the problem described by Eq.(12) completely, it is further necessary to state the properties of fluctuations of the nonequilibrium bath $\xi_{\rm neq}(t)$. We have first for Gaussian noise $$\begin{aligned} \langle \xi_{\rm neq} (t) \rangle = 0 \; \; .\end{aligned}$$ Also the essential properties of $\xi_{\rm neq} (t)$ explicitly depend on the nonequilibrium state of the intermediate oscillator modes $\{ q_k \}$ through ${\cal U}(\omega, t)$, the energy density distribution function at time $t$ in terms of the following fluctuation-dissipation relation$^{10}$ for the nonequilibrium bath, $$\begin{aligned} {\cal U}(\omega , t) & = & \frac{1}{4\gamma_{\rm neq}} \int_{-\infty}^{+\infty} d\tau \; \langle \xi_{\rm neq} (t) \xi_{\rm neq} (t+\tau) \rangle \; e^{i\omega \tau} \nonumber \\ & = & \frac{1}{2} KT \; + \; e^{-\gamma t/2} \; \left [ {\cal U}(\omega,0) - \frac{1}{2} KT \right ] \; \; ,\end{aligned}$$ $\left [ {\cal U}(\omega,0) - \frac{1}{2} KT \right ] $ is a measure of departure of energy density from thermal average at $t=0$. The exponential term implies that this deviation due to the initial excitation decays asymptotically to zero as $t\rightarrow \infty$, so that one recovers the usual fluctuation-dissipation relation for the thermal bath. With the above specification of correlation function of $\xi_{\rm neq}$ Eq.(15) thus attributes the nonstationary character of the {$q_k$}-subsystem. In passing, we stress that the above derivation$^{10}$ is based on the assumption that $\xi_{\rm neq}$ is effectively stationary on the fast correlation of the thermal modes. This is a necessary requirement for the systematic separation of time scales involved in the dynamics. We point out that the effective dynamics sets no choice on any special form of coupling $g(x)$ between the system mode and the relaxing mode and as such this may be of arbitrary nonsingular type for our problem we have considered here. **[III.The generalized Fokker-Planck description]{}** Eq.(12) is the required Langevin equation for the particle moving in a modified potential ${\tilde{V}}(x)$ \[Eq.(14)\] and damped by a coordinate-dependent friction $ \Gamma (x) $ \[Eq.(13)\] due to its linear coupling to a thermal bath and nonlinear coupling to the $\{q_k\}$-subsystem characterized by fluctuations $\xi_{\rm neq} (t)$. Before proceeding further a few pertinent points are to be noted to stress some distinct and important aspects of the model. First, depending on the system-{$q_k$}-subsystem coupling $g(x)$ both the modified potential ${\tilde{V}}(x)$ as well as $\Gamma(x)$ are, in general, nonlinear. So the stochastic differential equation (12) is nonlinear. Again, the stochasticity in Eq.(12) is composed of two parts : $\xi_{\rm eq} (t)$ is an additive noise due to thermal bath while $\xi_{\rm neq} g'(x)$ is a multiplicative contribution due to nonlinear coupling to {$q_k$}-subsystem. It is thus important to note that the presence of multiplicative noise and a fluctuating barrier are associated with nonlinearity in $g(x)$. Second, the Langevin equation (12) is non-Markovian. The origin of this non-Markovian nature lies in the decaying term in Eq.(15) where the decay explicitly expresses the initial nonequilibrium nature of the $\{ q_k\}$-subsystem following the sudden excitation at $t=0$. This non-Markovian feature is thus not to be confused with that arises due to the usual frequency dependence of the dissipation constant. Third, although the modification of $V(x)$ is due to the specific choice of the Debye model for the mode density which has so far been commonly used, the theory remains effectively unchanged as one goes over to more complicated spectrum. We now rewrite Eq.(12) in the form, $$\left.\begin{array}{l} \dot{u}_{1}=F_{1}(u_{1},u_{2},t ; \xi_{\rm neq},\xi_{\rm eq}) \dot{u}_{2}=F_{2}(u_{1},u_{2},t ; \xi_{\rm neq},\xi_{\rm eq}) \end{array}\right\}\hspace{0.2cm},$$ where we use the following abbreviations, $$\left.\begin{array}{l} u_{1}=x\\ u_{2}=v \end{array}\right\}$$ and $$\left.\begin{array}{l} F_{1}=v\\ F_{2}=-\Gamma(x)v-\tilde{V}^{'}(x)+ \xi_{\rm eq}(t)+g'(x) \xi_{\rm neq}(t)\end{array}\right\} \hspace{0.2cm}.$$ The vector $u$ with components $u_{1}$ and $u_{2}$ thus represents a point in a 2-dimensional ‘phase space’ and the Eq.(16) determines the velocity at each point in this phase space. The conservation of points now asserts the following linear equation of motion for density $\rho(u,t)$ in ‘phase space’, $$\begin{aligned} \frac{\partial}{\partial t}\rho(u,t)=-\sum_{n=1}^{2}\frac{\partial}{\partial u_{n}} F_{n}(u,t;\xi_{\rm neq},\xi_{\rm eq})\rho(u,t)\hspace{0.2cm},\end{aligned}$$ or more compactly $$\frac{\partial \rho}{\partial t}=-\nabla\cdot F\rho\hspace{0.2cm}.$$ Our next task is to find out a differential equation whose average solution is given by $\langle \rho \rangle$ where the stochastic averaging has to be performed over two noise processes $\xi_{\rm neq}$ and $\xi_{\rm eq}$. To this end we note that $\nabla \cdot F$ can be partitioned into two parts ; a constant part $\nabla \cdot F_0$ and a fluctuating part $\nabla \cdot F_1 (t)$, containing these noises. Thus we write $$\nabla \cdot F(u,t;\xi_{\rm neq},\xi_{\rm eq}) = \nabla\cdot F_0(u) + \epsilon \nabla \cdot F_1 (u,t;\xi_{\rm neq},\xi_{\rm eq}) \; \; ,$$ where $\epsilon$ is a parameter (we put it as an external parameter to keep track of the order of the perturbation expansion in $\epsilon \tau_c$, where $\tau_c$ is the correlation time of fluctuation of $\xi_{\rm neq} (t)$ ; we put $\epsilon=1$ at the end of calculation) and also note that $\langle F_1(t) \rangle =0$. Eq.(19) therefore takes the following form , $$\dot{\rho} (u,t) = (A_0 \; + \; \epsilon A_1) \; \rho (u,t) \; \; ,$$ where $A_0=-\nabla \cdot F_0$, $ A_1=-\nabla \cdot F_1$. The symbol $\nabla$ is used for the operator that differentiate everything that comes after it with respect to $u$. Making use of one of the main results for the theory of linear equation of the form (21) with multiplicative noise, we derive an average equation for $\rho$ \[$\langle \rho \rangle = p(u,t)$, the probability density of $u(t)$ ; for details refer to Van Kampen$^{12}$\], $$\dot{p} =\left \{ A_0 + \epsilon^2 \int_0^\infty \langle A_1(t) \; \exp (\tau A_0) \; A_1 (t-\tau) \rangle \; \exp (-\tau A_0) \right \} \; p \; \; .$$ The above result is based on second order cumulant expansion and is valid in the case that fluctuations are small but rapid and the correlation time $\tau_c$ is short but finite, i.e., $$\begin{aligned} \langle A_1(t) \; A_1(t') \rangle =0 \; \; {\rm for} \; |t-t'| > \tau_c \; \; .\end{aligned}$$ The Eq.(22) is exact in the limit correlation time $\tau_c$ tends to zero. Using the expressions for $A_0$ and $A_1$ we obtain $$\begin{aligned} \frac{\partial p(u,t)}{\partial t} = \{ -\nabla\cdot F_0 \; + \; \epsilon^2 \int_0^\infty d\tau \; \langle \nabla\cdot F_1(t) \; \exp(-\tau \nabla\cdot F_0) \; \nabla\cdot F_1(t-\tau) \rangle \nonumber \\ \exp(\tau\nabla\cdot F_0) \} \; p(u,t) \; \; .\end{aligned}$$ The operator $\exp(-\tau\nabla\cdot F_0)$ in the above equation provides the solution of the equation $$\frac{\partial f(u,t)}{\partial t} = - \nabla \cdot F_0 \; f(u,t) \; \; ,$$ ($f$ signifies the unperturbed part of $\rho$) which can be found explicitly in terms of characteristics curves. The equation $$\dot{u} = F_0(u)$$ for fixed $t$ determines a mapping from $u(\tau=0)$ to $u(\tau)$, i.e., $u\rightarrow u^\tau$ with inverse $(u^\tau)^{-\tau}=u$. The solution of Eq.(24) is $$f(u,t) = f(u^{-t},0) \left | \frac{d (u^{-t})}{d(u)} \right | = \exp(-t \nabla\cdot F_0) f(u,0) \; \; ,$$ $\left | \frac{d (u^{-t})}{d(u)} \right |$ being a Jacobian determinant. The effect of $\exp(-t\nabla\cdot F_0)$ on $f(u)$ is as follows ; $$\exp(-t\nabla\cdot F_0) \; f(u,0) = f(u^{-t},0) \left | \frac{d (u^{-t})}{d(u)} \right | \; \; .$$ The above simplification when put in Eq.(23) yields $$\begin{aligned} \frac{\partial}{\partial t}p(u,t)=\nabla\cdot \left\{-F_{0}+\epsilon^{2} \int_{0}^{\infty}\left|\frac{d(u^{-\tau})}{d(u)}\right| \langle F_{1}(u,t)\nabla_{-\tau}\cdot F_{1}(u^{-\tau},t-\tau) \rangle\right.\nonumber\\ \left.\left|\frac{d(u)}{d(u^{-\tau})}\right| d\tau\right\} p(u,t)\hspace{0.2cm}.\end{aligned}$$ $\nabla_{-\tau}$ denotes differentiation with respect to $u_{-\tau}$. We put $\epsilon = 1$ for the rest of the treatment. We now identify, $$\left.\begin{array}{l} u_{1}=x\\ u_{2}=v\\ F_{01}=v\hspace{0.2cm},\hspace{0.2cm}F_{11}=0\\ F_{02}=-\Gamma(x)v-\tilde{V}'(x)\hspace{0.2cm},\hspace{0.2cm} F_{12}=\xi_{\rm eq}(t)+g'(x)\xi_{\rm neq}(t) \end{array} \right \} \; \; .$$ In this notation Eq.(28) now reduces to $$\begin{aligned} \frac{\partial p}{\partial t}=-\frac{\partial}{\partial x}(vp)+\frac{\partial} {\partial v}\left\{\Gamma v+\tilde{V}'(x)\right\}p\hspace{7.5cm}\nonumber\\ \nonumber\\ +\frac{\partial}{\partial v}\int_{0}^{\infty}d\tau\langle\left [\xi_{\rm eq}(t) +g'(x)\xi_{\rm neq}(t)\right ]\left [\frac{\partial}{\partial v^{-\tau}}\{ \xi_{\rm eq}(t-\tau)+g'(x^{-\tau})\xi_{\rm neq}(t-\tau)\}\right]\rangle p \hspace{0.2cm},\end{aligned}$$ where we have used the fact that the Jacobian obey the equation $^{12}$ $$\frac{d}{dt}\log\left|\frac{d(x^{t},v^{t})}{d(x,v)}\right| =\frac{\partial}{\partial x}v+\frac{\partial}{\partial v}\{-\Gamma v+\tilde{V} '(x)\} =-\Gamma$$ so that Jacobian equals to $e^{-\Gamma t}$. As a next approximation we consider the ‘unpurterbed’ part of Eq.(16) and take the variation of $v$ during $\tau_{c}$ into account to first order in $\tau_{c}$. Thus we have $$x^{-\tau}=x-\tau v\hspace{0.2cm};\hspace{0.2cm}v^{-\tau}=v+\Gamma\tau v+\tau \tilde{V}'(x)\hspace{0.2cm}.$$ Neglecting terms ${\cal O}(\tau^{2})$ Eq.(32) yields, $$\frac{\partial}{\partial v^{-\tau}}=(1-\Gamma\tau)\frac{\partial}{\partial v} +\tau \frac{\partial}{\partial x}\hspace{0.2cm}.$$ Taking into consideration of Eq.(33), Eq.(30) can be simplified after some algebra to the following form, $$\begin{aligned} \frac{\partial}{\partial t}p(x,v,t)=-\frac{\partial}{\partial x}(vp)+ \frac{\partial}{\partial v}\left\{ \Gamma (x)v+\tilde{V}'(x)-2g'(x)g''(x) I_{nn}\right \}p\nonumber\\ \nonumber\\ +\left \{ I_{ee}+[g'(x)]^{2}I_{nn} \right \}\frac{\partial^{2} p}{\partial v\partial x}\hspace{5.0cm}\nonumber\\ \nonumber\\ +\left \{ J_{ee}-\Gamma(x)I_{ee}+[g'(x)]^{2}J_{nn}-\Gamma(x)[g'(x)]^{2}I_{nn} -vg'(x)g''(x)I_{nn}\right\}\frac{\partial^{2} p}{\partial v^{2}}\hspace{0.2cm},\end{aligned}$$ where, $$\left.\begin{array}{l} I_{ee}=\int_{0}^{\infty} d\tau \langle\xi_{\rm eq}(t)\xi_{\rm eq}(t-\tau)\rangle\tau \\ I_{nn}=\int_{0}^{\infty} d\tau \langle\xi_{\rm neq}(t)\xi_{\rm neq}(t-\tau)\rangle\tau \\ J_{ee}=\int_{0}^{\infty} d\tau \langle\xi_{\rm eq}(t)\xi_{\rm eq}(t-\tau)\rangle \\ J_{nn}=\int_{0}^{\infty} d\tau \langle\xi_{\rm neq}(t)\xi_{\rm neq}(t-\tau)\rangle \\ \end{array}\right\}\hspace{0.2cm}.$$ The subscripts $ee$ and $nn$ in the above expressions for the integrals over the correlation functions refer to equilibrium and nonequilibrium baths, respectively. In deriving the last Eq.(34) we have assumed that the two reservoirs are uncorrelated. Eq.(34) is the required generalized Fokker- Planck equation for our problem. In order to allow ourselves a fair comparison with Fokker-Planck equation of other forms$^{5,6,7}$, we first turn to the diffusion terms in Eq.(34). The coefficients are coordinate $(x)$ dependent. It is customary to get rid of this dependence by approximating the coefficients at the barrier top (say, $ x=0$) \[one may also use mean field or steady state solutions of Eq.(34) obtained by neglecting the fluctuation terms and putting appropriate stationary condition in the diffusion coefficients\]. The drift term in Eq.(34) refers to the presence of a dressed potential of the form, $$\begin{aligned} R(x)=\tilde{V}(x)-[g'(x)]^{2}\; I_{nn}\end{aligned}$$ or $$R(x)=V(x)-\frac{\omega_{c}}{\pi} \gamma_{\rm neq} g^{2}(x)-[g'(x)]^{2} \; I_{nn} \; \; .$$ The modification of the potential is essentially due to the nonlinear coupling of the system to the nonequilibrium modes. $I_{nn}$ is a non-Markovian small contribution and therefore the third term in (36) may be neglected without any loss of generality. For the rest of the treatment we use $R(x)\simeq \tilde{V}(x)$. At the vicinity of the barrier top $x=0, \tilde{V} '(x)$ may be approximated, as usual, by a parabolic potential, i.e., $$\tilde{V}(x)\simeq \bar{E}_{b}-\frac{1}{2}\bar{\omega}_{b}^{2}x^{2}$$ with $$V(x)\simeq E_{b}-\frac{1}{2}\omega_{b}^{2}x^{2}\hspace{0.2cm}.$$ For convenience, one may set $g(0) = 0$ in the Taylor series expansion for $g(x)$ (carried out at the barrier top $x=0$ ), without any loss of generality. And one obtains $$\bar{E_{b}} = E_{b}$$ and $$\bar{\omega_b}^{2} = {\omega_b}^{2} + \frac{2 \omega_{c} \gamma_{neq}}{\pi} [g'(0)]^{2}\; \; .$$ In the linearized description, the Fokker-Planck Eq.(34) is now reduced to the following form, $$\frac{\partial p}{\partial t}=-v\frac{\partial p}{\partial x}+\Gamma p+ [\Gamma v-\bar{\omega}_{b}^{2}x]\frac{\partial p}{\partial v}+ A\frac{\partial^{2} p}{\partial v^{2}}+B\frac{\partial^{2} p}{\partial v \partial x}\hspace{0.2cm},$$ where we have used the following abbreviations; $$A=J_{ee}-\Gamma(0)I_{ee}+[g'(0)]^{2}J_{nn}-\Gamma(0)[g'(0)]^{2}I_{nn}$$ and $$B=I_{ee}+[g'(0)]^{2}I_{nn}\hspace{0.2cm}.$$ From the last two relations we have $$A=\left[ J_{ee}+g'(0)^{2}J_{nn}\right]-\Gamma(0)B$$ Defining $A$ and $B$ as $$A=\bar{\gamma}KT \hspace{0.2cm}{\rm and}\hspace{0.2cm}B=\bar{\beta}KT$$ one obtains $$\begin{aligned} \frac{\partial p}{\partial t}=-v\frac{\partial p}{\partial x}-\bar{\omega}_{b} ^{2}x\frac{\partial p}{\partial v}+\Gamma\frac{\partial}{\partial v}(vp)+ \bar{\gamma}KT\frac{\partial^{2} p}{\partial v^{2}}\nonumber\\ \nonumber\\ +KT\left[ \frac{J_{ee}+g'(0)^{2}J_{nn}}{\Gamma(0) KT}-\frac{\bar{\gamma}} {\Gamma(0)}\right]\frac{\partial^{2} p}{\partial x\partial v}\hspace{0.2cm}.\end{aligned}$$ Identifying $$\bar{\Omega}^{2}=\Omega^{2}\left[ \frac{J_{ee}+g'(0)^{2}J_{nn}}{\Gamma(0) KT} \right]\hspace{0.2cm},$$ Eq.(46) may be rewritten as, $$\begin{aligned} \frac{\partial}{\partial t}p(x,v,t)=-v\frac{\partial p}{\partial x} -\bar{\omega}_{b}^{2}x\frac{\partial p}{\partial v} +\Gamma\frac{\partial}{\partial v}(vp)+ \bar{\gamma}KT\frac{\partial^{2} p}{\partial v^{2}}\nonumber\\ \nonumber\\ +KT\left[ \frac{\bar{\Omega}^{2}(t)}{\Omega^{2}}-\frac{\bar{\gamma}} {\Gamma(0)} \right ] \frac{\partial^{2} p}{\partial x\partial v}\hspace{0.2cm}.\end{aligned}$$ Here $\bar{\gamma}(t)$ and $\bar{\Omega}(t)$ are functions of time (due to the relaxation of the nonequilibrium modes) as defined by Eqs.(45) and (47). Or in other words nonstationary nature of the bath makes $\bar{\Omega}(t)$ time-dependent through $J_{nn}$ term which is essentially a non-Markovian modification. Now the fluctuation-dissipation relations for equilibrium and nonequilibrium baths stated in Sec.II may be invoked. For equilibrium baths as noted earlier we have the usual result; $$J_{ee}=\int_{0}^{\infty} d\tau \langle\xi_{\rm eq}(t)\xi_{\rm eq}(t-\tau) \rangle=\gamma_{\rm eq}KT$$ For the nonequilibrium version, Eq.(15) may be rearranged further to note that $$J_{nn}=\int_{0}^{\infty} d\tau \langle\xi_{\rm neq}(t)\xi_{\rm neq}(t-\tau) \rangle=\gamma_{\rm neq}KT(1+r e^{-\frac{\gamma}{2} t})$$ where $r$ is a measure of the deviation from equilibrium at the initial instant and is given by $r=\left \{ \frac{{\cal U}(\omega_{\rightarrow 0},0)}{2KT} -1 \right\}$. Here ${\cal U}(\omega,t)$ defines the energy density distribution at time $t$. Using (49) and (50) we obtain from Eq.(47) $$\frac{\bar{\Omega}^{2}(t)}{\Omega^{2}}=1+\frac{r \gamma_{\rm neq}e^{-\frac{\gamma}{2} t}} {\gamma_{\rm eq}+\gamma_{\rm neq}[g'(0)]^{2}}\hspace{0.2cm}.$$ In the long time limit the relation reduces to $$\left.{\cal L}t\right._{t \rightarrow \infty} \bar{\Omega}(t)=\Omega\hspace{0.2cm}.$$ It is interesting to note that with the replacement $\frac{\bar{\Omega}^{2} (t)}{\Omega^{2}} \sim \frac{\bar{\omega}_{b}^{2}}{\omega_{b}^{2}}$ (terms are of order $1+{\cal O}(\gamma)$) and $\Gamma(0) \sim \bar{\gamma}$ one recovers the Fokker-Planck equation in the Adelman’s form$^{5}$ (Eq.(8)). **[IV.Non-Markovian steady state Kramers rate]{}** We now proceed to analyze our generalized Fokker-Planck equation (48) and calculate the steady state current and the Kramers escape rate over the barrier. The procedure we follow in this section is similar to that of Kramers supplemented by Hänggi and Mojtabai’s earlier analysis$^{6}$. As usual we make the ansatz $$p(x,v,t)=F(x,v,t)\exp\left [ -\frac{\frac{v^{2}}{2}+\tilde{V}(x)}{KT}\right]$$ with $\tilde{V}(x)$ as approximated by a parabolic potential of the form \[ see Eqs. (37-40) \] $$\begin{aligned} \tilde{V}(x)\simeq \bar{E}_{b}-\frac{1}{2}\bar{\omega}_{b}^{2}x^{2}\end{aligned}$$ with$\hspace{5.0cm}\bar{E}_{b}=E_{b}$ and$\hspace{5.0cm}\bar{\omega}_{b}^{2}=\omega_{b}^{2}+ \frac{2 \; \omega_{c} \; \gamma_{\rm neq}}{\pi} [g'(0)]^{2}$ as stated earlier. We seek an equation for $F$ of the form $$F(x,v,t)=F(u,t)\hspace{0.2cm},\hspace{0.2cm}u=v+ax\hspace{0.2cm}.$$ Inserting (53) and (54) in Eq.(48) we obtain $$\begin{aligned} \frac{\partial F}{\partial t}=\left\{(\Gamma-\bar{\gamma})-\frac{1}{KT} (\Gamma-\bar{\gamma})v^{2}-\frac{\bar{\omega}_{b}^{2}}{KT}\Delta xv\right\}F \nonumber\\ \nonumber\\ +\left[\left\{(\Gamma-2\bar{\gamma})-a(1+\Delta)\right\}v-\bar{\omega}_{b}^{2} (1-\Delta)x\right]\frac{\partial F}{\partial u}\nonumber\\ \nonumber\\ +KT(\bar{\gamma}+\Delta a)\frac{\partial^{2}F}{\partial u^{2}}\hspace{0.2cm},\end{aligned}$$ where $$\Delta=\frac{\bar{\Omega}^{2}(t)}{\Omega^{2}}-\frac{\bar{\gamma}}{\Gamma(0 )}\hspace{0.2cm}.$$ Using (51), $\Delta$ may be rewritten as $$\Delta\simeq \frac{r \gamma_{\rm neq}e^{-\frac{\gamma}{2} t}}{\gamma_{\rm eq}+\gamma_{ \rm neq}[g'(0)]^{2}}$$ for $\Gamma\sim\bar{\gamma}$. Assuming $\frac{\Delta}{KT}$ and $(\Gamma-\bar{\gamma})$ to be very small we obtain $$\frac{\partial F}{\partial t} = KT\frac{\partial^{2} F}{\partial u^{2}} -\left[ \frac{\Gamma+a(1+\Delta)}{\Gamma+\Delta a}v+\frac{\bar{\omega}_{b} ^{2}(1-\Delta)}{\Gamma+\Delta a}x\right]\frac{\partial F}{\partial u} \hspace{0.2cm},$$ which may be written in the form $$\frac{\partial F}{\partial t}=KT\frac{\partial^{2} F}{\partial u^{2}}+ \bar{\alpha}u\frac{\partial F}{\partial u}\hspace{0.2cm},$$ with $$\bar{\alpha}=-\frac{\Gamma+a(1+\Delta)}{\Gamma+ \Delta a}\hspace{0.2cm},$$ and $a$ is a solution of the quadratic equation $$a^{2}(1+\Delta)+\Gamma a-\bar{\omega}_{b}^{2}(1-\Delta)=0\hspace{0.2cm}.$$ Since $\left.{\cal L}t\right._{t\rightarrow\infty}\Delta=0$, the long time or steady state solution of Eq.(58) is satisfied by $$KT\frac{\partial^{2} F}{\partial u^{2}}+ \bar{\alpha}u\frac{\partial F}{\partial u}=0$$ with $$\left.{\cal L}t\right._{t\rightarrow\infty}\bar{\alpha}(t)=-\frac{\Gamma+a} {\Gamma}=\alpha({\rm say})\hspace{0.2cm}.$$ Since the Eqs.(62) and (63) are identical in form to the expressions obtained in the usual Kramers theory one can have the usual expressions for the probability density $p(x,v)$ and the current $j_{s}$ as $$p(x,v,\infty)= N\left[\left(\frac{\pi KT}{2 \alpha}\right)^{\frac{1}{2}} +\int_{0}^{v-|a|x}dz \exp\left(-\frac{\alpha z^{2}}{2KT}\right)\right] \exp\left[ -\frac{\frac{v^{2}}{2}+\tilde{V}(x)}{KT}\right]$$ with $$\begin{aligned} F_{s} = N\left[\left(\frac{\pi KT}{2 \alpha}\right)^{\frac{1}{2}} +\int_{0}^{v-|a|x}dz \exp\left(-\frac{\alpha z^{2}}{2KT}\right)\right] \; \; ,\end{aligned}$$ (here the subscript $s$ in $F_s$ refers to steady state $F$) and $$j_{s}=\int_{-\infty}^{+\infty}dv \hspace{0.1cm}vp(x,v) =N(KT)^{\frac{3}{2}}\left(\frac{2\pi}{\alpha+1}\right)^{\frac{1}{2}} \exp\left(-\frac{E_b}{KT}\right)\hspace{0.2cm},$$ where we have used the linearized version of $\tilde{V}(x)$ near the top of the barrier at $x=0$, $$\begin{aligned} \tilde{V}(x)=\bar{E}_{b}-\frac{1}{2}\bar{\omega}_{b}^{2}x^{2}\hspace{0.2cm},\end{aligned}$$ with ${\bar{E}}_b = E_b$ and ${\bar{\omega}}_b$ is as given in Eq.(40) and $N$ is the normalization constant. Employing the asymptotic distribution (just before the system is subjected to the shock at $t=0$) of $P_{w}(x,v)$ for $x\rightarrow -\infty$ and at $t=0_-$ from $p(x,v,t)$, where $P_{w}(x,v)=p(x\rightarrow -\infty,v;t=0)$ \[see Sec. V for calculation of $p(x,v,t)$\], one obtains the total number of particles in the well, $$n_{a}=N \int_{-\infty}^{+\infty}dv \int_{-\infty}^{+\infty}dx P_{w}(x,v) =N \frac{2\pi KT}{\omega_{0}}\left(\frac{2\pi KT}{\alpha}\right)^{\frac{1}{2}} \hspace{0.2cm}.$$ Here $\omega_{0}$ is the frequency at the bottom of the left well. We have set the potential energy at the bottom of the left well equal to zero, for convenience. The final result for the rate of escape in the steady state is given by $$k=\frac{j_{s}}{n_{a}}=\frac{\omega_{0}\lambda}{2\pi\bar{\omega}_{b}} e^{-E_{b}/KT}\hspace{0.2cm},$$ where $$\lambda=\left[\left\{\left(\frac{\Gamma}{2}\right)^{2}+\bar{\omega}_{b} ^{2}\right\}^{\frac{1}{2}}-\frac{\Gamma}{2}\right]\hspace{0.2cm}.$$ It is evident that $\lambda$ is reminiscent of the ‘reactive frequency’ $\lambda_{r}$ of Grote and Hynes$^{4}$ . Microscopically the non-Markovian character of the dynamics in $\lambda$ enters through the explicit structure of $\Gamma$ and $\bar{\omega}_{b}$ which are given by $$\Gamma=\gamma_{\rm eq}+\gamma_{\rm neq}[g'(0)]^{2}$$ and $$\bar{\omega}_{b}^{2}=\omega_{b}^{2} \; + \; \frac{2\omega_c \gamma_{\rm neq}} {\pi} [g'(0)]^2 \; \; .$$ The appearance of the reactive frequency $\lambda$ is suggestive of the fact that the particle on the average is not moving on the bare barrier with frequency $\omega_{b}$ but on a dressed barrier frequency $\bar{\omega}_{b} $ corrected by $\lambda$. Pollak$^{8}$ has shown that the reactive frequency $\lambda$ is exactly an imaginary frequency of a barrier that has been modified by the bath modeled as a discrete set of harmonic oscillators linearly coupled to the system. The effect of $\lambda$ is to slow down the particle by friction near the barrier. In the present model where the generalized Langevin equation(12) describes the motion of the particle over a fluctuating barrier the essential modification of $\lambda$ and $\omega_{b}$ rests on the nonlinear coupling of the nonequilibrium relaxing modes with the system. Thus in addition to the properties of the bath, dynamic nature of the system-bath coupling is also significant in governing the barrier dynamics. We note in passing that the usual Markovian limit can be recovered if one puts $\gamma_{\rm neq}=0$ in Eq.(67) and associated quantities. Before closing this section one pertinent point need to be mentioned. A closer look into the derivation makes it clear that Eq.(67) results from an ansatz of the form (53) where we use $\tilde{V}(x)$ in the Boltzmann factor. This choice is basically guided by the fact that the potential $V(x)$ gets dressed at $t=0$ by initial excitation of nonequilibrium modes. This choice also makes the stationary current independent of position. However, if one uses the bare potential $V(x)$ and assume a weak dependence of $x$ on $j_{s}$, one obtains Eq.(67) with $\bar{\omega}_{b}$ in the denominator getting replaced by $\omega_{b}$ itself. The main lesson is that the modification of Kramers rate (67) is essentially due to $\lambda$, the reactive frequency of Grote-Hynes, which has been recognized as an important result in view of some experimental evidence$^{18}$ of relatively weak dependence of rate on damping in the large friction limit. **[V.Time-dependent solution of the generalized Fokker-Planck equation ; nonstationary Kramers rate ; nonexponential relaxation kinetics]{}** We now turn to Eq.(55). Rearranging the time-dependent $\Delta$-containing terms it may be rewritten as $$\frac{1}{\Gamma}\frac{\partial F}{\partial t}=-\left[\frac{ (\Gamma+a)v+\bar {\omega}_{b}^{2}x}{\Gamma}\right]\frac{\partial F}{\partial u}+KT \frac{\partial^{2} F}{\partial u^{2}}+\Delta\left[\frac{aKT}{\Gamma} \frac{\partial^{2} F}{\partial u^{2}}-\frac{(av-\bar{\omega}_{b}^{2}x)} {\Gamma}\frac{\partial F}{\partial u}\right]\hspace{0.2cm},$$ where $\Delta$ is defined in Eq.(57). Let us write $$\frac{ (\Gamma+a)v+\bar{\omega}_{b}^{2}x}{\Gamma}=-\alpha u$$ and $$\frac{(av-\bar{\omega}_{b}^{2}x)}{\Gamma}=-\lambda u$$ Here $\alpha$ is as defined in (63) and $\lambda$ is to be determined. In terms of the relations (72) and (73), Eq.(71) reduces to a more compact form. $$\frac{1}{\Gamma}\frac{\partial F}{\partial t}=\alpha u \frac{\partial F}{\partial u}+KT\frac{\partial^{2} F}{\partial u^{2}} +\Delta\left[\frac{aKT}{\Gamma} \frac{\partial^{2} F}{\partial u^{2}}+\lambda u \frac{\partial F}{\partial u}\right]\hspace{0.2cm}.$$ Eq.(72) may be used to calculate the value of $a$ as obtained from the solution of the algebraic equation $$a^{2}+\Gamma a-\bar\omega_{b}^{2}=0\hspace{0.2cm}.$$ Only the negative root of the above equation (say $a_{-}$) is the physically realizable solution corresponding to the steady state solution. This value of $a$ determines uniquely the value of $\lambda$ as defined in Eq.(73) to obtain $$\lambda=-\alpha\hspace{0.2cm}.$$ We now seek a solution $F(u,t)$ of Eq.(74) in the form $$F(u,t)=F_{s}(u)e^{-\phi(t)}\hspace{0.2cm},$$ where $F_{s}(u)$ is the steady state solution obtained in the earlier section, i.e., it satisfies $$\alpha u \frac{\partial F_{s}}{\partial u}+KT\frac{\partial^{2}F_{s}} {\partial u^{2}}=0\hspace{0.2cm}\hspace{0.2cm}.$$ We require further $$\left.{\cal L}t\right._{t\rightarrow\infty}\phi (t)=0\hspace{0.2cm}.$$ Substituting (77) in Eq.(74) it may be shown that the ‘space’ and the time part is separable. We obtain, $$-\frac{1}{\Gamma}\frac{\partial \phi}{\partial t} e^{\frac{\gamma}{2} t}=\frac{C} {F_{s}}\left[\lambda u\frac{\partial F_{s}}{\partial u}+\frac{aKT}{\Gamma} \frac{\partial^{2}F_{s}}{\partial u^{2}} \right]={\rm constant}=D({\rm say}) \hspace{0.2cm},$$ where we have made use of the Eq.(78) and also $$\begin{aligned} \Delta=Ce^{-\frac{\gamma}{2} t}\hspace{0.2cm} {\rm with} \hspace{0.2cm}C=\frac{r \gamma_{\rm neq}}{\gamma_{\rm eq}+\gamma_{ \rm neq}[g'(0)]^{2}}\hspace{0.2cm}.\end{aligned}$$ On integration over time we obtain from Eq.(80), the solution $$\phi (t)=2 D\frac{\Gamma}{\gamma}e^{-\frac{\gamma}{2} t}$$ where $D$ is determined by the initial condition. The time-dependent solution of Eq.(71) therefore reads as $$F(u,t)=F_{s}(u)\exp\left[-\frac{2D\Gamma}{\gamma} e^{-\frac{\gamma}{2} t}\right] \hspace{0.2cm}.$$ Thus the corresponding probability distribution is given by, $$\begin{aligned} p(x,v,t)=N\left[\left(\frac{\pi KT}{2\alpha}\right)^{\frac{1}{2}}+\int_{0}^ {v-|a|x}dz \exp\left(-\frac{\alpha z^{2}}{2KT}\right)\right]\nonumber\\ \nonumber\\ \exp\left[-\frac{\frac{v^{2}}{2}+\tilde{V}(x)}{KT}\right] e^{-\frac{2D\Gamma}{\gamma}\left[\exp(-\frac{\gamma}{2} t)\right]}\hspace{0.2cm}.\end{aligned}$$ To determine $D$ we now demand that just at the moment the system (and the nonthermal bath) is subjected to external excitation at $t=0$ and $x\rightarrow -\infty$ the distribution (75) must coincide with the usual Boltzmann distribution where the energy term in the Boltzmann factor in addition to usual kinetic and potential terms contains the initial fluctuation of energy density $\Delta {\cal U}$ \[$\Delta {\cal U}={\cal U}(\omega,0)-\frac{1}{2} KT$\] due to excitation of the system at $t=0$ \[see Eq.(15)\]. $$\begin{aligned} p(x,v,t)\stackrel{t\rightarrow 0} {\longrightarrow} N\left(\frac{2\pi KT}{\alpha}\right)^{\frac{1}{2}} e^{-2D\frac{\Gamma}{\gamma}} e^{-\frac{1}{KT}\left(\frac{v^{2}}{2}+\tilde{V}(x)\right)}\nonumber\\ \nonumber\\ =N\left(\frac{2\pi KT}{\alpha}\right)^{\frac{1}{2}} e^{-\frac{1}{KT}\left(\frac{v^{2}}{2}+{\tilde{V}}(x)+\Delta {\cal U}\right)}, \hspace{0.2cm}{\rm for} (x\rightarrow -\infty)\hspace{0.2cm}.\end{aligned}$$ The last equality demands that $$D=\frac{\gamma}{2\Gamma} \; \frac{\Delta {\cal U}}{KT}$$ \[for the current to be coordinate independent the parabolic approximation of $\tilde{V}(x)$ is to be used\]. $D$ is thus determined in terms of the relaxing mode parameters and fluctuations of the energy density distribution at $t=0$. The time-dependent probability density therefore allows us to construct nonstationary current, $$j(t)=\int_{-\infty}^{+\infty}dv\hspace{0.1cm}v\hspace{0.1cm}p(x,v,t)= j_{s}e^{-\frac{2D\Gamma}{\gamma}\exp(-\frac{\gamma}{2} t)}\hspace{0.2cm},$$ where $j_{s}$ is the stationary or steady state current as derived in the last section. By Eq.(74) we have, $$p_{w}(x,v)=p(x\rightarrow -\infty,v,t=0_{-})\hspace{0.2cm},$$ which was used to calculate the number of particles $n_{a}$ initially in the well just before the system was subjected to shock at $t=0$. Thus non-stationary Kramers rate of transition is given by $$k(t)=\frac{\omega_{0}}{2\pi\bar{\omega}_{b}}\left[\left\{\left(\frac{\Gamma} {2}\right)^{2}+\bar{\omega}_{b}^{2}\right\}^{\frac{1}{2}}-\frac{\Gamma}{2} \right]e^{-\frac{E_{b}}{KT}} e^{-\left[\frac{2D\Gamma}{\gamma}\exp(-\frac{\gamma}{2} t)\right]}\hspace{0.2cm},$$ or in terms of the steady state Kramers rate $k$ $$k(t)=k\exp\left[ -\frac{\Delta {\cal U}}{KT} e^{-\frac{\gamma}{2} t}\right]\hspace{0.2cm} \; \; ,$$ where $\Delta {\cal U}$ is a measure of the initial departure from the average energy density distribution due to the preparation of the nonstationary state of the intermediate bath modes as a result of excitation at $t=0$, and $k$ is given by $$k=\frac{\omega_0}{2\pi {\bar{\omega}}_b} \left[ \left \{ \left( \frac{\Gamma}{2} \right )^2 + {\bar{\omega}}_b^2 \right \}^{1/2} - \frac{\Gamma}{2} \right ] \; e^{-E_b/KT} \; \; .$$ The above result (88) illustrates a strong nonexponential relaxation of the system mode undergoing a nonstationary activated rate process. The origin of this is an initial preparation of nonequilibrium mode density distribution (with a deviation $\Delta {\cal U}$) which eventually relaxes to an equilibrium distribution. Eq. (88) implies that the initial transient rate is different from the asymptotic steady state Kramers rate. What is immediately apparent is that the sign of $\Delta {\cal U} [ = {\cal U}(\omega,0)-\frac{1}{2}KT ]$ determines whether the initial rate will be faster or slower than the steady state rate. When $\Delta {\cal U}$ is negative, i.e., the contribution of thermal energy dominates, the initial rate of thermal activation of the reaction co-ordinate gets enhanced as a consequence. On the other hand, when the sudden excitation of the nonequilibrium modes provides a positive deviation $\Delta {\cal U}$, the initial rate of activation becomes slower. This is because there likely to exist some time lag for the nonthermal energy gained by the few nonequilibrium modes by sudden excitation to be distributed over a range before it become available to the reaction co-ordinate as thermal energy for activation. It is also interesting to consider the zero and high temperature limits. When $T\rightarrow 0$ both the steady state Kramers rate $k$ as well as the time-dependent factor $exp[-\frac{\Delta {\cal U}}{KT} e^{-\gamma t/2} ]$ goes to zero. If $T=0$, then $k(t)$ is zero at all time. However, it seems intuitively that there should be a transient period during which the rate is finite. It may be noted that since the relaxation of the nonequilibrium bath modes (following the sudden excitation) is very slow compared to the rate of activation process, the particle undergoing barrier crossing cannot ‘sense’ this transient (ideally if the relaxation to equilibrium is adiabatic, i.e., the thermalization of the initial departure $\Delta{\cal U}$ is very slow, there should be no transient). We believe that the distinct separation of the two time-scales implied in the dynamics makes the transient unobservable. An interplay of overlapping time-scales pertaining to the relaxation of the bath and the activation of the system may give rise to transients in $k(t)$ at $T=0$. Evidently this is outside the scope of the present treatment. When $T$ is very high such that $\frac{1}{2} KT$ far exceeds ${\cal U}(\omega,0)$ the initial rate gets strongly enhanced (since $\Delta {\cal U}$ is negative) and the time-dependent exponential factor becomes roughly independent of temperature. In the limit $t\rightarrow \infty$ or $\Delta {\cal U}\rightarrow 0$ we recover steady state Kramers rate, as expected. The activation rate is thus consequently modified which effectively incorporates a secondary relaxation kinetics. The quasi-thermal excitations decay on the time scale $\frac{1}{\gamma}$, which is well separated from other internal time scales of the thermal bath. The dynamic nature of the coupling between the system and the nonequilibrium modes is responsible for fluctuating barrier. A closer look into the origin of the non-exponential kinetics makes it clear that the spiritual root of $D$-term is essentially the $\Delta$-containing term in Eq.(71) or $\frac{\partial^{2} p}{\partial x\partial v}$ term in Eq.(46) which is a non-Markovian contribution. We thus identify the non-exponential relaxation of the system mode as a typical non-Markovian dynamical feature. In the case of very small $\gamma$ one naturally recovers the exponential relaxation and Arrhenious rate of activation of the usual kinetic scheme. A relevant pertinent point regarding some of the related works need be considered here. Generalized Langevin equation (GLE) has been widely employed in various contexts, e.g., in the description of reactions in liquids. A search for realistic models began with the realization that friction exerted by the solvent on the solute is space dependent. A formally consistent approach to the problem of space and time dependent friction had been introduced early by Lindenberg and co-workers$^{20,21}$. Carmeli and Nitzan$^{22}$ have also derived a stochastic dynamical equation which is a generalization of GLE to the case of space and time dependent friction. Pollak and Berezhkorskii$^{25}$ have demonstrated that the space and time-dependent friction model is identical to a multidimensional anisotropic but Markovian friction problem in which the reaction co-ordinate is coupled to an additional co-ordinate which is governed by a Langevin type equation. A theory for treating spatially dependent friction in the classical activated rate processes has been considered and following the method of Pollak an effective Grote-Hynes reactive frequency for this case has been obtained as a transcendental equation$^{23}$. More recently a general theory for thermally activated rate constants influenced by spatially dependent and time correlated friction$^{24}$ has been proposed. While in the above problems one is concerned with the space and time dependent friction, which is essentially a characteristic of the solvent mode structure, in the present problem we deal with effect of a secondary relaxation of intermediate oscillator modes (following an initial excitation) on the primary kinetics of the system mode. The mode density function due to initial excitation differs from its equilibrium value - a feature which is marked in the nonequilibrium fluctuation-dissipation relation. Thus the exponential relaxation in Eq.(81) is not be confused with the exponential time-dependent friction employed in earlier instances. The origin of these two exponential terms are fundamentally different. The non-exponential kinetics is essentially an offshoot of a dynamic modification of the fluctuation-dissipation theorem appropriately carried over to a nonstationary regime. This nonequilibrium nature of activated process is reflected in the nonstationary kinetics that we derive here. The non - exponential relaxation kinetics had been explored earlier in different occasions in relation to disordered systems$^{13}$, viscous liquids$^{19}$, oxygen binding to h$\mbox{\ae}$moglobin$^{16}$, where phenomenological fluctuating barrier models have been employed (barriers arising from the collective motions of many degrees of freedom). The present model although oversimplified in many respects captures the essential nature of influence of an initial non-thermal mode density distribution on the relaxation kinetics of the system. **[VI.Conclusions]{}** In conclusion, we consider a simple microscopic system-nonequilibrium bath model to simulate nonstationary thermally activated processes. The nonequilibrium bath is effectively realized in terms of a semi-infinite dimensional broad-band reservoir which is subsequently kept in contact with a thermal reservoir which allows the nonthermal bath to relax with a characteristic time. A systematic separation of timescales is then used to construct the appropriate Langevin equation for the particle, which is nonlinear and non-Markovian in character. Based on a strategy of Van Kampen’s expansion in $\epsilon \tau_{c}$ of the relevant physical quantity where $\epsilon$ is the strength and $\tau_{c}$ is the correlation time of fluctuations of the relaxing modes, we show that this Langevin equation can be recast into the form of a generalized Fokker-Planck equation, when the correlation time is short but finite. Adelman’s form of the Fokker-Planck equation \[ Eq.(8) \] as well as the standard Markovian description can be recovered in the appropriate limits. We now summarize the main conclusions of this study: \(i) The model proposed here captures the essential features of Langevin dynamics with a fluctuating barrier. The present approach is equipped to deal with situations both in the non-stationary short time as well as stationary long time regimes. The origin of the short time non-exponential kinetics can be traced back in a non-stationary fluctuation-dissipation theorem. \(ii) We derive the expression for the steady state Kramers escape rate in the non-Markovian case and show that the Grote-Hynes ‘reactive frequency’ can be realized explicitly in terms of the microscopic parameters of the nonequilibrium relaxing modes and their arbitrary dynamic coupling to the system mode. \(iii) The central result of this paper is the derivation of a nonstationary Kramers rate in closed analytic form. This essentially illustrates the influence of an initial excitation and subsequent relaxation of the nonequilibrium bath modes on the system degree of freedom undergoing an activated process. The system mode is shown to follow strong non-exponential kinetics. The model considered in the present paper may be realized in a guest-host system embedded in a lattice where the immediate local neighborhood of the guest comprises intermediate oscillator modes whereas the lattice plays the role of a thermal bath. Appropriately identified reaction co-ordinate coupled to other degrees of freedom in a molecule embedded in a matrix may be another worthwhile candidate for such a scheme. Although simple, the model thus allows us explicit solutions and in view of the prototypical role played by the present model in several earlier investigations, we hope that the conclusions drawn here will find applications in some related experiments of physics and chemistry of complex systems. [**[Acknowledgments]{}**]{} : Partial financial support from the Department of Science and Technology (Govt. of India) is thankfully acknowledged. One of the authors (JRC) is thankful to Prof. J. K. Bhattacharjee (Dept. of Theoretical Physics, I.A.C.S) and to S. K. Banik for helpful discussions and suggestions. 1. H. A. Kramers, Physica [**7**]{}, 284 (1940). 2. See for example, a review by P. Hänggi, P. Talkner and M. Borkovec, Rev. Mod. Phys. [**62**]{}, 251 (1990) and the references given therein. 3. J. T. Hynes and J. M. Dentch, [*Physical Chemistry : An Advanced Treatise*]{}, vol. IIB (Academic, New York, 1975) ; S. A. Adelman and J. D. Doll, Acc. Chem. Res. [**10**]{}, 378 (1997) ; J. M. Dentch and I. Oppenheim, J. Chem. Phys. [**54**]{}, 3547 (1971). 4. R. F. Grote and J. T. Hynes, J. Chem. Phys. [**73**]{}, 2715 (1980). 5. S. A. Adelman, J. Chem. Phys. [**64**]{}, 124 (1976). 6. P. Hänggi and F. Mojtabai, Phys. Rev. A [**26**]{}, 1168 (1982) ; P. Hänggi, J. Stat. Phys. 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--- abstract: 'Spectroscopic studies play a key role in the identification and analysis of interstellar ices and their structure. Some molecules have been identified within the interstellar ices either as pure, mixed, or even as layered structures. Absorption band features of water ice can significantly change with the presence of different types of impurities (CO, $\rm {CO_2}$, $\rm{CH_3OH}$, $\rm{H_2CO}$, etc.). In this work, we carried out a theoretical investigation to understand the behavior of water band frequency, and strength in the presence of impurities. The computational study has been supported and complemented by some infrared spectroscopy experiments aimed at verifying the effect of HCOOH, $\rm{NH_3}$, and $\rm{CH_3OH}$ on the band profiles of pure $\rm{H_2O}$ ice. Specifically, we explored the effect on the band strength of libration, bending, bulk stretching, and free-OH stretching modes. Computed band strength profiles have been compared with our new and existing experimental results, thus pointing out that vibrational modes of $\rm{H_2O}$ and their intensities can change considerably in the presence of impurities at different concentrations. In most cases, the bulk stretching mode is the most affected vibration, while the bending is the least affected mode. HCOOH was found to have a strong influence on the libration, bending, and bulk stretching band profiles. In the case of NH$_3$, the free-OH stretching band disappears when the impurity concentration becomes 50%. This work will ultimately aid a correct interpretation of future detailed spaceborne observations of interstellar ices by means of the upcoming JWST mission.' author: - 'P. Gorai' - 'M. Sil' - 'A. Das' - 'B. Sivaraman' - 'S. K. Chakrabarti' - 'S. Ioppolo' - 'C. Puzzarini' - 'Z. Kanuchova' - 'A. Dawes' - 'M. Mendolicchio' - 'G. Mancini' - 'V. Barone' - 'N. Nakatani' - 'T. Shimonishi' - 'N. Mason' title: A Systematic Study on the Absorption Features of Interstellar Ices in Presence of Impurities --- **Keywords:** Astrochemistry, spectra, ISM: molecules, methods: numerical, experimental, infrared: Band strength, interstellar ice. Introduction {#sec:intro} ============ Interstellar grains mainly consist of nonvolatile silicate or carbonaceous compounds covered by icy mantle layers. Interstellar ices play a crucial role in the chemical enrichment of the interstellar medium (ISM). While the existence of interstellar ice was first proposed by @eddi37 in 1937, a turning point was marked, more than 40 years later, when @tiel82 introduced a combined gas-grain chemistry for the chemical evolution of the ISM. More recently, it has been demonstrated that even pre-biotic molecules can be produced in UV-irradiated astrophysical relevant ices [@woon02]. For instance, @nuev14 experimentally showed that nucleobases can be formed by UV irradiation of pyrimidine in H$_2$O-rich ice mixtures containing NH$_3$, CH$_3$OH, and CH$_4$. The composition of interstellar ices can be determined through their absorption spectra in the infrared (IR) region. Since the composition of ISM grain mantles strongly depends on physical conditions [@das08; @das10; @das11; @das16], the observed spectra can be very different in different astrophysical regions. $\rm{H_2O}$ is the most dominant ice component in dense molecular clouds [@gibb04], accounting for $60-70\%$ of the icy mantels [@whit03]. Water ice was firstly detected through the comparison of ground-based observations of its O-H stretching band at $3278.69$ cm$^{-1}$ ($3.05$ $\mu$m) toward Orion-KL [@gill73] and laboratory work by @irvi68. Since then, several ground-based observations were carried out to identify the signatures of water ice in different astrophysical environments, with further laboratory studies supporting such observations [@merr76; @lege79; @hage79]. More recently, water was detected by the space-borne Infrared Space Observatory (ISO) mission through its Short-Wavelength Spectrometer (SWS) and Long-Wavelength Spectrometer (LWS) in the mid- and far-infrared spectral region. In the mid-IR, along with its strong $\rm{O-H}$ stretching mode ($3.05$ $\mu$m), water shows weaker bending and combination bands at $1666.67$ cm$^{-1}$ (6.00 $\mu$m) and $2222.22$ cm$^{-1}$ (4.50 $\mu$m), respectively, and the libration mode at $769.23$ cm$^{-1}$ (13.00 $\mu$m), which is usually blended with the grain silicate spectroscopic features along the line of sight to star forming regions in the ISM [@gibb04]. After H$_2$, water is the second most abundant molecular species in the Universe and its gas-phase abundance in the ISM is even comparable to that of CO. Due to the high abundance of water in interstellar ices [@dart05], the amount of the other species is very often expressed in terms of the relative abundance with respect to $\rm{H_2O}$, and thus considered as impurities. Among other solid species, CO, CO$_2$, CH$_3$OH, H$_2$CO, HCOOH, NH$_3$, CH$_4$, and OCS have been unambiguously identified [@gibb04], while theoretical studies suggest that N$_2$ and O$_2$ might be trapped in the ice matrix as well [@vand93]. It should be noted that although homonuclear molecules are IR inactive, they can become IR active when embedded in ice matrices. Interstellar ice matrices are usually classified as (i) polar ices, if dominated by polar molecules like H$_2$O, CH$_3$OH, NH$_3$, OCS, H$_2$CO, HCOOH, and (ii) apolar ices, if they are dominated by molecules like CO, CO$_2$, CH$_4$, N$_2$, and O$_2$. Interstellar ices are believed to be a combination of both with a first polar (water-rich) layer and an apolar CO-dominated layer deposited on top of it during the catastrophic freeze-out of CO molecules in the cold core of molecular clouds [@boog15]. Infrared spectroscopy is a suitable technique for identifying interstellar species, particularly, in condensed phases. However, it requires that vibrations are IR active, condition which is fulfilled when the dipole moment changes during vibration. The IR spectrum of a water cluster is one of the primary tools to analyze the features of the aggregation processes in a water matrix [@ohno05; @bouw07; @ober07]. Moreover, four vibrational modes of water, namely libration, bending, bulk stretching, and free-OH stretching, are essential to obtain relevant information about the water cluster itself in various astrophysical environments [@gera95; @ohno05; @bouw07; @ober07]. However, there are some difficulties for the observation of interstellar ices in the mid-IR, such as the need for a background illuminating source being required for absorption, e.g. a protostar or a field star. Furthermore, peak positions, line widths, and intensities of molecular ice features need to be known and compared to laboratory spectra, which further depend on ice temperature, crystal structure of the ice, and mixing or layering with other species [@ehre97; @schu99; @cook16]. As a result, only a very limited number of species have been unambiguously detected in interstellar ices. CO is routinely observed from various ground-based facilities. In the solid phase, its abundance may vary from $3\%$ to $20\%$ of the water-ice. CO absorbance shows both polar and apolar band profiles. @soif79 reported the detection of the fundamental vibrational band of CO at 4.61 $\mu$m (2169.20 cm$^{-1}$) in absorption toward W33A, based on the laboratory work of @mant75. The corresponding band profile consists of a broad (polar) component peaking at $2136.75$ cm$^{-1}$ ($4.68$ $\mu$m) and a narrow (non-polar) component peaking at $2141.33$ cm$^{-1}$ ($4.67$ $\mu$m) [@chia95; @chia98]. $\rm{CO_2}$ was detected in absorption at $657.89$ cm$^{-1}$ (15.20 $\mu$m) toward several IRAS sources by @dhen89, based on their laboratory work. The presence of $\rm{CO_2}$ in ice mantles was found on very few astrophysical objects before the launch of ISO [@dart05], which allowed to firmly establish the ubiquitous nature of CO$_2$ [@degr96; @guer96; @gera99]. In the ice phase, CH$_3$OH abundance varies between 5% and 30% with respect to $\rm{H_2O}$. Its abundance can be even lower in some sources, such as Sgr A and Elias 16 [@gibb00]. From ground-based observations, the stretching band of methanol at $2832.80$ cm$^{-1}$ ($3.53$ $\mu$m) was detected in massive protostars [@baas88; @grim91; @alla92]. The first attempt of H$_2$CO observation was made by @schu96, based on the absorption feature at $2881.84$ cm$^{-1}$ ($3.47$ $\mu$m), towards the protostellar source GL 2136 using the United Kingdom Infrared Telescope (UKIRT). They estimated the abundance of $\rm{H_2CO}$ to be $\sim 7\%$ with respect to $\rm{H_2O}$; however, only a small fraction of $\rm{H_2O}$ is mixed with $\rm{H_2CO}$. Space-based ISO observations [@kess96; @kess03] estimated the formaldehyde abundance ranging between 1% and 3% in five high mass protostellar envelopes [@kean01]. HCOOH was detected both in the solid and gas phase [@schu99; @vand95; @iked01]. CH$_4$ was simultaneously detected in both gas and ice phases toward NGC 7538 IRS 9 [@lacy91]. Infrared spectra from the Spitzer Space Telescope show a feature corresponding to the bending mode of solid $\rm{CH_4}$ at $1298.70$ cm$^{-1}$ ($7.7$ $\mu$m) [@ober08]; in that work they derived its abundance to range from 2% to 8%, with the exception of some sources where abundances were found to be as high as $11-13\%$. @knac82 claimed the first identification of $\rm{NH_3}$ in interstellar grains from an IR absorption feature at $3367.00$ cm$^{-1}$ (2.97 $\mu$m) (NH stretching mode), a detection later proved to be wrong [@knac87]. Eventually, the detection of $\rm{NH_3}$ was reported by @lacy98, who assigned an absorption feature at $1109.88$ cm$^{-1}$ ($9.01$ $\mu$m) toward NGC 7538 IRS 9. @palu95 and @palu97, based on their laboratory work, identified toward a number of sources an absorption feature at $2040.82$ cm$^{-1}$ ($4.90$ $\mu$m) that can be assigned to OCS when mixed with $\rm{CH_3OH}$. Recently, the ROSINA mass spectrometer onboard the ESA’s Rosetta spacecraft has discovered an abundant amount of molecular oxygen, $\rm{O_2}$, in the coma of the 67P/Churyumov-Gerasimenko comet, thus deriving the ratio $\rm{O_2/H_2O}$ = $3.80\pm0.85\%$ [@biel15]. Neutral mass spectrometer data obtained during the ESA’s [*Giotto*]{} flyby are consistent with abundant amounts of $\rm{O_2}$ in the coma of comet 1P/Halley, the $\rm{O_2/H_2O}$ ratio being evaluated to be $3.70\pm1.7\%$ [@rubi15]. This makes $\rm{O_2}$ the third most abundant species. In the ISM, O$_2$ and N$_2$ are nearly absent in the gas phase because they are depleted on grains in the form of solid [@vand93]. Since N$_2$ and O$_2$ do not possess dipole moment, they cannot be detected using radio observations. However, $\rm{N_2}$ and $\rm{O_2}$ might be detected by their weak IR active fundamental transition in solid phase, which lies around $4.3 \ \mu$m for N$_2$ and around $6.4 \ \mu$m for O$_2$ [@sand01; @ehre97]. To date, infrared observations suggest that the ice mantles in molecular clouds are unambiguously composed of the few aforementioned molecules [@herb09]. However, more complex species, such as complex organic molecules (COMs), are also expected to be frozen on ice grains in dense cores. The low sensitivity or low resolution of available observations combined with spectral confusion in the infrared region can cause the weak features due to solid COMs to be hidden by those due to more abundant ice species. The upcoming NASA’s James Webb Space Telescope (JWST; <https://jwst.stsci.edu>) space mission set to explore the molecular nature of the Universe and the habitability of planetary systems promises to be a giant leap forward in our quest to understand the origin of molecules in space. The high-resolution of the spectrometers onboard the JWST will enable the search of new COMs in interstellar ices and will shed lights on different ice morphologies, thermal histories, and mixing environments. JWST will be able to map the sky and see right through and deep into massive clouds of gas and dust that are opaque in the visible. However, the large amount of spectral data provided by JWST could be analyzed only if extensive spectral laboratory and modeling datasets are available to interpret such data. The work presented here aims at gaining information on the effect of intermolecular interactions in interstellar relevant ices, thus providing some valuable new laboratory and computed absorption spectra of water-rich ices. These will be useful for the interpretation of future observations in the mid-infrared spectral region. In this paper, a detailed systematic study of the four fundamental vibrational modes of water in presence of various molecular species with different concentration ratios has been carried out. Since water is the major component of interstellar ice matrix, the latter is considered as composed of water molecules with the other compounds being impurities or pollutants. Since the hydrogen bonding network of pure water clusters in the solid state is strongly affected by increasing concentration of impurities, the spectra of the water pure ice is remarkably different from that of ice containing other species. Indeed, ice bands are very sensitive to intermolecular interactions [@sand98], with both strength and band profiles being affected [@knez05; @bouw07; @ober07]. The changes in the spectral behavior and band strengths are primarily due to molecular size, proton affinity, and polarity of the pollutants. [ To the best of our knowledge, there are not similar studies on liquid water systems as some of the impurity species chosen here are water-soluble. A large number of studies have been devoted to vibrational spectra of diluted aqueous solutions of species corresponding to the polar impurities we considered in the present study. However, the attention is usually focused on the variation of the vibrational properties of the solute and not of the solvent [@choi11; @capp11; @blas13]. In a broader context, some infrared studies are available for the whole solubility range of some species [@max03; @max04].]{} This paper is organized as follows. In Section \[sec:method\], we describe the methodology. In Section \[sec:experiment\], we briefly discuss the experimental details. Results and discussions are presented in Section \[sec:results\_discussions\], and finally, in Section \[sec:conclusions\], the concluding remarks are reported. Methodology {#sec:method} =========== There are three established structures of water ice (with a local density of $0.94$ g cm$^{-3}$) formed by vapor deposition at low-pressure. Two of them are crystalline (hexagonal and cubic) and one is a low-density amorphous form [@blak94]. High-density amorphous water ice (with a local density $1.07$ g cm$^{-3}$) also exists and can be formed by the vapor deposition at low temperatures [@blak94]. @prad12 experimentally analyzed the number of water molecules needed to generate the smallest ice crystal. According to that study, the appearance of crystallization is first observed for $275 \pm 25$ and $475 \pm 25$ water molecules, these aggregates showing the well-known band of crystalline ice around $3200$ cm$^{-1}$ (in the OH-stretching region). @blak94 found experimentally that the onset of crystallization occurs at $148$ K. Since we aim to validate our calculations for the low-temperature and low-pressure regime, we focus on the amorphous ices showing a peak in the IR spectrum at around $3400$ cm$^{-1}$. However, since it is not clear how many water molecules are necessary to mimic the amorphous nature of water ice, we considered pure water clusters of different sizes and studied their absorption spectra. For this purpose, we have optimized water clusters of increasing size at different levels of theory. The water clusters considered are: 2H$_2$O (dimer), 4H$_2$O (tetramer), 6H$_2$O (hexamer), 8H$_2$O (octamer), and 20H$_2$O, with their structures being optimized with three different methods (B3LYP, B2PLYP, and QM/MM) as explained in Section \[sec:comp\_details\]. The specific choice of these cluster models is based on experimental outcomes. Experimentally, it has been demonstrated that the water dimer has a nearly linear hydrogen bonded structure [@odut80]. The water clusters with 4 H$_2$O molecules are cyclic in the gas phase [@vian97]. Moving to a 6 H$_2$O cluster, different structures are available: a three-dimensional cage in the gas phase [@liu97] and cyclic (chair) in the liquid helium droplet [@naut00]. For 8H$_2$O, an octamer cube has been found in gaseous states [@ohno05]. Finally, the 20H$_2$O cluster has been considered to check the direct effect of the environment, with more details being provided in the computational detail section. Using the optimized structures of the series of clusters above, harmonic frequencies have been computed and the band strengths of the four fundamental modes have been calculated by assuming the integration bounds as shown in Table \[tab:integration\_bounds\]. Similar integration bounds (except for free-OH stretching mode) were considered in @bouw07 and @ober07. Similar absorption profiles of four fundamental modes of pure water have been obtained from our calculations, with their intensity, band positions and strengths varying for the different cluster sizes and levels of theory used. It is thus essential to find the best compromise between accuracy and computational cost. This means to understand which is the smallest cluster and the cheapest level of theory able to provide a reliable description of water ice. To this aim, we have compared the band positions and the corresponding band strengths of the four vibrational fundamental modes of water obtained with different cluster sizes and different methodologies, to experimental work. While the outcome of this comparison will be discussed later in the text, here we anticipate that the 4H$_2$O cluster in the c-tetramer configuration will be chosen as water ice unit. To investigate the effect of impurities, a number of impurity molecules have been added in order to obtain the desired ratio, as shown in Table \[tab:ice\_mixture\_composition\]. For example, in order to get 2:1 ratio of water:impurity(x), we considered $4$ water molecules hooked up with $2$ ‘$\rm{x}$’ molecules. However, for some systems, to have more realistic features of the water cluster, we needed to consider more water molecules. Since it is known that the water ice clusters containing six $\rm{H_2O}$ molecules are the form of all natural snow and ice on Earth [@abas05], we also present a case with six H$_2$O molecules together as a unit (see Figure \[fig:optimized\_structure\_6H2O\] in the Supporting Information, SI). The cyclic hexamer (chair) configuration of the water cluster containing six $\rm{H_2O}$ has been found the most stable [@ohno05], and considered in our calculations. \ 0.2 cm [**Notes.**]{} Contributions in percentage are provided in the parentheses. ![[Optimized structures for (a) pure water and for the $4:4$ concentration ratio: (b) $\rm{H_2O-HCOOH}$, (c) $\rm{H_2O-NH_3}$, (d) $\rm{H_2O-CH_3OH}$, (e) $\rm{H_2O-CO}$, (f) $\rm{H_2O-CO_2}$, (g) $\rm{H_2O-H_2CO}$, (h) $\rm{H_2O-CH_4}$, (i) $\rm{H_2O-OCS}$, (j) $\rm{H_2O-N_2}$, (k) $\rm{H_2O-O_2}$ clusters.]{}[]{data-label="fig:optimized_structure"}](4h2o_opt_x.pdf){width="\textwidth"} In Figure \[fig:optimized\_structure\]a, we present the optimized water clusters for the c-tetramer configuration. The same structure was considered by @ohno05 and others [@sil17; @das18; @nguy19]. Since four H atoms are available for interacting with the impurities by means of hydrogen bond, in our calculations we can reach up to a $1:1$ ratio between the water and the impurity (i.e., we can reach up to 50% concentration of the impurity in the ice mixture). In order to understand the effect of impurities on the band strengths of the four fundamental bands considered, we have calculated the area under the curve for each band for different mixtures of pure water and pollutants. The band strength has then been derived using the following relation (introduced in @bouw07 and @ober07): $$A_{H_2O:x=1:y}^{band} = \int_{band} I_{H_2O:x=1:y} \times \frac{A_{band}^{H_2O}}{ \int_{band} I_{H_2O}},$$ where $A_{H_2O:x}^{band}$ is the calculated band strength of the vibrational water mode in the $1:y$ mixture, $I_{H_2O:x=1:y}$ is its integrated area, $A_{band}^{H_2O}$ is the band strength of the water modes available from the literature, and $\int_{band} I_{H_2O}$ is the integrated area under the vibrational mode for pure water ice. The experimental absorption band strengths of the three modes of pure water ice are taken from @gera95, who carried out measurements with amorphous water at $14$ K. The adopted values are $2\times10^{-16}$, $1.2\times10^{-17}$, $3.1\times10^{-17}$ $\mathrm{cm \ molecule^{-1}}$, for the bulk stretching ($3280$ cm$^{-1}$), bending ($1660$ cm$^{-1}$), and libration mode ($760$ cm$^{-1}$), respectively. Our [*ab-initio*]{} calculations refer to the temperature at $0$ K. For the calculation of the band strengths, we are considering the strongest feature of that band. Since for the free-OH stretching mode no experimental values exist, we consider the result $A_{free-OH}^{H_2O}= 2.09 \times 10^{-17}$ and $2.52 \times 10^{-17}$ $\mathrm{cm \ molecule^{-1}}$ for the c-tetramer and hexamer water clusters, respectively. Computational details {#sec:comp_details} --------------------- As already mentioned, quantum-chemical calculations have been performed to evaluate the changes of the absorption features of four different fundamental modes, namely, (i) libration, (ii) bending, (iii) bulk stretching, and (iv) free-OH stretching of water in the presence of impurities (CO, $\rm{CO_2}$, $\rm{CH_3OH}$, $\rm{H_2CO}$, HCOOH, $\rm{CH_4}$, NH$_3$, OCS, $\rm{N_2}$, and $\rm{O_2}$). High-level quantum chemical calculations (such as CCSD(T) method and hybrid force field method) are proven to be the best suited for reproducing the experimental data [@puzz14; @baro15a]. However, due to the dimension of our targeted species, these levels of theory are hardly applicable. As already anticipated, different DFT functionals have been tested. Most computations have been carried out using the B3LYP hybrid functional [@beck88; @lee88] in conjunction with the 6-31G(d) basis set (Gaussian 09 package [@fris13]). Some test computations have also been performed by using the B2PLYP double-hybrid functional [@grim06] in conjunction with the the m-aug-cc-pVTZ basis set [@papa09], in which the $d$ functions have been removed on hydrogen atoms (maug-cc-pVTZ-$d$H). In this case, harmonic force fields have been obtained employing analytic first and second derivatives [@bicz10] available in the Gaussian 16 suite of programs [@fris16]. The reliability and effectiveness of this computational model in the evaluation of vibrational frequencies and intensities have been documented in several studies (see, for example, ref. ). We have also performed anharmonic calculations (at the B3LYP/6-31G(d) level) for the H$_2$O-CO and H$_2$O-NH$_3$ systems in order to check the effect of anharmonicity on the band strength profiles of the four water fundamental modes. The spectral features of the astrophysical ices can be altered in both active (direct) and passive (bulk) ways. Following a consolidated practice [@sanf20], to include the passive contribution of the bulk ice on the spectral properties of the ice mixtures considered, we embedded our explicit cluster in a continuum solvation field to represent local effects on the ice mixture. To this end, we resorted to the integral equation formalism (IEF) variant of the Polarizable Continuum Model (PCM) [@toma05]. The solute cavity has been built by using a set of interlocking spheres centered on the atoms with the following radii (in Å): $1.443$ for hydrogen, $1.925$ for carbon, $1.830$ for nitrogen, and $1.750$ for oxygen, each of them scaled by a factor of $1.1$, which is the default value in Gaussian. For the ice dielectric constant, that of bulk water ($\varepsilon=78.355$) has been used, although any dielectric constant larger than about $30$ would lead to very similar results. In addition, we have also performed QM/MM geometry optimizations of a pure water cluster containing 4 H$_2$O molecules, in which all but one molecule at the square vertexes were put in the MM layer (see Figure \[fig:qm-mm\], left panel). A pure water cluster system containing 20 H$_2$O molecules has also been considered. For this, we started from the coordinates of the full QM optimization and selected two alternative sets of four innermost molecules at the center of the cluster with a complete hydrogen bond network (determined with a geometric criterion [@pagl17]) with first neighbor water molecules; the remaining 16 molecules were described at the MM level (see Figure \[fig:qm-mm\], right panel). All QM/MM [@chun15] calculations were carried out with the Gaussian 16 [@fris16] code (rev. C01) using the hybrid B3LYP functional in conjunction with the 6-31 G(d) basis set. Atom types and force field parameters for water molecules in the MM layer were assigned according to the SPC-Fw flexible water model [@wu06]; the choice was driven by (i) the necessity for a flexible, 3-body classical water model and (ii) the accuracy with which the selected model reproduced ice I$_h$ properties. Solvent effects were mimicked by using PCM [@canc97]. The vibrational analysis results from QM/MM calculations are provided in Tables \[tab:4H2O\_MM-QM\], \[tab:20H2O\_MM-QM\_Conf\_1\], and \[tab:20H2O\_MM-QM\_Conf\_2\] in the SI. ![ [**Left:**]{} 4 water system; the single QM water molecule is depicted in ball and stick representation and the 3 MM molecules in licorice representation; O-H distances are indicated too. It is to be noted that the four H$_2$O can be considered equivalent, [**Right:**]{} Innermost water molecules described at QM level (ball and stick) and surrounding molecules described at MM level (lines) for the 20H$_2$O system. (a) Configuration 1. (b) Configuration 2.[]{data-label="fig:qm-mm"}](1QM_3MM.pdf){width="\textwidth"} ![ [**Left:**]{} 4 water system; the single QM water molecule is depicted in ball and stick representation and the 3 MM molecules in licorice representation; O-H distances are indicated too. It is to be noted that the four H$_2$O can be considered equivalent, [**Right:**]{} Innermost water molecules described at QM level (ball and stick) and surrounding molecules described at MM level (lines) for the 20H$_2$O system. (a) Configuration 1. (b) Configuration 2.[]{data-label="fig:qm-mm"}](ice20.pdf){width="\textwidth"} Experimental Methods {#sec:experiment} ==================== Literature laboratory data are here used whenever possible to constrain simulations [@bouw07; @ober07]. In the cases of formic acid, ammonia, and methanol in water ice, new experiments have been performed using the high vacuum (HV) Portable Astrochemistry Chamber (PAC) at the Open University (OU) in the United Kingdom. A detailed description of the system is reported elsewhere [@dawe16]. Briefly, the main chamber is a commercial conflat flange cube (Kimball Physics Inc.) connected to a turbo molecular pump (300 l/s), a custom made stainless steel dosing line through an all metal leak valve, a cold finger of a closed-cycle He cryostat (Sumitomo Cryogenics) and two ZnSe windows suitable for IR spectroscopy. During operation, the base pressure in the chamber is in the 10$^{-9}$ mbar range, and the base temperature of the cold finger is 20 K. In thermal contact with the cryostat, the substrate is a ZnSe window (20 mm x 2 mm). A DT-670 silicon diode temperature sensor (LakeShore Cryotronics) is connected to the substrate to measure its temperature, while a Kapton flexible heater (Omegalux) is used to change its temperature. Diode and heater are both connected to an external temperature controller (Oxford Instruments). Gaseous samples were prepared and mixed in a pre-chamber (dosing line) before being dosed into the main chamber through an all metal leak valve. A mass-independent pressure transducer was used to control the amount of gas components mixed in the pre-chamber. Chemicals were purchased at Sigma-Aldrich with the highest purity available \[HCOOH ($>$95%), NH$_3$ (99.95%), and CH$_3$OH (99.8%)\]. Ices were grown $in situ$ by direct vapour deposition onto the substrate at normal incidence via a 3 mm nozzle that is 20 mm away from the sample. Infrared spectroscopy was performed in transmission using a Fourier Transform infrared (FTIR; Nicolet Nexus 670) spectrometer with an external Mercury cadmium telluride (MCT) detector. A background spectrum comprising 512 co-added scans was acquired before deposition at 20 K and used as reference spectrum for all the spectra collected after deposition to remove all the infrared signatures along the beam pathway that were not originated by the ice sample. Each IR spectrum is a collection of 256 co-added scans. The IR path was purged with dry compressed air to remove water vapour. Results and Discussions {#sec:results_discussions} ======================= In this section, first of all, the pure water ice will be addressed in order to establish the best compromise between accuracy and computational cost for the description of the water ice unit cell. To this aim, we will resort on the comparison with experiment. Then, we will move to the ice containing impurities. To further proceed with the validation of our protocol, water ice containing HCOOH, NH$_3$, CH$_3$OH, CO, and CO$_2$ as impurities will be investigated, thus exploiting the comparison between experiment and computations. This will also involve, as mentioned above, new measurements. Finally, in the last part, our protocol will be extend to the study of water ices with H$_2$CO, CH$_4$, N$_2$, and O$_2$ as impurities. Part 1. Validation ------------------ ### Band strength of pure water {#band_strength_pure_water} In Table \[tab:band\_strength\_pure\_water\], the water band positions obtained with different methods and different sizes of the water cluster are compared with experimental data. Since computations provide several frequencies corresponding to a single mode of vibration, for the sake of comparison, we have reported the computed frequencies of the four fundamental modes after convolving them with a Gaussian function with an adequate width [@lica15] (all transition frequencies are collected in the Appendix, [ Table \[table:comparison-different-water-cluster\]]{}). The comparison of Table \[tab:band\_strength\_pure\_water\] is graphically summarized in Figure \[fig:histo-h2o-cluster-compare\]. The left panel shows the average deviation of the band position of three fundamental modes of water (libration, bending, and stretching) from the experimental counterpart [@gera95]. It is interesting to note that the band positions obtained using the tetramer configuration and the B3LYP/6-31G(d) level of theory provides the best agreement. The right panel shows the average deviation of the band strengths from experiments. QM/MM calculations for the 20 water-molecule cluster (as described in the computational details) show the minimum deviation from experimental data. The results obtained for the tetramer configuration, both at the B3LYP and B2PLYP level, also provide small deviations. Based on the results of the comparison carried out, the B3LYP/6-31G(d) level of theory and the tetramer configuration have been found to be a suitable combination to describe the water cluster with a limited computational cost. [ ]{} ![[Deviation of computed band positions (left panel) and band strengths (right panel) from experiments.]{}[]{data-label="fig:histo-h2o-cluster-compare"}](histo-h2o-cluster-compare.pdf "fig:"){width="8cm" height="6cm"} ![[Deviation of computed band positions (left panel) and band strengths (right panel) from experiments.]{}[]{data-label="fig:histo-h2o-cluster-compare"}](hist-band.pdf "fig:"){width="8cm" height="6cm"} [ In the following sections, the results for water ice with HCOOH and NH$_3$ as impurities are first reported and discussed, thereby exploiting the outcomes of new experiments. Then, we move to the CH$_3$OH-H$_2$O ice for which new experimental results have been obtained. For the last two cases addressed, namely CO-H$_2$O and CO$_2$-H$_2$O, the experimental data for the comparison have been taken from the literature. Unless otherwise stated, we use the c-tetramer configuration for the rest of our calculations.]{} ### HCOOH ice {#HCOOH_ice} Infrared spectra were measured for various mixtures of H$_2$O and HCOOH ice deposited at $20$ K, as explained in the experimental details section (see Section \[sec:experiment\]). These, normalized with respect to the O-H stretch, are shown in Figure \[fig:experiment\]a. A minor contamination due to CO$_2$ was detected in some experiments. In all experiments, the amount of CO$_2$ deposited in the ice was found to be between 1000 and more than 100 times less abundant than H$_2$O and HCOOH, respectively. Therefore, we do not expect that the CO$_2$ contamination affects the recorded IR spectra profiles. The mixture ratios were determined from the fit of the the spectrum of a selected mixture, the measurement of the area of the water band at $3333.33$ cm$^{-1}$ (3.00 $\mu$m), and the comparison with the pure water counterparts. For HCOOH, the absorption area is measured at 1700 cm$^{-1}$. In fact, HCOOH has the strongest mode at $1694.92$ cm$^{-1}$ ($5.90$ $\mu$m) which corresponds to its C=O stretching mode. But the feature overlaps with the position of the OH bending mode of solid water at $1666.67$ cm$^{-1}$ ($6.00$ $\mu$m). The contribution from the water bending mode at $\sim$1700 cm$^{-1}$ has been subtracted from the total area before the band strength mentioned above being used to calculate the amount of HCOOH in the ice mixture. The band strengths used here are $2.0\times10^{-16}$ for H$_2$O [@gera95] and $6.7\times10^{-17}$ for HCOOH [@mare87; @schu99]. Another relatively weaker mode of HCOOH at $1388.89$ cm$^{-1}$ ($7.20$ $\mu$m) was also considered because the corresponding region is free from interfering transitions [@schu99]. As seen in Figure \[fig:experiment\]a, the HCOOH:H$_2$O ratios cover the 0.05 to 3.46 range. In this respect, it is worthwhile noting that the abundances of solid phase HCOOH in the interstellar ices vary between 1% to 5% with respect to the $\rm{H_2O}$ ice [@biss07]. Moving to the computational study, Figure \[fig:optimized\_structure\]b shows how the HCOOH molecules are bonded to the water molecules to form the $4:4$ H$_2$O-$\rm{HCOOH}$ mixture used in our calculations. In the Appendix, the absorption band profiles of the H$_2$O-HCOOH clusters with different impurity concentrations are shown (see Figure \[fig:H2O-HCOOH\]). The transition frequencies and the corresponding strongest intensity values, obtained at the B3LYP/6-31G(d) level, are given in the Appendix (see Table \[tab:H2O\_X\]). Calculations have also been carried out using the B2PLYP functional, the results being summarized in Tables \[tab:4H2O\_B2PLYP\], \[tab:4H2O\_1HCOOH\_B2PLYP\], \[tab:4H2O\_2HCOOH\_B2PLYP\], \[tab:4H2O\_3HCOOH\_B2PLYP\], and \[tab:4H2O\_4HCOOH\_B2PLYP\] in the SI. To investigate how the band strength varies with impurity concentrations, the data are fitted with a linear function A$_{eff}$ = a$\cdot$\[X\] + b, where X = HCOOH, NH$_3$, $\rm{CH_3OH}$, CO, $\rm {CO_2}$, $\rm{H_2CO}$, CH$_4$, OCS, N$_2$, and O$_2$. The coefficient ‘a’ provides the information whether the band strength increases or decreases by increasing the concentration of X, \[X\], and the coefficient ‘b’ indicates the band strength of the vibration mode in the absence of impurities. The fitting coefficients, for all impurity considered, are provided in Table \[tab:linear\_coeff\]. In Figure \[fig:band\_strength\]a, the band strength profile as a function of the concentration of HCOOH is shown. ![(a) IR spectra for different HCOOH:H$_2$O ice mixtures deposited at T$=20$ K. (b) IR spectra for different NH$_3$:H$_2$O ice mixtures deposited at T$=20$ K. (c) IR spectra of different CH$_3$OH:H$_2$O ice mixtures deposited at T$=30$ K. The color legend is explained in the insets. All IR spectra are normalized with respect to the O-H stretching band.[]{data-label="fig:experiment"}](HCOOH.pdf "fig:"){width="8cm" height="6cm"} ![(a) IR spectra for different HCOOH:H$_2$O ice mixtures deposited at T$=20$ K. (b) IR spectra for different NH$_3$:H$_2$O ice mixtures deposited at T$=20$ K. (c) IR spectra of different CH$_3$OH:H$_2$O ice mixtures deposited at T$=30$ K. The color legend is explained in the insets. All IR spectra are normalized with respect to the O-H stretching band.[]{data-label="fig:experiment"}](NH3.pdf "fig:"){width="8cm" height="6cm"} 0.5cm ![(a) IR spectra for different HCOOH:H$_2$O ice mixtures deposited at T$=20$ K. (b) IR spectra for different NH$_3$:H$_2$O ice mixtures deposited at T$=20$ K. (c) IR spectra of different CH$_3$OH:H$_2$O ice mixtures deposited at T$=30$ K. The color legend is explained in the insets. All IR spectra are normalized with respect to the O-H stretching band.[]{data-label="fig:experiment"}](CH3OH.pdf "fig:"){width="8cm" height="6cm"} {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} ![[Band strengths of the four fundamental vibration modes of water for (a) $\rm{H_2O-HCOOH}$, (b) $\rm{H_2O-NH_3}$, (c) $\rm{H_2O-CH_3OH}$, (d) $\rm{H_2O-CO}$, (e) $\rm{H_2O-CO_2}$, (f) $\rm{H_2O-H_2CO}$, (g) $\rm{H_2O-CH_4}$, (h) $\rm{H_2O-OCS}$, (i) $\rm{H_2O-N_2}$, and (j) $\rm{H_2O-O_2}$ clusters with various concentrations. The water c-tetramer configuration was used for pure water.]{}[]{data-label="fig:band_strength"}](n2_auc_new.pdf){width="\textwidth"} ![[Band strengths of the four fundamental vibration modes of water for (a) $\rm{H_2O-HCOOH}$, (b) $\rm{H_2O-NH_3}$, (c) $\rm{H_2O-CH_3OH}$, (d) $\rm{H_2O-CO}$, (e) $\rm{H_2O-CO_2}$, (f) $\rm{H_2O-H_2CO}$, (g) $\rm{H_2O-CH_4}$, (h) $\rm{H_2O-OCS}$, (i) $\rm{H_2O-N_2}$, and (j) $\rm{H_2O-O_2}$ clusters with various concentrations. The water c-tetramer configuration was used for pure water.]{}[]{data-label="fig:band_strength"}](o2_auc_new.pdf){width="\textwidth"} ### NH$_3$ ice {#NH3_ice} Most of the intense modes of ammonia overlap with the dominant features due to water and silicates. However, when ammonia is mixed with $\rm{H_2O}$ ice, it forms hydrates that show an intense mode at $2881.84$ cm$^{-1}$ ($3.47$ $\mu$m) [@dart05], which lies in a relative clear region. Another characteristic feature of ammonia is the umbrella mode at $1111.11$ cm$^{-1}$ ($9.00$ $\mu$m), which is relatively intense, but it often overlaps with the $\rm{CH_3}$ rocking mode of methanol, thus leading to an overestimation of the abundance of ammonia. In this work, infrared spectra were recorded for various mixing ratios of H$_2$O-NH$_3$ ice deposited at 20 K. The IR spectra, normalized with respect to the most intense bend (i.e. the O-H stretching mode), are shown in Figure \[fig:experiment\]b. Mixing ratios were derived by measuring the areas of the selected bands for H$_2$O band (at 2220 cm$^{-1}$) [@mast09] and for NH$_3$ (umbrella mode band at 1070 cm$^{-1}$) [@dhen86], with a procedure analogous to that introduced in the previous section for HCOOH. Figure \[fig:optimized\_structure\]c shows the optimized geometry of the H$_2$O-$\rm{NH_3}$ system with a $4:4$ ratio as obtained from our quantum-chemical calculations. In the Appendix, Figure \[fig:H2O-NH3\] depicts the absorption band profiles of H$_2$O-$\rm{NH_3}$ mixtures with various concentrations. The transition frequencies and the corresponding intensity values are provided in the Appendix as well (see Table \[tab:H2O\_X\]). The vibrational analysis has also been carried out at a higher level of theory, thereby using the B2PLYP functional. The results are reported in Tables \[tab:4H2O\_1NH3\_B2PLYP\], \[tab:4H2O\_2NH3\_B2PLYP\], \[tab:4H2O\_3NH3\_B2PLYP\], and \[tab:4H2O\_4NH3\_B2PLYP\] in the SI. Figure \[fig:band\_strength\]b shows the band strengths as a function of the concentration of the impurity under consideration, i.e. NH$_3$. ### Comparison between experiment and simulations {#comparison} ![Comparison of computed and experimental IR spectra (0 - 4000 cm$^{-1}$) for pure water as well as water with HCOOH and NH$_3$ as impurities. We have used harmonic frequencies for the computed spectra and the intensity is scaled with a factor 1000 to have best match with the experimental one.[]{data-label="fig:comparison_IR"}](comparison_IR.pdf){width="\textwidth"} In Figure \[fig:comparison\_IR\], the comparison between experimentally obtained spectra and our computed spectra for pure water, H$_2$O-HCOOH mixture, and H$_2$CO-NH$_3$ mixture is shown. We note a good agreement between experimental and theoretical absorption spectra. Figure \[fig:HCOOH-NH3\_band\_strength\] shows the comparison between the experimental (dotted lines) and theoretical (solid and dashed lines) band strengths of the four water bands as a function of the concentration of HCOOH and NH$_3$. From Figure \[fig:HCOOH-NH3\_band\_strength\], it is evident that the experimental strength of the libration and bending modes increases by increasing the concentration of HCOOH. On the contrary, the strength of the stretching and free OH modes shows a decreasing trend. These behaviors should be compared with the B2PLYP/mug-cc-pVTZ (dashed) and B3PLYP/6-31G(d) (solid) trends. For the libration and bending band strength profiles, there is a qualitative agreement with experiments. In the case of the stretching and free OH modes, theoretical band strength profiles deviate from experimental work. The lack of experimental data in the 3600-4000 cm$^{-1}$ range (see Figure \[fig:experiment\]a) may have contributed to this disagreement. Concerning the comparison of the two levels of theory, it is noted that there is a rather good agreement. In case of H$_2$O-HCOOH mixture, HCOOH can act as both hydrogen bond donor and hydrogen bond acceptor. We considered both the interactions and noted that, if we consider HCOOH as H-bond acceptor, the band strength of three modes (libration, bending, and stretching) are lower with respect to case where HCOOH was treated as H-bond donor. But in the case of the free-OH mode, the band strength slope increases (See [ Figure \[fig:hcooh-donor-acceptor\]]{} in the Appendix). 0.8cm ![Comparison between the calculated and experimental band strength profiles with various concentration of HCOOH and NH$_3$.[]{data-label="fig:HCOOH-NH3_band_strength"}](HCOOH_Band_Strength_Comparison.pdf "fig:"){width="\textwidth"} ![Comparison between the calculated and experimental band strength profiles with various concentration of HCOOH and NH$_3$.[]{data-label="fig:HCOOH-NH3_band_strength"}](NH3_Band_Strength_Comparison.pdf){width="\textwidth"} Moving to ammonia, the experimental data of Figure \[fig:HCOOH-NH3\_band\_strength\] show that the band strength of the free-OH stretching mode nearly vanishes when a 50% concentration of the impurity (NH$_3$) is reached. This feature interestingly supports our calculated spectra shown in the Appendix (see [ Figure \[fig:H2O-NH3\]]{} last panel). Libration and bending modes have, instead, an opposite trend, with the band strength increasing by increasing the concentration of NH$_3$. The band strength of the stretching mode shows a slightly decreasing trend with the concentration of NH$_3$. From the inspection of Figure \[fig:HCOOH-NH3\_band\_strength\] it is evident that both sets of theoretical results (B3LYP and B2PLYP) are in reasonably good agreement with experimental data for the libration, bending, and free-OH modes. Interestingly, the results obtained using the lower level of theory are in better agreement with experiments. In [ Figure \[fig:harm-anharm-compare\]]{} (in the Appendix), the comparison of band strengths evaluated using (a) harmonic and (b) anharmonic calculations is shown. To investigate the effect of anharmonicity on the band strengths, we have only considered fundamental bands in the 0 to 3600 cm$^{-1}$ frequency range. From our experimental study on the H$_2$O-NH$_3$ system, as already mentioned, we obtained an increasing trend of the band strength for the libration, bending, and stretching modes with the increase in concentration of NH$_3$, whereas the band strength decreases for the free OH mode and tends to zero with 50% concentration of NH$_3$. When using harmonic calculations for all four fundamental modes, trends similar to what obtained from experiment were found. But, if we consider anharmonic calculations, only the behavior of the stretching mode is well reproduced. All other modes deviate from the experimental results. While not claiming that harmonic calculations are better than the anharmonic ones, this comparison seems to suggest that the former show a better error compensation. A similar outcome has been obtained for the H$_2$O-CO system and will be briefly addressed later in the text. Based on the comparisons discussed above, the B3LYP/6-31G(d) level of theory provides reliable results. Therefore, it has been employed in the following investigations. First of all, the comparison between computed and experimental band strengths for the H$_2$O-CH$_3$OH, CO-H$_2$O, and CO$_2$-H$_2$O mixtures will be considered to further support its suitability. ### CH$_3$OH ice {#CH3OH_ice} In this work, the effect of the CH$_3$OH concentration on the band profiles of water ice has been experimentally investigated. In the case of methanol, CO$_2$ gas is still present in the system (i.e., outside the vacuum chamber) in quantities that vary in time causing negative and/or positive contributions to CO$_2$ gas-phase absorption features with respect to the background spectrum, as evident in Figure \[fig:experiment\]c at $\sim$2340 cm$^{-1}$. Such contamination is most likely due to the dosing line, but its negligible amount should not affect the final results. Figure \[fig:experiment\]c shows the experimental absorption spectra for various CH$_3$OH-H$_2$O ice mixtures deposited at T$=30$ K. The spectra are normalized to 1 with respect to the maximum of the O-H stretch band. ![Comparison between calculated and experimental band strength profiles as a function of CH$_3$OH concentration. [Stars represent the experimental data points.]{}[]{data-label="fig:CH3OH-BS-COMP"}](BS-COMP-METHANOL.pdf){width="\textwidth"} Figure \[fig:optimized\_structure\]d shows the optimized structure of the $\rm{H_2O}$-$\rm{CH_3OH}$ mixture with a $4:4$ concentration ratio. It is noted that a weak hydrogen bond is expected to be formed. The simulated IR spectra for different concentrations are shown in [ Figure \[fig:H2O-CH3OH\]]{} (in the Appendix). Peak positions, integral absorption coefficients, and band assignments for various H$_2$O-CH$_3$OH mixtures are collected in [ Table \[tab:H2O\_X\]]{} in the Appendix. The computed band strengths as a function of different concentrations are shown in Figure \[fig:band\_strength\]c. The computed strength of the bending mode gradually increases with $\rm{CH_3OH}$ concentration (see Figure \[fig:CH3OH-BS-COMP\]; right panel), which is in qualitative agreement with the experimental results [@dawe16]. In case of the stretching mode, computationally, a slight increasing trend of the band strength is noted, whereas experimental results show an opposite trend (see Figure \[fig:CH3OH-BS-COMP\]; left panel). Because of the lack of experimental spectra, we cannot compare the band strength of the libration and free OH modes. In case of H$_2$O-CH$_3$OH mixtures, methanol can act as both hydrogen bond donor and hydrogen bond acceptor. We considered both possibilities and found that if we consider methanol as hydrogen bond donor, the band strength of all four modes show an increasing trend. On the other hand, if we consider methanol as hydrogen bond acceptor, the band strengths of three modes, namely libration, bending, and stretching, present trends similar to the previous case (where methanol acts as hydrogen bond donor), while the free-OH band shows a less pronounced behavior (see [ Figure \[fig:ch3oh-donor-acceptor\]]{} in the Appendix). ### CO ice {#CO_ice} Figure \[fig:optimized\_structure\]e depicts the H$_2$O-CO optimized structure with a $4:4$ concentration ratio: the four CO molecules interact with the H atoms of the water molecules not involved in the hydrogen bond (interaction of the O atom of CO with the hydrogen atom of water). However, for the H$_2$O-CO system, the interaction can take place through both O and C of CO with the hydrogen atom of H$_2$O [@zami18]. We have considered both types of interaction and evaluated their effects on the band strengths. However we did not find any significant difference. Thus, we only discuss the band strength of the H$_2$O-CO mixture with the interaction on the O side of CO. [For the sake of completeness, it should be mentioned that there is also another type of interaction, which occurs between the $\pi$ bond of CO and one water-hydrogen, and it gives rise to a T shaped complex [@coll14]. However, according to a computational study by Collings et al. [@coll14], this has a negligible effect on IR vibrational bands. As a consequence we have not investigated in detail this kind of complex.]{} The simulated IR absorption spectra of the four fundamental vibrational modes for various compositions are shown in the Appendix (see [Figure \[fig:H2O-CO\]]{}). The four fundamental frequencies of water ice change significantly by increasing the concentration of CO. The most intense peak positions and the corresponding integral abundance coefficients for different H$_2$O-CO mixtures are provided in the Appendix (see [Table \[tab:H2O\_X\]]{}). In Figure \[fig:band\_strength\]d, the integrated intensities of water vibrational modes are plotted as a function of the CO concentration. It is noted that the strength of the libration, bending, and stretching modes decreases with the concentration of CO. The free OH mode shows instead a sharp increase of the band strength when increasing the CO concentration. In Table \[tab:linear\_coeff\], the resulting linear fit coefficients are collected together with the available experimental values for H$_2$O-CO mixtures deposited at $15$ K [@bouw07]. It is noted that theoretical band strength slopes are in rather good agreement with experimental results [@bouw07]. For the H$_2$O-CO system, anharmonic calculations have also been carried out. While the band strengths of the bending and stretching modes have a similar trend as experimental data, a deviation is noted for the libration mode (see [Figure \[fig:harm-anharm-compare\]]{} in the Appendix). To check the effect of dispersion, B3LYP-D3/6-31G(d) calculations have been performed, with D3 denoting the correction for dispersion effects [@grim10]. B3LYP-D3 calculations have been carried out for H$_2$O-CO, H$_2$O-CH$_4$, H$_2$O-N$_2$, and H$_2$O-O$_2$ systems. In Figure \[fig:comp-norm-dis\]a, we have shown the comparison of the band strengths of different vibrational modes of water with and without the dispersion correction for the H$_2$O-CO system. The overall conclusion is that there is a good agreement with the experimental band strengths when dispersion effect is not considered. On the contrary, when the dispersion correction is included, our computed band strength profile shows a different trend. The libration and bending modes present a positive slope with the increase in impurity concentration, whereas experimental results show a negative slope. For the free OH mode, a slight increasing trend of the band strength is obtained, whereas the experimental band strength presents a sharp increase with the concentration of CO. The band strength of the stretching mode has a similar behaviour with dispersion and without dispersion, and in agreement with the experimental result [@bouw07](see Figure 3). Thus, in summary, while we are not claiming that the dispersion effects are not important for the systems investigated, we have noted that neglecting them we obtain a consistent description of the experimental behaviour (probably due to a fortuitous errors compensation). ![ [Comparison of the band strengths of the four fundamental modes of water for various mixtures of (a) H$_2$O-CO, (b) H$_2$O-CH$_4$, (c) H$_2$O-N$_2$, and (d) H$_2$O-O$_2$ by considering or not the dispersion effect.]{}[]{data-label="fig:comp-norm-dis"}](comp-norm-disp-co-bs.pdf "fig:"){width="8.1cm"} ![ [Comparison of the band strengths of the four fundamental modes of water for various mixtures of (a) H$_2$O-CO, (b) H$_2$O-CH$_4$, (c) H$_2$O-N$_2$, and (d) H$_2$O-O$_2$ by considering or not the dispersion effect.]{}[]{data-label="fig:comp-norm-dis"}](comp-norm-disp-ch4-bs.pdf "fig:"){width="8.1cm"} ![ [Comparison of the band strengths of the four fundamental modes of water for various mixtures of (a) H$_2$O-CO, (b) H$_2$O-CH$_4$, (c) H$_2$O-N$_2$, and (d) H$_2$O-O$_2$ by considering or not the dispersion effect.]{}[]{data-label="fig:comp-norm-dis"}](comp-norm-disp-n2-bs.pdf "fig:"){width="8.1cm"} ![ [Comparison of the band strengths of the four fundamental modes of water for various mixtures of (a) H$_2$O-CO, (b) H$_2$O-CH$_4$, (c) H$_2$O-N$_2$, and (d) H$_2$O-O$_2$ by considering or not the dispersion effect.]{}[]{data-label="fig:comp-norm-dis"}](comp-norm-disp-o2-bs.pdf "fig:"){width="8.1cm"} 0.2cm [**Notes.**]{} [Experimental values are provided in the parentheses. $^a$This work.]{} ### CO$_2$ ice {#CO2_ice} Figure \[fig:optimized\_structure\]f shows the optimized geometry of the $4:4$ mixture of H$_2$O:$\rm{CO_2}$. The absorption features of water ice for different $\rm{CO_2}$ concentrations are shown in the Appendix (see [Figure \[fig:H2O-CO2\]]{}). The most intense frequencies for the various H$_2$O-CO$_2$ mixtures are summarized in the Appendix as well (see [Table \[tab:H2O\_X\]]{}). The trend of the band strength as a function of CO$_2$ concentrations is shown in Figure \[fig:band\_strength\]e. For the free-OH mode, a rapid increase with CO$_2$ concentration is noted, which is in good agreement with the experimental results by @ober07. Computed band strengths of the libration and bending modes also increase by increasing the $\rm{CO_2}$ concentration, which is however in contrast with the available experimental data [@ober07]. The band strength of the bulk stretching mode decreases instead with $\rm{CO_2}$ concentration, in reasonable good agreement with the available experiments [@ober07]. FTIR spectroscopy of the matrix-isolated molecular complex H$_2$O-CO$_2$ shows that CO$_2$ does not form a weak hydrogen bond with H$_2$O [@tso85], but instead CO$_2$ destroys the bulk hydrogen bond network. This may cause a large decrease in the band strength of the bulk stretching mode, while the intermolecular O-H bond strength increases with the CO$_2$ concentration. Therefore, the disagreement between calculated and experimental band strengths could be thus due to the cluster size of water molecules. In Table \[tab:linear\_coeff\], the resulting linear fit coefficients are reported, together with the available experimental values for the H$_2$O-CO$_2$ mixture deposited at $15$ K [@ober07]. Part 2. Applications -------------------- The results discussed in previous sections suggest that the water c-tetramer structure together with harmonic B3LYP/6-31G(d) calculations are able to predict the experimental results presented here as well as literature data. Thus, to study the effect of other impurities ($\rm{H_2CO}$, CH$_4$, OCS, N$_2$, and O$_2$) on pure water ice, we have further exploited this methodology. Additionally, the effect of impurities on the band strengths of the four fundamental bands has also been studied by considering the c-hexamer (chair) structure and the corresponding results are provided in the SI (see Figure \[fig:6h2o\_x\_band\_strength\]). ### H$_2$CO ice {#H2CO_ice} The strongest modes of formaldehyde ($\rm{H_2CO}$) lie at $1724.14$ cm$^{-1}$ ($5.80$ $\mu$m) and $1497.01$ cm$^{-1}$ ($6.68$ $\mu$m). Figure \[fig:optimized\_structure\]g depicts the optimized structure of the $4:4$ $\rm{H_2O-H_2CO}$ mixture. The desired ratio is attained upon formation of the hydrogen bond between the O atom of $\rm{H_2CO}$ and the dangling H atoms of $\rm{H_2O}$. The effect of formaldehyde on water IR spectrum is shown in the Appendix (see [Figure \[fig:H2O-H2CO\]]{}). Frequencies, integral absorption coefficients, and mode assignments are reported in the Appendix as well (see [Table \[tab:H2O\_X\]]{}). The band strength profiles as a function of the concentration of H$_2$O are shown in Figure \[fig:band\_strength\]f. Similar to the methanol-water mixture, all band strengths are found to increase with the concentration of formaldehyde, the free-OH stretching mode being the most affected. ### CH$_4$ ice {#CH4_ice} $\rm{CH_4}$ cannot be observed by means of rotational spectroscopy since it has no permanent dipole moment. The optimized structure of the $\rm{H_2O-CH_4}$ system with a $4:4$ ratio is shown in Figure \[fig:optimized\_structure\]h. The absorption IR spectra for different $\rm{H_2O-CH_4}$ mixtures are depicted in the Appendix (see [Figure \[fig:H2O-CH4\]]{}). Peak positions, integral absorption coefficients, and band assignments are provided in the Appendix as well (see [Table \[tab:H2O\_X\]]{}). Figure \[fig:band\_strength\]g shows the band strength variations with the concentration of $\rm{CH_4}$. All band strengths marginally increase with the $\rm{CH_4}$ concentration. Figure \[fig:comp-norm-dis\]b shows the comparison of the band strengths with and without the incorporation of corrections for accounting for dispersion effects. For all the four fundamental modes, differences are minor. ### OCS ice {#OCS_ice} @garo10 proposed that carbonyl sulfide (OCS) is a key ingredient of the grain surface. Its abundance in ice phase may vary between 0.05 and $0.15$% [@dart05]. Figure \[fig:optimized\_structure\]i shows the optimized structure of the 4:4 H$_2$O-OCS. Since oxygen is more electronegative than sulfur, the O atom of the OCS molecule is hydrogen-bonded to the water free-hydrogens. In the Appendix, [Figure \[fig:H2O-OCS\]]{} shows the absorption IR band spectra for H$_2$O-OCS clusters with various concentrations. Figure \[fig:band\_strength\]h depicts the band strengths as a function of the concentration of OCS. Here, the free-OH mode is the most affected and its band strength increases with the concentration of OCS. All other modes roughly remain invariant by varying the amount of impurity. ### N$_2$ ice {#N2_ice} N$_2$ is a stable homonuclear molecule and, due to its symmetry, it is infrared inactive. However, when embedded in an ice matrix, the crystal field breaks the symmetry, and an infrared transition is activated around $2325.58$ cm$^{-1}$ ($4.30$ $\mu$m). Figure \[fig:optimized\_structure\]j shows the optimized geometry of the H$_2$O-$\rm{N_2}$ system with a $4:4$ ratio. The IR absorption spectra of water ice containing different amounts of $\rm{N_2}$ are shown in the Appendix (see [Figure \[fig:H2O-N2\]]{}). The corresponding peak frequencies and intensities are provided in the Appendix as well (see Table \[tab:H2O\_X\]). The dependence of the band strengths on the N$_2$ concentration is depicted in Figure \[fig:band\_strength\]i. It has been found that the slope of the band strength of the libration mode decreases, whereas the bending, stretching, and free OH modes show an increasing trend with the concentration of N$_2$. The linear fitting coefficients are provided in Table \[tab:linear\_coeff\]. Figure \[fig:comp-norm-dis\]c shows the comparison of band strengths with and without considering the dispersion effects. It is noted that the inclusion of dispersion effect leads to small changes. ### O$_2$ ice {#O2_ice} Analogously to $\rm{N_2}$, $\rm{O_2}$ is a homonuclear molecule, which is infrared inactive except when it is embedded in an ice matrix [@ehre92; @ehre98], thus giving rise to an absorption band around $1550.39$ cm$^{-1}$ ($6.45$ $\mu$m). $\rm{O_2}$ ice is not much abundant because the largest part of the oxygen budget in the dense molecular clouds is locked in the form of $\rm{CO_2}$, CO, water ice, and silicates. The optimized geometry of the $4:4$ H$_2$O-$\rm{O_2}$ ratio is shown in Figure \[fig:optimized\_structure\]k. IR spectra for different concentrations (Figure \[fig:H2O-O2\]) and the corresponding peak frequencies and intensities (Table \[tab:H2O\_X\]) are provided in the Appendix. The dependence of band strengths upon O$_2$ concentration is shown in Figure \[fig:band\_strength\]j. Similarly to the $\rm{N_2}$-water case, the free-OH mode is the most affected. The slope of the band strength of the libration and bending modes decreases, whereas the stretching and free-OH modes show an increasing trend with the concentration of O$_2$. The fitting coefficients for different H$_2$O-N$_2$ mixtures are provided in Table \[tab:linear\_coeff\]. Figure \[fig:comp-norm-dis\]d depicts the comparison of the band strengths with and without the inclusion of dispersion effects for the H$_2$O-O$_2$ system. It is evident that trend of the band strength with the impurity concentration slightly increases for the libration mode, whereas slightly decreases for the stretching mode when corrections for dispersion effects are present. In the case of the bending mode, the band strength rapidly increases, whereas the band strength rapidly decreases for the free OH mode. ### Comparison between various mixtures {#comparison_between_various_mixtures} 0.8cm ![Top panel: Effect of impurities on the four fundamental vibrational modes of water. Bottom panel: Comparison of the band strengths for the four fundamental vibrational modes as affected by impurities.[]{data-label="fig:comparison_between_four_modes"}](four_mode_x.pdf "fig:"){width="\textwidth"} 0.8cm ![Top panel: Effect of impurities on the four fundamental vibrational modes of water. Bottom panel: Comparison of the band strengths for the four fundamental vibrational modes as affected by impurities.[]{data-label="fig:comparison_between_four_modes"}](histogram.pdf "fig:"){width="\textwidth"} To compare the effect of all impurities considered in this study on the band strength, we have plotted the band profiles of the four fundamental modes of water ice as a function of the concentration of impurities, the results being shown in Figure \[fig:comparison\_between\_four\_modes\], top panel. For all fundamental modes, band strengths increase with the concentration of $\rm{CH_3OH}$, $\rm{H_2CO}$, HCOOH, $\rm{CH_4}$. To better understand their effect, in Figure \[fig:comparison\_between\_four\_modes\], bottom panel, we report the relative band strengths for the $4:4$ ratio mixtures. From this, it is clear that the libration, bending, and stretching modes are mostly affected by formic acid, while the free-OH mode is mostly affected by formaldehyde. An interesting feature is found for the free-OH mode for the $\rm{NH_3-H_2O}$ system. By increasing the $\rm{NH_3}$ concentration with respect to pure water, the band strength of the free-OH mode decreases and disappears when the 4 : 4 concentration ratio is reached. Figure \[fig:optimized\_structure\_6H2O\] (in the SI) depicts the optimized structures of the pure water c-hexamer (chair) configuration along with those obtained for a 6:1 concentration ratio. [Figure \[fig:6h2o\_x\_band\_strength\]]{} (in the SI) collects the results for the band strength variations for the c-hexamer (chair) water cluster configuration is considered, this being analogous to Figure \[fig:band\_strength\]. The geometries of water clusters containing $20$ water molecules with HCOOH as an impurity in various concentrations are shown in Figure \[fig:20H2O-HCOOH\] (in the SI) and the corresponding variations of the band strengths with increasing concentration of HCOOH are depicted in Figure \[fig:water\_clusters-HCOOH\] (in the SI). This figure also reports the comparison of band strength profiles for different water clusters. The structures of the 20-water-molecule cluster have been taken from @shim18, and were obtained by MD-annealing calculations using classical force-fields to reproduce a water cluster as a model of the ASW surface. The comparison shown in Figure \[fig:water\_clusters-HCOOH\] (in the SI) demonstrates that the 4H$_2$O model provides results similar to those obtained with 6 and 20 water molecules. This furthermore confirms the validity of our approach. Conclusions {#sec:conclusions} =========== Water ice is known to be the major constituent of interstellar icy grain mantles. Interestingly, there have been several astronomical observations [@boog00; @kean01] of the OH stretching and HOH bending modes at $3278.69$ cm$^{-1}$ ($3.05$ $\mu$m) and $1666.67$ cm$^{-1}$ ($6.00$ $\mu$m), respectively. It is noteworthy that the intensity ratio of these two bands is very different from what obtained in laboratory experiments for pure water ice. This suggested that the presence of impurities in water ice affects the spectroscopic features of water itself. For this reason, a series of laboratory experiments were carried out in order to explain the discrepancy between observations and experiments. Furthermore, these observations prompted us to perform an extensive computational investigation aiming to evaluate the effect of different amounts of representative impurities on the band strengths and absorption band profiles of interstellar ice. We selected the most abundant impurities ($\rm{HCOOH, \ NH_3, \ CH_3OH, \ CO, \ CO_2, \ H_2CO, \ CH_4, \ OCS, \ N_2}$, and $\rm{O_2}$) and studied their effect on four fundamental vibrational bands of pure water ice by employing different cluster models. Indeed, both the experimental and theoretical peak positions might differ from the astronomical observations. This is because the grain shape, size, and constituents, the surrounding physical conditions, and the presence of impurities play a crucial role in tuning the ice spectroscopic features.\ Although most of the computations were performed for a cluster containing only four water molecules as a model system (to find a trend in the absorption band strength), we demonstrated that increasing the size of the cluster would change the band strength profile only marginally. From the band strength profiles shown in Figure \[fig:CH3OH\_varied\_cluster\_size\] (in the SI), it is apparent that the stretching mode is the most affected and the bending mode is the least affected by the presence of impurities. Libration, bending, and bulk stretching modes were found to be most affected by HCOOH impurity, followed by $\rm{CH_3OH}$ and $\rm{H_2CO}$. Another interesting point to be noted is that the band strength of the free-OH stretching mode decreases with increasing concentration of $\rm{NH_3}$ and completely vanishes when the concentration of NH$_3$ becomes 50%. Most interestingly, the experimental free-OH band profile shows a decreasing trend when water is mixed with $\rm{NH_3}$ (Figure \[fig:HCOOH-NH3\_band\_strength\], right panel), similarly to that obtained computationally.\ Finally, our computed and laboratory absorption spectra of water-rich ices will be part of a larger infrared ice database in support of current and future observations. Understanding the effect of impurities in interstellar polar ice analogs will be pivotal to support the unambiguous identification of COMs in interstellar ice mantles by using future space missions such as JWST [@gibb04]. Acknowledgment ============== PG acknowledges the support of CSIR (Grant No. 09/904(0013) 2K18 EMR-I). MS gratefully acknowledges DST-INSPIRE Fellowship \[IF160109\] scheme. AD acknowledges ISRO respond (Grant No. ISRO/RES/2/402/16-17). This research was possible in part due to a Grant-In-Aid from the Higher Education Department of the Government of West Bengal. SI acknowledges the Royal Society for financial support. ZK was supported by VEGA – The Slovak Agency for Science, Grant No. 2/0023/18. This work was also supported by COST Action TD1308 – ORIGINS. Supporting Information (SI) =========================== Optimized structures of water clusters and impurities mixed with a 6:1 concentration ratio (Figure \[fig:optimized\_structure\_6H2O\]); band strengths of the four fundamental vibration modes of water clusters containing impurities with various concentrations (Figure \[fig:6h2o\_x\_band\_strength\]); structure of water clusters containing 20 H$_2$O molecules with HCOOH as impurity in different concentration ratio (Figure \[fig:20H2O-HCOOH\]); comparison of the band strength of various water clusters mixed with HCOOH (Figure \[fig:water\_clusters-HCOOH\]); effect of the cluster size on the band strength profile (Figure \[fig:CH3OH\_varied\_cluster\_size\]); harmonic infrared frequencies and intensities of the 4H$_2$O cluster (Table \[tab:4H2O\_B2PLYP\]), 4H$_2$O/HCOOH (Table \[tab:4H2O\_1HCOOH\_B2PLYP\]), 4H$_2$O/2HCOOH (Table \[tab:4H2O\_2HCOOH\_B2PLYP\]), 4H$_2$O/3HCOOH (Table \[tab:4H2O\_3HCOOH\_B2PLYP\]), 4H$_2$O/4HCOOH (Table \[tab:4H2O\_4HCOOH\_B2PLYP\]), 4H$_2$O/NH$_3$ (Table \[tab:4H2O\_1NH3\_B2PLYP\]), 4H$_2$O/2NH$_3$ (Table \[tab:4H2O\_2NH3\_B2PLYP\]), 4H$_2$O/3NH$_3$ (Table \[tab:4H2O\_3NH3\_B2PLYP\]), 4H$_2$O/4NH$_3$ (Table \[tab:4H2O\_4NH3\_B2PLYP\]) evaluated at the B2PLYP/maug-cc-pVTZ level; harmonic infrared frequencies and intensities of the H$_2$O$^{QM}$ + 3 H$_2$O$^{MM}$ complex (Table \[tab:4H2O\_MM-QM\]); harmonic infrared frequencies and intensities of the first 4 H$_2$O$^{QM}$+16 H$_2$O$^{MM}$: complex configuration 1 (Table \[tab:20H2O\_MM-QM\_Conf\_1\]) and complex configuration 2 (Table \[tab:20H2O\_MM-QM\_Conf\_2\]); geometric details of optimized structures of water clusters and impurities mixed with 4:4 and 6:1 concentration ratios (Optimized-Structures.zip). 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Solid filled squares represent the data sets where we considered anharmonic frequencies and the corresponding fitted results are the dotted lines.[]{data-label="fig:harm-anharm-compare"}](CO_BS_harm_anharm.pdf "fig:"){width="8cm" height="6cm"} ![The filled circles are the data points where we considered harmonic frequencies and the corresponding fitted profiles are the solid lines. Solid filled squares represent the data sets where we considered anharmonic frequencies and the corresponding fitted results are the dotted lines.[]{data-label="fig:harm-anharm-compare"}](NH3_BS_harm_anharm.pdf "fig:"){width="8cm" height="6cm"} ![ Band strength for H$_2$O-HCOOH mixtures: (a) HCOOH as hydrogen bond donor, and (b) HCOOH as hydrogen bond acceptor.[]{data-label="fig:hcooh-donor-acceptor"}](HCOOH_BS_donor.pdf "fig:"){width="8cm" height="6cm"} ![ Band strength for H$_2$O-HCOOH mixtures: (a) HCOOH as hydrogen bond donor, and (b) HCOOH as hydrogen bond acceptor.[]{data-label="fig:hcooh-donor-acceptor"}](HCOOH_BS_acceptor.pdf "fig:"){width="8cm" height="6cm"} ![Band strength for H$_2$O-CH$_3$OH mixtures: (a) CH$_3$OH as hydrogen bond donor, and (b) CH$_3$OH as hydrogen bond acceptor.[]{data-label="fig:ch3oh-donor-acceptor"}](CH3OH_BS_donor.pdf "fig:"){width="8cm" height="6cm"} ![Band strength for H$_2$O-CH$_3$OH mixtures: (a) CH$_3$OH as hydrogen bond donor, and (b) CH$_3$OH as hydrogen bond acceptor.[]{data-label="fig:ch3oh-donor-acceptor"}](CH3OH_BS_acceptor.pdf "fig:"){width="8cm" height="6cm"} ![Absorption spectra of the four modes for water ice for the five measured compositions, ranging from pure water ice (top) to 4:4 H$_2$O-HCOOH mixture (bottom). Black line represent the absorbance spectra of various concentration of H$_2$O-HCOOH, where HCOOH is used as hydrogen bond donor and for red line HCOOH is used as hydrogen bond acceptor.[]{data-label="fig:H2O-HCOOH"}](H2O-HCOOH.pdf){width="\textwidth"} ![Absorption spectra of the four modes for water ice for the five measured compositions, ranging from pure water ice (top) to 4:4 H$_2$O-NH$_3$ mixture (bottom).[]{data-label="fig:H2O-NH3"}](H2O-NH3.pdf){width="\textwidth"} ![Absorption spectra of the four modes for water ice for the five measured compositions, ranging from pure water ice (top) to 4:4 H$_2$O-CH$_3$OH mixture (bottom). Black line represent the absorbance spectra of various concentration of H$_2$O-CH$_3$OH, where CH$_3$OH is used as hydrogen bond donor and for red line CH$_3$OH is used as hydrogen bond acceptor.[]{data-label="fig:H2O-CH3OH"}](H2O-CH3OH.pdf){width="\textwidth"} ![Absorption spectra of the four modes for water ice for the five measured compositions, ranging from pure water ice (top) to 4:4 H$_2$O-CO mixture (bottom).[]{data-label="fig:H2O-CO"}](H2O-CO.pdf){width="\textwidth"} ![Absorption spectra of the four modes for water ice for the five measured compositions, ranging from pure water ice (top) to 4:4 H$_2$O-CO$_2$ mixture (bottom).[]{data-label="fig:H2O-CO2"}](H2O-CO2.pdf){width="\textwidth"} ![Absorption spectra of the four modes for water ice for the five measured compositions, ranging from pure water ice (top) to 4:4 H$_2$O-H$_2$CO mixture (bottom).[]{data-label="fig:H2O-H2CO"}](H2O-H2CO.pdf){width="\textwidth"} ![Absorption spectra of the four modes for water ice for the five measured compositions, ranging from pure water ice (top) to 4:4 H$_2$O-CH$_4$ mixture (bottom).[]{data-label="fig:H2O-CH4"}](H2O-CH4.pdf){width="\textwidth"} ![Absorption spectra of the four modes for water ice for the five measured compositions, ranging from pure water ice (top) to 4:4 H$_2$O-OCS mixture (bottom).[]{data-label="fig:H2O-OCS"}](H2O-OCS.pdf){width="\textwidth"} ![Absorption spectra of the four modes for water ice for the five measured compositions, ranging from pure water ice (top) to 4:4 H$_2$O-N$_2$ mixture (bottom).[]{data-label="fig:H2O-N2"}](H2O-N2.pdf){width="\textwidth"} ![Absorption spectra of the four modes for water ice for the five measured compositions, ranging from pure water ice (top) to 4:4 H$_2$O-O$_2$ mixture (bottom).[]{data-label="fig:H2O-O2"}](H2O-O2.pdf){width="\textwidth"} [ ]{} [ 0.2cm Notes: $^t$OH torsion; $^b$OH scissoring; $^s$OH stretching; $^f$free OH. For the conversion of km/mol to cm molecule$^{-1}$, intensity values need to multiply by a factor $\rm{1.6603\times10^{-19}}$.]{} ![Optimized structures of (a) pure water, (b) $\rm{H_2O-HCOOH}$, (c) $\rm{H_2O-NH_3}$, (d) $\rm{H_2O-CH_3OH}$, (e) $\rm{H_2O-CO}$, (f) $\rm{H_2O-CO_2}$, (g) $\rm{H_2O-H_2CO}$, (h) $\rm{H_2O-CH_4}$, (i) $\rm{H_2O-OCS}$, (j) $\rm{H_2O-N_2}$, and (k) $\rm{H_2O-O_2}$ clusters with a $6:1$ concentration ratio.[]{data-label="fig:optimized_structure_6H2O"}](6h2o_opt_x.pdf){width="\textwidth"} {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} ![Band strengths of the four fundamental vibration modes of water for (a) $\rm{H_2O-HCOOH}$, (b) $\rm{H_2O-NH_3}$, (c) $\rm{H_2O-CH_3OH}$, (d) $\rm{H_2O-CO}$, (e) $\rm{H_2O-CO_2}$, (f) $\rm{H_2O-H_2CO}$, (g) $\rm{H_2O-CH_4}$, (h) $\rm{H_2O-OCS}$, (i) $\rm{H_2O-N_2}$, and (j) $\rm{H_2O-O_2}$ clusters with various concentrations. The water c-hexamer (chair) configuration gas been used for pure water.[]{data-label="fig:6h2o_x_band_strength"}](6h2o_n2_auc_new.pdf){width="\textwidth"} ![Band strengths of the four fundamental vibration modes of water for (a) $\rm{H_2O-HCOOH}$, (b) $\rm{H_2O-NH_3}$, (c) $\rm{H_2O-CH_3OH}$, (d) $\rm{H_2O-CO}$, (e) $\rm{H_2O-CO_2}$, (f) $\rm{H_2O-H_2CO}$, (g) $\rm{H_2O-CH_4}$, (h) $\rm{H_2O-OCS}$, (i) $\rm{H_2O-N_2}$, and (j) $\rm{H_2O-O_2}$ clusters with various concentrations. The water c-hexamer (chair) configuration gas been used for pure water.[]{data-label="fig:6h2o_x_band_strength"}](6h2o_o2_auc_new.pdf){width="\textwidth"} ![Structure of water clusters containing 20 $\rm{H_2O}$ molecules with HCOOH as impurity in different concentration ratio: (a) pure water, (b) $\rm{H_2O:HCOOH=20:1}$, (c) $\rm{H_2O:HCOOH=10:1}$, (d) $\rm{H_2O:HCOOH=6.67:1}$, (e) $\rm{H_2O:HCOOH=5:1}$.[]{data-label="fig:20H2O-HCOOH"}](20_H2O_HCOOH.pdf){height="8cm" width="14cm"} ![Comparison of the band strength of the four fundamental vibrational modes for water clusters containing 20$\rm{H_2O}$, 6H$_2$O, and 4H$_2$O molecules with HCOOH as an impurity in different concentrations. Solid lines represent the band strength profiles for 20H$_2$O cluster, dotted lines for the water c-hexamer (chair) (6H$_2$O), and dashed lines for water c-tetramer (4H$_2$O).[]{data-label="fig:water_clusters-HCOOH"}](20h2o_comparison.pdf){height="7cm" width="10cm"} ![Effect of the cluster size on the band strength profile.[]{data-label="fig:CH3OH_varied_cluster_size"}](watercluster_methanol.pdf){width="\textwidth"} \ [**Notes.**]{} $^t$OH torsion; $^b$OH scissoring; $^s$OH stretching; $^f$free OH. \ [**Notes.**]{} $^t$OH torsion; $^b$OH scissoring; $^s$OH stretching; $^f$free OH; \*vibrations contaminated by HCOOH modes. \ [**Notes.**]{} $^t$OH torsion; $^b$OH scissoring; $^s$OH stretching; $^f$free OH; \*vibrations contaminated by HCOOH modes. \ [**Notes.**]{} $^t$OH torsion; $^b$OH scissoring; $^s$OH stretching; $^f$free OH; \*vibrations contaminated by HCOOH modes. \ [**Notes.**]{} $^t$OH torsion; $^b$OH scissoring; $^s$OH stretching; $^f$free OH; \*vibrations contaminated by HCOOH modes. \ [**Notes.**]{} $^t$OH torsion; $^b$OH scissoring; $^s$OH stretching; $^f$free OH; \*vibrations contaminated by NH$_3$ modes. \ [**Notes.**]{} $^t$OH torsion; $^b$OH scissoring; $^s$OH stretching; $^f$free OH; \*vibrations contaminated by NH$_3$ modes. \ [**Notes.**]{} $^t$OH torsion; $^b$OH scissoring; $^s$OH stretching; $^f$free OH; \*vibrations contaminated by NH$_3$ modes. \ [**Notes.**]{} $^t$OH torsion; $^b$OH scissoring; $^s$OH stretching; $^f$free OH; \*vibrations contaminated by NH$_3$ modes. \ [**Notes.**]{} $^t$OH torsion; $^b$OH scissoring; $^s$OH stretching; $^f$free OH \ [**Notes.**]{} $^t$OH torsion; $^b$OH scissoring; $^s$OH stretching; $^f$free OH. \ [**Notes.**]{} $^t$OH torsion; $^b$OH scissoring; $^s$OH stretching; $^f$free OH.

--- abstract: | This paper considers two important questions in the well-studied theory of graphs that are $F$-saturated. A graph $G$ is called $F$-saturated if $G$ does not contain a subgraph isomorphic to $F$, but the addition of any edge creates a copy of $F$. We first resolve the most fundamental question of minimizing the number of cliques of size $r$ in a $K_s$-saturated graph for all sufficiently large numbers of vertices, confirming a conjecture of Kritschgau, Methuku, Tait, and Timmons. We also go further and prove a corresponding stability result. Next we minimize the number of cycles of length $r$ in a $K_s$-saturated graph for all sufficiently large numbers of vertices, and classify the extremal graphs for most values of $r$, answering another question of Kritschgau, Methuku, Tait, and Timmons for most $r$. We then move on to a central and longstanding conjecture in graph saturation made by Tuza, which states that for every graph $F$, the limit $\lim_{n \rightarrow \infty} \frac{\operatorname{sat}(n, F)}{n}$ exists, where $\operatorname{sat}(n, F)$ denotes the minimum number of edges in an $n$-vertex $F$-saturated graph. Pikhurko made progress in the negative direction by considering families of graphs instead of a single graph, and proved that there exists a graph family $\mathcal{F}$ of size $4$ for which $\lim_{n \rightarrow \infty} \frac{\operatorname{sat}(n, \mathcal{F})}{n}$ does not exist (for a family of graphs $\mathcal{F}$, a graph $G$ is called $\mathcal{F}$-saturated if $G$ does not contain a copy of any graph in $\mathcal{F}$, but the addition of any edge creates a copy of a graph in $\mathcal{F}$, and $\operatorname{sat}(n, \mathcal{F})$ is defined similarly). We make the first improvement in 15 years by showing that there exist infinitely many graph families of size $3$ where this limit does not exist. Our construction also extends to the generalized saturation problem when we minimize the number of fixed-size cliques. We also show an example of a graph $F_r$ for which there is irregular behavior in the minimum number of $C_r$’s in an $n$-vertex $F_r$-saturated graph. author: - 'Debsoumya Chakraborti[^1]  and Po-Shen Loh[^2]' title: 'Minimizing the numbers of cliques and cycles of fixed size in an $F$-saturated graph' --- Introduction ============ Extremal graph theory focuses on finding the extremal values of certain parameters of graphs under certain natural conditions. One of the most well-studied conditions is $F$-freeness. For graphs $G$ and $F$, we say that $G$ is $F$-free if $G$ does not contain a subgraph isomorphic to $F$. This gives rise to the most fundamental question of finding the Turán number $\operatorname{ex}(n, F)$, which asks for the maximum number of edges in an $n$-vertex $F$-free graph. The asymptotic answer is known for most graphs $F$, with the exception of bipartite $F$ where the most intricate and unsolved cases appear (see, e.g., [@FS] and [@S] for nice surveys). Recently, Alon and Shikhelman [@AS] introduced a natural generalization of the Turán number. They systematically studied $\operatorname{ex}(n, H, F)$, which denotes the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. Note that the case $H = K_2$ is the standard Turán problem, i.e., $\operatorname{ex}(n, K_2, F) = \operatorname{ex}(n, F)$. While the Turán number asks for the maximum number of edges in an $F$-free graph, another very classical problem concerns the minimum number of edges in an $F$-free graph with a fixed number of vertices. This problem is not interesting as stated because the empty graph is the obvious answer. In much of the research, this issue is resolved by imposing the additional condition that adding any edge to $G$ will create a copy of $F$. With this additional condition, we say that $G$ is $F$-saturated. A moment’s thought will convince the reader that when maximizing the number of edges, this additional condition does not change the problem at all. On the other hand, this new condition makes the edge minimization problem very interesting, and this area of research is commonly known as graph saturation. Let the saturation function $\operatorname{sat}(n, F)$ denote the minimum number of edges in an $n$-vertex $F$-saturated graph. Erdős, Hajnal, and Moon [@EHM] started the investigation of this area with the following beautiful result. \[Erdős, Hajnal, and Moon 1964\] \[EHM\] For every $n \ge s \ge 2$, the saturation number $$\operatorname{sat}(n, K_s) = (s -2)(n-s+2) + \binom{s-2}{2}.$$ Furthermore, there is a unique $K_s$-saturated graph on $n$ vertices with $\operatorname{sat}(n, K_s)$ edges: the join of a clique with $s-2$ vertices and an independent set with $n-s+2$ vertices. The *join* $G_1 \ast G_2$ of two graphs $G_1$ and $G_2$ is obtained by taking the disjoint union of $G_1$ and $G_2$ and adding all the edges between them. Erdős, Hajnal, and Moon proved Theorem \[EHM\] by using a clever induction argument. A novel approach to prove this theorem is due to Bollobás [@B65], who developed an interesting tool based on systems of intersecting sets. Graph saturation has been studied extensively since Theorem \[EHM\] appeared half a century ago (see, e.g., [@FFS] for a very informative survey). Alon and Shikhelman’s generalization of the Turán number motivated Kritschgau, Methuku, Tait, and Timmons [@KMTT] to start the systematic study of the function $\operatorname{sat}(n, H, F)$, which denotes the minimum number of copies of $H$ in an $n$-vertex $F$-saturated graph. Here again note that $\operatorname{sat}(n, K_2, F) = \operatorname{sat}(n, F)$. Historically, a natural generalization of counting the number of edges ($K_2$) is to count the number of cliques ($K_r$) of a fixed size, see e.g., [@B76], [@E], and [@Z], where the authors answered the generalized extremal question of finding the maximum number of $K_r$’s in a $K_s$-free graph with fixed number of vertices. Towards generalizing Theorem \[EHM\] in a similar fashion, Kritschgau, Methuku, Tait, and Timmons proved the following lower and upper bounds, which differ by a factor of about $r-1$, and conjectured that the upper bound (achieved by the same construction given in Theorem \[EHM\]) is correct. \[Kritschgau, Methuku, Tait, and Timmons 2018\] \[tait\] For every $s > r \ge 3$, there exists a constant $n_{r,s}$ such that for all $n \ge n_{r,s}$, $$\begin{aligned} \max \left\{\frac{\binom{s-2}{r-1}}{r-1} \cdot n - 2 \binom{s-2}{r-1}, \frac{\binom{s-2}{r-1} + \binom{s-3}{r-2}}{r} \cdot n\right\} &\le \operatorname{sat}(n, K_r, K_s) \\ &\le (n - s + 2) \binom{s-2}{r-1} + \binom{s-2}{r} .\end{aligned}$$ Our first main contribution confirms their conjecture for sufficiently large $n$ by showing that the upper bound is indeed the correct answer. We also show that the natural construction is the unique extremal graph for this generalized saturation problem for large enough $n$. Furthermore, we prove a corresponding stability result for sufficiently large $n$ which shows that even if we allow up to some $cn$ more copies of $K_r$ than $\operatorname{sat}(n, K_r, K_s)$ in an $n$-vertex $K_s$-saturated graph, the extremal graph will still be the same and unique. It is worth noting that there are relatively few stability results in the area of graph saturation, essentially only [@AFGS] by Amin, Faudree, Gould, and Sidorowicz, and [@BFP] by Bohman, Fonoberova, and Pikhurko. In the notation of joins, the extremal graph in our problem is $K_{s-2} \ast \overline{K}_{n-s+2}$, i.e., the join of a clique with $s-2$ vertices and an independent set with $n-s+2$ vertices. \[sat\] For every $s > r \ge 2$, there exists a constant $n_{r,s}$ such that for all $n \ge n_{r,s}$, we have $\operatorname{sat}(n, K_r, K_s) = (n-s+2) \binom{s-2}{r-1} + \binom{s-2}{r}$. Moreover, there exists a constant $c_{r,s} > 0$ such that the only $K_s$-saturated graph with up to $\operatorname{sat}(n, K_r, K_s) + c_{r,s} n$ many copies of $K_r$ is $K_{s-2} \ast \overline{K}_{n-s+2}$. The moreover part of this theorem is tight in the sense that Theorem \[sat\] fails for $c_{r,s} = \binom{s-3}{r-2}$. To see that consider the graph $G$ on $n$ vertices which is the join of two graphs $G_1$ and $G_2$, where $G_1$ is $K_{s-1}$ minus an edge, and $G_2$ is an independent set on $n-s+1$ vertices. Clearly, $G$ is $K_s$-saturated, with $\left(2 \binom{s-3}{r-2} + \binom{s-3}{r-1}\right) (n-s+1) + 2\binom{s-3}{r-1} + \binom{s-3}{r}$ many copies of $K_r$. In the process of proving Theorem \[sat\], we consider a more general setting and prove and use an intermediate result, which may also be of independent interest. The condition that $G$ is $F$-saturated can be weakened by removing the condition that $G$ is $F$-free (as also studied in [@B78] and [@T92]). Perhaps counterintuitively, despite the fact that this is a weaker condition, the literature calls $G$ *strongly $F$-saturated* if adding any edge to $G$ creates a new copy of $F$. Following the notation in the literature, we write $\operatorname{ssat}(n, H, F)$ to denote the minimum number of copies of $H$ in an $n$-vertex strongly $F$-saturated graph. It is obvious that $\operatorname{ssat}(n, H, F) \le \operatorname{sat}(n, H, F)$. We have the following asymptotic result for the function $\operatorname{ssat}$ for cliques. \[ssat\] For every $s > r \ge 2$, we have $\operatorname{ssat}(n, K_r, K_s) = n \binom{s-2}{r-1} - o(n)$. Kritschgau, Methuku, Tait, and Timmons [@KMTT] showed an interesting result, which says that for any natural number $m$, there are graphs $H$ and $F$ such that $\operatorname{sat}(n,H,F) = \Theta(n^m)$. They showed this as an implication of the following bounds that they proved on the minimum number of $C_r$’s in an $n$-vertex $K_s$-saturated graph. \[Kritschgau, Methuku, Tait, and Timmons 2018\] \[tait2\] For $s \ge 5$ and $r \le 2s - 4$, $\operatorname{sat}(n, C_r, K_s) = \Theta(n^{\floor{\frac{r}{2}}})$. More precisely, $$\begin{aligned} & \left(1 - o(1)\right) \frac{n^k (s-2)_k}{4 k} \le \operatorname{sat}(n, C_r, K_s) \le \left(1 + o(1)\right) \frac{n^k (s-2)_k}{2 k} &&\text{ if } 2 \mid r \\& \left(1 - o(1)\right) \frac{n^k (s-2)_{k+1} (k-2)!}{r (r-3) (r)_k (s-1)} \le \operatorname{sat}(n, C_r, K_s) \le \left(1 + o(1)\right) \frac{n^k (s-2)_{k+1}}{2} &&\text{ if } 2 \nmid r\end{aligned}$$ where $k = \floor{\frac{r}{2}}$ and $(m)_k = m(m-1) \cdots (m-k+1)$. Note that the same construction as in Theorem \[EHM\] proves the upper bound in Theorem \[tait2\]. We explain the counting for the upper bound in the proof of our Theorem \[cycle\] in Section 4. We show that for all sufficiently large $n$ the same natural construction is indeed the unique extremal graph for most $r$. \[cycle\] For every $s \ge 4$ and odd $r$ with $r \ge 7$ or even $r$ with $r \ge 4 \sqrt{s-2}$, there exists a constant $n_{r,s}$ such that for all $n \ge n_{r,s}$, $K_{s-2} \ast \overline{K}_{n-s+2}$ has the minimum number of copies of $C_r$ among $n$-vertex $K_s$-saturated graphs. Moreover, when also $r \le 2s - 4$ this is the unique such graph. We remark here that for any $r,s$ that do not satisfy the assumptions that $s \ge 4$ and $ r \le 2s-4$, we have $\operatorname{sat}(n, C_r, K_s) = 0$, which can be seen from the same extremal graph $K_{s-2} \ast \overline{K}_{n-s+2}$. In Theorem \[cycle\], we could write the explicit value of $\operatorname{sat}(n, C_r, K_s)$, which is just the number of cycles of length $r$ in the graph $K_{s-2} \ast \overline{K}_{n-s+2}$. We chose not to do so because the explicit number is not particularly elegant. Also, it turns out that we are able to find the correct asymptotic answers for $r=4$ and $r=5$, which we include in the sections proving Theorem \[cycle\]. Next we turn our attention to a long-standing, yet very fundamental conjecture made by Tuza [@T86; @T88]. In contrast to the Turán number, one of the inherent challenges in studying the saturation number $\operatorname{sat}(n, H)$ for general graphs $H$ is that this function lacks monotonicity properties that one might hope for. For example, Pikhurko [@P] showed that there is a pair of connected graphs $F_1 \subset F_2$ on the same vertex set such that $\operatorname{sat}(n, F_1) > \operatorname{sat}(n, F_2)$ for large $n$, violating monotonicity in the second parameter. Regarding non-monotonicity in the first parameter, Kászonyi and Tuza [@KT] observed that $\operatorname{sat}(2k-1, P_3) = k+1 > k = \operatorname{sat}(2k, P_3)$ where $P_3$ is the path with $3$ edges. Moreover, Pikhurko showed a wide variety of examples of irregular behavior of the saturation function in [@P]. All of this non-monotonicity makes proving statements about the saturation function difficult, in particular because inductive arguments generally do not work. However, in order to find some smooth behavior of the saturation function Tuza conjectured the following. \[Tuza 1986\] \[con\] For every graph $F$, the limit $\lim_{n \rightarrow \infty} \frac{\operatorname{sat}(n, F)}{n}$ exists. Not much progress has been made towards settling the conjecture. The closest positive attempt was made by Truszczyński and Tuza [@TT], who showed that for every graph $F$, if $\lim \inf_{n \rightarrow \infty} \frac{\operatorname{sat}(n, F)}{n} < 1$, then $\lim_{n \rightarrow \infty} \frac{\operatorname{sat}(n, F)}{n}$ exists and is equal to $1 - \frac{1}{p}$ for some positive integer $p$. Pikhurko considered the saturation number for graph families to make progress in the negative direction of Conjecture \[con\]. For a family of graphs $\mathcal{F}$, the saturation number $\operatorname{sat}(n, \mathcal{F})$ is defined to be the minimum number of edges in an $n$-vertex $\mathcal{F}$-saturated graph, where a graph $G$ is called $\mathcal{F}$-saturated if $G$ does not contain a copy of any graph in $\mathcal{F}$ and adding any edge to $G$ will create a copy of a graph in $\mathcal{F}$. Pikhurko first showed in [@P01] that there exists an infinite family $\mathcal{F}$ of graphs for which $\lim_{n \rightarrow \infty} \frac{\operatorname{sat}(n, \mathcal{F})}{n}$ does not exist, and later in [@P] proved the same for a graph family of size only $4$. We make the first progress in 15 years, moving one step closer. \[pro\] There exist infinitely many graph families $\mathcal{F}$ of size $3$ such that the ratio $\frac{\operatorname{sat}(n,\mathcal{F})}{n}$ does not converge as $n$ tends to infinity. In the spirit of considering the generalized saturation number, it is natural to ask the more general question of whether $\lim_{n \rightarrow \infty} \frac{\operatorname{sat}(n, K_r, F)}{n}$ exists for every graph $F$. We remark that this problem is interesting since the order of $\operatorname{sat}(n, K_r, \mathcal{F})$ is linear in $n$ for every graph family $\F$, which can easily be shown by considering the same construction used by Kászonyi and Tuza in [@KT], who showed the same for $r=2$. We show that our construction of graph families of size $3$ can be extended to this scenario. \[construction\] For every $r \ge 3$, there exist infinitely many graph families $\mathcal{F}$ of size $3$ such that the ratio $\frac{\operatorname{sat}(n, K_r, \mathcal{F})}{n}$ does not tend to a limit as $n$ tends to infinity. We next show an example of a graph $F_r$ for which the function $\operatorname{sat}(n, C_r, F_r)$ behaves irregularly. To be precise, we show that for certain $F_r$, the value of the saturation function depends on certain divisibility conditions of $n$, and the sequence $\operatorname{sat}(n, C_r, F_r)$ oscillates. \[easy\_cons\] For every $r \ge 5$, there exists a graph $F_r$ such that $\operatorname{sat}(n, C_r, F_r)$ is zero for infinitely many values of $n$ and also positive infinitely often. The remainder of this paper is organized as follows. In the next section we prove an asymptotically tight lower bound on $\operatorname{ssat}(n, K_r, K_s)$. Then, we use the results and notations of that section to determine $\operatorname{sat}(n, K_r, K_s)$ exactly for sufficiently large $n$ in Section $3$. Next in Section $4$, we prove a few lemmas which will be useful for computing $\operatorname{sat}(n, C_r, K_s)$, i.e., Theorem \[cycle\]. We handle the cases of even and odd $r$ in Theorem \[cycle\] separately, and those will be proved in the subsequent two sections. In Section $7$, we construct infinitely many graph families $\mathcal{F}$ of size $3$ for which the ratio $\frac{\operatorname{sat}(n,\mathcal{F})}{n}$ does not converge. We then extend this construction, with the help of Theorem \[ssat\], in Section $8$ in order to prove Theorem \[construction\]. We prove Theorem \[easy\_cons\] in Section $9$. We finish with a few open problems and concluding remarks in Section $10$. Asymptotic result for $\boldsymbol{\operatorname{ssat}(n, K_r, K_s)}$ ===================================================================== In this section, we prove Theorem \[ssat\]. Let $G = (V,E)$ be an $n$-vertex strongly $K_s$-saturated graph such that the number of $K_r$’s in $G$ is $\operatorname{ssat}(n, K_r, K_s)$. Our aim is to find a lower bound on the number of $K_r$ in $G$. Note that if there is an edge $e \in E$ such that $e$ is not in a copy of $K_r$, then $e$ does not contribute to the number of copies of $K_r$. It turns out that a careful analysis of the edges which are in a copy of $K_r$ saves us the required factor of $r-1$ when we compare against the previous best result (Theorem \[tait\]). So, it is natural to split the edge set $E$ into two parts in the following manner. Let $E_1$ denote the set of edges which are at least in one copy of $K_r$. Let $E_2 = E \setminus E_1$ be the remaining edges in $G$. Now we will prove a simple but powerful lemma which will be useful throughout the current and next sections. \[trivial\] Every edge of $E_2$ would not be in a copy of $K_s$ even if any non-edge were added to $G$. Fix an arbitrary edge $uv$ of $E_2$ and an arbitrary non-edge $ab$ of $G$. Note that the sets $\{u,v\}$ and $\{a,b\}$ can overlap, but without loss of generality $b \not \in \{u,v\}$. Assume for the sake of contradiction that after adding the missing edge $ab$ we create a copy of $K_s$ containing both $u$ and $v$. Now if we remove the vertex $b$ from the created copy of $K_s$, we will find a copy of $K_{s-1}$ in $G$ which contains both $u$ and $v$. So $uv$ is in a copy of $K_{s-1}$ in $G$, which contradicts the fact that $uv$ is not in a copy of $K_r$, because $r \le s-1$. It will be convenient to define a couple of sets which we will use throughout this section and the next section. For $i = 1,2$, let $G_i$ denote the graph on the same vertex set $V$ with the edge set $E_i$. For a graph $H$, it will be convenient to use the notation $d_{H}(v)$ to denote the degree of $v$ in $H$. It will be useful to split the vertices according to their degree in $G_1$, so we define $$A = \{v \in V : d_{G_1}(v) \le n^{\frac{1}{3}}\}. \label{large}$$ We can observe that $A$ consists of almost all vertices of $G$, i.e., $|A| = n - o(n)$. This is because $|E_1| \le \binom{r}{2} \operatorname{ssat}(n, K_r, K_s) \le \binom{r}{2} \operatorname{sat}(n, K_r, K_s) = O(n)$, where the last equality follows from the upper bound in Theorem \[tait\], and so $|V \setminus A| = O(n^{\frac{2}{3}})$. Now our aim is to show that almost every vertex of $A$ is in a copy of $K_{s-1}$ which has only one vertex of $A$. Note that the extremal graph $K_{s-2} \ast \overline{K}_{n-s+2}$ has this property. Formally, we define the following: $$\begin{aligned} B = \{v \in A : \exists a_1, \dots, a_{s-2} \in V \setminus A \text{ such that } v, a_1, \dots, a_{s-2} \text{ induce a copy of } K_{s-1}\}. \label{select}\end{aligned}$$ \[main\] Almost all vertices are in $B$, in the sense that $|B| = n - o(n)$. Let $R$ denote the set of vertices in $A$ with degree more than $|A| - 2n^{\frac{2}{3}}$ in the induced subgraph of $G_2$ on $A$. Now we claim that $R$ has at most $2rn^{\frac{2}{3}}$ vertices. Assume for the sake of contradiction that $|R| > 2rn^{\frac{2}{3}}$; then with a simple greedy process we will find a copy of $K_r$ in $G_2$. Start with any vertex $v_1 \in R$, and let $R_1 \subseteq R$ denote the set of vertices in $R$ which are neighbors of $v_1$. Clearly, $|R_1| > 2(r-1)n^{\frac{2}{3}}$ because $v_1$ has less than $2n^{\frac{2}{3}}$ non-neighbors in $R$. For $2 \le i \le r$, we continue this process, i.e., at step $i$ we take a vertex $v_i \in R_{i-1}$, and let $R_i \subseteq R_{i-1}$ denote the set of vertices in $R_{i-1}$ which are neighbors of $v_i$. Clearly, $|R_i| > 2(r-i)n^{\frac{2}{3}}$. Now observe that $v_1, v_2, \dots, v_r$ induce a copy of $K_r$ in $G_2$ which is the desired contradiction. So $|R| \le 2rn^{\frac{2}{3}}$. Now our aim is to show that $A \setminus R \subseteq B$, which will be sufficient to finish the proof of this lemma. To this end, fix an arbitrary vertex $v \in A \setminus R$. We will first show that there is $w \in A$ such that $vw$ is not an edge of $G$ and there is no $z \in A$ such that $vz$ and $zw$ are both in $E_1$. This is because there are at most $|A| - 2n^{\frac{2}{3}}$ many $E_2$-neighbors of $v$ in $A$ (which follows from the definition of $R$), and in the induced graph of $G_1$ on $A$, there can be at most $n^{\frac{1}{3}} \left(n^{\frac{1}{3}} - 1\right) = n^{\frac{2}{3}} - n^{\frac{1}{3}}$ vertices at distance $2$ from $v$ (which follows from ). So, there are at least $$|A| - 1 - \left(|A| - 2n^{\frac{2}{3}}\right) - n^{\frac{1}{3}} - \left(n^{\frac{2}{3}} - n^{\frac{1}{3}}\right) = n^{\frac{2}{3}} - 1$$ choices for $w$. Fix such a vertex $w$. As $G$ is $K_s$-saturated, if we added the edge $vw$, then we would create a copy of $K_s$. Furthermore, that $K_s$ cannot contain any vertex from $A$ except $v$ and $w$, because if it contained some $z \in A$, then at least one of $vz$ or $zw$ is in $E_2$, contradicting Lemma \[trivial\]. Hence there is a copy of $K_{s-1}$ induced by $v$ together with $s-2$ vertices from $V \setminus A$, and so $v \in B$. Therefore, $|B| \ge |A| - |R| \ge n - o(n)$. For an arbitrary vertex $v \in B$, the number of $K_r$’s induced by $v$ together with $r-1$ vertices from $V \setminus A \subseteq V \setminus B$ is at least $\binom{s-2}{r-1}$ from . So by Lemma \[main\], the number of $K_r$’s in $G$ is at least $\binom{s-2}{r-1} |B| = \binom{s-2}{r-1}n - o(n)$. This matches the upper bound from Theorem \[tait\], completing the proof of Theorem \[ssat\]. Note that by defining the set $A$ in optimally, the best lower bound we can achieve with this argument is that $\operatorname{ssat}(n, K_r, K_s) \ge \binom{s-2}{r-1}n - O\left(\sqrt{n}\right)$. Also note that Theorem \[ssat\] already proves an asymptotically tight lower bound on $\operatorname{sat}(n, K_r, K_s)$, because: $$n \binom{s-2}{r-1} - o(n) \le \operatorname{ssat}(n, K_r, K_s) \le \operatorname{sat}(n, K_r, K_s) \le (n-s+2) \binom{s-2}{r-1} + \binom{s-2}{r}.$$ Exact result for $\boldsymbol{\operatorname{sat}(n, K_r, K_s)}$ =============================================================== In this section, we will find the exact value of $\operatorname{sat}(n, K_r, K_s)$ for all sufficiently large $n$, proving Theorem \[sat\]. The same argument will also show that the graph $K_{s-2} \ast \overline{K}_{n-s+2}$ is the unique extremal graph. Moreover, we will prove a stability result, i.e., the same graph is also the unique graph among $K_s$-saturated graphs even if we allow up to some $cn$ more copies of $K_r$ than $\operatorname{sat}(n, K_r, K_s)$. We will start with the structural knowledge we developed in the last section and successively deduce more structure to finally reach the exact structure. Define $c = \frac{1}{4r^2}$ and consider an $n$-vertex $K_s$-saturated graph $G$ with at most $\operatorname{sat}(n, K_r, K_s) + cn$ copies of $K_r$. By defining the sets $A$ and $B$ as in and and applying the same arguments we can make the same structural deductions about $G$ as in the last section. In particular, the number of $K_r$’s with one vertex in $B$ and $r-1$ vertices in $V \setminus A$ is at least $$n \binom{s-2}{r-1} - o(n). \label{count}$$ Next, define $$C = \{v \in B : d_{G_1} (v) > s-2\}. \label{problem}$$ For $v \in C$, fix $s-2$ neighbors of $v$ in $V \setminus A$ such that those neighbors along with $v$ induce a copy of $K_{s-1}$ in $G$. For each $v \in C$, pick an edge $vw \in E_1$ such that $w$ is not among the $s-2$ fixed neighbors. Note that the same edge $vw$ can be picked at most once more. Each of these particular edges is in $E_1$, hence these edges are contained in some $K_r$, which is not counted in . After counting for multiplicity, these extra edges will constitute at least an extra $\frac{|C|}{2\binom{r}{2}}$ many copies of $K_r$. Hence, for sufficiently large $n$, $\frac{|C|}{2\binom{r}{2}} \le 2cn$, which implies that $|C| \le \frac{n}{2}$. So, the set $B \setminus C$ is non-empty for large enough $n$. We will now prove two more structural lemmas. \[tight\] Let $v$ be an arbitrary vertex in $B \setminus C$, and suppose $x_1, x_2, \dots, x_{s-2}$ are vertices in $G$ such that $\{v, x_1, \dots, x_{s-2}\}$ induces a copy of $K_{s-1}$. Then for all $u \in V \setminus \{v\}$ such that $uv$ is not an edge, $u$ is adjacent to all of $x_1, \dots, x_{s-2}$. Since $\{v, x_1, \dots, x_{s-2}\}$ induces $K_{s-1}$ and $s-1 \geq r$, every edge $vx_i$ is in $E_1$. As $v \in B \setminus C$, $v$ has no more $E_1$-edges. If we add the non-edge $uv$, we must create a copy of $K_s$. If some vertex $w \not \in \{u, v, x_1, \dots, x_{s-2}\}$ participates in the created copy of $K_s$, then we know that $vw$ must be in $E_2$ since $v$ has no more $E_1$-edges, contradicting Lemma \[trivial\]. So, the only choice for the remaining $s-2$ vertices of the created copy of $K_s$ would be $x_1, \dots, x_{s-2}$. Thus $u$ must be adjacent to all of $x_1, \dots, x_{s-2}$. \[T\] All vertices of $B \setminus C$ have no incident edges from $E_2$. Assume for the sake of contradiction that $uv \in E_2$, where $v \in B \setminus C$. Since $G$ is $K_s$-saturated, $u$ is in a copy $S$ of $K_{s-1} \supseteq K_r$. Since $uv \in E_2$, $v \not \in S$ by Lemma \[trivial\]. Furthermore, $v$ cannot be adjacent to all the vertices in $S$, or else there would be a copy of $K_s$. Similarly, $v$ is in a copy of $K_{s-1}$, and $u$ is not adjacent to the full set of those vertices. Let $a_1, \dots, a_k$, $b_1, \dots, b_k$ and $c_{k+1}, \dots, c_{s-2}$ be distinct vertices such that $\{u, a_1, \dots, a_k, c_{k+1}, \dots, c_{s-2}\}$ and $\{v, b_1, \dots, b_k, c_{k+1}, \dots, c_{s-2}\}$ both induce $K_{s-1}$. The above argument shows that $k \ge 1$. Now we claim that there must be at least two non-edges between $v$ and the set $\{a_1, \dots, a_k\}$, otherwise the neighbors of $v$ in $\{a_1, \dots, a_k\}$ along with $u, v, c_{k+1}, \dots, c_{s-2}$ will induce a clique of order at least $s-1 \ge r$, which contradicts the fact that $uv \in E_2$. Without loss of generality, $v$ is not adjacent to both $a_1$ and $a_2$. Now by applying Lemma \[tight\] with $v \in B \setminus C$, and $b_1, \dots, b_k, c_{k+1}, \dots, c_{s-2}$ as $x_1, \dots, x_{s-2}$, and $a_1$ as $u$, we see that $a_1$ is adjacent to all of $b_1, \dots, b_k, c_{k+1}, \dots, c_{s-2}$. The same is true of $a_2$. So, $a_1, a_2, b_1, \dots, b_k, c_{k+1}, \dots, c_{s-2}$ induce a copy of $K_s$ in $G$ which is impossible. \[Proof of Theorem \[sat\]\] Fix any vertex $v \in B \setminus C$. There exists a set $S$ of $s-2$ vertices such that $S \cup \{v\}$ induces a copy of $K_{s-1}$. Since $v \in B \setminus C$, there are no more $E_1$ edges incident to $v$ other than those to $S$. By Lemma \[T\], there are no $E_2$ edges either. By Lemma \[tight\], every vertex $u \not \in S \cup \{v\}$ must be adjacent to all vertices in $S$. This is already the graph $K_{s-2} \ast \overline{K}_{n-s+2}$, which is $K_s$-saturated, so $G$ is precisely $K_{s-2} \ast \overline{K}_{n-s+2}$. Preparation to compute $\boldsymbol{\operatorname{sat}(n, C_r, K_s)}$ ===================================================================== In this section, we state a few lemmas which will be helpful to prove Theorem \[cycle\] in the subsequent two sections. Our proof is inspired by the proof in [@KMTT]. Compared to that paper, we count the number of cycles more carefully to avoid double-counting, which helps us to get the exact answer. We first find asymptotically the number of cycles of length $r$ in the graph $K_{s-2} \ast \overline{K}_{n-s+2}$. Let $\boldsymbol{k = \floor{\frac{r}{2}}}$ throughout the current and next two sections. There are $\binom{n-s+2}{k}$ many independent sets of order $k$ in the subgraph $\overline{K}_{n-s+2}$. If $r$ is even, then for an arbitrary $k$-vertex independent set $A$, the number of copies of $C_r$ containing $A$ is $\frac{(s-2)_k (k-1)!}{2}$, and each copy of $C_r$ is counted exactly once. If $r$ is odd, then for an arbitrary $k$-vertex independent set $A$, the number of copies of $C_r$ containing $A$ is $\frac{(s-2)_{k+1} k!}{2}$, and each copy of $C_r$ is counted exactly once. Furthermore, there is no copy of $C_r$ with more than $k$ vertices in $\overline{K}_{n-s+2}$ because the maximum independent set of $C_r$ has $k$ vertices. Hence, we have the upper bounds of Theorem \[tait2\], i.e., $$\begin{aligned} & \operatorname{sat}(n, C_r, K_s) \le \frac{(s-2)_k}{2 k} \cdot n^k + O(n^{k-1}) &&\text{ if } 2 \mid r \label{optimal1} \\& \operatorname{sat}(n, C_r, K_s) \le \frac{(s-2)_{k+1}}{2} \cdot n^k + O(n^{k-1}) &&\text{ if } 2 \nmid r \label{optimal2}\end{aligned}$$ We will use the standard notation $\Theta$ in the next few sections. For two functions $f(n)$ and $g(n)$, we call $f(n) = \Theta(g(n))$ if $0 < \lim\inf_{n \rightarrow \infty} \frac{g(n)}{f(n)} \le \lim\sup_{n \rightarrow \infty} \frac{g(n)}{f(n)} < \infty$. \[ramsey\] For every fixed $l$, there are $\Theta(n^l)$ independent sets of order $l$ in every $n$-vertex $K_s$-free graph. Consider an $n$-vertex $K_s$-free graph $G$. It is obvious that the number of independent sets of order $l$ in an $n$-vertex graph is at most $\binom{n}{l} = \Theta(n^l)$. From the most classical result [@ES] in Ramsey theory, we know that $R(l,s)$ exists, where $R(l,s)$ denotes the minimum number $N$ such that every graph of order $N$ contains an independent set of order $l$ or a clique of order $s$. So, for each $R(l,s)$-vertex subset $A$ of $G$, the subgraph induced by $A$ must contain an independent set of order $l$ because $A$ does not contain a copy of $K_s$. Now an independent set of order $l$ can be counted at most $\binom{n-l}{R(l,s)-l}$ times. Accounting for multiple-counts, the number of independent sets of order $l$ in $G$ is at least $\frac{\binom{n}{R(l,s)}}{\binom{n-l}{R(l,s)-l}} = \Theta(n^l)$. Next we give an upper bound on the number of edges of any $n$-vertex $K_s$-saturated graph minimizing the number of copies of $C_r$ with $r \le 2s-4$. It is shown in [@KMTT] that for every fixed even $r$, there are $o(n^2)$ many edges in an $n$-vertex $K_s$-saturated graph with minimal number of copies of $C_r$. Next we prove the same for all $r \le 2s-4$. We prove a stronger result for odd $r \le 2s-4$, and repeat the proof for even $r$ from [@KMTT] for the sake of completion. \[upperbound\] For every $n$-vertex $K_s$-saturated graph $G$ minimizing the number of copies of $C_r$, and for any function $f(n)$ such that $f(n) \rightarrow \infty$ as $n \rightarrow \infty$: - For odd $r \le 2s-4$, $G$ has $O\left(n f(n)\right)$ many edges. - For even $r$, $G$ has $o(n^2)$ many edges. In the case of even $r$, if we could prove that $G$ has $o\left(n^{\frac{3}{2}}\right)$ many edges, then we could follow the proof for odd $r$ and would not have the condition $r \ge 4 \sqrt{s-2}$ in Theorem \[cycle\]. We have briefly mentioned this again in the concluding remarks. *Case 1: $r$ is odd and $r \le 2s-4$.* We can assume that the function $f(n)$ is such that $f(n) = O(\log n)$. Let $G$ be an $n$-vertex $K_s$-saturated graph minimizing the number of copies of $C_r$. For the sake of contradiction, assume that $G$ has more than $n f(n)$ edges. Let $B$ denote the set of all vertices of $G$ with degree more than $f(n)$. A simple counting implies that $\sum_{v \in B} d(v) \ge n f(n)$. To prove Lemma \[upperbound\], it is enough to show that for all $v \in B$, there are at least $\Theta\left(n^{k-1} d(v)\right)$ cycles containing $v$. In this case, the total number of cycles will be at least $\Theta\left(\sum_{v \in B} n^{k-1} d(v)\right) \ge \Theta\left(n^k f(n)\right)$, contradicting for all sufficiently large $n$. To show this, consider a vertex $v \in B$. Consider an arbitrary independent set $I = \{v_1, \cdots, v_{k-1}\}$ of order $k-1$ in $V(G) \setminus \{v\}$. For every $i \in [k-2]$, choose a set $V_{i,i+1}$ of $s-2$ vertices such that adding the edge $v_iv_{i+1}$ would create a copy of $K_s$ on $\{v_i,v_{i+1}\} \cup V_{i,i+1}$. Let $V_1$ denote an empty set if $vv_1$ is an edge, else set it to be a set of $s-2$ vertices such that adding the edge $vv_1$ would create a copy of $K_s$ on $\{v,v_1\} \cup V_1$. Let $U = I \cup V_1 \cup V_{1,2} \cup \cdots \cup V_{k-2,k-1}$. Let $V'$ denote the set of neighbors of $v$ outside of $U$. Note that $|V'| \ge d(v) - ks \ge \frac{1}{2} \cdot d(v)$ for large enough $n$ (remember that $d(v) \ge f(n)$). For each $a \in V'$, we will show the existence of a cycle of length $r$ containing $a$, $v$, and all vertices in $I$, proving that there are at least $\frac{1}{2} \cdot d(v)$ many copies of $C_r$ containing $I$.\ *Subcase 1: $av_{k-1}$ and $vv_1$ both are edges.* Pick $k$ distinct vertices $u_1, u_1^*, u_1^{**} \in V_{1,2}, u_2 \in V_{2,3}, \cdots, u_{k-2} \in V_{k-2,k-1}$. This is clearly possible because $|V_{i,i+1}| = s-2$ for all $i$ and $k < s-2$. So, $v v_1 u_1 u_1^* u_1^{**} v_2 u_2 v_3 u_3 \cdots v_{k-2} u_{k-2} v_{k-1} a v$ forms a cycle of length $r$ in $G$.\ *Subcase 2: $av_{k-1}$ is an edge, but $vv_1$ is not an edge.* Pick $k$ distinct vertices $w, w^* \in V_1, u_1 \in V_{1,2}, \cdots, u_{k-2} \in V_{k-2,k-1}$. This is clearly possible because $|V_1| = s-2$, $|V_{i,i+1}| = s-2$ for all $i$, and $k < s-2$. So, $v w w^* v_1 u_1 v_2 u_2 \cdots v_{k-2} u_{k-2} v_{k-1} a v$ forms a cycle of length $r$ in $G$.\ *Subcase 3: $av_{k-1}$ is not an edge, but $vv_1$ is an edge.* Pick $k-1$ distinct vertices $u_1, u_1^* \in V_{1,2}, u_2 \in V_{2,3}, \cdots, u_{k-2} \in V_{k-2,k-1}$. This is clearly possible because $|V_{i,i+1}| = s-2$ for all $i$ and $k-1 < s-2$. Choose a set $S$ of $s-2$ vertices such that adding the edge $av_{k-1}$ would create a copy of $K_s$ on $\{a,v_{k-1}\} \cup S$. Now as $I$ is an independent set, no vertex from $I \setminus \{v_{k-1}\}$ can be in $S$, so there is a vertex $c \in S$ that is not in the set $I \cup \{v, u_1, u_1^*, \cdots, u_{k-2}\}$. Hence, $v v_1 u_1 u_1^* v_2 u_2 \cdots v_{k-2} u_{k-2} v_{k-1} c a v$ forms a cycle of length $r$ in $G$.\ *Subcase 4: $av_{k-1}$ and $vv_1$ both are not edges.* Pick $k-1$ distinct vertices $w \in V_1, u_1 \in V_{1,2}, \cdots, u_{k-2} \in V_{k-2,k-1}$. This is clearly possible because $|V_1| = s-2$, $|V_{i,i+1}| = s-2$ for all $i$, and $k-1 < s-2$. Choose a set $S$ of $s-2$ vertices such that adding the edge $av_{k-1}$ would create a copy of $K_s$ on $\{a,v_{k-1}\} \cup S$. Now as $I$ is an independent set, no vertex from $I \setminus \{v_{k-1}\}$ can be in $S$, so there is a vertex $c \in S$ that is not in the set $I \cup \{v, w, u_1, \cdots, u_{k-2}\}$. Hence, $v w v_1 u_1 v_2 u_2 \cdots v_{k-2} u_{k-2} v_{k-1} c a v$ forms a cycle of length $r$ in $G$.\ From Lemma \[ramsey\], we know that there are $\Theta(n^{k-1})$ many independent sets of order $k-1$ in the induced graph $G \setminus \{v\}$ for any vertex $v$, and for each $v \in B$ and such an independent set, we have $\frac{1}{2} \cdot d(v)$ many copies of $C_r$ containing $v$ and the independent set. It is clear that a copy of $C_r$ in $G$ can be counted at most only a constant (depending on $k$) times in this way. So, the number of $C_r$’s in $G$ is at least $\Theta\left(\sum_{v \in B} n^{k-1} d(v)\right) = \Theta\left(n^k f(n)\right)$, contradicting for all sufficiently large $n$.\ *Case 2: $r$ is even.* By Theorem $1^{**}$ in [@ES83], there exists $c, c' > 0$ such that for any graph $G$ with more than $c n^{2 - \frac{2}{r}}$ edges, there exists $$c'n^r \left(\frac{|E(G)|}{n^2}\right)^{\frac{r^2}{4}}$$ copies of $K_{\frac{r}{2},\frac{r}{2}}$. Therefore, if the number of edges of $G$ is $\epsilon n^2$ for some $\epsilon > 0$ and sufficiently large $n$, then there are $\Theta(n^r)$ copies of $C_r$, contradicting .\ Since all the cases give contradictions, we are done. It is also shown in [@KMTT] that for every even $r$, there are $(1 - o(1)) \binom{n}{k}$ many independent sets of order $k$ in an $n$-vertex $K_s$-saturated graph with minimal number of copies of $C_r$, with an application of the Moon-Moser theorem [@MM]. Next we prove the same for all $r \le 2s-4$ by using Lemma \[upperbound\] and the following lemma which is equivalent to the problem appeared in Exercise 40(b) in Chapter 10 of [@L]. \[mm\] Let $G$ be a graph on $n$ vertices with $\frac{1}{\tau} \binom{n}{2}$ many edges, where $\tau$ is a positive real number. Let $l$ be a positive integer such that $l \le \tau + 1$. Then, the number of independent sets of order $l$ in $G$ is at least $\binom{\tau}{l} \left(\frac{n}{\tau}\right)^l$. \[indep\] For every $n$-vertex $K_s$-saturated graph $G$ minimizing the number of copies of $C_r$ for some $r \le 2s-4$, $G$ has $(1 - o(1)) \binom{n}{k}$ many independent sets of order $k$. Consider an $n$-vertex $K_s$-saturated graph $G$ minimizing the number of copies of $C_r$. The number of edges in $G$ is $o(n^2)$ from Lemma \[upperbound\], so we can apply Lemma \[mm\] to conclude that $G$ has $(1 - o(1)) \binom{n}{k}$ many independent sets of order $k$. Notice that the arguments for the even cycles and the odd cycles are bit different in Lemma \[upperbound\]. It turns out that the proof of Theorem \[cycle\] for the cases of even and odd $r$ is very different. So, we split the cases in two subsequent sections. Few copies of $\boldsymbol{C_r}$ in $\boldsymbol{K_s}$-saturated graphs for odd $\boldsymbol{r}$ ================================================================================================ Let $G$ be an $n$-vertex $K_s$-saturated graph minimizing the number of copies of $C_r$. Similar to the proof of Theorem \[ssat\] in Section 2, we define $$A = \{ v \in V : d_G(v) \le n^{\frac{1}{3}}\}. \label{redefine}$$ We know that $G$ has $O\left(n \log n\right)$ edges from Lemma \[upperbound\], so $|A| = n - o(n)$. Recall that $r \le 2s-4$ and $k = \floor{\frac{r}{2}}$. Consider the collection of independent sets $I$ of order $k$ in $A$ such that for all $v_1, v_2 \in I$, there is no common neighbor of $v_1$ and $v_2$ in $A$. Denote this collection of such independent sets by $\mathcal{I}$. Clearly, there will be $(1 - o(1)) \binom{n}{k}$ independent sets in $\mathcal{I}$. Now consider an arbitrary independent set $I = \{v_1, \cdots, v_k\} \in \mathcal{I}$. For every $i,j \in [k]$, there exists a set $V_{i,j} \subseteq V \setminus A$ of $s-2$ vertices such that adding the edge $v_iv_j$ would create a copy of $K_s$ on $\{v_i,v_j\} \cup V_{i,j}$. Now an easy but cumbersome calculation (similar to the calculation for ) tells us that the number of copies of $C_r$ containing $I$ and $k+1$ vertices from $V \setminus A$ is at least $\frac{(s-2)_{k+1} k!}{2}$, where equality holds if and only if all $V_{i,j}$’s are the same and $v_i, v_j$ do not have any common neighbor in $(V \setminus A) \setminus V_{ij}$. At this point, we can conclude that the upper bound in equation is asymptotically tight for all odd $r$. Moreover, we can safely say that there are $(1 - o(1)) \binom{n}{k}$ independent sets $I = \{v_1, \cdots, v_k\} \in \mathcal{I}$ for which $V_{i,j}$’s are the same and $v_i, v_j$ do not have any common neighbor in $(V \setminus A) \setminus V_{ij}$, otherwise $G$ will have more copies of $C_r$ than the upper bound in , which is a contradiction. Let $\mathcal{J}$ denote the collection of independent sets for which the above holds. Although the statement of Theorem \[cycle\] is only for odd $r \ge 7$, the above argument actually asymptotically finds the value of $\operatorname{sat}(n, C_r, K_s)$ for $r = 5$ as well. \[maximum\_independent\] For odd $r$ with $7 \le r \le 2s-4$, there is an independent set of order $n - o(n)$ in $G$ such that there is a copy $T$ of $K_{s-2}$ in $G$ with the property that every vertex in $T$ is a neighbor of every vertex of the independent set. From the fact that $|\mathcal{J}| = \binom{n}{k} - o(n^k)$, we can say that there exist two vertices $u,v \in A$ such that there are $\binom{n}{k-2} - o(n^{k-2})$ independent sets in $\mathcal{J}$ where each of them contains both $u$ and $v$. Let $\mathcal{K}$ denote the collection of independent sets in $\mathcal{J}$ containing both $u$ and $v$. Let $T \subseteq V \setminus A$ be a set of $s-2$ vertices such that adding the edge $uv$ would create a copy of $K_s$ on $\{u,v\} \cup T$. By the definition of $\mathcal{J}$, all the vertices appearing in an independent set in $\mathcal{K}$ should be neighbors of all the vertices in $T$, hence they will form an independent set (because $G$ does not have a copy of $K_s$). For $r \ge 7$, equivalently for $k \ge 3$, it is easy to check that the number of such vertices is $n - o(n)$ (note that this is not true for $k=2$). So, we are done. Consider the maximum size independent set $I$ in $G$ such that there exists a copy $T$ of $K_{s-2}$ in $G$ such that every vertex in $T$ is a neighbor of every vertex of the independent set. Let $|I| = n - m$. We know that $m = o(n)$ from Lemma \[maximum\_independent\]. Let $S$ denote the set of all vertices outside of $I$ and $T$. For the sake of contradiction, assume that $S$ is non-empty. Now if we let $m' = |S|$, clearly $m' = m - s + 2 = o(n)$. We claim that any $v \in I$ has at least one neighbor in $S$, which will imply that there are at least $n-m$ edges between $I$ and $S$. If there is some $v \in I$ with no neighbor in $S$, then for any $u \in S$, the copy of $K_s$ created by adding the edge $uv$ cannot contain any vertex from $I$ or $S$ except $u$ and $v$, hence $u$ is neighbor of all the vertices of $T$, which in turn tells us that $u$ cannot have any neighbor in $I$, contradicting the maximal choice of $I$. Thus, every vertex in $I$ has at least one neighbor in $S$. Let $z$ be the number such that $(z)_k = \sqrt{n(m+k)} \cdot n^{k-1}$. We will show that all the vertices in $S$ can have at most $z$ neighbors in $I$ for sufficiently large $n$. Suppose for contradiction that $v \in S$ has more than $z = o(n)$ neighbors in $I$. We already know that the number of copies of $C_r$ in the induced subgraph on $T \cup I$ is at least $\frac{(s-2)_{k+1}}{2} \cdot (n-m)_k$. Now for any set of $k$ vertices from the neighbors of $v$, there is at least a copy of $C_r$ containing those vertices, together with the vertex $v$ and $k$ vertices from $T$. Clearly, there will be at least $\binom{z}{k}$ such copies of $C_r$. This implies that the number of copies of $C_r$ in $G$ is at least $\frac{(s-2)_{k+1}}{2} \cdot (n-m)_k + \binom{z}{k}$, which contradicts for all large $n$ because of the following. $$\begin{aligned} \frac{(s-2)_{k+1}}{2} &\cdot (n-m)_k + \binom{z}{k} \\ &\ge \frac{(s-2)_{k+1}}{2} \cdot (n-m-k)^k + \frac{(z)_k}{k!} \\ &\ge \frac{(s-2)_{k+1}}{2} \cdot n^k - \frac{(s-2)_{k+1}}{2} \cdot k (m+k) n^{k-1} + \frac{1}{k!} \sqrt{n(m+k)} \cdot n^{k-1} \\ &\ge \frac{(s-2)_{k+1}}{2} \cdot n^k + \frac{1}{2k!} \sqrt{n(m+k)} \cdot n^{k-1} \\ &\ge \frac{(s-2)_{k+1}}{2} \cdot n^k + \Theta\left(n^{k-\frac{1}{2}}\right)\end{aligned}$$ We now return to our analysis of $S$, which we assumed to be non-empty for the sake of contradiction. If $m \le \log n$, then the number of edges between $S$ and $I$ is at most $(m - s + 2) z < n - m$ for all sufficiently large $n$, which is a contradiction. The only case remaining to handle is when $m > \log n$. For a vertex $v \in S$, pick a set $B$ of $k-1$ vertices from $I$ which are not neighbors of $v$. Fix an order $v_1, v_2, \cdots, v_{k-1}$ of the vertices in $B$, and choose $(s-2)$-element sets $V_i$ such that adding $vv_i$ creates a copy of $K_s$ on $\{v,v_i\} \cup V_i$. Note that as $v$ is not adjacent to all the vertices in $T$ (this follows from the maximality of $I$), $V_i$ cannot be equal to $T$ for any $i$. Consider copies of $C_r$ containing $v, v_1, \cdots, v_{k-1}$ in that order such that the vertices (one or two) between $v$ and $v_1$ are from $V_1$, the vertices (one or two) between $v$ and $v_{k-1}$ are from $V_{k-1}$, and the rest of the vertices are from $T$. Call these cycles *good*. The number of good cycles is at least $\frac{(s-2)_{k+1} k}{2} + 1$. Then there are at least $\left(\frac{(s-2)_{k+1} k}{2} + 1\right) \cdot (k-1)! \ge \frac{(s-2)_{k+1} k!}{2} + 1$ many copies of $C_r$ of good type containing $v$ and all the vertices in $B$. So, if there is no over-counting, then the number of copies of $C_r$ containing $k-1$ vertices from $I$ and one vertex from $S$ is at least $\binom{n - z}{k-1} (m-s+2) \left(\frac{(s-2)_{k+1} k!}{2} + 1\right)$. To show that there is no over-counting, consider a vertex $v \in S$, $k-1$ non-neighbors $v_1, \cdots, v_{k-1}$ of $v$ in $I$ and sets $V_i$ for which adding $vv_i$ creates a copy of $K_s$ on $\{v,v_i\} \cup V_i$. The good cycles containing exactly one vertex from $V_1$ and one from $V_{k-1}$ cannot be counted twice because there is a unique independent set consisting of one vertex in $S$ and $k-1$ in $I$ in this kind of cycle. Now consider a good cycle with two vertices in $V_1$ or $V_{k-1}$. Without loss of generality, the cycle is of the form $vuu'v_1u_1v_2 \cdots u_{k-2}v_{k-1}wv$ with $u,u' \in V_1$. There is again a unique independent set consisting of one vertex in $S$ and $k-1$ in $I$ in this kind of cycle, because $u,v_1, \cdots, v_{k-1}$ cannot be an independent set due to the fact that $u$ and $v_1$ are adjacent. Hence, the total number of copies of $C_r$ in $G$ is at least $\frac{(s-2)_{k+1}}{2} \cdot (n-m)_k + \binom{n - z}{k-1} (m-s+2) \left(\frac{(s-2)_{k+1} k!}{2} + 1 \right)$. Noting that $z = o(n)$ and $m > \log n$, we can see that this contradicts due to the following: $$\begin{aligned} \begin{aligned}[t] &\frac{(s-2)_{k+1}}{2} \cdot (n-m)_k + \binom{n - z}{k-1} (m-s+2) \left(\frac{(s-2)_{k+1} k!}{2} + 1 \right) \\ &\ge \frac{(s-2)_{k+1}}{2} \cdot n^k - \frac{(s-2)_{k+1}}{2} \cdot k (m+k) n^{k-1} + \frac{(s-2)_{k+1}}{2} \cdot k m (n-z-k)^{k-1} \\ & \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; + \binom{n - z}{k-1} m - \Theta\left(n^{k-1}\right) \\ &\ge \frac{(s-2)_{k+1}}{2} \cdot n^k - \frac{(s-2)_{k+1}}{2} \cdot k m n^{k-1} - \Theta\left(n^{k-1}\right) + \frac{(s-2)_{k+1}}{2} \cdot k m n^{k-1} \\ & \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; - \Theta\left(mzn^{k-2}\right) + \Theta\left(mn^{k-1}\right) \\ &\ge \frac{(s-2)_{k+1}}{2} \cdot n^k + \Theta\left(mn^{k-1}\right). \end{aligned}\end{aligned}$$ This shows that $S$ is an empty set and so, $G$ is the union of $I$ (an independent set with maximum size) and $T$ (which is a $K_{s-2}$) where every vertex in $T$ is incident to every vertex in $I$. This finishes the proof of Theorem \[cycle\] for odd $r$. Few copies of $\boldsymbol{C_r}$ in $\boldsymbol{K_s}$-saturated graphs for even $\boldsymbol{r}$ ================================================================================================= Let $G$ be an $n$-vertex $K_s$-saturated graph minimizing the number of copies of $C_r$. Let $I$ be an arbitrary independent set of order $k = \frac{r}{2}$ in $G$. Our goal is to count the number of copies of $C_r$ in $G$ containing $I$. There are $\frac{(k-1)!}{2}$ circular permutations of the vertices of $I$, accounting for the directional symmetry of a cycle. Fix such an order $v_1, v_2, \cdots, v_k$. For every distinct $i,j \in [k]$, choose a set $V_{i,j}$ of $s-2$ vertices such that adding the edge $v_iv_j$ would create a copy of $K_s$ on $\{v_i,v_j\} \cup V_{i,j}$. For $i \in [k-1]$, we can iteratively pick a common neighbor $u_i$ of $v_i$ and $v_{i+1}$ among the $s-2$ vertices in $V_{i,i+1}$, and finally pick a common neighbor $u_k$ of $v_1$ and $v_k$ from $V_{1,k}$, thus forming a cycle $v_1 u_1 v_2 u_2 \cdots v_k u_k v_1$. Clearly, the number of ways to do this is at least $(s-2)_k$, so there are at least $\frac{(s-2)_k (k-1)!}{2}$ many copies of $C_r$ containing $I$. But there may be over-counting due to the fact that a cycle of length $r$ has two independent set of order $k$. So, to efficiently account for this double-counting, let us define a notion of ‘essential count’. The idea is to count a copy of $C_r$ containing two independent sets of order $k$ as half, so that the double-counting will make the count exactly one. So, we have two categories of $C_r$ containing $I$, (i) with two independent sets of order $k$, and (ii) with exactly one independent set of order $k$. Now, if there are $x$ copies of $C_r$ containing $I$ of type (i) and $y$ copies of $C_r$ containing $I$ with type (ii), then we say the essential count of the number of copies of $C_r$ containing $I$ is $\frac{x}{2} + y$. For a fixed $k$-independent set $I = \{v_1, \cdots, v_k\}$, we now want to find the essential count of the number of copies of $C_r$ containing $I$ in the order $v_1, v_2, \cdots, v_k$. As before, for every distinct $i,j \in [k]$, choose a set $V_{i,j}$ of $s-2$ vertices such that adding the edge $v_iv_j$ would create a copy of $K_s$ on $\{v_i,v_j\} \cup V_{i,j}$. Define the sets $A_j = V_{j,j+1} \setminus \bigcup_{i \neq j} V_{i,i+1}$, where $V_{k,k+1} = V_{k,1}$. Now observe that for all $j$ when we pick a common neighbor $u_j \in V_{j,j+1}$ of $v_j$ and $v_j$ to count the cycle $v_1u_1v_2 \cdots v_ku_kv_1$, the vertices $u_1, u_2, \cdots, u_k$ will form an independent set in $G$ if and only if $u_j \in A_j$ for all $j \in [k]$. So, if $s_j = |A_j|$, then the essential count is at least the following: $$f(s_1, s_2, \cdots, s_k) = \frac{1}{2} \prod_{j = 1}^k s_j + \sum_{\substack{J \subseteq [k] \\ |J| \neq 0}} \prod_{j \notin J} s_j \prod_{j \in J} (s-2-s_j - \iota(J,j)),$$ where $\iota(J,j)$ denotes the number of elements in $J$ smaller than $j$. \[calculus\] For any $k \ge 2 \sqrt{s-2}$, the function $f(s_1, s_2, \cdots, s_k)$ attains its minimum uniquely at $(0,0, \cdots, 0)$ over the region $\{0, 1, \cdots, s-2\}^k$. For a fixed $j$, if we fix all the variables except $s_j$ and vary $s_j$, $f$ is a linear function with respect to $s_j$. So, the minimum will occur either at $s_j = 0$ or $s_j = s-2$, when other variables are fixed. Hence, applying the same argument for all variables, we can conclude that the minimum can occur only at the vertices of the cube $[0,s-2]^k$. It is easy to check that if we evaluate $f$ at a vertex with at least one co-ordinate $0$ and one co-ordinate $s-2$, then the value will be strictly greater than $f(0,0, \cdots, 0)$. Now, the only thing we need to verify is that $f(s-2,s-2, \cdots, s-2) > f(0,0, \cdots, 0)$, which is equivalent to $\frac{1}{2} (s-2)^k > (s-2)_k$. This holds for $k \ge 2 \sqrt{s-2}$, because: $$\begin{aligned} \frac{(s-2)_k}{(s-2)^k} = 1 \left(1 - \frac{1}{s-2}\right) \cdots \left(1 - \frac{k-1}{s-2}\right) < e^{-\left(0 + \frac{1}{s-2} + \cdots + \frac{k-1}{s-2}\right)} = e^{-\frac{k(k-1)}{2(s-2)}} < \frac{1}{2}.\end{aligned}$$ The function $f$ takes strictly greater values at all vertices in $[0,s-2]^k$ than $f(0,0, \cdots, 0)$, so $f(s_1, s_2, \cdots, s_k)$ is strictly greater than $f(0,0, \cdots, 0)$ for all $(s_1, \cdots, s_k) \neq (0, \cdots, 0)$. As there are finitely many points in $\{0, 1, \cdots, s-2\}^k$, there exists some constant $\epsilon > 0$ (that does not depend on $n$, but may depend on $s$ and $k$) such that $f(s_1, s_2, \cdots, s_k) - f(0, 0, \cdots, 0) \ge \epsilon$ for all $(s_1, \cdots, s_k) \neq (0, \cdots, 0)$. The rest of the proof is similar to the odd $r$ case, and we provide an outline here for completeness. Corollary \[indep\] and Lemma \[calculus\] imply that the number of copies of $C_r$ in $G$ is at least $(1-o(1)) \frac{(s-2)_k}{2k} \cdot n^k$, which shows that is asymptotically tight for $r \ge 4 \sqrt{s-2}$. Now by a similar argument to the odd $r$ case, the number of independent sets $I = \{v_1, \cdots, v_k\}$ of order $k$, for which there is a set $V' \subseteq V(G)$ of size $s-2$ such that for all $i \neq j$, $v$ is a common neighbor of $v_i$ and $v_j$ if and only if $v \in V'$, is $(1-o(1))\binom{n}{k}$. Next we have the following lemma whose proof is the same as Lemma \[maximum\_independent\]. \[maximum\_independent1\] For even $r$ with $4 \sqrt{s-2} \le r \le 2s-4$, there is an independent set $I$ of order $n - o(n)$ in $G$ such that there is a copy $T$ of $K_{s-2}$ in $G$ with the property that every vertex in $T$ is a neighbor of every vertex of $I$. Define the sets $I$, $T$ and $S$, and the numbers $m$, $m'$ and $z$ as after Lemma \[maximum\_independent\]. Following the proof in the last section, we can show that all the vertices in $S$ can have at most $z$ neighbors in $I$ for sufficiently large $n$. For the sake of contradiction, assume that $S$ is non-empty. As before, the case when $m \le \log n$ leads to a contradiction, and $m > \log n$ remains the only case to resolve. For an arbitrary vertex $v \in S$ and an arbitrary set $B$ of $k-1$ vertices from $I$ that are not neighbors of $I$, consider the good cycles (as defined in the last section) containing $v$ and all the vertices in $B$. Like before, if there is no over-counting, then the number of copies of good $C_r$ containing $k-1$ vertices from $I$ and one vertex from $S$ is at least $\binom{n - z}{k-1} (m-s+2) \left(\frac{(s-2)_k (k-1)!}{2} + 1\right)$. To show that there is no over-counting, it turns out that the situation is simpler in this case compared to the odd $r$ case, which follows from the fact that the good cycles always have a unique independent set consisting of one vertex in $S$ and $k-1$ in $I$. Hence, the total number of copies of $C_r$ in $G$ is at least $\frac{(s-2)_k}{2k} \cdot (n-m)_k + \binom{n - z}{k-1} (m-s+2) \left(\frac{(s-2)_k (k-1)!}{2} + 1 \right) \ge \frac{(s-2)_k}{2k} \cdot n^k + \Theta\left(mn^{k-1}\right)$, contradicting . So, we have completed the proof of Theorem \[cycle\]. Having completed the proof of Theorem \[cycle\], we also solve the problem asymptotically for $r = 4$. For every $s \ge 4$, we have the following: $$\operatorname{sat}(n, C_4, K_s) = (1 + o(1)) \binom{s-2}{2} \binom{n}{2} = (1 + o(1)) \frac{n^2 (s-2)(s-3)}{4}.$$ The upper bound follows from . For the lower bound, consider an $n$-vertex $K_s$-saturated graph $G$ minimizing the number of copies of $C_4$. For a non-edge $uv \in E(G)$, choose a set $T \subseteq V(G)$ of $s-2$ vertices such that adding the edge $uv$ would create a copy of $K_s$ on $\{u,v\} \cup T$. Hence the number of copies of $K_4 \setminus e$ (which is the graph after removing an edge from a complete graph on 4 vertices) containing the non-edge $uv$ is at least $\binom{s-2}{2}$. By Lemma \[upperbound\], we can conclude that the number of copies of $K_4 \setminus e$ is at least $(1 - o(1)) \binom{s-2}{2} \binom{n}{2}$ (it is a routine to check that we are not doing any multiple-counting). Hence, $\operatorname{sat}(n, C_4, K_s) = (1 + o(1)) \binom{s-2}{2} \binom{n}{2}$. Family of size 3 with non-converging saturation ratio ===================================================== In this section, we prove Theorem \[pro\]. We begin by stating the families of graphs that we will use for the construction. \[def\] For every positive integer $m \ge 4$, let $\F_m$ be the family of the following three graphs. - Let $B_{m,m}$ be the disjoint union of two copies of $K_m$ plus one edge joining them (often called a “dumb-bell"). - Let $V_m$ be a copy of $K_m$ plus two more edges incident to a single vertex of the $K_m$. - Let $\Lambda_m$ be a copy of $K_m$ plus a single vertex with exactly two edges incident to the $K_m$. The proof of Theorem \[pro\] boils down to the fact that the behavior of $\operatorname{sat}(n, \F_m)$ depends on whether or not $n$ is divisible by $m$. The following two lemmas constitute the proof. \[easy\] For every $n$ divisible by $m$, we have $\operatorname{sat}(n, \F_m) \le \frac{n}{m} \binom{m}{2}$. Since $n$ is divisible by $m$, the graph $G$ consisting of the disjoint union of $\frac{n}{m}$ many copies of $K_m$ is clearly $\mathcal{F}_m$-saturated, and the number of edges in $G$ is $\frac{n}{m} \binom{m}{2}$, which proves the result. \[family\] For every $n \ge m \ge 4$ where $n$ is not divisible by $m$, we have $\operatorname{sat}(n, \F_m) \ge \frac{n-m}{m} \left(\binom{m}{2} + 1\right)$. The proof of Lemma \[family\] will easily follow from the next three lemmas about the structure of $\F_m$-saturated graphs. Let $G$ be an $\F_m$-saturated graph on $n$ vertices. Let $B$ be the set of all vertices of $G$ which are contained in any copy of $K_m$. \[B\] The subgraph induced by $B$ is only a disjoint union of $K_m$’s. First, note that no subgraph of $G$ is isomorphic to $F_{m,j}$ for any $j \in \{1, 2, \dots, m-1\}$, where $F_{m,j}$ denotes the union of $2$ copies of $K_m$ overlapping in exactly $j$ common vertices. This is because each $F_{m,j}$ contains a copy of $V_m$ or $\Lambda_m$. As $B$ does not have any copies of $F_{m,j}$ for all $j$, all copies of $K_m$ induced by $B$ are pairwise disjoint. Furthermore, the subgraph of $G$ that $B$ induces is just a disjoint union of $K_m$’s, because any other edge would create a copy of $B_{m,m}$ in $G$. Now let $A$ be the set of all vertices not in $B$. Since the structure in $B$ is so simple, our lower bound will follow by independently lower-bounding the number of edges induced by $A$, and the number of edges between $A$ and $B$. We start with $A$. \[A\] The set $A$ has at most $m$ vertices, or $A$ is $K_m$-saturated. If $A$ is complete, then the number of vertices in $A$ is at most $m$, or else $G$ contains a copy of $K_{m+1}$, and hence $G$ contains a copy of $\Lambda_m$, which is a contradiction. So, suppose $A$ is not complete. We claim that adding any edge to the induced graph on $A$ must create a copy of $K_m$ in $G$. Suppose for the sake of contradiction that there is a non-edge $uv$ with $u,v \in A$ such that adding $uv$ does not create a copy of $K_m$ in $G$. However, it must create a copy of one of the graphs $B_{m,m}$, $V_m$, or $\Lambda_m$, hence one of $u$ or $v$ must be in a copy of a $K_m$ in $G$ (because these three graphs have the property that for all edges $ab$, either $a$ or $b$ is in a copy of $K_m$), which contradicts the definition of $A$. Finally, we show that adding any edge to the induced graph on $A$ creates a copy of $K_m$ which entirely lies in $A$. Suppose for the sake of contradiction that there is a non-edge $uv$ in the induced graph on $A$ which, if added, would create a copy of $K_m$ which intersects $B$. Let $w \in B$ be a vertex which lies in a created copy of $K_m$ after adding the edge $uv$. That means that $G$ has the edges $uw$ and $vw$, and the copy of $K_m$ in $B$ containing $w$, together with the edges $uw$ and $vw$, creates a copy of $V_m$. So, this is not possible, and we conclude that the induced subgraph on $A$ is indeed $K_m$-saturated. It only remains to bound the number of edges between $A$ and $B$. We have the following structural lemma. \[AB\] If $A$ is non-empty, then each copy of $K_m$ in $B$ has at least one edge to $A$. Assume for the sake of contradiction that there is a copy $U$ of $K_m$ in $B$ which does not have an edge to $A$. Consider arbitrary vertices $u \in U$ and $v \in A$. Then one of the following situations must happen.\ *Case 1: Adding $uv$ creates a copy of $K_m$.* Then it is easy to check that $U$ has an edge to $A$, which is a contradiction.\ *Case 2: Adding $uv$ creates a copy of $B_{m,m}$ with $uv$ being the middle edge connecting the copies of $K_m$.* This would imply that $v$ is in a copy of $K_m$, which is a contradiction.\ *Case 3: Adding $uv$ creates a copy of $V_m$ or $\Lambda_m$ with $uv$ being one of the two extra edges outside of the copy of $K_m$.* Then if $uv$ becomes one of the extra edges, the other extra edge should already be there and will connect $U$ and $A$, giving a contradiction.\ Since all the cases give contradictions, we are done. We now combine the previous three lemmas to prove Lemma \[family\], which then finishes the proof of Theorem \[pro\]. Let $n$ and $m$ satisfy the conditions of Lemma \[family\]. Clearly $A$ must be non-empty because the number of vertices in $B$ is a multiple of $m$ by Lemma \[B\], and so Lemma \[AB\] implies that there are at least $k$ edges between $B$ and $A$, where $k$ is the number of disjoint copies of $K_m$ in $B$. Now from Lemma \[A\], we have two situations. When $A$ has at most $m$ vertices, using Lemmas \[B\] and \[AB\], the number of edges in $G$ is at least $\big\lfloor\frac{n}{m}\big\rfloor \binom{m}{2} + \big\lfloor\frac{n}{m}\big\rfloor \ge \frac{n-m}{m} \left(\binom{m}{2} + 1\right)$. Otherwise, $A$ is $K_m$-saturated, so Theorem \[EHM\] implies that for all $m \ge 4$ the number of edges in $G$ is at least: $$\begin{aligned} & k \binom{m}{2} + k + \left(n - (k+1)m + 2\right) (m-2) \\ & > (km) \frac{m-1}{2} + k + \left(n - (k+1)m\right) \left(\frac{m-1}{2} + \frac{1}{m}\right) \\ & = (km) \frac{m-1}{2} + \left(n - (k+1)m\right) \frac{m-1}{2} + k + \left(n - (k+1)m\right) \frac{1}{m} \\ & = \frac{n-m}{m} \left(\binom{m}{2} + 1\right).\end{aligned}$$ This completes the proof. Family of size 3 for generalized saturation ratio ================================================= Inspired by the construction for Theorem \[pro\], we extend the construction to prove Theorem \[construction\]. One of the key challenges is to find an appropriate extension of Theorem \[EHM\]. Fortunately, our Theorem \[ssat\] rescues us. We start by stating the families of graphs that we will use for the construction, which are not quite the straightforward generalizations of the families used in Theorem \[pro\]. For notational brevity, let $r \ge 2$ be a fixed integer for the remainder of this section. For every positive integer $m \ge 2r^2 + 2r$, let $\F_m$ be the family of the following three graphs. - Let $B_{m,m}$ be the same “dumb-bell" graph from Definition \[def\]. - Let $V_{m,r}$ be the union of a copy of $K_m$ and a copy of $K_{m-r+1}$ overlapping in exactly one common vertex. - Let $\Lambda_{m,r}$ be a copy of $K_m$ plus a single vertex with exactly $r$ edges incident to the $K_m$. Note that for $r = 2$, we have $\Lambda_{m,2} = \Lambda_m$. However, $V_{m,r}$ is not quite a generalization of $V_m$, and in fact $V_m$ is a subgraph of $V_{m,2}$. We considered $V_m$ instead of $V_{m,2}$ in the case of $r = 2$ to make the analysis simpler and more elegant. So, the above construction actually gives different families of three graphs with non-converging saturation ratio for $r = 2$. We proceed to the proof of Theorem \[construction\]. It turns out that the behavior of $\operatorname{sat}(n, K_r, \F_m)$ is similar to before, i.e., it depends on whether or not $n$ is divisible by $m$. The following two lemmas constitute the proof. For every $n$ divisible by $m$, we have $\operatorname{sat}(n, K_r, \F_m) \le \frac{n}{m} \binom{m}{r}$. The same graph used in the proof of Lemma \[easy\], i.e., the disjoint union of $\frac{n}{m}$ many copies of $K_m$, gives us the desired upper bound. \[family’\] For every $n \ge m \ge 2r^2 + 2r$ where $n$ is not divisible by $m$, we have that $\operatorname{sat}(n, K_r, \F_m) \ge \frac{n}{m} \left(\binom{m}{r} + 1\right) - o(n)$. Similarly to Lemma \[family\], the proof of Lemma \[family’\] will follow from the next couple of structural lemmas about $\F_m$-saturated graphs. Let $G$ be an $\F_m$-saturated graph on $n$ vertices. Let $B$ be the set of all vertices of $G$ which are contained in any copy of $K_m$. The subgraph induced by $B$ is only a disjoint union of $K_m$’s, by essentially the same proof as Lemma \[B\]. Now let $A$ be the set of all vertices not in $B$. Motivated by Lemma \[A\], we have the following lemma. \[A’\] $A$ has at most $m$ vertices, or $A$ is strongly $K_{m-r}$-saturated. This is in contrast to Lemma \[A\], which got that the induced graph on $A$ was $K_m$-saturated. Here we only get strongly $K_{m-r}$-saturated (recall that despite its counterintuitive name, strong saturation is a weaker condition), but we can later use our Theorem \[ssat\] to lower-bound the number of copies of $K_r$ in $A$. If $A$ is complete, then the number of vertices in $A$ is at most $m$, or else $G$ contains a copy of $K_{m+1}$, and hence $G$ contains a copy of $\Lambda_{m,r}$, which is a contradiction. So, suppose $A$ is not complete. Fix a non-edge $uv$ in the induced graph on $A$. We consider two cases.\ *Case 1: Adding $uv$ would create a copy of $K_m$ in $G$.* We will show that the copy of $K_m$ would lie entirely in $A$, giving the required $K_{m-r}$ in $A$. Indeed, assume for the sake of contradiction that there is a non-edge $uv$ in the induced graph on $A$ which, if added, would create a copy of $K_m$ which intersects $B$. That implies that there is a copy $T$ of $K_{m-1}$ which contains the vertex $u$ and intersects $B$. Clearly $T$ can intersect only a single copy $U$ of $K_m$ in $B$, because the induced graph on $B$ is just a disjoint union of $K_m$’s. Now, if $|T \cap U| \ge r$, then $T \cup U$ contains a copy of $\Lambda_{m,r}$, which is a contradiction. Otherwise, $|T \cap U| < r$, and so $T \cup U$ contains a copy of $V_{m,r}$, which is also a contradiction.\ *Case 2: Adding $uv$ would not create a copy of $K_m$ in $G$.* If adding $uv$ creates a copy of $B_{m,m}$ or $\Lambda_{m,r}$ in $G$, then one of $u$ or $v$ must be in a copy of a $K_m$ in $G$, which contradicts the definition of $A$. Alternatively, if adding $uv$ creates a copy of $V_{m,r}$ in $G$, then that copy of $V_{m,r}$ would contain a copy of $K_m$ in $B$, together with $m-r$ vertices in $A$. These $m-r$ vertices would clearly induce a copy of $K_{m-r}$ after adding $uv$. Hence we are done. Next, following the proof of Lemma \[family\], we bound the number of $K_r$’s that intersect both $A$ and $B$. \[AB’\] Suppose $m \ge 2r + 1$. If $A$ is non-empty, then for each copy $U$ of $K_m$ in $B$, there is at least one copy of $K_r$ intersecting both $U$ and $A$. Assume for the sake of contradiction that there is a copy $U$ of $K_m$ in $B$ for which there is no copy of $K_r$ intersecting both $U$ and $A$. Consider arbitrary non-adjacent vertices $u \in U$ and $v \in A$. One of the following situations must happen.\ *Case 1: Adding $uv$ creates a copy $T$ of $K_{m-r}$.* Then it is easy to check that there is a copy of $K_{m-r-1}$ (and hence a copy of $K_r$ if $m \ge 2r + 1$) intersecting both $U$ and $A$, which is a contradiction.\ *Case 2: Adding $uv$ creates a copy of $B_{m,m}$ with $uv$ being the middle edge connecting the copies of $K_m$.* This case is exactly the same as before, i.e., $v$ is in a copy of $K_m$, which is a contradiction.\ *Case 3: Adding $uv$ creates a copy of $\Lambda_{m,r}$ with $uv$ being one of the $r$ extra edges outside of the copy of $K_m$.* Then if $uv$ becomes one of the extra $r$ edges, the $r-1$ endpoints in $U$ of the remaining $r-1$ extra edges, together with the vertex $v$, induce a copy of $K_r$, giving a contradiction.\ Since all the cases give contradictions, we are done. Let $n$ and $m$ satisfy the conditions of Lemma \[family’\]. Clearly $A$ must be non-empty because the number of vertices in $B$ is a multiple of $m$, and so Lemma \[AB’\] implies that there are at least $k$ copies of $K_r$ intersecting both $B$ and $A$, where $k$ is the number of disjoint copies of $K_m$ in $B$. Now from Lemma \[A’\], we have two situations. When $A$ has at most $m$ vertices, by Lemma \[AB’\], the number of copies of $K_r$ in $G$ is at least $\big\lfloor\frac{n}{m}\big\rfloor \binom{m}{r} + \big\lfloor\frac{n}{m}\big\rfloor \ge \frac{n-m}{m} \left(\binom{m}{r} + 1\right)$. Otherwise, $A$ is strongly $K_{m-r}$-saturated, so Theorem \[ssat\] implies that $A$ induces at least $\binom{m-r-2}{r-1} (n-km) - o(n)$ many copies of $K_r$, and so for all $m \ge 2r^2 + 2r$ the number of copies of $K_r$ in $G$ is at least: $$\begin{aligned} k \binom{m}{r} + k + & \binom{m-r-2}{r-1} (n-km) - o(n). \label{binomial}\end{aligned}$$ To get the required lower bound, we next prove the simple claim that $\binom{m-r-2}{r-1} \ge \frac{1}{m} \left(\binom{m}{r} + 1\right)$ for all $m \ge 2r^2 + 2r$ and $r \ge 2$. The most convenient way to do this is to show that $m \binom{m-r-2}{r-1} > \binom{m}{r}$, since both sides of this last inequality are integers. Indeed, let $m$ and $r$ satisfy the conditions we just mentioned. Then, $$\frac{m-1}{m-r-2} \le \frac{m-2}{m-r-3} \le \cdots \le \frac{m-r+1}{m-2r} \le \frac{2r^2 + r + 1}{2r^2}.$$ Hence, $$\frac{\binom{m}{r}}{m \binom{m-r-2}{r-1}} \le \frac{1}{r} \left(1 + \frac{r + 1}{2r^2}\right)^{r-1} \le \frac{1}{r} \cdot e^{\frac{(r+1)(r-1)}{2r^2}} \le \frac{1}{r} \cdot \sqrt{e} < 1,$$ which establishes the claim that $\binom{m-r-2}{r-1} \ge \frac{1}{m} \left(\binom{m}{r} + 1\right)$. Using this, we get that is at least $k \binom{m}{r} + k + \frac{1}{m} \left(\binom{m}{r} + 1\right) (n-km) - o(n) \ge \frac{n}{m} \left(\binom{m}{r} + 1\right) - o(n)$, completing the proof. Irregular behavior of $\boldsymbol{\operatorname{sat}(n, C_r, F_r)}$ ==================================================================== In this section, we prove Theorem \[easy\_cons\]. In particular, we will prove that for every $r \ge 4$, $\lim \inf_{n \rightarrow \infty} \operatorname{sat}(n, C_{r+1}, B_{r,r}) = 0$, and $\lim \sup_{n \rightarrow \infty} \operatorname{sat}(n, C_{r+1}, B_{r,r}) > 0$, where $B_{r,r}$ is the same “dumb-bell” graph from Definition \[def\]. We remark here that this statement is false for $r = 2$ and $r = 3$, which we show in Proposition \[excess\] at the end of this section. The following two lemmas constitute the entire proof of Theorem \[easy\_cons\]. \[si\] For every $n$ divisible by $r$, we have $\operatorname{sat}(n, C_{r+1}, B_{r,r}) = 0$. The same graph used in the proof of Lemma \[easy\], i.e., the disjoint union of $\frac{n}{r}$ many copies of $K_r$, is $B_{r,r}$-saturated but has no copies of $C_{r+1}$, proving the lemma. \[cycle\_sat\] For every $n \ge 2r$, and $r \ge 4$ such that $n$ is not divisible by $r$, we have $\operatorname{sat}(n, C_{r+1}, B_{r,r}) \ge 1$. Let $G$ be a $B_{r,r}$-saturated graph on $n$ vertices. We show that there is a cycle of length $r+1$ in $G$ if the conditions of Lemma \[cycle\_sat\] are met. We divide the proof in two cases.\ *Case 1: There is a copy of $F_{r,j}$ in $G$ for some $j \in \{1, 2, \cdots, r-1\}$, where $F_{r,j}$ denotes the union of $2$ copies of $K_r$ overlapping in exactly $j$ common vertices.* It is easy to check that $F_{r,j}$ contains a copy of $C_{r+1}$ for every $j \ge 2$. Hence, if $G$ contains a copy of $F_{r,j}$ for some $j \ge 2$, then there is already a cycle of length $r+1$ in $G$. So, we can assume that $G$ contains a copy of $F_{r,1}$. Assume that $w, u_1, u_2, \cdots, u_{r-1}, v_1, v_2, \cdots, v_{r-1}$ are distinct vertices such that $\{w, u_1, \cdots, u_{r-1}\}$ and $\{w, v_1, \cdots, v_{r-1}\}$ both induce $K_r$. Note that if there is an edge $u_iv_j$ for some $i,j$, then it is easy to find a copy of $C_{r+1}$ using the edge $u_iv_j$. For example, if $u_1v_1$ is an edge, then $w v_1 u_1 u_2 \cdots u_{r-1} w$ forms a $C_{r+1}$. So, we can assume that there is no edge $u_iv_j$ for any $i,j$. Now one of the following situations must happen.\ *Subcase 1: Adding $u_1v_1$ creates a copy of $K_r$.* So, $u_1$ and $v_1$ must have at least $r-2$ common neighbors. If $r \ge 4$, then among $r-2 \ge 2$ common neighbors of $u_1$ and $v_1$, we can pick a vertex $x$ which is distinct from $w$. Now it is easy to check that $w v_1 x u_1 u_2 \cdots u_{r-2} w$ forms a cycle of length $r+1$.\ *Subcase 2: Adding $u_1v_1$ creates a copy of $B_{r,r}$ with $u_1v_1$ being the middle edge connecting the copies of $K_m$.* Hence, there is either a copy of $K_r$ containing $u_1$ and not containing any vertex in $\{w, v_1, \cdots, v_{r-1}\}$, or a copy of $K_r$ containing $v_1$ and not containing any vertex in $\{w, u_1, \cdots, u_{r-1}\}$. Due to symmetry, it is enough to check the first situation. If there is a copy of $K_r$ containing $u_1$ and not containing any vertex in $\{w, v_1, \cdots, v_{r-1}\}$, then that copy of $K_r$ along with the copy of $K_r$ induced by $\{w, v_1, \cdots, v_{r-1}\}$ and the edge $u_1w$ forms a copy of $B_{r,r}$, which is a contradiction.\ *Case 2: There is no copy of $F_{r,j}$ in $G$ for any $j \in \{1, 2, \cdots, r-1\}$.* Let $B$ be the set of all vertices of $G$ which are contained in any copy of $K_r$. Firstly note that $B$ cannot be empty, because there are two disjoint copies of $K_r$ in the graph $B_{r,r}$ and it is not possible to create two disjoint copies of $K_r$ by adding one edge to $G$. The subgraph induced by $B$ is only a disjoint union of $K_r$’s, by the same proof as Lemma \[B\]. Now let $A$ be the set of all vertices not in $B$. Clearly $A$ must be non-empty, because the number of vertices in $B$ is a multiple of $r$, and $r$ does not divide $n$. Fix a copy $T = \{v_1, v_2, \cdots, v_r\}$ of $K_r$ in $B$. Now one of the following situations must happen.\ *Subcase 1: There is at most one vertex in $T$ which has edges to $A$.* If there is no edge between $T$ and $A$, then some easy case-checking (similar to before) implies that adding an edge $v_1 a$ (where $a \in A$) would not create a copy of $B_{r,r}$. Now assume that there is exactly one vertex (without loss of generality $v_1$) in $T$ which has edges to $A$. Again some easy case-checking will tell us that adding $v_2 a$ for any $a \in A$ would not create a copy of $K_r$ (because $r \ge 4$), so, the only way to create a copy of $B_{r,r}$ would be to become the middle edge connecting the copies of $K_r$ in $B_{r,r}$, but that would contradict the fact that $a \in A$ (remember that no vertices in $A$ are in a copy of $K_r$). These are all contradictions.\ *Subcase 2: There are at least two vertices in $T$ which have edges to $A$.* If there is $a \in A$ such that $a$ is adjacent to at least two vertices in $T$, then one can find a cycle of length $r+1$ (for example, without loss of generality $a$ is adjacent to both $v_1$ and $v_2$, so, $v_1 a v_2 v_3 \cdots v_r v_1$ forms a copy of $C_{r+1}$). So, we can assume that for all $a \in A$, the vertex $a$ is adjacent to at most one vertex in $T$. Without loss of generality, $v_1, v_2 \in T$ have edges to $A$. Let $v_1 a_1$ and $v_2 a_2$ be edges for some $a_1, a_2 \in A$. If $a_1 a_2$ is an edge, then $v_1 a_1 a_2 v_2 v_3 \cdots v_{r-1} v_1$ forms a cycle of length $r+1$. Now if $a_1 a_2$ is a non-edge, then adding $a_1 a_2$ would create a copy of $K_r$ (because it must create a copy of $B_{r,r}$, but $a_1 a_2$ could not become the middle edge in the created copy of $B_{r,r}$ due to the fact that $a_1$ is not in a copy of $K_r$ in $G$). Note that $a_1$ and $a_2$ cannot have a common neighbor in $T$, so they must have a common neighbor $x \not \in T$, which implies that $v_1 a_1 x a_2 v_2 \cdots v_{r-2}$ forms a cycle of length $r+1$ in $G$.\ Since all the cases either find a cycle of length $r+1$ or give contradictions, we are done. \[excess\] For $r = 2$ and $r = 3$, we have $\operatorname{sat}(n, C_{r+1}, B_{r,r}) = 0$ for all $n \ge 2r$. In Lemma \[si\], we have already seen that $\operatorname{sat}(n, C_{r+1}, B_{r,r}) = 0$ when $n$ is divisible by $r$. So, we have to prove Proposition \[excess\] when $n$ is not divisible by $r$. The graph $B_{2,2}$ is the path with 3 edges, i.e., $P_3$. A graph which is a disjoint union of $P_2$’s and $P_1$’s is always $P_3$-saturated. If $n \ge 4$ is odd, then the graph consisting of the disjoint union of a copy of $P_2$ and $\frac{n-3}{2}$ many copies of $P_1$ is a $P_3$-saturated graph with no copy of $C_3$. So, we have $\operatorname{sat}(n, C_3, B_{2,2}) = 0$ for all $n \ge 4$. For $r = 3$, we split into two cases depending on the value of $n \pmod 3$. If $n$ is of the form $3m + 1$ for some integer $m$, then the graph consisting of $m$ disjoint copies of $K_3$ together with $m$ edges connecting an extra vertex to each copies of $K_3$, is a $B_{3,3}$-saturated graph without any copy of $C_4$. So, for $n \equiv 1 \pmod 3$, we have $\operatorname{sat}(n, C_4, B_{3,3}) = 0$. Now when $n$ is of the form $3m + 2$, we have a similar construction. Consider a graph $G$ on the vertex set $\{a,b\} \cup \{x_1, x_2, \cdots, x_m\} \cup \{y_1, \cdots, y_m\} \cup \{z_1, \cdots, z_m\}$, and the edge set $\{x_jy_j : j \in [m]\} \cup \{y_jz_j : j \in [m]\} \cup \{z_jx_j : j \in [m]\} \cup \{ax_j : j \in [m]\} \cup \{by_j : 2 \le j \le m\} \cup \{bx_1\}$. It is easy to verify that $G$ is $B_{3,3}$-saturated, and does not have a copy of $C_4$. Hence, we have $\operatorname{sat}(n, C_4, B_{3,3}) = 0$ for all $n \ge 6$. Concluding remarks ================== We end with some open problems. We determined the exact value of $\operatorname{sat}(n, K_r, K_s)$ for all sufficiently large $n$, but our arguments do not extend to find the value for small $n$. So, the following question still remains open. For $s > r \ge 3$, determine the exact value of $\operatorname{sat}(n, K_r, K_s)$ for all $n$. We have already made a remark on the maximum constant $c_r$ we can write in the stability result in Theorem \[sat\]. It would be interesting to determine that maximum constant. For $s > r \ge 3$, what is the second smallest number of copies of $K_r$ in an $n$-vertex $K_s$-saturated graph? It might be interesting to consider a more general problem of finding the spectrum (set of possible values) of the number of copies of $K_r$ in a $K_s$-saturated graph. The $r = 2$ case, i.e., the edge spectrum of $K_s$-saturated graphs, was completely solved in [@AFGS] and [@BCFF]. For $s > r \ge 3$, what are the possible numbers of copies of $K_r$ in an $n$-vertex $K_s$-saturated graph? We could not extend our method to find the exact value of $\operatorname{sat}(n, C_r, K_s)$ for the situations when $r = 5$, and when $r$ is an even number with $r = O(\sqrt{s})$. So, it will be interesting to find the values of $\operatorname{sat}(n, C_r, K_s)$ for all $r$. We conjecture that the extremal graph $K_{s-2} \ast \overline{K}_{n-s+2}$ is the unique graph minimizing the number of cycles of length $r$ among all $n$-vertex $K_s$-saturated graphs. It is worth mentioning that in Lemma \[upperbound\], if we can prove that any $n$-vertex $K_s$-saturated graph with the minimal number of copies of $C_r$ has $o(n^{\frac{3}{2}})$ edges for even $r$, then the proof of Theorem \[cycle\] for odd $r$ in Section 5 can be adapted for even $r$ as well, and it will prove our conjecture for even $r \ge 6$. For every $s \ge 4$ and $r \le 2s - 4$, compute the exact value of $\operatorname{sat}(n, C_r, K_s)$. Theorem \[sat\] and Theorem \[cycle\] motivate us to ask the following general question. Is there a graph $F$, for which $K_{s-2} \ast \overline{K}_{n-s+2}$ does not (uniquely) minimize the number of copies of $F$ among $n$-vertex $K_s$-saturated graphs for all sufficiently large $n$? As we mentioned earlier, Conjecture \[con\] is still wide open and likely needs new ideas to settle it. It would be interesting to figure out if the size of the family in Theorem \[pro\] can be further reduced to $2$. 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--- abstract: 'We study the effect of starlight from the first stars on the ability of other minihaloes in their neighbourhood to form additional stars. The first stars in the $\Lambda$CDM universe are believed to have formed in minihaloes of total mass $\sim 10^{5-6}\,M_\odot$ at redshifts $z\ga 20$, when molecular hydrogen ($\rm H_2$) formed and cooled the dense gas at their centres, leading to gravitational collapse. Simulations suggest that the Population III (Pop III) stars thus formed were massive ($\sim 100\,M_\odot$) and luminous enough in ionizing radiation to cause an ionization front (I-front) to sweep outward, through their host minihalo and beyond, into the intergalactic medium. Our previous work suggested that this I-front was trapped when it encountered other, nearby minihaloes, and that it failed to penetrate the dense gas at their centres within the lifetime of the Pop III stars ($\la 3\,\rm Myrs$). The question of what the dynamical consequences were for these target minihaloes, of their exposure to the ionizing and dissociating starlight from the Pop III star requires further study, however. Towards this end, we have performed a series of detailed, 1D, radiation-hydrodynamical simulations to answer the question of whether star formation in these surrounding minihaloes was triggered or suppressed by radiation from the first stars. We have varied the distance to the source (and, hence, the flux) and the mass and evolutionary stage of the target haloes to quantify this effect. We find: (1) trapping of the I-front and its transformation from R-type to D-type, preceded by a shock front; (2) photoevaporation of the ionized gas (i.e. all gas originally located outside the trapping radius); (3) formation of an $\rm H_2$ precursor shell which leads the I-front, stimulated by partial photoionization; and (4) the shock- induced formation of $\rm H_2$ in the minihalo neutral core when the shock speeds up and partially ionizes the gas. The fate of the neutral core is mostly determined by the response of the core to this shock front, which leads to molecular cooling and collapse that, when compared to the same halo without external radiation, is either: (a) expedited, (b) delayed, (c) unaltered, or (d) reversed or prevented, depending upon the flux (i.e. distance to the source) and the halo mass and evolutionary stage. When collapse is expedited, star formation in neighbouring minihaloes or in merging subhaloes within the host minihalo sometimes occurs [*within*]{} the lifetime of the first star. Roughly speaking, most haloes that were destined to cool, collapse, and form stars in the absence of external radiation are found to do so even when exposed to the first Pop III star in their neighbourhood, while those that would not have done so are still not able to. A widely held view that the first Pop III stars must exert either positive or negative feedback on the formation of the stars in neighbouring minihaloes should, therefore, be revisited.' author: - | Kyungjin Ahn[^1] and Paul R. Shapiro[^2]\ Department of Astronomy, The University of Texas at Austin, 1 University Station C1400, Austin, TX 78712, USA title: 'Does Radiative Feedback by the First Stars Promote or Prevent Second Generation Star Formation?' --- cosmology: large-scale structure of universe – cosmology: theory – early universe – stars: formation – galaxies: formation Introduction {#sec:Secondstar-Intro} ============ Cosmological minihaloes at high redshift – i.e. dark-matter dominated haloes with virial temperatures $T_{\rm vir} < 10^4 \,\rm K$, with masses above the Jeans mass in the intergalactic medium (IGM) before reionization ($10^4 \la M/M_\odot \la 10^8$) – are believed to have been the sites of the first star formation in the universe. To form a star, the gas inside these haloes must first have cooled radiatively and compressed, so that the baryonic component could become self-gravitating and gravitational collapse could ensue. For the neutral gas of H and He at $T < 10^4\,\rm K$ inside minihaloes, this requires that a sufficient trace abundance of $\rm H_2$ molecules formed to cool the gas by atomic collisional excitation of the rotational-vibrational lines of $\rm H_2$ . The formation of this trace abundance of $\rm H_2$ proceeds via the creation of intermediaries, $\rm H^-$ or $\rm H_{2}^{+}$, which act as catalysts, which in turn requires the presence of a trace ionized fraction, in the following two-step gas-phase reactions (see, e.g., @1968ApJ...154..891P [@1967Natur.216..976S; @1984ApJ...280..465L; @1987ApJ...318...32S]; @1994ApJ...427...25S, henceforth, “SGB94”; ): $$\begin{aligned} &&{\rm H + e^- \rightarrow H^- + \gamma},\nonumber \\ &&{\rm H^- + H \rightarrow H_2 + e^-}, \label{eq:solomon}\end{aligned}$$ and $$\begin{aligned} &&{\rm H + H^+ \rightarrow H_{2}^{+} + \gamma},\nonumber \\ &&{\rm H_{2}^{+} + H \rightarrow H_2 + H^+}. \label{eq:solomon2}\end{aligned}$$ Unless there is a strong destruction mechanism for $\rm H^-$ (e.g. cosmic microwave background at $z\ga 100$), the former (equation \[eq:solomon\]) is generally the dominant process for $\rm H_2$ formation. Gas-dynamical simulations of the Cold Dark Matter (CDM) universe suggest that the first stars formed in this way when the dense gas at the centres of minihaloes of mass $M \sim 10^{5 - 6}\, M_\odot$ cooled and collapsed gravitationally at redshifts $z \ga 20$ (e.g. @2000ApJ...540...39A [@2002Sci...295...93A]; @1999ApJ...527L...5B [@2002ApJ...564...23B]; @2003ApJ...592..645Y; @2001ApJ...548..509M [@2003MNRAS.338..273M]; @2006astro.ph..6106Y). This work and others further suggest that these stars were massive ($M_* \ga 100 \,M_\odot$), hot ($T_{\rm eff} \simeq 10^5 \,\rm K$), and short-lived ($t_* \la 3 \,\rm Myrs$), thus copious emitters of ionizing and dissociating radiation. These stars constitute the Population III (Pop III) stars, or zero metallicity stars, which are believed to have exerted a strong, radiative feedback on their environment. The details of this feedback and even the overall sign (i.e. negative or positive) are poorly understood. Once the ionizing radiation escaped from its halo of origin, it created H II regions in the IGM, beginning the process of cosmic reionization. The photoheating which accompanies this photoionization raises the gas pressure in the IGM, thereby preventing baryons from collapsing gravitationally out of the IGM into new minihaloes when they form inside the H II regions, an effect known as “Jeans-mass filtering” (SGB94; @1998MNRAS.296...44G; @2003MNRAS.346..456O). Inside the H II regions, whenever the I-fronts encounter pre-existing minihaloes, those minihaloes are subject to photoevaporation (@2004MNRAS.348..753S [henceforth, SIR]; @2005MNRAS.361..405I [henceforth, ISR]). A strong background of UV photons in the Lyman-Werner (LW) bands of $\rm H_2$ also builds up which can dissociate molecular hydrogen inside minihaloes even in the neutral regions of the IGM, thereby disabling further collapse and, thence, star formation (e.g. @1999ApJ...518...64O; @2000ApJ...534...11H; @2001ApJ...546..635O). This conclusion changes, however, if some additional sources of partial ionization existed to stimulate $\rm H_2$ formation without heating the gas to the usually high temperature of fully photoionized gas ($\sim 10^4 \,\rm K$) at which collisional dissociation occurs, such as X-rays from miniquasars [@1996ApJ...467..522H] or if stellar sources create a partially-ionized boundary layer outside of intergalactic H II regions [@2001ApJ...560..580R]. Such positive feedback effects, however, may have been only temporary, because photoheating would soon become effective as background flux builds up over time [@2006MNRAS.368.1301M]. The study of feedback effects has been limited mainly by technical difficulties. @2000ApJ...534...11H studied the feedback of LW, ultraviolet (UV), and X-ray backgrounds on minihaloes without allowing hydrodynamic evolution. @2001ApJ...560..580R studied the radiative feedback effect of stellar sources only on a static, uniform IGM. @2002ApJ...575...33R [@2002ApJ...575...49R] studied stellar feedback more self-consistently by performing cosmological hydrodynamic simulations with radiative transfer, but the resolution of these simulations is not adequate for resolving minihaloes. @2001ApJ...548..509M [@2003MNRAS.338..273M] also performed cosmological hydrodynamic simulations, with higher resolution, but radiative feedback was treated assuming the optically thin limit, which overestimates the ionization efficiency, especially in the high density regions which would initially be easily protected from ionizing radiation due to their high optical depth. The first self-consistent, radiation-hydrodynamical simulations of the feedback effect of external starlight on cosmological minihaloes were those of SIR and ISR, who studied the encounter between the intergalactic I-fronts that reionized the universe and individual minihaloes along their path. These simulations used Eulerian, grid-based hydrodynamics with radiative transfer and adaptive mesh refinement (AMR) to “zoom-in” with very high resolution, to demonstrate that the I-fronts from external ionizing sources are trapped when they encounter minihaloes, slowing down and transforming from weak, R-type to D-type, preceded by a shock. The gas on the ionized side of these I-fronts was found to be evaporated in a supersonic wind, and, if the radiative source continued to shine for a long enough time, the I-front eventually penetrated the minihaloes entirely and expelled all of the gas. These simulations elucidated the impact of the I-front and the physical effects of ionizing radiation on minihalo gas, quantifying the timescales and photon consumption required to complete the photoevaporation. They did not, however, address the aftermath of “interrupted” evaporation, when the source turns off before evaporation is finished. Recent studies by @2005ApJ...628L...5O, @2006ApJ...639..621A, and @2006astro.ph..4148M addressed this question for minihaloes exposed to the radiation from the first Pop III star in their neighbourhood, instead of the effect of either a steadily-driven I-front during global reionization or a uniform global background. The results of @2005ApJ...628L...5O and @2006astro.ph..4148M are seriously misleading, however, since they did not account properly for the optical depth to hydrogen ionizing photons. @2005ApJ...628L...5O assumed that the UV radiation from the first Pop III star that formed inside a minihalo in some region would fully ionize the gas in the neighbouring minihaloes. Using 3D hydrodynamics simulations, they found that, when the star turned off, $\rm H_2$ molecules formed in the dense gas that remained at the centre of the neighbouring minihalo, fast enough to cool the gas radiatively and cause gravitational collapse leading to more star formation. The $\rm H_2$ formation mechanism was the same as that described by @1987ApJ...318...32S, in which ionized gas of primordial composition at a temperature $T\ga 10^4 \,\rm K$ cools radiatively and recombines out of ionization equilibrium, enabling an enhanced residual ionized fraction to drive reaction (1) (and \[2\], as well) as the temperature falls below the level at which collisional dissociation suppresses molecule formation. As a result, @2005ApJ...628L...5O concluded that the radiative feedback of the first Pop III stars was positive, triggering a second generation of star formation in the minihaloes surrounding the one that hosted the first star. @2006astro.ph..4148M also used 3D hydrodynamics simulations to consider the fate of the gas in the relic H II regions created by the first Pop III stars. they concluded that the radiative feedback of the first stars could be either negative or positive and estimated a critical UV intensity which would mark the transition from negative to positive feedback. @2006astro.ph..4148M, however, studied this effect only in the optically thin limit, as had also been done by @2001ApJ...548..509M [@2003MNRAS.338..273M]. The main mechanisms of the positive feedback effect in @2005ApJ...628L...5O and @2006astro.ph..4148M are, therefore, identical. @2006ApJ...639..621A, on the other hand, performed a high-resolution ray-tracing calculation to track the position of the I-front created by the first Pop III star as it swept outward in the density field of a 3D cosmological SPH simulation of primordial star formation in the $\Lambda$CDM universe over the lifetime of the star. When this I-front encountered the minihaloes in the neighbourhood of the one which hosted the first Pop III star, it was trapped by the minihalo gas before it could reach the high-density region (core), due to the minihalo’s high column density of neutral hydrogen. This is consistent with the results of SIR and ISR mentioned above. According to @2006ApJ...639..621A, in fact, the lifetime of the Pop III star is less than the evaporation times determined by SIR and ISR for the relevant minihalo masses and flux levels in this case, so the neutral gas in the core is never ionized by the I-front. It seems that the initial assumption of full ionization of nearby haloes by @2005ApJ...628L...5O and the optically thin limit assumed by @2006astro.ph..4148M are invalid. The final fate of this protected neutral core, however, is still unclear, because the I-front tracking calculations by @2006ApJ...639..621A did not include the hydrodynamical response of the minihalo gas to its ionization, a full treatment of radiative transfer or the primordial chemistry involving $\rm H_2$. One might naively expect that the nett effect would be negative, because heating from photoionization would ultimately expel most of gas from minihaloes, although the results of SIR and ISR, again, show that this minihalo evaporation would not be complete within the lifetime of the Pop III star. On the other hand, partial ionization beyond the I-front by hard photons from a Pop III star might be able to [*promote*]{} ${\rm H_2}$ formation, once the dissociating UV radiation from the star is turned off, which would then lead to a cooling and collapsing core. This issue can be addressed only by a fully coupled calculation of radiative transfer, chemistry, and hydrodynamics, which will be the focus of this paper. We shall attempt to answer the following questions: Does the light from the first Pop III star in some neighbourhood promote or prevent the formation of more Pop III stars in the surrounding minihaloes? More specifically, do the neutral cores of these nearby minihaloes, which are shielded from the ionizing radiation from the external Pop III star, subsequently cool and collapse gravitationally, as they must in order to form stars, or are they prevented from doing so? Towards this end, we simulate the evolution of these target haloes under the influence of an external Pop III star using the 1-D spherical, Lagrangian, radiation-hydrodynamics code we have developed. We adopt a $120\,M_\odot$ Pop III star as a source, and place different mass haloes at different distances to explore a wide range of the parameter space for this problem. Masses of target haloes are chosen to span the range from those too low for haloes to cool and collapse by ${\rm H_2}$ cooling without external radiation to those massive enough to do so on their own. Our calculation is the first self-consistent gas-dynamical calculation of the feedback effects of a single Pop III star on nearby haloes. A similar approach by 1-D radiation-hydrodynamics calculation has been performed by @2001MNRAS.326.1353K. Their work, however, focuses on the effect of a steady global background from quasars and from stars with surface temperatures $T_{*} \sim 10^{4} \,{\rm K}$, rather than a single, short-lived Pop III star with $T_{*} \sim 10^{5} \,{\rm K}$. In addition, while we were preparing this manuscript, a study which is similar to our work was reported by @2006ApJ...645L..93S, where a 3D radiation-hydrodynamics calculation with SPH particles was performed[^3]. A major difference of their work from ours is that they focus on the subclumps of the halo which hosts the first Pop III star, while we focus on external minihaloes in the neighbourhood of such a host halo. We also apply a more accurate treatment of ${\rm H_2}$ self-shielding, as well as a more complete chemistry network of neutral and ionic species of H, He, and ${\rm H_2}$. A more fundamental difference from these previous studies is our finding of a novel ${\rm H_2}$ formation mechanism: [*collisional ionization of pre-I-front gas by a shock detached from a D-type I-front*]{}. This mechanism occurs at the centre of target haloes, which would otherwise remain very neutral. This mechanism creates new electrons abundant enough to promote further ${\rm H_2}$ formation, which can even expedite the core collapse. In Section \[sec:code\] we describe the details of the 1-D spherical radiation-hydrodynamics code we have developed. Some details left out in Section \[sec:code\] will be described in Appendices. In Section \[sec:Initial-Setup\], we describe the initial setup of our problem. We briefly describe a test case in Section \[sec:opt-thin\], where we let a minihalo evolve from an initially ionized state, to show that our code reproduces the result of @2005ApJ...628L...5O in that case. In Section \[sec:mcm\] and Section \[sec:2star-Result\], we present the main results of our full radiation-hydrodynamics calculation. We summarize our results in Section \[sec:2star-Discussion\]. Throughout this paper, we use the $\Lambda$CDM cosmological parameters, ($\Omega_\Lambda$, $\Omega_0$, $\Omega_b$, $h$) = ($0.73$, $0.27$, $0.043$, $0.7$), consistent with the [*WMAP*]{} first-year data [@2003ApJS..148..175S][^4]. Numerical Method: 1-D spherical, radiation-hydrodynamics with primordial chemistry network {#sec:code} ========================================================================================== In this section, we describe in detail the 1-D spherical, Lagrangian, radiation-hydrodynamics code we have developed for both dark and baryonic matter. We describe how hydrodynamics, dark matter dynamics, radiative transfer, radiative heating and cooling, and finally the nonequilibrium chemistry are handled. The finite differencing scheme, reaction rates, and certain other details not treated in this section will be described in Appendices. We include the neutral and ionic species of H, He and ${\rm H_{2}}$, namely H, $\rm H^+$, He, ${\rm He^{+}}$, ${\rm He^{++}}$, $\rm H^-$, $\rm H_2$, $\rm H_{2}^+$ and $e^-$, in order to treat the primordial chemistry fully. As deuterium and lithium exist in a negligible amount, we neglect ${\rm D}$ and ${\rm Li}$ species[^5]. Hydrodynamic Conservation Equations {#sub:2star-Hydrodynamics} ----------------------------------- The baryonic gas obeys inviscid fluid conservation equations, $$\frac{\partial\rho}{\partial t}+\frac{\partial}{r^{2}\partial r}(r^{2}(\rho u))=0, \label{eq:realhydro_mass}$$ $$\frac{\partial}{\partial t}(\rho u)+\frac{\partial}{\partial r}(p+\rho u^{2})+\frac{2}{r}\rho u^{2}=-\rho\frac{Gm}{r^{2}}, \label{eq:realhydro_momentum}$$ $$\frac{De}{Dt}=-\frac{p}{\rho}\frac{\partial}{r^{2}\partial r}(r^{2}u)+\frac{\Gamma-\Lambda}{\rho}, \label{eq:realhydro_energy}$$ where $e\equiv(3p)/(2\rho)$ is the internal energy per unit baryon mass, $\Gamma$ is the external heating rate, and $\Lambda$ is the radiative cooling rate. Note that all the variables in equations (\[eq:realhydro\_mass\]) - (\[eq:realhydro\_energy\]) denote baryonic properties, except for $m$, the mass enclosed by a radius $r$, which is composed of both dark and baryonic matter. We do not change the adiabatic index $\gamma$ throughout the simulation. As long as monatomic species, H and He, dominate the abundance, $\gamma=5/3$ is the right value to use. This ratio of specific heats, $\gamma$, can change significantly, however, if a large fraction of H is converted into molecules. For example, the three-body ${\rm H_2}$ formation process, $${\rm H+H+H \rightarrow H_2 + H}, \label{threebody}$$ will occur vigorously when $n_{\rm H}\ga 10^8 \,{\rm cm\, s^{-1}}$ and $T\la 10^{3} \,{\rm K}$, which will invalidate the use of a constant $\gamma$. To circumvent such a problem, when such high density occurs, we simply stop the simulation. This process is, nevertheless, important in forming the protostellar molecular cloud (e.g. @2002Sci...295...93A). This issue will be further discussed in Section \[sec:2star-Result\], when we define the criterion for the collapse of cooling regions. The shock is treated using the usual artificial viscosity technique (e.g. @artvis). The pressure $p$ in equations (\[eq:realhydro\_momentum\]) and (\[eq:realhydro\_energy\]) contains the artificial viscosity term. The details of this implementation are described in Appendix A. Dark Matter Dynamics {#sub:fluid-approx} -------------------- Gravity is contributed both by the dark matter and the baryonic components. Let us first focus on the dark matter component. In order to treat the dark matter gravity under spherical symmetry, almost all previous studies have used either a frozen dark matter potential or a set of self-gravitating dark matter shells in radial motion only (e.g. @1995ApJ...442..480T). Both methods have their own limitations. The frozen potential approximation cannot address the effect of a possible evolution of the gravitational potential. The radial-only dark matter approximation suffers from the lack of any tangential motion, producing a virialized structure whose central density profile is much steeper ($\rho\propto r^{-\beta}$ with $\beta \ge 2$; see e.g. ) than that of haloes in cosmological, 3-D N-body simulations ($\beta \approx 1$, as found in @1997ApJ...490..493N). In order to treat the dynamics of dark matter more accurately than these previous treatments, we use the the fluid approximation we have developed and reported elsewhere [@2005MNRAS.363.1092A]. We briefly summarize its derivation here; for a detailed description, see @2005MNRAS.363.1092A. Collisionless CDM particles are described by the collisionless Boltzmann equation. When integrated, it yields an infinite set of conservation equations, which is called the BBGKY hierarchy (e.g. @1987gady.book.....B). However, CDM N-body simulations show that virialized haloes are well approximated by spherical symmetry. These simulations also show that the velocity dispersions are highly isotropic: radial dispersion is almost the same as the tangential dispersion. These two conditions make it possible to truncate the hierarchy of equations to a good approximation, which then yields only three sets of conservation equations. Amazingly enough, these equations are identical to the normal fluid conservation equations for the adiabatic index $\gamma=5/3$ gas: $$\frac{\partial\rho_d}{\partial t}+\frac{\partial}{r^{2}\partial r}(r^{2}(\rho_d u_d))=0, \label{eq:DM_mass}$$ $$\frac{\partial}{\partial t}(\rho_d u_d)+\frac{\partial}{\partial r}(p_d+\rho_d u_{d}^{2})+\frac{2}{r}\rho_d u_{d}^{2}=-\rho_d \frac{Gm}{r^{2}}, \label{eq:DM_momentum}$$ $$\frac{De_d}{Dt}=-\frac{p_d}{\rho_d}\frac{\partial}{r^{2}\partial r}(r^{2}u_d), \label{eq:DM_energy}$$ where the subscript $d$ represents dark matter, the effective pressure $p_d \equiv \rho_d \left\langle u_d - \left\langle u_d\right\rangle \right\rangle^2$ is the product of the dark matter density and the velocity dispersion at a given radius, and the effective internal energy per dark matter mass $e_d \equiv 3 p_d/ 2 \rho_d$. We use these effective fluid conservation equations (equation \[eq:DM\_mass\], \[eq:DM\_momentum\], \[eq:DM\_energy\]) to handle the motion of dark matter particles. Note that dark matter shells in this code represent a collection of dark matter particles in spherical bins, in order to describe “coarse-grained” properties such as density ($\rho_d$) and the effective pressure ($p_d$). As these coarse-grained variables follow the usual fluid conservation equations, the hyperbolicity of these equations leads to the formation of an effective “shock.” The location of this shock will determine the effective “post-shock” region. This post-shock region corresponds to the dark matter shell-crossing region. Because of the presence of this effective shock, we also use the artificial viscosity technique. This collisional behaviour of our coarse-grained dark matter shells originates from our choice of physical variable. For further details, the reader is referred to @2005MNRAS.363.1092A and @2003RMxAC..18....4A for description and application of our fluid approximation. The mass enclosed by a dark matter shell of radius $r$, $$m(<r)=m_{{\rm DM}}(<r)+m_{{\rm bary}}(<r), \label{eq:totalmass}$$ enters equations (\[eq:realhydro\_momentum\]) and (\[eq:DM\_momentum\]). When computing $m(<r)$, we properly take account of the mismatch of the location of dark matter shells and baryon shells. Radiative transfer ------------------ A full, multi-frequency, radiative transfer calculation is performed in the code. Since ${\rm H_2}$ cooling is of prime importance here, we first pay special attention to calculating the optical depth to UV dissociating photons in the LW bands and the corresponding ${\rm H_2}$ self-shielding function. We then describe how we calculate the optical depth associated with any other species depending upon the location of the radiation source. The finite difference scheme for the calculation of radiative rates is described in the Appendix A. ### Photodissociation of $\rm H_2$ and Self-Shielding {#sub:ss} Hydrogen molecules are photodissociated when a UV photon in the LW bands between $11\,\rm eV$ and $13.6\,\rm eV$ excites $\rm H_2$ to an excited electronic state from which dissociation sometimes occurs. When the column density of ${\rm H_2}$ becomes high enough ($N_{\rm H_2}\ga 10^{14} \,\rm cm^{-2}$), the optical depth to photons in these Lyman-Werner bands can be high, so ${\rm H_2}$ can “self-shield” from dissociating photons. Exact calculation of this self-shielding requires a full treatment of all 76 Lyman-Werner lines, even when only the lowest energy level transitions are included. Such a calculation is feasible under simplified conditions such as a radiative transfer problem through a static medium (e.g. @2000ApJ...534...11H; @2001ApJ...560..580R). Unfortunately, for combined calculations of radiative transfer and hydrodynamics, such a full treatment is computationally very expensive. Under certain circumstances, however, one can use a pre-computed self-shielding function expressed in terms of the molecular column density $N_{{\rm H_{2}}}$ and the temperature $T$ of gas, which saves a great amount of computation time. In a *cold, static* medium, for instance, one can use a self-shielding function provided by @1996ApJ...468..269D: $$F_{{\rm shield}}={\rm min}\left[1,\left(\frac{N_{{\rm H_{2}}}}{10^{14}{\rm cm^{-2}}}\right)^{-3/4}\right]. \label{eq:DB_shield_factor_cold}$$ The photodissociation rate is then given by $$k_{{\rm H_{2}}}=1.38\times10^{9}\left(J_{\nu}\right)_{h\nu=12.87{\rm eV}}F_{{\rm shield}}, \label{eq:DB_rate}$$ where $\left(J_{\nu}\right)_{h\nu=12.87{\rm eV}}$ ($\rm erg\,s^{-1}\,cm^{-2}\,Hz^{-1}\,sr^{-1}$) is the mean intensity in the spectral region of the LW bands. This approximation has been widely used in the study of high redshift structure formation (e.g. @2001MNRAS.326.1353K [@2001MNRAS.321..385G; @2003ApJ...592..645Y; @2004ApJ...613..631K]). The problem with equation (\[eq:DB\_shield\_factor\_cold\]) is that when the gas temperature is high or gas has motion along the line of sight to the source, the thermal and velocity broadening of the LW bands caused by the Doppler effect can significantly reduce the optical depth. A better treatment for thermal broadening is also given by @1996ApJ...468..269D, now in terms of the molecular column density $N_{\rm H_{2}}$ and the velocity-spread parameter $b$ of the gas: $$\begin{aligned} F_{{\rm shield}}= \frac{0.965}{(1+x/b_5)^2} + \frac{0.035}{(1+x)^{0.5}} \nonumber \\ \times \exp[-8.5\times 10^{-4}(1+x)^{0.5}], \label{eq:DB_shield_factor}\end{aligned}$$ where $x\equiv N_{\rm H_2}/5\times 10^{14}\, {\rm cm}^{-2}$, $b_5\equiv b/10^5 {\rm cm \, s^{-1}} $, and $b=1.29 \times 10^4 \left(T_K/A\right)^{1/2} {\rm cm \, s^{-1}},$ where $A$ is the atomic weight [@1978ppim.book.....S]. For ${\rm H_2}$, $b=9.12 \,{\rm km \, s^{-1}} \left(T/10^4 \, {\rm K}\right)^{1/2}$. In the problem treated in this paper, we frequently find $T\approx 10^3 - 5\times 10^3 \, {\rm K}$ in the gas parcel (shell) which contributes most of the ${\rm H_2}$ column density. We also find that this gas parcel usually moves at $v \approx 2-5 \, {\rm km \, s^{-1}}$ (see Section \[sub:H2shell\]). The combined effect of the thermal broadening and the Doppler shift on the shielding function, then, may be well approximated by a thermally broadened shielding function with $T\approx 10^4\,{\rm K}$. Throughout this paper, therefore, we use equation (\[eq:DB\_shield\_factor\]) with $T= 10^4\,{\rm K}$ to calculate the self-shielding. For the photo-dissociation rate, we use equation (\[eq:DB\_rate\]). We show in Fig. \[fig-DB\] how much the static, cold shielding function (equation \[eq:DB\_shield\_factor\_cold\]) may overestimate the self-shielding in our problem, by comparing this to the thermally-broadened shielding function (equation \[eq:DB\_shield\_factor\]) at $T= 10^4\,{\rm K}$. The biggest discrepancy between these two shielding functions exists for $N_{\rm H_2} \approx 10^{14} - 10^{16} \, {\rm cm}^{-2}$. Interestingly enough, the ${\rm H_2}$ column density in our problem usually resides in this regime. It is crucial, therefore, to take into account the effects of thermal broadening and Doppler shift carefully, as we do in this paper. ### External Source {#sub:External-source} Since our calculations are 1-D, spherically-symmetric, we have assumed the external radiation source contributes a radial flux $F_{\nu}^{{\rm ext}}(r)$ at frequency $\nu$ and radius $r$, measured from the minihalo centre, given by $$% % F_{\nu}^{{\rm ext}}(r)= \frac{L_{\nu}^{{\rm ext}}}{4\pi D^{2}}e^{-\tau_{\nu}(>r)}, \label{eq:fnu_ext}$$ where $L_{\nu}^{{\rm ext}}$ is the source luminosity, and $\tau_{\nu}(>r)$ is the optical depth along the radial direction from radius $r$ to the source located at a distance $r=D$. The radiative rate of species $i$ at radius $r$ is then given by $$k_{i}(r)=\int_{0}^{\infty}d\nu\frac{\sigma_{i,\nu}4\pi J_{\nu}(r)}{h\nu} =\int_{0}^{\infty}d\nu\frac{\sigma_{i,\nu}F_{\nu}^{{\rm ext}}(r)}{h\nu}, \label{eq:rad_rate_ext_body}$$ where we have used the fact that $4\pi J_{\nu} = F_{\nu}^{\rm ext}$, as long as the external radiation can be approximated as a 1D planar flux. In practice, one calculates this rate in a given grid-cell – i.e. spherical shell – with finite thickness. If such a grid-cell has a small optical depth, $F_{\nu}^{{\rm ext}}$ is almost constant across the grid, so one could take the grid-centred value of $F_{\nu}^{{\rm ext}}$ to calculate $k_{i}(r)$. This naive scheme, however, does not yield an accurate result when a grid-cell is optically thick, where $F_{\nu}^{{\rm ext}}$ may vary significantly over the cell width. This problem occurs frequently for solving radiative transfer through optically thick media, where individual cells have large optical depth. In order to resolve this problem, we use a “photon-conserving” scheme like that described by @1999MNRAS.309..287R and @1999ApJ...523...66A. The details of our implementation of this scheme are described in the Appendix A. Heating and Cooling {#sub:heating-cooling} ------------------- ### Photoheating {#sub:photo-heating} Photoheating results from thermalization of the residual kinetic energy of electrons after they are photoionized. In general, the photoheating function is described by $$\begin{aligned} \Gamma=\sum_{i}\Gamma_{i}&=&\sum_{i}n_{i}\int_{0}^{\infty}d\nu\frac{4\pi J_{\nu}\sigma_{\nu}}{h\nu}(h\nu-h\nu_{i,{\rm th}}) \nonumber \\ &=&\sum_{i}n_{i}\int_{0}^{\infty}d\nu\frac{ F_{\nu}^{\rm ext}\sigma_{\nu}}{h\nu}(h\nu-h\nu_{i,{\rm th}}), \label{eq:generic_photo_heat}\end{aligned}$$ where $h \nu_{i,{\rm th}}$ is the threshold energy over which the residual photon energy is converted into the kinetic energy of electrons, and the nett heating function $\Gamma$ is the sum of individual heating functions ({$\Gamma_{i}$}). In finite-differencing equation (\[eq:generic\_photo\_heat\]), we also use the photon-conserving scheme as we do for equation (\[eq:rad\_rate\_ext\_body\]). This prevents cells with large optical depth from obtaining unphysically high heating rates. See Appendix A for details. ### Radiative cooling Cooling occurs through various processes. For atomic species, it comes from collisional excitation, collisional ionization, recombination, free-free emission, and CMB photons scattering off free electrons (Compton cooling/heating). For atomic H and He, cooling is dominated by collisional excitation (for $T\la 2\times 10^{5}{\rm K}$) and free-free emission (for $T\ga 2\times 10^{5}{\rm K}$). The atomic cooling rate decreases rapidly at $T\la 10^{4}{\rm K}$, as there are no collisions energetic enough to cause excitation. It is difficult, therefore, to cool gas below $T\approx10^{4}{\rm K}$ solely by atomic cooling of primordial gas. Molecular hydrogen (${\rm H_{2}}$), however, is able to cool gas below $T\approx10^{4}{\rm K}$, down to $T\approx100{\rm K}$, by collisional excitation of rotational-vibrational lines by H atoms. An important question to address is how much ${\rm H_{2}}$ is created, maintained, or destroyed under the influence of an ionizing and dissociating radiation field. Even a small fraction, $n_{\rm H_{2}}/n_{\rm H}\ga 10^{-4}$, is sometimes enough to cool gas below $10^{4}{\rm K}$ (e.g. see @1987ApJ...318...32S). We use cooling rates in the parametrized forms given by @1997NewA....2..209A, except for the hydrogen molecular cooling. For ${\rm H_2}$ cooling, we use the fit given by , where the low density cooling rate has been updated significantly from the previously used rate by @1984ApJ...280..465L, which suffers from the uncertainties associated with the only collisional coefficients available at that time. At low densities, $n_{\rm H} \la 10^2 {\rm cm^{-3}}$, the cooling rate of @1984ApJ...280..465L is bigger by an order of magnitude than that of at $T\approx 1000 K$. Nonequilibrium chemistry {#sub:noneq_chem} ------------------------ The general rate equation for the abundance of species $i$ is given by $$\frac{\partial n_{i}}{\partial t}=C_{i}(T,\{ n_{j}\})-D_{i}(T,\{ n_{j}\})n_{i}, \label{eq:verygeneric_rate_eq}$$ where $C_{i}$ is the collective source term for the creation of species $i$, and the second term is the collective “sink” term for the destruction of species $i$. The processes included and adopted are shown in Table \[table:rates\] in Appendix B. Most of the rate coefficients are those from the fits by @1987ApJ...318...32S, with a few updates. We also adopt the rate solving scheme proposed by @1997NewA....2..181A. It is well known that coupled rate equations in the form of equation (\[eq:verygeneric\_rate\_eq\]) are “stiff” differential equations, whose numerical solution suffers from instability if explicit ODE solvers are used. @1997NewA....2..181A show that their implicit, backward difference scheme provides enough stability. Accuracy of the solution is achieved by updating each species in some specific order, rather than updating all species simultaneously from their values at the last time step. In addition, the abundance of the relatively fast reactions of ${\rm H^{-}}$ and ${\rm H_{2}^{+}}$ are approximated by their equilibrium values, which are expressed by simple algebraic equations. See the Appendix A for the corresponding finite-differencing scheme. We will frequently quote our results in terms of the fractional number density of species $i$, $y_{i}\equiv\frac{n_{i}}{n_{{\rm H}}}$, where $n_{{\rm H}}$ is the number density of the total atomic hydrogen atoms. We use $x$, however, to denote the fractional electron number density, $y_{e}$, which is a measure of the ionized fraction. Code tests {#sec:codetest} ---------- We tested our code against the following problems which have analytic solutions: \(A) the self-similar, spherical, cosmological infall and accretion shock resulting from a point-mass perturbation in an Einstein-de Sitter universe of gas and collisionless dark matter [@1985ApJS...58...39B]; \(B) the self-similar blast wave which results from a strong, adiabatic point explosion in a uniform gas – the Sedov solution (@1959sdmm.book.....S) \(C) the propagation of an I-front from a steady point-source in a uniform, static medium \(D) the gas-dynamical expansion of an H II region from a point source in a uniform gas [@1966ApJ...143..700L] \(E) the gas-dynamical expansion-phase of the H II region from a point-source in a nonuniform gas whose density varies with distance $r$ from the source as $r^{-w}$, $w=3/2$ [@1990ApJ...349..126F]. Our code passed all the tests described above with an acceptable accuracy. Test results are described in Appendix C. The Simulations =============== Initial Setup {#sec:Initial-Setup} ------------- We now describe the initial setup for the problem of radiative feedback effects of Pop III stars on nearby haloes at $z\approx20$. The first stars form inside rare, high density peaks at high redshift. We place target haloes of different mass $M=[2.5\times 10^{4}, \, 5\times 10^{4},\, 10^{5},\, 2\times 10^{5},\, 4\times 10^{5},\, 8\times 10^{5}]\, M_{\odot}$ at different locations from the source, with proper distance $D=\{180,\,\,360,\,\,540,\,\,1000\}\,{\rm pc}$, which are all assumed to be affected directly by the radiation field from the source Pop III star of mass $M_{*}=120\, M_{\odot}$[^6]. We expose the target halo to this radiation field for the lifetime of the star, $t_{*}(120\, M_{\odot})\simeq2.5\,{\rm Myrs}$ (). The source Pop III star is assumed to be located in a halo of mass $M\simeq10^{6}M_{\odot}.$ Time is measured from the arrival of the stellar radiation at the location of the target minihalo. This setup is well justified by the cosmological simulations by @2006ApJ...639..621A. A cosmological gas and N-body simulation of structure formation in the $\Lambda$CDM universe on small scales by a GADGET/SPH code was used to identify the site at which the first Pop III star would form. This occurred at $z=20$, at the location of the highest density SPH particle in the simulation box, located within a halo of mass $M\simeq10^{6}M_{\odot}.$ This provided the initial density field for the I-front tracking calculations in @2006ApJ...639..621A. The I-front from this first star escaped from the host halo quickly with high escape fraction, traveling as a supersonic, weak R-type front. By the end of the lifetime of the star ($\sim[3-2]\,{\rm Myrs}$) for stellar masses in the range $M_* \sim[80-200]M_{\odot}$, the star’s H II region had reached a maximum radius of about $3\,{\rm kpc}$. We approximate the spectral energy distribution (SED) of the source star by a blackbody spectrum. A Pop III star of mass $M_{*}\approx120\, M_{\odot}$, according to , has the time-average effective temperature $T_{{\rm eff}}\approx10^{5}{\rm K}$ and luminosity $L=\int_{0}^{\infty}d\nu L_{\nu}\approx10^{6.243} L_{\odot}$. The corresponding ionizing photon luminosity with this blackbody spectrum is $Q_* \equiv \int_{\nu_{\rm H}}^{\infty}d\nu L_{\nu}/h \nu =1.5\times 10^{50} \rm s^{-1}$, where $h\nu_{\rm H}\equiv 13.6\,{\rm eV}$ is the hydrogen ionization threshold energy. We assume that the source radiates with these time-averaged values throughout its lifetime, then stops. As the photons escape in a time scale short compared to the lifetime of the star and the escape fraction is high, we simply ignore the effect of the intervening gas (e.g. optical depth from the host halo and the IGM) and assume that the bare radiation field hits the edge of target haloes directly. As we fix the luminosity of the source, different distances correspond to different fluxes. We express the frequency-integrated ionizing photon flux, $F$ in units of $\rm 10^{50}\, s^{-1}\, kpc^{-2}$, to give the dimensionless flux, $F_0 \equiv N_{\rm ph,50}/D_{\rm kpc}^2 = N_{\rm ph,56}/D_{\rm Mpc}^2$, where $N_{\rm ph,50}$ is the ionizing photon luminosity (in units of $\rm 10^{50}\,s^{-1}$) and $D_{\rm kpc}$ ($D_{\rm Mpc}$) is the distance in units of kpc (Mpc), respectively. The value $F_0\approx 1$ is typical for minihaloes encountered by intergalactic I-fronts during global reionization (e.g. see @2004MNRAS.348..753S). Interestingly enough, $F_0$ for our “small-scale” problem has a similar value. The Pop III star in our problem has $N_{\rm ph,50}\equiv Q_*/10^{50}\,s^{-1}=1.5$. For distances $180\,{\rm pc}$, $360\,{\rm pc}$, $540\,{\rm pc}$ and $1000\,{\rm pc}$, $F_0$ corresponds to 46.3, 11.6, 5.14 and 1.5, respectively. Initial Halo Structure {#sub:IHS} ---------------------- For the initial halo structure, we adopt the minimum-energy truncated isothermal sphere (TIS) model (@1999MNRAS.307..203S; @2001MNRAS.325..468I), which will be described further in Section \[sub:phase1\]. The thermodynamic properties and chemical abundances of the gas in these target haloes, however, is somewhat ambiguous. The density and virial temperature of these haloes are higher than those of the IGM in general, which drives their chemical abundances to change from the IGM equilibrium state to a new equilibrium state. The most notable feature is the change of $y_{\rm H_2}$ and $x$. The IGM equilibrium value of the electron abundance, $x\approx 10^{-4}$, is high enough to promote ${\rm H_2}$ formation inside minihaloes to yield a high molecule fraction, $y_{\rm H_2}\approx 10^{-4} - 10^{-3}$. At the density of gas in the halo core, this newly created ${\rm H_2}$ is capable of cooling the minihalo gas to $T\approx 100\, {\rm K}$, and depending on the virial temperature, the minihalo may, therefore, undergo a runaway collapse. The time for this evolution of the target halo gas is short compared to the age of the universe when the first star forms in their neighbourhood. As a result, it is likely that the target haloes are exposed to the ionizing and dissociating radiation from that first star as they are in the midst of evolving, with fine-tuning required to catch all of them in a particular stage of this evolution. As the evolutionary “phase” of our target haloes is uncertain, we adopt two different phases as our representative initial conditions. In Phase I, chemical abundances have not yet evolved away from their IGM equilibrium values. This stage is characterized by low $\rm H_2$ fraction, $y_{\rm H_2}\sim 2\times 10^{-6}$ and high electron fraction, $x\sim 10^{-4}$. Phase II is the state which is reached, after allowing the Phase I minihalo to evolve chemically, thermodynamically and hydrodynamically for a few million years (a small fraction of a Hubble time, $t_{\rm H}=186\,\rm Myrs$ at $z=20$), until the electron fraction has decreased to $x\sim 10^{-5}$. Phase II is characterized by high $\rm H_2$ fraction, $y_{\rm H_2}\sim 10^{-4} - 10^{-3}$, and cooling-induced compression of the core relative to Phase I, by a factor between 1 and 20, higher for higher minihalo mass. ### Phase I: Unevolved Halo with IGM chemical abundance in hydrostatic equilibrium {#sub:phase1} The first phase we choose is the initial state we assumed above, namely the nonsingular TIS structure with IGM chemical abundances. This phase is characterized by gas in hydrostatic equilibrium, with the truncation radius (outer boundary of the halo) $$\begin{aligned} r_{t}&=&102.3 \left(\frac{\Omega_{0}}{0.27}\right)^{-1/3} \left(\frac{h}{0.7}\right)^{-2/3} \nonumber \\ &&\times \left(\frac{M}{2\cdot10^{5}M_{\odot}}\right)^{1/3} \left(\frac{1+z}{1+20}\right)^{-1}\,{\rm pc},\end{aligned}$$ the virial temperature $$\begin{aligned} T&=&593.5 \left(\frac{\mu}{1.22}\right) \left(\frac{\Omega_{0}}{0.27}\right)^{1/3} \left(\frac{h}{0.7}\right)^{2/3} \nonumber \\ &&\times \left(\frac{M}{2\cdot10^{5}M_{\odot}}\right)^{2/3} \left(\frac{1+z}{1+20}\right)\,{\rm K}, \label{eq:tvir}\end{aligned}$$ where $\mu$ is the mean molecular weight (1.22 for neutral gas and 0.59 for ionized gas) and the central density $$\rho_{0}=4.144\times10^{-22} \left(\frac{\Omega_{0}}{0.27}\right) \left(\frac{h}{0.7}\right)^{2} \left(\frac{1+z}{1+20}\right)^{3}\, {\rm g\,\, cm^{-3}},$$ which can also be expressed in terms of the hydrogen number density by $$\begin{aligned} n_{{\rm H,0}}&=&\frac{X(\Omega_{b}/\Omega_{0})\rho_{0}}{m_{{\rm H}}} \nonumber \\ &=&30\left(\frac{X}{0.76}\right)\left(\frac{\Omega_{b}}{0.043}\right)\left(\frac{h}{0.7}\right)^{2}\left(\frac{1+z}{1+20}\right)^{3}{\rm \, cm^{-3}},\end{aligned}$$ where $X$ is the hydrogen mass fraction in the baryon component. This central density is about $1.8\times 10^4 \,\overline{\rho}(z),$ where $\overline{\rho}(z)$ is the mean matter density at redshift $z$, while at $r=r_{\rm tr}$, $\rho=35\,\overline{\rho}(z)$. For more details, see @1999MNRAS.307..203S and @2001MNRAS.325..468I. We assign chemical abundances that reflect the IGM equilibrium state, which is characterized by high electron fraction – high enough to promote ${\rm H_2}$ formation under the right conditions – and low ${\rm H_2}$ fraction – low enough to contribute negligible molecular cooling. We adopt $y_{\rm H}=1$, $y_{{\rm He}}=0.0789$, $x\simeq y_{{\rm H^{+}}}=10^{-4}$, $y_{{\rm H_{2}}}=2\times10^{-6}$, and $\{y_{i}\}=0$ for other species (see, e.g. SGB94; @2001ApJ...560..580R). ### Phase II: Evolved Halo with Recombining and Cooling Core {#sub:phase2} The second initial condition we choose is the evolved state (Phase II) reached by allowing the system to evolve from Phase I initial conditions before the arrival of radiation from the Pop III star. In particular, we follow this evolution until the central electron fraction has dropped to $10^{-5}$ by recombination from Phase I. We choose this condition because it is now characterized by high molecule and low electron fraction, contrary to Phase I. The fate of this halo will then mainly be determined by how easily this abundant ${\rm H_2}$ is protected against dissociating radiation after the star turns on. The answer will also depend upon how much change has occurred hydrodynamically, because in some cases the halo core may have cooled and collapsed significantly enough to be unaffected by the feedback from [*late*]{} irradiation. The time to reach Phase II is different for different mass haloes because of different gas properties. Initially, as we start from the TIS density profile whose central density is independent of the halo mass, the recombination rate is higher for smaller mass haloes, because hydrogen recombines according to the following: $$\label{eq:k2_dep} \frac{dx}{dt} \propto n_{\rm H} n_{e^-} T^{-0.7}.$$ The situation becomes complicated, however, once evolution begins and density changes. The ${\rm H_2}$ cooling and collapse in the central region of the haloes is increasingly effective as halo mass increases, because of the increasingly large difference between the virial temperature and the ${\rm H_2}$ cooling temperature plateau, $\sim 100\,{\rm K}$. The corresponding rapid collapse and cooling in massive haloes can easily offset the initial temperature dependence by obtaining high density and low temperature, as is seen in equation (\[eq:k2\_dep\]). Phase II for large mass haloes represents haloes that have already started their cooling and collapse. In Fig. \[fig-init\], we show halo profiles in Phase I and Phase II for different halo masses. We also show how much time it takes for the haloes to evolve from Phase I to Phase II. The times for gas at the halo centre to recombine to $x=10^{-5}$ are in the range $7 \le \Delta t_{\rm I,\,II}(\rm Myrs) \le 24$ for halo masses $0.25 \le M/(10^5\,M_\odot) \le 8$, peaked at $\Delta t_{\rm I,\,II}=24 \,\rm Myrs$ for $M=5\times 10^4 \, M_\odot$. In all cases, $\Delta t_{\rm I,\,II} \ll t_{\rm H}=186\,\rm Myrs$, the age of the universe at $z=20$. Halo Evolution from Fully-Ionized Initial Conditions: The Consequences of Irradiation Without Optical Depth {#sec:opt-thin} =========================================================================================================== Before describing the results of our full radiative transfer, hydrodynamics calculation, we describe an experiment designed to show the effect of neglecting the optical depth of the minihalo to ionizing radiation from the external star during the star’s lifetime on the minihalo’s evolution after the star shuts off. For this purpose, we assume the target minihalo is initially fully-ionized and heated to the temperature of a photoionized gas as it would be if it were instantaneously flash-ionized by starlight in the optically-thin limit. Such a setup is equivalent to that used by @2005ApJ...628L...5O, where they find that second-generation star formation is triggered when the ionization of the minihalo caused by the nearby Pop III star leads to cooling by ${\rm H_{2}}$. The high initial electron fraction is present because of the assumption of full ionization allows quick formation of ${\rm H_{2}}$, which then cools the central region before it reaches the escape velocity. For this experiment, we initialized ionized fractions as following: $y_{\rm H I}=6.4\times 10^{-4}$, $x=1.15$, $y_{\rm H II} = 1$, $y_{\rm He I}=6.8\times 10^{-6}$, $y_{\rm He II}=8.9\times 10^{-3}$, $y_{\rm He III}=7\times 10^{-2}$, $y_i =0$ for other species. Without disturbing the halo density profile – we use the TIS halo model, which is described in Section \[sub:phase1\] –, we also assigned a high initial temperature appropriate for photoionized gas, $T=2\times 10^{4} {\rm K}$. These abundance and temperature values roughly mimic the condition found in typical H II regions. We find that such an initial condition leads to the collapse of the core region, when the formation of ${\rm H_{2}}$ stimulated by the high initial electron fraction enables $\rm H_2$ cooling. Gas in the outskirts evaporates from the halo, however, because pressure forces accelerate the gas to escape velocity before it can form ${\rm H_{2}}$ and cool. The ${\rm H_{2}}$ cooling and adiabatic cooling which happen later in this outflowing gas do not reverse the evaporation (Fig. \[fig-zap\]). Our results for this case agree with the outcome of @2005ApJ...628L...5O. This led those authors to suggest that the first stars exerted a positive feedback effect on their surroundings, triggering a second generation of star formation. A question arises, however, as to whether this fully-ionized initial condition of nearby minihaloes is actually achieved by the first Pop III star to form in their neighbourhood. As already mentioned in Section \[sec:Secondstar-Intro\], @2006ApJ...639..621A found that the I-front from this Pop III star gets trapped in those minihaloes and cannot reach the central region before the star dies. In this paper, we will confirm that the fully-ionized initial condition of @2005ApJ...628L...5O is never achieved when one considers the coupled radiative and hydrodynamic processes more fully. We will also show that, if any protostellar region is to form in the target halo, it does so in the neutral core region which the ionizing photons do not penetrate. Minimum Halo Mass for Collapse: the case without radiative feedback {#sec:mcm} =================================================================== When a minihalo forms as a nonlinear, virialized, gravitationally-bound structure out of the linearly perturbed IGM, a change of chemical abundance occurs due to the change of gas properties. Most importantly, the hydrogen molecule fraction changes from the IGM equilibrium value, $y_{\rm H_2}\sim 2\times 10^{-6}$, to a new equilibrium value, $y_{\rm H_2}\ga 10^{-4}$. Even with such a small fraction, ${\rm H_2}$ can cool gas to $T_{\rm H_2}\simeq 100\,{\rm K}$, where $T_{\rm H_2}$ represents the temperature “plateau” that gas in primordial composition can reach by $\rm H_2$ cooling. There exists a minimum collapse mass of minihaloes, $M_{\rm c,min}$, above which haloes, in the absence of external radiation, can form cooling and collapsing cores within the Hubble time at a given redshift. The gap between the $\rm H_2$ cooling plateau temperature, $T_{\rm H_2}$, and the minihalo virial temperature, $T_{\rm vir}$, given by equation (\[eq:tvir\]) is a useful indicator of the success or failure of collapse. For instance, at $z\approx 20$, $T_{\rm vir}\sim 160\,{\rm K}$ for $M=2.5\times 10^4\,M_\odot$. As $T_{\rm vir}\simeq T_{\rm H_2}$ , even after gas cools to $T_{\rm H_2}$, it cannot collapse fast enough to serve as a site for star formation. On the other hand, $T_{\rm vir}\sim 10^3 \,\rm K $ for $M=4\times 10^5\,M_\odot$, and the temperature’s cooling down to $T_{\rm H_2}\approx 100\,\rm K$ will make the gas gravitationally unstable, which will lead to runaway collapse. This argument is supported by the results of @1996ApJ...464..523H, for example, that collapse can occur only in haloes with $T_{\rm vir}\ga 100\,\rm K$. We model the initial minihalo structure by the TIS model as described in Section \[sub:phase1\] and let it evolve in the absence of radiation, starting from the IGM chemical abundance and minihalo virial temperature (Phase I). We determine $M_{\rm c,min}$ by the criterion $$t_{\rm coll}=t_{\rm H},$$ where $t_{\rm coll}$ is the time at which the central density reaches $n_{\rm H}=10^8\,\rm cm^{-3}$ (the density suitable for initiating three-body $\rm H_2$ formation; see e.g. @2000ApJ...540...39A), and $t_{\rm H}$ is the Hubble time at a given redshift. We find that $M_{\rm c,min}\simeq 7\times 10^4\,M_\odot$ at $z=20$ (see Fig. \[fig-freeevol\]). We have plotted the evolution of minihalo centres in the absence of radiation, where each run starts from Phase I. This is in rough agreement with $M_{\rm c,min}\simeq 1.25\times 10^5\,M_\odot$, the value found by @2001ApJ...548..509M. The discrepancy is larger with results by @2000ApJ...544....6F and @2003ApJ...592..645Y, where they obtain $M_{\rm c,min}\simeq 7\times 10^5\,M_\odot$. The biggest contrast exists with @1997ApJ...474....1T, where they find $M_{\rm c,min}\simeq 2\times 10^6\,M_\odot$ at $z\approx 20$, almost 30 times as large as our findings. We argue that this discrepancy in minimum collapse mass results primarily from how well the minihalo structure is resolved. Unless the centre, which gains the highest molecule formation rate due to the highest density, is fully resolved, one could be misled by a poor numerical resolution such that certain low-mass haloes, which can cool and collapse in reality, are in hydrostatic equilibrium in the simulation. The resolution becomes poorer in the following sequence: @2001ApJ...548..509M, which gives the best agreement with our result, used an adaptive mesh refinement (AMR) scheme, resolving baryonic mass down to $M_b\sim 5\,M_\odot$. Such high resolution is suitable to resolve even the central part of the smallest minihaloes whose total baryonic mass content is roughly $ 2-3\times 10^3\,\rm M_\odot$. @2000ApJ...544....6F and @2003ApJ...592..645Y, on the other hand, used the smoothed particle hydrodynamics (SPH) scheme, using SPH particles of mass $M_b \sim 40-140\times 10^2\,\rm M_\odot$. Finally, @1997ApJ...474....1T used a uniform top-hat model, where there is no radial variation in gas properties such as density and temperature, thus the central region is, in effect, completely unresolved. In addition, some of the rates used in @1997ApJ...474....1T were not accurate [@2000ApJ...544....6F]. We believe that $M_{\rm c,min}\simeq 7\times 10^4\,M_\odot$ at $z=20$ is close to reality, because our 1-D spherical setup is based upon the TIS model which is a highly concentrated structure, and the resolution of our code is superior to previous calculations[^7]. It is not our objective, however, to settle the exact value of $M_{\rm c,min}$. This estimate is based upon our specific criterion described in this section, and is subject to change under different criteria. This may also change if one adopts a more realistic halo formation history to account, for instance, for dynamical heating by accretion (see @2003ApJ...592..645Y). As the haloes we choose are rather conservatively divided into successful collapse (for $M\ge 10^5\,M_\odot$) and failure (for $M < 10^5\,M_\odot$), agreeing with AMR simulation result by @2001ApJ...548..509M, we shall proceed with our choice of parameter space and [*see how this fate of minihaloes changes as a result of external radiation from a Pop III star.*]{} Results: Radiative Feedback on Nearby Minihaloes by an External Pop III Star {#sec:2star-Result} ============================================================================ As described in Section \[sec:Initial-Setup\], we expose target haloes of different mass to the radiation from a Pop III star whose spectrum is approximated as a $10^5\,\rm K$ blackbody radiation field and whose flux is attenuated by the geometrical factor $\left(\frac{D}{R_{*}}\right)^{-2}$ for different values of $D$. In this section, we summarize the simulation results for both the Phase I (early irradiation) and the Phase II (late irradiation) initial conditions. I-front trapping and photo-evaporation -------------------------------------- In all cases, even in the presence of evaporation, we find no evidence of penetration of ionizing radiation into the halo core. This is consistent with the results of @2006ApJ...639..621A for the H II regions of the first Pop III stars and of @2004MNRAS.348..753S and @2005MNRAS.361..405I for the encounters between intergalactic I-fronts and minihaloes during reionization. There are two main reasons for this behaviour. First, the total intervening hydrogen column density is initially high enough to trap the I-front outside the core. Second, the lifetime of the source is short compared to the evaporation time. If the source lived longer than the evaporation time, the I-front would eventually have reached the centre of the halo. In that case, @2004MNRAS.348..753S find that the minihalo gas is completely evaporated. In our problem, however, the slow evaporation does not allow the I-front to reach the centre within the lifetime of a Pop III star. The I-front entering the minihaloes propagates as a weak R-type front in the beginning. The I-front then makes the transition to the D-type, after reaching the R-critical state. This R-critical state is reached when the I-front velocity $v_{\rm I}$ satisfies the following condition: $$v_{\rm I}=c_{\rm I,2}+(c_{\rm I,2}^2 - c_{\rm I,1}^2)^{0.5},$$ where $c_{\rm I}$ is the isothermal sound speed, $c_{\rm I}\equiv \sqrt{p/\rho}$, and subscripts 1 and 2 represent pre-front and post-front, respectively. When the I-front propagates into a cold region ($T\ll 10^{4} \, {\rm K}$), as in our problem, this condition is approximately $v_{\rm I}\approx 2\,c_{\rm I,2} \approx 20\,{\rm km\,s^{-1}}.$ In all cases, we find that this transition occurs in times less than the lifetime of the source star, 2.5 Myrs. After reaching the R-critical state, gas in front of the I-front forms a shock, which then detaches from the slowed I-front. As an example, we plot in Fig. \[fig-Rcrit\] the profiles of Phase I, $4\times 10^5\,M_\odot$ halo at $t=t_{\rm R-crit}$ under different fluxes. All of the post-front (ionized) gas, initially undisturbed, eventually evaporates away, accelerated outward by a large pressure gradient. As the line-of-sight is cleared by this evaporation, ionizing radiation penetrates deeper, until the source turns off. See Figs \[fig-Rcrit\], \[fig-midpoint\] and \[fig-endpoint\] for the evolution of the I-front. This result invalidates the initial condition adopted by @2005ApJ...628L...5O and @2006astro.ph..4148M which led them to find that $\rm H_2$ formed in the core region after it was ionized and then cooled while recombining, once the source turned off. As we show, the core remains neutral before and after the source is turned off, so the mechanism explored by @2005ApJ...628L...5O does not work. This neutral core, therefore, must find a different way to cool and collapse if star formation is to happen in the target minihalo. What happens to the initially ionized gas after the star turns off? This gas recombines as it cools radiatively and by adiabatic expansion, even forming $\rm H_2$ molecules. We find that this cooling cannot reverse the evaporation, however. Gas is simply carried away with the initial momentum given to it when it was in an ionized state. In Table \[table:ionized\], we list the fraction of the baryonic halo mass which is ionized during the lifetime of the star. This mass serves as a crude estimate of the mass lost from these haloes by evaporation. We found no major difference between Phase I and Phase II in this matter, so we provide only one table. -------------------- ------------ --------------------- ----------- ----------- ----------- ------------ $D$ (pc) \[$F_0$\] $0.25 $ $0.5 \cdot 10^{4} $ $1 $ $2 $ $4 $ $8 $ $(0.043 )$ $(0.086 )$ $(0.17 )$ $(0.34 )$ $(0.69 )$ $(1.371 )$ 180 \[46.3\] 0.95 0.92 0.88 0.84 0.82 0.79 360 \[11.6\] 0.85 0.81 0.77 0.74 0.70 0.67 540 \[5.14\] 0.78 0.74 0.70 0.66 0.62 0.59 1000 \[1.5\] 0.66 0.60 0.55 0.50 0.47 0.43 -------------------- ------------ --------------------- ----------- ----------- ----------- ------------ Formation of ${\rm H_2}$ precursor shell in Front of the I-Front {#sub:H2shell} ---------------------------------------------------------------- We find that a thin shell of ${\rm H_{2}}$ is formed just ahead of the I-front, with peak abundance $y_{{\rm H_{2}}}\approx10^{-4}$. It happens mainly because the increased electron fraction across the I-front promotes the formation of $\rm H_2$. More precisely, the gas ahead of the I-front is ionized to the extent that the electron abundance is large enough to form ${\rm H_{2}}$, but at the same time too low to drive significant collisional dissociation of $\rm H_2$. The width of this ${\rm H_{2}}$ shell and the amount of $\rm H_2$ in this region is determined by the hardness of the energy spectrum of the source: the width of the I-front is of the order of the mean free path of the ionizing photons. Pop III stars, in general, produce a large number of hard photons due to their high temperature, which can penetrate deeper into the neutral region than soft photons. This precursor ${\rm H_2}$ shell feature is evident in Figs \[fig-Rcrit\], \[fig-midpoint\], and \[fig-endpoint\]. We show the detailed structure of these ${\rm H_{2}}$ shells in Fig. \[fig-midtrap\], where we plot the radial profile of the abundance of different species for the case of $M=4\times 10^5 \,M_\odot$, Phase I, $D=540\,\rm pc$ ($F_0=5.14$) at $t=0.5\, t_{*}$. We note the similarity between our results and those of @2001ApJ...560..580R for an I-front in a uniform, static IGM at the mean density (see Fig. 3 in @2001ApJ...560..580R) which also show a precursor $\rm H_2$ shell. A similar effect was reported by @2006ApJ...645L..93S, as well. What is the importance of this $\rm H_2$ shell in protecting the central region of haloes from dissociating radiation? The molecular column density obtained by this $\rm H_2$ shell sometimes reaches $\sim 10^{16}\,\rm cm^{-2}$, which provides an appreciable amount of self-shielding. The self-shielding due to the $\rm H_2$ shell, however, is not the major factor that determines whether or not the $\rm H2$ in the core region is protected. A more important factor is which evolutionary phase the target halo is in when it is irradiated. Roughly speaking, when a target halo is irradiated early in its evolution (Phase I), the precursor $\rm H_2$ shell dominates the total $\rm H_2$ column density available to shield the central region, but this shielding is not sufficient to prevent photodissociation there anyway. On the other hand, if the halo is irradiated later in its evolution (Phase II), the $\rm H_2$ column density of the shell is only a small part of the total $\rm H_2$ column density, so shielding is successful independent of the precursor shell. We describe this in more detail as follows. In order to understand quantitatively the importance of the $\rm H_2$ shell in protecting the central $\rm H_2$ fraction, we have performed simulations with a source SED that is identical to the Pop III SED below 13.6 eV, but zero above 13.6 eV. As the radiation is now incapable of ionizing the halo gas, the $\rm H_2$ shell formation by partial ionization will not occur. This enables us to compare our results where the $\rm H_2$ shell is present to those cases without an $\rm H_2$ shell. We describe a specific case of $M=2\times 10^{5} \,M_\odot$ as an illustration. Roughly speaking, the ${\rm H_2}$ shell which forms only in the presence of ionizing radiation compensates for the amount by which the initial molecular column density, $N_{\rm H_2}$, is reduced when molecules in the ionized region are destroyed by collisional dissociation. The nett column density in the case where the $\rm H_2$ shell is present even exceeds that in the case without the $\rm H_2$ shell (Figs \[fig-nh2-C1\] and \[fig-nh2-C4\]). The nett effect is the increase of the self-shielding. Such an increase of the self-shielding, however, is not too dramatic. In the case of $M=2\times 10^{5} \,M_\odot$ with Phase I initial conditions, $y_{\rm H_2}\approx 10^{-5.3}$ at the centre, about an order of magnitude higher than the central $y_{\rm H_2}$ of the case without ionizing photons (Fig. \[fig-nh2-C1\]). This molecule fraction is still too low, however, to cool the gas. On the other hand, in the case of $M=2\times 10^{5} \,M_\odot$ with Phase II initial conditions, $y_{\rm H_2}\approx 10^{-3.5}$ at the centre throughout the lifetime of the Pop III source, whether or not the ${\rm H_2}$ shell is formed. The depth (radius) of penetration of dissociating photons differs by a factor of 2 if the shell is included, but the central ${\rm H_2}$ is still protected because of the high ${\rm H_2}$ column density [*apart*]{} from the precursor shell (Fig. \[fig-nh2-C4\]). The major factor that determines the fate of the central $\rm H_2$ fraction is instead the evolutionary phase of a target halo when it is irradiated. The short lifetime of a Pop III star plays an important role of either reconstituting or protecting molecules in the core, depending upon the evolutionary phase of the halo, as will be described in Section \[sub:Collapse\]. Note that in all cases, we use equation (\[eq:DB\_shield\_factor\]), the shielding function for thermally-broadened lines with $T=10^{4}\,{\rm K}$. This is justified by the fact that the ${\rm H_2}$ shell moves inward with $v\approx 2-5 \,{\rm km\,s^{-1}}$ and the shell achieves $T=T_{\rm sh}\approx 10^{3}-5\times 10^{3}\,{\rm K}$, where $T_{\rm sh}$ denotes the temperature of the shell. If we take this peculiar velocity as sound speed, $v\approx 2-5 \,{\rm km\,s^{-1}}$ corresponds to $T=T_p\equiv v^2 \mu m_{\rm H}/k=6\times 10^2 - 3.7\times 10^3\,{\rm K}$, where the subscript $p$ denotes the peculiar velocity. A crude way to imitate both effects by thermal broadening is to use the sum of these two temperatures ($T_{\rm sh}$ and $T_p$). We take the most conservative stand – the least self-shielding effect – in order not to overestimate the self-shielding, and use $T=10^{4}\,{\rm K}$ as the temperature responsible for the nett thermal broadening of the molecular LW bands. Formation of shock and Evolution of core {#sub:shock} ---------------------------------------- After the I-front decelerates as it enters the target halo, transforming from R-type to D-type, a shock front forms to lead the D-type front. The neutral gas in the core is strongly affected by this shock front as it propagates. This shock plays an important role in providing both positive and negative feedback effects. By identifying successive evolutionary stages of the shock, we now describe how the core responds to the shock and evolves accordingly. ### Stage I: Formation and acceleration of Shock {#sub:shockstageI} A shock starts to form as the I-front, initially moving supersonically as an R-type, slows down and turns into a D-type. The pre-front gas – neutral gas ahead of the I-front – can respond to the I-front before it is swept by the I-front, because the D-type front moves subsonically into the neutral gas. It is easier to understand the formation of the shock by using the I-front jump conditions: the pre-front gas speed in the rest frame of the I-front, $v_1$, derived from the I-front jump conditions, should satisfy either $v_1\ge v_R \equiv c_{\rm I,2}+(c_{\rm I,2}^2 - c_{\rm I,1}^2)^{0.5},$ or $v_1\le v_D \equiv c_{\rm I,2}-(c_{\rm I,2}^2 - c_{\rm I,1}^2)^{0.5}$, where $c_{\rm I,1}$ and $c_{\rm I,2}$ are the isothermal sound speeds of the pre-front and post-front gas, respectively. $v_R$ and $v_D$ have a gap of $2(c_{\rm I,2}^2 - c_{\rm I,1}^2)^{0.5}$, which is nonzero in general. As the I-front slows down and $v_1$ starts to cross $v_R$, $v_1$ encounters a value which is not allowed mathematically. This paradox is resolved, however, because the pre-front gas now “prepares” a new hydrodynamic condition by forming a shock. The shock wave increases $\rho_1$ and thereby reduces $v_1$ and increases $v_D$, making it possible to satisfy the D-type condition, $v_1\le v_D$. This shock-front then propagates inward, separating from the I-front, due to the discrepancy between the speed of the shock-front and the speed of the I-front. As the shock-front enters the flat-density core, the shock front starts to accelerate, leaving behind the post-shock gas with ever increasing temperature (e.g. see time steps 4 and 5 in Fig. \[fig-evol\], where the post-shock temperature increases as the radius $r$ decreases). As the shock boosts the density and temperature in the neutral, post-shock gas, the ${\rm H_2}$ formation rate there increases, boosting the $\rm H_2$ column density even further. We can understand the evolution of $y_{\rm H_2}$ in the presence of this shock quantitatively by using its equilibrium value, $y_{\rm H_2,eq}$. The increase of density and temperature due to this shock promotes $\rm H_2$ formation, as follows. When there is no significant $\rm H^-$ destruction mechanism, the dominant $\rm H_2$ formation mechanism is through $\rm H^-$ (equation \[eq:solomon\]), and the $\rm H_2$ formation rate becomes equivalent to the $\rm H^-$ formation rate. Photo-dissociation dominates over collisional dissociation in destroying $\rm H_2$, which occurs when $x\la 4\times 10^{-3}\,T_{\rm K}^{1/2}$ and $n_{\rm H}\ga 0.045 \times (F_{\rm LW}/10^{-21}\,\rm erg\,s^{-1}\,cm^{-2}\,Hz^{-1})$ (e.g. @2001MNRAS.321..385G). Using the $\rm H^-$ formation rate coefficient [@1972AA....20..263D] $$k_{\rm H^-}=10^{-18}\,T_{\rm K}\,{\rm cm}^{3}\,{\rm s}^{-1},$$ and the photo-dissociation rate coefficient $k_{\rm H_2}$ given by equation (\[eq:DB\_rate\]), we obtain $$\begin{aligned} y_{\rm H_2, eq}&=&4.1\times 10^{-5} \left( \frac{T}{5000\,{\rm K}}\right)\left( \frac{x}{10^{-4}}\right)\nonumber \\ && \times \left( \frac{n_{\rm H}}{30\,{\rm cm}^{-3}}\right) \left( F_0\cdot F_{\rm shield}\right)^{-1}, \label{eq:yh2}\end{aligned}$$ where we have used the fact that one can scale $F_{\rm LW}$ by $F_0$ according to the following: $$F_{\rm LW}\approx 3.25\times 10^{-21} \,{\rm erg \,s^{-1}\,cm^{-2}\,Hz^{-1}}\, F_0, \label{eq:flw}$$ if one adopts a black-body spectrum with $T=10^{5}\,\rm K$. As seen in equation (\[eq:yh2\]), both the high temperature ($\sim 1000-5000\,\rm K$) and increased density ($\times 4$ in the case of strong shock) of the post-shock gas contributes to boosting the $\rm H_2$ fraction. As $y_{\rm H_2}\propto F_{\rm shield}^{-1}$, molecular self-shielding also plays an important role in determining $y_{\rm H_2}$. If the shock boosts the formation rate of $\rm H_2$ and $y_{\rm H_2}$ increases, so will $N_{\rm H_2}$, and with it the shielding. These two effects, therefore, amplify each other. There is an additional mechanism to create molecules: the shock-induced molecule formation (SIMF). The acceleration of the shock-front accompanied by an increasing post-shock temperature, leads to a partial ionization of the post-shock gas in many cases, when the right condition ($T\ga 10^4\,\rm K$) is met to trigger collisional ionization – see, for example, step 5 in Fig. \[fig-evol\]: the centre is shock-heated above $10^4\,\rm K$, with a boost in $x$. The electron fraction $x$ now reaches $\sim 10^{-4} - 10^{-2}$, which promotes further ${\rm H_2}$ formation. This mechanism is indeed identical to the $\rm H_2$ formation mechanism in a gas that has been shock-heated to temperatures above $10^4\,\rm K$ [@1987ApJ...318...32S; @1992ApJ...386..432K]. When a gas cools radiatively from a temperature well above $10^4\,\rm K$, it cools faster than it recombines. As a result, the recombination is out of equilibrium, and an enhanced electron fraction exists at temperatures even below $10^4\,\rm K$ compared to the equilibrium value. This electron fraction triggers the formation of $\rm H_2$ through the gas-phase reactions (equations \[eq:solomon\] and \[eq:solomon2\]). SIMF does not always occur, however. The shock-front can accelerate when the pre-shock density remains almost constant (e.g. Fig. \[fig-evol\]). If the density increases faster than the shock propagates, on the other hand, the shock-front will encounter an ever increasing density “hill” and it will never accelerate to generate post-shock temperature above $10^4\,\rm K$ (e.g. Fig. \[fig-evol2e5C4\]). The dependence of SIMF on the halo mass, source flux, and the initial phase will be described in Section \[sub:Collapse\]. ### Stage II: Cooling and Compression of Core As the shock-front approaches the centre of the halo, the post-shock gas there becomes more concentrated and denser than the pre-shock gas. This shock-induced compression leads to a very fast molecular cooling in the core and further compression in almost a runaway fashion, as follows. Molecular cooling occurs very rapidly at a high density and temperature condition. Assuming that the pre-shock gas of the halo core remains unchanged before the shock-front arrives – as is usually the case in Phase I – and the shock is strong, the post-shock density of the core becomes 4 times higher than that of the pre-shock, namely $n_{\rm HI}\approx 4\times 30\,\rm cm^{-3}=120\,\rm cm^{-3}$ in a TIS halo core at $z=20$. At the same time, post-shock temperature can be as high as $10^4\,\rm K$. The molecular cooling time, $t_{\rm cool,H_2}\equiv T/(dT/dt)$, is $$t_{\rm cool,H_2}=\frac{kT}{X\mu(\gamma-1)y_{\rm H_2}n_{\rm HI}\Lambda_{\rm H_2}}, \label{eq:tcool}$$ where $X=0.75$ is the hydrogen mass fraction, and $\Lambda_{\rm H_2}$ is the molecular cooling rate. For a gas with $n_{\rm HI}=120\,\rm cm^{-3}$ and $T=10^4\,\rm K$, $\Lambda_{\rm H_2}\approx 3.4\times 10^{-22} \,\rm erg\,cm^{-3}\,s^{-1}$, and thus $$t_{\rm cool,H_2}\approx 1.8\times 10^3\,{\rm yr}\, \left(\frac{y_{\rm H_2}}{10^{-3}}\right)^{-1}. \label{eq:tool-num}$$ With such a rapid cooling, the isothermal shock jump condition ($T_2=T_1$) is a good approximation, and the post-shock density becomes even higher than that of the adiabatic strong shock, because $\rho_{b,2}/\rho_{b_1}\approx M_{I,1}^2$ now. Such a strong compression of the core is observed very frequently in our parameter space of different halo masses and source fluxes. For example, Fig. \[fig-M2e5-ing\] shows how the centre of a halo with $M=2\times 10^4\,M_\odot$ evolves in response to the shock. As the shock hits the centre, density increases by many orders of magnitude. Does this compression eventually lead to the core collapse? As the shock carries the kinetic energy as well as the thermal energy, the shock will bounce off the centre after it hits the centre. In the following section, we describe this final stage of the shock propagation and show how it will affect the core collapse. ### Stage III: Bounce of Shock and Collapse of Core After the shock hits the centre, the shock wave will be reflected and propagate outward. In our 1D calculation, this reflection will mimic the transmission of the shock wave through the centre. This bouncing shock will try to disrupt the gas. The core that is undergoing cooling and compression due to the positive feedback effects mentioned so far will be affected by this negative feedback effect, as well. The final fate of the core depends on how well the core endures such a disruption. As the shock bounces off the centre, density starts to decrease. If this bounce is weak, the core quickly reassembles, cools, and finally collapses. If this bounce is strong, the core will take a longer time to collapse and, in some cases, the core will never collapse within the Hubble time. Haloes of smaller mass seem to be more susceptible to this shock-bounce than those of larger mass (see Figs \[fig-M1e5-ing\] and \[fig-M8e5-ing\] for comparison). If the core finally takes the collapse route, the central hydrogen number density increases to $\sim 10^{4}\,\rm cm^{-3}$, at which point the ro-vibrational levels of $\rm H_2$ are populated at their equilibrium values and the molecular cooling time becomes independent of density (e.g. @2002Sci...295...93A). Since then, adiabatic heating dominates over the molecular cooling, and the temperature increases as collapse proceeds. Finally, when $n_{\rm HI}$ reaches $\sim 10^{8}\,\rm cm^{-3}$, the three-body hydrogen reaction ensues and converts most hydrogen atoms into the molecules, which will undergo a further collapse and form a proto-star. Feedback of Pop III starlight on Nearby Minihaloes: parameter dependence of core collapse {#sub:Collapse} ----------------------------------------------------------------------------------------- We now summarize the outcome of our full parameter study of radiative feedback effects of Pop III starlight on nearby minihaloes. As we have described in the previous section, positive and negative feedback effects of the shock compete and produce a nett effect which can be either 1) an expedited collapse, 2) delayed collapse, 3) neutral (unaffected) collapse, or 4) a disruption. Overall, the radiative feedback effect of a Pop III star is not as destructive as naively expected. Minihaloes with $M\ga [1-2]\times 10^5\,M_\odot$, which can cool and collapse without radiation, are still able to form cooling and collapsing clouds at their centre even in the presence of Pop III starlight. The quantitative results are summarized in Tables \[table:case1\], \[table:case2\] and Fig. \[fig-coll\]. The relatively short lifetime of a Pop III star, compared to the recombination timescale in the core, is a key to understanding this behaviour. One of the necessary conditions for the core collapse is that $\rm H_2$ molecular cooling should occur in the core. As this requires a sufficient molecular fraction, namely $y_{\rm H_2}\ga 10^{-4}$, it is crucial to understand how molecules are created at such a level. In Phase I (low $y_{\rm H_2}$ and high $x$), radiation can easily dissociate $\rm H_2$ while the source is on, but after the source dies, the high electron fraction stimulates $\rm H_2$ formation. This is possible because the recombination time in the TIS core is longer than the lifetime of the source Pop III star. On the contrary, in Phase II (high $y_{\rm H_2}$ and low $x$), $\rm H_2$ is more easily protected against the dissociating radiation because the higher $\rm H_2$ column density provides self-shielding and compression increases the formation rate. Because the source irradiates these haloes for a short period of time, the [ *dissociation front*]{} does not reach the centre, and its high molecule fraction is preserved throughout the Pop III stellar lifetime. ### Phase I When haloes start their evolution from Phase I – IGM chemical abundance and the TIS structure –, other than the change of collapse times, there is no reversal of collapse. In other words, haloes that were destined to cool and collapse would do so even when exposed to the first Pop III star in the neighbourhood. Minihaloes with $M\ga 10^5 \,M_\odot$ are able to collapse without radiation, while those with $M < 10^5 \,M_\odot$ are not. In the presence of radiation, haloes with $M\ga 10^5 \,M_\odot$ are still able to collapse, while those with $M < 10^5 \,M_\odot$ are still unable to do so, even with the help of shock-induced molecule formation (Fig. \[fig-coll\]; Table \[table:case1\]). The core collapse in Phase I occurs mostly as an expedited collapse (Table \[table:case1\]). The shock plays a major role in driving such an expedited collapse: the $\rm H_2$ fraction becomes boosted by the higher density and high temperature delivered by the shock. Whether or not SIMF has occurred, such a boost in $y_{\rm H_2}$ is sufficient to expedite the core collapse. There is one delayed collapse case at the low mass and the high flux end. For $M=10^5\,\rm M_\odot$ at $F_0=46.3$, the boosted molecule formation is not sufficient to bring the core to an immediate collapse. As the shock bounces, the momentum carries gas away from the centre until it cools and recollapses. The unchanged collapses occur at the high mass and the low flux end. For $M=8\times 10^5\,M_\odot$ at $F_0=[1.5,\,5.14]$, the shock propagates into the already collapsing core. The shock energy delivered in these cases is not significant enough to change the course of collapse. -------------------- ----------- ----------- ----------------------- ----------------------- ----------------------- ----------------------- $D$ (pc) \[$F_0$\] $0.25 $ $0.5 $ $1 $ $2 $ $4 $ $8 $ $(\cdot)$ $(\cdot)$ $(88.82)$ $(31.02)$ $(14.61)$ $(8.66)$ 180 pc \[46.3\] $\cdot$ $\cdot$ $1.455$ $7.288 \cdot 10^{-2}$ $1.838 \cdot 10^{-1}$ $4.712 \cdot 10^{-1}$ 360 pc \[11.6\] $\cdot$ $\cdot$ $1.935 \cdot 10^{-1}$ $1.308 \cdot 10^{-1}$ $3.597 \cdot 10^{-1}$ $8.177 \cdot 10^{-1}$ 540 pc \[5.14\] $\cdot$ $\cdot$ $3.427 \cdot 10^{-1}$ $2.093 \cdot 10^{-1}$ $4.919 \cdot 10^{-1}$ $1.000$ 1000 pc \[1.5\] $\cdot$ $\cdot$ $9.497 \cdot 10^{-1}$ $4.525 \cdot 10^{-1}$ $7.144 \cdot 10^{-1}$ $1.241$ -------------------- ----------- ----------- ----------------------- ----------------------- ----------------------- ----------------------- ### Phase II The overall effect of radiation from a Pop III star on neighbouring minihaloes in Phase II is similar to the effect on the minihaloes in Phase I: haloes that were destined to cool and collapse would do so even when exposed to the first Pop III star in the neighbourhood. A slight shift of the trend exists, however, in Phase II (Fig. \[fig-coll\]; Table \[table:case2\]). When haloes start their evolution from Phase II, those with $M\ga 10^5 \,M_\odot$ are able to collapse without radiation, while those with $M \la 2 \times 10^5 \,M_\odot$ are not. The collapse in Phase II is reversed (halted) for the low mass end: for $M=10^5\,M_\odot$, the shock disrupts the core and it never recollapses. SIMF occurs at $F_0>1.5$ for $M=10^5\,M_\odot$, but this does not prevent such a destructive process from happening. As haloes start their evolution from Phase II, in which the halo cores are already cooling and collapsing, the neutral (unaffected) collapse cases occur more frequently than in Phase I. At high and intermediate masses, the collapse time hardly changes from the case without radiation. Haloes with $M=8\times 10^5 \,M_\odot$ collapse [*before*]{} the source dies, as they do without radiation, simply because the shock wave does not affect the core. In this case, shock propagates into the centre after collapse has advanced significantly. There is one delayed collapse case: compared to the delayed collapse in Phase I, which occurred at low mass/high flux end ($M=10^5\,M_\odot$ at $F_0=46.3$), this now occurs at an intermediate mass/high flux end ($M=2\times 10^5\,M_\odot$ at $F_0=46.3$). Otherwise, for intermediate mass, collapse is either neutral or expedited. -------------------- ----------- ----------- ----------- ----------------------- ----------------------- ------------------------ $D$ (pc) \[$F_0$\] $0.25 $ $0.5 $ $1 $ $2 $ $4 $ $8 $ $(\cdot)$ $(\cdot)$ $(65.66)$ $(14.49)$ $(4.23)$ $(1.65)$ 180 pc \[46.3\] $\cdot$ $\cdot$ $\cdot$ $4.269 $ $7.151 \cdot 10^{-1}$ $9.541 \cdot 10^{-1}$ 360 pc \[11.6\] $\cdot$ $\cdot$ $\cdot$ $4.997 \cdot 10^{-1}$ $1.155 $ $1.002 $ 540 pc \[5.14\] $\cdot$ $\cdot$ $\cdot$ $6.740 \cdot 10^{-1}$ $9.794 \cdot 10^{-1}$ $9.964 \cdot 10^{-1}$ 1000 pc \[1.5\] $\cdot$ $\cdot$ $\cdot$ $5.794 \cdot 10^{-1}$ $9.926 \cdot 10^{-1}$ $9.994 \cdot 10^{-1} $ -------------------- ----------- ----------- ----------- ----------------------- ----------------------- ------------------------ The structure of haloes at the moment of collapse {#sub:onset} ------------------------------------------------- The structure of halo at collapse determines how a protostar evolves into a star and how the starlight will later propagate through the host halo. We first show how halo profiles at collapse vary for different mass without radiation. We then describe how halo structure is affected by the Pop III starlight. We note that halo structure shows a strong dependence on the halo mass. For radius $r\ga 10^{-2}\,\rm pc$, density profiles of haloes without radiation are well fit by a power law, $\rho \propto r^{-w}$. The value of $w$, however, is dependent upon the mass of the halo. We find that $w=2.5$, 2.4, 2.3, and 2.2 for haloes of mass $M=10^5$, $M=2\times 10^5$, $M=4\times 10^5$, and $M=8\times 10^5\,M_\odot$, respectively. In all cases, the temperature is somewhat flat with $T\sim 10^{2.5} - 10^3\,\rm K$. The temperature at $r\approx 10^{-2}\,\rm pc$, where $\rho\approx 3\times 10^{-16} \rm g\,cm^{-3}$ (or $n_{\rm H}\approx 10^8\,\rm cm^{-3}$), is about $800 \,\rm K$ in all cases. The universality of these core properties seems to originate from the fact that the dominant process, $\rm H_2$ cooling, causes loss memory of the initial condition (e.g. different virial temperatures for different virial masses). The outer part of these haloes, however, still retain the memory virial equilibrium because radiative cooling is negligible. Overall, as mass decreases, density slope increases (see Fig. \[fig-onset\]). The radiative feedback effect of the starlight on final halo profiles is found to be negligible in most cases. The region that has been photo-ionized during the stellar lifetime is obviously strongly affected. The neutral region, however, is almost indistinguishable from the case without radiation in most cases. The variance of temperature profile exists only at the low-mass end, $M=10^5\,M_\odot$, or the high-flux end, $F_0=46.3$ ($D=180\,\rm pc$). Such variance completely disappears at the high-mass end, $M=8\times 10^5\,M_\odot$, because collapse is mostly unaffected (Fig. \[fig-onset\]). This result indicates that the mass of secondary Pop III stars would be almost identical to that of the Pop III stars which form without radiative feedback effect. A more fundamental variance may exist, however, due to the environmental variance of star forming regions: @2006astro.ph..7013O show that temperature variance of different regions result in the variance of protostellar masses, due to the corresponding variance of mass infall rate. As our simulation does not advance beyond $n_{\rm H}=10^8\,\rm cm^{-3}$, where three-body collision can produce copious amount of $\rm H_2$ molecules and change the adiabatic index of the gas, we are unable to quantify the final mass of the protostar at this stage. Feedback of Pop III Starlight on Merging Haloes and Subclumps {#sub:abel} ------------------------------------------------------------- While we were preparing this manuscript, two preprints were posted describing simulations of the radiative feedback of the first Pop III star on dense gas clumps even closer to the star than the external minihaloes we have considered so far, for the case of subclumps [@2006ApJ...645L..93S] and the case of a second minihalo undergoing a major merger with the minihalo that hosts the first star [@2006astro.ph..6019A]. The centre of the target halo or clumps in this case is well within the virial radius of the halo which hosts the first star, and, these authors find that secondary star formation occurs in these subhaloes. @2006astro.ph..6019A, for instance, report that the first star forms inside a minihalo of mass $M=4\times 10^5\,M_\odot$ as it merges with a second minihalo of mass $M=5.5\times 10^5\,M_\odot$ (the target halo). The centre of this target halo is at a distance of only 50 parsecs from the first star. Cooling and collapse leading to the formation of a protostar is found to occur inside the target halo about 6 Myrs [*after*]{} the first star has died. We ask the same question that whether or not a halo would collapse to form a secondary Pop III star if a nearby Pop III star irradiates the halo at a distance of 50 pc. Note that the target halo we consider now would collapse anyway if there were no radiation, in $\sim 11$ Myrs for Phase I and $\sim 3$ Myrs for Phase II (see Table \[table:collsub\]). This problem requires us to extend our parameter space beyond what has been considered so far, because of the short distance (high flux) between the source and the target. We have attempted to reproduce the result of @2006astro.ph..6019A using our code for a target halo of mass $M=5.5\times 10^5\,M_\odot$ and $D=50\,\rm pc$, corresponding to the ionizing flux $F_0=600$. Note that the LW band flux is very high: $F_{\rm LW}\sim 2000\times 10^{-21}\,\rm erg\,s^{-1}\,cm^{-2}\,Hz^{-1}$ (equation \[eq:flw\]). As $D$ is smaller than the virial radius of the target halo, we truncated the halo profile at 50 pc. To be consistent with our previous calculations, we neglect the geometrical variation of the flux with position inside the target halo. Surprisingly enough, contrary to the outcome of @2006astro.ph..6019A, we find that collapse is expedited, occurring [*within the lifetime of the first star*]{}, for both Phase I and Phase II initial conditions. The main mechanism was SIMF: initially, $\rm H_2$ is completely wiped out by a strong dissociating radiation, but as the SIMF occurs, newly created molecules lead to cooling and collapsing. This result is in disagreement with the result of @2006astro.ph..6019A, which shows that the second star forms [*after the star has died*]{}. This puzzling result shows the importance of $\rm H_2$ self-shielding. @2006astro.ph..6019A performed an optically-thin calculation for Lyman-Werner bands, neglecting the $\rm H_2$ self-shielding, while our calculation took the self-shielding into account. In order to mimic their calculation more consistently, we artificially performed an optically-thin calculation for Lyman-Werner bands. We found that, if the target halo is irradiated without $\rm H_2$ self-shielding, the core collapse is delayed and occurs [*after the star dies*]{} both in Phase I and Phase II. In our simulations without $\rm H2$ self-shielding, the core bounced and recollapsed in $\sim$44 Myrs and $\sim$111 Myrs after the star has turned off in Phase I and Phase II, respectively (Table \[table:collsub\]). Qualitatively, our calculation without $\rm H_2$ self-shielding agrees with the result of @2006astro.ph..6019A, that collapse in the target halo occurs after the source dies. We find that SIMF is the main mechanism for the formation of $\rm H_2$. Initially, the strong LW band photons destroy molecules in the core. As the shock propagates inward, however, boosted density and temperature of the post-shock gas enhances the molecule fraction (equation \[eq:yh2\]), and increases the $\rm H_2$ column density. As the shock front accelerates, SIMF occurs, and newly created $\rm H_2$ is protected from the LW band photons because of increased self-shielding. If self-shielding is not accounted for, however, this $\rm H_2$ is destroyed and never restored, so collapse does not proceed during the lifetime of the source. We conclude, therefore, that neglecting $\rm H_2$ self-shielding in calculation explains why @2006astro.ph..6019A observes a delayed collapse. The quantitative disagreement between our collapse times (when we neglect self-shielding) and theirs may originate from the difference in the structure and chemical abundances of the target halo when the source irradiates it. How do our results compare with those of @2006ApJ...645L..93S? A fundamental difference exists other than the fact that their work is limited to subclumps of a halo that hosts a Pop III star. They interpret the shock only as a carrier of negative feedback effect, while the shock, in our case, delivers both the positive and negative feedback effects. In their shock-driven evaporation (Model C) case, the collapsing core eventually fails to collapse, because the shock heats the core before it finishes collapse. Their successful collapse case (Model B) is simply an unaltered collapse: an already collapsing core finishes collapse before the shock front reaches the centre. On the other hand, we have observed expedited collapses as well as delayed or failed collapse. Such expedited collapses we observe are truly positive feedback effects. Quantitatively, because of their limited interpretation of the role of the shock, they argue that only regions with hydrogen number density $n_{\rm H}\ga 10^{2-3} \, \rm cm^{-3}$, high enough to finish collapse before the shock front reaches the centre, can collapse under the influence of Pop III starlight. On the contrary, we find, for instance, that regions with $n_{\rm H}\sim 30 \, \rm cm^{-3}$ – core density of TIS haloes in Phase I – can cool and collapse even after the shock front has reached the centre. As the shock-front accelerates and delivers strong positive feedback effects in the small core region, high resolution is required to produce this mechanism in simulations. The relatively poor resolution of SPH simulations by @2006ApJ...645L..93S might have prevented them from fully resolving the shock structure in the core, and potentially producing the positive feedback effects. Our result indicates that secondary star formation may occur even in subclumps of the host halo, which are subject to much stronger radiative feedback than isolated, nearby minihaloes. We have shown in this section that $\rm H_2$ self-shielding is important even at this high level of ionizing ($F_0=600$) and dissociating ($F_{\rm LW}= 2\times 10^{-18}\,\rm erg\,s^{-1}\,cm^{-2}\,Hz^{-1}$) fluxes. It is even more surprising because the collapse is expedited and [*coeval formation*]{} of Pop III stars in the same neighbourhood is possible. The naive expectation of negative feedback effect of a Pop III star in its neighbourhood, therefore, should be revisited. no radiation self-shielding no self-shielding ---------- -------------- ---------------- ------------------- Phase I 11.2 1.1 47 Phase II 2.7 1.3 114 : Collapse time (in units of Myrs) of a subclump with $M=5.5\times 10^5\,M_\odot$ irradiated by a Pop III star at distance $D=50\,\rm pc$ ($F_0=600$). For both Phase I and Phase II, we show how a case with a proper treatment of $\rm H_2$ self-shielding (2nd column) differs in collapse time from a case without self-shielding (3rd column) and a case without radiation. When $\rm H_2$ self-shielding is properly treated, collapse occurs in $\sim 1\,\rm Myr$, [*before*]{} the neighbouring Pop III turns off, while when $\rm H_2$ self-shielding is neglected, collapse occurs [*after*]{} the star turns off, which is qualitatively consistent with the simulation results by @2006astro.ph..6019A. []{data-label="table:collsub"} Summary/Discussion {#sec:2star-Discussion} ================== We have studied the radiative feedback effects of the first stars (i.e. Pop III stars) on their nearby minihaloes, by solving radiative transfer and hydrodynamics self-consistently using the 1-D spherical, radiation-hydrodynamics code we have developed. The results can be summarized as follows: - We identified the minimum collapse mass, namely the mass of minihaloes which are able to have a core which cools and collapses in the absence of external radiation. We find that $M_{\rm c,min}\sim 7\times 10^4\,M_\odot$ at $z=20$. In determining $M_{\rm c,min}$, we applied two criteria. First, the collapsing region should reach $n_{\rm H}=10^8\,\rm cm^{-3}$ to be considered as a collapse. Second, this should occur within the Hubble time. The minimum collapse mass we find roughly agrees with that of @2001ApJ...548..509M, where the AMR scheme they used seems to have resolved the inner structure of minihaloes. - Minihaloes could have been in very different stages of their evolution when they were irradiated by a Pop III star. We used two different initial conditions to represent such phase differences. In Phase I, chemical abundances have not yet evolved away from their IGM equilibrium values. This stage is characterized by low $\rm H_2$ fraction, $y_{\rm H_2}\sim 2\times 10^{-6}$ and high electron fraction, $x\sim 10^{-4}$ at the centre. Haloes can be irradiated in Phase II, which is the state of these haloes evolved from Phase I, where $x$ has dropped to $10^{-5}$ by recombination. Phase II is characterized by high $\rm H_2$ fraction $y_{\rm H_2}\sim 10^{-4} - 10^{-3}$, low electron fraction $x= 10^{-5}$, and core density higher than that of Phase I. - Within our parameter space, the I-front is trapped before reaching the core in all cases. Ionized gas evaporates, and a shock-front develops ahead of the I-front and travels into the core. The shock front leads to both positive and negative feedback effects. A boost in density and temperature by a shock increases the $\rm H_2$ formation rate. In some cases, the shock accelerates and obtains a temperature above $10^4\,\rm K$, which is high enough to drive collisional ionization, which then leads to a further boost in $\rm H_2$ fraction. The high temperature and kinetic energy delivered by the shock, on the other hand, tries to disrupt the gas. The nett effect is either 1) an expedited collapse, 2) delayed collapse, 3) neutral (unaffected) collapse, or 4) a disruption, depending upon the flux, halo mass, and the initial condition when irradiated. - At the moment of collapse, halo profiles under radiation are almost identical to those without radiation. Density profiles of different mass haloes are well fit by different power-law profiles, $\rho\propto r^{-w}$, where $w=2.5$, 2.4, 2.3, and 2.2 for $M=10^5$, $2\times 10^5$, $4\times 10^5$, and $8\times 10^5\,M_\odot$, respectively. Some variation in temperature profile exists at the low-mass end, $M=10^5\,M_\odot$, and the high-flux end $F_0=46.3$ ($D=180\,\rm pc$). - Overall, the radiative feedback effect of Pop III stars is not as destructive as naively expected. Minihaloes with $M\ga [1-2]\times 10^5\,M_\odot$ are still able to form cooling and collapsing clouds at their centres even in the presence of radiation. A simple explanation is possible for such behaviour. In Phase I (low $y_{\rm H_2}$ and high $x$), radiation can easily dissociate $\rm H_2$ while the source is on, but after the source dies, high electron fraction allows $\rm H_2$ formation. On the contrary, in Phase II (high $y_{\rm H_2}$ and low $x$), $\rm H_2$ is more easily protected against the dissociating radiation because the higher $\rm H_2$ column density provides self-shielding and compression increases the formation rate. The situation becomes more complicated, however, by other feedback effects which will be described in the following bullets. - Within our parameter space, haloes that are irradiated at Phase I experience expedited collapse predominantly for $10^5 \la M/M_\odot \la 8\times 10^5$, except for the delayed or neutral collapses occurring at the low mass/high flux and the high mass/low flux extremes (e.g. for $M=10^5\,M_\odot$ at $F_0=46.3$ and for $M=8\times 10^5\,M_\odot$ at $F_0=[1.5,\,5.14]$). - Haloes that are irradiated at Phase II show a more complicated behaviour. In this case, unaffected collapse is more frequent, in general, at high and intermediate masses, while for $M=10^5\,M_\odot$, core collapse is now reversed at any $F_0$. Delayed collapse occurs for $M=2\times 10^5\,M_\odot$ at $F_0=46.3$. Unaffected collapse occurs for $M=8\times 10^5\,M_\odot$ for any $F_0$, and for $M=4\times 10^5\,M_\odot$ at $F_0\la 11.6$. Otherwise, for intermediate mass, collapse is either neutral or expedited. - We first find in this paper that coeval formation of Pop III stars is possible even under the influence of ionizing and dissociating radiation from a first star. This occurs either as an expedited collapse or an unaffected collapse. Among those parameters explored in this paper, expedited collapse occurs during the lifetime of the source star when a halo of mass $M=2\times 10^5 \,M_\odot$ in Phase I is irradiated by a Pop III star at a distance $D=180 \,\rm pc$ ($F_0=46.3$). Unaffected collapse occurs for haloes of mass $M=8\times 10^5$ in Phase II during the lifetime of the source star for all different distances (fluxes). - Extending our parameter space to include a specific case studied by @2006astro.ph..6019A, a minihalo merging with a halo hosting a Pop III star, we find that the coeval formation of Pop III stars is possible even in this high ionizing ($F_0\approx 600$) and dissociating ($F_{\rm LW}\sim 2\times 10^{-18}\,\rm erg\,s^{-1}\,cm^{-2}\,Hz^{-1}$) flux case. While @2006astro.ph..6019A find that the secondary star formation in this target halo occurs after the first star dies because of $\rm H_2$ destruction by photodissociation, we find that the minihalo core collapse is expedited to form a star in $\sim 1\,\rm Myr$, long before the first star dies, due to the SIMF and $\rm H_2$ self-shielding. This discrepancy comes from the fact that we account for the effect of $\rm H_2$ self-shielding, while they do not. A proper treatment of $\rm H_2$ self-shielding is important even for such a high flux regime, because the central $\rm H_2$ fraction can reach $y_{\rm H_2}\ga 10^-3$ due to the SIMF and strong $\rm H_2$ self-shielding is possible due to newly created $\rm H_2$. We find the minimum collapse mass $M_{\rm c,min}\sim 7\times 10^4\,M_\odot$ at $z=20$ without radiation. While our result agrees roughly with that of the 3D AMR simulation by @2001ApJ...548..509M, discrepancy becomes larger with those of 3D SPH simulation results (e.g. @2000ApJ...544....6F; @2003ApJ...592..645Y) and a semi-analytical calculation using a uniform-sphere model [@1997ApJ...474....1T]. This implies that the central region of haloes should be resolved well in order to quantify the minimum collapse mass exactly. What does the result of our paper imply for the “first” H II region created by Pop III stars? Because a significant fraction of nearby minihaloes can host second generation stars within the first H II region, it is possible that such a subsequent star formation may at least keep the first H II regions ionized. It may even be possible that individual H II regions grow and overlap, thus finishing the first cosmological reionization. A semi-analytic calculation of minihalo clustering around high density peaks, for example, might allow us to quantify how fast and how big such bubbles can grow. Without secondary star formation, this would simply be a relic H II region in which gas recombines and cools after the source star dies, possibly with metal enrichment from supernova explosion (e.g. @2003ApJ...596L.135B). We found that the minimum collapse mass is $\sim 1-2\times 10^5 \,M_\odot$ even in the presence of Pop III starlight. Such a low value may affect the reionization history significantly. @2006ApJ...639..621A estimates that the instantaneous ionized mass fraction at $z=20$ is $\sim 0.1$, if individual $\sim 10^6 \,M_\odot$ haloes host one $\sim 100 \,M_\odot$ Pop III star each. If the typical mass scale of host haloes is $\sim 10^5 \,M_\odot$ instead, as the number density of haloes would be roughly 10 times as big as that for $M\sim 10^6 \,M_\odot$, Pop III stars alone would be able to finish cosmological reionization at $z \sim 20$[^8]. New reionization sources will form later in more massive haloes with $T_{\rm vir}\ga 10^4 \,\rm K$, which will host a region cooling by the hydrogen atomic cooling. Depending upon how fast such transition occurs, the global reionization history will have different characteristics (e.g. monotonic growth of ionization fraction vs. double reionization). In this paper, we have considered only the radiative feedback effect. Pop III stars, however, may exert additional feedback effects. The H II region developed by a Pop III star inside the host halo breaks out as a “champagne flow” inside the host minihalo, where the I-front separates from the shock-front and runs ahead, transforming from D-type to R-type. The shock front left behind also expands into the IGM and nearby minihaloes would be encountered by this shock-front ultimately. Other feedback effects will come from supernova explosions. If the first star dies and explodes as a supernova, both dynamical and chemical feedback effects would alter the fate of nearby minihaloes, as well. How would the additional presence of $\rm H_2$ dissociating background radiation affect our results? In this paper, we have considered the effect of the radiation from an individual nearby Pop III star, whose SED takes a black body form for a short lifetime ($\sim 2.5$ Myrs). This is the case appropriate to the earliest star formation. It is valid whenever a minihalo resides in a place and time where the background from other, more distant stars is negligible. On average, however, the mean free path to $\rm H_2$ dissociating radiation is greater than that for ionizing radiation prior to reionization, so the situation can arise in which the ionizing radiation from distant sources is filtered out but the UV radiation in the LW bands is not. Suppose a minihalo is under the influence of both Pop III starlight from a nearby star and a persistent background radiation field in the LW bands. In the absence of the nearby star, the dissociating background can only hinder the formation of $\rm H_2$ and its cooling. As such, the $\rm H_2$ fraction inside the minihalo when the nearby Pop III star starts to irradiate it would be lower than it would have been without the background. In this case, even if the background were intense enough on its own to prevent the minihalo from cooling and collapsing, the minihalo could still host a cooling core if $\rm H_2$ formed by the positive feedback from the Pop III star, despite the presence of the background. Indeed, this could occur frequently, because we find that a high electron fraction – and, thus a high $\rm H_2$ fraction – can be achieved by collisional ionization in the postshock region in many cases (SIMF; see Section \[sub:shockstageI\]). This newly created $\rm H_2$ will then be easily protected from the dissociating background by self-shielding, since our simulation results show that this SIMF $\rm H_2$ survives even the much larger – albeit short-lived – flux of $\rm H_2$ dissociating radiation from a nearby star in our most extreme case, $F_{\rm LW}\approx 2000 \times 10^{-21} \rm erg \, s^{-1}\,cm^{-2}\, Hz^{-1}$, as has been shown in Section \[sub:abel\]. Thus, the background would then only prevent those haloes that cannot “host” this SIMF mechanism from cooling and forming stars. We will address this issue further in the future. As the focus of our paper is the fate of neutral cores of target haloes, in which the ionized fraction never exceeds $\sim 10^{-2}$, we neglected processes which are relevant only when gas achieves high ionized fraction, such as HD cooling and charge exchange between $\rm He^+$ ($\rm He$) and $\rm H$ ($\rm H^+$) (see e.g. @2006astro.ph..6106Y). These processes may be important, however, in the relic H II region outside the target minihalos. For instance, HD cooling may cool gas down below the $\rm H_2$ cooling temperature plateau, $T_{\rm H_2}\sim 100\,\rm K$, if $\rm H_2$ formation and cooling start from a highly ionized initial state (e.g. @2006MNRAS.366..247J). We chose two different evolutionary phases of nearby minihaloes as our initial conditions. A more natural way to address this problem is to use the structure and chemical composition of minihaloes and IGM from 3-D, chemistry-hydrodynamics calculation. We intend to extend our study in a more consistent manner by combining a 3-D, chemistry-hydrodynamics simulation and the 1-D, radiation-hydrodynamics simulation in the future. In this paper, we simply adopted a model for virialized haloes (TIS profile). In the future, we will also implement a more realistic growth history of haloes (e.g. @2002ApJ...568...52W) to account for the dynamical effect of mass accretion. Acknowledgments {#acknowledgments .unnumbered} =============== We thank M. Alvarez, T. Abel, S. Glover, I. Iliev, B. O’shea, H. Susa, and D. Whalen for helpful discussions. We also acknowledge the Institute for Nuclear Theory at the University of Washington for their support and hospitality. This work was supported by NASA Astrophysical Theory Program grants NAG5-10825, NAG5-10826, NNG04G177G. Numerical Method and Code Tests {#finite_appendix} =============================== Here we describe the finite-difference scheme used for our 1-D spherical, radiation-hydrodynamics code. The subscript, unless noted otherwise, denotes the position of a shell. The superscript denotes the time. For instance, $\rho_{j+1/2}^{n+1}$ is the zone-centred density of shell $j+1$ at time $t^{n+1}$, and $r_{j}^{n}$ is the zone-edge-centred radius of shell $j$ at time $t^{n}$. The Gas Dynamical Conservation Equations ---------------------------------------- Hydrodynamic conservation equations for the baryonic component (eqs. \[\[eq:realhydro\_mass\]\] - \[\[eq:realhydro\_energy\]\]) are solved following the finite-difference scheme by @1995ApJ...442..480T. We first update the velocity and position using the so-called “leap-frog” scheme, so that the velocity and the position are staggered in time: $$v_{j}^{n+1/2} = v_{j}^{n-1/2} -\left[ 4\pi(r_{j}^{n})^{2}\frac{p_{j+1/2}^{n} -p_{j-1/2}^{n}}{dm_{j}} +\frac{m_{j}^{n}}{(r_{j}^{n})^{2}}\right] dt^{n}, \label{eq:leapfrog1}$$ and $$r_{j}^{n+1} = r_{j}^{n} + v_{j}^{n+1/2}dt^{n+1/2}, \label{eq:leapfrog2}$$ which are second-order accurate. As the mass of each shell is conserved for such a Lagrangian scheme, density is updated following $$\rho_{j+1/2}^{n+1} = \frac{dm_{j+1/2}}{(4/3) \pi[(r_{j+1}^{n+1})^{3} -(r_{j}^{n+1})^{3}]}.$$ In these equations, $$dt^{n}=\frac{1}{2}(dt^{n-1/2}+dt^{n+1/2}),$$ and $$dm_{j}=\frac{1}{2}(dm_{j-1/2}+dm_{j+1/2}).$$ We then advance the energy by $$\begin{aligned} e_{i+1/2}^{n+1} &=& e_{i+1/2}^{n} - p_{i+1/2}^{n}\left(\frac{1}{\rho_{i+1/2}^{n+1}} - \frac{1}{\rho_{i+1/2}^{n}}\right) \nonumber \\ && +\frac{(\Gamma-\Lambda)_{i+1/2}^{n}}{\rho_{i+1/2}^{n+1}}dt^{n+1/2}. \label{eq:energy_tw}\end{aligned}$$ Shocks are treated with the usual artificial viscosity technique. The pressure in the momentum and energy conservation equations is replaced by $P=p+q$, where $$\begin{aligned} q_{i+1/2}^{n+1} = -c_{q}\frac{2}{1/\rho_{i+1/2}^{n+1} -1/\rho_{i+1/2}^{n}} \left|v_{i+1}^{n+1/2}-v_{i}^{n+1/2} \right| \nonumber \\ \times (v_{i+1}^{n+1/2}-v_{i}^{n+1/2}), \label{eq:arti_viscos}\end{aligned}$$ if $v_{i+1}^{n+1/2}-v_{i}^{n+1/2}<0$, and $q=0$ otherwise. We use $c_{q}=4$, which spreads the shock fronts over four or five cells. Dark matter shells are also updated according to equations (\[eq:leapfrog1\]) - (\[eq:arti\_viscos\]) – note that we use fluid approximation as described in Section \[sub:fluid-approx\] –, except that the heating/cooling term is zero in equation (\[eq:energy\_tw\]). Note that the dark matter shells are allowed to have effective shock in our fluid approximation, and therefore we need to compute the artificial viscosity when dark matter shells are converging (equation \[eq:arti\_viscos\]), as in the case of the baryonic gas component. Time Steps ---------- Time step for the finite-differencing is chosen such that important fluid variables do not change abruptly. The relevant time scales are the dynamical, sound-crossing (Courant), cooling(heating), and species-change time scales. In addition, to ensure that the fluid shells do not cross, we also adopt a shell-crossing time. $$dt=\min\{ dt_{{\rm dyn}},\, dt_{{\rm Cour}},\, dt_{{\rm cool}},\, dt_{{\rm spec}},\, dt_{vel}\}$$ $$dt_{{\rm dyn}}=\min\left\{ c_{d}\sqrt{\frac{\pi^{2}r_{j}^{3}}{4m_{j}}}\right\} ,$$ $$dt_{{\rm Cour}}=\min\left\{ c_{{\rm C}}\left|\frac{r_{j}-r_{j-1}}{\sqrt{\gamma(\gamma-1)u_{j}}}\right|\right\} ,$$ $$dt_{{\rm cool}}=\min\left\{ c_{c}\left|\frac{u_{j}\rho_{j}}{(\Gamma-\Lambda)_{j}}\right|\right\} ,$$ $$dt_{{\rm spec}}=\min\left\{ c_{{\rm sp}}\left|\frac{x_{j}}{dx_{j}/dt}\right|,\, c_{{\rm sp}}\left|\frac{y_{{\rm H I},\, j}}{dy_{{\rm H I,}\, j}/dt}\right|\right\}$$ $$dt_{{\rm vel}}=\min\left\{ c_{v}\left|\frac{r_{j}-r_{j-1}}{v_{j}-v_{j-1}}\right|\right\} ,$$ where $c_{d}$, $c_{\rm C}$, $c_{c}$, $c_{\rm sp}$, and $c_{v}$ are coefficients that ensure accurate calculation of the finite difference equations. We use $c_{d}=0.1$, $c_{\rm C}=0.1$, $c_{c}=0.1$, $c_{\rm sp}=0.1$, and $c_{v}=0.05$. In practice, we frequently find that $dt_{\rm dyn}$ can be very small compared to other time scales. We sometimes disregard $dt_{\rm dyn}$ in order to achieve computational efficiency. We confirmed, especially in our problem, that such a treatment does not produce any significant discrepancy from a calculation with $dt_{\rm dyn}$ considered. When the virial temperature of a halo is close to the cooling temperature plateau, for instance, $dt_{\rm dyn}$ must be irrelevant because gas would be almost hydrostatic. Radiative Transfer ------------------ For the radiation field generated from a point source at the centre, the radiative rate coefficient of species $i$ at radius $r$ is given by equation (\[eq:rad\_rate\_ext\_body\]). Finite-differencing this rate coefficient, however, requires some caution. For the baryonic shell at position $j$ (smaller $j$ means closer to the centre) whose inner edge and outer edge have radii $r_{j-1/2}$ and $r_{j+1/2}$, respectively, the incident differential flux at the outer edge is $F_{\nu}^{{\rm int}}(r_{j+1/2})$, and one could naively calculate the rate coefficient of species $i$ by $$k_{i}(r_{j})=\int_{0}^{\infty}d\nu\frac{\sigma_{i,\nu}F_{\nu}^{{\rm ext}}(r_{j+1/2})}{h\nu}. \label{nonconserving_k_ext2}$$ As mentioned already in Section \[sub:External-source\] and Section \[sub:photo-heating\], however, this expression may not yield an accurate result when the shell $k$ is optically thick. In this case, $F_{\nu}$ may change substantially over the shell width, and equation (\[nonconserving\_k\_ext2\]) might overpredict the ionization rate by applying a constant flux over the shell width ($\Delta r_{j}\equiv r_{j+1/2}-r_{j-1/2}$). One may, in principle, choose to set up the initial condition such that all shells are optically thin. However, such a scheme can be very expensive computationally, especially when collapsed haloes are treated. In order to resolve this problem, we use the “photon-conserving scheme” by @1999MNRAS.309..287R and @1999ApJ...523...66A. In this treatment, the number of photons that are absorbed in a shell is the same as the number of ionization events. Equation (\[nonconserving\_k\_ext2\]) can then be re-written as $$\begin{aligned} k_{i}(r_{j}) &=& \int_{0}^{\infty}d\nu\frac{L_{\nu}^{{\rm ext}}(r_{j+1/2})-L_{\nu}^{{\rm ext}}(r_{j-1/2})}{h\nu}\cdot\frac{1}{n_{i}V_{{\rm shell},j}} \nonumber \\ &\simeq& \int_{0}^{\infty}d\nu\frac{F_{\nu}^{{\rm ext}}(r_{j+1/2})}{h\nu} \cdot\frac{1-e^{-\Delta\tau_{i,\nu}(r_{j})}}{n_{i}\Delta r_{j}}, \label{eq:conserving_k_ext2}\end{aligned}$$ where $L_{\nu}^{{\rm ext}}(r)=4\pi r^{2}F_{\nu}^{{\rm ext}}(r)$, $\Delta\tau_{i,\nu}(r_{j})\equiv n_{i}\Delta r_{j}\sigma_{i,\nu}$ is the optical depth of a shell $k$ on a species $i$, and $V_{{\rm shell},j}\simeq4\pi r_{j}^{2}\Delta r_{j}$ is the volume of the shell. Note that when $\Delta\tau_{\nu}\ll1$, equation (\[eq:conserving\_k\_ext2\]) becomes equivalent to equation (\[nonconserving\_k\_ext2\]). For each species, the corresponding radiative reaction rate is calculated by quadrature, by summing the integrand in equation (\[eq:conserving\_k\_ext2\]), then summing over the frequency to obtain the nett radiative reaction rate. Nonequilibrium Chemistry {#nonequilibrium-chemistry} ------------------------ As described in Section \[sub:noneq\_chem\], in order to update the abundance of species $i$, we adopt the finite difference scheme by @1997NewA....2..181A. Based upon equation (\[eq:verygeneric\_rate\_eq\]), each species $i$ is updated by $$n_{i}^{n+1}=\frac{C_{i}^{n+1}(T,\{ n_{j}\}) dt^{n+1/2} + n_{i}^{n}}{1+D_{i}^{n+1}(T,\{ n_{j}\})dt^{n+1/2}}, \label{eq:backward-diff}$$ where the species $\{ n_{j}\}$ is the previously updated value in the order given by @1997NewA....2..181A (note that the letter $n$ ($n+1/2$, $n+1$) in superscript denotes the time $t^{n}$ ($t^{n+1/2}$, $t^{n+1}$). The order they find to be optimal is H, ${\rm H^{+}}$, He, ${\rm He^{+}}$, ${\rm He^{++}}$ and ${\rm e^{-}}$, followed by the algebraic equilibrium expressions for ${\rm H^{-}}$ and ${\rm H^{+}}$, and finally ${\rm H_{2}}$, again by equation (\[eq:backward-diff\]). Numerical resolution {#sub:resolution} -------------------- In practice, we use $500$ dark matter and $1000$ fluid shells sampled uniformly (in radius) from the centre to the truncation radius $r_{{\rm tr}}$. We put a small reflecting core at the centre with negligible size, namely $r_{{\rm core}}=10^{-4}r_{{\rm tr}}$. Such a core is found to be useful in reducing undesirable numerical instability at the centre. Our choice is conservative enough not to affect the overall answer. A wide range of radiation frequency (energy), $h\nu\sim[0.7\,-\,7000]\,{\rm eV}$, is covered by $100$, logarithmically spaced bins, $\Delta E/E \approx 0.04$, together with additional, linearly-spaced bins where radiative cross sections change rapidly as frequency changes. About a dozen linearly spaced bins at each of those rapidly changing points turned out to produce reliable results. Rate coefficients {#sub:rates} ================= In Table \[table:rates\], we list the chemical reaction rates we implemented in our code and the corresponding references. The rate coefficients (1-19) and radiative cross sections (20-26) are mostly from the fit by @1987ApJ...318...32S, except for a few updates. Reactions Reference ---- ------------------------------------------------------ -------------------------------------------- 1 $\rm H + e^- \rightarrow H^+ + 2e^- $ @1987ephh.book.....J 2 $\rm H^+ + e^- \rightarrow H + \gamma $ Case B; @1989agna.book.....O 3 $\rm He + e^- \rightarrow He^+ + 2e^- $ @1987ephh.book.....J 4 $\rm He^+ + e^- \rightarrow He + \gamma $ @1973AA....25..137A 5 $\rm He^+ + e^- \rightarrow He^{++} + 2e^- $ AMDIS Database; @1997NewA....2..181A 6 $\rm He^{++} + e^- \rightarrow He^+ + \gamma $ @1978ppim.book.....S 7 $\rm H + e^- \rightarrow H^- + \gamma $ @1972AA....20..263D [@1987ApJ...318...32S] 8 $\rm H^- + H \rightarrow H_2 + e^- $ @bieniek 9 $\rm H + H^+ \rightarrow H_{2}^{+} + \gamma $ @1976PhRvA..13...58R 10 $\rm H_{2}^{+} + H \rightarrow H_2 + H^+ $ @1979JChPh..70.2877K 11 $\rm H_2 + H \rightarrow 3H $ @1986ApJ...311L..93D 12 $\rm H_2 + H^+ \rightarrow H_{2}^{+} + H $ @2004ApJ...606L.167S 13 $\rm H_2 + e^- \rightarrow 2H + e^- $ @1983ApJ...266..646M 14 $\rm H^- + e^- \rightarrow H + 2e^- $ @1987ephh.book.....J 15 $\rm H^- + H \rightarrow 2H + e^- $ @1984SvA....28...15I 16 $\rm H^- + H^+ \rightarrow 2H $ @1984inch.book.....D 17 $\rm H^- + H^+ \rightarrow H_{2}^{+} + e^- $ @1978JPhB...11L.671P 18 $\rm H_{2}^{+} + e^- \rightarrow 2H $ @1994ApJ...424..983S 19 $\rm H_{2}^{+} + H^- \rightarrow H + H_2 $ @1987IAUS..120..109D 20 $\rm H + \gamma \rightarrow H^+ + e^- $ @1989agna.book.....O 21 $\rm He^+ + \gamma \rightarrow He^{++} + e^- $ @1989agna.book.....O 22 $\rm He + \gamma \rightarrow He^{+ } + e^- $ @1989agna.book.....O 23 $\rm H^- + \gamma \rightarrow H + e^- $ @1972AA....20..263D [@1987ApJ...318...32S] 24 $\rm H_{2}^{+} + \gamma \rightarrow H + H^+ $ @1968PhRv..172....1D 25 $\rm H_2 + \gamma \rightarrow H_{2}^{+} + e^- $ @1978JChPh..69.2126O 26 $\rm H_{2}^{+} + \gamma \rightarrow 2H^{+} + e^- $ @1968JPhB....1..543B 27 $\rm H_2 + \gamma \rightarrow 2H $ Section \[sub:ss\]; @1996ApJ...468..269D Code tests {#code-tests} ========== We now extend the description of our code test problems in Section \[sec:codetest\] and show the results. \(A) The self-similar, spherical, cosmological infall problem [@1985ApJS...58...39B]: A point mass, if placed in an unperturbed Einstein-de Sitter universe, will make all particles around it to be gravitationally bound, leading to a successive turnaround and collapse of spherically shells. Infalling matter will be shocked and form a virialized structure, whose profiles are well described by a self-similar solution. We restrict ourselves to purely baryonic fluid with the ratio of specific heats $\gamma = 5/3$. The turnaround radius $r_{\rm ta}$, at which the Lagrangian proper velocity of a shell is zero, evolves as $$r_{\rm ta}(t)=\left( \frac{3\pi}{4}\right)^{-8/9} \left({\delta_{i} R_{i}^{3}}\right)^{1/3} (t/t_{i})^{8/9}, \label{eq:bert-rta}$$ where $\delta_i R_{i}^{3}$ defines the seed mass $\delta m$ added to the Einstein-de Sitter universe, $$\delta m = \frac{4}{3}\pi \rho_{{\rm H},i} \delta_i R_{i}^{3}, \label{eq:bert-seed}$$ where the initial cosmic mean density $\rho_{{\rm H},i}=1/(6\pi G t_{i}^{2})$ at $t=t_{i}$. The shock radius $r_s$ is a constant fraction of $r_{\rm ta}$: $r_s (t)= 0.338976 \,r_{\rm ta} (t)$ for $\gamma = 5/3$. The dimensionless radius $\lambda \equiv r/r_{\rm ta}$ and the dimensionless density $D\equiv \rho / \rho_{\rm H}$, where the cosmic mean density $\rho_{\rm H}=1/(6\pi G t^2)$, satisfy the unique Bertschinger solution. In Fig. \[fig-test\], we show the density profiles and $r_{\rm s}(t)$, obtained from the simulation with $\delta_i R_{i}^{3} = 1.84\times 10^{71} \,\rm cm^3$, $t_{i}=5.572\times 10^{14} \,\rm s$. \(B) The self-similar blast wave from a strong, adiabatic point explosion in a uniform gas [@1959sdmm.book.....S]: A point explosion drives a self-similar blast wave through the initially static, uniform medium. A strong shock is generated, and $r_{\rm s} (t) = \xi_0 \left( \frac{E}{\rho_0} \right)^{1/5} t^{2/5}$, where $E$ is the thermal energy of explosion, $\rho_0$ is the initial density, and $\xi_0$ is a dimensionless constant determined by $\gamma$. For $\gamma=5/3$, $\xi_0 = 1.152$. We use $E=1.053\times 10^{61} \,\rm erg$, $\gamma=5/3$, and $\rho_0 = 2.5626\times 10^{-24} \,\rm cm^{-3}$ for simulation results displayed in Fig. \[fig-test\]. \(C) The propagation of an I-front from a steady point-source in a uniform, static medium: This is the case where the classical description of the Strömgren radius is plausible, since gas is forced to remain static, and photoionization and recombination are the only physical processes determining the ionized fraction. The I-front from a point source with $N_*$ number of ionizing photons evolves as $$r_{\rm I} (t) = R_{\rm S} \left(1-\exp(-t/t_{\rm rec}) \right)^{1/3}, \label{eq:pure-ift}$$ where $R_{\rm S}\equiv \left[ 3N_{*}/(4\pi n_{\rm H}^2 \alpha) \right]^{1/3}$ is the Strömgren radius, $t_{\rm rec}\equiv 1/(n_{\rm H} \alpha)$ is the recombination time, and $\alpha$ is the recombination rate coefficient. We adopt $N_* = 10^{47} \,\rm s^{-1}$, $n_{\rm H}=10 \,\rm cm^{-3}$, and $\alpha = 1.05\times 10^{-13} \,\rm cm^{3} \,s^{-1}$. For this test, we use a monochromatic light whose frequency is slightly above the hydrogen ionization threshold. \(D) the gas-dynamical expansion of an H II region from a point source in a uniform gas [@1966ApJ...143..700L]: The I-front, initially propagating as a weak R-type front into a uniform medium, slows down and travels as a D-type front, developing a shock front ahead of it. The I-front evolves as $$r_{\rm I} (\tilde{t}) = R_{\rm S,I} \left(1+\frac{7}{4} \frac{\tilde{t}}{t_{\rm sc}}\right)^{4/7}, \label{eq:lasker}$$ where $R_{\rm S,I}$ is the initial Strömgren radius, $\tilde{t}\equiv t-t_c$ is the time measured from the moment $t_c$ when $dr_{\rm I}/dt = c_{\rm I}$, and $c_{\rm I}\equiv (p/\rho)^{1/2}$ is the isothermal sound speed of the ionized gas [@1978ppim.book.....S]. We adopt $N_* = 2.45\times 10^{48} \,\rm s^{-1}$ and $n_{\rm H}=6.4\,\rm cm^{-3}$. Following @1966ApJ...143..700L, we force temperature of the ionized gas to be $10^4 \,\rm K$, which gives $c_{\rm I}=12.86 \,\rm km/s$. \(E) The gas-dynamical expansion-phase of the H II region from a point-source in a nonuniform gas whose density varies with distance $r$ from the source as $r^{-w}$, $w=3/2$ [@1990ApJ...349..126F]: This case is similar to the case (D), except that the density follows a power law, $n_{\rm H} \propto r^{-w}$. Inside the core radius $r_c$, the density is constant at $n_{{\rm H},c}$. The I-front evolves as $$r_{\rm I} (t) = R_{w} \left[1+\frac{7-2w}{4}\left(\frac{12}{9-4w}\right)^{1/2}\frac{c_{\rm I} t}{R_w} \right], \label{eq:franco}$$ where $R_w$ is the size of the initial H II region obtained by equating the ionization rate and the recombination rate. For instance, when $w=3/2$, $$R_{3/2} = r_c \exp \left\{\frac{1}{3} \left[ \left( \frac{R_{\rm S}}{r_c} \right)^3 -1 \right] \right\},$$ where $R_{\rm S}\equiv \left[ 3N_{*}/(4\pi n_{{\rm H},c}^2 \alpha) \right]^{1/3}$. If $w\le 3/2$, the shock front always travels ahead of the I-front. If $w> 3/2$, however, the shock front is overtaken by the I-front, which soon runs to infinity in this “champagne” phase. We restrict ourselves to this critical exponent $w=3/2$. From equation (\[eq:franco\]), we obtain $r_{\rm I} (t) = R_{3/2}\left( 1+2c_{\rm I}/ R_{3/2}\right)$. In our simulation, we use $n_{{\rm H},c}=2\times 10^6 \,\rm cm^{-3}$, $r_c = 2.1\times 10^{16} \,\rm cm$, $N_* = 5\times 10^{49}\,\rm s^{-1}$, and $\alpha = 2.6\times 10^{-13}\,\rm cm^3 s^{-1}$. Temperature of the ionized gas is set at $T=8000\,\rm K$, such that $c_{\rm I} = 11.5\,\rm km/s$. 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We will discuss this further in Section \[sub:abel\] [^4]: As we do not perform a statistical study, our result is independent of the cosmic density power spectrum. The three-year [*WMAP*]{} data does not show a big discrepancy in the set of cosmological parameters of the interest in this paper [@2006astro.ph..3449S]. The change in $\sigma_8$ and the index of the primordial power spectrum $n$ would translate to $\sim 1.4$ redshift delay of structure formation and reionization [@2006ApJ...644L.101A] [^5]: D and Li components have usually been neglected due to their relatively low abundance, hence the negligible contribution to cooling (e.g. @1984ApJ...280..465L [@1987ApJ...318...32S]). Recent studies by @2005MNRAS.364.1378N and @2006MNRAS.366..247J, however, show that enough HD is generated in strongly-shocked, ionized primordial gas which then can cool below the temperature of $\sim 100 \,\rm K$ already achieved by $\rm H_2$ cooling alone, down to the temperature of the CMB. As the HD cooling process is negligible if gas remains neutral (e.g. @2006MNRAS.366..247J), however, we may neglect the HD cooling process in our calculation as long as we are interested in the centre of target haloes which remains mostly neutral at any time. We will discuss this issue further in Section \[sec:2star-Discussion\]. [^6]: The additional case of $D=50\,\rm pc$, $F_0=600$, $M=5.5\times 10^5 \,M_\odot$, will be discussed separately in Section \[sub:abel\] with regard to the case in which the target minihalo is merging with the minihalo which hosts the star, separated by less than its virial radius from the star [^7]: After this paper was written a new preprint was posted which is consistent with our description here, finding $M_{\rm c,min}\approx 10^5\, M_\odot$ [@2006astro.ph..7013O]. [^8]: This argument is based upon the fact that the comoving number density of haloes, $M\,dn/dM$, is roughly proportional to $M^{-1}$. The minihalo population, however, might have been severely reduced by the “Jeans-mass filtering” inside ionized bubbles created around rare, but more massive objects (e.g. @2006astro.ph..7517I), in which case sources hosted by minihaloes would make negligible contribution to cosmic reionization.

--- abstract: 'In this paper we study the smoothness properties of solutions to the KP-I equation. We show that the equation’s dispersive nature leads to a gain in regularity for the solution. In particular, if the initial data $\phi$ possesses certain regularity and sufficient decay as $x \rightarrow \infty$, then the solution $u(t)$ will be smoother than $\phi$ for $0 < t \leq T$ where $T$ is the existence time of the solution.' author: - 'Julie Levandosky[^1] Mauricio Sep’[u]{}lveda[^2] Octavio Vera Villagr’[a]{}n[^3]' title: '**Gain of Regularity for the KP-I Equation**' --- : KP-I equation, gain in regularity, weighted Sobolev space. Introduction ============ The KdV equation is a model for water wave propagation in shallow water with weak dispersive and weak nonlinear effects. In 1970, Kadomtsev & Petviashvili [@KP] derived a two-dimensional analog to the KdV equation. Now known as the KP-I and KP-II equations, these equations are given by $$u_{tx} + u_{xxxx} + u_{xx} + \epsilon u_{yy} + (uu_x)_x = 0$$ where $\epsilon = \mp 1$. In addition to being used as a model for the evolution of surface waves [@AC], the KP equation has also been proposed as a model for internal waves in straits or channels of varying depth and width [@Sn], [@DLW]. The KP equation has also been studied as a model for ion-acoustic wave propagation in isotropic media [@PY]. In this paper we consider smoothness properties of solutions to the KP-I equation $$\begin{aligned} \label{e101}& & (u_{t} + u_{xxx} + u_{x} + u\,u_{x})_{x} - u_{yy} =0,\qquad (x,\,y)\in\mathbb{R}^{2},\quad t\in\mathbb{R}\\ \label{e102}& & u(x,\,y,\,0)=\phi(x,\,y).\end{aligned}$$ Certain results concerning the Cauchy problem for the KP-I equation include the following. Ukai [@Uk] proved local well-posedness for both the KP-I and KP-II equations for initial data in $H^s(\mathbb R^2)$, $s \geq 3$, while Saut [@Sa] proved some local existence results for generalized KP equations. More recently, results concerning global well-posedness for the KP-I equation have appeared. In particular, see the works of Kenig [@Ke] and Molinet, Saut, and Tzvetkov [@MST]. Here we consider the question of gain of regularity for solutions to the KP-I equation. A number of results concerning gain of regularity for various nonlinear evolution equations have appeared. This paper uses the ideas of Cohen [@Co], Kato [@Ka], Craig and Goodman [@CG] and Craig, Kappeler, and Strauss [@CKS]. Cohen considered the KdV equation, showing that “box-shaped" initial data $\phi \in L^2(\mathbb R^2)$ with compact support lead to a solution $u(t)$ which is smooth for $t > 0$. Kato generalized this result, showing that if the initial data $\phi$ are in $L^2((1+e^{\sigma x})\,dx)$, the unique solution $u(t) \in C^\infty(\mathbb R^2)$ for $t > 0$. Kruzhkov and Faminskii [@KF] replaced the exponential weight function with a polynomial weight function, quantifying the gain in regularity of the solution in terms of the decay at infinity of the initial data. Craig, Kappeler, and Strauss expanded on the ideas from these earlier papers in their treatment of highly generlized KdV equations. Other results on gain of regularity for linear and nonlinear dispersive equations include the works of Hayashi, Nakamitsu, and Tsutsumi [@HNT1], [@HNT2], Hayashi and Ozawa [@HO], Constantin and Saut [@CS], Ponce [@Po], Ginibre and Velo [@GV], Kenig, Ponce and Vega [@KPV], Vera [@thesi1], [@Ve] and Ceballos, Sepulveda and Vera [@CSV]. In studying propagation of singularities, it is natural to consider the bicharacteristics associated with the differential operator. For the KdV equation, it is known that the bicharacteristics all point to the left for $t > 0$, and all singularities travel in that direction. Kato [@Ka] makes use of this uniform dispersion, choosing a nonsymmetric weight function decaying as $x \rightarrow -\infty$ and growing as $x \rightarrow \infty$. In [@CKS], Craig, Kappeler and Strauss also make use of a unidirectional propagation of singularities in their results on infinite smoothing properties for generalized KdV-type equations for which $f_{u_{xxx}} \geq c > 0$. For the two-dimensional case, Levandosky [@Le1] proves smoothing properties for the KP-II equation. This result makes use of the fact that the bicharacteristics all point into one half-plane. Subsequently, in [@Le2], Levandosky considers generalized KdV-type equations in two-dimensions, proving that if all bicharacteristics point into one half-plane, an infinite gain in regularity will occur, assuming sufficient decay at infinity of the initial data. In this paper, we address the question regarding gain in regularity for the KP-I equation. Unlike the KP-II equation, the bicharacteristics for the KP-I equation are not restricted to a half-plane but span all of $\mathbb R^2$. As a result, singularities may travel in all of $\mathbb R^2$. However, here we prove that if the initial data decays sufficiently as $x \rightarrow \infty$, then we will gain a finite number of derivatives in $x$ (as well as mixed derivatives). In order to state a special case of our gain in regularity theorem, we first introduce certain function spaces we will be using.\ We define $$\begin{aligned} \label{e106}X^{0}(\mathbb{R}^{2})= \left\{u:\;u,\;\xi^{3}\widehat{u},\;\frac{\eta^{2}}{\xi}\,\widehat{u}\in L^{2}(\mathbb{R}^{2})\right\}\end{aligned}$$ equipped with the natural norm. On the space $$\begin{aligned} \label{e107}\widetilde{X}^{0}(\mathbb{R}^{2})= \left\{u:\;\frac{1}{\xi}\,\widehat{u}(\xi,\,\eta)\in L^{2}(\mathbb{R}^{2})\right\}\end{aligned}$$ we define the operator $\partial_{x}^{-1}$ by $\widehat{\partial_{x}^{-1}u}\equiv\frac{1}{i\,\xi}\,\widehat{u}.$ Therefore, in particular, we can write the norm of $X^{0}(\mathbb{R}^{2})$ as $$\begin{aligned} \label{e108}||u||_{X^{0}(\mathbb{R}^{2})}^{2}=\int_{\mathbb{R}^{2}}[\,u^{2} + u_{xxx}^{2} + (\partial_{x}^{-1}u_{yy})^{2}\,]\,dx\,dy<+\infty\end{aligned}$$ On this space of functions $X^{0}(\mathbb{R}^{2}),$ it makes sense to rewrite - as $$\begin{aligned} \label{e109}& & u_{t} + u_{xxx} + u_{x} + u\,u_{x} - \partial_{x}^{-1}u_{yy} =0,\qquad (x,\,y)\in\mathbb{R}^{2},\quad t\in\mathbb{R}\\ \label{e110}& & u(x,\,y,\,0)=\phi(x,\,y)\end{aligned}$$ and consider weak solutions $u\in X^{0}(\mathbb{R}^{2}).$\ \ [*Definition.*]{} Let N be a positive integer. We define the space of functions $X^{N}(\mathbb{R}^{2})$ as follows $$\begin{aligned} \label{e111}X^{N}=\left\{u:\;u\in L^{2}(\mathbb{R}^{2}),\;{\cal F}^{-1}(\xi^{3}\,\widehat{u})\in H^{N}(\mathbb{R}^{2}),\,{\cal F}^{-1}\left(\frac{\eta^{2}}{\xi}\,\widehat{u}\right)\in H^{N}(\mathbb{R}^{2})\right\}\end{aligned}$$ equipped with the norm $$\begin{aligned} \label{e112}||u||_{X^{N}(\mathbb{R}^{2})}^{2} = \int_{\mathbb{R}^{2}}\left(u^{2} + \sum_{|\alpha|\leq N}[\,(\pa u_{xxx})^{2} + (\pa \partial_{x}^{-1}u_{yy})^{2}\,]\right)\,dx\,dy<+\,\infty\end{aligned}$$ where $\alpha=(\alpha_1,\,\alpha_2)\in\mathbb{Z}^{+}\times\mathbb{Z}^{+}$ and $|\alpha|=\alpha_1 + \alpha_2.$\ \ \ (1) If we consider $|\alpha| = 2L-1$ above, then $\alpha = (2L-1,0)$ in which case the result states that $$\sup_{0 \leq t \leq T} \int_{\mathbb R^2} t^{L-1} (x_+ + e^{\sigma x_-})(\partial_x^{2L-1}u)^2 + \int_0^T \int_{\mathbb R^2} t^{L-1} (1 + e^{\sigma x_-})(\partial_x^{2L}u)^2 < \infty.$$ In particular, this result shows a [*gain*]{} in $L$ derivatives in $x$. (2) While we gain $x$ derivatives and mixed derivatives, we do not gain pure $y$ derivatives. However, we do not require any weighted estimates on $\partial_y^L u$. In addition, we do not require any weighted estimates on $u$. The results on the KP-II equation include gains in pure $y$ derivatives, but also require weighted estimates on $\partial_y^L u$. (3) The assumptions on $u$ are reasonable and shown to hold in section 6. The main idea of the proof is the following. We use an inductive argument where on each level $|\alpha|$, we apply the operator $\pa = \partial_x^{\alpha_1} \partial_y^{\alpha_2}$ to , multiply the differentiated equation by $2f_\alpha\pa u$ where $f_\alpha$ is our weight function, to be specified later, and integrate over $\mathbb R^2$. Doing so, we arrive at the following inequality $$\begin{split} \label{main-eq} & \partial_t \int f (\pa u)^2 + 3 \int f_x (\pa u_x)^2 \leq \int f_x (\pa \partial_x^{-1} u_y)^2 \\ & \qquad + \int [f_t + f_{xxx} + f_x] (\pa u)^2 + \left|2 \int f (\pa u)\pa (uu_x)\right| \end{split}$$ where $\int = \int_{\mathbb R^2} dx dy$. Assuming $f_x > 0$, the second term on the left-hand side has a positive sign, thus allowing us to prove a gain in regularity. We notice that the first term on the right-hand side is of order $|\alpha|$. By choosing appropriate weight functions for each $\alpha$, we have a bound on that term from the previous step of the induction. After proving estimates involving the nonlinear term on the right-hand side of the equation, we apply Gronwall’s inequality to prove the bounds on the terms on the left-hand side of the equation. The plan of the paper is the following. In section 2 we show the derivation of . In sections 3 and 4 we prove an existence result showing that for initial data $\phi \in X^N(\mathbb R^2)$ there exists a smooth solution $u \in L^\infty([0,T]; X^N(\mathbb R^2))$ for a time $T$ depending only on $||\phi||_{X^0}$. In section 5 we prove estimates for the terms on the right-hand side of . In section 6 we prove a priori estimates showing that the solution $u$ found in section 4 also satisfies for the same time $T$ as long as $$\int (\phi^2 + (\partial_y^L \phi)^2 + (1+x_+)^L(\partial_x^L \phi)^2) < \infty.$$ Once we have found the solution $u$ in the appropriate weighted space as well as bounds for terms on the right-hand side of , in section 7, we can state and prove our main gain in regularity result. This proof uses an inductive argument along with the main estimates proven in section 5.\ We will be using non-symmetric weight functions. In particular, we will be using weight functions $f(x,t) \in C^\infty$ which behave roughly like powers of $x$ for $x > 1$ and decay exponentially for $x < -1$. We define our weight classes as follows.\ A function $f = f(x,\,t)$ belongs to the weight class $W_{\sigma \;i\;k}$ if it is a positive $C^{\infty }$ function on $\mathbb{R}\times [0,\,T]$ and there are constants $c_{j},\,0\leq j\leq 5$ such that $$\begin{aligned} \label{e103}&0<c_{1}\leq t^{-\,k}\,e^{-\,\sigma \,x}\,f(x,\,t) \leq c_{2}\qquad \forall \;x<-1,\quad 0<t<T.&\\ \label{e104}&0<c_{3}\leq t^{-\,k}\,x^{-\,i}\,f(x,\,t)\leq c_{4}\qquad \forall \;x>1,\quad 0<t<T.&\\ \label{e105}&\left(t\mid \partial_{t}f\mid + \mid \partial_{x}^{r}f\mid \right)/f\leq c_{5}\qquad \forall \;(x,\,t)\in \mathbb{R}\times [0,\,T],\quad \forall \;r\in \mathbb{N}.&\end{aligned}$$ Thus $f$ looks like $t^{k}$ as $t\rightarrow 0,$ like $x^{i}$ as $x\rightarrow +\infty$ and like $e^{\sigma\,x}$ as $x\rightarrow -\infty.$\ \ Before proceeding, we introduce some other function spaces we will be using.\ Let $N$ be a positive integer. Let $\widetilde H^{N}_x(W_{\sigma\;i\;k})$ be the space of functions $$\begin{aligned} \label{e113}\widetilde H^{N}_x(W_{\sigma \;i\;k})= \left\{v\colon \mathbb{R}^{2}\rightarrow \mathbb{R}\,:\,\;\,||v||_{\widetilde H^{N}_x(W_{\sigma \;i\;k})}^{2}= \int \sum_{|\alpha| \leq N} [(\pa v)^2 + f|\partial_x^{N} v|^{2}] <+\,\infty \,\right\}\end{aligned}$$ with $f\in W_{\sigma \;i\;k}$ fixed.\ \ [**Remarks.**]{} (1) We note that although the norm above depends on $f,$ all choices of $f$ in this class lead to equivalent norms. (2) The usual Sobolev space is $\,H^{N}(\mathbb{R}^{2})$ without a weight. For fixed $f\in W_{\sigma \;i\;k}$ define the space ($N$ be a positive integer) $$\begin{aligned} \lefteqn{L^{2}([0,\,T]:\,\widetilde H_x^{N}(W_{\sigma \;i\;k}))}\nonumber \\ \label{e114}& = & \left\{v(x,\,y,\,t):\; ||v||_{L^{2}([0,\,T]:\,\widetilde H_x^{N}(W_{\sigma \;i\;k}))}^{2}= \int_{0}^{T}||v(\,\cdot \,,\,\cdot\,,\,t)||_{\widetilde H_x^{N}(W_{\sigma \;i\;k})}^{2}dt<+\,\infty \,\right\}\end{aligned}$$ $$\begin{aligned} \lefteqn{L^{\infty }([0,\,T]:\,\widetilde H_x^{N}(W_{\sigma \;i\;k})) } \nonumber \\ \label{e115}& = & \left\{\,v(x,\,y,\,t):\;||v||_{L^{\infty }([0,\,T]:\,\widetilde H_x^{N}(W_{\sigma \;i\;k}))}= \sup_{t\in [0,\,T]}\,||v(\,\cdot\,,\,\cdot \,,\,t)||_{\widetilde H_x^{N}(W_{\sigma \;i\;k})}<+\,\infty \,\right\}.\end{aligned}$$ For simplicity, let $$\begin{aligned} \label{e118}{\cal Z}_{L} = X^{1}(\mathbb{R}^{2})\bigcap \widetilde{H}_x^{L}(W_{0\;L\;0}).\end{aligned}$$ With this notation, ${\cal Z}_{L}$ consists of those functions $u$ such that $$\begin{aligned} \label{e119}||u||_{{\cal Z}_{L}}^{2}=\int_{\mathbb{R}^{2}}[u^2 + u_{xxxx}^{2} + (\partial_{x}^{-1}u_{yy})^{2} + u_{yy}^2 + \sum_{|\alpha| \leq L} (\pa u)^2 + f(\partial_x^L u)^2 \,]\,dx\,dy\end{aligned}$$ for some $f\in W_{0\;L\;0}.$\ \ We now state a lemma describing one of the types of bounds we will be using for our a priori estimates.\ $$\begin{aligned} \label{e120}||u||_{L^{\infty}(\mathbb{R}^{2})}\leq c\left(\int_{\mathbb{R}^{2}}[\,1 + |\xi|^{p} + |\eta|^{q}\,]\,|\widehat{u}|^{2}\,d\xi\,d\eta\right)^{1/2}.\end{aligned}$$ [*Proof.*]{} The proof follows from writing $u$ in terms of its inverse Fourier transform and using the fact that $$\begin{aligned} \int_{\mathbb{R}^{2}}\frac{1}{1 + |\xi|^{p} + |\eta|^{q}}\,d\xi\,d\eta <+\infty\end{aligned}$$ for $p,\,q$ satisfying our hypothesis. $\square$\ \ In particular, we have: $$\begin{aligned} \label{e121}||u||_{L^{\infty}(\mathbb{R}^{2})}\leq c\left(\int_{\mathbb{R}^{2}}[\,u^{2} + u_{xx}^{2} + u_{y}^{2}\,]\,dx\,dy\right)^{1/2}.\end{aligned}$$ Main Equality ============= We consider the KP-I equation $$\begin{aligned} \label{e201}& & u_{t} + u_{xxx} + u_{x} + u\,u_{x} - \partial_{x}^{-1}u_{yy} =0,\qquad (x,\,y)\in\mathbb{R}^{2},\quad t\in\mathbb{R}\\ \label{e202}& & u(x,\,y,\,0)=\phi(x,\,y).\end{aligned}$$ [**Lemma 2.1.**]{} [*Let $u$ be a solution of - with enough Sobolev regularity and with sufficient decay at infinity. Let $f=f(x,t)$. Then*]{} $$\begin{aligned} \lefteqn{\partial_{t}\int_{\mathbb{R}^{2}}f (\pa u)^{2}\,dx\,dy + \int_{\mathbb{R}^{2}}g\,(\pa u_x)^{2}\,dx\,dy}\nonumber \\ \label{e203}& & +\int_{\mathbb{R}^{2}}\theta\, (\pa u)^{2}\,dx\,dy + \int_{\mathbb{R}^{2}}\theta_{1}\, (\pa \partial_x^{-1} u_y)^{2}\,dx\,dy + \int_{\mathbb{R}^{2}}R_{\alpha}\,dx\,dy = 0\end{aligned}$$ [*such that*]{} $$\begin{aligned} g & = & 3\,f_{x}\\ \theta & = & -\;[f_{t} + f_{xxx} + f_{x}\,]\\ \theta_{1} & = & -\;f_{x}\\ R_{\alpha} & = & 2\sum_{n=0}^{\alpha_1}\sum_{m=0}^{\alpha_2} {\alpha_1\choose n}{\alpha_2\choose m}f\,(\pa u)\,(\partial_{x}^{n}\partial_{y}^{m}u) \,(\partial_{x}^{\alpha_1 + 1 - n}\partial_{y}^{\alpha_2 - m}u).\end{aligned}$$ [*Proof.*]{} Applying the operator $\pa$ to , we have $$\begin{aligned} & & \pa u_{t} + \pa u_{xxx} + \pa u_x + \pa (u\,u_x) - \pa \partial_x^{-1} u_{yy} =0.\end{aligned}$$ Multiplying by $2\,f\, \pa u$ and integrating over $\mathbb{R}^{2},$ we have $$\begin{split} \label{e204} & 2\int f\,(\pa u)\,(\pa u)_{t} + 2\int f\,(\pa u)\,(\pa u_{xxx}) + 2\int f\,(\pa u)\,(\pa u_x) \\ & \qquad + 2\int f\, (\pa u)\,\pa (u\,u_{x}) -\;2\int f\, (\pa u)\,(\pa \partial_{x}^{-1} u_{yy}) =0. \end{split}$$ Each term in is calculated separately integrating by parts $$\begin{aligned} 2\int f\, (\pa u)\,(\pa u)_{t} = \partial_{t}\int f\,(\pa u)^{2} - \int f_{t}\,(\pa u)^{2}.\end{aligned}$$ $$\begin{aligned} 2\int f\,(\pa u)\,(\pa u_{xxx}) =3\int f_{x}\,(\pa u_x)^{2} - \int f_{xxx}\,(\pa u)^{2}.\end{aligned}$$ $$\begin{aligned} 2\int f\,(\pa u)\,(\pa u_x) = -\int f_{x}\,(\pa u)^{2}\end{aligned}$$ $$\begin{aligned} -\;2\int f\, (\pa u)\,(\pa \partial_{x}^{-1}u_{yy}) =-\int f_{x}\, (\pa \partial_{x}^{- 1}u_y)^{2}.\end{aligned}$$ $$\begin{aligned} 2\int f\, (\pa u)\,\pa (u\,u_{x}) = \;2\sum_{n=0}^{\alpha_1}\sum_{m=0}^{\alpha_2}{\alpha_1\choose n}{\alpha_2\choose m}\int f\,(\pa u)\,(\partial_{x}^{n}\partial_{y}^{m}u)\,(\partial_{x}^{\alpha_1 + 1 - n}\partial_{y}^{\alpha_2 - m}u).\end{aligned}$$ Replacing in we obtain $$\begin{split} & \partial_{t}\int f\,(\pa u)^{2} + 3\int f_{x}\,(\pa u_x)^{2} \\ & - \int [f_{t} + f_{xxx} + f_{x}]\,(\pa u)^{2} - \int f_{x}\,(\pa \partial_{x}^{- 1}u_y)^{2} \\ & +\;2\sum_{n=0}^{\alpha_1}\sum_{m=0}^{\alpha_2}{\alpha_1\choose n}{\alpha_2\choose m}\int f\,(\pa u)\,(\partial_{x}^{n} \partial_{y}^{m}u)\,(\partial_{x}^{\alpha_1 + 1 - n}\partial_{y}^{\alpha_2 - m}u) = 0. \end{split}$$ Therefore, we obtain the [**Main Equality**]{}, $$\begin{split} & \partial_{t}\int f (\pa u)^{2} + 3\int f_{x}\,(\pa u_x)^{2} + \int \theta\, (\pa u)^{2} + \int \theta_{1}\, (\pa \partial_{x}^{- 1}u_y)^{2} + \int R_{\alpha} = 0 \end{split}$$ such that $$\begin{aligned} \theta & = & -\;[f_{t} + f_{xxx} + f_{x}\,]\\ \theta_{1} & = & -\;f_{x}\\ R_{\alpha} & = & 2\sum_{n=1}^{\alpha_1}\sum_{m=1}^{\alpha_2}{\alpha_1\choose n}{\alpha_2 \choose m}f\,(\pa u)\,(\partial_{x}^{n}\partial_{y}^{m}u)\,(\partial_{x}^{\alpha_1 + 1 - n}\partial_{y}^{\alpha_2 - m}u).\end{aligned}$$ An a priori estimate ==================== In section four we prove a basic local-in-time existence theorem for -. The proof relies on approximating by a sequence of linear equations. In this section, we prove an existence theorem for linear equations as well as an a priori estimate on those solutions which will be necessary for our main existence theorem in the next section. We begin by approximating by the linear equation $$\begin{aligned} \label{e305}& & u_{t}^{(n)} + u_{xxx}^{(n)} + u_{x}^{(n)} + u^{(n - 1)}\,u_{x}^{(n)} - \partial_{x}^{-1}u_{yy}^{(n)} =0\end{aligned}$$ where the initial condition is given by $u^{(n)}(x,\,y,\,0)=\phi(x,\,y)$ and the first approximation is given by $u^{(0)}(x,\,y,\,t)=\phi(x,\,y).$ The linear equation which is to be solved at each iteration is of the form $$\begin{aligned} \label{e306}& & u_{t} + u_{xxx} + u_{x} + b\,u_{x} - \partial_{x}^{-1}u_{yy} =0.\end{aligned}$$ where $b$ is a smooth bounded coefficient. Below we show that this equation can be solved in any interval of time in which the coefficient is defined.\ (Existence for linear equation). [*Given initial data $\phi\in H^{\infty}(\mathbb{R}^{2})=\bigcap_{N\geq 0}H^{N}(\mathbb{R}^{2})$ and $\partial_{x}^{-1}\phi_{yy}\in\bigcap_{N\geq 0}H^{N}(\mathbb{R}^{2})$ there exists a unique solution of . The solution is defined in any time interval in which the coefficients are defined.*]{}\ \ [*Proof.*]{} Let $T>0$ be arbitrary and $M>0$ be a constant. Let $$\begin{aligned} {\cal L} = \partial_{t} + \partial_{x}^{3} + \partial_{x} + b\,\partial_{x} - \partial_{x}^{-1}\partial_{y}^{2}\end{aligned}$$ be defined on those functions $u\in X^{0}(\mathbb{R}^{2}).$ Recall that $u\in X^{0}(\mathbb{R}^{2})$ means $u,\,u_{xxx},\,\frac{\eta^{2}}{\xi}\widehat{u}\in L^{2}(\mathbb{R}^{2}).$ We consider the bilinear form ${\cal B}:{\cal D}\times {\cal D}\rightarrow \mathbb{R},$ $$\begin{aligned} {\cal B}(u,\,v) = \left<u,\,v\right>=\int_{0}^{T}\int_{\mathbb{R}^{2}}e^{-M\,t}\,u\,v\,dx\,dy\,dt\end{aligned}$$ where ${\cal D}=\{u\in C([0,\,T]:\,L^{2}(\mathbb{R}^{2})):\,u(x,\,y,\,0)=0\}.$ By integration by parts, we see that $$\begin{aligned} \int_{\mathbb{R}^{2}}{\cal L}u\cdot u\,dx\,dy & = & \frac{1}{2}\,\partial_{t}\int_{\mathbb{R}^{2}}u^{2}\,dx\,dy - \frac{1}{2}\int_{\mathbb{R}^{2}}b_{x}\,u^{2}\,dx\,dy \\ & \geq & \frac 12 \partial_t \int_{\mathbb R^2} u^2 \,dx\,dy - \frac 12 \int_{\mathbb R^2} c u^2 \,dx\,dy\end{aligned}$$ We multiply by $e^{-M\,t}$ and integrate in time to obtain for $u\in C([0,\,T]:\,X^{0}(\mathbb{R}^{2}))$ with $u(x,\,y,\,0)=0$ $$\begin{aligned} \label{e307}\left<{\cal L}u,\,u\right> \geq e^{-M\,t}\int_{\mathbb{R}^{2}}u^{2}\,dx\,dy + (M - c)\int_{\mathbb{R}^{2}}e^{-M\,t}\,u^{2}\,dx\,dy.\end{aligned}$$ Thus, $\left<{\cal L}u,\,u\right>\geq \left<u,\,u\right>$ provided $M$ is chosen large enough. Similarly, $\left<{\cal L}^{*}v,\,v\right>\geq \left<v,\,v\right>$ for all $v\in C([0,\,T]:\,X^{0}(\mathbb{R}^{2}))$ such that $v(x,\,y,\,T)=0$ where ${\cal L}^{*}$ denotes the formal adjoint of ${\cal L}.$ Therefore, $\left<{\cal L}^{*}v,\,{\cal L}^{*}u\right>$ is an inner product on ${\cal D}^{*}=\{v\in C([0,\,T]:\,X^{0}(\mathbb{R}^{2})):\,v(x,\,y,\,T)=0\}.$ Denote by $Y$ the completion of ${\cal D}^{*}$ with respect to this inner product. By the Riesz representation theorem, there exists a unique solution $V\in Y,$ such that for any $v\in {\cal D}^{*},$ $\left<{\cal L}^{*}V,\,{\cal L}^{*}v\right>=(\phi,\,v(x,\,y,\,0))$ where we used the fact that $(\phi,\,v(x,\,y,\,0))$ is a bounded linear functional on ${\cal D}^{*}.$ Then $w={\cal L}^{*}V$ is a weak solution of ${\cal L}w=0,$ $w_{0}=\phi$ with $w\in L^{2}(\mathbb{R}^{2}\times [0,\,T]).$ [*Remark.*]{} To obtain higher regularity of the solution, we repeat the proof with higher derivatives included in the inner product. It is a standard approximation procedure to obtain a result for general initial data. $\square$\ \ Next, we need to introduce a new function space. Let $$\begin{aligned} \label{e303}{Z}_{T}^{N}=\{u:\;u\in L^{\infty}([0,\,T]:\,H^{(N + 3,\,N + 2)}(\mathbb{R}^{2})),\;u_{t}\in L^{\infty}([0,\,T]:\,H^{N}(\mathbb{R}^{2}))\}\end{aligned}$$ where $H^{(\alpha_1,\,\alpha_2)}(\mathbb{R}^{2})=\{u:\;u,\, \partial_{x}^{\alpha_1}u,\,\partial_{y}^{\alpha_2}u\in L^{2}(\mathbb{R}^{2})\}$ with the accompanying norm $$\begin{aligned} \label{e304}||u||_{{Z}_{T}^{N}}^{2}=\sup_{t\in [0,\,T]}\int_{\mathbb{R}^{2}}\left(u^{2} + \sum_{|j|=N}[\,(\partial^{j}u_{xxx})^{2} + (\partial^{j}u_{yy})^{2}\,]\right)+ \int_{\mathbb{R}^{2}}\left(u_{t}^{2} + \sum_{|j|=N}(\partial^{j}u_{t})^{2}\right).\end{aligned}$$ Using this function space and the linearized equation , we consider the mapping $\Pi: Z_{T}^{N} \rightarrow Z_{T}^{N}$ such that $u^{(n)} = \Pi(u^{(n-1)})$ and our first approximation is given by $u^{(0)}(x,y,t) = \phi(x,y)$. In Lemma 3.2 below, we show an a priori estimate which will be used on our sequence of solutions $\{u^{(n)}\}$ in our main existence theorem in section four.\ \ [**Lemma 3.2.**]{} [*Let $v,\,w$ be a pair of functions in ${Z}_{t}^{N}$ for all $N$ and all $t\geq 0,$ such that $v,\,w$ are solutions to*]{} $$\begin{aligned} \label{e308}& & v_{t} + v_{xxx} + v_{x} + w\,v_{x} - \partial_{x}^{-1}v_{yy} =0.\end{aligned}$$ [*Then for all $N\geq 0,$ the following inequality holds: $$\begin{aligned} \label{e309}||v||_{{Z}_{t}^{N}}^{2} \leq ||v(\,\cdot\,,\,\cdot\,,\,0)||_{H^{(N + 3,\,N + 2)}(\mathbb{R}^{2})}^{2} + ||v_{t}(\,\cdot\,,\,\cdot\,,\,0)||_{H^{N}(\mathbb{R}^{2})}^{2} + c\,t\,||w||_{{Z}_{t}^{N}}\;||v||_{{Z}_{t}^{N}}^{2}\end{aligned}$$ for all $t\geq 0.$*]{}\ \ [*Proof.*]{} We will show that for each $j,$ $|j|\geq 0$ and $0\leq \widetilde{t}\leq t,$ $$\begin{aligned} \partial_{t}\int[\,(\partial^{j}v(\,\cdot\,,\,\cdot\,,\,\widetilde{t}))^{2} + & (\partial^{j}\partial_{x}^{3}v(\,\cdot\,,\,\cdot\,,\,\widetilde{t}))^{2} + (\partial^{j}\partial_{y}^{2}v(\,\cdot\,,\,\cdot\,,\,\widetilde{t}))^{2} + (\partial^{j}v_{t}(\,\cdot\,,\,\cdot\,,\,\widetilde{t}))^{2}\,] \\ & \leq c\,||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ We begin by taking $j$ derivatives of . We have $$\begin{aligned} \label{e310}& & \partial^{j}v_{t} + \partial^{j}v_{xxx} + \partial^{j}v_{x} + \partial^{j}(w\,v_{x}) - \partial^{j}\partial_{x}^{-1}v_{yy} =0.\end{aligned}$$ Multiply by $2\,\partial^{j}v$ and integrate over $\mathbb{R}^{2}.$ Hence $$\begin{aligned} \partial_{t}\int (\partial^{j}v(\,\cdot\,,\,\cdot\,,\,\widetilde{t}))^{2} & \leq & c\,\left|\int\partial^{j}(w\,v_{x})\,(\partial^{j}v) \right|\\ & \leq &\left|\int[\,(\partial^{j}w)\,v_{x}\;+\, \ldots\,+\;w\,(\partial^{j}v_{x})\,]\,(\partial^{j}v) \right|.\end{aligned}$$ The remainder terms can be bounded as follows: $$\begin{aligned} \left|\int(\partial^{j}w)\,v_{x}\,(\partial^{j}v) \right| & \leq & ||v_{x}||_{L^{\infty}(\mathbb{R}^{2})} \left(\int(\partial^{j}w)^{2} \right)^{1/2} \left(\int(\partial^{j}v)^{2} \right)^{1/2}\\ & \leq & c\left(\int[\,v_{x}^{2} + v_{xxx}^{2} + v_{xy}^{2}\,] \right)^{1/2}||w||_{H^{|j|}(\mathbb{R}^{2})}\; ||v||_{H^{|j|}(\mathbb{R}^{2})}\\ & \leq & c\;||w||_{{Z}_{t}^{|j|}}\; ||v||_{{Z}_{t}^{|j|}}^{2}\end{aligned}$$ and $$\begin{aligned} \left|\int w\,(\partial^{j}v_{x})\,(\partial^{j}v) \right| & \leq & \left|\int w_{x}\,(\partial^{j}v)^{2} \right| \\ & \leq & c\;||w_{x}||_{L^{\infty}(\mathbb{R}^{2})}\int (\partial^{j}v)^{2} \\ & \leq & c\left(\int[\,w_{x}^{2} + w_{xxx}^{2} + w_{xy}^{2}\,] \right)^{1/2}||v||_{{Z}_{t}^{|j|}}^{2}\\ & \leq & c\,||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ Therefore, we obtain $$\begin{aligned} \partial_{t}\int(\partial^{j}v(\,\cdot\,,\,\cdot\,,\,\widetilde{t}))^{2} \leq c\,||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ Next, we take three $x$ derivatives of , multiply by $2\,\partial^{j}v_{xxx}$ and integrate over $\mathbb{R}^{2}.$ Our inequality becomes $$\begin{aligned} \partial_{t}\int\partial^{j}v_{xxx} & \leq c\left|\int(\partial^{j}(w\,v_{x})_{xxx}) \,(\partial^{j}v_{xxx}) \right|\\ & \leq c\left|\int \partial^{j}(w_{xxx}\,v_{x} + 2\,w_{xx}\,v_{xx} + 2\,w_{x}\,v_{xxx} + w\,v_{xxxx}) (\partial^{j}v_{xxx}) \right|\\ & \leq c\left|\int \partial^{j}(w_{xxx}\,v_{x})\,(\partial^{j}v_{xxx}) \right| + c\left|\int\partial^{j}(w_{xx}\,v_{xx})\, (\partial^{j}v_{xxx}) \right|\\ & \qquad +\; c\left|\int\partial^{j}(w_{x}\,v_{xxx})\, (\partial^{j}v_{xxx}) \right| + c \left|\int\partial^{j}(w\,v_{xxxx})\,(\partial^{j}v_{xxx}) \right|\\ & \leq I_{1} + I_{2} + I_{3} + I_{4}.\end{aligned}$$ We will look at terms $I_{k},\;k=1,\,2,\,3,\,4$ below. For $I_{1}$ we have $$\begin{aligned} \lefteqn{\left|\int \partial^{j}(w_{xxx}\,v_{x})\,(\partial^{j}v_{xxx}) \right|}\\ & = & \left|\int [\,\partial^{j}w_{xxx})\,v_{x}\,(\partial^{j}v_{xxx})\;+ \,\ldots \,+\; w_{xxx}\,(\partial^{j}v_{x})\,(\partial^{j}v_{xxx})\,] \right|\\ & \leq & ||v_{x}||_{L^{\infty}(\mathbb{R}^{2})} \left(\int(\partial^{j}w_{xxx})^{2} \right)^{1/2} \left(\int(\partial^{j}v_{xxx})^{2} \right)^{1/2}\\ & & +\, \cdots\, + ||\partial^{j}v_{x}||_{L^{\infty}(\mathbb{R}^{2})} \left(\int w_{xxx}^{2} \right)^{1/2} \left(\int(\partial^{j}v_{xxx})^{2} \right)^{1/2}\\ & \leq & ||v||_{{Z}_{t}^{0}}\,||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}} +\,\cdots \, + ||v||_{{Z}_{t}^{|j|}}\,||w||_{{Z}_{t}^{0}}\,||v||_{{Z}_{t}^{|j|}}\\ & \leq & c\;||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ For $I_{2},$ $$\begin{aligned} \left|\int\partial^{j}(w_{xx}\,v_{xx}) \,(\partial^{j}v_{xxx}) \right| = \left|\int[\,(\partial^{j}w_{xx})\,v_{xx} +\,\ldots\,+ w_{xx}(\partial^{j}v_{xx})\,]\,(\partial^{j}v_{xxx}) \right|.\end{aligned}$$ To bound these terms, we will use the following anisotropic imbedding in [@BIN]. For $2\leq n<6,$ $$\begin{aligned} \label{e311}\left(\int_{\mathbb{R}^{2}}|u|^{n} \right)^{1/n}\leq \left(\int_{\mathbb{R}^{2}}[\,u^{2} + u_{x}^{2} + (\partial_{x}^{-1}u_{y})^{2}\,]\right)^{1/2}.\end{aligned}$$ We will look at the most difficult terms to bound below. $$\begin{aligned} \lefteqn{\left|\int (\partial^{j}w_{xx})\,v_{xx}\,(\partial^{j}v_{xxx}) \right|}\\ & \leq & \left(\int(\partial^{j}w_{xx})^{4} \right)^{1/4} \left(\int(v_{xx})^{4} \right)^{1/4} \left(\int(\partial^{j}v_{xxx})^{2} \right)^{1/2}\\ & \leq & \left(\int[\,(\partial^{j}\,w_{xx})^{2} + (\partial^{j}\,w_{xxx})^{2} + (\partial^{j}\,w_{xy})^{2}\,] \right)^{1/2}\\ & & \times \left(\int[\,(v_{xx})^{2} + (v_{xxx})^{2} + (v_{xy})^{2}\,] \right)^{1/2} \left(\int(\partial^{j}v_{xxx})^{2} \right)^{1/2}\\ & \leq & ||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2},\end{aligned}$$ while, $$\begin{aligned} \lefteqn{\left|\int w_{xx}\,(\partial^{j}v_{xx}) \,(\partial^{j}v_{xxx}) \right|}\\ & = & c\left|\int w_{xxx}\,(\partial^{j}v_{xx})^{2} \right|\\ & \leq & c\left(\int w_{xxx}^{2} \right)^{1/2} \left(\int(\partial^{j}v_{xx})^{4} \right)^{1/2}\\ & \leq & c\;||w||_{{Z}_{t}^{0}}\left(\int [\,(\partial^{j}v_{xx})^{2} + (\partial^{j}v_{xxx})^{2} + (\partial^{j}v_{xy})^{2}\,] \right)^{1/2}\\ & \leq & c\;||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ For $I_{3},$ $$\begin{aligned} \lefteqn{\left|\int \partial^{j}(w_{x}\,v_{xxx})\,(\partial^{j}v_{xxx}) \right|}\\ & = & \left|\int[\,(\partial^{j}w_{x})\,v_{xxx} +\,\ldots\,+ w_{x}\,(\partial^{j}v_{xxx})\,]\,(\partial^{j}v_{xxx}) \right|\\ & \leq & ||\partial^{j}w_{x}||_{L^{\infty}(\mathbb{R}^{2})} \left(\int v_{xxx}^{2} \right)^{1/2} \left(\int(\partial^{j}v_{xxx})^{2} \right)^{1/2}\\ & & +\,\ldots\,+ ||w_{x}||_{L^{\infty}(\mathbb{R}^{2})} \left(\int(\partial^{j}v_{xxx})^{2} \right)^{1/2}\\ & \leq & c\,||w||_{{Z}_{t}^{|j|}}\,||v_{{Z}_{t}^{0}}\,||v||_{{Z}_{t}^{|j|}} +\,\ldots\,+ c\,||w||_{{Z}_{t}^{0}}\,||v||_{{Z}_{t}^{|j|}}^{2}\\ & \leq & c\,||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ Lastly, for $I_{4},$ $$\begin{aligned} \left|\int\partial^{j}(w\,v_{xxxx}) \,(\partial^{j}v_{xxx}) \right| = \left|\int[\,(\partial^{j}w)\,v_{xxxx} + \,\ldots\, + w\,(\partial^{j}v_{xxxx})\,]\,(\partial^{j}v_{xxx}) \right|.\end{aligned}$$ The first term is handled below. If $j=(0,\,0),$ then $$\begin{aligned} \left|\int(\partial^{j}w) \,v_{xxxx}\,(\partial^{j}v_{xxx}) \right| & = & \left|\int w\,v_{xxxx}\,v_{xxx} \right|\\ & = & c\left|\int w_{x}\,v_{xxx}^{2} \right|\\ & \leq & c\,||w_{x}||_{L^{\infty}(\mathbb{R}^{2})} \left(\int v_{xxx}^{2} \right)\\ & \leq & c\,||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{0}}^{2},\end{aligned}$$ while, if $|j|>0,$ then $$\begin{aligned} \left|\int(\partial^{j}w)\,w_{xxxx} \,(\partial^{j}v_{xxx}) \right| & \leq & ||\partial^{j}w||_{L^{\infty}(\mathbb{R}^{2})} \left(\int v_{xxxx}^{2}\right)^{1/2} \left(\int(\partial^{j}v_{xxx})^{2}\right)^{1/2}\\ & \leq & ||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ The last term in $I_{4}$ is handled below $$\begin{aligned} \left|\int w\,(\partial^{j}v_{xxxx}) \,(\partial^{j}v_{xxx}) \right| & = & c\left|\int w_{x}\,(\partial^{j}v_{xxx})^{2} \right|\\ & \leq & c\,||w_{x}||_{L^{\infty}(\mathbb{R}^{2})} \left(\int(\partial^{j}v_{xxx})^{2} \right)\\ & \leq & c\,||w||_{{Z}_{t}^{0}}\,||v||_{{Z}_{t}^{|j|}}^{2}\\ & \leq & c\,||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ Consequently, we conclude $$\begin{aligned} \partial_{t}\int(\partial^{j}v_{xxx})^{2} \leq c\,||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ Next we take two $y$ derivatives of , multiply by $2\,(\partial^{j}v_{yy}),$ and integrate over $\mathbb{R}^{2}.$ Therefore, we have $$\begin{aligned} \partial_{t}\int(\partial^{j}v_{yy})^{2} & \leq c\,\left|\int\partial^{j}(w\,v_{x})_{yy} \,(\partial^{j}v_{yy}) \right|\\ & \leq c\,\left|\int\partial^{j}(w_{yy}\,v_{x} + 2\,w_{y}\,v_{xy} + w\,v_{xyy}) \,(\partial^{j}v_{yy}) \right|\\ & \leq c\,\left|\int\partial^{j}(w_{yy}\,v_{x})\,(\partial^{j}v_{yy}) \right| + c\left|\int\partial^{j}(w_{y}\,v_{xy})\,(\partial^{j}v_{yy}) \right| \\ & \qquad + c\left|\int\partial^{j}(w\,v_{xyy}) \,(\partial^{j}v_{yy}) \right|\\ & \leq I_{5} + I_{6} + I_{7}.\end{aligned}$$ First, we look at $I_{5},$ $$\begin{aligned} \lefteqn{\left|\int\partial^{j}(w_{yy}\,v_{x}) \,(\partial^{j}v_{yy}) \right|}\\ & = & \left|\int[\,(\partial^{j}w_{yy})\,v_{x} + \,\ldots\, + w_{yy}\,(\partial^{j}v_{x})\,]\,(\partial^{j}v_{yy}) \right|\\ & \leq & c\,||v_{x}||_{L^{\infty}(\mathbb{R}^{2})} \left(\int(\partial^{j}w_{yy})^{2} \right)^{1/2} \left(\int(\partial^{j}v_{yy})^{2} \right)^{1/2}\\ & & +\,\ldots\,+ c\,||\partial^{j}v_{x}||_{L^{\infty}(\mathbb{R}^{2})} \left(\int w_{yy}^{2} \right)^{1/2} \left(\int(\partial^{j}v_{yy})^{2} \right)^{1/2}\\ & \leq & c\,||v||_{{Z}_{t}^{0}}\,||w||_{{Z}_{t}^{|j|}}\,||v||_{Z_{t}^{|j|}} +\,\ldots\,+ c\,||v||_{{Z}_{t}^{|j|}}\,||w||_{{Z}_{t}^{0}}\\ & \leq & c\,||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ For $I_{6},$ $$\begin{aligned} \lefteqn{\left|\int\partial^{j}(w_{y}\,v_{xy}) \,(\partial^{j}v_{yy}) \right|}\\ & = & \left|\int[\,(\partial^{j}w_{y})\,v_{xy} + \,\ldots\, + w_{y}\,(\partial^{j}v_{xy})\,]\,(\partial^{j}v_{yy}) \right|\\ & \leq & c\,||w_{y}||_{L^{\infty}(\mathbb{R}^{2})} \left(\int(\partial^{j}v_{xy})^{2} \right)^{1/2} \left(\int(\partial^{j}v_{yy})^{2} \right)^{1/2}\\ & & +\,\ldots\,+ c\,||\partial^{j}w_{y}||_{L^{\infty}(\mathbb{R}^{2})} \left(\int v_{xy}^{2} \right)^{1/2} \left(\int(\partial^{j}v_{yy})^{2} \right)^{1/2}\\ & \leq & c\,||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{0}}\,||v||_{{Z}_{t}^{|j|}} +\,\ldots\,+ c\,||w||_{{Z}_{t}^{0}}\,||v||_{{Z}_{t}^{0}}\\ & \leq & c\,||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ Lastly, for $I_{7},$ $$\begin{aligned} \left|\int\partial^{j}(w\,v_{xyy})\,(\partial^{j}v_{yy}) \right| = \left|\int[\,(\partial^{j}w)\,v_{xyy} + \,\ldots\,+ w\,(\partial^{j}v_{xyy})\,]\,(\partial^{j}v_{yy}) \right|.\end{aligned}$$ We will look at the first and last of these terms below. The rest of these terms are handled similarly. For the first term, if $j=(0,\,0),$ then we have $$\begin{aligned} \left|\int(\partial^{j}w) \,v_{xyy}\,(\partial^{j}v_{yy}) \right| & = & \left|\int w\,v_{xyy}\,v_{yy} \right|\\ & = & c\left|\int w_{x}\,v_{yy}^{2} \right|\\ & \leq & c\,||w_{x}||_{L^{\infty}(\mathbb{R}^{2})} \int v_{yy}^{2} \\ & \leq & c\,||w||_{{Z}_{t}^{0}}\,||v||_{{Z}_{t}^{|j|}}^{2},\end{aligned}$$ while for $|j|>0,$ $$\begin{aligned} \left|\int (\partial^{j}w)\,v_{xyy}\,(\partial^{j}v_{yy}) \right| & \leq & ||\partial^{j}w||_{L^{\infty}(\mathbb{R}^{2})} \left(\int v_{xyy}^{2}\right)^{1/2} \left(\int(\partial^{j}v_{yy})^{2}\right)^{1/2}\\ & \leq & ||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ The last term for $I_{7}$ is bounded as follows, $$\begin{aligned} \left|\int w\,(\partial^{j}v_{xyy})\,(\partial^{j}v_{yy}) \right| & = & c\left|\int w_{x}\,(\partial^{j}v_{yy})^{2} \right|\\ & \leq & c\,||w_{x}||_{L^{\infty}(\mathbb{R}^{2})}\int(\partial^{j}v_{yy})^{2} \\ & \leq & c\,||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ Now apply one $t$ derivative to , multiply by $2\,(\partial^{j}v_{t})$ and integrate over $\mathbb{R}^{2}.$ We arrive at the following inequality, $$\begin{aligned} \partial_{t}\int(\partial^{j}v_{t}) & \leq & c\left|\int(\partial^{j}(w\,v_{x})_{t}) \,(\partial^{j}v_{t}) \right|\\ & \leq & c\left|\int\partial^{j}(w_{t}\,v_{x})\,(\partial^{j}v_{t}) \right| + c\left|\int\partial^{j}(w\,v_{xt})\,(\partial^{j}v_{t})\,dx\,dt\right|\\ & = & I_{9} + I_{10}.\end{aligned}$$ For $I_{9},$ we have $$\begin{aligned} \lefteqn{\left|\int\partial^{j}(w_{t}\,v_{x}) \,(\partial^{j}v_{t}) \right|}\\ & \leq & c\left|\int(\partial^{j}w_{t})\,v_{x}\,(\partial^{j}v_{t}) \right| +\,\ldots\,+ c\left|\int w_{t}\,(\partial^{j}v_{x})\,(\partial^{j}v_{t}) \right|\\ & \leq & c\,||v_{x}||_{L^{\infty}(\mathbb{R}^{2})} \left(\int(\partial^{j}w_{t})^{2} \right)^{1/2} \left(\int(\partial^{j}v_{t})^{2} \right)^{1/2}\\ & & +\,\ldots\,+ ||\partial^{j}v_{x}||_{L^{\infty}(\mathbb{R}^{2})} \left(\int(w_{t})^{2} \right)^{1/2} \left(\int(\partial^{j}v_{t})^{2} \right)^{1/2}\\ & \leq & c\,||v||_{{Z}_{t}^{0}}\,||w||_{Z_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}} +\,\ldots\,+ c\,||v||_{{Z}_{t}^{|j|}}\,||w||_{{Z}_{t}^{0}}\,||v||_{{Z}_{t}^{|j|}}\\ & \leq & c\,||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ Next we look at $I_{10}.$ If $j=(0,\,0),$ we have $$\begin{aligned} \left|\int w\,v_{xt}\,v_{t} \right| & = & \left|\int w_{x}\,v_{t}^{2} \right|\\ & \leq & ||w_{x}||_{L^{\infty}(\mathbb{R}^{2})}\int v_{t}^{2} \\ & \leq & c\,||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ If $j\neq (0,\,0),$ we have $$\begin{aligned} \left|\int\partial^{j}(w\,v_{xt})\,(\partial^{j}v_{t}) \right| & = c\left|\int(\partial^{j}w)\,v_{xt}\,(\partial^{j}v_{t}) \right| +\,\ldots \\ & \qquad + c\left|\int w\,(\partial^{j}v_{xt})\,(\partial^{j}v_{t}) \right|\\ & = I_{10}(a) +\,\ldots\,+ I_{10}(\widetilde{a}).\end{aligned}$$ Now for $I_{10}(a),$ we use the following estimate $$\begin{aligned} \left|\int(\partial^{j}w)\,v_{xt}\,(\partial^{j}v_{t}) \right| & \leq & c\,||\partial^{j}w||_{L^{\infty}(\mathbb{R}^{2})} \left(\int v_{xt}^{2} \right)^{1/2} \left(\int(\partial^{j}v_{t})^{2} \right)^{1/2}\\ & \leq & c\,||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ While for $I_{10}(\widetilde{a}),$ we use the following estimate $$\begin{aligned} \left|\int w\,(\partial^{j}v_{xt})\,(\partial^{j}v_{t}) \right| & = & c\left|\int w_{x}\,(\partial^{j}v_{t})^{2} \right|\\ & \leq & c\,||w_{x}||_{L^{\infty}(\mathbb{R}^{2})}\int(\partial^{j}v_{t})^{2} \\ & \leq & c\,||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ Therefore, for $0\leq\widetilde{t}\leq t,$ we conclude that $$\begin{aligned} & \partial_{t}\int[\,(\partial^{j}v(\,\cdot\,,\,\cdot\,,\,\widetilde{t}))^{2} + (\partial^{j}v_{xxx}(\,\cdot\,,\,\cdot\,,\,\widetilde{t}))^{2} + (\partial^{j}v_{yy}(\,\cdot\,,\,\cdot\,,\,\widetilde{t}))^{2} + (\partial^{j}v_{t}(\,\cdot\,,\,\cdot\,,\,\widetilde{t}))^{2}\,] \\ & \qquad \leq c\,||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ Integrating with respect to $t,$ we obtain $$\begin{aligned} ||v||_{Z_{t}^{|j|}}^{2}\leq ||v(\,\cdot\,,\,\cdot\,,\,0))||_{H^{(|j| + 3,\,|j| + 2)}(\mathbb{R}^{2})}^{2} + ||v(\,\cdot\,,\,\cdot\,,\,0))||_{H^{|j|}(\mathbb{R}^{2})}^{2} + c\,t\,||w||_{Z_{t}^{|j|}}\,||v||_{Z_{t}^{|j|}}^{2},\end{aligned}$$ as desired. $\square$ Uniqueness and Existence of a local solution ============================================ In this section, we will prove that for $\phi\in X^{N}(\mathbb{R}^{2})$ there exists a unique solution of - in $L^{\infty}([0,\,T]:\,X^{N}(\mathbb{R}^{2})),$ where the time $T$ depends only $||\phi||_{X^{0}(\mathbb{R}^{2})}.$ First we prove uniqueness of solutions.\ \ [**Theorem 4.1**]{} (Uniqueness). [*Let $\phi\in X^{0}(\mathbb{R}^{2})$ and $0<T<+\infty.$ Then there is at most one solution of - in $L^{\infty}([0,\,T]:\,X^{0}(\mathbb{R}^{2}))$ with initial data $u(x,\,y,\,0)=\phi(x,\,y).$*]{}\ \ [*Proof.*]{} Assume that $u,\,$ $v\in L^{\infty}([0,\,T]:\,X^{0}(\mathbb{R}^{2}))$ are two solutions of - with $u_{t},\,$ $v_{t}\in L^{\infty}([0,\,T]:\,L^{2}(\mathbb{R}^{2})),$ so all integrations below are justified and with the same initial data, in fact, with $(u - v)(x,\,y,\,0)=0.$ Then $$\begin{aligned} \label{e401}& & (u - v)_{t} + (u - v)_{xxx} + (u - v)_{x} + (u\,u_{x} - v\,v_{x}) - \partial_{x}^{-1}(u - v)_{yy} =0.\end{aligned}$$ By , $$\begin{aligned} \label{e402}& & (u - v)_{t} + (u - v)_{xxx} + (u - v)_{x} + (u - v)\,u_{x} + (u - v)_{x}\,v - \partial_{x}^{-1}(u - v)_{yy} =0.\end{aligned}$$ Multiplying by $2\,(u - v)$ and integrating with respect to $(x,\,y)$ over $\mathbb{R}^{2},$ $$\begin{aligned} \label{e403}& & 2\int (u - v)\,(u - v)_{t} + 2\int (u - v)\,(u - v)_{xxx} + 2\int (u - v)\,(u - v)_{x} \\ & & +\;2\int (u - v)^{2}\,u_{x} + 2\int (u - v)\,(u - v)_{x}\,v - 2\int (u - v)\,\partial_{x}^{-1}(u - v)_{yy} =0.\nonumber\end{aligned}$$ Integrating by parts each term in we obtain $$\begin{aligned} \partial_{t}\int (u - v)^{2} & = & -\;2\int (u - v)^{2}\,u_{x} + \int (u - v)^{2}\,v_{x} \nonumber \\ & \leq & c\left(||u_{x}||_{L^{\infty}(\mathbb{R}^{2})} + ||v_{x}||_{L^{\infty}(\mathbb{R}^{2})}\right)\int(u - v)^{2} \nonumber \\ \label{e404}& \leq & c\left(\,||u||_{X^{0}(\mathbb{R}^{2})} + ||v||_{X^{0}(\mathbb{R}^{2})}\,\right)\int (u - v)^{2}\end{aligned}$$ Using Gronwall’s inequality and the fact that $(u - v)$ vanishes at $t=0,$ it follows that $u=v.$ This proves the uniqueness of the solution. $\square$\ \ Now we consider existence of solutions to -. Our plan is to show that for $\phi \in X^N(\mathbb R^2)$ there exists a solution $u \in L^\infty([0,T]:X^N(\mathbb R^2))$ for a time $T$ depending only on $||\phi||_{X^0}$. In order to prove this we must first prove a preliminary result by introducing the following function space. Let $$\begin{aligned} \label{e301}Y^{N}(\mathbb{R}^{2}) = \left\{u:\;u,\,u_{xxx},\,u_{yy},\, \frac{\eta^{2}}{\xi}\,\widehat{u}\in H^{N}(\mathbb{R}^{2})\right\}\end{aligned}$$ with the accompanying norm $$\begin{aligned} \label{e302}||u||_{Y^{N}(\mathbb{R}^{2})}^{2}=\int_{\mathbb{R}^{2}}\left(u^{2} + \sum_{|j|\leq N}[\,(\partial^{j}u_{xxx})^{2} + (\partial_{x}^{-1}\partial^{j}u_{yy})^{2} + (\partial^{j}u_{yy})^{2}\,]\right)dx\,dy\end{aligned}$$ where $j=(\alpha_1,\,\alpha_2)$ and $|j| = \alpha_1 + \alpha_2.$ We will begin by showing that, for $\phi \in Y^N(\mathbb R^2)$, there exists a solution $u$ of such that $u \in L^\infty([0,T]: Y^N(\mathbb R^2))$ for a time $T$ depending only on $||\phi||_{Y^0}$. Then we will prove a differential inequality of the form $$\partial_t \left(\int u^2 + u_{xxx}^2 + (\partial_x^{-1}u_{yy})^2\right) \leq \left(\int u^2 + u_{xxx}^2 + (\partial_x^{-1}u_{yy})^2\right)^{3/2},$$ to show that in fact the solution $u$ obtained in Theorem 4.2 is in $L^\infty([0,T']; X^0(\mathbb R^2))$ for a time $T'$ depending only on $||\phi||_{X^0(\mathbb R^2)}$. With these ideas in mind we state our existence theorem.\ (Existence). [*Let $k_{0}>0$ and $N$ be an integer $\geq 0.$ Then there exists a time $0<T<+\infty$ depending only on $k_{0}$ such that for all $\phi\in Y^{N}(\mathbb{R}^{2})$ with $||\phi||_{Y^{0}(\mathbb{R}^{2})}\leq k_{0}$ there exists a solution of , $u\in L^{\infty}\left([0,\,T]:\,Y^{N}(\mathbb{R}^{2})\right)$ such that $u(x,\,y,\,0)=\phi(x,\,y).$*]{}\ \ The method of proof is as follows. As discussed in section 3, we begin by approximating by the linear equation . We construct the mapping $$\begin{aligned} \Pi : Z_{T}^{N} \rightarrow Z_{T}^{N}\end{aligned}$$ where the initial condition is given by $u^{(n)}(x,\,y,\,0)=\phi(x,\,y)$ and the first approximation is given by $u^{(0)}(x,y,t) = \phi(x,y)$. Subsequent approximations are given by $u^{(n)} = \Pi(u^{(n-1)})$ for $n \geq 1$. Equation is a linear equation which by Lemma 3.1 can be solved at each iteration. We show that the sequence of solutions $\{u^{(n)}\}$ to our linear equation is bounded in $L^\infty([0,T];Y^0(\mathbb R^2))$ for a time $T$ depending only on $||\phi||_{Y^0}$. We then show that there is a subsequence of solutions to our approximate equations which converges to a solution $u \in L^\infty([0,T]; Y^0(\mathbb R^2))$ of . Lastly, we show that if $\phi \in Y^N(\mathbb R^2)$ for $N > 0$, then our solution $u \in L^\infty([0,T];Y^N(\mathbb R^2))$ where the time $T$ depends only on $||\phi||_{Y^0}$.\ \ It suffices to prove this result for $\phi\in\bigcap_{N\geq 0}H^{N}(\mathbb{R}^{2})$ and $\partial_{x}^{-1}\phi_{yy}\in\bigcap_{N\geq 0}H^{N}(\mathbb{R}^{2}).$ We can then use the same approximation procedure as before to prove the result for general initial data. Let $u^{(n)}$ be a solution of with initial data $u^{(n)}(x,y,0) = \phi(x,y)$ and where the first approximation is given by $u^{(0)}(x,y,t) = \phi(x,y)$. By Lemma 3.2, we know that $$\begin{aligned} \label{e405}||u^{(n)}||_{{Z}_{t}^{0}}^{2} \leq ||u^{(n)}(\,\cdot\,,\,\cdot\,,\,0)||_{H^{(3,\,2)}(\mathbb{R}^{2})}^{2} + ||u_{t}^{(n)}(\,\cdot\,,\,\cdot\,,\,0)||_{L^{2}(\mathbb{R}^{2})}^{2} + c\,t\,||u^{(n - 1)}||_{{Z}_{t}^{0}}\;||u^{(n)}||_{{Z}_{t}^{0}}^{2}.\end{aligned}$$ Further, using the fact that $||\phi||_{Y^{0}}\leq k_{0}$, we have $$\begin{aligned} \lefteqn{||u^{(n)}(\,\cdot\,,\,\cdot\,,\,0)||_{H^{(3,\,2)}(\mathbb{R}^{2})}^{2} + ||u_{t}^{(n)}(\,\cdot\,,\,\cdot\,,\,0)||_{L^{2}(\mathbb{R}^{2})}^{2}} \\ & = & ||u^{(n)}(\,\cdot\,,\,\cdot\,,\,0)||_{H^{(3,\,2)}(\mathbb{R}^{2})}^{2}\\ & & + \int [\,u_{xxx}^{(n)}(\,\cdot\,,\,\cdot\,,\,0) + u_{x}^{(n)}(\,\cdot\,,\,\cdot\,,\,0) - \partial_{x}^{-1}u_{yy}^{(n)}(\,\cdot\,,\,\cdot\,,\,0) + u^{(n - 1)}(\,\cdot\,,\,\cdot\,,\,0)u_{x}^{(n)}(\,\cdot\,,\,\cdot\,,\,0)\,]^{2} \\ & \leq & ||\phi||_{Y^{0}(\mathbb{R}^{2})}^{2} + c\int [\,\phi_{xxx}^{2} + \phi_{x}^{2} + (\partial_{x}^{-1}\phi_{yy})^{2} + (\phi\,\phi_{x})^{2}\,] \\ & \leq & C\,||\phi||_{Y^{0}(\mathbb{R}^{2})}^{2} \leq C\,k_{0}^{2},\end{aligned}$$ where $\;C\,$ is independent of $n.\;$ Define $c_{0}=\left(\frac{C}{2}k_{0}^{2} + 1\right).\;$ Let $T_{0}^{(n)}$ be the maximum time such that $||u^{(j)}||_{Z_{t}^{0}}\leq c_{0}$ for $0\leq t\leq T_{0}^{(n)},$ $0\leq j\leq n.$ That is $$\begin{aligned} T_{0}^{(n)}=\sup\{t\in [0,\,T_{0}^{(n)}]:\;||u^{(j)}||_{{Z}_{t}^{0}}\leq c_{0}\quad\mbox{for}\quad 0\leq j\leq n\}.\end{aligned}$$ Therefore, $$\begin{aligned} \label{e406}||u^{(n)}||_{{Z}_{t}^{0}}^{2} & \leq & ||u^{(n)}(\,\cdot\,,\,\cdot\,,\,0)||_{H^{(3,\,2)}(\mathbb{R}^{2})}^{2} + ||u_{t}^{(n)}(\,\cdot\,,\,\cdot\,,\,0)||_{L^{2}(\mathbb{R}^{2})}^{2} + c\,t\,||u^{(n - 1)}||_{{Z}_{t}^{0}}\;||u^{(n)}||_{{Z}_{t}^{0}}^{2} \nonumber \\ & \leq & C\,k_{0}^{2} + c\,t\,c_{0}^{3}.\end{aligned}$$ [*Claim:*]{} $T_{0}^{(n)}$ does not approach $0.$\ \ On the contrary, assume that $T_{0}^{(n)}\rightarrow 0.$ Since $||u^{(n)}(\,\cdot\,,\,\cdot\,,\,t)||_{{Z}_{t}^{0}}$ is continuous for $t\geq 0,$ there exists $\tau\in [0,\,T]$ such that $c_{0}=||u^{(j)}(\,\cdot\,,\,\cdot\,,\,\tau)||_{{Z}_{\tau}^{0}}$ for $0\leq\tau\leq T_{0}^{(n)},$ $0\leq j\leq n.$ Then, by we have $$\begin{aligned} \label{e407}c_{0}^{2} \leq C\,k_{0}^{2} + c\,T_{0}^{(n)}\,c_{0}^{3}.\end{aligned}$$ As $n\rightarrow \infty,$ we have $$\begin{aligned} \label{e408}\left(\frac{C}{2}\,k_{0}^{2} + 1\right)^{2} \leq C\,k_{0}^{2}\quad \Longrightarrow \quad \frac{C^{2}}{4}\,k_{0}^{4} + 1 \leq 0\end{aligned}$$ which is a contradiction. Consequently $T_{0}^{(n)}\not\rightarrow 0.$ Choosing $T=T(c_0)$ sufficiently small, and $T$ not depending on $n,$ one concludes that $$\begin{aligned} \label{e409}||u^{(n)}||_{{Z}_{t}^{0}}^{2} \leq c\quad\mbox{for}\quad 0\leq t\leq T.\end{aligned}$$ This show that $T_{0}^{(n)}\geq T.$ Hence from we see that there exists a bounded sequence of solutions $u^{(n)}\in {Z}_{T}^{0}$ and therefore a subsequence $u^{(n_{j})}\equiv u^{(n)}$ such that $$\begin{aligned} u^{(n)}\stackrel{*}\rightharpoonup u\quad \mbox{weakly in}\quad L^{\infty}([0,\,T]:\,H^{(3,\,2)}(\mathbb{R}^{2}))\end{aligned}$$ $$\begin{aligned} u_{t}^{(n)}\stackrel{*}\rightharpoonup u_{t}\quad \mbox{weakly in}\quad L^{\infty}([0,\,T]:\,L^{2}(\mathbb{R}^{2})).\end{aligned}$$ Therefore, by Lions-Aubin’s compactness theorem there is a subsequence $u^{(n_{j})}\equiv u^{(n)}$ such that $u^{(n)}\rightarrow u$ strongly on $L^{\infty}([0,\,T]:\,H_{loc}^{1}(\mathbb{R}^{2})).$ Now it remains to show that each term in converges to its correct limit. First, $u_{xxx}^{(n)}\stackrel{*}\rightharpoonup u_{xxx}$ weakly on $L^{\infty}([0,\,T]:\,L^{2}(\mathbb{R}^{2})).$ Similarly $u_{t}^{(n)}\rightarrow u_{t}$ and $u_{x}^{(n)}\rightarrow u_{x}$ weak$^{*}$ in $L^{\infty}([0,\,T]:\,L^{2}(\mathbb{R}^{2})).$ Now we will show that the nonlinear term converges to its correct limit. First, $u^{(n - 1)}\rightarrow u$ strongly in $L^{\infty}([0,\,T]:\,H_{loc}^{1}(\mathbb{R}^{2})).$ Moreover, $u_{x}^{(n - 1)}\stackrel{*}\rightharpoonup $ weakly in $L^{\infty}([0,\,T]:\,L^{2}(\mathbb{R}^{2})).$ Therefore, $$\begin{aligned} u^{(n - 1)}\,u_{x}^{(n)}\stackrel{*}\rightharpoonup u\,u_{x}\quad \mbox{weakly in}\quad L^{\infty}([0,\,T]:\,L^{2}(\mathbb{R}^{2})).\end{aligned}$$ Consequently, $$\begin{split} \partial_{x}^{-1}u_{yy}^{(n)} & = u_{t}^{(n)} + u_{xxx}^{(n)} + u_{x}^{(n)} + u^{(n)}\,u_{x}^{(n)}\\ & \stackrel{*}\rightharpoonup u_{t} + u_{xxx} + u_{x} + u\,u_{x}\quad \mbox{weakly in} \quad L^{\infty}([0,\,T]:\,L^{2}(\mathbb{R}^{2})). \end{split}$$ But, also note that $$\begin{aligned} u_{yy}^{(n)}\stackrel{*}\rightharpoonup u_{yy}\quad \mbox{weakly in}\quad L^{\infty}([0,\,T]:\,L^{2}(\mathbb{R}^{2})).\end{aligned}$$ Therefore $$\begin{aligned} \partial_{x}^{-1}u_{yy}^{(n)} \stackrel{*}\rightharpoonup \partial_{x}^{-1}u_{yy}\quad \mbox{weakly in}\quad L^{\infty}([0,\,T]:\,L^{2}(\mathbb{R}^{2}))\end{aligned}$$ and consequently $u$ is a solution to . Now, we prove that there exists a solution to with $u\in L^{\infty}([0,\,T]:\,Y^{N}(\mathbb{R}^{2}))$ for the time $T$ chosen above. We already know that there is a solution $u\in L^{\infty}([0,\,T]:\,Y^{0}(\mathbb{R}^{2})).$ Therefore, it suffices to show that the approximating sequence $u^{(n)}$ is bounded in ${Z}_{T}^{N}$ and thus, by the convergence arguments above, our solution $u$ is in $L^{\infty}([0,\,T]:\;Y^{N}(\mathbb{R}^{2})).$ Again, by Lemma 3.1, we know our linearized equation can be solved in any interval of time in which the coefficients are defined. Therefore, for each iterate, $||u^{(n)}||_{{Z}_{t}^{N}}$ is continuous in $t\in [0,\,T].$ By Lemma 3.2, it follows that $$\begin{aligned} ||u^{(n)}||_{{Z}_{t}^{N}}^{2} & \leq & ||u^{(n)}(\,\cdot\,,\,\cdot\,,\,0)||_{H^{(N + 3,\,N + 2)}(\mathbb{R}^{2})}^{2} \nonumber \\ \label{e410}& & +\; ||u_{t}^{(n)}(\,\cdot\,,\,\cdot\,,\,0)||_{H^{N}(\mathbb{R}^{2})}^{2} + c\,t\,||u^{(n - 1)}||_{{Z}_{t}^{N}}\;||u^{(n)}||_{{Z}_{t}^{N}}^{2}.\end{aligned}$$ On the other hand, as before and using $||\phi||_{Y^{N}}\leq k_{N}$ we obtain $$\begin{aligned} ||u^{(n)}(\,\cdot\,,\,\cdot\,,\,0)||_{H^{(N + 3,\,N + 2)}(\mathbb{R}^{2})}^{2} + ||u_{t}^{(n)}(\,\cdot\,,\,\cdot\,,\,0)||_{H^{N}(\mathbb{R}^{2})}^{2} &\leq C\,k_{N}^{2},\end{aligned}$$ where $k_{N}$ is independent of $n.$ Define $c_{N}=\left(\frac{C}{2}\,k_{N}^{2} + 1\right).$ Let $T_{N}^{(n)}$ be the largest time that $||u^{(j)}||_{{Z}_{t}^{N}}\leq c_{N}$ for $0\leq t\leq T_{N}^{(n)},$ $0\leq j\leq n.$ That is, $$\begin{aligned} T_{N}^{(n)}=\sup\{t\in [0,\,T_{N}^{(n)}]:\;||u^{(j)}||_{{Z}_{t}^{N}}\leq c_{N}\quad\mbox{for}\quad 0\leq j\leq n\}.\end{aligned}$$ Therefore, for $0\leq t\leq T_{N}^{(n)},$ $$\begin{aligned} \label{e411}||u^{(n)}||_{{Z}_{t}^{N}}^{2}\leq C\,k_{N}^{2} + c\,t\,c_{N}^{3}.\end{aligned}$$ [*Claim:*]{} $T_{N}^{(n)}$ does not approach $0.$\ \ On the contrary, assume that $T_{N}^{(n)}\rightarrow 0.$ Since $||u^{(n)}(\,\cdot\,,\,\cdot\,,\,t)||_{{Z}_{t}^{N}}$ is continuous for $t\geq 0,$ there exists $\tau\in [0,\,T]$ such that $c_{N}=||u^{(j)}(\,\cdot\,,\,\cdot\,,\,\tau)||_{{Z}_{\tau}^{N}}$ for $0\leq\tau\leq T_{N}^{(n)},$ $0\leq j\leq n.$ Then, by we have $$\begin{aligned} \label{e412}c_{N}^{2} \leq C\,k_{N}^{2} + c\,T_{N}^{(n)}\,c_{N}^{3}.\end{aligned}$$ As $n\rightarrow \infty,$ we have $$\begin{aligned} \label{e413}\left(\frac{C}{2}\,k_{N}^{2} + 1\right)^{2} \leq C\,k_{N}^{2}\quad \Longrightarrow \quad \frac{C^{2}}{4}\,k_{N}^{4} + 1 \leq 0\end{aligned}$$ which is a contradiction. Consequently $T_{N}^{(n)}\not\rightarrow 0.$ Choosing $T_N$ sufficiently small, and $T_N$ not depending on $n,$ one concludes that $$\begin{aligned} \label{e414}||u^{(n)}||_{{Z}_{t}^{N}}^{2} \leq c\quad\mbox{for}\quad 0\leq t\leq T_N.\end{aligned}$$ This show that $T_{N}^{(n)}\geq T_N.$ Now, let $$\begin{aligned} T_{N}^{*} =\sup\{t\in [0,\,T_{N}^{*}]:\;u\in {Z}_{t}^{N}\}.\end{aligned}$$ We claim that $T_{N}^{*}\geq T$ and therefore, a time of existence can be chosen depending only on $||\phi||_{Y^{0}}.$ By Lemma 3.1 the linear equation can be solved in any interval of time in which the coefficients are defined, and thus $T_{N}^{*}\geq T.$ $\square$\ \ Now we want to improve our existence theorem. In particular, we want to show that the solution $u\in L^{\infty}([0,\,T]:\,Y^{N}(\mathbb{R}^{2}))$ found in Theorem 4.2 is in $L^{\infty}([0,\,T']:\,X^{N}(\mathbb{R}^{2}))$ for a time $T'$ depending only on $||\phi||_{X^{0}(\mathbb{R}^{2})}.$ In order to do so, we first prove a differential inequality.\ \ [**Lemma 4.3.**]{} [*Let $u$ be the solution to our main equation in $L^{\infty}([0,\,T]:\,Y^{N}(\mathbb{R}^{2})).$ Then for any $0\leq t\leq T,$ we have*]{} $$\begin{aligned} \lefteqn{\partial_{t}\int_{\mathbb{R}^{2}}\left(u^{2} + \sum_{|j|\leq N}[\,(\partial^{j}u_{xxx})^{2} + (\partial^{j}(\partial_{x}^{-1}u_{yy}))^{2}\,]\right)dx\,dy}\nonumber \\ \label{e415}& \leq & c\left[\int_{\mathbb{R}^{2}}\left(u^{2} + \sum_{|j|\leq N}[\,(\partial^{j}u_{xxx})^{2} + (\partial^{j}(\partial_{x}^{-1}u_{yy}))^{2}\,]\right)dx\,dy\right]^{3/2}.\end{aligned}$$ [*Proof.*]{} We use a priori estimates on smooth solutions $u.$ Multiplying by $u$ and integrating over $\mathbb{R}^{2},$ it is straightforward to see that the $L^{2}(\mathbb{R}^{2})$-norm is conserved. Therefore, we only need to show that $$\begin{aligned} \lefteqn{\partial_{t}\int \sum_{|j|\leq N}[\,(\partial^{j}u_{xxx})^{2} + (\partial^{j}(\partial_{x}^{-1}u_{yy}))^{2}\,]}\nonumber \\ \label{e416}& \leq & c\left[\int \left(u^{2} + \sum_{|j|\leq N}[\,(\partial^{j}u_{xxx})^{2} + (\partial^{j}(\partial_{x}^{-1}u_{yy}))^{2}\,]\right) \right]^{3/2}.\end{aligned}$$ We consider the case $j=(0,\,0).$ The case $j\neq (0,\,0)$ is handled in a similar way.\ \ Applying $\partial_{x}^{3}$ to we obtain $$\begin{aligned} \label{e417}& & u_{xxxt} + u_{xxxxxx} + u_{xxxx} + (u\,u_{x})_{xxx} - u_{xxyy} = 0.\end{aligned}$$ Multiplying by $2\,u_{xxx}$ and integrating over $\mathbb{R}^{2}$ we obtain $$\begin{aligned} & & 2\int u_{xxx}\,u_{xxxt} + 2\int u_{xxx}\,u_{xxxxxx} + 2\int u_{xxx}\,u_{xxxx} \nonumber \\ \label{e418}& & +\;2\int u_{xxx}\,(u\,u_{x})_{xxx} - 2\int u_{xxx}\,u_{xxyy} = 0.\end{aligned}$$ Using in straightforward integration by parts, we obtain $$\begin{aligned} \lefteqn{\partial_{t}\int u_{xxx}^{2} = -\;2\int u_{xxx}\,(u\,u_{x})_{xxx}} \\ & = & -\;2\int [\,3\,u_{xx}^{2} + 4\,u_{x}\,u_{xxx} + u\,u_{xxxx}\,]\,u_{xxx} \\ & = & -\;7\int u_{x}\,u_{xxx}^{2} \leq 7\,||u_{x}||_{L^{\infty}(\mathbb{R}^{2})} \int_{\mathbb{R}^{2}}u_{xxx}^{2} \\ & \leq & c\left(\int [\,u_{x}^{2} + u_{xxx}^{2} + u_{xy}^{2}\,]\right)^{1/2}\int_{\mathbb{R}^{2}}u_{xxx}^{2} \\ & \leq & c\left(\int [\,u^{2} + u_{xxx}^{2} + (\partial_{x}^{-1}u_{yy}^{2})^{2}\,] \right)^{3/2}.\end{aligned}$$ In a similar way, but now apply $\partial_{x}^{-1}\partial_{y}^{2}$ to instead of $\partial_{x}^{3}$ and multiply by $2\,\partial_{x}^{-1}u_{yy}$ instead of $2\,u_{xxx}$ we get $$\begin{aligned} \lefteqn{\partial_{t}\int (\partial_{x}^{-1}u_{yy})^{2}}\\ & \leq & c\left|\int (u^{2})_{yy}\,(\partial_{x}^{-1}u_{yy})\right| = c\left|\int [\,u_{yy}\,u + u_{y}^{2}\,]\,(\partial_{x}^{-1}u_{yy})\right|\\ & \leq & c\left|\int u_{x}\,(\partial_{x}^{-1}u_{yy})^{2}\right| + \left|\int u_{y}^{2}\,(\partial_{x}^{-1}u_{yy})\right|\\ & \leq & c||u_{x}||_{L^{\infty}(\mathbb{R}^{2})} \int (\partial_{x}^{-1}u_{yy})^{2} + \left(\int u_{y}^{4} \right)^{1/2} \left(\int (\partial_{x}^{-1}u_{yy})^{2} \right)^{1/2}\\ & \leq & c||u_{x}||_{L^{\infty}(\mathbb{R}^{2})} \int (\partial_{x}^{-1}u_{yy})^{2} + \left(\int [\,u_{y}^{2} + u_{xy}^{2} + (\partial_{x}^{-1}u_{yy})^{2}\,] \right) \left(\int (\partial_{x}^{-1}u_{yy})^{2} \right)^{1/2}\\ & \leq & c\left(\int [\,u^{2} + u_{xxx}^{2} + (\partial_{x}^{-1}u_{yy})^{2}\,]\right)^{3/2}.\end{aligned}$$ The lemma follows. $\square$\ \ [**Corollary 4.4.**]{} [*Let $u$ be the solution to with initial data $\phi\in Y^{N}(\mathbb{R}^{2}).$ Denote by $0<T<+\infty$ the life span of this solution in $Y^{N}(\mathbb{R}^{2}).$ Then there exists $0<T'\leq T,$ depending only on the norm of $\phi\in X^{0}(\mathbb{R}^{2})$ such that $u\in L^{\infty}([0,\,T']:\,X^{N}(\mathbb{R}^{2})).$*]{}\ \ [*Proof.*]{} Let $$\begin{aligned} h(t) = \int_{\mathbb{R}^{2}}\left(u^{2} + \sum_{|j|\leq N}[\,(\partial^{j}u_{xxx})^{2} + (\partial^{j}(\partial_{x}^{-1}u_{yy}))^{2}\,]\right)dx\,dy\equiv ||u||_{X^{N}}^2.\end{aligned}$$ Using we have $h'(t)\leq c\;[\,h(t)\,]^{3/2}.$ Integrating this inequality with respect to $t$, we obtain that $h(t)^{1/2}\leq c/(h(0)^{-1/2} - t)$ and therefore, we get a lower bound on the time of existence of $h(t)$ depending only on $h(0).$ $\square$\ \ [**Corollary 4.5.**]{} [*Let $\phi\in X^{N}(\mathbb{R}^{2})$ for some $N\geq 0$ and let $\phi^{(n)}$ be a sequence converging to $\phi$ in $X^{N}(\mathbb{R}^{2}).$ Let $u$ and $u^{(n)}$ be the corresponding unique solutions, given by Theorems 4.1 and 4.2 and Corollary 4.4, in $L^{\infty}([0,\,T]:\,X^{N}(\mathbb{R}^{2}))$ for a time $T$ depending only on $\sup_{n}||\phi^{(n)}||_{X^{0}(\mathbb{R}^{2})}.$ Then* ]{} $$\begin{aligned} \label{e419}u^{(n)}\stackrel{*}\rightharpoonup u\quad \mbox{weakly in}\quad L^{\infty}([0,\,T]:\,X^{N}(\mathbb{R}^{2})).\end{aligned}$$ [*Proof.*]{} By assumption $u\in L^{\infty}([0,\,T]:\,X^{N}(\mathbb{R}^{2})),$ then there exists a weak$^{*}$ convergent subsequence, still denoted $u^{(n)}$ such that $$\begin{aligned} u^{(n)}\stackrel{*}\rightharpoonup u\quad \mbox{weakly in}\quad L^{\infty}([0,\,T]:\,X^{N}(\mathbb{R}^{2}))\hookrightarrow L^{\infty}([0,\,T]:\,H^{1}(\mathbb{R}^{2})).\end{aligned}$$ Moreover, by equation , $u^{(n)}\in L^{\infty}([0,\,T]:\,X^{N}(\mathbb{R}^{2}))$ implies $u_{t}^{(n)}\in L^{\infty}([0,\,T]:\,L^{2}(\mathbb{R}^{2})).$ By The Lions-Aubin compactness theorem, $$\begin{aligned} u^{(n)}\rightarrow u\quad \mbox{strongly in}\quad L^{\infty}([0,\,T]:\,H_{loc}^{1/2}(\mathbb{R}^{2})).\end{aligned}$$ Now we just need to show that each term in converges to its correct limit, and $u_{t}^{(n)}\rightarrow u_{t}$ for $u\in L^{\infty}([0,\,T]:\,X^{N}(\mathbb{R}^{2})).$\ \ The only thing we need to show is that the nonlinear term converges to its correct limit, namely that $u^{(n)}\,u_{x}^{(n)}\rightarrow u\,u_{x}.$ We know that $u_{x}^{(n)}\stackrel{*}\rightharpoonup u_{x}$ weakly in $ L^{\infty}([0,\,T]:\,H^{1}(\mathbb{R}^{2}))$ and $u^{(n)}\rightarrow u$ strongly in $ L^{\infty}([0,\,T]:\,H_{loc}^{1/2}(\mathbb{R}^{2})).$ Therefore, their product converges in $L^{2}([0,\,T]:\,L_{loc}^{1}(\mathbb{R}^{2})).$ Clearly, the linear terms also converge in $L^{2}([0,\,T]:\,L_{loc}^{1}(\mathbb{R}^{2}))$ and therefore, we conclude that $u_{t}^{(n)}\rightarrow u_{t}$ in $L^{2}([0,\,T]:\,L_{loc}^{1}(\mathbb{R}^{2})).$ The proof follows. $\square$ Estimate of error terms ======================= In this section we prove the main estimates used in our gain of regularity theorem.\ \ \ \ \ The idea of the proof is the following. For a given $\alpha$ satisfying the hypotheses above, we choose a weight function $f_\alpha \approx t^{|\alpha|-L} x^{2L-|\alpha|-\alpha_2}$ for $x > 1$ and $f_\alpha \approx t^{|\alpha|-L}e^{\sigma x}$ for $x < -1$. Then with this choice of weight function, we apply the operator $\partial^\alpha$ to , multiply by $f_\alpha\pa u$ and integrate over $\mathbb R^2$ to obtain the main equality stated in . In this theorem, we bound the last three terms on the left-hand side of by terms of the form and .\ For each $\alpha$ we apply the operator $\partial^\alpha$ to (2.1), multiply our differentiated equation by $2 \fa (\pa u)$ where we take $$\fa(x,t) = \int_{-\infty}^x g_\alpha(z,t) \,dz \quad \text{for } g_\alpha \in W_{\sigma, 2L-|\alpha|-\alpha_2-1,|\alpha|-L},$$ and integrate over $\mathbb R^2 \times [0,t]$ for $0 \leq t \leq T$. As stated in Lemma 2.1, we arrive at our main equality $$\begin{aligned} & \partial_t \int \fa (\pa u)^2 + 3 \int (\fa)_x (\pa u_x)^2 - \int [(\fa)_t + (\fa)_{xxx} + (\fa)_x] (\pa u)^2 \\ & \qquad - \int (\fa)_x (\pa \partial_x^{-1}u_y)^2 + 2 \int \fa (\pa u)\pa(uu_x) = 0.\end{aligned}$$ Using (1.5) and $f(\cdot,0) = 0$ we get the following identity after integrating with respect to $t$, $$\label{before-estimating} \begin{split} & \int \fa (\pa u)^2 + 3 \int_0^T \int (\fa)_x (\pa u_x)^2 \\ & \leq \int_0^T \int (\fa)_x (\pa \partial_x^{-1} u_y)^2 + C \int_0^T \int \fa (\pa u)^2 + 2 \left|\int_0^T \int \fa (\pa u) \pa (uu_x) \right|. \end{split}$$ We notice that the first term on the right-hand side of can be written as $$\label{inverse-term} \int_0^T \int (\fa)_x (\pa \partial_x^{-1} u_y)^2 = C \int_0^T \int t g_\gamma (\partial^\gamma u_x)^2$$ for some $g_\gamma \in W_{\sigma,2L-|\gamma|-\gamma_2-1,|\gamma|-L}$ where $\gamma = (\alpha_1-2,\alpha_2+1)$. Further, we notice that $2L-|\gamma|-\gamma_2 \geq 1$ and $\alpha_1 \geq 2$ since $2L-|\alpha|-\alpha_2\geq 1$ and $L+1 \leq |\alpha|$. Therefore, is of the form specified by . Therefore, $$\label{after-inverse-term} \begin{split} & \int \fa (\pa u)^2 + 3 \int_0^T \int (\fa)_x (\pa u_x)^2 \\ & \qquad \leq C + C \int_0^T \int \fa (\pa u)^2 + 2 \left|\int_0^T \int \fa (\pa u) \pa (uu_x) \right| \end{split}$$ where $C$ depends only on terms of the form . We now need to estimate the term $$\left|\int_0^T \int \fa (\pa u)\pa (uu_x)\right|.$$ Each term is of the form $$\left|\int_0^T \int \fa (\pa u)(\partial^r u)(\partial^s u_x)\right|.$$ where $r_1 + s_1 = \alpha_1$, $r_2 + s_2 = \alpha_2$. Below we consider all terms of level $|\alpha|$. In this case, $|r|=0$, and we have $$\begin{aligned} \left|\int_0^T \int \fa (\pa u)(\partial^r u)(\partial^s u_x)\right| & = \left|\int_0^T \int \fa (\pa u) u (\pa u_x)\right| \\ & = C \left|\int_0^T \int (\fa u)_x (\pa u)^2\right| \\ & \leq C ||u||_{X^0} \int_0^T \int \fa (\pa u)^2.\end{aligned}$$ In this case, $|r|=1$ giving us the following two subcases: \(a) [**The subcase $r = (1,0)$.**]{} In this case, we have $$\begin{aligned} \left|\int_0^T \int \fa (\pa u)(\partial^r u)(\partial^s u_x)\right| & = \left|\int_0^T \int \fa (\pa u)u_x (\pa u)\right| \\ & \leq C||u||_{X^0} \int_0^T \int \fa (\pa u)^2.\end{aligned}$$ \(b) [**The subcase $r = (0,1)$.**]{} We note that this case will only occur if $\alpha_2 \geq 1$. In this case, we have $$\begin{aligned} \left|\int_0^T \int \fa (\pa u)(\partial^r u)(\partial^s u_x)\right| & = \left|\int_0^T \int \fa (\pa u) u_y (\partial_x^{\alpha_1} \partial_y^{\alpha_2-1}u_x)\right| \\ & \leq C||u||_{X^1} \left(\int_0^T \int \fa (\pa u)^2\right)^{1/2} \left(\int_0^T \int \fa (\partial_x^{\alpha_1+1}\partial_y^{\alpha_2-1}u)^2\right)^{1/2}\end{aligned}$$ Since $f_\alpha \approx x^{2L-|\alpha|-\alpha_2}$ as $x \rightarrow \infty$, it is clear that $f_\alpha \leq C f_{\alpha_1+1,\alpha_2-1}$ We have three subcases to consider. \(a) [**The subcase $r = (2,0)$.**]{} In this case, we have $$\begin{aligned} \left|\int_0^T \int \fa (\pa u)(\partial^r u)(\partial^s u_x)\right| & = \left|\int_0^T \int \fa (\pa u)u_{xx} (\partial_x^{\alpha_1-2}\partial_y^{\alpha_2} u_x)\right| \\ & \leq C||u_{xx}||_{L^\infty} \left(\int_0^T \int \fa (\pa u)^2\right)^{1/2} \left(\int_0^T \int \fa (\partial_x^{\alpha_1-1}\partial_y^{\alpha_2}u)^2\right)^{1/2} \\ & \leq C ||u||_{X^1}\left(\int_0^T \int \fa (\pa u)^2\right)^{1/2} \left(\int_0^T \int \fa (\partial_x^{\alpha_1-1}\partial_y^{\alpha_2}u)^2\right)^{1/2}.\end{aligned}$$ The last term on the right-hand side above is of order $|\alpha|-1 \geq L$. The weight function $f_\alpha \in W_{\sigma, 2L-|\alpha|-\alpha_2,|\alpha|-L}$. Since $2L-|\alpha|-\alpha_2 < 2L-(\alpha_1-1+\alpha_2) -\alpha_2$, we see that this term is bounded by a term of the form . \(b) [**The subcase $r=(1,1)$.**]{} In this case, we have $$\begin{aligned} \left|\int_0^T \int \fa (\pa u)(\partial^r u)(\partial^s u_x)\right| & = \left|\int_0^T \int \fa (\pa u)u_{xy} (\partial_x^{\alpha_1-1}\partial_y^{\alpha_2-1}u_x)\right| \\ & \leq C||u_{xy}||_{L^\infty} \left(\int_0^T \int \fa (\pa u)^2\right)^{1/2} \left(\int_0^T \int \fa (\partial_x^{\alpha_1}\partial_y^{\alpha_2-1} u)^2\right)^{1/2} \\ & \leq C||u||_{X^1} \left(\int_0^T \int \fa (\pa u)^2\right)^{1/2} \left(\int_0^T \int \fa (\partial_x^{\alpha_1}\partial_y^{\alpha_2-1} u)^2\right)^{1/2}\end{aligned}$$ Using the fact that $\fa \in W_{\sigma,2L-|\alpha|-\alpha_2,|\alpha|-L}$ and $2L-|\alpha| -\alpha_2 < 2L-(\alpha_1+\alpha_2-1) - (\alpha_2-1)$, we conclude that the last term is bounded by a term of the form . \(c) [**The subcase $r=(0,2)$.**]{} In this case, we have $$\begin{aligned} \left|\int_0^T \int \fa (\pa u)(\partial^r u)(\partial^s u_x)\right| & = \left|\int_0^T \int \fa (\pa u)u_{yy} (\partial_x^{\alpha_1} \partial_y^{\alpha_2-2}u_x)\right| \\ & \leq \left(\int_0^T \int \fa (\pa u)^2\right)^{1/2} \left(\int_0^T \int u_{yy}^4 \right)^{1/4} \\ & \qquad \times \left(\int_0^T \int (\fa^{1/2} \partial_x^{\alpha_1+1}\partial_y^{\alpha_2-2}u)^4 \right)^{1/4}.\end{aligned}$$ Now $$\left(\int_0^T \int u_{yy}^4\right)^{1/4} \leq C \left(\int_0^T \int [u_{yy}^2 + u_{xyy}^2 + (\partial_x^{-1} u_{yyy})^2]\right)^{1/2} \leq C||u||_{X^1}.$$ Further, $$\begin{aligned} & \left(\int_0^T \int (\fa^{1/2} \partial_x^{\alpha_1+1}\partial_y^{\alpha_2-2}u)^4 \right)^{1/4}\\ & \leq C \left(\int_0^T \int \fa (\partial_x^{\alpha_1+1} \partial_y^{\alpha_2-2}u)^2 + \fa (\partial_x^{\alpha_1+2}\partial_y^{\alpha_2-2}u)^2 + \fa (\partial_x^{\alpha_1} \partial_y^{\alpha_2-1}u)^2 \right)^{1/2}\end{aligned}$$ Now the first and third terms in the integrand are of order $|\alpha|-1$ and are clearly bounded by terms of the form . The second term in the integrand is of order $|\alpha|$. It will be bounded using Gronwall’s inequality (and using the fact that the order of the $y$ derivative is less than $\alpha_2$ and therefore this terms can handle an even greater power of $x$.) In this case $|r|=3$. First, we consider the case in which $L \geq 4$. Since $|\alpha| \geq L+1$, we note that $|s|+2 = |\alpha|-1 \geq L$. Using this fact, we bound as follows. $$\begin{aligned} \left|\int_0^T \int \fa (\pa u)(\partial^r u)(\partial^s u_x)\right| & \leq \left(\int_0^T \int \fa (\pa u)^2\right)^{1/2} \left(\int_0^T \int (\partial^r u)^4\right)^{1/4} \\ & \qquad \times \left(\int_0^T \int (\fa^{1/2} \partial^s u_x)^4\right)^{1/4}\end{aligned}$$ Now $$\begin{aligned} \left(\int_0^T \int (\partial^r u)^4\right)^{1/4} \leq C\left(\int_0^T \int [(\partial^r u)^2 + (\partial^r u_x)^2 + (\partial^r u_y)^2]\right)^{1/2}.\end{aligned}$$ Since $|r| = 3$, each of these terms is at most of order $4 \leq L \leq |\alpha|-1$, and, therefore, bounded by terms of the form . Similarly, $$\label{s-alpha-3} \left(\int_0^T \int (\fa^{1/2} \partial^s u_x)^4\right)^{1/4} \leq C \left(\int_0^T \int \fa [(\partial^s u_x)^2 + (\partial^s u_{xx})^2 + (\partial^s u_{xy})^2]\right)^{1/2}$$ We notice that the last two terms on the right-hand side of are of order $|s|+2=|\alpha|-1\geq L$. In order to verify that we have the correct power of $x$, we note that $$2L-|\alpha|-\alpha_2 \leq 2L-(|s|+2)-(s_2+1)$$ since $|s|= |\alpha|-3$. Therefore, we can conclude that each of those terms is bounded by a term of the form . Finally, we look at the first term on the right-hand side of . If $|s|+1 \geq L$, then this term is bounded by as the other two terms. If $|s|+1 < L$, then using the fact that $|s|+2 \geq L$, we conclude that $|s| = L-2$, and, therefore, $$\begin{aligned} 2L-|\alpha|-\alpha_2 & \leq 2L-(|s|+2)-(s_2+1) \\ &\leq s_1+1\end{aligned}$$ Therefore, we conclude that the first term above is bounded by a term of the form of order $|s|+1 < L$. We now look at the cases when $L=2$ or $L=3$. In either case, if $|\alpha| \geq 5$, then we can handle as above. We first consider the case when $|\alpha|=4$ ($L=2$ or $L=3$). In this case, using the fact that $|r|=3$ and $|s|=1$, we have $$\begin{aligned} \left|\int_0^T \int \fa (\pa u)(\partial^r u)(\partial^s u_x)\right| & \leq C||\partial^s u_x||_{L^\infty} \left(\int_0^T \int \fa (\pa u)^2\right)^{1/2} \left(\int_0^T \int \fa (\partial^r u)^2 \right)^{1/2} \\ & \leq C||u||_{X^1} \left(\int_0^T \int \fa (\pa u)^2\right)^{1/2} \left(\int_0^T \int \fa (\partial^r u)^2\right)^{1/2}.\end{aligned}$$ Since $|r|=3=|\alpha|-1$ and $2L-|\alpha|-\alpha_2 \leq 2L-|r|-r_2$, we see that the last term above is bounded by a term of the form . Last, we consider $|\alpha|=3$. In this case, we must have $L=2$, $|r|=3$ and $|s|=0$. Therefore, $r = \alpha$. We bound as follows: $$\begin{aligned} \left|\int_0^T \int \fa (\pa u)(\partial^r u)(\partial^s u_x) \right| & \leq C||u_x||_{L^\infty} \int_0^T \int \fa (\pa u)^2 \\ & \leq C||u||_{X^1} \int_0^T \int \fa (\pa u)^2.\end{aligned}$$ We consider the set $A = \{x: x > 1\}$. The set $A_{-1} = \{x < -1\}$ can be handled similarly. We have $$\begin{aligned} & \left|\int_0^T \int_A \fa (\pa u)(\partial^r u)(\partial^s u_x)\right| \\ & \leq C T^M|| t^{\nu_s}\partial^s u_x||_{L^\infty(A)} \left(\int_0^T \int_A \fa (\pa u)^2\right)^{1/2} \left(\int_0^T \int_A t^{\nu_r} x^{2L-|\alpha|-\alpha_2}(\partial^r u)^2\right)^{1/2}\end{aligned}$$ where $\nu_s = \frac{(|s|+3-L)^+}2$ and $\nu_r = \frac{(|r|-L)^+}2$. First, we must verify that $M \geq 0$. We see that $$\begin{aligned} M & = \frac {|\alpha|-L}2 - \nu_s - \nu_r \\ & = \frac{|\alpha|-L-(|s|+3-L)^+-(|r|-L)^+}2 \\ & \geq \frac {|\alpha|-|s|-|r|-3+L}2 \\ & = \frac {L-3}2 \geq 0\end{aligned}$$ as long as $L \geq 3$. By assumption, $L \geq 2$. If $L = 2$, then $|\alpha| = 3$. In that case, we cannot have $|s| \leq |\alpha| - 4$. Therefore, we conclude that $M \geq 0$. Further, $$\begin{aligned} ||t^{\nu_s}\partial^s u_x||_{L^\infty} & \leq C \left(\int t^{2\nu_s} [(\partial^su_x)^2 + (\partial^s u_{xxx})^2 + (\partial^s u_{xy})^2]\right)^{1/2}.\end{aligned}$$ Each of those terms is of order at most $|\alpha|-1$, and therefore, bounded by terms of the form . Further, $2L-|\alpha|-\alpha_2 \leq 2L-|r|-r_2$. Therefore, for $|r| \leq |\alpha|-1$, the last term above is bounded by terms of the form . In the case that $|r|=|\alpha|$, we have $|s| = 0$, and therefore $$\begin{aligned} \left|\int_0^T \int \fa (\pa u)(\partial^r u)(\partial^s u_x)\right| & \leq C ||u_x|| _{L^\infty} \int_0^T \int \fa (\pa u)^2 \\ & \leq C ||u||_{X^0} \int_0^T \int \fa (\pa u)^2.\end{aligned}$$ Combining our estimates above on $$\left|\int_0^T \int \fa (\pa u)(\partial^r u)(\partial^s u_x)\right|$$ with , we see that $$\begin{split} \label{power-1} \int \fa (\pa u)^2 + 3 \int_0^T \int (\fa)_x (\pa u_x)^2 & \leq C + C\sum_{\stackrel{|\gamma| = |\alpha|}{ 2L-|\gamma|-\gamma_2 \geq 1}} \int_0^T \int f_\gamma (\partial^\gamma u)^2 \end{split}$$ where the constant $C$ depends only on terms of the form and . Using the above estimate for all derivatives $\gamma$ of order $|\alpha|$ such that $2L-|\gamma|-\gamma_2 \geq 1$, we see that $$\begin{split} \sum_{\stackrel{|\gamma|=|\alpha|}{2L-|\gamma|-\gamma_2 \geq 1}}& \left[\int f_\gamma (\pa u)^2 + 3 \int_0^T \int (f_\gamma)_x (\pa u_x)^2\right] \\ & \leq C + \sum_{\stackrel{|\gamma|=|\alpha|}{2L-|\gamma|-\gamma_2 \geq 1}}\int_0^T \int f_\gamma (\partial^\gamma u)^2. \end{split}$$ where $C$ depends only on terms of the form and . Applying Gronwall’s inequality, we get the desired estimate. $\square$\ \ Persistence Theorem =================== In section four we proved the existence of a solution $u$ to in $L^\infty([0,T]; X^N(\mathbb R^2))$ for given initial data $\phi \in X^N(\mathbb R^2)$. In this section, we prove that if, in addition, our initial data $\phi$ lies in the weighted space $\widetilde H_x^K(W_{0\;K\;0})$ for some $K \geq 0$, then the solution $u$ also lies in $L^\infty([0,T];\widetilde H_x^K(W_{0\;K\;0}))$. This property is known as a “persistence" property of the initial data. This property provides a basis for starting the induction in our Gain of Regularity theorem in Section 7.\ \ We use induction on $j=|\alpha|$ for $1 \leq j \leq K$. The case that $j = 0$ follows from conservation of $L^2$ norm. We derive formally some a priori estimates for the solution where the bound involves only the norms of $u \in L^\infty([0,T]:X^1(\mathbb R^2))$ and the norms of $\phi \in \widetilde H_x^K( W_{0\;K\;0})$. Then, we can apply convergence arguments to show that the result holds true for general solutions. In order to do so, we need to approximate general solutions $u \in X^1(\mathbb R^2)$ by smooth solutions and approximate general weight functions $f \in W_{0\;j\;0}$ by smooth, bounded weight functions. The first of these procedures has already been discussed, so we will concentrate on the second. For a fixed $i$, we begin by taking a sequence of bounded weight functions $g_{i,\delta}$ which decay as $|x| \rightarrow \infty$ and which approximate $g_i \in W_{\sigma\; i-1\; 0}$ with $\sigma > 0$ from below, uniformly on any half-line $(-\infty,c)$. Define the weight functions $$f_{i,\delta}(x,t) = 1 + \int_{-\infty}^x g_{i,\delta}(z,t) \,dz.$$ Therefore, the functions $f_{i,\delta}$ are bounded weight functions approximating $f_i \in W_{0,i,0}$ from below, uniformly on compact sets. &gt;From (5.3) and using the fact that $\partial_t (\fid) \leq c \fid$ and $\partial_x (\fid) \leq c \fid$, we have $$\begin{aligned} & \int \fid (\pa u)^2 + 3 \int_0^T \int (\fid)_x (\pa u_x)^2 \,dt \leq \int_0^T (\fid)_x (\pa \partial_x^{-1} u_y)^2 \,dt \\ & \qquad + C\int_0^T \int \fid (\pa u)^2 \,dt + 2 \left|\int_0^T \int \fid (\pa u) \pa (uu_x) \,dt \right|.\end{aligned}$$ \(a) [**The subcase $\alpha = (1,0)$.**]{} Defining $g_{1,\delta}$ and $f_{1,\delta}$ as above, we see that $f_{1,\delta}$ will approximate $f_1 \in W_{0\;1\;0}$ from below. Differentiating (2.1) in the $x-$variable, multiplying by $2f_{1,\delta}$ and integrating over $\mathbb R^2$, we have $$\begin{split} \label{alpha1beta0} & \partial_t \int \foned u_x^2 + 3 \int(\foned)_xu_{xx}^2 \\ & \qquad = \int (\foned)_x u_y^2 + \int [\partial_t\foned + \partial_x^3\foned + \partial_x \foned]u_x^2 - 2 \int \foned u_x(uu_x)_x \\ & \leq C \int u_y^2 + C \int \foned u_x^2 + 2 \left|\int \foned u_x(uu_x)_x\right| \end{split}$$ Moreover $$\begin{aligned} \left|\int \foned u_x(uu_x)_x \right| & = \left|\int \foned u_x[u_x^2 + uu_{xx}]\right| \\ & = \left|\int \foned [u_x^3 + uu_{x}u_{xx}] \right| \\ & \leq c ||u_x||_{L^\infty} \int \foned u_x^2 + \frac 12\left|\int (\foned u)_x u_x^2 \right| \\ & \leq c(||u||_{L^\infty} + ||u_x||_{L^\infty})\int \foned u_x^2 \\ & \leq C ||u||_{X^0}\int \foned u_x^2.\end{aligned}$$ Combining this estimate with , we conclude that for $0 \leq t \leq T$, $$\int \foned(\cdot,t) u_x^2 + 3 \int_0^T \int (\foned)_x u_{xx}^2 \leq C \int \foned(\cdot,0) \phi_x^2 + C \int_0^T \int u_y^2 + C \int_0^T \int \foned u_x^2.$$ Applying Gronwall’s inequality, we conclude that $$\sup_{0 \leq t \leq T} \int \foned u_x^2 + 3 \int_0^T \int (\foned)_x u_{xx}^2 \leq C$$ where $C$ does not depend on $\delta$ but only on $T$ and the norm of $\phi \in X^1(\mathbb R^2) \cap \widetilde H_x^1(W_{0\;1\;0})$. Taking the limit as $\delta\rightarrow \infty$, we conclude that $$\sup_{0 \leq t \leq T} \int f_1 u_x^2 + 3 \int_0^T \int g_1 u_{xx}^2 \leq C,$$ as claimed. \(b) [**The subcase $\alpha = (0,1)$.**]{} Here, our weight function $f_0 \in W_{0\;0\;0}$. Differentiating (2.1) in the $y$-variable, multiplying by $2\fzerod u_y$ and integrating over $\mathbb R^2$, we have $$2 \int \fzerod u_y u_{yt} + 2 \int \fzerod u_y u_{xxxy} - 2 \int \fzerod u_y \partial_x^{-1} u_{yyy} + 2 \int \fzerod u_y u_{xy} + 2 \int \fzerod u_y (uu_x)_y = 0.$$ Integrating each term by parts gets $$\begin{split} \label{alpha0beta1} & \partial_t \int \fzerod u_y^2 + 3 \int (\fzerod)_x u_{xy}^2 \\ & \qquad \leq \int (\fzerod)_x (\partial_x^{-1}u_{yy})^2 + C \int \fzerod u_y^2 + 2 \left|\int \fzerod u_y (uu_x)_y\right| \\ & \leq C \int (\partial_x^{-1} u_{yy})^2 + C \int \fzerod u_y^2 + 2 \left| \int \fzerod u_y (uu_x)_y\right|. \end{split}$$ Moreover, $$\begin{aligned} \left|\int \fzerod u_y (uu_x)_y\right| & = \left|\int \fzerod u_y (u_yu_x + uu_{xy})\right| \\ & \leq C ||u_x||_{L^\infty} \int \fzerod u_y^2 + C ||u||_{L^\infty} \int (\fzerod)_x u_y^2 \\ & \leq C ||u||_{X^0}\int \fzerod u_y^2.\end{aligned}$$ Combining this estimate with , we conclude that for $0 \leq t \leq T$, $$\int \fzerod(\cdot,t)u_y^2 + 3 \int_0^T \int (\fzerod)_x u_{xy}^2 \leq \int \fzerod(\cdot,0) \phi_y^2 + C \int_0^T \int (\partial_x^{-1}u_{yy})^2 + C \int_0^T \int \fzerod u_y^2.$$ Applying Gronwall’s inequality, we conclude that $$\sup_{0 \leq t \leq T} \int \fzerod u_y^2 + 3 \int_0^T \int (\fzerod)_x u_{xy}^2 \leq C$$ where $C$ does not depend on $\delta$, but only on $T$ and the norm of $\phi \in X^0(\mathbb R^2) \cap \widetilde H_x^1(W_{0\;1\;0})$. Passing to the limit, we conclude that $$\sup_{0 \leq t \leq T} \int f_0 u_y^2 + 3 \int_0^T \int g_0 u_{xy}^2 \leq C.$$ \(a) [**The subcase $\alpha = (2,0)$.**]{} In this case, $\ftwod$ will approximate $f_2 \in W_{0\;2\;0}$. In a similar way as above, we have $$\begin{aligned} & 2 \int \ftwod u_{xx} u_{xxt} + 2 \int \ftwod u_{xx}u_{xxxxx} - 2 \int \ftwod u_{xx} u_{xyy} \\ & \qquad + 2 \int \ftwod u_{xx}u_{xxx} + 2 \int \ftwod u_{xx}(uu_x)_{xx} = 0.\end{aligned}$$ Integrating each term by parts gets $$\begin{aligned} & \partial_t \int \ftwod u_{xx}^2 + 3 \int (\ftwod)_xu_{xxx}^2 \\ & \qquad = c \int (\ftwod)_x u_{xy}^2 + \int [\partial_t \ftwod + \partial_x^3 \ftwod + \partial_x \ftwod]u_{xx}^2 - 2 \int \ftwod u_{xx} (uu_x)_{xx} \\ & \leq \int \ftwod u_{xy}^2 + c \int \ftwod u_{xx}^2 + 2 \left|\int \ftwod u_{xx} (uu_x)_{xx} \right|.\end{aligned}$$ Moreover $$\begin{aligned} \left|\int \ftwod u_{xx}(uu_x)_{xx} \right| & = \left|\int \ftwod u_{xx}[3u_{x} u_{xx} + uu_{xxx}] \right| \\ & = \left|\int \ftwod[3u_{x} u_{xx}^2 + uu_{xx} u_{xxx}] \right| \\ & \leq c||u_x||_{L^\infty} \int \ftwod u_{xx}^2 + \frac{1}{2}\left|(\ftwod u)_x u_{xx}^2 \right| \\ & \leq c(||u||_{L^\infty} + ||u_x||_{L^\infty})\int \ftwod u_{xx}^2 \\ & \leq C ||u||_{X^0} \int \ftwod u_{xx}^2.\end{aligned}$$ Therefore, $$\partial_t \int \ftwod u_{xx}^2 + 3 \int (\ftwod)_x u_{xxx}^2 \leq \int (\ftwod)_x u_{xy}^2 + C \int \ftwod u_{xx}^2,$$ where $C$ depends only on the norm of $\phi \in X^0(\mathbb R^2)$. We will combine this estimate with the estimate below. \(b) [**The subcase $\alpha = (1,1)$**]{} Applying $\partial_x\partial_y$ to (2.1), multiplying by $2 \foned u_{xy}$ where $\foned$ approximates $f_1 \in W_{0\;1\;0}$, and integrating over $\mathbb R^2$, we have $$\begin{aligned} & 2 \int \foned u_{xy} u_{xyt} + 2 \int \foned u_{xy}u_{xxxxy} - 2 \int \foned u_{xy}u_{yyy} \\ & \qquad + 2 \int \foned u_{xy}u_{xxy} + 2 \int \foned u_{xy}(uu_x)_{xy} = 0.\end{aligned}$$ Integrating each term by parts gets $$\begin{aligned} & \partial_t \int \foned u_{xy}^2 + 3 \int (\foned)_x u_{xxy}^2 \\ & \qquad = \int (\foned)_x u_{yy}^2 + \int [\partial_t \foned + \partial_x^3 \foned + \partial_x \foned]u_{xy}^2 - 2 \int \foned u_{xy}(uu_x)_{xy} \\ & \leq \int (\foned)_x u_{yy}^2 + c \int \foned u_{xy}^2 + 2 \left|\int \foned u_{xy} (uu_x)_{xy}\right|.\end{aligned}$$ Moreover, $$\begin{aligned} \left|\int \foned u_{xy}(uu_x)_{xy}\right| & = \left|\int \foned u_{xy}(2u_x u_{xy} + u_{xx}u_y + uu_{xxy})\right| \\ & \leq C(||u_x||_{L^\infty} + ||u||_{L^\infty})\int \foned u_{xy}^2 + \left|\int \foned u_{xx}u_{xy} u_y\right| \\ & \leq C||u||_{X^0}\int \foned u_{xy}^2 + C||u_y||_{L^\infty} (\int \foned u_{xx}^2 + \int \foned u_{xy}^2) \\ & \leq C||u||_{X^0} \int \foned u_{xy}^2 + C ||u||_{X^1} \int \foned (u_{xx}^2 + u_{xy}^2)\end{aligned}$$ since $$\begin{aligned} ||u_y||_{L^\infty} & \leq \left(\int u_y^2 + u_{xxy}^2 + u_{yy}^2\right)^{1/2} \leq ||u||_{X^1}.\end{aligned}$$ Therefore, $$\partial_t \int \foned u_{xy}^2 + 3 \int (\foned)_x u_{xxy}^2 \leq \int (\foned)_x u_{yy}^2 + C \int \foned u_{xx}^2 + C \int \foned u_{xy}^2.$$ \(c) [**The subcase $\alpha = (0,2)$.**]{} Applying $\partial_y^2$ to (2.1), multiplying by $u_{yy}$ and integrating over $\mathbb R^2$, we have $$\begin{aligned} & 2 \int u_{yy} u_{yyt} + 2 \int u_{yy}u_{xxxyy} - 2 \int u_{yy} \partial_x^{-1}u_{yyyy} \\ & \qquad + 2 \int u_{yy} u_{xyy} + 2 \int u_{yy}(uu_x)_{yy} = 0.\end{aligned}$$ Integrating by parts gets $$\partial_t \int u_{yy}^2 \leq 2 \left|\int u_{yy}(uu_x)_{yy}\right|.$$ Now $$\begin{aligned} \left|\int u_{yy} (uu_x)_{yy}\right| & = \left|\int u_{yy}(u_{yy}u_x + 2u_y u_{xy} + uu_{xyy}) \right| \\ & \leq ||u_x||_{L^\infty} \int u_{yy}^2 + ||u_y||_{L^\infty} \left(\int u_{yy}^2 + u_{xy}^2 \right) \\ & \leq ||u||_{X^0}\int u_{yy}^2 + C||u||_{X^1} \int (u_{yy}^2 + u_{xy}^2).\end{aligned}$$ Now combining these estimates from (a), (b) and (c) above, we have $$\begin{aligned} & \partial_t \int (\ftwod u_{xx}^2 + \foned u_{xy}^2 + u_{yy}^2) + 3 \int \left[(\ftwod)_x u_{xxx}^2 + (\foned)_x u_{xxy}^2\right] \\ & \qquad \leq C \int (\ftwod + \foned) u_{xx}^2 + C \int ((\ftwod)_x + \foned + 1) u_{xy}^2 + \int ((\foned)_x + 1) u_{yy}^2,\end{aligned}$$ where $C$ depends only on the norm of $\phi \in X^1(\mathbb R^2)$. Since $\ftwod$ approximates $f_2 \in W_{0\;2\;0}$ and $\foned$ approximates $f_1 \in W_{0\;1\;0}$, we can choose $\ftwod, \foned$ such that $(\ftwod)_x \leq C \foned$, etc. Therefore, $$\begin{aligned} & \partial_t \int (\ftwod u_{xx}^2 + \foned u_{xy}^2 + u_{yy}^2) + 3 \int \left[(\ftwod)_x u_{xxx}^2 + (\foned)_x u_{xxy}^2\right] \\ & \qquad \leq C \int (\ftwod u_{xx}^2+ \foned u_{xy}^2 + u_{yy}^2).\end{aligned}$$ Integrating with respect to $t$, we have $$\begin{aligned} & \int (\ftwod(\cdot,t) u_{xx}^2 + \foned(\cdot,t) u_{xy}^2 + u_{yy}^2) + 3 \int_0^t\int \left[(\ftwod)_x u_{xxx}^2 + (\foned)_x u_{xxy}^2\right] \\ & \leq \int (\ftwod(\cdot,0) \phi_{xx}^2 + \foned(\cdot,0) \phi_{xy}^2 + \phi_{yy}^2) + C \int_0^t \int (\ftwod u_{xx}^2+ \foned u_{xy}^2 + u_{yy}^2).\end{aligned}$$ Further, integrating by parts and using the fact that $f_1$ approximates $\foned \approx x$ for $x>1$, we note that $$\int \foned \phi_{xy}^2 \leq C \int \ftwod \phi_{xx}^2 + C \int \phi_{yy}^2.$$ Therefore, by Gronwall’s inequality $$\begin{aligned} & \sup_{0 \leq t \leq T} \int (\ftwod(\cdot,t) u_{xx}^2 + \foned(\cdot,t) u_{xy}^2 + u_{yy}^2) + 3 \int_0^T\int \left[(\ftwod)_x u_{xxx}^2 + (\foned)_x u_{xxy}^2\right] \leq C\end{aligned}$$ where $C$ does not depend on $\delta$ but only on $T$ and the norm of $\phi \in X^1(\mathbb R^2) \cap \widetilde H_x^2(W_{0\;2\;0})$. Consequently, we can pass to the limit and conclude that $$\begin{aligned} & \sup_{0 \leq t \leq T} \int (f_2(\cdot,t) u_{xx}^2 + f_1(\cdot,t) u_{xy}^2 + u_{yy}^2) + 3 \int_0^T\int \left[g_2 u_{xxx}^2 + g_1 u_{xxy}^2\right] \leq C.\end{aligned}$$ \(a) [**The subcase $\alpha = (3,0)$.**]{} We choose our weight functions such that $\fthreed$ approximates $f_3 \in W_{0\;3\;0}$. Applying $\partial_x^3$ to (2.1), multiplying by $\fthreed u_{xxx}$ and integrating over $\mathbb R^2$, we have $$\begin{aligned} & 2 \int \fthreed u_{xxx}u_{xxxt} + 2 \int \fthreed u_{xxx}u_{xxxxxx} - 2 \int \fthreed u_{xxx}u_{xxyy} \\ & \qquad + 2 \int \fthreed u_{xxx}u_{xxxx} + 2 \int \fthreed u_{xxx}(uu_x)_{xxx} = 0.\end{aligned}$$ Integrating by parts gets $$\begin{aligned} & \partial_t \int \fthreed u_{xxx}^2 + 3 \int (\fthreed)_x u_{xxxx}^2 \\ & \qquad = \int (\fthreed)_x u_{xxy}^2 + \int [\partial_t \fthreed + \partial_x^3 \fthreed + \partial_x \fthreed]u_{xxx}^2 - 2 \int \fthreed u_{xxx}(uu_x)_{xxx} \\ & \qquad \leq \int (\fthreed)_x u_{xxy}^2 + c \int \fthreed u_{xxx}^2 + 2 \left| \int \fthreed u_{xxx} (uu_x)_{xxx} \right|.\end{aligned}$$ Moreover $$\begin{aligned} \left|\int \fthreed u_{xxx}(uu_x)_{xxx} \right| & = \left|\int \fthreed [3u_{xx}^2 + 4u_x u_{xxx} + uu_{xxxx}] u_{xxx} \right| \\ & \leq 3 \left|\int \fthreed u_{xx}^2 u_{xxx} \right| + 4 \left|\int \fthreed u_x u_{xxx}^2 \right| + \left|\int \fthreed u u_{xxx} u_{xxxx} \right| \\ & \leq 3 \left|\int \fthreed u_{xx}^2 u_{xxx}\right| + c ||u_x||_{L^\infty}\int \fthreed u_{xxx}^2 + c \left|\int (\fthreed u)_x u_{xxx}^2 \right| \\ & \leq 3 \left|\int \fthreed u_{xx}^2 u_{xxx} \right| + c ||u||_{X^0} \int \fthreed u_{xxx}^2 \\ & \qquad + c (||u||_{L^\infty(\mathbb R^2)} + ||u_x||_{L^\infty(\mathbb R^2)})\int \fthreed u_{xxx}^2 \\ & \leq C \left|\int \fthreed u_{xx}^2 u_{xxx}\right| + c||u||_{X^0(\mathbb R^2)}\int \fthreed u_{xxx}^2.\end{aligned}$$ Now we estimate the first term on the right-hand side. Let $A_1 = \left\{x \in \mathbb R: x > 1\right\} \times \mathbb R \subseteq \mathbb R^2$. $$\begin{aligned} \left|\int_{A_1} \fthreed u_{xx}^2 u_{xxx} \right| & = C \left|\int_{A_1} (\fthreed)_x u_{xx}^3 \right| \\ & \leq C ||u_{xx}||_{L^\infty(A_1)} \int_{A_1} (\fthreed)_x u_{xx}^2 \\ & \leq C \left(\int_{A_1} u_{xx}^2 + u_{xxxx}^2 + u_{xxy}^2\right)^{1/2} \int_{A_1} \ftwod u_{xx}^2 \\ & \leq \epsilon \int_{A_1} (u_{xx}^2 + u_{xxxx}^2 + u_{xxy}^2) + C\left(\int_{A_1} \ftwod u_{xx}^2\right)^2.\end{aligned}$$ Now the terms involving $u_{xx}$ have been bounded by the previous step in the induction. Therefore, we conclude that $$\left|\int_{A_1} \fthreed u_{xx}^2 u_{xxx} \right| \leq C + \epsilon \int_{A_1} (u_{xxxx}^2 + u_{xxy}^2).$$ $x < -1$. Let $A_{-1} = \left\{x \in \mathbb R: x < -1\right\} \times \mathbb R \subseteq \mathbb R^2$. We use the fact that $\fthreed \approx c$ to show $$\begin{aligned} \left|\int_{A_{-1}} \fthreed u_{xx}^2 u_{xxx} \right| & \leq c \left|\int_{A_{-1}} u_{xx}^2 u_{xxx} \right| \\ & \leq c \left(\int_{A_{-1}} u_{xx}^4\right)^{1/2}\left(\int_{A_{-1}} u_{xxx}^2\right)^{1/2} \\ & \leq C \left(\int_{A_{-1}}[u_{xx}^2 + u_{xxx}^2 + u_{xy}^2]\right)\left(\int_{A_{-1}} u_{xxx}^2 \right)^{1/2} \\ & \leq c ||u||_{X^0}^3.\end{aligned}$$ Combining these estimates for the subcase $\alpha = (3,0)$, yields $$\begin{aligned} & \int \fthreed(\cdot,t) u_{xxx}^2 + 3 \int_0^t \int (\fthreed)_x u_{xxxx}^2 \\ & \leq \int \fthreed(\cdot,0) \phi_{xxx}^2 + C + \int_0^t\int (\fthreed)_x u_{xxy}^2 + c \int_0^t \int \fthreed u_{xxx}^2 + \epsilon \int_0^t \int_{A_1} u_{xxxx}^2 \\ & \leq \int \fthreed(\cdot,0) \phi_{xxx}^2 + C + \int_0^t\int (\fthreed)_x u_{xxy}^2 + c \int_0^t \int \fthreed u_{xxx}^2 + \epsilon \int_0^t \int (\fthreed)_x u_{xxxx}^2\end{aligned}$$ Therefore, $$\begin{aligned} & \int \fthreed(\cdot,t) u_{xxx}^2 + 3 \int_0^t \int (\fthreed)_x u_{xxxx}^2 \\ & \leq \int \fthreed(\cdot,0) \phi_{xxx}^2 + C + \int_0^t\int (\fthreed)_x u_{xxy}^2 + c \int_0^t \int \fthreed u_{xxx}^2 \\ & \leq \int \fthreed(\cdot,0) \phi_{xxx}^2 + C + C \int_0^t \int \ftwod u_{xxy}^2 + C \int_0^t \int \fthreed u_{xxx}^2.\end{aligned}$$ \(b) [**The subcase $\alpha = (2,1)$.**]{} In this case, we take $\ftwod$ approximating $f_2 \in W_{0\;2\;0}$. Apply $\partial_x^2 \partial_y$ to (2.1), multiply by $\ftwod u_{xxy}$ and integrating over $\mathbb R^2$, we have $$\begin{aligned} & \partial_t \int \ftwod u_{xxy}^2 + 3 \int (\ftwod)_x u_{xxxy}^2 \\ & \leq \int (\ftwod)_x u_{xyy}^2 + \int \ftwod u_{xxy}^2 + 2\left|\int \ftwod u_{xxy} (uu_x)_{xyy}\right|.\end{aligned}$$ Now $$\begin{aligned} \left|\int \ftwod u_{xxy}(uu_x)_{xxy} \right| & = \left|\int \ftwod u_{xxy} [3u_{xy}u_{xx} + 3u_x u_{xxy} + u_y u_{xxx} + uu_{xxxy}]\right| \\ & \leq C \int \ftwod u_{xxy}u_{xy} u_{xx} + \int \ftwod u_{xxy}u_y u_{xxx} + C||u||_{X^0} \int \ftwod u_{xxy}^2.\end{aligned}$$ The second term on the right-hand side satisfies $$\begin{aligned} \int \ftwod u_{xxy} u_y u_{xxx} & \leq C ||u_y||_{L^\infty} \int \ftwod (u_{xxx}^2 + u_{xxy}^2) \\ & \leq C ||u||_{X^1} \int (\fthreed u_{xxx}^2 + \ftwod u_{xxy}^2).\end{aligned}$$ For the first term on the right-hand side, we consider two cases. First, for $A_1$, $$\begin{aligned} \int_{A_1} \ftwod u_{xxy}u_{xy} u_{xx} & \leq ||u_{xy}||_{L^\infty(A_1)} \left(\int_{A_1} \ftwod u_{xxy}^2\right)^{1/2} \left(\int_{A_1} \ftwod u_{xx}^2\right)^{1/2} \\ & \leq \epsilon \left(\int_{A_1} u_{xy}^2 + u_{xxxy}^2 + u_{xyy}^2\right) + C \int_{A_1} \ftwod u_{xxy}^2 \\ & \leq C + \epsilon \int (\ftwod)_x u_{xxxy}^2 + C \int \foned u_{xyy}^2 + C \int \ftwod u_{xxy}^2.\end{aligned}$$ Then for $A_{-1}$, $$\begin{aligned} \int_{A_{-1}} \ftwod u_{xxy} u_{xy} u_{xx} & = \int_{A_{-1}} u_{xxy} u_{xy} u_{xx} \\ & = C \int_{A_{-1}} u_{xy}^2 u_{xxx} \\ & \leq C \left(\int_{A_{-1}} u_{xy}^4\right)^{1/2} \left(\int_{A_{-1}} u_{xxx}^2\right)^{1/2} \\ & \leq C \left(\int_{A_{-1}} u_{xy}^2 + u_{xxy}^2 + u_{yy}^2\right)\left(\int_{A_{-1}} u_{xxx}^2 \right)^{1/2} \\ & \leq C(||u||_{X^0}) \left(1 + \int \ftwod u_{xxy}^2\right)\end{aligned}$$ Combining these estimates and integrating with respect to $t$, we have $$\begin{aligned} \int \ftwod(\cdot,t) u_{xxy}^2 + 3 \int_0^t \int (\ftwod)_x u_{xxxy}^2 & \leq C + \int \ftwod(\cdot,0) \phi_{xxy}^2 + \epsilon \int_0^t \int (\ftwod)_x u_{xxxy}^2 \\ & \qquad + \int_0^t \int \ftwod u_{xxy}^2 + C \int_0^t \int \foned u_{xyy}^2.\end{aligned}$$ Therefore, $$\int \ftwod(\cdot,t) u_{xxy}^2 + 3 \int_0^t \int (\ftwod)_x u_{xxxy}^2 \leq C + \int \ftwod(\cdot,0) \phi_{xxy}^2 + \int_0^t \int \ftwod u_{xxy}^2 + C \int_0^t \int \foned u_{xyy}^2.$$ \(c) [**The subcase $\alpha = (1,2)$.**]{} We take our weight function $\foned$ approximating $f_1 \in W_{0\;1\;0}$. Applying $\partial_x\partial_y^2$ to (2.1), multiplying by $\foned u_{xyy}$ and integrating over $\mathbb R^2$, we get $$\begin{aligned} & \partial_t \int \foned u_{xyy}^2 + 3 \int (\foned)_x u_{xxyy}^2 \\ & \leq c \int (\foned)_x u_{yyy}^2 + \int \foned u_{xyy}^2 + 2 \left|\int \foned u_{xyy} (uu_x)_{xyy}\right|.\end{aligned}$$ Now $$\begin{aligned} 2 \left|\int \foned u_{xyy} (uu_x)_{xyy} \right| & = 2 \left|\int \foned u_{xyy} (2u_{xy}^2 + 2u_x u_{xyy} + u_{yy}u_{xx} + 2u_y u_{xxy} + uu_{xxyy})\right|.\end{aligned}$$ The first term on the right-hand side satisfies $$\int \foned u_{xyy}u_{xy}^2 = C \int \foned (u_{xy}^3)_y = 0,$$ since $(\foned)_y = 0$. Integrating by parts, it is clear that the second and fifth terms on the right-hand side are bounded by $$C ||u||_{X^0}\int \foned u_{xyy}^2.$$ The fourth term on the right-hand side is bounded by $$C||u_y||_{L^\infty} \left(\int \foned u_{xyy}^2 + \int \foned u_{xxy}^2\right).$$ For the third-term on the right-hand side, we consider the cases when $x > 1$ and $x < -1$ separately. First, for $x > 1$, we have $$\begin{aligned} \int_{A_1} \foned u_{xyy} u_{yy} u_{xx} & \leq ||u_{yy}||_{L^\infty(A_1)} \left(\int_{A_1} \foned u_{xyy}^2\right)^{1/2} \left(\int_{A_1} \foned u_{xx}^2\right)^{1/2} \\ & \leq C \left(\int_{A_1} u_{yy}^2 + u_{xxyy}^2 + u_{yyy}^2\right)^{1/2} \left(\int_{A_1} \foned u_{xyy}^2\right)^{1/2} \\ & \leq C + \epsilon \int_{A_1} u_{xxyy}^2 + C \int_{A_1} \foned u_{xyy}^2 + C \int_{A_1} u_{yyy}^2\end{aligned}$$ where we have used the fact that $\int \foned u_{xx}^2$ was bounded on the previous step of the induction. We will bound the $\epsilon$ term back on the left-hand side. For $x < -1$, we have $$\begin{aligned} \int_{A_{-1}} \foned u_{xyy}u_{yy}u_{xx} & \approx \int \chi_{[x < -1]}u_{yy}^2 u_{xxx} \\ & \leq \left(\int_{A_{-1}} u_{yy}^4\right)^{1/2} \left(\int_{A_{-1}} u_{xxx}^2\right)^{1/2} \\ & \leq C\int_{A_{-1}} u_{yy}^2 + u_{xyy}^2 + (\partial_x^{-1} u_{yy})^2 \\ & \leq C\end{aligned}$$ where $C$ depends only on the norm of $u \in X^1(\mathbb R^2)$. Combining these estimates and integrating with respect to $t$, we have $$\begin{aligned} \int \foned(\cdot,t) u_{xyy}^2 + 3 \int_0^t \int (\foned)_x u_{xxyy}^2 \leq \int \foned(\cdot,0) \phi_{xyy}^2 + C \int_0^t \int \foned u_{xyy}^2 + C \int_0^t \int u_{yyy}^2.\end{aligned}$$ \(d) [**The subcase $\alpha = (0,3)$.**]{} In this case we apply $\partial_y^3$ to (2.1), multiply by $u_{yyy}$ and integrate over $\mathbb R^2$. We have $$\partial_t \int u_{yyy}^2 \leq \int u_{yyy}^2 + 2 \left|\int u_{yyy}(uu_x)_{yyy}\right|.$$ Now $$\left|\int u_{yyy}(uu_x)_{yyy} \right| = \left|\int u_{yyy}(u_xu_{yyy} + 3u_{yy}u_{xy} + 3u_y u_{xyy} + uu_{xyyy})\right|$$ Integrating by parts as necessary, we see that the first and fourth terms on the right-hand side are bounded by $$||u||_{X^0} \int u_{yyy}^2.$$ The second term on the right-hand side is bounded by $$\begin{aligned} & \left(\int u_{yy}^4\right)^{1/4} \left(\int u_{xy}^4\right)^{1/4} \left(\int u_{yyy}^2 \right)^{1/2} \\ & \leq \left(\int u_{yy}^2 + u_{xyy}^2 + (\partial_x^{-1}u_{yyy})^2\right)^{1/2} \left(\int u_{xy}^2 + u_{xxy}^2 + u_{yy}^2\right)^{1/2} \left(\int u_{yyy}^2\right)^{1/2} \\ & \leq C||u||_{X^1(\mathbb R^2)}\left(\int u_{yyy}^2 \right)^{1/2} \\ & \leq C + C \int u_{yyy}^2.\end{aligned}$$ The third term on the right-hand side is bounded by $$\begin{aligned} & C||u_y||_{L^\infty} \left(\int u_{xyy}^2\right)^{1/2} \left(\int u_{yyy}^2\right)^{1/2} \\ & \leq C \left(\int u_y^2 + u_{xxy}^2 + u_{yy}^2 \right)^{1/2} \left(\int u_{xyy}^2\right)^{1/2} \left(\int u_{yyy}^2\right)^{1/2} \\ & \leq C ||u||_{X^1}^2 \left(\int u_{yyy}^2\right)^{1/2} \\ & \leq C + C \int u_{yyy}^2.\end{aligned}$$ Combining the estimates above and integrating with respect to $t$, we have $$\begin{aligned} \int u_{yyy}^2 \leq \int \phi_{yyy}^2 + C + C \int_0^t \int u_{yyy}^2\end{aligned}$$ where $C$ depends only on the norm of $\phi$ in $X^1(\mathbb R^2)$. Combining the estimates above for (a), (b), (c) and (d) and applying Gronwall’s inequality, we conclude that $$\begin{aligned} & \sup_{0 \leq t \leq T}\int (\fthreed u_{xxx}^2 + \ftwod u_{xxy}^2 + \foned u_{xyy}^2 + u_{yyy}^2) \\ & + 3 \int_0^T \int ((\fthreed)_x u_{xxxx}^2 + (\ftwod)_x u_{xxxy}^2 + (\foned)_x u_{xxyy}^2) \leq C\end{aligned}$$ where $C$ does not depend on $\delta$, but only on $T$ and the norm of $\phi \in X^1(\mathbb R^2) \cap \widetilde H_x^3(W_{0\;3\;0})$. Consequently, we can pass to the limit and conclude that $$\sup_{0 \leq t \leq T} \int (f_3 u_{xxx}^2 + f_2 u_{xxy}^2 + f_1 u_{xyy}^2 + u_{yyy}^2) + 3 \int_0^T \int (g_3 u_{xxxx}^2 + g_2 u_{xxxy}^2 + g_1 u_{xxyy}^2) \leq C.$$ In this case, we take $f_{\alpha_1,\delta}$ approximating $\faone \in W_{0\;\alpha_1\;0}$. We apply $\pa$ to (2.1), multiply by $f_{\alpha_1,\delta}\pa u$ and integrate over $\mathbb R^2$. We need to get a bound on $$\left|\int_0^T \int f_{\alpha_1,\delta} (\pa u) \pa (uu_x)\right|.$$ In Lemma 6.2 below, we prove that $$\left|\int_0^t \int \fad (\pa u)\pa(uu_x)\right| \leq C + C \sum_{\gamma_1 + \gamma_2 = j} \int_0^t \int \fgd (\partial^\gamma u)^2$$ where $C$ depends only on terms bounded in the previous step of the induction. Consequently, we have that $$\sup_{0\leq t\leq T} \sum_{|\alpha|=j} f_{\alpha_1,\delta} (\pa u)^2 + \sum_{|\alpha|=j} \int_0^T \int (f_{\alpha_1,\delta})_x (\pa u_x)^2 \leq C,$$ where $C$ does not depend on $\delta$, but only on $T$ and the norm of $\phi \in X^1(\mathbb R^2) \cap \widetilde H_x^j(W_{0\;j\;0})$. Passing to the limit, we get the desired estimate, namely, $$\sup_{0\leq t\leq T} \sum_{|\alpha|=j} \faone (\pa u)^2 + \sum_{|\alpha|=j} \int_0^T \int \gaone (\pa u_x)^2 \leq C.$$ $\square$ In order to get bounds on the left-hand side of , we use the fact that every term in the integrand is of the form $$\fad (\pa u)(\partial^r u)(\partial^s u_x)$$ where $r_i + s_i = \alpha_i$. Before showing the bounds on each of the terms in the integrand we point one bound we will be using frequently: $$||\partial^\gamma u||_{L^\infty} \leq C \left(\int (\partial^\gamma u)^2 + (\partial^\gamma u_{xx})^2 + (\partial^\gamma u_y)^2\right)^{1/2}.$$ In this case, $|r| = 0$ and $s = \alpha$. Therefore, $$\begin{aligned} \int \faoned (\pa u)(\partial^r u)(\partial^s u_x) & = \int \faoned u (\pa u)(\pa u_x) \\ & = - \frac 12 \int [\faoned u]_x (\pa u)^2 \\ & \leq C ||u||_{X^0} \int \faoned (\pa u)^2.\end{aligned}$$ Therefore, $|r| = 1$. We have two subcases below: \(a) [**The subcase $r = (1,0)$.**]{} In this case, $$\begin{aligned} \int \faoned (\pa u)(\partial^r u)(\partial^s u_x) & = \int \faoned (\pa u)u_x (\pa u) \\ & \leq C||u_x||_{L^\infty} \int \faoned (\pa u)^2.\end{aligned}$$ \(b) [**The subcase $r = (0,1)$.**]{} In this case, $$\begin{aligned} \int \faoned (\pa u)(\partial^r u)(\partial^s u_x) & = \int \faoned (\pa u)u_y (\partial_x^{\alpha+1} \partial_y^{\beta-2} u) \\ & \leq C||u_y||_{L^\infty} \left(\int \faoned (\pa u)^2\right)^{1/2} \left(\int \faoned (\partial_x ^{\alpha_1+1}\partial_y^{\alpha_2-1}u)^2\right)^{1/2} \\ & \leq C||u||_{X^1} \left[\int \faoned (\pa u)^2 + \int f_{\alpha+1,\delta}(\partial_x^{\alpha_1+1, \alpha_2-1}u)^2\right].\end{aligned}$$ We consider three subcases below: \(a) [**The subcase $r = (2,0)$.**]{} In this case $$\begin{aligned} \int \faoned (\pa u)(\partial^r u)(\partial^s u_x) & = \int \faoned (\pa u)u_{xx} (\partial_x^{\alpha_1-1} \partial_y^{\alpha_2} u) \\ & \leq C||\foned u_{xx}||_{L^\infty} \left(\int \faoned (\pa u)^2\right)^{1/2} \left(\int f_{\alpha_1-1,\delta}(\partial_x^{\alpha_1-1}\partial_y^{\alpha_2}u)^2\right)^{1/2}.\end{aligned}$$ Now $$\foned u_{xx} \approx \left\{ \begin{array}{ll} x u_{xx} & \quad x > 1 \\ u_{xx} & \quad x < -1. \end{array} \right.$$ For $x < -1$, we use the fact that $$\begin{aligned} ||u_{xx}||_{L^\infty(A_{-1})} & \leq C \left(\int_{A_{-1}} (u_{xx}^2 + u_{xxxx}^2 + u_{xxy}^2) \right)^{1/2} \\ & \leq C ||u||_{X^1}.\end{aligned}$$ For $x > 1$, we use the fact that $$\begin{aligned} \foned u_{xx} & = (\foned u)_{xx} - (\foned)_{xx} u - 2(\foned)_x u_x \\ & = (\foned u)_{xx} - (\foned)_{xx} u - 2 ((\foned)_xu)_x + 2 (\foned)_{xx} u.\end{aligned}$$ Therefore, $$\begin{aligned} ||\foned u_{xx} ||_{L^\infty} & \leq ||(\foned u)_{xx}||_{L^\infty} + C ||((\foned)_x u)_x||_{L^\infty} + C||(\foned)_{xx}u||_{L^\infty} \\ & \leq C \left(\int ((\foned u)_{xx})^2 + ((\foned u)_{xxxx})^2 + ((\foned u)_{xxy})^2\right)^{1/2} \\ & + C \left(\int (((\foned)_xu)_x)^2 + (((\foned)_x u)_{xxx})^2 + (((\foned)_x u)_{xy})^2 \right)^{1/2} \\ & + C\left(\int ((\foned)_{xx}u)^2 + (((\foned)_{xx}u)_{xx})^2 + (((\foned)_{xx}u)_y)^2 \right)^{1/2} \\ & \leq C + C\int u^2 + u_x^2 + \foned u_{xx}^2 + u_{xxx}^2 + \foned u_{xxxx}^2 + u_y^2 + u_{xy}^2 + \foned u_{xxy}^2 \\ & \leq C + C \sum_{|\gamma| \leq j} \int \fgd (\partial^\gamma u)^2.\end{aligned}$$ \(b) [**The subcase $r=(1,1)$.**]{} Then $$\begin{aligned} \int \faoned (\pa u)(\partial^r u)(\partial^su_x) & = \int \faoned (\pa u)u_{xy} (\partial_x^{\alpha_1} \partial_y^{\alpha_2-1} u) \\ & \leq ||u_{xy}||_{L^\infty} \left(\int \faoned (\pa u)^2\right)^{1/2} \left(\int \faoned (\partial_x^{\alpha_1} \partial_y^{\alpha_2-1}u)^2\right)^{1/2} \\ & \leq C||u||_{X^1} \left(\int \faoned (\pa u)^2\right)^{1/2} \left(\int \fad (\partial_x^{\alpha_1} \partial_y^{\alpha_2-1}u)^2\right)^{1/2} \\ & \leq C + C \sum_{|\gamma| \leq j} \int \fgd (\partial^\gamma u)^2\end{aligned}$$ where $C$ depends only on the bounds in the statement of the theorem. \(c) [**The subcase $r=(0,2)$.**]{} Then $$\begin{aligned} \int \faoned (\pa u)(\partial^r u)(\partial^su_x) & = \int \faoned (\pa u)u_{yy} (\partial_x^{\alpha_1+1}\partial_y^{\alpha_2-2}u) \\ & \leq ||u_{yy}||_{L^\infty} \left(\int \faoned (\pa u)^2\right)^{1/2} \left( \int \faoned (\partial_x^{\alpha_1+1} \partial_y^{\alpha_2-2} u)^2\right)^{1/2} \\ & \leq C ||u_{yy}||_{L^\infty} \left(\int \faoned (\pa u)^2\right)^{1/2} \\ & \leq C \left(\int u_{yy}^2 + u_{xxyy}^2 + u_{xyy}^2\right)^{1/2} \left(\int \faoned (\pa u)^2\right)^{1/2} \\ & \leq C + C\sum_{|\gamma| = j}\int \fgd (\partial^\gamma u)^2\end{aligned}$$ where $C$ depends only on the bounds in the statement of the theorem. In this case we consider $x > 1$ and $x < -1$ separately. First, for $x < -1$, we have $$\begin{aligned} \int_{A_{-1}} \faoned (\pa u)(\partial^r u)(\partial^s u_x) & \leq \int \left(\int_{A_{-1}} (\pa u)^2 \right)^{1/2} \left(\int_{A_{-1}} (\partial^r u)^4\right)^{1/4} \left(\int_{A_{-1}} (\partial^s u_x)^4\right)^{1/4}.\end{aligned}$$ Now $$\left(\int_{A_{-1}} (\partial^r u)^4\right)^{1/4} \leq \left(\int_{A_{-1}} (\partial^r u)^2 + (\partial^r u_x)^2 + (\partial^r u_y)^2\right)^{1/2}$$ and $|r| = 3$ implies each of these terms is bounded by $C$ where $C$ depends only on $$\int \fgd (\partial^\gamma u)^2 \qquad \text{for } |\gamma| \leq j-1$$ since $j \geq 5$ and each of these terms has derivatives of order $\leq 4$. Also, $$\left(\int_{A_{-1}} (\partial^s u_x)^4 \right)^{1/4} \leq \left(\int_{A_{-1}} (\partial^su_x)^2 + (\partial^su_{xx})^2 + (\partial^s u_{xy})^2 \right)^{1/2}$$ and $|s| = j-3$. Therefore, each of these terms has order at most $j-1$ and thus bounded by $$C \sum_{|\gamma| \leq j-1} \int \fgd (\partial^\gamma u)^2.$$ Therefore, for $x < -1$, we have $$\int_{A_{-1}} \faoned (\pa u)(\partial^r u)(\partial^s u_x) \leq C + C \int_{A_{-1}} \faoned (\pa u)^2,$$ where $C$ depends only on the terms in the statement of the theorem. Now for $x > 1$, we have $$\begin{aligned} \int_{A_1} \faoned (\pa u)(\partial^r u)(\partial^s u_x) & \leq \left(\int_{A_1} \faoned (\pa u)^2 \right)^{1/2} \left(\int_{A_1} (f_{r_1/2,\delta} \partial^ru)^4\right)^{1/4} \\ & \times \left(\int_{A_1} (f_{(s_1+1)/2,\delta} \partial^su_x)^4\right)^{1/4}.\end{aligned}$$ since $r_1 + s_1 = \alpha_1$. Now $$\begin{aligned} \left(\int_{A_1} (f_{r_1/2,\delta} \partial^r u)^4 \right)^{1/4} & \leq \left(\int_{A_1} (f_{r_1/2,\delta} \partial^r u)^2 + ((f_{r_1/2,\delta} \partial^ru)_x)^2 + ((f_{r_1/2,\delta} \partial^r u)_y)^2\right)^{1/2} \\ & \leq C \left(\int_{A_1} f_{r_1,\delta} (\partial^ru)^2 + f_{r_1,\delta} (\partial^r u_x)^2 + f_{r_1,\delta}(\partial^ru_y)^2\right)^{1/2}.\end{aligned}$$ Since $|r| = 3$, each of the terms above has order at most four, and, therefore, is bounded by a constant $C$ which depends only on $$\int \fgd (\partial^\gamma u)^2 \qquad \text{for } |\gamma| \leq 4 \leq j-1,$$ since here we are assuming $j \geq 5$. Similarly, $$\begin{aligned} & \left(\int (f_{(s_1+1)/2,\delta}(\partial^s u_x))^4\right)^{1/4} \\ & \leq C \left(\int (f_{(s_1+1)/2,\delta}(\partial^s u_x))^2+((f_{(s_1+1)/2,\delta}(\partial^su_x))_x)^2 + ((f_{(s_1+1)/2,\delta}(\partial^su_x))_y)^2\right)^{1/2} \\ & \leq C \left(\int f_{s_1+1,\delta}[(\partial^su_x)^2 + (\partial^su_{xx})^2+ (\partial^s u_{xy})^2]\right)^{1/2}.\end{aligned}$$ Since $|s| = j-3$ each of these terms is of order at most $j-1$. Therefore, each of these terms is bounded by $$C \sum_{|\gamma| \leq j-1} \int \fgd (\partial^\gamma u)^2.$$ In this case, $|s| = 1$. Therefore, either $s = (1,0)$ or $s = (0,1)$. For $s = (1,0)$, we have $$\begin{aligned} \int \faoned (\pa u)(\partial^r u) (\partial^s u_x) & = \int \faoned (\pa u)(\partial^r u) u_{xx} \\ & \leq \left(\int \faoned (\pa u)^2\right)^{1/2} \left(\int (f_{r_1/2,\delta} \partial^r u)^4 \right)^{1/4} \left(\int (\foned u_{xx})^4\right)^{1/4}.\end{aligned}$$ Now $$\begin{aligned} \left(\int (f_{r_1/2,\delta}(\partial^r u))^4\right)^{1/4} & \leq \left(\int f_{r_1,\delta} [(\partial^r u)^2+ (\partial^r u_x)^2 + (\partial^r u_y)^2]\right)^{1/2}.\end{aligned}$$ We note that $|r| = 3$. Therefore, each of these terms is at most of order $4 = j$. Further, $$\begin{aligned} \left(\int (\foned u_{xx})^4\right)^{1/4} & \leq C \left(\int \ftwod[u_{xx}^2 + u_{xxx}^2 + u_{xxy}^2]\right)^{1/2}.\end{aligned}$$ We note that each of these terms is of order at most $j-1$. Combining these estimates, we conclude that $$\int \faoned (\pa u)(\partial^r u) (\partial^s u_x) \leq C + C \sum_{|\gamma| = j} \int \fgd (\partial^\gamma u)^2,$$ where $C$ depends only on $$\sum_{|\gamma| \leq j-1} \int \fgd (\partial^\gamma u)^2.$$ The case in which $s = (0,1)$ is handled similarly. In this case, we bound the terms as follows: $$\begin{aligned} \int \faoned (\pa u)(\partial^r u)(\partial^s u_x) & \leq ||f_{(s_1+1)/2,\delta}\partial^su_x ||_{L^\infty} \left(\int \faoned(\pa u)^2\right)^{1/2} \left(\int f_{r_1,\delta} (\partial^r u)^2 \right)^{1/2}.\end{aligned}$$ Now $$\begin{aligned} ||f_{(s_1+1)/2,\delta}(\partial^su_x)||_{L^\infty} & \leq C\left(\int f_{s_1+1,\delta}[ (\partial^s u_x)^2 + (\partial^s u_{xxx})^2 + (\partial^s u_{xy})^2]\right)^{1/2}.\end{aligned}$$ Since $|s| \leq j-4$, all of these terms are of order at most $j-1$. Therefore, each of these terms is bounded by $$C \sum_{|\gamma| \leq j-1} \int \fgd (\partial^\gamma u)^2$$ and, therefore, $$\int \faoned (\pa u) (\partial^r u)(\partial^s u_x) \leq C + C \sum_{|\gamma| = j} \int \fgd (\partial^\gamma u)^2.$$ where $C$ depends only on $$C \sum_{|\gamma| \leq j-1} \int \fgd (\partial^\gamma u)^2.$$$\square$ Main Theorem ============ In this section we state and prove our main theorem, which states that if the initial data $\phi$ possesses certain regularity and sufficient decay at infinity, then the solution $u(t)$ will be smoother than $\phi$. In particular if the initial data satisfies $$\int \phi^2 + (1 + x_+^L) (\partial_x^L \phi)^2 + (\partial_y^L \phi)^2 < \infty,$$ then the solution will [*gain*]{} $L$ derivatives in $x$. More specifically, $$\int_0^T \int t^{L-1}(1+e^{\sigma x_-}) (\partial_x^{2L} u)^2 < \infty$$ for $\sigma > 0$ arbitrary, where $T$ is the existence time of the solution.\ (Main Theorem). [*Let $T>0$ and let $u$ be the solution of in the region $[0,\,T]\times\mathbb{R}^{2}$ such that $u\in L^{\infty}([0,\,T]:\,{\cal Z}_{L})$ for some $L\geq 2.$ Then*]{} $$\begin{aligned} \label{e701}\sup_{0\leq t\leq T}\int_{\mathbb{R}^{2}}f_\alpha\,(\pa u)^{2}\,dx\,dy + \int_{0}^{T}\int_{\mathbb{R}^{2}}g_\alpha\,(\pa u_x)^{2}\,dx\,dy\,dt \leq C\end{aligned}$$ [*for $L+1\leq |\alpha| \leq 2L - 1$, $2L-|\alpha|-\alpha_2 \geq 1$ where $f_\alpha \in W_{\sigma,\;2L - |\alpha|-\alpha_2,\;|\alpha|-L}$ and $g_\alpha\in W_{\sigma,\;2L - |\alpha| - \alpha_2 - 1,\;|\alpha|-L},$ $\sigma>0$ arbitrary.*]{}\ \ [*Proof.*]{} By assumption, $u\in L^{\infty}([0,\,T]:\,{\cal Z}_{L}).$ Therefore $u_{t}\in L^{\infty}([0,\,T]:\,L^{2}(\mathbb{R}^{2})),$ then $u\in C([0,\,T]:\,L^{2}(\mathbb{R}^{2}))\cap C_{w}([0,\,T]:\,{\cal Z}_{L}).$ Hence $u:[0,\,T]\rightarrow {\cal Z}_{L}$ is a weakly continuous function. In particular, $u(\,\cdot\,,\,\cdot\,,t)\in {\cal Z}_{L}$ for every $t.$ Let $t_{0}\in (0,\,T)$ and $u(\,\cdot\,,\,\cdot\,,t_{0})\in {\cal Z}_{L},$ then there are $\{\phi^{(n)}\}\subseteq C_{0}^{\infty}(\mathbb{R}^{2})$ such that $\partial_{x}^{-1}\phi_{yy}^{(n)}$ are in $C_{0}^{\infty}(\mathbb{R}^{2})$ and $\phi^{(n)}(\,\cdot\,,\,\cdot\,)\rightarrow u(\,\cdot\,,\,\cdot\,,t_{0})$ in ${\cal Z}_L.$ Let $u^{(n)}$ be the unique solution of with initial data $\phi^{(n)}(x,\,y)$ at time $t=t_{0}.$ By Corollary 4.4, the solution $u^{(n)}\in L^{\infty}([t_{0},\,t_{0} + \delta]:\,X^{1}(\mathbb{R}^{2}))$ for a time interval $\delta$ which not depend on $n.$ By Theorem 6.1, $u^{(n)}\in L^{\infty}([t_{0},\,t_{0} + \delta]:\,{\cal Z}_L)$ and $$\begin{aligned} \label{e702}\int_{t_{0}}^{t_{0} + \delta}\int_{\mathbb{R}^{2}}g_{\alpha_1}\,(\pa u_x)^{2}\,dx\,dy\,dt\leq C\end{aligned}$$ for $|\alpha|=L$, $\alpha_1 \neq 0$, where $g_{\alpha_1}\in W_{\sigma,\;\alpha_1 - 1,\;0}$ and $C$ depends only on the norm of $\phi^{(n)}\in {\cal Z}_{L}.$ Also by Theorem 6.1, we have (non-uniform) bounds on $$\begin{aligned} \label{e703}\sup_{[t_{0},\,t_{0} + \delta]}\sup_{(x,y)}\left[(1 + x^{+})^{k}|\partial_x^{\alpha_1} u^{(n)}(x,\,y,\,t)| + |\partial_y^{\alpha_2} u^{(n)}(x,\,y,\,t)|\right] <+\infty\end{aligned}$$ for each $n,\,k$ and $\alpha_1,\,\alpha_2.$ Therefore, the a priori estimates in Lemma 5.1, are justified for each $u^{(n)}$ in the interval $[t_{0},\,t_{0} + \delta].$\ \ We start our induction with $|\alpha|=L+1$. In this case, we take $g_\alpha \in W_{\sigma,\;L - 2 - \alpha_2,\;1}$ and let $f_\alpha =\frac{1}{3}\int_{-\infty}^{x} g_\alpha(z,\,t)\,dz.$ We note that $2L-|\alpha|-\alpha_2 \geq 1$ by assumption. Therefore, $L-2-\alpha_2 \geq 0$. As shown in Lemma 5.1, we have the following bounds on the higher derivatives of $u^{(n)},$ $$\begin{aligned} \label{e704}\sup_{[t_{0},\,t_{0} + \delta]}\int_{\mathbb{R}^{2}}f_\alpha \,(\pa u^{(n)})^{2}\,dx\,dy + \int_{t_{0}}^{t_{0} + \delta}\int_{\mathbb{R}^{2}}g_\alpha \,(\pa u^{(n)}_x)^{2}\,dx\,dy\,dt\leq C\end{aligned}$$ where $C$ depends only on the norm of $u^{(n)}\in L^{\infty}([0,\,T]:\,{\cal Z}_{L})$ and the term in . We conclude, therefore, that the constant $C$ in depends only on $||\phi^{(n)}||_{{\cal Z}_{L}}.$ We continue this procedure inductively. For the $|\alpha|^{th}$ step, let $g_\alpha \in W_{\sigma,\;2L - |\alpha| -\alpha_2 - 1,\;|\alpha|-L}$ for $\alpha_2 \leq 2L-|\alpha|-1$ and define $f_\alpha =\frac{1}{3}\int_{-\infty}^{x}g_\alpha(z,\,t)\,dz.$ The non-uniform bounds on $u^{(n)}$ in allows us to use Lemma 5.1 and our inductive hypothesis to conclude that $$\begin{aligned} \sup_{[t_{0},\,t_{0} + \delta]}\int_{\mathbb{R}^{2}}f_\alpha \,(\pa u^{(n)})^{2}\,dx\,dy + \int_{t_{0}}^{t_{0} + \delta}\int_{\mathbb{R}^{2}}g_\alpha \,(\pa u^{(n)}_x)^{2}\,dx\,dy\,dt\leq C\end{aligned}$$ where again $C$ does not depend on $n,$ but only on the norm of $\phi^{(n)}\in {\cal Z}_{L}.$ By Corollary 4.5, $$\begin{aligned} u^{(n)}\stackrel{*}\rightharpoonup u\quad \mbox{weakly in}\quad L^{\infty}([t_{0},\,t_{0} + \delta]:\,X^{1}(\mathbb{R}^{2})).\end{aligned}$$ Therefore, we can pass to the limit and conclude that $$\begin{aligned} \label{e706}\sup_{[t_{0},\,t_{0} + \delta]}\int_{\mathbb{R}^{2}}f_\alpha \,(\pa u)^{2}\,dx\,dy + \int_{t_{0}}^{t_{0} + \delta}\int_{\mathbb{R}^{2}}g_\alpha \,(\pa u)^{2}\,dx\,dy\,dt\leq C.\end{aligned}$$ This proof is continued inductively up to $|\alpha| =2L - 1.$ Since $\delta$ does not depend on $n,$ this result is valid over the whole interval $[0,\,T].$ $\square$ [00]{} M.J. 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--- abstract: 'We investigate Monte Carlo Markov Chain (MCMC) procedures for the random sampling of some one-dimensional lattice paths with constraints, for various constraints. We will see that an approach inspired by *optimal transport* allows us to efficiently bound the mixing time of the associated Markov chain. The algorithm is robust and easy to implement, and samples an “almost” uniform path of length $n$ in $n^{3+{\varepsilon}}$ steps. This bound makes use of a certain *contraction property* of the Markov chain, and is also used to derive a bound for the running time of Propp-Wilson’s *Coupling From The Past* algorithm.' author: - Lucas Gerin title: 'Random sampling of lattice paths with constraints, via transportation' --- Lattice Paths with Constraints ============================== Lattice paths arise in several areas in probability and combinatorics, either in their own interest (as realizations of random walks, or because of their interesting combinatorial properties: see [@Ban] for the latter) or because of fruitful bijections with various families of trees, tilings, words. The problem we discuss here is to efficiently sample uniform (or *almost* uniform) paths in a family of paths with constraints. There are several reasons for which one may want to generate uniform samples of lattice paths: to make and try conjectures on the behaviour of a large “typical” path, test algorithms running on paths (or words, trees,...). In view of random sampling, it is often very efficient to make use of the combinatorial structure of the family of paths under study. In some cases, this yields linear-time (in the length of the path) *ad-hoc* algorithms [@MBM; @Duc]. However, the nature of the constraints makes sometimes impossible such an approach, and there is a need for robust algorithms that work in lack of combinatorial knowledge. Luby,Randall and Sinclair [@LRS] design a Markov chain that generate sets of non-intersecting lattice paths. This was motivated by a classical (and simple, see illustrations in [@Des; @Wilson]) correspondence between dimer configurations on an hexagon, rhombae tilings of this hexagon and families of non-intersecting lattice paths. As the first step for the analysis of this chain, Wilson [@Wilson] introduces a peak/valley Markov chain (see details below) over some simple lattice paths and obtain sharp bounds for its mixing time. We present in this paper a variant of this Markov chain, which is valid for various constraints and whose analysis is simple. It generates an “almost” uniform path of length $n$ in $n^{3+{\varepsilon}}$ steps, this bound makes use of a certain *contraction property* of the chain. Appart from the algorithmic aspect, the peak/valley process seems to have a physical relevancy as a simplified model for the evolution of *quasicrystals* (see a discussion on a related process in the introduction of [@Des]). In particular, the mixing time of this Markov seems to have some importance. Notations {#notations .unnumbered} --------- {width="40mm"} We fix three integers $n,a,b>0$, and consider the paths of length $n$, with steps $+a/-b$, that is, the words of $n$ letters taken in the alphabet ${\left\{a,-b\right\}}$. Such a word $s=(s_1,s_2,\dots,s_n)$ is identified to the path $S=(S_1,\dots,S_n):=(s_1,s_1+s_2,\dots,s_1+s_2+\dots +s_n)$. To illustrate the methods and the results, we focus on some particular sub-families ${\mathcal{A}_{n}}\subset {\left\{a,-b\right\}}^n$: 1. Discrete *meanders*, denoted by ${\mathcal{M}_{n}}$, which are simply the non-negative paths: $S\in{\mathcal{M}_{n}}$ if for any $i\leq n$ we have $S_i\geq 0$. This example is mainly illustrative because the combinatorial properties of meanders make it possible to perform exact sampling very efficiently (an algorithm running in $\mathcal{O}(n^{1+{\varepsilon}})$ steps is given in [@MBM], an order that we cannot get in the present paper). 2. Paths with *walls*. A path with a wall of height $h$ between $r$ and $s$ is a path such that $S_i\geq h$ for any $r\leq i\leq s$ (see Fig. \[Fig:CheminMur\] for an example). These are denoted by ${\mathcal{W}_{n}}={\mathcal{W}_{n}}(h,r,s)$, they appear in statistical mechanics as toy models for the analysis of random interfaces and polymers (see examples in [@Walls]). 3. *Excursions*, denoted by ${\mathcal{E}_{n}}$, which are non-negative paths such that $S_n=0$. In the case $a=b=1$, these correspond to well-parenthesed words and are usually called Dyck words. In the general case, Duchon [@Duc] proposes a rejection algorithm which generates excursions in linear time. 4. *Culminating paths* of size $n$, denoted further by ${\mathcal{C}_{n}}$, which are non-negative paths whose maximum is attained at the last step: for any $i$ we have $0\leq S_i\leq S_n$. They have been introduced in [@MBM], motivated in particular by the analysis of some algorithms in bioinformatics. ![A path of steps $+1/-2$, with a wall of height $h=6$ between $i=10$ and $j=15$.[]{data-label="Fig:CheminMur"}](CheminMur.eps){width="65mm"} Sampling with Markov chains {#Sec:Sampling} =========================== We will consider Markov chains in a family ${\mathcal{A}_{n}}$, where all the probability transitions are symmetric. For a modern introduction to Markov chains, we refer to [@Hagg]. Hence we are given a transition matrix $(p_{i,j})$ of size $|{\mathcal{A}_{n}}|\times|{\mathcal{A}_{n}}|$ with $$\begin{aligned} p_{i,j} &=p_{j,i} \mbox{ whenever }i\neq j,\\ p_{i,i} &= 1-\sum_{j\neq i}p_{i,j}.\end{aligned}$$ \[Lem:Unif\] If such a Markov chain is irreducible, then it admits as unique stationary distribution the uniform distribution $\pi=\pi({\mathcal{A}_{n}})$ on ${\mathcal{A}_{n}}$. The equality $\pi(i) p_{i,j}= \pi(j) p_{j,i}$ holds for any two vertices $i,j$. This shows that the probability distribution $\pi$ is reversible for $(p_{i,j})$, and hence stationary. It is unique if the chain is irreducible. This lemma already provides us with a scheme for sampling an almost uniform path in ${\mathcal{A}_{n}}$, without knowing much about ${\mathcal{A}_{n}}$. To do so, we define a “flip” operator on paths, this is an operator $$\begin{array}{r c c c} \phi: & {\mathcal{A}_{n}}\times {\left\{1,\dots,n\right\}}\times {\left\{\downarrow,\uparrow\right\}}\times {\left\{+,-\right\}} &\to &{\mathcal{A}_{n}}\\ & (\mathbf{S},i,{\varepsilon},\delta) &\mapsto & \phi(\mathbf{S},i,{\varepsilon},\delta). \end{array}$$ When $i\in{\left\{1,2,\dots,n-1\right\}}$ the path $\phi(\mathbf{S},i,\uparrow,\delta)$ is defined as follows : if $(s_i,s_{i+1})=(-b,a)=$ {width="7mm"} then these two steps are changed into $(a,-b)=$ {width="7mm"}. The $n-2$ other steps remain unchanged. If $(s_i,s_{i+1})\neq (-b,a)$ then ${\phi(\mathbf{S},i,\uparrow)}{\delta}=\mathbf{S}$. Note that in the case $i\in{\left\{1,2,\dots,n-1\right\}}$ the value of $\phi$ does not depend on $\delta$. For the case $i=n$, if $\delta=+$, we define ${\phi(\mathbf{S},n,{\varepsilon})}{\delta}$ as before as if there would be a $+a$ as the end if the path. For instance, in the case where $S_n=-b$, the path ${\phi(\mathbf{S},n,\uparrow)}{+}$, the $n$-th step is turned into $a$. The path ${\phi(\mathbf{S},i,\downarrow)}{\delta}$ is defined equally: if $i<n$ and $(s_i,s_{i+1})=$ {width="7mm"}, it turns into {width="7mm"}. When $\delta=-$, one flips as if there would be a $-b$ at the end of the path. For culminating paths, we have to take another definition of ${\phi(\mathbf{S},n,\uparrow)}{\delta},{\phi(\mathbf{S},n,\downarrow)}{\delta}$, see Section \[Sec:Analysis\]. We are also given a probability distribution $\mathbf{p}=(p_i)_{1\leq i\leq n}$, and we assume that $p_i>0$ for each $i$. We will consider a particular sequence $\mathbf{p}$ later on, but at this point we can take the uniform distribution in ${\left\{1,\dots,n\right\}}$. We describe the algorithm below in Algorithm \[Algo:CM\]. initialize $\mathbf{S}=(+a,+a,+a,\dots,+a)$ $I_{1},I_{2},\dots\leftarrow$ i.i.d. r.v. with law $\mathbf{p}$ ${\varepsilon}_{1},{\varepsilon}_{2},\dots\leftarrow$ i.i.d. uniform r.v. in ${\left\{\uparrow,\downarrow\right\}}$ $\delta_{1},\delta_{2},\dots\leftarrow$ i.i.d. uniform r.v. in ${\left\{+,-\right\}}$ $\mathbf{S}\leftarrow {\phi(\mathbf{S},I_t,{\varepsilon}_t)}{\delta_t}$ In words, this algorithm performs the Markov chain in ${\mathcal{A}_{n}}$ with transition matrix $P=\left(P_{\mathbf{R},\mathbf{S}}\right)_{\mathbf{R},\mathbf{S}\in{\mathcal{A}_{n}}}$ defined as follows: $$\begin{cases} P_{\mathbf{R},\mathbf{S}}&=p_i/2, \mbox{ if } \mathbf{S}\neq\mathbf{R}\text{ and } \mathbf{S}={\phi(\mathbf{R},i,{\varepsilon})}{\delta} \mbox{ for some }i,{\varepsilon}, \delta\\ P_{\mathbf{R},\mathbf{S}}&=0\text{ otherwise,}\\ P_{\mathbf{R},\mathbf{R}}&=1-\sum_{\mathbf{S}\neq \mathbf{R}} P_{\mathbf{R},\mathbf{S}}.\\ \end{cases}$$ Denote by $S(t)$ the random path obtained after the $t$-th run of the loop in Algorithm \[Algo:CM\]. When $t\to\infty$, the sequence $S(t)$ converges in law to the uniform distribution in ${\mathcal{A}_{n}}$. Moreover, the execution of Algorithm \[Algo:CM\] until time $T$ is linear in $T$. For the first claim, we have to check that the chain is aperiodic and irreducible. Aperiodicity comes from the (many) loops. Irreducibility will follow from Lemma \[Lem:Geodesique\]. For the second claim, notice that the time needed for the test “${\phi(\mathbf{S},I_t,{\varepsilon}_t)}$ is in ${\mathcal{A}_{n}}$” can be considered as constant, since for the families ${\mathcal{M}_{n}}$ and ${\mathcal{E}_{n}}$ we only have to compare $0,S_i$ while for the family ${\mathcal{W}_{n}}$ we only have to compare $S_i$ with the height of the wall at $i$. For the case of the culminating paths, see below in Section \[Sec:Analysis\]. We now choose the distribution $(p_i)$. Instead of $p_i=1/n$, we will use the probability distribution defined by$$\label{Eq:Poids} p_i:=i(2n-i)\kappa_0 +a\quad (\mbox{ for }i=1,\dots,n),$$ where $$\begin{aligned} \kappa_0&=\frac{3}{2n^2(n+1)}\\ a &=1/4n^3.\end{aligned}$$ We let the reader check that $(p_i)_{i\leq n}$ is indeed a probability distribution. The reason for which we use this particular distribution will appear in the proof of Proposition \[Lem:Courbure\]. We will then need the following relation: for each $1\leq i\leq n-1$, $$\label{Eq:kappa} p_i-p_{i-1}/2-p_{i+1}/2 = \kappa_0.$$ For Algorithm \[Algo:CM\] to be efficient, we need to know how $S(T)$ is close in law to $\pi$. This question is related to the spectral properties of the matrix $P$. In particular, the speed of convergence is governed by the spectral gap ([[*i.e.*]{} ]{}$1-\lambda$, where $\lambda$ is the largest of the modulus of the eigenvalues different from one, see [@Mix] for example), but this quantity is not known in general. Some geometrical methods [@Dia] allow to bound from below $1-\lambda$, but they assume a precise knowledge of the structure of the graph defined by the chain $P$. It seems that such results do not apply here. Instead, we will study the metric properties of the chain $P$ with respect to a natural distance on ${\mathcal{A}_{n}}$, and show that it satisfies a certain *contraction property*. The variant of Algorithm \[Algo:CM\] for culminating paths {#Sec:Analysis} ---------------------------------------------------------- Unchanged, our Markov chain $P$ cannot generate culminating paths since the path $\mathbf{S}=(a,a,\dots,a)$ would then be isolated: it has no peak/valley and ${\phi(\mathbf{S},n,\downarrow)}{-}=(a,a,\dots,-b)$ which is not culminating. Thus we propose a slight modification for the family ${\mathcal{C}_{n}}$. We only change the definition of ${\phi(\mathbf{S},i,{\varepsilon})}{\delta}$ when $i=n$ (it won’t depend on $\delta$). Since the maximum is reached at $n$, the $\lceil b/a\rceil +1$ last steps are necessarily $$(a,a,\dots,a) \mbox{ or } (-b,a,\dots,a).$$ We thus define ${\phi(\mathbf{S},n,\uparrow)}{\delta}$ as the path obtained by changing the $\lceil b/a\rceil +1$ last steps into $(a,a,\dots,a)$ (regardless of their initial values in $\mathbf{S}$) and ${\phi(\mathbf{S},n,\downarrow)}{\delta}$ as the path obtained by changing the $\lceil b/a\rceil +1$ last steps into $(-b,a,\dots,a)$. Notice that despite this change the execution time of each loop of Algorithm \[Algo:CM\] is still a $\mathcal{O}(1)$: - If $I_t< n$, the time needed for the test “${\phi(\mathbf{S},I_t,{\varepsilon}_t)}{\delta_t}$ is in ${\mathcal{A}_{n}}$” can be considered as constant, since we only have to compare $0,S_i,S_n$. - If $I_t=n$, the new value $S_n$ is compared with the maximum of $S$, which can be done in $\mathcal{O}(n)$. Fortunately, this occurs with probability $p_n=\mathcal{O}(1/n)$, so that the time-complexity of each loop is, on average, a $\mathcal{O}(1)$. Error estimates with contraction {#Sec:Ricci} ================================ Going back to a more general setting, we consider a Markov chain in a finite set $V$, endowed with a metric $d$. For a vertice $x\in V$ and a transition matrix $P$, we denote by $P\delta_x$ (resp. $P^t\delta_x$) the law of the Markov chain associated with $P$ at time $1$ (resp. $t$), when starting from $x$. For $x,y\in V$, the main assumption made on $P$ is that there is a coupling between $P\delta_x,P\delta_y$ (that is, a random variable $(X_1,Y_1)$ with $X_1\stackrel{\mbox{law}}=P\delta_x,Y_1\stackrel{\mbox{law}}=P\delta_y$) such that $$\label{Eq:Courbure} \mathbb{E}\left[d(X_1,Y_1)\right]\leq (1-\kappa)d(x,y),$$ for some $\kappa >0$, which is called the *Ricci curvature* of the chain, by analogy with the Ricci curvature in differential geometry[^1]. If the inequality holds, then it implies ([@Mix],p.189) that $P$ admits a unique stationary measure $\pi$ and that, for any $x$, $$\label{Eq:Mixing} \parallel P^t\delta_x -\pi\parallel_{\mathrm{TV}} \leq (1-\kappa)^t\mathrm{diam}(V),$$ where $\mathrm{diam}(V)$ is the diameter of the graph with vertices $V$ induced by the Markov chain. The notation $\parallel .\parallel_{\mathrm{TV}}$ stands, as usual, for the *Total Variation* distance over the probability distributions on $V$ defined by $$\parallel \mu_1 -\mu_2\parallel_{\mathrm{TV}}:= \sup_{A\subset V} \left|\mu_1(A)-\mu_2(A)\right|.$$ Hence, a positive Ricci curvature gives the exponential convergence to the stationary measure, with an exact ([[*i.e.*]{} ]{} is non-asymptotic in $t$) bound. In many situations, a smart choice for the coupling between $X_1,X_2$ gives a sharp rate of convergence in eq. (see some striking examples in [@Olli]). Metric properties of $P$ ------------------------ To apply the Ricci curvature machinery, we endow each ${\mathcal{A}_{n}}$ with the $L^1$-distance $$d_1(S,S')=\frac{1}{a+b}\sum_{i=0}^n |S_i-S_i'|.$$ (Notice that $|S_i-S_i'|$ is always a multiple of $a+b$.) For our purpose, it is fundamental that this metric space is *geodesic*. A Markov chain $P$ in a finite set $V$ is said to be *geodesic* with respect to the distance $d$ on $V$ if for any two $x,y\in V$ with $d(x,y)=k$, there exist $k+1$ vertices $x_0=x,x_1,\dots,x_k=y$ in $V$ such that for each $i$ - $d(x_i,x_{i+1})=1$ ; - $x_i$ and $x_{i+1}$ are neighbours in the Markov chain $P$ ([[*i.e.*]{} ]{}$P(x_i,x_{i+1})>0$). This implies in particular that $P$ is irreducible and that the diameter of $P$ is smaller than $\max_{x,y}d(x,y)$. \[Lem:Geodesique\] For each family ${\mathcal{C}_{n}}$,${\mathcal{W}_{n}}$,${\mathcal{E}_{n}}$,${\mathcal{M}_{n}}$ the Markov chain of Algorithm \[Algo:CM\] is geodesic with respect to $d_1$. The proof goes by induction on $k$. We fix $S\neq T$ (and denote by $s,t$ the corresponding words) ; we want to decrease $d_1(S,T)$ by one, by applying the operator $\phi$ with proper $i,{\varepsilon}$. We denote by $i_0\in{\left\{1,\dots,n\right\}}$ the first index for which $S\neq T$. For instance we have $T_{i_0}=S_{i_0}+a+b$. Let $j$ be the position of the left-most peak in $T$ in ${\left\{i_0+1,i_0+2,\dots,n\right\}}$, if such a peak exists. Then $S':={\phi(\mathbf{T},j,\downarrow)}{\delta}$ is also in ${\mathcal{A}_{n}}$: it is immediate for the families ${\mathcal{M}_{n}},{\mathcal{W}_{n}},{\mathcal{C}_{n}},{\mathcal{E}_{n}}$. We have $d_1(S,S')=k-1$. If there is no peak in $T$ after $i_0$, then $(t_{i_0+1},t_{i_0+2},\dots,t_n)=(a,a,\dots,a)$. Hence we try to increase the final steps of $S$ by one. To do so, we choose $S':={\phi(\mathbf{S},n,\uparrow)}{\delta}$ if $S\neq {\phi(\mathbf{S},n,\uparrow)}{\delta}$, or $S'={\phi(\mathbf{S},j,\uparrow)}{\delta}$ where $j$ is the position of the right-most $-b$ otherwise (we choose the right-most one to ensure that ${\phi(\mathbf{S},j,\uparrow)}{\delta}$ remains culminating in the case where ${\mathcal{A}_{n}}={\mathcal{C}_{n}}$.). For meanders, excursions and walls, we will show that the Ricci curvature of $P$ with respect to the distance $d_1$ is (at least) of order $1/n^3$. \[Lem:Courbure\] For the three families ${\mathcal{M}_{n}},{\mathcal{E}_{n}},{\mathcal{W}_{n}}$, the Ricci curvature of the associated Markov chain, with weights $(p_i)$ defined as in , is larger than $\kappa_0$. Fix $\mathbf{S},\mathbf{T}$ in ${\mathcal{A}_{n}}\in{\left\{{\mathcal{M}_{n}},{\mathcal{E}_{n}},{\mathcal{W}_{n}}\right\}}$, we first assume that $\mathbf{S},\mathbf{T}$ are neighbours, for instance $\mathbf{T}={\phi(\mathbf{S},i,\uparrow)}$ for some $i$. {width="35mm"} Let $(\mathbf{S}^1,\mathbf{S}^2)$ be the random variable in ${\mathcal{A}_{n}}\times{\mathcal{A}_{n}}$ whose law is defined by $$(\mathbf{S}^1,\mathbf{S}^2)\stackrel{\mbox{(law)}}{=} \left(\phi(\mathbf{S},\mathcal{I},{E}),\phi(\mathbf{T},\mathcal{I},E)\right),$$ where $\mathcal{I}$ is a r.v. taking values in ${\left\{1,\dots,n\right\}}$ with distribution $\mathbf{p}$ and $E$ is uniform in ${\left\{\uparrow,\downarrow\right\}}$. In other words, we run one loop of Algorithm \[Algo:CM\] simultaneously on both paths. We want to show that $\mathbf{S}^1,\mathbf{S}^2$ are, on average, closer than $\mathbf{S},\mathbf{T}$. Different cases may occur, depending on $\mathcal{I}$ and on the index $i$ where $\mathbf{S},\mathbf{T}$ differ. [**  **]{} [**  **]{} This occurs with probability $p_i$ and, no matter the value of $E$, we have $\mathbf{S}^1=\mathbf{S}^2$. [**  **]{} We consider the case $i-1$. Since $\mathbf{S}$ and $\mathbf{T}$ coincide everywhere but in $i$, we necessarily have one of these two cases: - there is a peak in $\mathbf{S}$ at $i-1$ and neither a peak nor a valley in $\mathbf{T}$ at $i-1$ (as in the figure on the right) ; - there is a valley in $\mathbf{T}$ at $i-1$ and neither a peak nor a valley in $\mathbf{S}$ at $i-1$. In the first case for instance, then we may have $d_1(\mathbf{S}^1,\mathbf{S}^2)=2$ if $E=\downarrow$, while the distance remains unchanged if $E=\uparrow$. The case $\mathcal{I}=i+1$ is identical. This shows that with a probability smaller than $p_{i-1}/2+p_{i+1}/2$ we have $d_1(\mathbf{S}^1,\mathbf{S}^2)=2$. [**  **]{} In this case, $\mathbf{S}$ and $\mathbf{T}$ are possibly modified in $\mathcal{I}$, but if there is a modification it occurs in both paths. It is immediate since for the families ${\mathcal{M}_{n}}$,${\mathcal{W}_{n}}$ and ${\mathcal{E}_{n}}$ since the constraints are local. [**  **]{} In this case, it is easy to check that, because of our definition of ${\phi(\mathbf{S},n,{\varepsilon})}{\delta}$, we have $$\mathbb{E}\left[d_1(\mathbf{S}^1,\mathbf{S}^2)\right] \leq 1-p_{n-1}+p_{n-2}/2+p_{n}/2 =1-\kappa_0.$$ [**  **]{} We have $$\mathbb{E}\left[d_1(\mathbf{S}^1,\mathbf{S}^2)\right] \leq 1+p_{n-1}/2-p_{n}/2=1-\kappa_0.$$ Thus, we have proven that when $\mathbf{S},\mathbf{T}$ only differ at $i$ $$\begin{aligned} \mathbb{E}\left[d_1(\mathbf{S}^1,\mathbf{S}^2)\right] &\leq 2\times(p_{i-1}/2+p_{i+1}/2) +0\times p_i+1\times(1-p_i-p_{i-1}/2-p_{i+1}/2)\label{Eq:Accroissement}\\ &\leq (1-\kappa_0)\times 1=(1-\kappa_0)d_1(S,T)\notag.\end{aligned}$$ What makes Ricci curvature very useful is that if this inequality holds for pairs of neighbours then it holds for any pair, as noticed in [@Bub]. Indeed, take $k+1$ paths $S_0=S,S_1,\dots,S_k=T$ as in Lemma \[Lem:Geodesique\] and apply the triangular inequality for $d_1$: $$\begin{aligned} \mathbb{E}\left[d_1(\phi(S,\mathcal{I},{E}),\phi(T,\mathcal{I},E))\right] &\leq \sum_{i=0}^{k-1} \mathbb{E}\left[d_1(\phi(S_i,\mathcal{I},{E}),\phi(S_{i+1},\mathcal{I},E))\right]\\ &\leq (1-\kappa_0)k=(1-\kappa_0)d_1(S,T).\end{aligned}$$ It is easy to exhibit some $S,T$ such that ineq. is in fact an equality. In the case where $p_i=1/n$, this equality reads $\mathbb{E}\left[d_1(\mathbf{S}^1,\mathbf{S}^2)\right]=d_1(S,T)$, and we cannot obtain a positive Ricci curvature (though this does not prove that there is not another coupling or another distance for which we could get a $\kappa >0$ in the case $p_i=1/n$.). We recall that for each family ${\mathcal{A}_{n}}$, $\mathrm{diam}({\mathcal{A}_{n}})= \max d_1(\mathbf{S},\mathbf{T}) \leq n(n+1)/2$. Hence, combining Proposition \[Lem:Courbure\]with Eq. gives our main result: \[Th:Mix\] For meanders, excursions and path with walls, Algorithm \[Algo:CM\] returns an almost uniform sample of $\pi$, as soon as $T \gg n^3$. Precisely, for any itinialization of Algorithm \[Algo:CM\], $$\parallel \mathbf{S}(T) -\pi\parallel_{\mathrm{TV}} \leq \mathrm{diam}({\mathcal{A}_{n}})(1-\kappa)^T \leq \frac{n(n+1)}{2}\exp\left(-\frac{3}{2n^2(n+1)}T\right).$$ Another formulation of this result is that the mixing time of the associated Markov chain, defined as usual by $$\label{Eq:tmix} t_{\mbox{mix}}:={\left\{\inf\ t\geq 0\ ;\ \sup_{v\in V} \parallel P^t\delta_v -\pi\parallel_{\mathrm{TV}}\leq e^{-1}\right\}}$$ ($e^{-1}$ is here by convention), is smaller than $n^2(n+1)\log n$. For culminating paths, the argument of Case 1c fails and does not hold, we are not able to prove such a result as Theorem \[Th:Mix\]. However, it seems empirically that the mixing time is also of order $n^3\log n$ (with a constant strongly dependent on $a,b$). A way to prove this could be the following observation: take $(\mathbf{S}^0,\mathbf{T}^0)=(\mathbf{S},\mathbf{T})$ two any culminating paths, and define $$(\mathbf{S}^{t+1},\mathbf{T}^{t+1})=(\phi(\mathbf{S}^t,I_t,{\varepsilon}_t,\delta_t),\phi(\mathbf{T}^t,I_t,{\varepsilon}_t,\delta_t)),$$ where $I_t,{\varepsilon}_t,\delta_t$ are those in Algorithm \[Algo:CM\]. The sequence $ \left(\parallel \mathbf{S}^t-\mathbf{T}^t\parallel_\infty \right)_t $ is decreasing throughout the process. Unfortunately we cannot get a satisfactory bound for the time needed for this quantity to decrease by one. Related works {#Sec:Related} ------------- Bounding mixing times via a contraction property over the transportation metric is quite a standard technique, the main ideas dating back to Dobrushin (1950’s). A modern introduction is made in [@Mix]. For geodesic spaces, this technique has been developped in [@Bub] under the name *path coupling*. As mentioned in the introduction, the Markov chain $P$ on lattice paths with uniform weights $p_i=1/n$ has in fact already been introduced for paths starting and ending at zero (sometimes called *bridges*) in [@LRS], and its mixing time has been estimated in [@Wilson]. Wilson also proves a mixing time of order $n^3\log n$, by showing that holds with a different distance (namely, a kind of Fourier transform of the heights of the paths)[^2]. This is the concavity of this Fourier transform which gives a good mixing time, exactly as the concavity of our $p_i$’s speeds up the convergence of our chain. Wilson’s method is developped only for bridges in [@Wilson] and it is not completely straightforward to use it when the endpoints are not fixed. For instance, take $n=7$ and $a=b=1$, and consider the paths $+++--++$ and $---++--$. There are more “bad moves” (moves that take away these paths) than “good moves”. *Coupling From The Past* with $P$ {#Sec:ProppWilson} ================================= Propp-Wilson’s Coupling From The Past (CFTP) [@PW] is a very general procedure for the exact sampling of the stationary distribution of a Markov chain. It is efficient if the chain is monotonous with respect to a certain order relation $\preceq$ on the set $V$ of vertices, with two extremal points denoted $\hat{0},\hat{1}$ ([[*i.e.*]{} ]{}such that $\hat{0}\preceq x\preceq\hat{1}$ for any vertex $x$). This is the case here for each family ${\mathcal{C}_{n}}$,${\mathcal{W}_{n}}$,${\mathcal{E}_{n}}$,${\mathcal{M}_{n}}$ , with the partial order $$\mathbf{S}\preceq \mathbf{T} \mbox{ iff } S_i\leq T_i \mbox{ for any }i.$$ For the family ${\mathcal{M}_{10}}$ with $a=1,b=-2$ for instance, we have $$\begin{aligned} \hat{0}=\hat{0}_{\mbox{meanders}}&=(1,1,-2,1,1,-2,1,1,-2,1),\\ \hat{1}=\hat{1}_{\mbox{meanders}}&=(1,1,1,1,1,1,1,1,1,1).\end{aligned}$$ We describe CFTP, with our notations, in Algorithm \[Algo:CFTP\]. $\mathbf{S}\leftarrow\hat{0}$, $\mathbf{T}\leftarrow\hat{1}$ $\dots,I_{-2},I_{-1}\leftarrow$ i.i.d. r.v. with law $\mathbf{p}$ $\dots,{\varepsilon}_{-2},{\varepsilon}_{-1}\leftarrow$ i.i.d. uniform r.v. in ${\left\{\uparrow,\downarrow\right\}}$ $\dots,\delta_{-2},\delta_{-1}\leftarrow$ i.i.d. uniform r.v. in ${\left\{+,-\right\}}$ $\tau=1$ $\mathbf{S}\leftarrow\hat{0}$, $\mathbf{T}\leftarrow\hat{1}$ ${\phi(\mathbf{S},I_t,{\varepsilon}_t)}$ is in ${\mathcal{A}_{n}}$ [**then** ]{} $\mathbf{S}\leftarrow {\phi(\mathbf{S},I_t,{\varepsilon}_t)}{\delta_t}$ ${\phi(\mathbf{T},I_t,{\varepsilon}_t)}$ is in ${\mathcal{A}_{n}}$ [**then** ]{} $\mathbf{T}\leftarrow {\phi(\mathbf{T},I_t,{\varepsilon}_t)}{\delta_t}$ $\tau\leftarrow 2\tau$ We refer to ([@Hagg],Chap.10) for a very clear introduction to CFTP, and we only outline here the reasons why this indeed gives an exact sampling of the stationary distribution. - The output of the algorithm (if it ever ends!) is the state of the chain $P$ that has been running “since time $-\infty$”, and thus has reached stationnarity. - The exit condition $\mathbf{S}=\mathbf{T}$ ensures that it is not worth running the chain from $T$ steps earlier, since the trajectory of any lattice path $\hat{0}\preceq \mathbf{R}\preceq\hat{1}$ is “sandwiched” between those of $\hat{0},\hat{1}$, and therefore ends at the same value. ![A sketchy representation of CFTP : trajectories starting from $\hat{0},\hat{1}$ at time $-T/2$ don’t meet before time zero, while those starting at time $-T$ do.[]{data-label="Fig:ProppWilson"}](ProppWilson.eps){width="65mm"} \[Prop:CFTP\] Algorithm \[Algo:CFTP\] ends with probability $1$ and returns an exact sample of the uniform distribution over ${\mathcal{A}_{n}}$. For the families ${\mathcal{W}_{n}}$,${\mathcal{E}_{n}}$,${\mathcal{M}_{n}}$, this takes on average $\mathcal{O}(n^3(\log n)^2)$ time units. Let us mention that in the case where the mixing time is not rigorously known, Algorithm \[Algo:CFTP\] (when it ends) outputs an exact uniform sample and therefore is of main practical interest compared to MCMC. It is shown in [@PW] that Algorithm \[Algo:CFTP\] returns an exact sampling in $\mathcal{O}(t_{\mbox{mix}}\log H)$ runs of the chain, where $t_{\mbox{mix}}$ is defined in and $H$ is the length of the longest chain of states between $\hat{0}$ and $\hat{1}$. It is a consequence of the proof of Lemma \[Lem:Geodesique\] that $H=\mathcal{O}(n^2)$. We have seen that $t_{\mbox{mix}}=\mathcal{O}(n^3\log n)$. (Recall that each test in Algorithm \[Algo:CFTP\] takes, on average, $\mathcal{O}(1)$ time units.) We recall that CFTP has a major drawback compared to MCMC. For the algorithm to be correct, we have to reuse the same random variables $I_t,{\varepsilon}_t,\delta_t$, so that space-complexity is in fact linear in $n^3(\log n)^2$. This may become an issue when $n$ is large. Concluding remarks and simulations ================================== [**1.**]{} In Fig.\[Fig:Simus\], we show simulations of the three kinds of paths, for $a=1,b=2,n=600$. We observe that the final height of the culminating path is very low (about $30$), it would be interesting to use our algorithm to investigate the behaviour of this height when $n\to\infty$ ; this question was left open in [@MBM]. ![(Almost) uniform paths of length $600$, with $a=1,b=2$. From top to bottom: a culminating path, a meander, a path with wall (shown by an arch).[]{data-label="Fig:Simus"}](SimuCulmi600.eps "fig:"){width="140mm"}\ ![(Almost) uniform paths of length $600$, with $a=1,b=2$. From top to bottom: a culminating path, a meander, a path with wall (shown by an arch).[]{data-label="Fig:Simus"}](SimuPositif600.eps "fig:"){width="140mm"}\ ![(Almost) uniform paths of length $600$, with $a=1,b=2$. From top to bottom: a culminating path, a meander, a path with wall (shown by an arch).[]{data-label="Fig:Simus"}](SimuWall600.eps "fig:"){width="140mm"}\ [**2.**]{} One may wonder to what extent this work applies to other families ${\mathcal{A}_{n}}$ of paths. The main assumption is that the family of paths should be a geodesic space w.r.t. distance $d_1$. This is true for example if the following condition on ${\mathcal{A}_{n}}$ is fulfilled: $$\left(R,T\in{\mathcal{A}_{n}} \mbox{ and }R\preceq S\preceq T\right) \Rightarrow S\in{\mathcal{A}_{n}}.$$ Notice however that this is quite a strong requirement, and it is not verified for culminating paths for instance. [**3.**]{} A motivation to sample random paths is to make and test guesses for some functionals of these paths, taken on average over ${\mathcal{A}_{n}}$. Consider a function $f:{\mathcal{A}_{n}}\to\mathbb{R}$, we want an approximate value of $\pi(f):={\mathrm{card}}({\mathcal{A}_{n}})^{-1}\sum_{s\in{\mathcal{A}_{n}}}f(s)$, if the exact value is out of reach by calculation. We estimate this quantity by $$\label{Eq:Chapeau} \hat{\pi}(f):=\frac{1}{T} \sum_{t=1}^T f \left(\mathbf{S}(t)\right),$$ (recall that $S(t)$ is the value of the chain at time $t$). For Algorithm \[Algo:CM\] to be efficient in practice, we have to bound $$\label{Eq:AMajorer} \mathbb{P}\left(\left|\pi(f)-\hat{\pi}(f)\right|>r\right),$$ for any fixed $r>0$, by a non-asymptotic (in $T$) quantity. This can be done with ([@JouOlli], Th.4-5), in which one can find concentration inequalities for . The sharpness of these inequalities depends on $\kappa$ and on the geometrical structure of ${\mathcal{A}_{n}}$. [**Aknowledgements.**]{} Many thanks to Frédérique Bassino and the other members of <span style="font-variant:small-caps;">Anr Gamma</span> for the support ; I also would like to thank Élie Ruderman for the English corrections. A referee raised a serious error in the first version of this paper, I am grateful to them. [99]{} C. Banderier and Ph. Flajolet. Basic Analytic Combinatorics of Directed Lattice Paths, *Theoretical Computer Science* [**281**]{} (1):37-80 (2002). M.Bousquet-Mélou and Y. Ponty. Culminating paths, *Discrete Math and Theoretical Computer Science* [**10**]{} (2):125-152 (2008). R.Bubley and M. Dyer. Path coupling: A technique for proving rapid mixing in Markov chains. Proceedings of the *38th Annual Symposium on Foundations of Computer Science*, p.223-231 (1997). N.Destainville. Flip dynamics in octagonal rhombus tiling sets, *Physical Review Letters* [**88**]{} (2002). P.Diaconis and D.W.Stroock. Geometric Bounds for Eigenvalues of Markov Chains, *Annals of Applied Probability* [**1**]{} (1):36-61 (1991). Ph.Duchon. On the enumeration and generation of generalized Dyck words. *Discrete Mathematics* [**225**]{} (1-3)121–135 (2000). M.E.Fisher. Walks, Walls, Wetting, and Melting. *Journal of Statistical Physics* [**34**]{} (5):667-729 (1984). O.Häggström. *Finite Markov Chains and Algorithmic Applications*. London Mathematical Society (2002). A.Joulin and Y.Ollivier. Curvature, concentration, and error estimates for Markov chain Monte Carlo. To appear in *Annals of Probability*, `arXiv:0904.1312` (2009). D.A.Levin, Y.Peres, and E.L.Wilmer. *Markov Chains and Mixing Times*. American Mathematical Society (2009). M.Luby, D.Randall, A.Sinclair. Markov chain algorithms for planar lattice structures. *SIAM Journal on Computing*, [**31**]{} (1):167–192 (2001). Y.Ollivier. Ricci curvature of Markov chains on metric spaces. *Journal of Functional Analysis* [**256**]{} (3):810-864 (2009). J.G. Propp and D.B. Wilson. Exact sampling with coupled Markov chains and applications to statistical mechanics. *Random Structures and Algorithms*, [**9**]{} (1):223-252, (1996). D.B.Wilson. Mixing times of Lozenge tiling and card shuffling Markov chains. *Annals of Applied Probability* [**14**]{} (1):274–325 (2004). [^1]: The Ricci curvature is actually the largest positive number such that holds, for all the couplings of $P\delta_x,P\delta_y$ ; here we should rather say that Ricci curvature is larger than $\kappa$. [^2]: Notice that $a,b$ do not have the same meaning in Wilson’s paper: $a$ (resp. $b$) stands for the number of positive (resp. negative) steps.

--- abstract: 'A loop whose inner mappings are automorphisms is an *automorphic loop* (or *A-loop*). We characterize commutative (A-)loops with middle nucleus of index $2$ and solve the isomorphism problem. Using this characterization and certain central extensions based on trilinear forms, we construct several classes of commutative A-loops of order a power of $2$. We initiate the classification of commutative A-loops of small orders and also of order $p^3$, where $p$ is a prime.' address: - 'Department of Mathematics, Faculty of Engineering, Czech University of Life Sciences, Kamýcká 129, 165 21 Prague 6–-Suchdol, Czech Republic' - 'Department of Mathematics, University of Denver, 2360 S Gaylord St, Denver, Colorado, 80208, U.S.A.' - 'Department of Mathematics, University of Denver, 2360 S Gaylord St, Denver, Colorado, 80208, U.S.A.' author: - Přemysl Jedlička - 'Michael K. Kinyon' - Petr Vojtěchovský title: Constructions of commutative automorphic loops --- Introduction ============ A *loop* is a groupoid $(Q,\cdot)$ with neutral element $1$ such that all left translations $L_x:Q\to Q$, $y\mapsto xy$ and all right translations $R_x:Q\to Q$, $y\mapsto yx$ are bijections of $Q$. Given a loop $Q$ and $x$, $y\in Q$, we denote by $x\ld y$ the unique element of $Q$ satisfying $x(x\ld y) = y$. In other words, $x\ld y = L_x^{-1}(y)$. To reduce the number of parentheses, we adopt the following convention for term evaluation: $\ld$ is less binding than juxtaposition, and $\cdot$ is less binding than $\ld$. For instance $xy\ld u\cdot v\ld w$ is parsed as $((xy)\ld u)(v\ld w)$. The *inner mapping group* $\inn{Q}$ of a loop $Q$ is the permutation group generated by $$L_{x,y} = L_{yx}^{-1}L_yL_x,\quad R_{x,y} = R_{xy}^{-1}R_yR_x,\quad T_x = L_x^{-1}R_x,$$ where $x$, $y\in Q$. A subloop of $Q$ is *normal* if it is invariant under all inner mappings of $Q$. A loop $Q$ is an *automorphic loop* (or *A-loop*) if $\inn{Q}\le\aut{Q}$, that is, if every inner mapping of $Q$ is an automorphism of $Q$. Hence a commutative loop is an A-loop if and only if all its left inner mappings $L_{y,x}$ are automorphisms, which can be expressed by the identity $$\label{Eq:A} xy\ld x(yu)\cdot xy\ld x(yv) = xy\ld x(y\cdot uv).\tag{\textsc{A}}$$ Note that the class of commutative A-loops contains commutative groups and commutative Moufang loops. We assume that the reader is familiar with the terminology and notation of loop theory, cf. [@Bruck] or [@Pflugfelder]. This paper is a companion to [@JKV], where we have presented a historical introduction and many new structural results concerning commutative $A$-loops, including: 1. commutative A-loops are power-associative (see already [@BP]), 2. for a prime $p$, a finite commutative A-loop $Q$ has order a power of $p$ if and only if every element of $Q$ has order a power of $p$, 3. every finite commutative A-loop is a direct product of a loop of odd order (consisting of elements of odd order) and a loop of order a power of $2$, 4. commutative A-loops of odd order are solvable, 5. the Lagrange and Cauchy theorems hold for commutative A-loops, 6. every finite commutative A-loop has Hall $\pi$-subloops (and hence Sylow $p$-subloops), 7. if there is a nonassociative finite simple commutative A-loop, it is of exponent $2$. Despite these deep results, the theory of commutative A-loops is in its infancy. As an illustration of this fact, the present theory is not sufficiently developed to classify commutative A-loops of order $8$ without the aid of a computer, commutative A-loops of order $pq$ (where $p<q$ are primes), nor commutative A-loops of order $p^3$ (where $p$ is an odd prime). The two main problems for commutative A-loops stated in [@JKV] were: *For an odd prime $p$, is every commutative A-loop of order $p^k$ centrally nilpotent?* *Is there a nonassociative finite simple commutative A-loop, necessarily of exponent $2$ and order a power of $2$?* For an example of a commutative A-loop of order $8$ that is not centrally nilpotent, see Subsection \[Ss:8\]. In the meantime, we have managed to solve the first problem of [@JKV] in the affirmative, but we neither use nor prove the result here—it will appear elsewhere. The second problem remains open and the many constructions of commutative A-loops of exponent $2$ obtained here can be seen as a step toward solving it. One of the most important concepts in the investigation of commutative A-loops appears to be the middle nucleus $N_\mu(Q)$, since, by [@BP], $N_\lambda(Q)\le N_\mu(Q)$, $N_\rho(Q)\le N_\mu(Q)$ and $N_\mu(Q)\unlhd Q$ is true in any A-loop $Q$. In §\[Sc:Index2\] we characterize all commutative loops with middle nucleus of index $2$, solve the isomorphism problem, and then characterize all commutative A-loops with middle nucleus of index $2$. In §\[Sc:AppsIndex2\] we classify commutative A-loops of order $8$, among other applications of §\[Sc:Index2\]. Central extensions of commutative A-loops are described in §\[Sc:Extensions\]. A broad class of such extensions is obtained from trilinear forms that are symmetric with respect to an interchange of (fixed) two arguments. As an application, we characterize all parameters $(k,\ell)$ with the property that there is a nonassociative commutative A-loop of order $2^k$ with middle nucleus of order $2^\ell>1$. §\[Sc:p3\] uses another class of central extensions partially based on the overflow in modular arithmetic that yields many (conjecturally, all) nonassociative commutative A-loops of order $p^3$, where $p$ is an odd prime. A classification of commutative A-loops of small orders based on the theory and computer computations can be found in §\[Sc:Enumeration\]. Commutative loops with middle nucleus of index $2$ {#Sc:Index2} ================================================== Throughout this section, we denote by $\ov{X} = \{\ov{x};\;x \in X\}$ a disjoint copy of the set $X$. Let $G$ be a commutative group and $f$ a bijection of $G$. Then $G(f)$ will denote the groupoid $(G\cup \ov{G},*)$ with multiplication $$\label{Eq:Gf} x*y = xy,\quad x*\ov{y} = \ov{xy},\quad \ov{x}*y=\ov{xy},\quad \ov{x}*\ov{y}=f(xy),$$ for $x$, $y\in G$. Note that $G(f)$ is a loop with neutral element $1$. \[Lm:PropertiesGf\] Let $G$ be a commutative group, $f$ a bijection of $G$ and $(Q,\cdot) = G(f) = (G\cup \ov{G},*)$. Then: 1. $Q$ is commutative. 2. $x\ld y=x^{-1}y$, $x\ld\ov{y}=\ov{x^{-1}y}$, $\ov{x}\ld y = \ov{x^{-1}f^{-1}(y)}$, $\ov{x}\ld\ov{y} = x^{-1}y$ for every $x$, $y\in G$. 3. $G\le\mnuc{Q}$. 4. $Q$ is a group if and only if $f$ is a translation of the group $G$. 5. $N_\lambda(Q)\cap G = N_\rho(Q)\cap G = Z(Q)\cap G = \{x\in G;\;f(xy)=xf(y)\text{ for every }y\in G\}$. When $Q$ is not a group $($that is, $G=\mnuc{Q})$, then $N_\lambda(Q)=N_\rho(Q)=Z(Q)\le G$. Part (i) follows from the definition of $G(f)$. Part (ii) is straightforward, for instance, $x*\ov{x^{-1}y} = \ov{xx^{-1}y}=\ov{y}$ shows that $x\ld\ov{y}=\ov{x^{-1}y}$. For (iii), let $x$, $y$, $z\in G$ and verify that $$\begin{aligned} &x*(y*z) = (x*y)*z,\\ &\ov{x}*(y*z)=\ov{x}*yz = \ov{xyz} = \ov{xy}*z = (\ov{x}*y)*z,\\ &x*(y*\ov{z}) = x*\ov{yz} = \ov{xyz} = xy*\ov{z} = (x*y)*\ov{z},\\ &\ov{x}*(y*\ov{z}) = \ov{x}*\ov{yz} = f(xyz) = \ov{xy}*\ov{z} = (\ov{x}*y)*\ov{z}.\end{aligned}$$ This shows $G\le \mnuc{Q}$. \(iv) An easy calculation shows that $\ov{1}\in \mnuc{Q}$ (that is, $Q$ is a group) if and only if $f(xy)=xf(y)=f(x)y$ for every $x$, $y\in G$. With $y=1$ we deduce that $f(x)=xf(1)$ for every $x$. On the other hand, if $f(x)=xf(1)$ for every $x$ then $f(xy)=xf(y)=f(x)y$. \(v) We have $x*(y*z) = (x*y)*z$, $x*(\ov{y}*z) = \ov{xyz} = (x*\ov{y})*z$, $x*(y*\ov{z}) = \ov{xyz} = (x*y)*\ov{z}$, and $x*(\ov{y}*\ov{z}) = xf(yz)$, while $(x*\ov{y})*\ov{z}= f(xyz)$. Hence $x\in\lnuc{Q}$ if and only if $xf(yz)=f(xyz)$ for every $y$, $z\in G$, which holds if and only if $xf(y) = f(xy)$ for every $y\in G$. By commutativity, $\lnuc{Q}=\rnuc{Q}$. By (iii), $\lnuc{Q}\cap G = Z(Q)\cap G$. Assume that $Q$ is not a group. Suppose that $\ov{x}\in\lnuc{Q}$. Then $f(xyz) = \ov{x} *\ov{yz} = \ov{x}*(\ov{y} *z) = (\ov{x}*\ov{y})*z = f(xy)*z= f(xy)z$ for every $y$, $z\in G$, and hence (with $y=x^{-1}$), $f(z)=f(1)z$ for every $z\in G$. By (iv), $Q$ is a group, a contradiction. Thus $\lnuc{Q}\le G$. Let $Q$ be a commutative loop with subloop $G$ satisfying $G\le \mnuc{Q}$, $[Q:G]=2$. Then $G$ is a commutative group and there exists a bijection $f$ of $G$ such that $Q$ is isomorphic to $G(f)$. The commutative loop $G$ is a group by $G\le\mnuc{Q}$. Denote by $\ov{1}$ a fixed element of $Q\setminus G$, and define $\ov{x}=\ov{1}x=x\ov{1}$ for every $x\in G$. Note that $\ov{1}$ is well-defined, $G\cap \ov{G}=\emptyset$ and $Q=G\cup\ov{G}$. Moreover, $x\ov{y} = x\cdot y\ov{1} = xy\cdot \ov{1} = \ov{xy}$ and $\ov{x}y = \ov{1}x\cdot y = \ov{1}\cdot xy = \ov{xy}$ for every $x$, $y\in G$, using $G\le\mnuc{Q}$ again. Finally, if $x_1$, $y_1$, $x_2$, $y_2\in G$ satisfy $x_1y_1=x_2y_2$ then $$\ov{x_1}\ov{y_1} = \ov{1}x_1\cdot y_1\ov{1} = \ov{1}(x_1\cdot y_1\ov{1}) = \ov{1}(x_1y_1\cdot\ov{1}) = \ov{1}(x_2y_2\cdot \ov{1}) = \ov{x_2}\ov{y_2}.$$ Thus the multiplication in the quadrant $\ov{G}\times\ov{G}$ mimics that of $G\times G$, except that the elements are renamed according to the permutation $f:G\to G$, $x\mapsto \ov{1}\cdot x\ov{1}$. \[Cr:Gf\] Let $Q$ be a commutative loop possessing a subgroup of index $2$. Then $[Q:\mnuc{Q}]\le 2$ if and only if there exists a commutative group $G$ and a bijection $f$ of $G$ such that $Q$ is isomorphic to $G(f)=(G\cup\ov{G},*)$ defined by . The assumption that $Q$ possesses a subgroup of index $2$ in Corollary $\ref{Cr:Gf}$ is needed only when $Q$ is a group. We now solve the isomorphism problem for nonassociative commutative loops with middle nucleus of index $2$ in terms of the associated bijections: \[Pr:IsoGf\] Let $G$ be a commutative group and $f_1$, $f_2$ bijections of $G$ such that $G(f_1)$, $G(f_2)$ are not groups. Then $G(f_1)\cong G(f_2)$ if and only if there is $\psi\in\aut{G}$ such that $$\label{Eq:IsoGf} f_2^{-1}\psi f_1(x) = f_2^{-1}\psi f_1(1)\cdot \psi(x)\quad \text{for all $x\in G$},$$ and $f_2^{-1}\psi f_1(1)$ is a square in $G$. Denote by $*$ the multipication in $G(f_1)$, and by $\circ$ the multiplication in $G(f_2)$. Assume that $\varphi:G(f_1)\to G(f_2)$ is an isomorphism. Since $G(f_1)$, $G(f_2)$ are not groups, $\varphi$ maps $\mnuc{G(f_1)}=G$ onto $\mnuc{G(f_2)}=G$ by Lemma \[Lm:PropertiesGf\](iii), and hence $\psi = \varphi|_G$ is a bijection of $G$. Then $$\psi(xy) = \varphi(xy) = \varphi(x*y) = \varphi(x)\circ\varphi(y) = \psi(x)\circ\psi(y) = \psi(x)\psi(y)$$ for every $x$, $y\in G$, so $\psi\in\aut{G}$. Define $\rho:G\to G$ by $\ov{\rho(x)} = \varphi(\ov{x})$. We have $$\ov{\rho(x)} = \varphi(\ov{x}) = \varphi(x*\ov{1}) = \varphi(x)\circ\varphi(\ov{1}) = \psi(x)\circ\ov{\rho(1)} = \ov{\psi(x)\rho(1)},$$ so $\rho(x) = \rho(1)\psi(x)$ for every $x\in G$. Using this observation, we have $$\psi(f_1(xy)) = \varphi(f_1(xy)) = \varphi(\ov{x}*\ov{y}) = \varphi(\ov{x})\circ \varphi(\ov{y}) = \ov{\rho(x)}\circ \ov{\rho(y)} = f_2(\rho(x)\rho(y)) = f_2(\rho(1)^2\psi(xy)).$$ Equivalently, $f_2^{-1}\psi f_1(x) = \rho(1)^2\psi(x)$ for every $x\in G$. With $x=1$, we deduce that $\rho(1)^2 = f_2^{-1}\psi f_1(1)$ is a square in $G$, and that holds. Conversely, assume that holds for some $\psi\in\aut{G}$, and that $u^2 = f_2^{-1}\psi f_1(1)$ is a square in $G$. Define $\varphi:G(f_1)\to G(f_2)$ by $\varphi(x)=\psi(x)$, $\varphi(\ov{x}) = \ov{u\psi(x)}$. Then $$\begin{gathered} \varphi(x*y) = \varphi(xy) = \psi(xy) = \psi(x)\psi(y) = \psi(x)\circ\psi(y) = \varphi(x)\circ\varphi(y),\\ \varphi(\ov{x}*y) = \varphi(\ov{xy}) = \ov{u\psi(xy)} = \ov{u\psi(x)\psi(y)} = \ov{u\psi(x)}\circ\psi(y) = \varphi(\ov{x})\circ \varphi(y),\end{gathered}$$ and, similarly, $\varphi(x*\ov{y}) = \varphi(x)\circ\varphi(\ov{y})$ for every $x$, $y\in G$. Finally, using to obtain the third equality below, we have $$\varphi(\ov{x}*\ov{y}) = \varphi(f_1(xy)) = \psi(f_1(xy)) = f_2(u^2\psi(xy)) = \ov{u\psi(x)}\circ \ov{u\psi(y)} = \varphi(\ov{x})\circ\varphi(\ov{y})$$ for every $x$, $y\in G$. Thus $G(f_1)\cong G(f_2)$. We say that two bijections $f_1$, $f_2$ of $G$ are *conjugate in $\aut{G}$* if there is $\psi\in\aut{G}$ such that $f_2 = \psi f_1\psi^{-1}$. The following specialization of Proposition \[Pr:IsoGf\] will be useful in the classification of commutative A-loops of order $8$. \[Cr:IsoGf\] Let $G$ be a commutative group, and let $f_1$, $f_2$ be bijections of $G$ such that $G(f_1)$, $G(f_2)$ are not groups. 1. If $f_1$, $f_2$ are conjugate in $\aut{G}$ then $G(f_1)\cong G(f_2)$. 2. If $f_1(1)=1=f_2(1)$ then $G(f_1)\cong G(f_2)$ if and only if $f_1$, $f_2$ are conjugate in $\aut{G}$. 3. If $f_2\in\aut{G}$, $t$ is a square in $G$ and $f_1(x)=f_2(x)t$ for every $x\in G$ then $G(f_1)\cong G(f_2)$. \(i) Let $\psi\in\aut{G}$ be such that $f_2=\psi f_1\psi^{-1}$. Then $f_2^{-1}\psi f_1 = \psi$, so $f_2^{-1}\psi f_1(1) = \psi(1) = 1$ is a square in $G$ and holds. \(ii) Assume that $G(f_1)\cong G(f_2)$. Then there is $\psi\in\aut{G}$ such that holds. Since $f_2^{-1}\psi f_1(1) = f_2^{-1}\psi(1) = f_2^{-1}(1) = 1$, we deduce from that $f_1$, $f_2$ are conjugate in $\aut{G}$. The converse follows by (i). \(iii) Let $\psi$ be the trivial automorphism of $G$. Then $\eqref{Eq:IsoGf}$ becomes $f_2^{-1}f_1(x) = f_2^{-1}f_1(1)\cdot x$, and it is our task to check this identity and that $f_2^{-1}f_1(1)$ is a square in $G$. Now, $f_2^{-1}f_1(1) = f_2^{-1}(f_2(1)t) = f_2^{-1}(f_2(1))f_2^{-1}(t) = f_2^{-1}(t)$ is a square in $G$ since $t$ is. Moreover, $f_1(1) = f_2(1)\cdot t = t$, so $f_1(x)=f_1(1)f_2(x)$, and follows upon applying $f_2^{-1}$ to this equality. Finally, we describe all commutative A-loops with middle nucleus of index $2$. \[Pr:AGf\] Let $Q$ be a commutative loop possessing a subgroup of index $2$. Then the following conditions are equivalent: 1. $Q$ is an A-loop and $[Q:\mnuc{Q}]\le 2$. 2. $Q=G(f)$, where $G$ is a commutative group, $[Q:G]=2$, and $f$ is a permutation of $G$ satisfying $$\begin{aligned} &f(xy) = f(x)f(y)f(1)^{-1},\label{Eq:f1}\tag{$P_1$}\\ &f(x^2) = x^2f(1),\label{Eq:f2}\tag{$P_2$}\\ &f^2(x)^2f(x)^{-2}=f^2(1)\label{Eq:f3}\tag{$P_3$}\end{aligned}$$ for every $x$, $y\in G$. 3. $Q=G(f)$, where $G$ is a commutative group, $[Q:G]=2$, and $f$ is a permutation of $G$ satisfying , and $f^2(1) = f(1)^2$. 4. $Q=G(f)$, where $G$ is a commutative group, $[Q:G]=2$, $f(x)=g(x)t$ for every $x\in G$, $g\in\aut{G}$, $g(x^2)=x^2$ for every $x\in G$, and $t$ is a fixed point of $g$. By Corollary \[Cr:Gf\], we can assume that $Q=G(f)=(G\cup \ov{G},*)$, where $G\le \mnuc{Q}$ is a commutative group and $f$ is a bijection of $G$. Let us establish the equivalence of (i) and (ii). Denote by $\alpha(a,b,c,d)$ the $*$ version of , namely $$(a*b)\ld (a*(b*(c*d))) = [(a*b)\ld (a*(b*c))]*[(a*b)\ld (a*(b*d))],$$ where $a$, $b$, $c$, $d$ are taken from $G\cup\ov{G}$, and where $\ld$ is understood in $(Q,*)$. With the exception of the variables $a$, $b$, $c$, $d$, we implicitly assume that variables without bars are taken from $G$, while variables with bars are taken from $\ov{G}$. Then $\alpha(x,y,u,v)$ holds in $G(f)$, as the evaluation of $\alpha(x,y,u,v)$ takes place in the group $G$. Since $y\in \mnuc{Q}$, $\alpha(a,y,c,d)$ holds. By commutativity of $*$, $\alpha(a,b,c,d)$ holds if and only if $\alpha(a,b,d,c)$ holds. Hence it remains to investigate the identities $\alpha(x,\ov{y},u,v)$, $\alpha(x,\ov{y},u,\ov{v})$, $\alpha(x,\ov{y},\ov{u},\ov{v})$, $\alpha(\ov{x},\ov{y},u,v)$, $\alpha(\ov{x},\ov{y},u,\ov{v})$, and $\alpha(\ov{x},\ov{y},\ov{u},\ov{v})$. Straightforward calculation with and Lemma \[Lm:PropertiesGf\] shows that $\alpha(\ov{x},\ov{y},u,\ov{v})$ holds if and only if $$\label{Eq:Case1} f(yuv) = f(xy)^{-1}f(xyu)f(yv).$$ Using $x=y=1$, reduces to . On the other hand, already implies , and so $\alpha(\ov{x},\ov{y},u,\ov{v})$ is equivalent to . From now on, we will assume that holds and denote $f(1)$ by $t$. The identity $\alpha(x,\ov{y},\ov{u},\ov{v})$ is then equivalent to $$\label{Eq:Case2} x^{-1}t^{-1}=f(x^{-2})f(y^{-2})f(y)^2xt^{-5},$$ and since $t=f(yy^{-1})=f(y)f(y^{-1})t^{-1}$ yields $$\label{Eq:fInverse} f(y^{-1})=f(y)^{-1}t^2,$$ we can rewrite as $f(x)^2=x^2t^2$, or, equivalently (using ), as . Finally, note that and imply $$\label{Eq:ff} f^2(uv) = f(f(uv)) = f(f(u)f(v)t^{-1}) = f^2(u)f^2(v)f(t^{-1})t^{-2} = f^2(u)f^2(v)f(t)^{-1}.$$ Using and , we see, after a lengthy calculation, that the identity $\alpha(\ov{x},\ov{y},\ov{u},\ov{v})$ is equivalent to . We leave it to the reader to check that the identities $\alpha(x,\ov{y},u,v)$, $\alpha(x,\ov{y},u,\ov{v})$, $\alpha(x,\ov{y},\ov{u},\ov{v})$ imply no additional conditions on $f$ beside –, and, conversely, that if – are satisfied then the identities $\alpha(x,\ov{y},u,v)$, $\alpha(x,\ov{y},u,\ov{v})$, $\alpha(x,\ov{y},\ov{u},\ov{v})$ hold. We have proved the equivalence of (i) and (ii). Assume that (ii) holds. With $x=1$ in we have $f^2(1)^2f(1)^{-2}=f(t)$, or $f(t)^2t^{-2}=f(t)$, or $f(t)=t^2$, so (iii) holds. Conversely, assume that (iii) holds. Then, $f^2(x)^2f(t)^{-1} = f^2(x)^2t^{-2} = f(f(x))f(f(x))t^{-2} = f(f(x)f(x))t^{-1}=f(f(x)^2)t^{-1} = f(x)^2$, which is , so (ii) holds. Assume that (iii) holds and define $g$ by $g(x)=f(x)t^{-1}$, where $t=f(1)$. Then $g(xy)=f(xy)t^{-1} = f(x)f(y)t^{-2} = f(x)t^{-1}f(y)t^{-1} = g(x)g(y)$ by , $g(x^2) = f(x^2)t^{-1} = x^2$ by , and $g(t) = f(t)t^{-1} = t$ by $f(t)=t^2$. Conversely, assume that (iv) holds, $f(x)=g(x)t$, $g\in\aut{G}$, where $g(x^2)=x^2$ and $t$ is a fixed point of $g$ (not necessarily satisfying $t=f(1)$). Then $f(1) = g(1)t=t$, $f(xy) = g(xy)t = g(x)g(y)t = g(x)tg(y)tt^{-1} = f(x)f(y)t^{-1}$, $f(x^2) = g(x^2)t = x^2t$, and $f(t)=g(t)t = t^2$, proving (iii). Constructions of commutative A-loops with middle nucleus of index $2$ {#Sc:AppsIndex2} ===================================================================== As an application of Proposition \[Pr:AGf\], we classify all commutative A-loops of order $8$ and present a class of commutative A-loops of exponent $2$ with trivial center and middle nucleus of index $2$. Commutative A-loops of order $8$ {#Ss:8} -------------------------------- It is not difficult to classify all commutative A-loops of order $8$ up to isomorphism with a finite model builder, such as Mace4 [@Mace4]. It turns out that there are $4$ nonassociative commutative A-loops of order $8$. All such loops have middle nucleus of index $2$; a fact for which we do not have a human proof. But using this fact, we can finish the classification by hand with Proposition \[Pr:IsoGf\], Corollary \[Cr:IsoGf\] and Proposition \[Pr:AGf\]. \[Lm:TranslatedCenters\] Let $G$ be a commutative loop, $1\ne g\in\aut{G}$ and $t\in G$. Let $f$ be a bijection of $G$ defined by $f(x)=g(x)t$. Then $Z(G(f)) = Z(G(g))$ as sets, and $Z(G(g)) = \{x\in G;\;g(x)=x\}$. Since $g$ is not a translation of $G$, neither is $f$. Hence both $G(g)$ and $G(f)$ are nonassociative, by Lemma \[Lm:PropertiesGf\](iv). By Lemma \[Lm:PropertiesGf\](v), $Z(G(f)) = \{x\in G;\;f(xy)=xf(y)\text{ for every }y\in G\} = \{x\in G;\;g(xy)t = xg(y)t\text{ for every }y\in G\} = \{x\in G;\;g(xy)=xg(y)\text{ for every }y\in G\} = Z(G(g))$ and it is also equal to $\{x\in G;\;g(x)=x\}$ since $g(xy)=g(x)g(y)$. Let $Q$ be a nonassociative commutative A-loop of order $8$, necessarily with a middle nucleus of index $2$. By Proposition \[Pr:AGf\], $Q=G(f)$, where $G$ is a commutative group of order $4$ and $f(x)=g(x)t$ for some $g\in\aut{G}$ and $t\in G$ such that $g(x^2)=x^2$ and $g(t)=t$. Let $G=\mathbb Z_4=\langle a\rangle$ be the cyclic group of order $4$. The two automorphisms of $G$ are the trivial automorphism $g=1$ and the transposition $g=(a,a^3)$; both fix all squares of $G$. Let $g=1$ and $f(x)=g(x)t=xt$ for some $t\in G$. Then $G(f)$ is a commutative group by Lemma \[Lm:PropertiesGf\](iv). Assume that $g=(a,a^3)$. Then $G(g)$ is a nonassociative commutative A-loop. The only nontrivial fixed point of $g$ is $a^2$. Let $f(x) = g(x)a^2$. By Corollary \[Cr:IsoGf\](iii), $G(f)\cong G(g)$. Now let $G=\mathbb Z_2\times\mathbb Z_2 = \langle a\rangle \times \langle b\rangle$ be the Klein group. Then $\aut{G} = \{1$, $(a,b)$, $(a,ab)$, $(b,ab)$, $(a,b,ab)$, $(a,ab,b)\}\cong S_3$. The only square in $G$ is $1$ and it is trivially fixed by all $g\in\aut{G}$. If $g=1$ and $f(x)=g(x)t=xt$ for some $t\in G$, $G(g)$ is a commutative group by Lemma \[Lm:PropertiesGf\](iv). Let $g_1=(a,b)$. The choices for $t$ are $t=1$, $t=ab$. Let $f_1(x)=g_1(x)ab$. Then $G(g_1)$, $G(f_1)$ are nonassociative commutative A-loops. Since $g_1(xx) = g_1(1) = 1$, $G(g_1)$ has exponent $2$. Since $f_1(xx) = f_1(1) = ab$, $G(f_1)$ does not have exponent $2$. Hence $G(g_1)\not\cong G(f_1)$. Let $g_2=(a,ab)$, and note that the choices for $t$ are $t=1$, $t=b$. Let $f_2(x) = g_2(x)b$. Since all transpositions of $S_3$ are conjugate in $S_3$, $G(g_1)\cong G(g_2)$ by Corollary \[Cr:IsoGf\](i). Note that $f_1 = \psi^{-1} f_2 \psi$ with $\psi = (b,ab)$. Hence $G(f_1)\cong G(f_2)$ by Corollary \[Cr:IsoGf\]. Similarly, no new nonassociative commutative A-loop of order $8$ is obtained with $g_3=(b,ab)$. Let $g_4 = (a,b,ab)$. Then $t=1$ is the only choice, and $G(g_4)$ is a nonassociative commutative A-loop. By Lemma \[Lm:TranslatedCenters\], $Z(G(g_4))=1$ and $Z(G(f_1)) = Z(G(g_1))\cong \mathbb Z_2$. Thus $G(g_4)$ is a new nonassociative commutative A-loop. Finally, let $g_5 = (a,ab,b)$. Since $g_4$, $g_5$ are conjugate in $\aut{G}$, $G(g_4)\cong G(g_5)$ by Corollary \[Cr:IsoGf\](i). A class of commutative A-loops of exponent $2$ with trivial center and middle nucleus of index $2$ {#Ss:Index2} -------------------------------------------------------------------------------------------------- Let $\gf{2}$ be the two-element field and let $V$ be a vector space over $\gf{2}$ of dimension $n\ge 2$. Let $G=(V,+)$ be the corresponding elementary abelian $2$-group. Let $\{e_1,\dots,e_n\}$ be a basis of $V$. Define an automorphism of $G$ by $$g(e_1)=e_2,\quad g(e_2)=e_3,\quad g(e_{n-1})=e_n,\quad g(e_n)=e_1+e_n.$$ Since $g(x+x) = g(0) = 0 = g(x)+g(x)$, the equivalence of (i) and (iv) in Proposition \[Pr:AGf\] with $f=g$ shows that $Q_n = G(f)$ is a commutative A-loop of order $2^{n+1}$ with nucleus of index at most $2$. We claim that $g$ has no fixed points besides $0$. Indeed, for $x=\sum_{i=1}^n\alpha_ie_i$ we have $$g(x) = \alpha_ne_1 + \alpha_1e_2+\cdots \alpha_{n-2}e_{n-1} + (\alpha_{n-1}+\alpha_n)e_n,$$ so $x=g(x)$ if and only if $$\alpha_1=\alpha_n,\quad \alpha_2=\alpha_1,\quad \alpha_{n-1}=\alpha_{n-2},\quad \alpha_n=\alpha_{n-1}+\alpha_n,$$ or, $\alpha_1=\cdots = \alpha_n=0$. Thus Lemma \[Lm:TranslatedCenters\] implies that $Z(Q_n)=1$, and $[Q_n:\mnuc{Q_n}]=2$ follows. Finally, $x*x=x+x=0$ and $\ov{x}*\ov{x} = g(x+x) = 0$ for every $x\in G$, so $Q_n$ has exponent two. Central extensions based on trilinear forms {#Sc:Extensions} =========================================== Let $Z$, $K$ be loops. We say that a loop $Q$ is an *extension* of $Z$ by $K$ if $Z\unlhd Q$ and $Q/Z\cong K$. If $Z\le Z(Q)$, the extension is said to be *central*. It is well-known that central extensions of an abelian group $Z$ by a loop $K$ are precisely the loops $K\ltimes_\theta Z$ defined on $K\times Z$ by $$(x,a)(y,b) = (xy,\,ab\theta(x,y)),$$ where $\theta:K\times K\to Z$ is a *(loop) cocycle*, that is, a mapping satisfying $\theta(x,1) = \theta(x,1)=1$ for every $x\in K$. In [@BP Theorem 6.4], Bruck and Paige described all central extensions of an abelian group $Z$ by an A-loop $K$ resulting in an A-loop $Q$. The cocycle identity they found is rather complicated, and despite some optimism of Bruck and Paige, it is by no means easy to construct cocycles that conform to it. In the commutative case, we deduce from [@BP Theorem 6.4]: \[Cr:CentralExtension\] Let $Z$ be an abelian group and $K$ a commutative A-loop. Let $\theta:K\times K\to Z$ be a cocycle satisfying $\theta(x,y)=\theta(y,x)$ for every $x$, $y\in K$ and $$\label{Eq:Cocycle} F(x,y,z)F(x',y,z)\theta(R_{y,z}(x),R_{y,z}(x')) = F(xx',y,z)\theta(x,x')$$ for every $x$, $y$, $z$, $x'\in K$, where $$F(x,y,z) = \theta(R_{y,z}(x),yz)^{-1}\theta(y,z)^{-1}\theta(xy,z)\theta(x,y).$$ Then $K\ltimes_\theta Z$ is a commutative A-loop. Conversely, every commutative A-loop that is a central extension of $Z$ by $K$ can be represented in this manner. Let $Z$ be an elementary abelian $2$-group and $K$ a commutative A-loop of exponent two. Let $\theta:K\times K\to Z$ be a cocycle satisfying $\theta(x,y) = \theta(y,x)$ for every $x$, $y\in K$, $\theta(x,x)=1$ for every $x\in K$, and $$\begin{gathered} \label{Eq:Cocycle2} \theta(x,y)\theta(x',y)\theta(xx',y)\theta(x,x')\theta(xy,z)\theta(x'y,z)\theta(y,z)\theta((xx')y,z)=\\ \theta(R_{y,z}(x),yz)\theta(R_{y,z}(x'),yz)\theta(R_{y,z}(xx'),yz)\theta(R_{y,z}(x),R_{y,z}(x'))\end{gathered}$$ for every $x$, $y$, $z$, $x'\in K$. Then $K\ltimes_\theta Z$ is a commutative A-loop of exponent two. Conversely, every commutative A-loop of exponent two that is a central extension of $Z$ by $K$ can be represented in this manner. When $K$ is an elementary abelian $2$-group, the cocycle identity can be rewritten as $$\begin{aligned} &\theta(x,y)\theta(x',y)\theta(xx',y)\notag\\ &\theta(xy,z)\theta(x'y,z)\theta(xx',z)\label{Eq:CocycleLines}\\ &\theta(x,yz)\theta(x',yz)\theta(xx',yz)\notag\\ &\theta(y,z)\theta(xx',z)\theta((xx')y,z)=1.\notag\end{aligned}$$ Since every line above is of the form $\theta(u,w)\theta(v,w)\theta(uv,w)$, it is tempting to try to satisfy by imposing $\theta(u,w)\theta(v,w)\theta(uv,w)=1$ for every $u$, $v$, $w\in K$. However, that identity already implies associativity. A nontrivial solution to the cocycle identity for commutative A-loops of exponent two can be obtained as follows: \[Pr:Trilinear\] Let $Z=\gf{2}$ and let $K$ be an elementary abelian $2$-group. Let $g:K^3\to\gf{2}$ be a trilinear form such that $g(x,xy,y) = g(y,xy,x)$ for every $x$, $y\in K$. Define $\theta:K^2\to\gf{2}$ by $\theta(x,y) = g(x,xy,y)$. Then $Q=K\ltimes_\theta Z$ is a commutative A-loop of exponent $2$. Moreover, $(y,b)\in \mnuc{Q}$ if and only if for every $x$, $z\in K$ we have $g(y,x,z)=g(x,z,y)$. Trilinearity alone implies that $\theta(u,w)\theta(v,w)\theta(uv,w)=g(u,v,w)g(v,u,w)$. The left-hand side of can then be rewritten as $$g(x,x',y)g(x',x,y)g(xy,x'y,z)g(x'y,xy,z)g(x,x',yz)g(x',x,yz)g(y,xx',z)g(xx',y,z),$$ which reduces to $1$ by trilinearity. We have $(y,b)\in\mnuc{Q}$ if and only if $\theta(x,y)\theta(xy,z) = \theta(y,z)\theta(x,yz)$ for every $x$, $z\in K$, and the rest follows from trilinearity of $g$. Let $V=\gf{2}^n$. Call a $3$-linear form $g:V\to\gf{2}$ *$(1,3)$-symmetric* if $g(x,y,z) = g(z,y,x)$ for every $x$, $y$, $z\in V$. By Proposition \[Pr:Trilinear\], a $(1,3)$-symmetric trilinear form gives rise to a commutative A-loop $Q$ of exponent $2$, and $(y,b)\in \mnuc{Q}$ if and only if $g(y,x,z)=g(y,z,x)$ for every $x$, $z$, that is, if and only if the induced bilinear form $g(y,-,-):V^2\to \gf{2}$ is symmetric. \[Lm:NewForms\] Let $V$ be a vector space over $\gf{2}$ of dimension at least $3$. Then there exists a trilinear form $g:V\to\gf{2}$ such that for any $0\ne x\in V$ the induced bilinear form $g(x,-,-):V^2\to \gf{2}$ is not symmetric. Let $\{e_1,\dots,e_n\}$ be a basis of $V$. The trilinear form $g$ is determined by the values $g(e_i,e_j,e_k)\in\gf{2}$, for $1\le i$, $j$, $k\le n$. Set $g(e_i,e_i,e_{i+1})=1$ for every $i$ (with $e_{n+1}=e_1$) and $g(e_i,e_j,e_k)=0$ otherwise. Let $x = \sum\alpha_j e_j$ be such that $\alpha_i\ne 0$ for some $i$. Then $g(x,e_i,e_{i+1}) = \sum \alpha_j g(e_j,e_i,e_{i+1}) = \alpha_i g(e_i,e_i,e_{i+1}) = \alpha_i\ne 0$, while, similarly, $g(x,e_{i+1},e_i)=0$. \[Ex:SmallMiddleNucleus\] By Lemma $\ref{Lm:NewForms}$, for every $n\ge 3$ there is a commutative A-loop $Q$ of exponent $2$ and order $2^{n+1}$ with $\mnuc{Q}=Z(Q)$, $|Z(Q)|=2$. Let $Q$ be a finite commutative A-loop of exponent $2$. By results of [@JKV], $|Q|=2^k$ for some $k$. Let $|\mnuc{Q}|=2^\ell$. We show how to realize all possible pairs $(k,\ell)$ with $\ell>0$. \[Lm:PossibleMiddleNuclei\] Let $k\ge \ell>0$. Then there is a nonassociative commutative A-loop of order $2^k$ with middle nucleus of order $2^\ell$ if and only if: either $d=k-\ell\ge 3$, or $d\ge 1$ and $\ell\ge 2$. If $d\ge 3$, consider the loop $Q$ of order $2^{d+1}$ with middle nucleus of order $2$ from Example \[Ex:SmallMiddleNucleus\]. Then $Q\times(\mathbb Z_2)^{k-(d+1)}$ achieves the parameters $(k,\ell)$. Assume that $d=2$. The parameters $(3,1)$ are not possible by §\[Sc:AppsIndex2\], and the parameters $(4,2)$ are possible (see §\[Sc:Enumeration\]). Then $(k,\ell)$ can be achieved using the appropriate direct product. Finally, assume that $d=1$. Then we are done by Subsection \[Ss:Index2\]. We obviously must have $\ell\ge 2$, else $|Q|=2^k\le 4$. We remark that Lemma \[Lm:NewForms\] cannot be improved: \[Lm:LowDimension\] Let $V=\gf{2}^n$ and let $g:V^3\to\gf{2}$ be a $(1,3)$-symmetric trilinear form. If $n<3$ then there is $0\ne x\in V$ such that the induced form $g(x,-,-)$ is symmetric. There is nothing to show when $n=1$, so assume that $n=2$ and $\{e_1$, $e_2\}$ is a basis of $V$. The form $g$ is determined by the $6$ values $g(e_1,e_1,e_1)$, $g(e_1,e_1,e_2)$, $g(e_1,e_2,e_1)$, $g(e_1,e_2,e_2)$, $g(e_2,e_1,e_2)$ and $g(e_2,e_2,e_2)$. Suppose that no induced form $g(x,-,-)$ is symmetric, for $0\ne x\in V$. Then $g(e_1,e_1,e_2)\ne g(e_1,e_2,e_1)$, else $g(e_1,-,-)$ is symmetric. Similarly, $g(e_2,e_1,e_2)\ne g(e_2,e_2,e_1)$. But then $g(e_1+e_2,e_1,e_2) = g(e_1,e_1,e_2)+g(e_2,e_1,e_2) = g(e_1,e_2,e_1)+g(e_2,e_2,e_1) = g(e_1+e_2,e_2,e_1)$, hence $g(e_1+e_2,-,-)$ is symmetric, a contradiction. The many examples presented so far might suggest that $Q/\mnuc{Q}$ is a group in every commutative A-loop. This is not so: Consider a commutative Moufang loop $Q$. Then $Q$ is a commutative A-loop, and $\mnuc{Q}=Z(Q)$ since the three nuclei of $Q$ coincide. So the statement “$Q/\mnuc{Q}$ is a group” is equivalent to “$Q/Z(Q)$ is an abelian group”, i.e., to “$Q$ has nilpotency class at most $2$”. There are commutative Moufang loops of nilpotency class $3$. Find a smallest commutative A-loop $Q$ in which $Q/\mnuc{Q}$ is not a group. Adding group cocycles --------------------- Let $Z$ be an abelian group and $K$ a loop. Then a loop cocycle $\theta:K\times K\to Z$ is said to be a *group cocycle* if it satisfies the identity $$\label{Eq:GroupCocycle} \theta(x,y)\theta(xy,z)=\theta(y,z)\theta(x,yz).$$ Note that if $K$ is a group and $\theta$ is a group cocycle then $K\ltimes_\theta Z$ is a group, too. \[Lm:AddGroupCocycle\] Let $Z$ be an abelian group, $K$ a group and $\theta$, $\mu:K\times K\to Z$ loop cocycles such that $\nu=\theta\mu^{-1}:(x,y)\mapsto \theta(x,y)\mu(x,y)^{-1}$ is a group cocycle. Then the left inner mappings in $K\ltimes_\theta Z$ and $K\ltimes_\mu Z$ coincide. Calculating in $K\ltimes_\theta Z$, we have $$\begin{aligned} (x,a)(y,b) &= (xy,ab\theta(x,y)),\\ (x,a)\ld (y,b) &= (x\ld y, a^{-1}b\theta(x,x\ld y)^{-1}).\end{aligned}$$ Then $$\begin{gathered} \label{Eq:SameCoset} (x,a)(y,b)\ld (x,a)((y,b)(z,c)) = (xy,ab\theta(x,y))\ld (xyz,abc\theta(x,yz)\theta(y,z)\\ = (z,c\theta(x,yz)\theta(y,z)\theta(x,y)^{-1}\theta(xy,z)^{-1}).\end{gathered}$$ Thus the left inner mappings in $K\ltimes_\theta Z$ and $K\ltimes_\mu Z$ coincide if and only if $$\theta(x,yz)\theta(y,z)\theta(x,y)^{-1}\theta(xy,z)^{-1} = \mu(x,yz)\mu(y,z)\mu(x,y)^{-1}\mu(xy,z)^{-1}$$ for every $x$, $y$, $z\in K$, which happens precisely when $\nu = \theta\mu^{-1}$ is a group cocycle. \[Lm:AddGroupCocycle2\] Let $Z$ be an abelian group, $K$ a group and $\theta:K\times K\to Z$ a cocycle such that $K\ltimes_\theta Z$ is a commutative A-loop. Let $\mu:K\times K\to Z$ be a group cocycle satisfying $\mu(x,y)=\mu(y,x)$ for every $x$, $y\in K$. Then $K\ltimes_{\mu\theta} Z$ is a commutative A-loop with the same (left) inner mappings as $K\ltimes_\theta Z$. Both $Q_\theta = K\ltimes_\theta Z$, $Q_{\mu\theta}=K\ltimes_{\mu\theta} Z$ are commutative loops. Since $\mu\theta\theta^{-1}$ is a group cocycle, $Q_{\mu\theta}$ has the same (left) inner mappings as $Q_\theta$, by Lemma \[Lm:AddGroupCocycle\]. It therefore remains to show that every left inner mapping of $Q_{\mu\theta}$ is an automorphism. Let $(x,a)$, $(y,b)\in K\times Z$ and let $\varphi$ be a left inner mapping of $Q_{\mu\theta}$ (and hence of $Q_\theta$). Denote by $\cdot$ the multiplication in $Q_\theta$ and by $*$ the multiplication in $Q_{\mu\theta}$. Then $$\varphi((x,a)*(y,b)) = \varphi((x,a)\cdot(y,b)\cdot(1,\mu(x,y))) = \varphi((x,a))\cdot \varphi((y,b))\cdot (1,\mu(x,y)),$$ because $(1,\mu(x,y))\in Z$ is a central element. The equation in fact shows that $\varphi((x,a)) = (x,a')$ for some $a'$, and similarly, $\varphi((y,b)) = (y,b')$ for some $b'$. Thus $$\varphi((x,a))\cdot\varphi((y,b))\cdot(1,\mu(x,y)) = (x,a')\cdot (y,b')\cdot (1,\mu(x,y)) = (x,a')*(y,b') = \varphi((x,a))*\varphi((y,b)),$$ proving $\varphi\in\aut{Q_{\mu\theta}}$. A class of commutative A-loops of order $p^3$ {#Sc:p3} ============================================= Let $Q$ be a commutative A-loop of odd order. Equivalently, let $Q$ be a finite commutative A-loop in which the mapping $x\mapsto x^2$ is a bijection of $Q$ (cf. [@JKV Lemma 3.1]). For $x\in Q$, denote by $x^{1/2}$ the unique element of $Q$ such that $(x^{1/2})^2 = x$. Define a new operation $\circ$ on $Q$ by $$x\circ y = (x^{-1}\ld xy^2)^{1/2}.$$ By [@JKV Lemma 3.5], $(Q,\circ)$ is a Bruck loop. By [@JKV Corollary 3.11], $(Q,\circ)$ is commutative if and only if it is isomorphic to $Q$. \[Pr:p2\] Let $p$ be an odd prime, and let $Q$ be a commutative A-loop of order $p$, $2p$, $4p$, $p^2$, $2p^2$ or $4p^2$. Then $Q$ is an abelian group. Loops of order less than $5$ are abelian groups. By the Decomposition Theorem mentioned in the introduction, it remains to prove that commutative A-loops of order $p$ and $p^2$ are abelian groups. For $|Q|=p$, this follows from the Lagrange Theorem and power-associativity. Assume that $|Q|=p^2$. Then $(Q,\circ)$ is a Bruck loop of order $p^2$, in particular a Bol loop of order $p^2$. Burn showed in [@Burn] that all Bol loops of order $p^2$ are groups, and hence $(Q,\circ)$ is an abelian group. Consequently, $Q$ is an abelian group. In this section we initiate the study of nonassociative commutative A-loops of order $p^3$. We conjecture that the class of loops constructed below accounts for all such loops. There is no commutative A-loop with center of prime index. For a contradiction, let $Q$ be a commutative A-loop such that $|Q/Z(Q)|=p$ for some prime $p$. By Proposition \[Pr:p2\], $Q/Z(Q)$ is the cyclic group of order $p$. Let $x\in Q\setminus Z(Q)$. Then $|xZ(Q)|=p$ and every element of $Q$ can be written as $x^iz$, where $0\le i<p$ and $z\in Z(Q)$. With $0\le i$, $j$, $k<p$ and $z_1$, $z_2$, $z_3\in Z(Q)$ we have $$(x^iz_1\cdot x^jz_2)\cdot x^kz_3 = (x^ix^j)x^k\cdot z_1z_2z_3 = x^i(x^jx^k)\cdot z_1z_2z_3 = x^iz_1\cdot (x^jz_2\cdot x^kz_3)$$ by power-associativity, so $Q$ is an abelian group with center of prime index, a contradiction. Hence a nonassociative commutative A-loop of order $p^3$ has center of size $1$ or $p$. (By the result announced in the introduction, we know, in fact, that the center must have size $p$ if $p$ is odd.) Let $n\ge 1$. The *overflow indicator* is the function $(-,-)_n:\mathbb Z_n\times\mathbb Z_n\to \{0,1\}$ defined by $$(x,y)_n = \left\{\begin{array}{ll} 1,\text{ if $x+y\ge n$,}\\ 0,\text{ otherwise}. \end{array}\right.$$ Denote by $\oplus$ the addition in $\mathbb Z_n$, and note that for $x$, $y\in\mathbb Z_n$ we have $x\oplus y = x+y-n(x,y)_n$, and thus $$\label{Eq:Indicator} (x,y)_n = \frac{x+y-(x\oplus y)}{n}.$$ \[Lm:IndicatorGroup\] We have $$\label{Eq:IndicatorGroup} (x,y)_n + (x\oplus y,z)_n = (y,z)_n + (x,y\oplus z)_n$$ for every $x$, $y$, $z\in\mathbb Z_n$. Using , the identity can be rewritten as $$x+y-(x\oplus y) + (x\oplus y)+z - (x\oplus y\oplus z) = y+z - (y\oplus z) + x + (y\oplus z) - (x\oplus y\oplus z),$$ which holds. From now on we write $+$ for the addition in $\mathbb Z_n$, too. For $n\ge 1$ and $a$, $b\in\mathbb Z_n$, define $\terg{\mathbb Z_n}{a}{b}$ on $\mathbb Z_n\times\mathbb Z_n\times\mathbb Z_n$ by $$\label{Eq:Terg} (x_1,x_2,x_3)(y_1,y_2,y_3) = (x_1+y_1+(x_2+y_2)x_3y_3 + a(x_2,y_2)_n + b(x_3,y_3)_n, x_2+y_2,x_3+y_3).$$ Then $\terg{\mathbb Z_n}{a}{b}$ can be seen as a central extension of $\mathbb Z_n$ by $\mathbb Z_n\times\mathbb Z_n$ via the loop cocycle $\theta((x_2,x_3),(y_2,y_3)) = (x_2+y_2)x_3y_3+a(x_2,y_2)_n+b(x_3,y_3)_n$, and hence $\terg{\mathbb Z_n}{a}{b}$ is a commutative loop with neutral element $(0,0,0)$. Note that we can write $\theta$ as $\theta=\mu+\nu$, where $\mu((x_2,y_2),(x_3,y_3)) = (x_2+y_2)x_3y_3$ and $\nu((x_2,y_2),(x_3,y_3)) = a(x_2,y_2)_n + b(x_3,y_3)_n$. By Lemma \[Lm:IndicatorGroup\], $\nu$ is a group cocycle. \[Pr:Terg\] Let $n\ge 2$ and $a$, $b\in\mathbb Z_n$. Let $Q=\terg{\mathbb Z_n}{a}{b}$ and $x=(x_1,x_2,x_3)$, $y=(y_1,y_2,y_3)$, $z=(z_1,z_2,z_3)\in Q$. Then: 1. $x\ld y = (y_1-x_1-(y_3-x_3)x_3y_2-a(x_2,y_2-x_2)_n - b(x_3,y_3-x_3)_n,y_2-x_2,y_3-x_3)$, 2. $L_{y,x}(z) = xy\ld x(yz) = (z_1+y_3(x_3z_2-x_2z_3),z_2,z_3)$, 3. $Q$ is a nonassociative commutative A-loop of order $n^3$, 4. $\lnuc{Q} = Z(Q) = \mathbb Z_n\times 0\times 0$, $\mnuc{Q} = \mathbb Z_n\times\mathbb Z_n\times 0$ as subsets of $Q$, 5. $Q/Z(Q)\cong\inn{Q}\cong \mathbb Z_n\times\mathbb Z_n$, and $\inn{Q} = \{L_{u,v};\;u,v\in Q\}$, 6. for every $m\ge 0$, $x^m = (mx_1+2\binom{m+1}{3}x_2x_3^2 + at_2+bt_3,mx_2,mx_3)$, where $t_i = \sum_{k=1}^{m-1}(x_i,kx_i)_n$. (As usual, the summation is considered empty and the binomial coefficient vanishes when $m<2$.) Part (i) follows from the multiplication formula . Let $Q_0 = \terg{\mathbb Z_n}{0}{0}$. By Lemma \[Lm:AddGroupCocycle\], it suffices to verify the formula (ii) for $Q_0$ instead of $Q$. Now, calculating in $Q_0$, $$x(yz) = (x_1+y_1+z_1+(y_2+z_2)y_3z_3 + (x_2+y_2+z_2)x_3(y_3+z_3),x_2+y_2+z_2,x_3+y_3+z_3),$$ so (i) for $Q_0$ implies that $xy\ld x(yz)$ is equal to $$(z_1+(y_2+z_2)y_3z_3+(x_2+y_2+z_2)x_3(y_3+z_3)-(x_2+y_2)x_3y_3 -z_3(x_3+y_3)(x_2+y_2+z_2),z_2,z_3),$$ which simplifies in a straightforward way to (ii). By Lemma \[Lm:AddGroupCocycle2\], to verify that left inner mappings of $Q$ are automorphisms of $Q$, it suffices to check that the left inner mappings of $Q_0$ are automorphisms of $Q_0$. With $u=(u_1,u_2,u_3)$, $v=(v_1,v_2,v_3)$, use (ii) to see that $$\begin{aligned} &xy\ld x(yu)\cdot xy\ld x(yv) \\ &=(u_1+y_3(x_3u_2-x_2u_3),u_2,u_3)(v_1+y_3(x_3v_2-x_2v_3),v_2,v_3)\\ &=(u_1{+}v_1{+}y_3(x_3(u_2{+}v_2){-}x_2(u_3{+}v_3)){+}(u_2{+}v_2)u_3v_3{+}a(u_2,v_2)_n{+}b(u_3,v_3)_n, u_2{+}v_2, u_3{+}v_3)\\ &=xy\ld x(y\cdot uv).\end{aligned}$$ Hence $Q$ is a commutative A-loop of order $n^3$. To calculate the middle nucleus, we can once again resort to the loop $Q_0$, since the group cocycle will not play any role in identities that are consequences of associativity. We have $$\begin{aligned} y\cdot (x_1,x_2,0)z &= y(x_1+z_1,x_2+z_2,z_3)\\ &=(x_1+y_1+z_1+(x_2+y_2+z_2)y_3z_3,x_2+y_2+z_2,y_3+z_3)\\ &=(x_1+y_1,x_2+y_2,y_3)z = y(x_1,x_2,0)\cdot z,\end{aligned}$$ so $\mathbb Z_n\times\mathbb Z_n\times 0\le \mnuc{Q_0}$. On the other hand, $$(0,0,x_3)(x_1,x_2,0) = (x_1,x_2,x_3),$$ so to prove that $(x_1,x_2,x_3)\not\in\mnuc{Q_0}$ whenever $x_3\ne 0$, it suffices to show that $(0,0,x_3)\not\in\mnuc{Q_0}$ whenever $x_3\ne 0$. Now, $$\begin{gathered} (0,0,1)\cdot (0,0,x_3)(0,1,0) = (0,0,1)(0,1,x_3) = (x_3,1,1+x_3)\\ \ne (0,1,1+x_3) = (0,0,1+x_3)(0,1,0) = (0,0,1)(0,0,x_3)\cdot (0,1,0)\end{gathered}$$ shows just that. Similarly, $$\begin{aligned} (x_1,0,0)\cdot yz &= (x_1,0,0)(y_1+z_1+(y_2+z_2)y_3z_3,y_2+z_2,y_3+z_3)\\ &=(x_1+y_1+z_1+(y_2+z_2)y_3z_3,y_2+z_2,y_3+z_3)\\ &=(x_1+y_1,y_2,y_3)z = (x_1,0,0)y\cdot z\end{aligned}$$ proves that $\mathbb Z_n\times 0\times 0\le \lnuc{Q_0}$, and, for $x_2\ne 0$, $$\begin{gathered} (x_1,x_2,0)\cdot (0,0,1)(0,0,1) = (x_1,x_2,0)(0,0,2) = (x_1,x_2,2)\\ \ne (x_1+x_2,x_2,2) = (x_1,x_2,1)(0,0,1) = (x_1,x_2,0)(0,0,1)\cdot (0,0,1)\end{gathered}$$ implies that $\lnuc{Q} = \mathbb Z_n\times 0\times 0$ (recall that $\lnuc{Q}\le\mnuc{Q}$ in any A-loop $Q$). Consider the mapping $\varphi:Q\to\inn{Q}$ defined by $$\varphi(x_1,x_2,x_3) = L_{(0,0,1),(0,x_2,x_3)}.$$ Then $$\begin{aligned} &\varphi(x_1,x_2,x_3)\varphi(y_1,y_2,y_3)(z_1,z_2,z_3)\\ &=\varphi(x_1,x_2,x_3)(z_1+y_3z_2-y_2z_3,z_2,z_3) = (z_1+y_3z_2-y_2z_3+x_3z_2-x_2z_3,z_2,z_3)\\ &=\varphi((x_1,x_2,x_3)(y_1,y_2,y_3))(z_1,z_2,z_3)\end{aligned}$$ and $\varphi$ is a homomorphism. Its kernel consists of all $(x_1,x_2,x_3)\in Q$ such that $x_3z_2-x_2z_3=0$ for every $z_2$, $z_3\in Q$. Thus $\ker{\varphi} = \{(x_1,0,0);\;x_1\in \mathbb Z_n\}$. To prove (v), it remains to show that $\varphi$ is onto $\inn{Q}$. By (ii), $$L_{(y_1,y_2,y_3),(x_1,x_2,x_3)} = L_{(0,0,y_3),(0,x_2,x_3)} = L_{(0,0,1),(0,y_3x_2,y_3x_3)}.$$ This means that $\im\varphi$ contains a generating subset of $\inn{Q}$, and hence it is equal to $\inn{Q}$. In fact, purely on the grounds of cardinality, we have $\inn{Q} = \{L_{u,v};\;u,\,v\in Q\}$. The identity of (vi) clearly holds when $m=0$. Assume that it holds for some $m\ge 0$. Let $t_i^m = \sum_{k=1}^m (x_i,kx_i)_n$. By power-associativity, we have $$\begin{aligned} x^{m+1} &= xx^m = x(mx_1+2\binom{m+1}{3}x_2x_3^2 + at_2^{m-1} + bt_3^{m-1}, mx_2,mx_3)\\ &= ((m{+}1)x_1{+}2\binom{m+1}{3}x_2x_3^2{+}(m{+}1)x_2mx_3^2{+}at_2^m{+}bt_3^m, (m{+}1)x_2,(m{+}1)x_3),\end{aligned}$$ Since $2\binom{m+1}{3}+(m+1)m = 2\binom{m+2}{3}$, we are through. \[Lm:IsoExp\] Let $p$ be a prime and $a$, $b\in\mathbb Z_p$. Let $Q=\terg{\mathbb Z_p}{a}{b}$. Then: 1. if $(a,b)=(0,0)$ and $p\ne 3$ then $Q$ has exponent $p$, 2. if $(a,b)\ne(0,0)$ or $p=3$ then $Q$ has exponent $p^2$, 3. if $a=0$ then $\mnuc{Q}\cong\mathbb Z_p\times\mathbb Z_p$, 4. if $a\ne 0$ then $\mnuc{Q}\cong\mathbb Z_{p^2}$. By [@JKV], every element of $Q$ has order a power of $p$, so $Q$ has exponent $p$, $p^2$ or $p^3$. Since $Q$ is nonassociative by Proposition \[Pr:Terg\], the exponent is either $p$ or $p^2$. Assume that $(a,b)=(0,0)$. Then by Proposition \[Pr:Terg\](vi), $$(x_1,x_2,x_3)^p = (2\binom{p+1}{3}x_2x_3^2,0,0).$$ The integer $2\binom{p+1}{3}$ is divisible by $p$ if and only if $p\ne 3$. This proves (i). To show (ii), it remains to prove that $Q$ has exponent $p^2$ if $(a,b)\ne (0,0)$. Assume that $a\ne 0$, and note that, by Proposition \[Pr:Terg\](vi), $$(0,1,0)^p = (a\sum_{k=1}^{p-1}(1,k)_p,0,0) = (a(1,p-1)_p,0,0) = (a,0,0).$$ This means that $Q$ does not have exponent $p$, and it also shows, by Proposition \[Pr:Terg\](iv), that $\mnuc{Q}\cong\mathbb Z_{p^2}$. Similarly, when $b\ne 0$, use $$(0,0,1)^p = (b\sum_{k=1}^{p-1}(1,k)_p,0,0) = (b,0,0)$$ to conclude that $Q$ does not have exponent $p$. Finally, when $a=0$, we have $(x_1,x_2,0)^p=0$ by Proposition \[Pr:Terg\](vi), so $\mnuc{Q}\cong\mathbb Z_p\times\mathbb Z_p$ by Proposition \[Pr:Terg\](iv). As usual, denote by $\mathbb Z_n^*$ the set of all invertible elements of $\mathbb Z_n$. \[Lm:TergIso1\] Let $n>0$. If $b$, $c\in \mathbb Z_n^*$ then $\terg{\mathbb Z_n}{0}{b} \cong \terg{\mathbb Z_n}{0}{c}$. Define $\varphi:\terg{\mathbb Z_n}{0}{b}\to\terg{\mathbb Z_n}{0}{c}$ by $(x_1,x_2,x_3)\mapsto ((c/b)x_1,(c/b)x_2,x_3)$, and note that $\varphi$ is a bijection since $b$, $c$ are invertible. Denote by $\cdot$ the multiplication in $\terg{\mathbb Z_n}{0}{b}$ and by $*$ the multiplication in $\terg{\mathbb Z_n}{0}{c}$. Then $$\begin{aligned} &\varphi((x_1,x_2,x_3)\cdot(y_1,y_2,y_3)) = \varphi((x_1+y_1+(x_2+y_2)x_3y_3+b(x_3,y_3)_n,x_2+y_2,x_3+y_3))\\ &=(\frac{c}{b}(x_1+y_1+(x_2+y_2)x_3y_3+b(x_3,y_3)_n),\frac{c}{b}(x_2+y_2),x_3+y_3)\\ &=(\frac{c}{b}x_1,\frac{c}{b}x_2,x_3)*(\frac{c}{b}y_1,\frac{c}{b}y_2,y_3) = \varphi((x_1,x_2,x_3))*\varphi((y_1,y_2,y_3)).\end{aligned}$$ Let $p$ be an odd prime. Recall that $a\in\mathbb Z_p^*$ is a *quadratic residue modulo $p$* if there is $x\in\mathbb Z_p^*$ such that $x^2\equiv a\pmod p$. Else $a$ is a *quadratic nonresidue modulo $p$*. Also recall that $ab^{-1}$ is a quadratic residue if and only if either both $a$, $b$ are quadratic residues or both $a$, $b$ are quadratic nonresidues. \[Lm:IsoQuadraticResidue\] Let $p$ be an odd prime and $a_1$, $a_2\in\mathbb Z_p^*$. If $a_1$, $a_2$ are either both quadratic residues or both quadratic nonresidues then $\terg{\mathbb Z_p}{a_1}{0}\cong\terg{\mathbb Z_p}{a_2}{0}$. Since $a_1a_2^{-1}$ is a quadratic residue, there is $u$ such that $a_2=a_1u^2$. Define $\varphi:\terg{\mathbb Z_p}{a_1}{0}\to\terg{\mathbb Z_p}{a_2}{0}$ by $(x_1,x_2,x_3)\mapsto (u^2x_1,x_2,ux_3)$. Then $\varphi$ is a bijection. Denote by $\cdot$ the multiplication in $\terg{\mathbb Z_p}{a_1}{0}$ and by $*$ the multiplication in $\terg{\mathbb Z_p}{a_2}{0}$. Then $$\begin{aligned} &\varphi((x_1,x_2,x_3)\cdot (y_1,y_2,y_3)) = \varphi((x_1+y_1+(x_2+y_2)x_3y_3+a_1(x_2,y_2)_p,x_2+y_2,x_3+y_3))\\ &= (u^2(x_1+y_1+(x_2+y_2)x_3y_3+a_1(x_2,y_2)_p),x_2+y_2,u(x_3+y_3))\\ &= (u^2x_1+u^2y_1+(x_2+y_2)ux_3uy_3+a_2(x_2,y_2)_p, x_2+y_2, u(x_3+y_3))\\ &= (u^2x_1,x_2,ux_3)*(u^2y_1,y_2,uy_3) = \varphi((x_1,x_2,x_3))*\varphi((y_1,y_2,y_3)).\end{aligned}$$ \[Lm:Isof\] For a prime $p$, let $Q_1=\terg{\mathbb Z_p}{a}{b}=(Q_1,\cdot)$, $Q_2=\terg{\mathbb Z_p}{a}{c}=(Q_2,*)$ and let $f:Q_1\to Q_2$ be an isomorphism that pointwise fixes the middle nucleus of $Q_1$ $($i.e., $f$ is identical on $\mathbb Z_p\times\mathbb Z_p\times 0)$. Then there are $A$, $B\in\mathbb Z_p$ and $C\in\mathbb Z_p^*$ such that $$\label{Eq:Isof} f(x_1,x_2,x_3) = (x_1,x_2,0)*(A,B,C)^{x_3}$$ for every $(x_1,x_2,x_3)\in Q_1$. In addition, every mapping $f:Q_1\to Q_2$ defined by with $A$, $B\in\mathbb Z_p$ and $C\in \mathbb Z_p^*$ is a bijection that pointwise fixes $\mnuc{Q_1}$. Let $f:Q_1\to Q_2$ be an isomorphism that pointwise fixes $\mnuc{Q_1}$. As $Q_1/\mnuc{Q_1}$ is a cyclic group, $f$ is determined by the image of any element in $Q_1\setminus \mnuc{Q_1}$. Let $f(0,0,1)=(A,B,C)$. We must have $C\ne 0$, else $f$ is not a bijection. Since $(x_1,x_2,x_3)=(x_1,x_2,0)(0,0,x_3)$ and $(0,0,x_3) = (0,0,1)^{x_3}$ by Proposition \[Pr:Terg\](vi), we have $$f(x_1,x_2,x_3) = f(x_1,x_2,0)*f(0,0,1)^{x_3} = (x_1,x_2,0)*(A,B,C)^{x_3}.$$ Conversely, define $f:Q_1\to Q_2$ by , where $C\ne 0$. Then $f$ obviously pointwise fixes $\mnuc{Q_1}$. To show that $f$ is a bijection, assume that $f(x_1,x_2,x_3) = f(y_1,y_2,y_3)$. Since the last coordinate of $(x_1,x_2,0)*(A,B,C)^{x_3}$ is $Cx_3$, we conclude that $x_3=y_3$. The second coordinate of $(x_1,x_2,0)*(A,B,C)^{x_3}$ is $x_2+Bx_3$, and we conclude that $x_2=y_2$. Then $x_1=y_1$ follows from the multiplication formula for $Q_2$ and from Proposition \[Pr:Terg\](vi). \[Lm:IsoSameA\] Let $p\ne 3$ be a prime and assume that $a$, $b$, $c\in\mathbb Z_p$ are such that $a+c\equiv b\pmod p$. Let $Q_1=\terg{\mathbb Z_p}{a}{b} = (Q_1,\cdot)$ and $Q_2=\terg{\mathbb Z_p}{a}{c} = (Q_2,*)$. Then $f:Q_1\to Q_2$ defined by with $(A,B,C) = (0,1,1)$ is an isomorphism. For $x\in \mathbb Z_p$, let $x' = (x-1)x(x+1)/3$. By Lemma \[Lm:Isof\], $f$ is a bijection onto $Q_2$ that pointwise fixes $\mnuc{Q_1}$. Upon expanding the formula , we see that $$f(x_1,x_2,x_3) = (x_1+x_3'+ a(x_2,x_3)_p, x_2+x_3, x_3),$$ since the expression $\sum_{k=1}^{x_3-1}(1,k)_p$ vanishes for every $x_3<p$. Let $$(u_1,u_2,u_3) = f(x_1,x_2,x_3)*f(y_1,y_2,y_3)$$ and $$(v_1,v_2,v_3) = f((x_1,x_2,x_3)\cdot (y_1,y_2,y_3)).$$ A quick calculation then shows that $$(u_2,u_3)=(v_2,v_3)=(x_2+x_3+y_2+y_3,x_3+y_3),$$ $u_1$ is equal to $$x_1{+}x_3'{+}a(x_2,x_3)_p{+}y_1{+}y_3'{+}a(y_2,y_3)_p{+}(x_2{+}x_3{+}y_2{+}y_3)x_3y_3{+}a(x_2{+}x_3,y_2{+}y_3)_p{+}c(x_3,y_3)_p,$$ while $v_1$ is equal to $$x_1{+}y_1{+}(x_2{+}y_2)x_3y_3{+}a(x_2,y_2)_p{+}b(x_3,y_3)_p{+}(x_3{+}y_3)'{+}a(x_2{+}y_2,x_3{+}y_3)_p.$$ Now, $x_3'+y_3' = (x_2+y_2)x_3y_3 + (x_3+y_3)'$. Using , it is easy to see that $$(x_2,x_3)_p + (y_2,y_3)_p+(x_2+x_3,y_2+y_3)_p = (x_2,y_2)_p + (x_2+y_2,x_3+y_3)_p + (x_3,y_3)_p.$$ Hence we are done by $a+c\equiv b\pmod p$. \[Cr:IsoExp\] Let $p\ne 3$ be a prime, $a\in\mathbb Z_p^*$ and $b$, $c\in\mathbb Z_p$. Then $\terg{\mathbb Z_p}{a}{b}$ is isomorphic to $\terg{\mathbb Z_p}{a}{c}$. By Lemma \[Lm:IsoSameA\] we have $\terg{\mathbb Z_p}{a}{0}\cong \terg{\mathbb Z_p}{a}{a}\cong\terg{\mathbb Z_p}{a}{2a}$, and so on. Ring construction ----------------- Note that for $a=b=0$, the construction makes sense over any commutative ring $R$, not just over $\mathbb Z_n$. We can summarize the most important features of the construction as follows: Let $R\ne 0$ be a commutative ring. Let $Q=\ter{R}$ be defined on $R\times R\times R$ by $$(x_1,x_2,x_3)(y_1,y_2,y_3) = (x_1+y_1+(y_2+x_2)x_3y_3,x_2+y_2,x_3+y_3).$$ Then $Q$ is a commutative A-loop satisfying $\lnuc{Q}=Z(Q)=R\times 0\times 0$ and $\mnuc{Q} = R\times R\times 0$. See the relevant parts of the proof of Proposition \[Pr:Terg\]. Towards the classification of commutative A-loops of order $p^3$ ---------------------------------------------------------------- The results obtained up to this point come close to describing the isomorphism types of all loops $\terg{\mathbb Z_p}{a}{b}$ for all primes $p\ne 3$. Fix $p\ne 3$. The loop $\terg{\mathbb Z_p}{0}{0}$ is of exponent $p$ and is not isomorphic to any other loop $\terg{\mathbb Z_p}{a}{b}$, by Lemma \[Lm:IsoExp\]. By Lemma \[Lm:IsoExp\] and Corollary \[Cr:IsoExp\], the loops $\{\terg{\mathbb Z_p}{0}{b};\;0<b<p\}$ form an isomorphism class. By Lemmas \[Lm:IsoQuadraticResidue\] and \[Lm:Isof\], each of the two sets $I_r = \{\terg{\mathbb Z_p}{a}{b};\;a>0$ is a quadratic residue modulo $p$ and $0\le b<p\}$ and $I_n = \{\terg{\mathbb Z_p}{a}{b};\;a>0$ is a quadratic nonresidue modulo $p$ and $0\le b<p\}$ consist of pairwise isomorphic loops. However, we did not manage to establish the following: \[Cr:IsoQuadraticResidue\] Let $p>3$ be a prime, let $a_1\in\mathbb Z_p^*$ be a quadratic residue and $a_2\in\mathbb Z_p^*$ be a quadratic nonresidue. Then $\terg{\mathbb Z_p}{a_1}{0}$ is not isomorphic to $\terg{\mathbb Z_p}{a_2}{0}$. We have verified the conjecture computationally with the GAP [@GAP] package LOOPS [@LOOPS] for $p=5$, $7$. It appears that one of the distinguishing isomorphism invariants is the multiplication group $\mlt{Q} = \langle L_x,\,R_x;\;x\in Q\rangle$. The loops $\terg{\mathbb Z_p}{a}{b}$ behave differently for $p=3$ due to the fact that $3$ is the only prime $p$ for which $p$ does not divide $2\binom{p+1}{3}$. Denote by $f_{(A,B,C)}$ the bijection defined by . It can be verified by computer that $f_{(0,1,1)}$ is an exceptional isomorphism $\terg{\mathbb Z_3}{0}{0}\to\terg{\mathbb Z_3}{0}{1}$, $f_{(0,0,2)}$ is an isomorphism $\terg{\mathbb Z_3}{1}{1}\to\terg{\mathbb Z_3}{1}{2}$, $f_{(0,1,2)}$ is an isomorphism $\terg{\mathbb Z_3}{2}{0}\to\terg{\mathbb Z_3}{2}{1}$ and $f_{(0,1,1)}$ is an isomorphism $\terg{\mathbb Z_3}{2}{0}\to\terg{\mathbb Z_3}{2}{2}$. The loops $\terg{\mathbb Z_3}{0}{0}$, $\terg{\mathbb Z_3}{1}{0}$, $\terg{\mathbb Z_3}{1}{1}$ and $\terg{\mathbb Z_3}{2}{0}$ contain precisely $12$, $6$, $24$ and $18$ elements of order $9$, respectively, so no two of them are isomorphic. Altogether, Figure \[Fg:IsoTypes\] depicts the isomorphism classes of loops $\terg{\mathbb Z_p}{a}{b}$ as connected components, for $p\in\{2,3,5,7\}$ and $a$, $b\in\mathbb Z_p$. Moreover, if Conjecture \[Cr:IsoQuadraticResidue\] is true, the pattern established by $p=2$, $5$ and $7$ continues for all primes $p>7$. It is reasonable to ask whether, for an odd prime $p$, there are nonassociative commutative A-loops of order $p^3$ not of the form $\terg{\mathbb Z_p}{a}{b}$. Using a linear-algebraic approach to cocycles (see Subsection \[Ss:32\]), we managed to classify all nonassociative commutative A-loops of order $p^3$ with nontrivial center, for $p\in\{2,3,5,7\}$. It turns out that all such loops are of the type $\terg{\mathbb Z_p}{a}{b}$. In particular, $p=3$ is the only prime for which there is no nonassociative commutative A-loop of order $p^3$ and exponent $p$. Let $p$ be an odd prime and $Q$ a nonassociative commutative A-loop of order $p^3$. Is $Q$ isomorphic to $\terg{\mathbb Z_p}{a}{b}$ for some $a$, $b\in\mathbb Z_p$? Enumeration {#Sc:Enumeration} =========== We believe that future work will benefit from an enumeration of small commutative A-loops. The results are summarized in Table \[Tb:Enum\], which lists all orders $n\le 32$ for which there exists a nonassociative commutative A-loop. $$\begin{array}{r||c|c|c|c|c|c|c|c} \begin{array}{r} n \end{array} &8&15&16&21&24&27&30&32\\ \hline\hline \begin{array}{r} \text{commutative groups} \end{array} &3&1&5&1&3&3&1&7\\ \hline \begin{array}{r} \text{commutative nonassociative A-loops} \end{array} &4(3)&1&46(38)&1&4(3)&4&1&?\\ \hline \begin{array}{r} \text{commutative nonassociative A-loops}\\ \text{with nontrivial center} \end{array} &3(2)&0&44(37)&0&4(3)&4&1&?\\ \hline \begin{array}{r} \text{commutative nonassociative A-loops}\\ \text{of exponent $p$} \end{array} &2&-&12(11)&-&-&0&-&?\\ \hline \begin{array}{r} \text{commutative nonassociative A-loops}\\ \text{of exponent $p$}\\ \text{with nontrivial center} \end{array} &1&-&10&-&-&0&-&211(210) \end{array}$$ If there is only one number in a cell of the table, it is both the number of isomorphism classes and the number of isotopism classes. If there are two numbers in a cell, the first one is the number of isomorphism classes and the second one (in parentheses) is the number of isotopism classes. All computations were done with the finite model builder Mace4 and with the GAP package LOOPS on a Unix machine with a single $2$ GHz processor, with computational times for individual orders ranging from seconds to hours. Comments on commutative A-loops of order $8$ -------------------------------------------- For classification up to isomorphism, see Section \[Sc:AppsIndex2\]. \[Lm:GfItp\] Let $G$ be a commutative loop, $g\in \aut{G}$, and let $t_1$, $t_2$ be fixed points of $g$. Define $f_i(x)=g(x)t_i$, for $i=1$, $2$. If there is $z\in G$ such that $g(z)=z^{-1}t_1^{-1}t_2$, then $G(f_1)$, $G(f_2)$ are isotopic. Denote by $*$ the multiplication in $G(f_1)$ and by $\circ$ the multiplication in $G(f_2)$. For $x\in G$, define $\alpha(x)=x$, $\alpha(\ov{x}) = \ov{xz^{-1}}$, $\beta(x)=zx$, $\beta(\ov{x})=\ov{x}$, $\gamma(x)=zx$, and $\gamma(\ov{x})=\ov{x}$. Then $$\begin{aligned} \alpha(x)\circ\beta(y) &= x\circ zy = xzy = \gamma(xy) = \gamma(x*y),\\ \alpha(x)\circ\beta(\ov{y}) &= x\circ\ov{y} = \ov{xy} = \gamma(\ov{xy}) = \gamma(x*\ov{y}),\\ \alpha(\ov{x})\circ\beta(y) &= \ov{xz^{-1}}\circ zy = \ov{xy} = \gamma(\ov{xy}) = \gamma(\ov{x}*y),\\ \alpha(\ov{x})\circ\beta(\ov{y}) &= \ov{xz^{-1}}\circ \ov{y} = g(xz^{-1}y)t_2 = zg(xy)t_1 = \gamma(g(xy)t_1) = \gamma(\ov{x}*\ov{y}),\end{aligned}$$ where we have used $g(z)=z^{-1}t_1^{-1}t_2$ in the last line. Let $G=\mathbb Z_2\times\mathbb Z_2 = \langle a\rangle \times \langle b\rangle$ be the Klein group. Consider the transposition $g=(a,b)$ with fixed points $t_1=1$, $t_2=ab$. Let $f_i(x) = g(x)t_i$, for $i=1$, $2$. Then $b = g(a) = a^{-1}t_1^{-1}t_2$, so $G(f_1)$, $G(f_2)$ are isotopic by Lemma \[Lm:GfItp\]. Comments on commutative A-loops of order $15$ and $21$ ------------------------------------------------------ \[Lm:pq\] Let $Q$ be a nonassociative commutative A-loop of order $p_0p_1$, where $p_0\ne p_1$ are odd primes. Then there is $0\le i\le 1$ such that $Q$ contains a normal subloop $S$ of order $p_i$, and all elements in $Q\setminus S$ have order $p_{i+1}$, where the subscript is calculated modulo $2$. We will use results of [@JKV] mentioned in the introduction without further reference. Since $Q$ is of odd order, it is solvable. Since $Q$ is also nonassociative, there is a normal subloop $S$ of $Q$ such that $1\ne S\ne Q$. By the Lagrange Theorem, $|S|=p_i$ for some $0\le i\le 1$. Without loss of generality, let $|S|=p_0$. Let $y\in Q\setminus S$ and let $T$ be the preimage of the subloop $\langle yS\rangle$ of $Q/S$. By the Lagrange Theorem again, $y^{p_1}=1$, as the only other alternative $|y|=p_0p_1$ would mean that $Q$ is a group by power-associativity. The information afforded by Lemma \[Lm:pq\] is sufficient to construct all nonassociative commutative A-loops of order $15$ and $21$ by the finite model builder Mace4. It turns out that in each case there is a unique such loop. These two loops were constructed already by Drápal [@Drapal Proposition 3.1]. Nevertheless the following problem remains open: Classify commutative A-loops of order $pq$, where $p<q$ are odd primes. We have some reasons to believe that there is no nonassociative commutative A-loop of order $35$. Comments on commutative A-loops of order $16$ {#Ss:16} --------------------------------------------- Among the $12$ nonassociative commutative A-loops of order $16$ and exponent $2$, three have inner mapping groups of orders that are not a power of $2$, namely $12$, $56$ and $56$. We now construct the two nonassociative commutative A-loops of order $16$ and exponent $2$ with inner mapping groups of order $56$, and we show that they are isotopic. Let $G=\mathbb Z_4\times\mathbb Z_2$. Define $g\in\aut{G}$ by $g(i,j) = (i,i+j\mod 2)$. Note that $t_1=(0,0)$, $t_2=(2,1)$ are fixed points of $g$, and let $f_i(x)=g(x)+t_i$. Then $G(f_1)$, $G(f_2)$ are the two announced loops, and they are isotopic by Lemma \[Lm:GfItp\], since $g(1,0)=(1,1)$ and $-(1,0)-(0,0)+(2,1)=(1,1)$. Comments on commutative A-loops of order $32$ and exponent $2$ with nontrivial center {#Ss:32} ------------------------------------------------------------------------------------- The methods developed in [@NV] in order to classify Moufang loops of order $64$ can be adopted to other classes of loops. Using the cocycle formula of Corollary \[Cr:CentralExtension\] and the classification of commutative A-loops of order $16$ from Subsection \[Ss:16\], we were able to classify all commutative A-loops of order $32$ and of exponent $2$ with nontrivial center. We now briefly describe the search, following the method of [@NV] closely. For more details, see [@NV]. Let $Q$ be a commutative A-loop of order $32$ and exponent $2$ with nontrivial center. Then $Z(Q)$ is obviously an elementary abelian $2$-group, and hence it possesses a $2$-element central subgroup $Z=(Z,+,0)$. Then $Q/Z=K$ is a commutative A-loop of order $16$ and exponent $2$. The loop cocycles $\theta:K\times K\to Z$ form a vector space $V$ over $Z=\gf{2}$ with respect to addition $(\theta+\mu)(x,y) = \theta(x,y)+\mu(x,y)$. The vector space $V$ has basis $\{\theta_{u,v};\;1\ne u\in K,\,1\ne v\in K\}$, where $$\theta_{u,v}(x,y) = \left\{\begin{array}{ll} 1,&\text{if $(u,v)=(x,y)$},\\ 0,&\text{otherwise.} \end{array}\right.$$ The extension $K\ltimes_\theta Z$ will be a commutative A-loop of exponent $2$ if and only if $\theta$ belongs to the subspace $C = \{\theta\in V;\;\theta$ satisfies , $\theta(x,x)=0$ for every $x\in K$ and $\theta(x,y)=\theta(y,x)$ for every $x$, $y\in K\}$. For every $x$, $y$, $z$, $x'\in K$, the equation can be viewed as a linear equation over $\gf{2}$ in variables $\theta_{u,v}$. Similarly, for every $x$, $y\in K$ we obtain linear equations from the condition $\theta(x,y)=\theta(y,x)$, and from $\theta(x,x)=0$. Upon solving this system of linear equations, we obtain (a basis of) $C$, and it is in principle possible to construct all extensions $K\ltimes_\theta Z$ for $\theta\in C$. The two computational problems we face are: (i) the dimension of $C$ can be large, (ii) it is costly to sort the resulting loops up to isomorphism. In order to overcome these obstacles, we take advantage of coboundaries and of an induced action of $\aut{K}$ on $C$. Let $\tau:K\times Z$ be a mapping satisfying $\tau(1)=0$. Then $\delta\tau:K\times K\to Z$ defined by $$\delta\tau(x,y) = \tau(xy)-\tau(x)-\tau(y)$$ is a *coboundary*. Coboundaries form a subspace $B$ of $V$. In fact, $B$ is a subspace of $C$. This can be proved explicitly by verifying that every coboundary $\theta = \delta\tau$ satisfies the identity , $\theta(x,y)=\theta(y,x)$ and $\theta(x,x)=0$. The verification of is a bit unpleasant, so it is worth realizing that every coboundary $\theta$ satisfies the group cocycle identity $$\theta(x,y)+\theta(xy,z) = \theta(y,x)+\theta(x,yz),$$ and hence also any cocycle identity that follows from associativity, in particular . Moreover, if $\theta$, $\mu:K\times K\to Z$ are two cocycles such that $\theta-\mu$ is a coboundary, then $K\ltimes_\theta Z$ is isomorphic to $K\ltimes_\theta Z$, cf. [@NV Lemma 9]. It therefore suffices to construct loops $K\ltimes_\theta Z$, where $\theta\in D$, $C=B\oplus D$. Given $\theta\in V$ and $\varphi\in\aut{K}$, we define $\theta_\varphi\in V$ by $$\theta_\varphi(x,y) = \theta(\varphi(x),\varphi(y)).$$ This action of $\aut{K}$ on $V$ induces an action on $D$. Moreover, by [@NV Lemma 14], $K\ltimes_\theta Z$ is isomorphic to $K\ltimes_{\theta_\varphi}Z$. It therefore suffices to construct loops $K\ltimes_\theta Z$, where we take one $\theta$ from each orbit of $\aut{K}$ on $D$. Using each of the $13$ commutative A-loops of order $16$ and exponent $2$ as $K$ (the elementary abelian group of order $16$ must also be taken into account), the above search finds $355$ commutative A-loops of order $32$ and exponent $2$ within several minutes. The final isomorphism search takes several hours with LOOPS. The lone isotopism $\mathbb Z_2\times Q_1 \to \mathbb Z_2\times Q_2$ is induced by the isotopism $Q_1\to Q_2$ described in Subsection \[Ss:16\]. Acknowledgement =============== We thank the anonymous referee for the nice proof of Lemma \[Lm:NewForms\]. [99]{} R. H. Bruck, A survey of binary systems, third printing, corrected, *Ergebnisse der Mathematik und Ihrer Grenzgebiete*, new series, volume **20**, Springer, 1971. R. H. Bruck and L. J. Paige, *Loops whose inner mappings are automorphisms*, Ann. of Math. (2) **63** (1956), 308–323. R. P. Burn, *Finite Bol loops*, Math. Proc. Cambridge Philos. Soc. **84** (1978), no. **3**, 377–385. A. Drápal, *A class of commutative loops with metacyclic inner mapping groups*, submitted. The GAP Group, *GAP – Groups, Algorithms, and Programming, Version 4.4*; 2004, `http://www.gap-system.org`. P. Jedlička, M. K. Kinyon and P. Vojtěchovský, *The structure of commutative automorphic loops*, to appear in Trans. Amer. Math. Soc. W. McCune, *Prover9* and *Mace4*, `http://www.prover9.org`. G. P. Nagy and P. Vojtěchovský, *LOOPS: Computing with quasigroups and loops*, version 2.0.0, package for GAP, `http://www.math.du.edu/loops`. G. P. Nagy and P. Vojtěchovský, *The Moufang loops of order $64$ and $81$*, J. Symbolic Computation **42** (2007), no. **9**, 871–883. H. O. Pflugfelder, Quasigroups and Loops: Introduction, *Sigma Series in Pure Mathematics* **7**, Heldermann Verlag Berlin, 1990.

--- abstract: 'Interesting effects arise in cyclic machines where both heat and ergotropy transfer take place between the energising bath and the system (the working fluid). Such effects correspond to unconventional decompositions of energy exchange between the bath and the system into heat and work, respectively, resulting in efficiency bounds that may surpass the Carnot efficiency. However, these effects are not directly linked with quantumness, but rather with heat and ergotropy, the likes of which can be realised without resorting to quantum mechanics.' author: - Arnab Ghosh - Victor Mukherjee - Wolfgang Niedenzu - Gershon Kurizki title: 'Are quantum thermodynamic machines better than their classical counterparts?' --- The term “quantum thermodynamic machines” may be understood in two different ways. One is that these are machines ruled by laws that are specific to quantum thermodynamics (QTD), an emerging field that attempts to combine quantum mechanics and thermodynamics [@scovil1959three; @pusz1978passive; @lenard1978thermodynamical; @alicki1979quantum; @kosloff1984quantum; @scully2003extracting; @allahverdyan2004maximal; @erez2008thermodynamic; @delrio2011thermodynamic; @horodecki2013fundamental; @correa2014quantum; @skrzypczyk2014work; @brandao2015second; @pekola2015towards; @uzdin2015equivalence; @campisi2016power; @rossnagel2016single; @kosloff2013quantum; @gelbwaser2015thermodynamics; @goold2016role; @vinjanampathy2016quantum; @kosloff2017quantum]. Such laws must depend on quantumness to qualify for QTD. The other possible meaning is that these machines are comprised of quantum systems: either all or some of their ingredients are describable quantum-mechanically, but this does not imply that these machines function in a quantum fashion. Here we argue, based on our research over the past six years [@gelbwaser2013minimal; @gelbwaser2013work; @gelbwaser2014heat; @gelbwaser2015thermodynamics; @niedenzu2016operation; @dag2016multiatom; @mukherjee2016speed; @ghosh2017catalysis; @ghosh2018two; @niedenzu2018quantum; @ghosh2019thermodynamics], that quantum thermodynamic machines either conform to the latter meaning and do not rely on quantumness [@niedenzu2016operation; @niedenzu2018quantum] or they are truly quantum, exhibit “quantum advantage” [@delcampo2014more] but do not contradict the second law of thermodynamics [@gelbwaser2013work; @ghosh2017catalysis; @ghosh2019thermodynamics]. The first machine we analysed [@gelbwaser2013minimal] was deemed to be the minimal or simplest heat machine based on a quantum system — a qubit. The qubit with resonance frequency $\omega_0$ is the working fluid (WF) of the machine. It is permanently coupled to cold and hot thermal baths with different, non-overlapping spectra. The qubit is driven periodically by a classical field which acts as a piston that causes periodic modulation of the qubit frequency $\omega(t)$ (Fig. \[1\]). The modulation period $2\pi/\Delta$ constitutes the machine cycle time. The merit of this model is that it is amenable to a complete analysis within the weak-coupling, Markovian approximation for the system-bath dynamics [@gelbwaser2013minimal; @gelbwaser2015thermodynamics]. This analysis yields the result that the machine may act as a refrigerator (or heat pump) under the condition $$\begin{aligned} \label{heat-pump-cond} {n^{\mathrm{C}}}(\omega_0-\Delta) > {n^{\mathrm{H}}}(\omega_0+\Delta),\end{aligned}$$ and as a heat engine under the converse condition $$\begin{aligned} \label{heat-engine-cond} {n^{\mathrm{C}}}(\omega_0-\Delta) < {n^{\mathrm{H}}}(\omega_0+\Delta).\end{aligned}$$ Here ${n^{\mathrm{C}}}(\omega_0-\Delta)$ and ${n^{\mathrm{H}}}(\omega_0+\Delta)$ are the cold and hot bath thermal occupancies at the downshifted and upshifted transition frequencies, respectively. These conditions characterise the optimal scenario wherein the qubit at the upshifted frequency only couples to the hot bath and at the downshifted frequency to the cold bath. Equivalently to Eqs. -, the machine acts as a heat engine whose piston extracts power (${\mathcal{P}}<0$) provided that the (positive) modulation frequency $\Delta$ is bounded (from above) by $$\begin{aligned} \label{delta-<-delta-cr} {\Delta_{\mathrm{cr}}}=\omega_0\frac{{T_{\mathrm{H}}}-{T_{\mathrm{C}}}}{{T_{\mathrm{H}}}+{T_{\mathrm{C}}}},\end{aligned}$$ ${T_{\mathrm{H}}}$ and ${T_{\mathrm{C}}}$ being the hot and cold bath temperatures, respectively. The efficiency, defined as the ratio of the extracted power $-{\mathcal{P}}$ to the input heat input current ${J_{\mathrm{H}}}$ from the hot bath, grows with $\Delta$ until the Carnot bound is attained at ${\Delta_{\mathrm{cr}}}$, $$\begin{aligned} \label{carnot-bound} \eta=\frac{-{\mathcal{P}}}{{J_{\mathrm{H}}}}=\frac{2\Delta}{\omega_0+\Delta}\leq 1-\frac{{T_{\mathrm{C}}}}{{T_{\mathrm{H}}}}.\end{aligned}$$ As the modulation frequency exceeds the critical value, i.e., $\Delta > {\Delta_{\mathrm{cr}}}$, the machine becomes a refrigerator for the cold bath. It consumes power (${\mathcal{P}}>0$) from the piston and converts it into cold current ${J_{\mathrm{C}}}$ as characterised by the coefficient of performance (COP) that reaches its maximal value at $\Delta={\Delta_{\mathrm{cr}}}$, $$\begin{aligned} \label{cop-bound} \mathrm{COP}=\frac{{J_{\mathrm{C}}}}{{\mathcal{P}}}=\frac{\omega_0-\Delta}{2\Delta} \leq \frac{{T_{\mathrm{C}}}}{{T_{\mathrm{H}}}-{T_{\mathrm{C}}}}.\end{aligned}$$ These lucid, simple results show clearly that although the WF is a qubit, there is nothing uniquely quantum-mechanical about the machine performance, which adheres to the standard thermodynamic bound. Yet, the field of quantum-thermodynamic machines has been propelled by ingenious porposals to benefit from quantum resources embodied by non-thermal baths [@scully2003extracting; @dillenschneider2009energetics; @huang2012effects; @abah2014efficiency; @rossnagel2014nanoscale; @niedenzu2016operation; @manzano2016entropy; @hardal2015superradiant; @klaers2017squeezed; @agarwalla2017quantum; @niedenzu2018quantum]. Schematically, such machines have the same ingredients as conventional Carnot heat engines (Fig. \[3\]). However, at least the hot bath, which is the source of energy, may have non-thermal properties that stem from its quantum-mechanical preparation. The question has been posed whether a cycle energised by such a bath must abide by the Carnot efficiency bound derived in 1824 for steam engines [@carnotbook], $$\begin{aligned} \label{carnot-bound-1824} \eta=\frac{-W}{{Q_{\mathrm{H}}}}\leq 1-\frac{{T_{\mathrm{C}}}}{{T_{\mathrm{H}}}}=:\eta_\mathrm{C},\end{aligned}$$ where the efficiency is the ratio of the work output $-W$ to the heat input ${Q_{\mathrm{H}}}$. Two crucial assumptions have been made in Eq. : (i) that the input from the “hot” (better: energising) bath is indeed heat, and (ii) that this bath has a “temperature” ${T_{\mathrm{H}}}$, although a non-thermal bath need not have one. Before addressing these issues, we consider the specific setups which have promoted our general investigation of these issues [@niedenzu2016operation; @niedenzu2018quantum]. The first setup, whose study by Scully et. al. [@scully2003extracting] pioneered the field, consists of an engine that is energised by “phaseonium” fuel. The latter are three-level atoms whose lower two near-degenerate levels are coherently superimposed with a phase $\phi$ (Fig. \[phaseonium\]). The consecutive interactions of these atoms with the WF (a single cavity-field mode) conform to the micromaser model whereby the bath *appears* to be at temperature ${T_{\mathrm{H}}}(\phi)$, which is now $\phi$-dependent. Consequently, whenever a $\phi$ is chosen such that ${T_{\mathrm{H}}}(\phi)$ exceeds ${T_{\mathrm{H}}}$ in the absence of coherence, then the resulting Carnot bound becomes higher than the standard (incoherent) Carnot bound. Is this a quantum advantage? Not from the point of view of the WF (cavity field mode) that interacts with a bath at a temperature ${T_{\mathrm{H}}}(\phi)$ — the WF has no other bath except the phaseonium. We next turn to another setup that was proposed by Roßnagel *et al.* [@rossnagel2014nanoscale]: an Otto cycle that is energised by a squeezed-thermal bath. In this cycle the strokes are realized by adiabatic compression and expansion of the WF, its consecutive coupling to and decoupling from the cold and the hot baths, the only difference from the standard cycle being that the hot bath is not in a thermal state (at temperature $T_\mathrm{H}$) but rather in a squeezed-thermal state (described by the temperature $T_\mathrm{H}$ and the squeezing parameter $r$). How does the squeezing affect the cycle efficiency? The tendency of most works on the subject [@huang2012effects; @abah2014efficiency; @rossnagel2014nanoscale; @manzano2016entropy; @klaers2017squeezed] has been to identify the entire energy delivered by the hot bath to the WF as heat. If we attribute to this definition and the extra (squeezing) energy to an effective temperature ${T_{\mathrm{H}}}(r)$ that increases with the squeezing parameter, then we may again deduce from this analysis that the cycle may surpass the standard Carnot bound provided ${T_{\mathrm{H}}}(r)$ is higher than ${T_{\mathrm{H}}}$ without squeezing [@rossnagel2014nanoscale; @manzano2016entropy]. However, there is a missing piece in this puzzle: can we really claim that heat and squeezing are interchangeable resources? To answer this question, we have to properly unravel the energy exchange between a quantum system and a quantum bath, namely, reach better understanding of the *first law of thermodynamics* in the quantum domain. To this end, we start from Alicki’s pioneering decomposition [@alicki1979quantum] of the change in the mean energy of a quantum system that is both driven by a time-dependent system Hamiltonian $H(t)$ and is coupled to a quantum bath. Such a change in the mean system energy $$\begin{aligned} \label{mean-energy} E(t)={\mathrm{Tr}}[\rho(t)H(t)]\end{aligned}$$ for the reduced system density operator $\rho(t)$ was decomposed by Alicki as \[eq\_defs\_Ediss\_work\] $$\label{eq_first_law} \Delta E(t)=\mathcal{E}(t)+W(t),$$ and split into heat $$\label{eq_def_DeltaEdiss} \mathcal{E}(t){\mathrel{\mathop:}=}\int_0^t{\mathrm{Tr}}[\dot\rho(t^\prime)H(t^\prime)]{\mathrm{d}}t^\prime$$ and work $$\label{eq_def_work} W(t){\mathrel{\mathop:}=}\int_0^t{\mathrm{Tr}}[\rho(t^\prime)\dot H(t^\prime)]{\mathrm{d}}t^\prime.$$ Here $\rho(t)$ is the reduced density operator of the system obtained by tracing out the bath degrees of freedom and $H(t)$ is the controlled Hamiltonian for the system. In Refs. [@pusz1978passive; @lenard1978thermodynamical], Eq.  is interpreted as work because it is associated with a change $\dot H(t)$ in the driving Hamiltonian. Alternative definitions of quantum work in the literatures include, for example, the concept of “work operator” in the context of quantum measurement processes [@talkner2016aspects; @baeumler2018fluctuating; @PhysRevE.75.050102; @Hanggi2015; @RevModPhys.83.771; @RevModPhys.83.1653; @PhysRevLett.102.210401; @PhysRevLett.118.070601; @PhysRevLett.93.048302; @PhysRevE.71.066102; @PhysRevE.78.011116]. Here, we are interested in the question whether necessarily correspond to entropy change (in the regime of weak coupling between the system and the baths, neglecting correlations between the system and the baths [@breuerbook]). On the face of it, it does, because it arises from a change in the state of the system $\dot\rho(t)$ that may contribute to entropy change. However, as shown below, this is not always the case. In fact, there are isentropic processes associated with $\dot{\rho}(t)\neq 0$, which physically correspond to work rather than heat exchange. As a simple example of such a process, which may constitute part of a cycle, consider a single-mode cavity field initially prepared in a coherent state $|\alpha_0\rangle$. It leaks out of the cavity through the front mirror, until at long times the cavity-field state becomes the vacuum $|0\rangle$ (Fig. \[4\]). The point is that the state has changed and so has its mean energy $E(t)$ but not its purity or entropy. This is an example of a system that is transformed from a *non-passive* state, here a coherent state with non-zero amplitude $|\alpha(t) \neq 0\rangle$, to a passive state, here the vacuum state $|0\rangle$. A passive state is a state that does not allow to extract work from the system under cyclic unitary transformations [@pusz1978passive; @lenard1978thermodynamical]. As long as the Hamiltonian is non-degenerate, there is a unique passive state for each non-passive state: the two are unitarily related. Thus, the leakage of the field from the cavity is a change in the *degree of non-passivity* but not in entropy. Such a change has been dubbed *ergotropy* change [@allahverdyan2004maximal]. We note that ergotropy cannot be readily measured, as *any* measurement would cause back-action which is typically not unitary [@talkner2016aspects]. It would thus be interesting to study the issue of back-action on ergotropy [@venkatesh2015quantum]. Passivity of a quantum state implies a monotonically-decreasing distribution of energy eigenvalues. For details please see Refs. [@pusz1978passive; @lenard1978thermodynamical]. As shown in Fig. \[5\], a unitary transformation from a Fock state $|n\neq 0\rangle$ to the vacuum $|0\rangle$ (Fig. \[5\]a) or from a non-monotonic distribution of Fock states to a reshuffled monotonic distribution with the same Fock-state ingredients (Fig. \[5\]b) releases all the ergotropy of the initial non-passive state $\rho$, resulting in the passive state $\pi$. Being isentropic, it should be distinguishable from dissipative change in passive energy. Such distinction is effected by the decomposition of the energy exchanged between the system and the bath into \[eq\_defs\_heat\_passive\_work\] $$\label{eq_DeltaEdiss_decomposition} \mathcal{E}(t)=\int_0^t{\mathrm{Tr}}[\dot\rho(t^\prime)H(t^\prime)]{\mathrm{d}}t^\prime =\mathcal{Q}(t)+\Delta\mathcal{W}|_\mathrm{diss}(t),$$ where the non-unitary changes in passive energy and system ergotropy are given by [@niedenzu2018quantum] $$\label{eq_def_heat} \mathcal{Q}(t){\mathrel{\mathop:}=}\int_{0}^{t} {\mathrm{Tr}}[\dot{\pi}(t^\prime)H(t^\prime)]{\mathrm{d}}t^\prime$$ and $$\label{eq_def_DeltaW_diss} \Delta\mathcal{W}|_\mathrm{diss}(t){\mathrel{\mathop:}=}\int_0^t{\mathrm{Tr}}\Big[\big(\dot\rho(t^\prime)-\dot\pi(t^\prime)\big)H(t^\prime)\Big]{\mathrm{d}}t^\prime,$$ respectively. The instantaneous passive state $\pi(t)$ and its derivative $\dot{\pi}(t)$ are obtained by unitary transformations from the time evolution of $\rho(t)$. Hence, Eq.  is always associated with a change in the von Neumann entropy $\mathcal{S}(\rho(t))=-{k_\mathrm{B}}{\mathrm{Tr}}[\rho(t)\ln \rho(t)]$ because $\mathcal{S}$ of a non-passive state $\rho(t)$ is same as that of its unitarily transformed passive state $\pi(t)$. In accordance with the existing literature, we consider the von-Neumann entropy to be relevant in thermodynamic settings, whether in or out of equilibrium [@spohn1978entropy; @alicki1979quantum; @kosloff2013quantum; @binder2019thermodynamics; @vinjanampathy2016quantum]. Thus, in analogy to thermodynamics, we hereafter refer to Eq.  as *heat* transfer because of its entropy-changing character. The above decomposition of the energy exchange with the bath into heat and ergotropy exchange has led us to suggest the following inequalities for the entropy change over long times $t \rightarrow \infty$ of the *system* (assuming the bath is *thermal*, i.e., its temperature $T$ to be immutable) [@niedenzu2018quantum; @ghosh2019thermodynamics], \[our-inequality-nat-comm\] $$\label{eq_DeltaS_QdT} \Delta\mathcal{S}\geq\frac{\mathcal{Q}}{T}$$ for a system governed by a constant Hamiltonian or $$\label{eq_DeltaS_Qth} \Delta\mathcal{S}\geq\frac{\mathcal{E}^\prime}{T},$$ for a system driven by a time-dependent Hamiltonian, where $\mathcal{E}^\prime$ is the energy which would be exchanged with the bath under such driving if the initial state were the passive counterpart $\pi(0)$ of the actual initial state $\rho(0)$. It is clear, particularly from , that $\Delta \mathcal{S}$ is here bounded from below only by heat exchange. Our suggested inequality must be contrasted with Spohn’s [@spohn1978entropy] $$\label{eq_DeltaS_QdT_DeltaW} \Delta\mathcal{S}\geq\frac{\mathcal{E}}{T}=\frac{\mathcal{Q}+\left.\Delta\mathcal{W} \right|_\mathrm{diss}}{T},$$ where $\Delta \mathcal{S}$ is bounded from below by the sum of heat and ergotropy exchange. Although inequality is as much consistent with the second law (the Clausius inequality [@clausiusbook1]) as our inequality , the latter is much tighter than the former, because, for an initial non-passive state, $\Delta\mathcal{W}|_\mathrm{diss} \leq 0$ in a relaxation process of the system towards its thermal steady-state. These considerations have been used by us to derive a generalised efficiency bound for cycles in which the WF may be energised by either a thermal or a non-thermal bath [@niedenzu2016operation; @niedenzu2018quantum]. As shown schematically in Fig. \[6\], such a cycle consists of time-dependent periodic change of the coupling between the WF and the energising (thermal or non-thermal) bath (right ellipse) and entropy dumping into a cold thermal bath (left circle). Our general efficiency bound [@niedenzu2018quantum] is $$\label{eq_etamax_gen} \eta\leq 1-\frac{{T_{\mathrm{C}}}}{{T_{\mathrm{H}}}}\frac{\mathcal{E}^\prime_\mathrm{h}}{\mathcal{E}_\mathrm{h}}{=\mathrel{\mathop:}}\eta_\mathrm{max} \le 1.$$ On the r.h.s, the temperature ratio of the two baths is multiplied by the ratio of the heat exchange $\mathcal{E}^\prime_\mathrm{h}$ to the total energy exchange $\mathcal{E}_\mathrm{h}$, which combines both heat and ergotropy exchange. Here $\mathcal{E}^\prime_\mathrm{h}$ is the heat that the WF would have obtained in the same cycle if the hot bath was thermal. It then follows that in the absence of ergotropy transfer from the bath to the WF, as for a thermal bath, the Carnot bound is recovered. By contrast, if the heat exchange with the energised bath is much less than total energy exchange, because ergotropy transfer dominates, the efficiency bound approaches $1$ and surpasses the Carnot efficiency. This, however, should in no way imply a *surpassing* of the Carnot bound: it means that the Carnot bound is inapplicable to such a scenario, wherein ergotropy rather than heat is exchanged with the bath. In fact, the second law, from which the Carnot bound follows, only relates to entropy-changing processes, whereas ergotropy exchange is isentropic. Here we have not considered the cost of producing the state of the bath. By the same token, it is customary not to consider the cost of producing thermal baths for conventional heat engines [@callenbook]. Can the cycle described above, wherein ergotropy and non-passivity play a central role, be deemed genuinely quantum, or at least exhibit quantum advantage? Not necessarily, since the energising bath in question is in a squeezed-thermal state and such a state exhibits genuine quantumness only if the temperature ${T_{\mathrm{H}}}$ associated with its thermal component is low enough, such that the state approximates the squeezed-vacuum state [@niedenzu2016operation; @kim1989properties]. Although we have explicitly considered above “semiclassical” engines whose cycle is effected by a classical periodic driving field (“piston”), similar conclusions hold for fully quantised machines, wherein the quantum state of the piston is explicitly accounted for [@gelbwaser2014heat; @ghosh2017catalysis; @ghosh2018two]: In the latter class of machines it is the ergotropy (non-passivity) of the piston state and its heating (thermalisation), which are not exclusively related to quantumness, that determine the efficiency and power output of the machine. Since our discussion has revolved around the decomposition of system-bath energy exchange into work and heat transfer, it should be stressed that we cannot identify any alternative, physical, quantifiers of these processes. Namely, we find the physical arguments in favour of the uniqueness of the present quantifiers to be compelling. To sum up, interesting effects arise when one considers cyclic machines where both heat and ergotropy transfer take place. Such effects correspond to unconventional decompositions of energy exchange between the bath and the WF into heat and work, resulting in efficiency bounds that may exceed Carnot’s. This is allowed because the thermodynamic Carnot bound is only valid for machines energised by heat transfer. However, these effects are not directly linked with quantumness, but rather with heat and ergotropy exchange, the likes of which can be constructed without resorting to quantum mechanics. These conclusions have to be revised when multiple quantum-correlated (e.g. entangled) machines are compared to their classical counterparts [@niedenzu2018cooperative]. **Acknowledgements:** We acknowledge the support of the DFG, ISF and SAERI (G.K.), VATAT (V.M.) and the ESQ fellowship of the Austrian Academy of Sciences (ÖAW) (W.N.). **Author contribution statement:** Arnab Ghosh, Victor Mukherjee and Wolfgang Niedenzu (listed alphabetically) have contributed to the research results and physical insights involved in this paper. Gershon Kurizki has contributed to the formulation of the overall physical picture and has mostly written the article. [63]{}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 “,” [****,  ()](\doibase 10.1103/PhysRevLett.2.262) “,” [****,  ()](\doibase 10.1007/BF01614224) “,” [****,  ()](\doibase 10.1007/BF01011769) “,” [****,  ()](\doibase 10.1088/0305-4470/12/5/007) “,” [****,  ()](\doibase 10.1063/1.446862) “,” [****,  ()](\doibase 10.1126/science.1078955) “,” [****,  ()](\doibase 10.1209/epl/i2004-10101-2) “,” [****,  ()](\doibase 10.1038/nature06873) “,” [****,  ()](\doibase 10.1038/nature10123) “,” [****,  ()](\doibase 10.1038/ncomms3059) “,” [****,  ()](\doibase 10.1038/srep03949) “,” [****,  ()](\doibase 10.1038/ncomms5185) “,” [****,  ()](\doibase 10.1073/pnas.1411728112) “,” [****,  ()](\doibase 10.1038/nphys3169) “,” [****,  ()](\doibase 10.1103/PhysRevX.5.031044) “,” [****, ()](\doibase 10.1038/ncomms11895) “,” [****,  ()](\doibase 10.1126/science.aad6320) “,” [****,  ()](\doibase 10.3390/e15062100) “,” [****,  ()](\doibase 10.1016/bs.aamop.2015.07.002) “,” [****,  ()](\doibase 10.1088/1751-8113/49/14/143001) “,” [****,  ()](\doibase 10.1080/00107514.2016.1201896) “,” [****,  ()](\doibase 10.3390/e19040136) “,” [****, ()](\doibase 10.1103/PhysRevE.87.012140) “,” [****,  ()](\doibase 10.1209/0295-5075/103/60005) “,” [****,  ()](\doibase 10.1103/PhysRevE.90.022102) “,” [****,  ()](\doibase 10.1088/1367-2630/18/8/083012) “,” [****,  ()](\doibase 10.3390/e18070244) “,” [****, ()](\doibase 10.1103/PhysRevE.94.062109) “,” [****,  ()](\doibase 10.1073/pnas.1711381114) “,” [****, ()](\doibase 10.1073/pnas.1805354115) “,” [****,  ()](\doibase 10.1038/s41467-017-01991-6) in [**](https://books.google.co.il/books?id=IQlouQEACAAJ),  (, ) pp.  “,” [****,  ()](\doibase 10.1038/srep06208) “,” [****,  ()](\doibase 10.1103/PhysRevLett.109.090601) “,” [****, ()](\doibase 10.1209/0295-5075/88/50003) “,” [****,  ()](\doibase 10.1103/PhysRevE.86.051105) “,” [****,  ()](\doibase 10.1209/0295-5075/106/20001) “,” [****,  ()](\doibase 10.1103/PhysRevLett.112.030602) “,” [****,  ()](\doibase 10.1103/PhysRevE.93.052120) “,” [****,  ()](\doibase 10.1038/srep12953) “,” [****,  ()](\doibase 10.1103/PhysRevX.7.031044) “,” [****,  ()](\doibase 10.1103/PhysRevB.96.104304) @noop [**]{} (, , ) “,” [****,  ()](\doibase 10.1103/PhysRevE.93.022131) “,” [ ()](https://arxiv.org/abs/1805.10096) “,” [****,  ()](\doibase 10.1103/PhysRevE.75.050102) “,” [****,  ()](https://doi.org/10.1038/nphys3167) “,” [****,  ()](\doibase 10.1103/RevModPhys.83.771) “,” [****,  ()](\doibase 10.1103/RevModPhys.83.1653) “,” [****,  ()](\doibase 10.1103/PhysRevLett.102.210401) “,” [****,  ()](\doibase 10.1103/PhysRevLett.118.070601) “,” [****,  ()](\doibase 10.1103/PhysRevLett.93.048302) “,” [****,  ()](\doibase 10.1103/PhysRevE.71.066102) “,” [****,  ()](\doibase 10.1103/PhysRevE.78.011116) @noop [**]{} (, ) “,” [****,  ()](\doibase 10.1088/1367-2630/17/7/075018) “,” [****,  ()](\doibase 10.1063/1.523789) [**](https://books.google.co.il/books?id=IQlouQEACAAJ), Fundamental Theories of Physics (, ) @noop [**]{},  ed. (, , ) @noop [**]{},  ed. 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--- author: - 'S. Sabari$^{1,2}$ and R. Kishor Kumar$^{3}$' title: 'Effect of an oscillating Gaussian obstacle in a Dipolar Bose-Einstein condensate' --- Introduction {#sec1} ============ The remarkable observation of Bose-Einstein condensates (BECs) in $^{52}$Cr [@Lahaye:2007; @Griesmaier:2006], $^{164}$Dy [@Lu:2011; @Youn:2010], and $^{168}$Er [@Aikawa:2012; @rev1] with both dipole-dipole interaction (DDI) and *s-wave* contact interaction has opened a wholly new exciting field that continues to thrive [@dbec1; @dbec2; @FetterRMP]. Contrary to the short-range contact interaction, the DDI is a long-range anisotropic interaction that can be either repulsive or attractive. The *s-wave* contact interaction, $a_s$, is experimentally controllable by Feshbach resonance [@FBR]. It is therefore appealing to study the properties of dipolar BECs in variable short-range contact interaction regimes. However, the DDI is also inherently controllable, either via the magnitude of the external electric field, or by modulating the external aligning field in time, which allows to tune the magnitude and sign of the DDI [@tuneDDI]. Due to the long-range nature and anisotropic character of the DDI, the dipolar BEC possesses many distinct features and new phenomena such as the new dispersion relations of elementary excitations [@Wilson:2010; @Ticknor:2011], unusual equilibrium shapes, the roton-maxon character of the excitation spectrum [@Santos:2000; @Yi:2003; @Ronen:2007; @Parker:2008], quantum phases including supersolid and checkerboard phases [@Tieleman:2011; @Zhou:2010], anisotropic solitons [@equ2Db; @solRK; @solPM], vortices [@rev4; @rev5], hidden vortices [@Sabari2017] and distinct vortex lattices including crater-like structure, square lattices [@vor1; @vor2; @vor3]. The modern optical techniques help to control the parameters of the condensate and visualize topological defects such as rarefaction pulses and quantized vortices in BECs [@FetterRMP; @equ2Db; @solRK; @solPM; @rev4; @rev5; @vor1; @vor2; @vor3]. Recently, vortex tangles caused by an oscillatory perturbation were observed experimentally. The vortex tangle configuration is a signature of the presence of a quantum turbulent regime in the BEC cloud [@Henn09]. Moreover, recent studies on quantum turbulence are still concentrating on understanding the dynamics of quantized vortices [@PLTP]. Vibrating structures such as spheres, grids, and wires are used in superfluid $^3$He and $^4$He to create quantum turbulence [@PLTP; @Hanninen07]. Despite the differences between these structures, the experiments show surprisingly similar behavior. Introducing an oscillating potential in atomic dipolar BECs will be helpful to analyze the intrinsic nucleation of topological defects and synergy dynamics of vortices and rarefaction pulses. Also, this technique suggests a powerfull method for making quantum turbulence in trapped dipolar BECs, in addition to the other methods that have been used so far [@Henn09; @Berloff02; @Kobayashi07] in alkali BECs. Eventually, the dynamics of vortices and rarefaction pulses can be visualized in atomic dipolar BECs, which enabling experimental and theoretical challenges for further analysis. Up to now, vortex dipoles caused by oscillating potentials in alkali BECs have been observed in experiments and compared to theoretical models [@osc1; @osc2; @Jackson00; @Raman99; @Onofrio20; @Neely10; @rev2; @rev3]. Nonlinear dynamical behaviors, critical velocity for vortex dipoles, hydrodynamic flow, vortices, rarefaction pulses and other interesting perspectives have been carried out in alkali BECs using the oscillating Gaussian potential [@osc1; @osc2; @Jackson00; @Raman99; @Onofrio20; @Neely10]. Inspite of the many experiments that have been carried out on $^{164}$Dy and $^{168}$Er condensates there has still been no experimental observation of vortices in dipolar BECs. So, investigating the dynamics of vortex dipoles in a dipolar BEC by introducing an oscillating potential will be a fascinating experimental exploration. Thus, this model will be helpful to perform new experiments with the aim of observing vortices in dipolar BECs. In the present work, we are interested in studying the nucleation and dynamics of vortex dipoles and rarefaction pulses. The next sections are organized as follows. In Sec. \[sec:frame\], we present the general three-dimensional mean-field equation for the dipolar BECs and the corresponding two-dimensional (2D) reduction. In Sec. \[sec:numerical\], we present our numerical results, where we include plots on the critical velocity for the nucleation and dynamics of vortex-dipoles. Further, in this section, we show the rarefaction pulses due to the annihilation of vortex-dipoles. Finally, in Sec. \[sec:con\], we present a summary of our conclusions and perspectives. The mean-field formalism {#sec:frame} ======================== At ultralow temperatures, a dipolar BEC is described by the time-dependent GP equation with a nonlocal integral corresponding to the DDI [@dbec1; @dbec2; @lasPM; @rev5; @rev6] $$\begin{aligned} i\hbar\frac{\partial \phi({\mathbf r},t)}{\partial t}& =\Big(-\frac{\hbar^2}{2m}\nabla^2+V({\mathbf r},t) + g \left\vert \phi({\mathbf r},t)\right\vert^2 \Big)\phi({\mathbf r},t)\notag \\ &+N \int U_{\mathrm{dd}}({\mathbf r}-{\mathbf r}')\left\vert\phi({\mathbf r}',t)\right\vert^2 d{\mathbf r}'\phi({\mathbf r},t), \label{eqn:dgpe}\end{aligned}$$ with $({\bf r},t)=({\bf \rho},t)$ and the radial coordinate being $\rho=\sqrt{x^2+y^2}$. The trapping potential $V({\mathbf r},t) = V_{ext} + V_G$ contains a cylindrically symmetric harmonic trap in addition to a Blue detuned Gaussian obstacle. The cylindrically symmetric trap is $V_{ext}({\mathbf r})=\frac{1}{2} m (\omega_\rho^2 (x^2+y^2)+ \omega_z^2 z^2)$, with $\omega_x = \omega_y = \omega_\rho$ and $\omega_z$ being the radial and axial trap frequencies respectively. The trap aspect ratio of the harmonic trap is $\lambda=\omega_{z}/\omega_{\rho}$. The Gaussian obstacle is $$V_{G}(\rho,t) = V_{0} \exp\left(-\frac{\left[x-x_0(t)\right]^2+y^2}{w_0^2}\right),$$ where $V_0$, $x_0(t)$ and $w_0^2$ are the height, position and width of the Gaussian obstacle. The position of the obstacle $x_0(t)=\epsilon \sin(\omega t)$ provides parametric resonance with respect to the oscillating frequency $\omega$. One can control the velocity ($v=\epsilon\omega$) of oscillation of the obstacle with respect to the amplitude $\epsilon$ and the frequency $\omega$. In the present study, $\epsilon = 10 \mu$m and $\omega = 60 /s$. However, the velocity of the obstacle also depends on $V_{0}$ and $w_0$, and we keep these fixed: $V_0=80\,\hbar\omega_\rho$ and $w_{0}=0.25\,\mu m$. The two-body contact interaction strength is $g=4\pi$ $\hbar^2a_s N/m$ where $a_s$, $m$, and $N$ are atomic scattering length, mass of the atom and number of atoms respectively. We consider that the magnetic dipoles are polarized along $z$ direction and the corresponding dipolar interaction term is $ U_{\mathrm{dd}}(\bf R)=(\mu_0 \mu^2/4\pi)(1-3\cos^2 \theta/ \vert {\bf R} \vert ^3)$, where the relative position of the dipoles is ${\bf R= r -r'}$, $\theta$ is the angle between ${\bf R}$ and the direction of polarization $z$, $\mu_0$ is the permeability of free space and $\mu$ is the dipole moment of the condensate atom. In the present study, we consider the $^{168}$Er and $^{164}$Dy atoms, their corresponding dipole moments being $\mu=7\mu_B$ and $10\mu_B$ respectively. The normalization of the mean-field wavefunction is $\int d{\bf r}\vert\phi({\mathbf r},t)\vert ^2=1.$ It is convenient to use the GP equation (\[eqn:dgpe\]) in a dimensionless form. For this purpose, we introduce the dimensionless variables ${\bar {\bf r}}= {\bf r}/l,{\bar {\bf R}}={\bf R}/l, \bar a_s=a_s/l, \bar a_{\mathrm{dd}}=a_{\mathrm{dd}}/l, \bar t=t\bar \omega_\rho$, $\bar x=x/l, \bar y=y/l, \bar z=z/l, \bar \phi=l^{3/2}\phi$, $l=\sqrt{\hbar/(m \omega_\rho)}$. Eq. (\[eqn:dgpe\]) can be rewritten (after removing the overhead bar from all the variables) as $$\begin{aligned} i \frac{\partial \phi({\mathbf r},t)}{\partial t} & = \Big(-\frac{1}{2}\nabla^2+V({\mathbf r},t) +g \vert {\phi({\mathbf{r}},{t})} \vert^2 \Big) \phi({\mathbf r},t) \\ \nonumber &+ 3N a_{\mathrm{dd}}\int \frac{1-3\cos^2\theta}{\vert \bf{R}\vert^3} \vert \phi({\mathbf{r}}',t) \vert^2 d{\mathbf{r}}' \phi({\mathbf r},t),\label{gpe3d} \end{aligned}$$ To compare the dipolar and contact interactions, often it is useful to introduce the length scale $a_{\mathrm{dd}}\equiv \mu_0 \mu^2 m/(12\pi \hbar^2 l)$ and its values for $^{164}$Er and $^{168}$Dy are $66a_0$ and $131a_0$ respectively, with $a_0$ being the Bohr radius [@dbec1]. The stability of dipolar BEC depends on the external trap geometry, *e.g.* a dipolar BEC is stable or unstable depending on whether the trap is pancake- or cigar-shaped, respectively. The instability usually can be overcome by applying a strong pancake trap and applying repulsive two-body contact interaction. The external trap helps to stabilize the dipolar BEC by imprinting anisotropy to the density distribution. Hence, we carry out the present study with axially-symmetric pancake-shaped magnetic trap. So, we assume that the dynamics of the dipolar BEC in the axial direction is strongly confined with the ground state $\phi(z)=\exp(-z^2 /2d_z^2)/(\pi d_z^2)^{1/4}, \quad d_z= \sqrt{1/\lambda},$ and the wave function $$\begin{aligned} \phi({\bf r})\equiv \phi(z) \psi(\rho,t)=(1/(\pi d_z^2)^{\frac{1}{4}})\exp(-z^2/2d_z^2) \psi(\rho,t),\nonumber \end{aligned}$$ where axial harmonic oscillator length $d_z=\sqrt{1/(\lambda)}$ and ${\bf\rho} \equiv (x,y)$. Therefore, it is more suitable to consider the system in quasi two-dimensions. One can obtain the effective 2D equation for the pancake-shaped dipolar BEC by integrating the equation (\[gpe3d\]) over the $z$ direction using the above wave function $\phi({\bf r})$ [@equ2Da; @equ2Db; @lasPM] $$\begin{aligned} i\frac{\partial \psi(\rho,t) }{\partial t}& =\Big(-\frac{1}{2}\nabla_\rho^2+ V_{2D} + g_{2} \, \vert \psi(\rho,t) \vert^2 \Big) \psi(\rho,t) \\ \notag &+g_{d}\int \frac{d{\bf k}_\rho}{(2\pi)^2} e^{-i{\bf k}_\rho.{\bf\rho}}\widetilde n({\bf k}_\rho,t)h_{2D}\left(\frac{k_\rho d_z}{\sqrt{2}}\right) \psi(\rho,t).\label{gpef}\end{aligned}$$ where, two-body contact interaction $g_{2}= 4\pi Na_s/(\sqrt{2\pi}d_z)$, dipolar interaction $g_{d}=4\pi Na_{\mathrm{dd}}/(\sqrt{2\pi}d_z)$, and $k_\rho=(k_x^2+k_y^2)^{1/2}$. The dipolar term has been written in the Fourier space [@equ1; @equ2]. Dynamics of vortices {#sec:numerical} ==================== The results in this section were obtained within a full numerical approach to solve 2D nonlinear differential Eq. (4). When we have dipolar interactions, we have to combine the split-step Crank-Nicholson method and Fast Fourier Transform (FFT) for evaluating dipolar integrals in momentum space as in Ref. [@equ1; @equ2; @CUDA]. {width="14.0cm"} For looking at the dynamics of vortex dipoles, the 2D numerical simulations were carried out in real time with a grid size having $512$ points for each dimension, where we have $\Delta x = \Delta y = 0.05$ for the space-steps and $\Delta t = 0.00025$ for the time-step. Also, the results were verified by doubling the aforementioned grid sizes. ![(color online) Dynamics of the vortices for (upper panel) $^{168}$Er BEC ($a_{dd}=66a_0$) and (lower panel) $^{164}$Dy BEC ($a_{dd}=131a_0$) and $a_s=50a_0$ for all cases. From left to right $t=0\,$ms, $t=0.4\,$ms, $t=2.0\,$ms, $t=2.6\,$ms, $t=4.9\,$ms, $t=5.9\,$ms, $t=6.7\,$ms, and $t=7.5\,$ms, respectively. []{data-label="f2"}](cs.eps "fig:"){width="9cm" height="3.5cm"} ![(color online) Dynamics of the vortices for (upper panel) $^{168}$Er BEC ($a_{dd}=66a_0$) and (lower panel) $^{164}$Dy BEC ($a_{dd}=131a_0$) and $a_s=50a_0$ for all cases. From left to right $t=0\,$ms, $t=0.4\,$ms, $t=2.0\,$ms, $t=2.6\,$ms, $t=4.9\,$ms, $t=5.9\,$ms, $t=6.7\,$ms, and $t=7.5\,$ms, respectively. []{data-label="f2"}](dy.eps "fig:"){width="9cm" height="3.5cm"} First, we calculate the critical velocity for the nucleation of vortex dipoles with respect to scattering length as shown in in Fig. (\[f1\]) (a) for the three different cases, non-dipolar $a_{dd}=0\,a_0$ (Green line with circles), $^{168}$Er dipolar $a_{dd}=66\,a_0$ (Blue line with triangles), and $^{164}$Dy dipolar $a_{dd}=131\,a_0$ (Magenta line with stars) condensates, respectively. If, the velocity of the Gaussian potential ($V_p=\epsilon\,\, \omega$) is larger than the critical velocity $V_c$, vortex dipoles are nucleated in the condensate. As shown in Fig. (\[f1\]) (a), the critical velocity ($V_c$) for the nucleation of vortex dipoles increases with respect to increasing two-body contact and dipolar interaction strengths. This happens due to increasing interaction strengths leading to increase in the chemical potential of the condensate. Usually the height of the obstacle is determined with respect to the chemical potential. One needs to increase the height of the obstacle when the chemical potential increases. In this case, the critical velocity for nucleating vortices also depends on the height of the Gaussian obstacle. Also, the chemical potentials for the three different cases, alkali BEC (non-dipolar $a_{dd}=0$, $^{168}$Er dipolar BEC, and $^{164}$Dy dipolar BEC are 36.77, 44.65 and 64.55, respectively. Figure (\[f1\]) (a) shows that if we fix the amplitude of the obstacle then the critical velocity increases with respect to the interaction strengths. Usually, in rotating magnetic trap, for the nucleation of single vortex, the critical rotation frequency decreases with respect to increasing contact and dipolar interaction strengths. In the rotating trap, the vortex with same charge circulation is created whereas while stirring with an obstacle, we observe vortices with opposite charges circulations. Stirring beyond the critical velocity nucleates more vortices. This will be helpful to study the dynamics of multiple vortex dipoles. Another feature to look in Fig. (\[f1\]) (a) is the stability of the $^{164}$Dy dipolar BECs. We show the critical velocity of the $^{164}$Dy BEC from $35\,a_0$, because below this scattering length the obstacle creates the local collapse. On the other hand, $^{168}$Er BEC is stable from $20\,a_0$. Figure (\[f1\]) (b) illustrates the plot of the $V_c/c_s$ Vs $a_s$ for the three different cases, nondipolar (Green line with circles) [@ref1], $^{168}$Er (Blue line with triangles), and $^{164}$Dy (Magenta line with stars) condensates, respectively. The speed of sound ($c_s$) depends on the scattering length and dipolar interaction strength ($c=\sqrt{2 n_0 \sqrt{2\pi} (a+a_{dd})/d_z}$). ![(Color online) Density profile for nucleation of a vortex pair by the oscillating potential at (a) $t$ = 0 ms, (b) $t$ = 0.2 ms, and (c) $t$ = 0.4 ms are shown. The symbols $-$ and $+$ denote a vortex with clockwise or counterclockwise circulation, respectively. The black arrows indicate the direction of motion of the potential. A ghost vortex pair nucleates inside the potential (a), exits it (b), and finally fully leaves the potential (c).[]{data-label="f3"}](nucleationF.eps){width="0.7\linewidth"} The dynamics of vortices in which they experience a lengthy migration are shown in Figs. (\[f2\]) for two different cases. Following the destiny of vortices nucleated by the oscillating potential enables us to survey their dynamics. The initial state in the static Gaussian potential in Fig. (\[f2\]) is obtained by an imaginary time step of the GP equation. A vortex pair is nucleated behind the Gaussian potential in Fig. (\[f2\])(b) as the potential starts to move. Then the oscillating potential nucleates vortex pairs whose impulses alternately change direction. They reconnect with each other to make new vortex pairs, leaving the potential in two cases. This phenomenon is not observed for the case of uniform motion of the potential, but only for an oscillating potential. Reaching the surfaces, the vortex pairs interact with ghost vortices, which are vortices in the low-density region. Then the vortices head toward the bow of the condensate along the surfaces. A vortex coming up from the left side reaches the bow to meet one from the right side, thus making a new vortex pair. Finally, the pair comes back to the center of the condensates. Thus the vortices nucleated by the potential enjoy a lengthy migration in the “sea” of BEC; the vortices are nucleated from the potential, reconnect, move away from it, reach the surface, head toward the bow and come back to the center. In the following, we illustrate elementary processes related to the synergy dynamics. ![(Color online) Density profile for reconnection of vortex pairs near the oscillating potential at (a) $t$ = 0.8 ms, (b) $t$ = 1.2 ms, and (c) $t$ = 2.3 ms are shown. The density profile before the collision between the potential and the vortex pair are shown in (a). Thereafter, another ghost vortex pair nucleates in (b), exiting the potential through the collision, which causes reconnection of the vortices. As a result, two pairs appear in (c).[]{data-label="f4"}](reconnection2.eps){width="0.7\linewidth"} An oscillating potential creates vortex pairs, causes reconnection of pairs characterized by the oscillation, and causes the new pairs to leave for the surface of the condensate. Consequently, the surface becomes filled with vortices having positive and negative circulation, which leads to nucleation of rarefaction pulses and the migration of vortices. We call this sequence synergy dynamics of vortices and rarefaction pulses, which often occurs in cases where the amplitude of the oscillation is larger than the size of the oscillating potential. Ghost vortices, namely quantized vortices in a low density region, are important for the nucleation of the usual vortices in the bulk density region since nucleation requires seeds of vortices. In rotating BECs, ghost vortices are nucleated outside the condensate, entering it through the excitation of the surface waves, leading to the creation of usual vortices [@Tsubota02; @Kasamatsu03]. Thus, the periphery of the condensate provides seeds of topological defects. In our system, the oscillating potential provides seeds within itself. The potential starts to move, inducing a velocity field like back-flow, emitting phonons, and a ghost vortex pair is nucleated inside the potential as shown in Fig. (\[f3\])(a). The ghost pair tends to move away from the potential in Fig. (\[f3\])(b), and a usual vortex pair appears in the condensate in Fig. (\[f3\])(c). Thus, the ghost vortices work as seeds of usual vortices. Reconnection of vortex pairs occurs near the oscillating potential. The new vortex pair has an impulse in the same direction as that of the potential. Then, the potential changes the direction of the velocity. Thus, the potential will collide with the pair in Fig. (\[f4\])(a). Then, a new ghost vortex pair is nucleated inside the potential whose impulse is opposite to that of the usual vortex pair, reconnecting with it as shown in Fig. (\[f4\])(b). Thus, two new vortex pairs appear in the condensate in Fig. (\[f4\])(c). Thereafter, the pairs move away from the potential, leaving for the surface of the condensate. This reconnection is characteristic of the oscillating potential because the potential repeatedly emits vortices of positive and negative circulation in opposite directions, which is not seen for potentials of uniform motion. This leads to nucleation of rarefaction pulses, as shown in the following. ![(Color online) Density profile for nucleation of rarefaction pulses at (a) $t$ = 2.5 ms, (b) $t$ = 3.2 ms, (c) $t$ = 3.8 ms and (d) $t$ = 4.7 ms are shown. Some vortices sit near the surface in (a) and reconnection of the vortices occurs in (b). While the new pairs move toward the center of the condensate, the pairs annihilate, which leads to nucleation of rarefaction pulses in (c) and (d).[]{data-label="f5"}](soliton2.eps){width="9cm" height="4.5cm"} The vortex pairs separate as they approach the surface of the condensate shown in Fig. (\[f5\]). This behavior is qualitatively understood by applying the idea of an image vortex, which is often used in dynamics. The vortices induce a circular velocity field in a uniform system, but the field is distorted in a nonuniform system. This effect is strongly evident near the surface of the condensate where the density profile rapidly varies. As the vortices arrive at the surface, the normal component of the velocity field is suppressed. This situation is approximately equal to the relation between a vortex and a solid wall, so that the dynamics of vortices near the surface in Fig. (\[f5\]) can be shown by the image vortex. Note that this idea only gives a qualitative understanding since the surface is not exactly a solid wall. The vortices near the surface of the condensate have two fates. One is that a vortex pair transforms into rarefaction pulses through the annihilation of the pairs. The other is that the vortices migrate in the condensate. We show these dynamics in the following. Many vortices have accumulated near the surface of the condensate in Fig. (\[f5\])(a) since the oscillating potential continues to make vortices with positive and negative circulation. Hence, the vortices near the surface can reconnect with each other as shown in Fig. (\[f5\])(b), where we enclose the new vortex pairs with square dotted lines. These pairs have impulse toward the center of the condensate. As a pair approaches the center, the size of the pair diminishes. Consequently, pair annihilation of vortices occurs, making the rarefaction pulses shown in Figs. (\[f5\])(c) and (d). The low density parts in Figs. (\[f5\])(c) and (d) have these properties and hence we can identify them as rarefaction pulses. This kind of nucleation of rarefaction pulses is characteristic of an oscillating potential since it is caused by the potential emitting vortices in opposite directions. Conclusion {#sec:con} ========== We have reported the dynamics of vortices in a dipolar Bose-Einstein condensate by solving the two-dimensional, nonlocal, Gross-Pitaevskii equation numerically. We have calculated the critical velocity for vortex nucleation and found that the critical velocity increases when we increase the strength of the dipolar interaction in the atomic BEC. We have showed the formation of the rarefaction pulses during the dynamics of the vortices with opposite rotations. 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--- abstract: 'We study the structure of the load-based spanning tree (LST) that carries the maximum weight of the Erdös-Rényi (ER) random network. The weight of an edge is given by the edge-betweenness centrality, the effective number of shortest paths through the edge. We find that the LSTs present very inhomogeneous structures in contrast to the homogeneous structures of the original networks. Moreover, it turns out that the structure of the LST changes dramatically as the edge density of an ER network increases, from scale-free with a cutoff, scale-free, to a star-like topology. These would not be possible if the weights are randomly distributed, which implies that topology of the shortest path is correlated in spite of the homogeneous topology of the random network.' author: - 'Dong-Hee Kim and Hawoong Jeong' title: Inhomogeneous substructures hidden in random networks --- Complex network theories have attracted much attention because of the usefulness to analyze diverse complex systems in the real world [@Albert1; @Dorogovtsev1]. The most representative measure characterizing a network is the *degree distribution*, $P(k)$, which indicates probability for a vertex to be directly connected to $k$ neighboring vertices with edges. Specifically, while the power-law distribution, $P(k) \sim k^{-\gamma}$, is widely found in the most of real-world networks including technological, biological, and social networks, such as the Internet [@Faloutsos1], the World Wide Web [@Albert2], the metabolic networks [@Jeong1], the protein interaction networks [@Jeong2], and the coauthorship networks [@Newman1], there also exist homogeneous networks, such as the US highway network and the US power-grid network, which are explained by the bell-shaped and exponential degree distributions. Recently it has been claimed that an apparent scale-free network can be originated from a sampling result of a underlying homogeneous network [@Clauset1]. While the degree distribution gives valuable knowledge of local structures of networks around us, it is also necessary to know global structures of networks to understand dynamics on the networks properly. The information transport between two vertices occurs along an optimal path connecting them, defined as the path minimizing the total cost [@Braunstein1; @Sreenivasan1; @Buldyrev1], which is usually determined by using the global knowledge of the network. For instance, full information of connections in the network is required to determine a shortest path defined as a path consisting of a minimum number of edges, which would be an optimal path if the costs of all edges are identical. The scale-free network has been revealed to have very inhomogeneous shortest path topology so that one can find extremely important vertices or edges that huge number of shortest paths are passing through, which has been supported by the power-law distribution of the betweenness centrality (BC)  [@Goh1] and the existence of the transport skeleton structure [@dhkim1] that makes it possible to understand the origin of the difference between the BC exponent classes of real-world networks [@dhkim1; @Goh1] and the universal properties of the fractal scaling [@Goh2; @Song1]. However, in the non-scale-free networks, even though it has been known that there are no such vertices or edges used heavily in the shortest paths, the topology of the shortest paths has not been intensively studied so far. Our main interest is to find out how the topology of shortest paths is correlated with the network topology in the Erdös-Rényi (ER) random network model [@Erdos1], where two arbitrary vertices in the network are randomly connected to each other by an edge with a given probability $p$, which gives the Poisson degree distribution. In order to systematically study the shortest path topology, it is necessary to treat the network as a weighted network in which the contribution of the shortest paths on each edge is assigned as the weight of the edge. We use the *edge-betweenness centrality* (edge-BC) [@Freeman1; @Girvan1; @Newman2] to represent the contribution of the shortest paths on each edge of the network, which is a convenient quantity counting the effective number of shortest paths through the edge and thereby gives the average traffic though the edges. The edge-BC of an edge $e_{ij}$ between vertices $i$ and $j$ is the total contribution of the edge on the shortest paths between all possible pairs of vertices, which is defined as follows: $$b(e_{ij}) = \sum_{m \neq n} b(m,n;i,j) = \sum_{m \neq n} \frac{c(m,n;i,j)}{c(m,n)},$$ where $c(m,n;i,j)$ denotes the number of shortest paths from a vertex $m$ to $n$ through the edge $e_{ij}$, and $c(m,n)$ is the total number of shortest paths from $m$ to $n$. In this weighted networks, one useful way to study the spatial correlation of the weight is to investigate the *load-based spanning tree* (LST) [@dhkim1], which consists of a set of selected edges to maximize its total weights, which corresponds to the skeleton of the network  [@dhkim1]. From the degree distribution of the LST, we can check whether the spatial distribution of the weights is correlated with the original network topology. If the weights are randomly distributed on the edges of the network, the degree distribution of the original network would be preserved in its LST [@Szabo1]. On the other hand, if there exists significant topological correlation in the distribution of the weights on the edges, the degree distribution is expected to show a large deviation from the Poisson distribution, the degree distribution of the ER network. In this paper, we investigate the structure properties of the LST of the ER model in this respect. We find that the LSTs show very inhomogeneous structures in contrast to the homogeneous structures of the original networks. It is found that the degree distribution of the LST shows rich characteristics depending on the connection probability and the size of the ER network, which turns out to be very different from the Poisson distribution. Figure \[fig:pk\](a) shows an illustration of the inhomogeneous LST structure obtained from the ER network with $N=100$ vertices and $p=0.1$, in which the hub vertices having a significant number of degree are found. The detailed characteristics of degrees can be found in Fig. \[fig:pk\](b) which displays the degree distributions of the LSTs obtained from the ER networks generated with various connection probabilities. It is the most interesting feature that the right-skewed degree distributions are observed in the LSTs for the wide range of connection probability $p$ of the ER network whose degree distributions follow the narrow Poisson distribution. For the examined LSTs, it is found that a power-law with a cutoff fits well to the degree distribution of the LST, where the exponent and the cutoff degree depend on $p$. The emergence of these inhomogeneous degree distributions in the LSTs indicates that there exist non-negligible correlations between the weights of neighboring edges sharing a common vertex at their ends, since a vertex shared by the edges having higher weights becomes a hub in the LST. If we remove the correlation by the random redistribution [@shuffle] of the weights on the edges, the degree distributions of the LSTs become the Poisson distribution as expected \[see Fig. \[fig:pk\](c)\]. These imply that the shortest paths, which give the weights of its constituting edges, are not randomly distributed on the ER network but strongly correlated enough to generate an inhomogeneity in spite of the homogeneous topology of the ER network. The topological correlation of the shortest path that gives rise to the inhomogeneous LST structure can be specified by the correlation between the degree of the vertex and the weights of the edges connected to the vertex. In order to find out more about the spatial correlation in the distribution of the weights, we measure the average weight rank $R_k$, an averaged value of weight-rank $r$ over the edges attached to a vertex having degree $k$. The rank $r_{ij}$ of the edge between $i$ and $j$ is graded for its weight $w_{ij}$, i.e., the largest weight gives $r=1$, the second largest weight gives $r=2$, and so on. Mathematically $R_{k}$ is defined as follows: $$R_k = \bigg\langle \frac{1}{|\mathbf{V}_k|} \sum_{i \in \mathbf{V}_k} \frac{1}{k} \sum_{j} r_{ij} a_{ij} \bigg\rangle ,$$ where $\mathbf{V}_k$ and $|\mathbf{V}_k|$ are the set of vertices having degree $k$ and the number of those vertices, respectively, $a_{ij}$ is the adjacency matrix element; $a_{ij}=1$ if $i$ and $j$ is connected and $a_{ij}=0$ otherwise, and $\langle \ldots \rangle$ denotes an average over network ensembles. Consequently, the small (large) value of $R$ indicates that a vertex has the edge with high (low) ranks. The reason why we attach great importance to $R_k$ is that it gives an insight into how the structure of the network changes in the LST because the edges of the network are picked in the rank order for the LST. While the average value of the weights can give similar information, since the ER network has a rather homogeneous distribution of weight values, the average weight rank shows more clear changes depending on the spatial distribution of the weights as compared with the average value of the weights. In counting ranks, we do not admit a tie in ranks for the edges with the same weights but assign ranks according to random priority if the values of weights are the same. However, we note that averaging over the network ensembles removes the dependence on the counting method of the tie in ranks. In order to figure out how the weights are distributed on the network, we compare the average weight rank $R_{k}$ with that for the uncorrelated one in which the weights are randomly redistributed on the edges to remove the correlation between the edge and the weight on it. For instance, if $R_k$ is smaller than the value for uncorrelated weights, the vertex with degree $k$ can be regarded to have an edge with a higher weight as compared with the uncorrelated situation. This comparison gives an insight into the topological correlation of the weights on the network. From Fig. \[fig:corr\](a), we find that the extreme vertices, which belong to the tail of either the smaller or larger degree part in the Poisson degree distribution, have significantly higher average weight ranks (small value of $R_k$) than the average rank for the uncorrelated spatial distribution of the weights, given by $\sim L/2$, where $L$ is the total number of edges of the original network. This indicates that these extreme vertices are more likely to preserve their degree in the LSTs. More directly, we also measure the degree correlation between the LST and its original network. In Fig. \[fig:corr\](b), for the extreme vertices, it is confirmed that the degrees of the LST show strong correlations with the degrees of the original network. Consequently, the extreme vertices become hubs that constitute the heavy tail of the inhomogeneous degree distribution of the LST. The structural homogeneity of the ER network varies with the edge density represented by using the connection probability $p$. As well as the structural properties, the characteristics of the shortest path topology also depend on the edge density of the ER network. Specifically, the change of the connection probability $p$ in the ER network gives rise to the change of the degree distribution of its LST. At small values of $p$’s, as seen in Fig. \[fig:pk\](b), the degree distribution of the LST can be characterized by the power-law exponent $\gamma$ and the average maximum degree $k_{max}$. Thus, we can monitor the structural change of the LST by looking at the dependence of $\gamma$ and $k_{max}$ on the connection probability $p$ and the network size $N$. As $p$ increases, $\gamma$ monotonically increases, but interestingly, $k_{max}$ shows non-trivial behavior of decreasing even though $k_{max}$ has a chance to increase further as $p$ increases because a higher $p$ generates a larger average degree in the original ER network. To find a universal description of this non-trivial behavior of the degree distributions of the LSTs, we perform a size-scaling on data obtained from the ER networks of various sizes and connection probabilities and find out a common scaling feature; $k_{max}$ monotonically increases to the maximum value of $\sim N^{1/2}$ until $p$ reaches at $\sim N^{-1/2}$, then it decreases as $p$ increases further while $\gamma$ monotonically increases \[see Fig. \[fig:pdep\]\]. This $p$ dependence of $\gamma$ and $k_{max}$ strongly implies that there exists a critical $p$ value at which a clear scale-free degree distribution emerges by satisfying the balance between $\gamma$ and $k_{max}$. More precisely, because the natural cutoff $k_{cutoff}$ scales with $\sim N^{1/(\gamma -1)}$ in the finite sized network of $N$ vertices with power-law degree distribution $P(k)\sim k^{-\gamma}$ [@Cohen1; @Dorogovtsev2], we can test the *scale-freeness* of the degree distribution of the LST depending on $p$ by comparing $k_{max}$ with the natural cutoff. From the plot of $k_{max} / N^{1/(\gamma -1)}$ as a function of $pN^{1/2}$ shown in Fig. \[fig:pdep\](c), we find that $k_{max}$ of the LST becomes comparable with the natural cutoff $N^{1/(\gamma-1)}$ when $p$ reaches at $N^{-1/2}$ so that the cutoff diminishes in the degree distribution, which leads to the emergence of the scale-free spanning tree. The exponent $\gamma$ of the power-law degree distribution increases as $p$ increases in the small $p$ region as shown in Fig. \[fig:pdep\], but the Poisson degree distribution of the original ER network is not recovered even if $p$ gets much larger. Interestingly, the average maximum degree $k_{max}$ increases when $p$ becomes far larger than $\sim N^{-1/2}$ [@comment1]. For intermediate values of $p$’s, the structures of LSTs show very diverse degree distribution depending on the ensemble of the ER network generation since there can exist lots of edges that have same value of the edge-BC (weights), which gives large degeneracy in constructing LSTs. However, we find that a *star-like* structure finally emerges in the LST \[see Fig. \[fig:star\]\] as the ER network gets very denser to be a nearly fully-connected network. In the limiting cases, this can be understood easily. Let us consider a nearly fully connected network in which only a single pair of vertices are not connected while all the other vertex pairs are directly connected by edges [@comment2]. In this case, the shortest path connecting these two vertices must pass through one of their common neighbor vertices, and consequently this shortest path gives additional contributions to the edge-BCs of the edges connected to these two vertices having smaller degrees \[see Fig. \[fig:star\](a)\]. Therefore, in the edge selection for constructing LSTs, the edges connected to the smaller degree vertices are chosen with higher priorities, which leads the emergence of the star-like LSTs. Finally, we note that the emergence of the scale-free spanning tree in the ER network also has been reported in the two recent works. Clauset *et al.* [@Clauset1] reported that the spanning tree constructed by using traceroutes from a single source has power-law degree distribution with cutoff. Kalisky *et al.* [@Kalisky1] presented that the minimum spanning tree of the percolation cluster in the ER network shows the power-law degree distribution. Together with these works, our findings indicate that the consideration of the transport pathways or the weights of the edges can drastically changes the original homogeneous topology of the ER network. In conclusion, we have investigated the structural properties of the LST of the weighted ER network in which the weight of an edge is given by the edge-BC. We have found that the degree distribution of the LST shows very inhomogeneous distribution, which can be described as power-law with a cutoff, scale-free, or star-like depending on the edge density of the ER network. The emergence of these inhomogeneous degree distributions in the LST implies that the shortest paths are not homogeneously distributed on the ER network but spatially correlated in spite of the homogeneous topology of the ER network. This work was supported by Grant No. R14-2002-059-01002-0 from the KOSEF-ABRL Program (D.H.K.) and by Grant No. R01-2005-000-1112-0 from the Basic Research Program of the Korea Science & Engineering Foundation (H.J.). [10]{} R. Albert and A.-L. Barabási, Rev. Mod. Phys. [**74**]{}, 47 (2002). S. N. Dorogovtsev and J. F. F. Mendes, Adv. Phys. [**51**]{}, 1079 (2002). M. Faloutsos [*et al.*]{}, Comput. Commun. Rev. [**29**]{}, 251 (1999). R. Albert [*et al.*]{}, Nature (London) [**401**]{}, 130 (1999). H. Jeong [*et al.*]{}, Nature (London) [**411**]{}, 41 (2000). H. Jeong [*et al.*]{}, Nature (London) [**407**]{}, 651 (2000). M. E. J. Newman, Proc. Natl. Acad. Sci. U.S.A. [**98**]{}, 404 (2001). A. Clauset and C. Moore, Phys. Rev. Lett. [**94**]{}, 018701 (2005). L. A. Braunstein [*et al.*]{}, Phys. Rev. Lett. [**91**]{}, 168701 (2003). S. Sreenivasan [*et al.*]{}, Phys. Rev. E [**70**]{}, 046133 (2004). S. V. Buldyrev [*et al.*]{}, Phys. Rev. E 70, 035102(R) (2004). K.-I. 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Marc Massar [^1] and Jan Troost [^2]\ [*Theoretische Natuurkunde, Vrije Universiteit Brussel*]{}\ [*Pleinlaan 2, B-1050 Brussel, Belgium*]{}\ ABSTRACT > We study a configuration in matrix theory carying longitudinal fivebrane charge, i.e. a D0-D4 bound state. We calculate the one-loop effective potential between a D0-D4 bound state and a D0–anti-D4 bound state and compare our results to a supergravity calculation. Next, we identify the tachyonic fluctuations in the D0-D4 and D0–anti-D4 system. We analyse classically the action for these tachyons and find solutions to the equations of motion corresponding to tachyon condensation. Introduction ============ Matrix theory [@BFSS] [@Su] [@Se] is the M-theory interpretation of U(N) supersymmetric quantum mechanics which has passed many stringent tests. The brane content of matrix theory was determined in [@BSS]. Amongst other branes, the longitudinal fivebrane was identified [^3]. Two types of representation for the longitudinal fivebrane were proposed. One in terms of an instanton gauge field, which was used in [@CT1] to calculate one loop effective potentials between the D0-D4 bound state and other objects in matrix theory. Another representation was proposed in terms of two pairs of canonical conjugate variables. We use this representation to calculate one-loop effective potentials (see e.g. [@AB] [@L2] [@CT1] [@CT2]) between this object and a graviton, another D0-D4 bound, and a D0–anti-D4 system. Naturally, we find agreement with [@CT1] for the cases studied there and with an extra supergravity calculation for the D0-D4 and D0-anti-D4 system. In [@Ja] a first step towards the understanding of Sen’s tachyon condensation mechanism [@Sen] in matrix theory was taken, by analyzing the tachyon in the D0-D2 and D0–anti-D2 system. We concentrate on the D0-D4 and D0–anti-D4 system. We identify the tachyonic fluctuations in the D0-D4 and D0–anti-D4 background and analyse the classical action for these fluctuations in the spirit of [@Ja]. We find solutions to the action representing condensation to a vacuum filled with D0-branes and gravitons. The first section concentrates on a discussion of the classical solution of matrix theory corresponding to a D0-D4 bound state system. In the second section some effective potentials are calculated in detail to get acquainted with the representation of the longitudinal fivebrane in terms of canonical conjugate variables. We add a remark about the spectrum of the fluctuations around one longitudinal fivebrane. The next section deals with an analysis of the tachyonic fluctuations. Then we analyse possible solutions to the action for the tachyonic fluctuations. Finally, we add remarks on the results and open problems. Preliminary discussion of the classical solution ================================================ The lagrangian of matrix theory is given by $U(N)$ supersymmetric quantum mechanics, namely the dimensional reduction of tendimensional ${\cal N}=1$ $U(N)$ Super Yang-Mills theory to $0+1$ dimensions. It reads [@BFSS]: $$\begin{aligned} {\cal L} &=& \frac{T_0}{2} Tr \left( (D_0 X_I)^2+ \frac{1}{2} \left[ X_{I} , X_J \right]^2 + 2 \theta^{T} D_0 \theta - 2 \theta^{T} \g^I \left[ \theta, X_I \right] \right) \end{aligned}$$ where we take $ 2 \p \a' = 1 $ and $ T_0 = \frac{\sqrt{2 \p}}{g} $. Furthermore we have $D_0 = \partial_t - i \left[A_0, . \right] $ and $I=1,2, \dots, 9$. All fields are in the adjoint of $U(N)$. The fermions are Majorana-Weyl. The equations of motion for static configurations with trivial $A_0$ and vanishing fermions are: $$\begin{aligned} \left[ X_I, \left[ X_I, X_J \right] \right] &=& 0. \end{aligned}$$ We study especially a background configuration ($X_I = B_I$) corresponding to a D0-D4 bound state, or longitudinal fivebrane, satisfying the following commutation rules [@BSS]: $$\begin{aligned} \left[ B_1, B_2 \right] &=& - i c \, \s_3 \otimes I_{\frac{N}{2} \times \frac{N}{2}} \nonumber \\ \left[ B_3, B_4 \right] &=& - i c \, \s_3 \otimes I_{\frac{N}{2} \times \frac{N}{2}}, \end{aligned}$$ and the other matrices and commutators zero. Here $ \s_3 $ is the third Pauli matrix and $c$ is a constant. We take the infinite background matrices to be blockdiagonal such that this configuration solves the equations of motion. We will use two representations for this solution. The first one is in terms of two ’canonical conjugate’ pairs: $$\begin{aligned} \left[ P_1, Q_1 \right] &=& - i c \nonumber \\ \left[ P_2, Q_2 \right] &=& - i c \nonumber \\ B_1 &=& \left( \begin{array}{cc} P_1 & 0 \\ 0 & P_1 \end{array} \right) \nonumber \\ B_2 &=& \left( \begin{array}{cc} Q_1 & 0 \\ 0 & -Q_1 \end{array} \right) \nonumber \\ B_3 &=& \left( \begin{array}{cc} P_2 & 0 \\ 0 & P_2 \end{array} \right) \nonumber \\ B_4 &=& \left( \begin{array}{cc} Q_2 & 0 \\ 0 & -Q_2 \end{array} \right) \label{bl} \end{aligned}$$ This representation makes it easy to interpret the brane content of the configuration. Clearly, this solution as a whole carries no membrane charge since $q_2=- \frac{i}{2 \p} Tr \left[ B_I , B_J \right] = 0 $. It carries longitudinal fivebrane charge in the $1,2,3,4$ directions though: $$\begin{aligned} q_5=-\frac{1}{8 \p^2} \e^{IJKL} Tr \left[ B_I B_J B_K B_L \right] & = & N \frac{c^2}{4 \p^2} \end{aligned}$$ We refer to [@KK] for a clear and detailed analysis of the charges of the configuration, which yields the fact that the configuration you build in this way represents at least two D0-D4 bound states. That can be understood from the following reasoning. When we focus on the left upper block, it clearly has membrane charge in directions $1,2$ and $3,4$, as well as longitudinal fivebrane charge. It represents a D0-D4-D2-D2 bound state. Zooming in on the right lower block we see a D0-D4-anti-D2-anti-D2 bound state. If we formally superimpose the two parts we find two D0-D4 bound states, the 2-brane charge cancelling out. Thinking naively, one might be worried that this superposition is unstable, in particular, one might expect a tachyonic off-diagonal mode in the background configuration, representing a string stretching between a D2-brane and an anti-D2-brane. We will come back to this point and show that there is no such tachyonic mode. Moreover, the configuration was shown in [@BSS] to preserve 1/4 supersymmetry, as expected from D0-D4 bound states. An alternative representation of the background configuration in terms of gauge fields, discussed in detail in [@KK] will come in handy later on. It is given by: $$\begin{aligned} B^1 &=& c \left ( \begin{array}{cc} -i \partial_{x_1} & 0 \\ 0 & -i \partial_{x_1} \end{array} \right) \nonumber \\ B^2 &=& c \left ( \begin{array}{cc} -i \partial_{x_2} + \frac{ x_1}{c} & 0 \\ 0 & -i \partial_{x_2} - \frac{ x_1}{c} \end{array} \right) \nonumber \\ B^3 &=& c \left ( \begin{array}{cc} -i \partial_{x_3} & 0 \\ 0 &-i \partial_{x_3} \end{array} \right) \nonumber \\ B^4 &=& c \left ( \begin{array}{cc} -i \partial_{x_4} + \frac{ x_3}{c} & 0 \\ 0 & -i \partial_{x_4} - \frac{ x_3}{c} \end{array} \right) \end{aligned}$$ Note that we introduce the same four coordinates on the two D0-D4 bound states. This indicates our intention of treating them as a single object. Indeed, we will only analyse interaction potentials and fluctuations where the two D0-D4 bound states move as one. We define the left-upper and the right-lower part to be made up of $\frac{N_0}{2}$ D0-branes and denote the D0-brane charge density as $ \r_0 = \frac{N_0}{2 A_4} $, where $A_4$ represents the (possibly infinite) area of the coinciding D4-D0 bound states. Then we can derive the following relation [@KK] : $$\begin{aligned} c^2 &=& \frac{A_4 N_4}{(2 \p)^2 N_0} \label{c} \end{aligned}$$ where $N_4$ is the number of fourbranes and $N_0$ the total number of D0-branes in the bound state. Calculating effective potentials in matrix theory at one loop [^4] ==================================================================== In this section we calculate some interaction potentials between the D0-D4 bound state [^5] and other objects explicitly. In the literature (e.g. [@AB] [@L2] [@CT1] [@CT2]), some of these potentials have already been calculated using the representation in terms of an instanton background gauge field [@CT1]. But in the next section we will need a more detailed analysis of the fluctuations, when we identify the tachyonic ones. Moreover there are a few new technicalities in calculating the spectrum of the fluctuations when a single object is represented by ’two-by-two’ matrices, which have not been discussed in the literature yet. Therefore we find it useful to first redo some of the calculations in the literature in our representation, next to treat the new case of the D0-D4 and D0–anti-D4 interaction in detail. Because one object is sometimes represented by ’two-by-two’ matrices, we need some new conventions and nomenclature, which we will take to be as follows. In this section, the first object will have extent $n_0$, the second object $N_0$. When one object is represented by a ’two-by-two’ matrix, the submatrices will have half the extent of the object, e.g. $\frac{n_0}{2}$. Moreover, suppose we have two objects in the background represented by blockdiagonal ’two-by-two’ matrices. Then we will take the following nomenclature for the different parts of the coordinate matrices: $$\begin{aligned} X_I &=& \left ( \begin{array}{cccc} \mbox{block (1)} & 0 & \mbox{sector 13} & \mbox{sector 14} \\ 0 & \mbox{block (2)} & \mbox{sector 23} & \mbox{sector 24} \\ \mbox{sector 13}^\dagger & \mbox{sector 23}^\dagger & \mbox{block (3)} & 0 \\ \mbox{sector 14}^\dagger & \mbox{sector 24}^\dagger & 0 & \mbox{block (4)} \end{array} \right) \end{aligned}$$ The off-diagonal modes have been divided up into four different sectors. Other cases to be discussed are simpler and the nomenclature will be analogous in an obvious way. The technique to calculate the one-loop effective potential between two objects in matrix theory is standard by now [@AB] [@L2]. To calculate the potential, we determine the spectrum of the off-diagonal fluctuations corresponding to strings stretching from one object to the other. Their mass matrix is easily determined by expanding the action of matrix theory around the relevant background. This is slightly more involved when objects are represented by two-by-two matrices, but the general formulae in for instance [@CT1] [@KT] can easily be adapted to our case, essentially because the background matrices are block diagonal. We do not give the details of the calculation, but summarize for each case the result. In the following three subsections, we will discuss three different cases. For the first object we always take the D0-D4 bound state. For the second object we take respectively a graviton, a D0-D4 bound state, or a D0–anti-D4 bound state. The second object will always be taken to be at a distance $b$ of the first object in some transverse direction ($"8"$) and it will be moving with a velocity $v$ relative to the first object in another transverse direction ($"9"$). This is incorporated by choosing the background matrices corresponding to these transverse coordinates to be: $$\begin{aligned} B_8 &=& \left ( \begin{array}{cc} 0_{n_0 \times n_0} & 0 \\ 0 & b \, I_{N_0 \times N_0} \end{array} \right) \nonumber \\ B_9 &=& \left ( \begin{array}{cc} 0_{n_0 \times n_0} & 0 \\ 0 & v t \, I_{N_0 \times N_0} \end{array} \right) \label{X89} \end{aligned}$$ Finally, to make the interaction energies finite, we wrap the fourbranes on a four-torus. This hardly influences the calculation. It is moreover convenient to take the four-torus to have self-dual radii $R_i = \sqrt{\a'}$. It is straightforward to again add in the dependence on the compactification radii in the final formulae. See for instance [@CT1]. Interaction potential between a D0-D4 bound state system and a graviton ----------------------------------------------------------------------- For the first case, namely the D0-D4 bound state interacting with a moving graviton, the non-trivial background matrices are (recall also the separation matrices $B_8$ and $B_9$ given in (\[X89\])): $$\begin{aligned} \left[ P_1, Q_1 \right] &=& - i c \nonumber \\ \left[ P_2, Q_2 \right] &=& - i c \nonumber \\ B_1 &=& \left( \begin{array}{ccc} P_1 & 0 & 0 \\ 0 & P_1 & 0 \\ 0 & 0 & 0 \end{array} \right) \nonumber \\ B_2 &=& \left( \begin{array}{ccc} Q_1 & 0 & 0 \\ 0 & -Q_1 & 0 \\ 0 & 0 & 0 \end{array} \right) \nonumber \\ B_3 &=& \left( \begin{array}{ccc} P_2 & 0 & 0 \\ 0 & P_2 & 0 \\ 0 & 0 & 0 \end{array} \right) \nonumber \\ B_4 &=& \left( \begin{array}{ccc} Q_2 & 0 & 0 \\ 0 & -Q_2 & 0 \\ 0 & 0 & 0 \end{array} \right) \end{aligned}$$ For the quantum fluctuations we find identical spectra [^6] in the two sectors of extent $\frac{n_0}{2} \times N_0$. We define the hamiltonian: $$\begin{aligned} H &=& P_1^2 + Q_1^2 + P_2^2 + Q_2^2 +b^2 + v^2 t^2 \label{H}, \end{aligned}$$ corresponding to two non-interacting harmonic oscillators with frequency $c$ and a trivial extra part. After diagonalization, we find for the mass operators of the real bosons $ 4: H \pm 2iv;$ $ 8: H \pm 2 c ;$ $ 4 : H $ and for the fermions $8: H \pm iv; $ $ 8: H \pm 2 c \pm iv $, where we always state the number of fields first, and then the mass operator that corresponds to them. For instance $ 4: H \pm 2 c $ corresponds to 2 fields with mass operator $ H + 2c $ and 2 fields with mass operator $H - 2c $. The spectrum of these mass operators is easily determined. Following [@AB] [@L2] it is then straightforward to calculate the phase shift due to the interactions, and to approximate the phase shift at large distances $ b^2 \gg c $ and small velocities $v \ll b^2 $: $$\begin{aligned} \d &=& 2 \, {\cal N} \int_0^{\infty} \frac{ds}{s} \frac{e^{-b^2 s}}{8 \sin{vs} \sinh^2{c s}} \times \nonumber \\ & & (2+ 2 \cos{2vs}+4\cosh{2cs} \nonumber \\ && -4 \cos{vs}-4 \cos{vs} \cosh{2cs}) \\ & \approx & (\frac{n_4 N_0 v}{2 b^2}+ \frac{v^3 n_0 N_0}{8 b^2}) \end{aligned}$$ We denoted the degeneracy of the energy levels by $ {\cal N}$ which in this case is given by: $$\begin{aligned} {\cal N} &=& c^2 \frac{n_0}{2} N_0 \label{deg} \\ c^2 &=& \frac{n_4}{n_0} \end{aligned}$$ and we have used formula (\[c\]) at self-dual radii ($A_4=(2 \p)^2$). Determining the degeneracy of the spectrum has been done for the equivalent problem in the Landau model – a charged particle in a magnetic field. The degeneracy for the Landau levels was determined in [@AC]. Translating the formula for the degeneracy to our problem and carefully keeping track of normalization factors, we find the following heuristic for the degeneracy in general: $$\begin{aligned} {\cal N} &=& \mbox{Dimension fluctuation-matrix} \times \\ & & \mbox{Product of frequencies of the harmonic oscillators in H}. \end{aligned}$$ This rule is applied in (\[deg\]), $\frac{n_0 N_0}{2}$ being the dimension of the fluctuations and $c^2$ being the product of the harmonic oscillator frequencies in $H$ (\[H\]). The end result for the phase shift matches with the supergravity calculation in the relevant regime (\[sg1\]), and with the result obtained in a different manner in [@CT1]. The phase shift starts at order $v$ because the background configuration preserves $ \frac{1}{4} $ supersymmetry. D0-D4 and D0-D4 interaction --------------------------- In the second case the background matrices are: $$\begin{aligned} \left[ P_1, Q_1 \right] &=& - i c_1 \nonumber \\ \left[ P_2, Q_2 \right] &=& - i c_1 \nonumber \\ \left[ P_3, Q_3 \right] &=& - i c_3 \nonumber \\ \left[ P_4, Q_4 \right] &=& - i c_3 \nonumber \\ B^1 &=& \left ( \begin{array}{cccc} P_1 & 0 & 0 & 0 \\ 0 & P_1 & 0 & 0 \\ 0 & 0 & P_3 & 0 \\ 0 & 0 & 0 & P_3 \end{array} \right) \nonumber \\ B^2 &=& \left ( \begin{array}{cccc} Q_1 & 0 & 0 & 0 \\ 0 & -Q_1 & 0 & 0 \\ 0 & 0 & Q_3 & 0 \\ 0 & 0 & 0 & -Q_3 \end{array} \right) \nonumber \\ B^3 &=& \left ( \begin{array}{cccc} P_2 & 0 & 0 & 0 \\ 0 & P_2 & 0 & 0 \\ 0 & 0 & P_4 & 0 \\ 0 & 0 & 0 & P_4 \end{array} \right) \nonumber \\ B^4 &=& \left ( \begin{array}{cccc} Q_2 & 0 & 0 & 0 \\ 0 & -Q_2 & 0 & 0 \\ 0 & 0 & Q_4 & 0 \\ 0 & 0 & 0 & -Q_4 \end{array} \right) \end{aligned}$$ Here we find four sectors of extent $\frac{n_0}{2} \times \frac{N_0}{2}$, two by two identical, namely sector $(13)=(24)$ and sector $(23)=(14)$. We define the two relevant hamiltonians $$\begin{aligned} H^{(13)} &=& (P_1+P_3)^2 + (Q_1-Q_3)^2 + (P_2+P_4)^2 + (Q_2-Q_4)^2 +b^2 + v^2 t^2 \nonumber \\ H^{(23)} &=& (P_1+P_3)^2 + (Q_1+Q_3)^2 + (P_2+P_4)^2 + (Q_2+Q_4)^2 +b^2 + v^2 t^2. \nonumber \end{aligned}$$ Each describes a system of two decoupled harmonic oscillators. The diagonalized mass operators are: in sector (13) for the bosons $4: H \pm 2iv ;$ $ 8: H \pm 2 (c_1-c_3) ; $ $ 2 : H $ and for the fermions $8: H \pm iv ; $ $ 8 : H \pm iv \pm 2 (c_1-c_3)$. In sector (23) they read for the bosons $4: H \pm 2iv; $ $8 : H \pm 2 (c_1+c_3) ;$ $ 4 : H $ and for the fermions $8: H \pm iv ;$ $ 8 : H \pm iv \pm 2(c_1+c_3) $. The spectrum is again easily determined, and the phase shift now gets two different contributions: $$\begin{aligned} \d_{(13)+(24)} &=& 2 \, {\cal N}_{13} \int_0^{\infty} \frac{ds}{s} \frac{e^{-b^2 s}}{8 \sin{vs} \sinh^2{(c_1-c_3) s}} \times \nonumber \\ & & (2+ 2 \cos{2vs}+4\cosh{2(c_1-c_3)s} \nonumber \\ & & -4 \cos{vs}- 4 \cos{vs} \cosh{2(c_1-c_3)s}) \\ & \approx & {\cal N}_{13} (\frac{v}{ b^2}+ \frac{v^3 }{4 (c_1-c_3)^2 b^2}) \\ \d_{(23)+(14)} &=& 2 \, {\cal N}_{23} \int_0^{\infty} \frac{ds}{s} \frac{e^{-b^2 s} }{8 \sin{vs} \sinh^2{(c_1+c_3) s}} \nonumber \times \\ & & (2+ 2 \cos{2vs}+4\cosh{2(c_1+c_3)s} \nonumber \\ & & -4 \cos{vs}- 4 \cos{vs} \cosh{2(c_1+c_3)s}) \\ & \approx & {\cal N}_{23} (\frac{v}{ b^2}+ \frac{v^3 }{4 (c_1+c_3)^2 b^2}) \end{aligned}$$ giving a total phase shift $$\begin{aligned} \d & \approx & \frac{ (n_0 N_4+N_0 n_4) v}{2 b^2}+ \frac{n_0 N_0 v^3 }{8 b^2} \label{d4d4} \end{aligned}$$ where we have used the following formulae: $$\begin{aligned} {\cal N}_{13} &=& (c_1-c_3)^2 \frac{n_0}{2} \frac{N_0}{2} \\ {\cal N}_{23} &=& (c_1+c_3)^2 \frac{n_0}{2} \frac{N_0}{2} \\ c_1^2 &=& \frac{n_4}{n_0} \\ c_3^2 &=& \frac{N_4}{N_0} . \end{aligned}$$ The fact that the phase shift starts at order $v$ is due to the fact that the background configuration preserves 1/4 supersymmetry. The endresult matches with the supergravity calculation (\[sg2\]) [@CT1]. D0-D4 and D0-anti-D4 interaction -------------------------------- In the third case the background matrices are: $$\begin{aligned} \left[ P_1, Q_1 \right] &=& - i c_1 \nonumber \\ \left[ P_2, Q_2 \right] &=& - i c_1 \nonumber \\ \left[ P_3, Q_3 \right] &=& - i c_3 \nonumber \\ \left[ P_4, Q_4 \right] &=& - i c_3 \nonumber \\ B^1 &=& \left ( \begin{array}{cccc} P_1 & 0 & 0 & 0 \\ 0 & P_1 & 0 & 0 \\ 0 & 0 & -P_3 & 0 \\ 0 & 0 & 0 & -P_3 \end{array} \right) \nonumber \\ B^2 &=& \left ( \begin{array}{cccc} Q_1 & 0 & 0 & 0 \\ 0 & -Q_1 & 0 & 0 \\ 0 & 0 & Q_3 & 0 \\ 0 & 0 & 0 & -Q_3 \end{array} \right) \nonumber \\ B^3 &=& \left ( \begin{array}{cccc} P_2 & 0 & 0 & 0 \\ 0 & P_2 & 0 & 0 \\ 0 & 0 & P_4 & 0 \\ 0 & 0 & 0 & P_4 \end{array} \right) \nonumber \\ B^4 &=& \left ( \begin{array}{cccc} Q_2 & 0 & 0 & 0 \\ 0 & -Q_2 & 0 & 0 \\ 0 & 0 & Q_4 & 0 \\ 0 & 0 & 0 & -Q_4 \end{array} \right) \end{aligned}$$ Note the partial sign change in the first background matrix, turning the second object into a D0–anti-D4 bound state. We find four sectors of extent $\frac{n_0}{2} \times \frac{N_0}{2}$, all with identical spectra, when we ignore the origin in terms of the different coordinates [^7]. The relevant hamiltonian is: $$\begin{aligned} H^{(13)} &=& (P_1-P_3)^2 + (Q_1-Q_3)^2 + (P_2+P_4)^2 + (Q_2-Q_4)^2 +b^2 + v^2 t^2, \nonumber \end{aligned}$$ corresponding to a system of two harmonic oscillators. We will always suppose that $ c_1 - c_3 $ is positive, the other case being fully equivalent. The mass operators are for each sector for the bosons $4: H \pm 2iv ;$ $ 4 : H \pm 2 (c_1+c_3); $ $ 4 : H \pm 2 (c_1-c_3) ;$ $ 4 : H $ and for the fermions $ 16: H \pm iv \pm (c_1+c_3) \pm (c_1-c_3) $. The potential is then : $$\begin{aligned} {\cal V} &=& \frac{2}{\sqrt{\p}} \, {\cal N} \int_0^{\infty} \frac{ds}{s} \frac{e^{-b^2 s}}{4 s^{1/2} \sinh{(c_1-c_3) s} \sinh{(c_1+c_3) s}} \times \nonumber \\ & & (2+ 2 \cos{2vs}+2\cosh{2(c_1-c_3)s}+2\cosh{2(c_1+c_3)s} \nonumber \\ & &-8 \cos{vs} \cosh{(c_1+c_3)s} \cosh{(c_1-c_3)s})\\ & \approx & \frac{ n_4 N_4}{ b^3}+ \frac{(n_0 N_4+N_0 n_4) v^2}{4 b^3}+ \frac{n_0 N_0 v^4 }{16 b^3} \end{aligned}$$ Compared to the previous case (\[d4d4\]), there is an extra interaction between the D4-brane and the anti-D4-brane. The interaction potential is non-trivial at zero velocity and the background fully breaks supersymmetry. The end result is reproduced by our supergravity calculation (\[sg3\]) in the appendix. Clearly, the formula for the potential breaks down at small distances $b^2 \le 2 c_3$. Then there is a tachyon in the spectrum of the bosons since the lowest energy mode has mass: $E=(c_1-c_3) + (c_1+c_3)+b^2 - 2 (c_1+c_3)= b^2- 2 c_3$. We will treat the system at short distances in section 5. Summary ------- The conclusions we draw from these calculations are the following. At the level we are probing the system, the representation of the D0-D4 system that we use is equivalent to the instanton gauge field representation used in [@CT1]. We found full agreement when we compared the long range one loop potentials with supergravity results, also for the case of the D0-D4 and the D0–anti-D4 system, as expected. Moreover, we showed that it makes perfect sense to divide the off-diagonal modes into different sectors and treat them separately, which will be important in the second part of our paper. Remark on the fluctuations around one longitudinal fivebrane ============================================================ We refer to [@KK] for an analysis of the effective action for the fluctuations around the D0-D4 bound state system, but we add a remark that fits well into the context of our paper. As we mentioned in section 2, you might expect a tachyonic off-diagonal mode in the coordinate matrices spanning the fivebranes (\[bl\]), since they could correspond to strings stretching from a D2-brane to an anti-D2-brane. That this does not happen is shown by a small calculation. The relevant mass matrix for these modes is, for instance for the fluctuations in the coordinates $X_1$ and $X_2$: $$\begin{aligned} M_{12} &=& \left ( \begin{array}{cc} H & -2 i c \\ 2 i c & H \end{array} \right) \end{aligned}$$ where $$\begin{aligned} H &=& P_1^2 + Q_1^2 + P_2^2 + Q_2^2. \end{aligned}$$ Diagonalizing the mass matrix and determing the spectrum yields two kinds of fluctuations with the following energies: $$\begin{aligned} E &=& c (2n+1) + c (2m+1) + 2c \\ E' &=& c (2 n'+1) + c(2 m'+1) -2c \end{aligned}$$ Note that for the last kind of fluctuation, we find a massless mode, and not a tachyonic one. This is due to the quantummechanical zero point energies coming from the object spanning in the 1,2 as well as the 3,4 direction. For a membrane–anti-membrane system this mode would be tachyonic [@AB] [@Ja]. The action for the tachyonic fluctuations ========================================= The tachyonic fluctuations -------------------------- From now on, we will consider the D0-D4 system and the D0–anti-D4 system to lie on top of each other, so we put the background matrices $B_8$ and $B_9$ (\[X89\]) to zero. Then, when we compute the mass matrix for the fluctuations in the coordinate matrices $X_1$ and $X_2$, we find the following matrix for sector 13: $$\begin{aligned} M^{(13)}_{12} &=& \left ( \begin{array}{cc} H^{(13)} & -2 i (c_1+c_3) \\ 2 i (c_1+c_3) & H^{(13)} \end{array} \right) \end{aligned}$$ where $$\begin{aligned} H^{(13)} &=& (P_1-P_3)^2 + (Q_1-Q_3)^2 + (P_2+P_4)^2 + (Q_2-Q_4)^2 . \end{aligned}$$ We diagonalize the mass matrix and determine the spectrum for the diagonal fluctuations [^8]: $$\begin{aligned} E &=& (c_1+c_3) (2n+1) + (c_1-c_3) (2m+1) + 2(c_1+c_3) \label{spec} \\ E' &=& (c_1+c_3) (2n'+1) + (c_1-c_3) (2 m'+1) -2 (c_1+c_3) \end{aligned}$$ From the second line, we find a tachyonic mode, as expected, with mass $-2 c_3$. Note that for $c_1 < 2 c_3 $ you find several tachyonic modes. When you follow the simple diagonalization procedure in detail, you find that the tachyonic fluctuation [^9] is: $$\begin{aligned} \f &=& \frac{y^{(13)}_2 - i y^{(13)}_1}{\sqrt{2}} \end{aligned}$$ where $y^{(mn)}_I$ denotes the fluctuation in sector $(mn)$ and coordinate matrix $X_I$. The fluctuation $$\begin{aligned} \bar{\f} &=& \frac{y^{(13)}_2 + i y^{(13)}_1}{\sqrt{2}} \label{if} \end{aligned}$$ corresponds to (\[spec\]) and is never tachyonic. In the other sectors the computation goes analogously for a total of four tachyonic fields that correspond to strings stretching between the two D4 branes and the two anti-D4 branes (in the presence of the D0-branes). They are given by: $$\begin{aligned} \f &=& \frac{y^{(13)}_2 - i y^{(13)}_1}{\sqrt{2}} \nonumber \\ \f' &=& \frac{y^{(24)}_2 + i y^{(24)}_1}{\sqrt{2}} \nonumber \\ \chi &=& \frac{y^{(14)}_4 - i y^{(14)}_3}{\sqrt{2}} \nonumber \\ \chi' &=& \frac{y^{(23)}_4 + i y^{(23)}_3}{\sqrt{2}} \end{aligned}$$ The action ---------- Next we turn to the analysis of the action for the tachyonic fluctuations in the spirit of [@Ja]. We expand the classical action around the D0-D4 and D0–anti-D4 background, only keeping track of the tachyonic fluctuations and the gauge fields of the unbroken gauge group $U(1)^4$ under which the tachyons are charged. All fields we believe to be irrelevant, we put to zero, for instance (\[if\]). For simplicity, we take the number of D0-D4 bound states and D0-anti-D4 bound states to be equal, i.e. $ c_1 = c_3 = c$. Now the second representation introduced in section 2 comes in handy. Under the preceding assumptions, the coordinate matrices are given by: $$\begin{aligned} X^1 &=& c \left ( \begin{array}{cccc} - i \partial_{x_1} + A^{(1)}_{x_1}+a^{(1)}_{x_1} & 0 & i \frac{\f}{\sqrt{2} c} & 0 \\ 0 & - i \partial_{x_1} + A^{(2)}_{x_1}+a^{(2)}_{x_1} & 0 & -i\frac{\f'}{\sqrt{2}c} \\ -i \frac{\f^{\ast}}{\sqrt{2}c} & 0 & - i \partial_{y_1} + A^{(3)}_{y_1}+a^{(3)}_{y_1} & 0 \\ 0 & i \frac{{\f '}^{\ast}}{\sqrt{2}c} & 0 & - i \partial_{y_1} + A^{(4)}_{y_1}+a^{(4)}_{y_1} \end{array} \right) \nonumber \\ X^2 &=& c \left ( \begin{array}{cccc} - i \partial_{x_2} + A^{(1)}_{x_2}+a^{(1)}_{x_2} & 0 & \frac{\f}{\sqrt{2}c} & 0 \\ 0 & - i \partial_{x_2} + A^{(2)}_{x_2}+a^{(2)}_{x_2} & 0 & \frac{\f'}{\sqrt{2}c} \\ \frac{\f^{\ast}}{\sqrt{2}c} & 0 & - i \partial_{y_2} + A^{(3)}_{y_2}+a^{(3)}_{y_2} & 0 \\ 0 & \frac{{\f '}^{\ast}}{\sqrt{2}c} & 0 & - i \partial_{y_2} + A^{(4)}_{y_2}+a^{(4)}_{y_2}) \end{array} \right) \nonumber \\ X^3 &=& c \left ( \begin{array}{cccc} - i \partial_{x_3} + A^{(1)}_{x_3}+a^{(1)}_{x_3} & 0 & 0 & i \frac{\chi}{\sqrt{2}c} \\ 0 & - i \partial_{x_3} + A^{(2)}_{x_3}+a^{(2)}_{x_3} & -i\frac{\chi'}{\sqrt{2}c} & 0 \\ 0 & i\frac{{\chi '}^{\ast}}{\sqrt{2}c} & - i \partial_{y_3} + A^{(3)}_{y_3}+a^{(3)}_{y_3} & 0 \\ -i\frac{\chi^{\ast}}{\sqrt{2}c} & 0 & 0 & - i \partial_{y_3} + A^{(4)}_{y_3}+a^{(4)}_{y_3} \end{array} \right) \nonumber \\ X^4 &=& c \left ( \begin{array}{cccc} - i \partial_{x_4} + A^{(1)}_{x_4}+a^{(1)}_{x_4} & 0 & 0 & \frac{\chi}{\sqrt{2}c} \\ 0 & - i \partial_{x_4} + A^{(2)}_{x_4}+a^{(2)}_{x_4} & \frac{\chi'}{\sqrt{2}c} & 0 \\ 0 & \frac{{\chi '}^{\ast}}{\sqrt{2}c} & - i \partial_{y_4} + A^{(3)}_{y_4}+a^{(3)}_{y_4} & 0 \\ \frac{\chi^{\ast}}{\sqrt{2}c} & 0 & 0 & - i \partial_{y_4} + A^{(4)}_{y_4}+a^{(4)}_{y_4} \end{array} \right) \nonumber \end{aligned}$$ where $A$ is the background gauge field and $a$ the gauge field fluctuation. The background is invariant under $U(1)^4$, each $U(1)$ has its own upper index. We choose the background gauge fields such that the appropriate commutation relations between the background matrices are satisfied: $$\begin{aligned} A^{(1)}_{x_2} = - A^{(2)}_{x_2} &=& \frac{x_1}{c} \nonumber \\ A^{(1)}_{x_4} = - A^{(2)}_{x_4} &=& \frac{x_3}{c} \nonumber \\ A^{(3)}_{y_2} = -A^{(4)}_{y_2} &=& - \frac{y_1}{c} \nonumber \\ A^{(3)}_{y_4} = - A^{(4)}_{y_4} &=& \frac{y_3}{c} \label{bg}, \end{aligned}$$ and the rest zero. Each tachyonic mode is charged under two of the abelian gauge symmetries, with opposite charges, as can easily be seen by looking at the transformation properties of the full coordinate matrix. To represent the action in terms of an integral over the worldvolume of the branes, we use the rules of [@BSS], improved in [@KK] and elaborated upon in [@Ja]. We rally some of the technical details to appendix B. The following definitions come in handy in writing down the endresult. The non-center-of-mass coordinates are: $$\begin{aligned} u_i &=& \frac{ x_i+y_i}{2}. \end{aligned}$$ Covariant derivatives and field strengths are defined as (Upper indices label the gauge symmetries, lower indices $w_i =(x_i,y_i)$ label coordinates.) : $$\begin{aligned} \nabla^{(\pm m)}_{w_i} &=& \partial_{w_i} \pm iA^{(m)}_{w_i} \pm i a^{(m)}_{w_i} \nonumber \\ F^{(m)}_{w_i w_j} &=& i \left[ \nabla^{(m)}_{w_i} , \nabla^{(m)}_{w_j} \right] \nonumber \\ \nabla^{(m,\pm n)}_{u_i} &=& \nabla^{(m)}_{x_i} + \nabla^{(\pm n)}_{y_i} \nonumber \\ F^{(m, \pm n)}_{u_i u_j} &=& i \left[ \nabla^{(m, \pm n)}_{u_i} , \nabla^{(m, \pm n)}_{u_j} \right] \nonumber \\ &=& F^{(m)}_{x_i x_j} \pm F^{(n)}_{y_i y_j} \label{def} \end{aligned}$$ By a small $f$ we will denote the field strength $F$ without the background gauge fields contribution. The relevant part of the action for the fluctuations that we consider is then given by $S=\int d^4 u \, {\cal L} $, and the lagrangian by (up to an overall factor) : $$\begin{aligned} -{\cal L} &=& ( \frac{c^2}{2} f^{(1,-3)}_{u_1 u_2} - c + |\f|^2)^2+ ( \frac{c^2}{2} f^{(1,-4)}_{u_3 u_4} - c + |\chi|^2)^2 \nonumber \\ & & + ( \frac{c^2}{2} f^{(2,-4)}_{u_1 u_2} + c - |\f'|^2)^2+ ( \frac{c^2}{2} f^{(2,-3)}_{u_3 u_4} + c - |\chi'|^2)^2 \nonumber \\ & & +\frac{c^2}{2} ( |(\nabla^{(1,-3)}_{u_2}+ i \nabla^{(1,-3)}_{u_1}) \f|^2 + 2 |\nabla_{u_3}^{(1,-3)} \f|^2+ 2 |\nabla_{u_4}^{(1,-3)} \f|^2 \nonumber \\ & & +|(\nabla^{(1,-4)}_{u_4}+ i \nabla^{(1,-4)}_{u_3}) \chi|^2 + 2 |\nabla_{u_1}^{(1,-4)} \chi|^2 + 2 |\nabla_{u_2}^{(1,-4)} \chi|^2 \nonumber \\ & & +|(\nabla^{(2,-4)}_{u_2}- i \nabla^{(2,-4)}_{u_1}) \f'|^2 + 2 |\nabla_{u_3}^{(2,-4)} \f'|^2+ 2 |\nabla_{u_4}^{(2,-4)} \f'|^2 \nonumber \\ & & +|(\nabla^{(2,-3)}_{u_4}- i \nabla^{(2,-3)}_{u_3}) \chi'|^2 + 2 |\nabla_{u_1}^{(2,-3)} \chi'|^2 + 2 |\nabla_{u_2}^{(2,-3)} \chi'|^2 ) \nonumber \\ & & + \frac{c^4}{4} (f^{(1,3)^2}_{u_1 u_3}+ f^{(1,-3)^2}_{u_1 u_3} + f^{(2,4)^2}_{u_1 u_3}+ f^{(2,-4)^2}_{u_1 u_3} + f^{(1,3)^2}_{u_1 u_3}+ f^{(1,-3)^2}_{u_2 u_3} + f^{(2,4)^2}_{u_2 u_3}+ f^{(2,-4)^2}_{u_2 u_3} \nonumber \\ & & + f^{(1,3)^2}_{u_1 u_4}+ f^{(1,-3)^2}_{u_1 u_4} + f^{(2,4)^2}_{u_1 u_3}+ f^{(2,-4)^2}_{u_1 u_4} + f^{(1,3)^2}_{u_2 u_4}+ f^{(1,-3)^2}_{u_2 u_4} + f^{(2,4)^2}_{u_2 u_4}+ f^{(2,-4)^2}_{u_2 u_4} \nonumber \\ & & + f^{(1,3)^2}_{u_1 u_2}+ f^{(2,4)^2}_{u_1 u_2} + f^{(1,4)^2}_{u_3 u_4} + f^{(2,3)^2}_{u_3 u_4} ) \nonumber \\ & & + |(\phi {\chi '}^{\ast } - \chi {\phi '}^{\ast })|^2 + |(\phi \chi^{\ast } - \chi' {\phi '}^{\ast })|^2 \label{action} \end{aligned}$$ where all fields only depend on the non-center-of-mass coordinates. Note that it is the Lagrangian you expect, with the usual kinetic terms for the gauge fields, the appropriate covariant derivatives hitting the tachyons and a Higgs potential for the tachyons. There are some interactions between the tachyons and the gauge fields [@Ja], and an interaction potential between the different tachyons. Boundary conditions ------------------- The background gauge fields corresponding to the diagonal U(1)’s (\[bg\]) can be rewritten as follows: $$\begin{aligned} {\cal A}_{u_1} &=& 0 \nonumber \\ {\cal A}_{u_2} &=& \left ( \begin{array}{cccc} A_{u_2}^{(1)} & 0 & 0 & 0 \\ 0 & A_{u_2}^{(2)}& 0 & 0 \\ 0 & 0 & A_{u_2}^{(3)} & 0 \\ 0 & 0 & 0 & A_{u_2}^{(4)} \end{array} \right) \nonumber \\ &=& \frac{u_1}{c} \left ( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right) \nonumber \\ {\cal A}_{u_3} &=& 0 \nonumber \\ {\cal A}_{u_4} &=& \frac{u_3}{c} \left ( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \end{array} \right) \end{aligned}$$ The non-zero background gauge fields appearing in the covariant derivatives in the kinetic terms for the tachyons are: $$\begin{aligned} {\cal A}^{(1,-3)}_{u_2} &=& \frac{2 u_1}{c} \nonumber \\ {\cal A}^{(2,-4)}_{u_2} &=& -\frac{2 u_1}{c} \nonumber \\ {\cal A}^{(2,-3)}_{u_4} &=& -\frac{2 u_3}{c} \nonumber \\ {\cal A}^{(1,-4)}_{u_4} &=& \frac{2 u_3}{c} \end{aligned}$$ Taking the background gauge fields to live on a four-torus with radii $R_{u_i}$, they satisfy ’t Hooft’s twisted boundary conditions [@H]. They read in direction $u_1$ : $$\begin{aligned} {\cal A}_{u_i} (R_{u_1},u_2,u_3,u_4) &=& - i \O_{u_1} \partial_{u_i} \O_{u_1}^{-1} + \O_{u_1} {\cal A}_{u_i} (0,u_2,u_3,u_4) \O_{u_1}^{-1} \end{aligned}$$ and analogous for the other directions, where $ \O_{u_i} $ are the transition functions. The transition functions can be choosen to be: $$\begin{aligned} \O_{u_1} &=& \exp{[ -i u_2 \frac{R_{u_1}}{c} \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right)]} \nonumber \\ \O_{u_2} &=& 1 \nonumber \\ \O_{u_3} &=& \exp{[-i u_4 \frac{R_{u_3}}{c} \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \end{array} \right)]} \nonumber \\ \O_{u_4} &=& 1 \label{tm} \end{aligned}$$ These boundary conditions are due to the presence of the *background *field, i.e. due to the magnetic field made up of the D0-branes, representing the background objects. For the full background matrix this implies: $$\begin{aligned} B_{I} (R_{u_1},u_2,u_3,u_4) &=& \O_{u_1} B_I (0,u_2,u_3,u_4) \O_{u_1}^{-1} \label{bbc} \end{aligned}$$ and analogously for the other directions.** The boundary conditions for the tachyons that are trivial *with respect to the background *can be read of from (\[bbc\]): $$\begin{aligned} \f (u_1=R_1) &=& \f(u_1=0) e^{-2 i u_2 R_1 /c} \nonumber \\ \f' (u_1=R_1) &=& \f'(u_1=0) e^{2 i u_2 R_1 /c} \nonumber \\ \chi (u_3=R_3) &=& \chi(u_3=0) e^{-2 i u_4 R_3 / c} \nonumber \\ \chi' (u_3=R_3) &=& \chi'(u_3=0) e^{2 i u_4 R_3 / c} \label{tbbc} \end{aligned}$$ and the other background boundary conditions are trivial.** A solution to the equations of motion ===================================== First we look for a solution to the equations of motion where the total Lagrangian (\[action\]) vanishes and the background boundary conditions are satisfied. We make the following ansatz: $$\begin{aligned} \phi &=& {\phi '} ^{ \ast} (u_1, u_2) \nonumber \\ \chi &=& {\chi '}^{\ast} (u_3,u_4) \label{ans}. \end{aligned}$$ Then we find we can take: $$\begin{aligned} & & a^{(1)}_{u_{1,2}} = - a^{(2)}_{u_{1,2}} = - a^{(3)}_{u_{1,2}} = a^{(4)}_{u_{1,2}} \nonumber \\ & & a^{(1)}_{u_{3,4}} = - a^{(2)}_{u_{3,4}} = a^{(3)}_{u_{3,4}} = - a^{(4)}_{u_{3,4}}. \label{gs} \end{aligned}$$ The remaining non-trivial equations are: $$\begin{aligned} \frac{c^2}{2} f^{(1,-3)}_{u_1 u_2} - c + |\f|^2 &=& 0 \nonumber \\ (\nabla^{(1,-3)}_{u_2}+ i \nabla^{(1,-3)}_{u_1}) \f &=& 0 \nonumber \\ \frac{c^2}{2} f^{(1,-4)}_{u_3 u_4} - c + |\chi|^2 &=& 0 \nonumber \\ (\nabla^{(1,-4)}_{u_4}+ i \nabla^{(1,-4)}_{u_3}) \chi &=& 0 \end{aligned}$$ Under the assumption (\[ans\]), we get two copies of the Bogomolny equations. These have been studied in the context of Chern-Simons theory in detail [@JP] [@O] and we only summarize some main features. We can find magnetic soliton solutions to these equations with the background boundary conditions (\[tbbc\]). Since the spatial worldvolume of the D4-brane is fourdimensional, and the tachyons have non-trivial winding number around a circle at infinity, the magnetic solitons are twodimensional. The boundary conditions are treated in detail in [@Ja]. Using the solutions, we calculate the D0-brane charge from the worldvolume action of the D4-branes: $$\begin{aligned} N &=& \frac{1}{8 \p^2} \int d^4 u \left(F^{(1)} F^{(1)} + F^{(2)} F^{(2)} - F^{(3)} F^{(3)} -F^{(4)} F^{(4)} \right) \\ &=& \frac{1}{4 \p^2} \int d^4 u F^{(1,-3)}_{u_1 u_2} F^{(1,-4)}_{u_3 u_4} \nonumber \\ &=& \frac{A_4}{c^2 \p^2}, \end{aligned}$$ which is the original D0-brane charge. The D0 charge is concentrated at the intersections of the orthogonal twodimensional solitons. Moreover, from (\[gs\]) we find that the D2-brane charge cancels. This is consistent with the fact that we find, from the commutators (\[com\]) and (\[com2\]), and the supersymmetry variations $$\begin{aligned} \d \theta &=& \frac{1}{2} \left( D_0 X^I \g_I +\frac{1}{2} \left[X^I,X^J \right] \g_{IJ} \right) \e + \e' \label{susy}\end{aligned}$$ that the tachyon condensation restores all dynamical supersymmetry. We conclude that the end products after tachyon condensation are the original D0-branes, and extra gravitons as argued in [@Ja]. Remarks and conclusion ====================== In the previous section, we considered tachyon condensation where the tachyons had trivial boundary conditions relative to the background. We can consider more general possibilities, where the tachyons satisfy different boundary conditions. In the case of a membrane–anti-membrane configuration, this amounts to the following. By choosing the topological sector of the tachyon on the D2-brane anti-D2-brane to be non-trivial, one can add or subtract D0-brane charge. After condensation, this gives an arbitrary number of D0-branes. Technically, this is a trivial extension of [@Ja]. In particular, the approximate solution to the equations of motion in [@Ja] remains practically unchanged. In the case of the D0-D4 and D0-anti-D4, we have more possibilities. For instance, by changing the topological sectors of the four tachyons simultaneously, we can modify the amount of D0-brane charge in the end product in a fairly obvious manner (keeping the condition (\[ans\])). It is clear that for a more general choice of topological sectors, the end product will have D2-brane charge. It would be interesting to study such condensation in detail. In this paper we have studied the interactions between a D0-D4 bound state and a D0-anti-D4 bound state in matrix theory. First, we calculated the interaction potential at large distances and succesfully compared the result to an equivalent supergravity calculation. Next, we looked at a coinciding D0-D4 and D0–anti-D4 bound state system and identified the tachyonic fluctuations. We derived the classical action for these tachyonic fluctuations and found solutions to the equations of motion corresponding to tachyon condensation to D0-branes. [**Acknowledgments**]{}: We would like to thank Richard Corrado, Ben Craps, Shiraz Minwalla, Frederik Roose, Alex Sevrin and Walter Troost for useful discussions. This work was supported in part by the European Commission TMR programme ERBFMRX-CT96-0045 in which the authors are associated to K.U.Leuven. The probe-background calculation ================================ The standard technique for calculating the interaction potential (or corresponding phase shift) between two objects from the Born-Infeld action and supergravity approach is the following. You treat one object as the background and take the corresponding solution of the supergravity equations of motion. Next, you consider the worldvolume action of the other object in this background and calculate the potential it feels due to the background. This has been done for many situations in the literature (see for instance [@CT1] [@CT2]). We state the results of these calculations for comparison with the results obtained for matrix theory in the body of the paper. We take the conventions of [@CT1] ($2 \p \a'=1$) and we work at self-dual radius ($R_i = \sqrt{\a'}$) for the compactified directions. We moreover approximate the potential at large distances and small relative velocities between the two objects. For the interaction between a D0-brane bound state and a D0-D4 bound state we find the following phase shift $ \d $ [@CT1]: $$\begin{aligned} \d & \approx & \frac{1}{2 b^2} N_0 \left[ n_4 v +\frac{1}{4} (n_0 +2 n_4) v^3 \right] + O (\frac{1}{b^5},v^5). \label{sg1} \end{aligned}$$ For the interaction between two D0-D4 bound states, we find [@CT1]: $$\begin{aligned} \d & \approx & \frac{1}{2 b^2} \left[(n_0 N_4 + N_0 n_4) v +\frac{1}{4}( n_0 N_0 + n_4 N_4 +2 n_0 N_4 + 2 n_4 N_0) v^3 \right] \nonumber \\ & & +O(\frac{1}{b^5},v^5). \label{sg2} \end{aligned}$$ For the interaction between a D0-D4 bound state and a D0-anti-D4 bound state we generalize the calculation in [@CT1], to find the potential: $$\begin{aligned} {\cal V} & \approx & \frac{1}{4 b^3} \left[4 n_4 N_4 + (n_0 N_4 + N_0 n_4 )v^2 + \frac{1}{4} ( n_0 N_0 + n_4 N_4 + 2 n_0 N_4+ 2 N_0 n_4 ) v^4 \right] \nonumber \\ & & +O(\frac{1}{b^6},v^6). \label{sg3} \end{aligned}$$ The results agree with the matrix theory calculation at large $N_0$ and $n_0$. Note that the results for potentials (or corresponding phase shifts ($\d = \int dt \, {\cal V} (\sqrt{b^2+ (vt)^2})$) in matrix theory can also be compared directly to string theory calculations [@L2] [@L4] . Technical details ================= Some of the technical details for determining the action (\[action\]) are assembled here. We refer to [@KK] and [@Ja] for the rules to convert matrices into functions and traces into integrals. We only keep the relevant terms and consider static configurations only. First we define the non-center-of-mass coordinates – the center of mass coordinates just describe overall movements of the system in which we are not interested –: $$\begin{aligned} u_i &=& \frac{x_i+y_i}{2}. \end{aligned}$$ Using the definitions given in the body of the text (\[def\]), we can write down the commutators of the coordinate fields relevant to the problem in a reasonably compact form[^10]: $$\begin{aligned} \left[ X^1,X^2 \right] = & \qquad & \hspace{10 cm}\end{aligned}$$ $$\begin{aligned} \left[ X^3,X^4 \right] = & \qquad & \hspace{10 cm} \label{com} \end{aligned}$$ The other relevant commutators are all analogous to the following one: $$\begin{aligned} \left[ X^1,X^3 \right] &=& \left ( \begin{array}{cccc} i c^2 f^{(1)}_{u_1 u_3} & -\frac{1}{2} (\f {\chi '}^{\ast} - \chi {\f '}^{ \ast} ) & -\frac{c}{\sqrt{2}} \nabla_{u_3}^{(1,-3)} \f & \frac{c}{\sqrt{2}} \nabla_{u_1}^{(1,-4)} \chi \\ \ast & i c^2 f^{(2)}_{u_1 u_3} & -\frac{c}{\sqrt{2}} \nabla_{u_1}^{(2,-3)} \chi ' & \frac{c}{\sqrt{2}} \nabla_{u_3}^{(2,-4)} \f '\\ \ast & \ast & i c^2 f^{(3)}_{u_1 u_3} & \frac{1}{2} (\chi \f^{\ast} - \f ' {\chi '}^{\ast} ) \\ \ast & \ast & \ast & i c^2 f^{(4)}_{u_1 u_3} \end{array} \right) \nonumber \\ & & \label{com2}\end{aligned}$$ The commutators are antihermitian. We then simplify the action by concentrating on the non-center-of-mass fluctuations of the tachyon (compare [@Ja]): $$\begin{aligned} \phi (x_i,y_i) &=& \phi(u_i) \sqrt{\d (x_1-y_1) \d(x_2-y_2) \d(x_3-y_3) \d(x_4-y_4 )} \\ \chi (x_i,y_i) &=& \chi(u_i) \sqrt{\d (x_1-y_1) \d(x_2-y_2) \d(x_3-y_3) \d(x_4-y_4 )} \end{aligned}$$ Then the action reduces to (\[action\]), the integration running over four variables only. [99]{} T. Banks, W. Fischler, S. Shenker and L. Susskind, Phys. Rev. [**D55** ]{}(1997) 5112, hep-th/9610043 L. Susskind, hep-th/9704080 N. Seiberg, Phys. Rev. Lett. [**79** ]{} (1997) 3577, hep-th/9710009 ; A. Sen, Adv. Theor. Math. Phys. [**2**]{} (1998) 51, hep-th/9709220 T. Banks, N. Seiberg and S. Shenker, Nucl. Phys. [**B490** ]{}(1997) 91, hep-th/9612157 G. Lifschytz, Phys. Lett. [**B409**]{} (1997) 124, hep-th/9703201; E. Halyo, hep-th/9704086; M. Berkooz and M. Douglas Phys. Lett. [**B395**]{} (1997) 196, hep-th/9610236 O. Aharony and M. Berkooz, Nucl. Phys. 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[^9]: By abuse of language, we take ’tachyonic fluctuation’ to mean that the field includes a tachyonic mode. [^10]: We leave out the factors of the zero brane density $\r_0$ to avoid cluttering the formulas even more. They can easily be added in [@KK] [@Ja]

--- abstract: 'We establish a link between the maximization of Kolmogorov Sinai entropy (KSE) and the minimization of the mixing time for general Markov chains. Since the maximisation of KSE is analytical and easier to compute in general than mixing time, this link provides a new faster method to approximate the minimum mixing time dynamics. It could be interesting in computer sciences and statistical physics, for computations that use random walks on graphs that can be represented as Markov chains.' author: - 'M. Mihelich' - 'B. Dubrulle' - 'D. Paillard' - 'Q. Kral' - 'D. Faranda' bibliography: - 'biblio.bib' title: ' Maximum Kolmogorov-Sinai entropy vs minimum mixing time in Markov chains' --- Many modern techniques of physics, such as computation of path integrals, now rely on random walks on graphs that can be represented as Markov chains. Techniques to estimate the number of steps in the chain to reach the stationary distribution (the so-called “mixing time”), are of great importance in obtaining estimates of running times of such sampling algorithms [@bhakta2013mixing] (for a review of existing techniques, see e.g. [@guruswami2000rapidly]). On the other hand, studies of the link between the topology of the graph and the diffusion properties of the random walk on this graph are often based on the entropy rate, computed using the Kolmogorov-Sinai entropy (KSE)[@gomez2008entropy]. For example, one can investigate dynamics on a network maximizing the KSE to study optimal diffusion [@gomez2008entropy], or obtain an algorithm to produce equiprobable paths on non-regular graphs [@burda2009localization]. In this letter, we establish a link between these two notions by showing that for a system that can be represented by Markov chains, **a non trivial relation exists between the maximization of KSE and the minimization of the mixing time**. Since KSE are easier to compute in general than mixing time, this link provides a new faster method to approximate the minimum mixing time that could be interesting in computer sciences and statistical physics and gives a physical meaning to the KSE. We first show that on average, the greater the KSE, the smaller the mixing time, and we correlated this result to its link with the transition matrix eigenvalues. Then, we show that the dynamics that maximises KSE is close to the one minimizing the mixing time, both in the sense of the optimal diffusion coefficient and the transition matrix. Consider a network with $m$ nodes, on which a particle jumps randomly. This process can be described by a finite Markov chain defined by its adjacency matrix $A$ and its transition matrix $P$. $A(i,j)=1$ if and only if there is a link between the nodes $i$ and $j$ and 0 otherwise. $P=(p_{ij})$ where $p_{ij}$ is the probability for a particle in $i$ to hop on the $j$ node. Let us introduce the probability density at time $n$ $\mu_n=(\mu_n^i)_{i=1...m}$ where $\mu_n^i$ is the probability that a particle is at node $i$ at time $n$. Starting with a probability density $ \mu_0$, the evolution of the probability density writes: $\mu_{n+1}=P^t\mu_{n}$ where $P^t$ is the transpose matrix of $P$.\ Within this paper, we assume that the Markov chain is irreducible and thus has a unique stationary state. Let us define: $$\label{eqdn} d(n)= max{ || (P^t)^n\mu - \mu_{stat}|| \text{ } \forall \text{ } \mu },$$ where $||.||$ is a norm on $\mathbb{R}^n$. For $ \epsilon > 0$, the mixing time, which corresponds to the time such that the system is within a distance $\epsilon$ from its stationary state is defined as follows: $$\label{eq:mix1} t(\epsilon)= \min{ n, \, d(n) \leq \epsilon}.$$ For a Markov chain the KSE takes the analytical form [@billingsley1965ergodic]: $$\label{eqhks} h_{KS}=-\sum_{ij} \mu_{stat_{i}}p_{ij}\log(p_{ij}).$$ Random $m$ size Markov matrices are generated by assigning to each $p_{ij}$ ($i\neq j$) a random number between $0$ and $ \frac{1}{m}$ and $p_{ii}= 1-\sum_{j\neq i} p_{ij}$. The mean KSE is plotted versus the mixing time (Fig. \[fig:KS1\]) by working out $h_{KS}$ and $t(\epsilon)$ for each random matrix. (Fig. \[fig:KS1\]) shows that KSE is on average a decreasing function of the mixing time. ![Averaged KSE versus mixing time (top) for $10^6$ random $m=10$ size matrices and averaged $\lambda(P)$ versus mixing time (bottom) for $10^6$ random $m=10$ size matrices in curve blue and $f(t)=\epsilon^{1/t}$ in red. $\epsilon=10^{-3}$ and the norm is chosen to be the euclidian one.[]{data-label="fig:KS1"}](KSfuncmixtime1m10gmax1064.jpg){width="10cm"} We stress the fact that this relation is only true on average. We can indeed find two special Markov chains $P1$ and $P2$ such that $h_{KS}(P1) \leq h_{KS}(P2)$ and $t_1(\epsilon) \leq t_2(\epsilon)$. We illustrate this point further. The link between the mixing time and the KSE can be understood via their dependence as a function of the transition matrix eigenvalues. A general irreducible transition matrix $P$ is not necessarily diagonalizable on $\mathbb{R}$. However, since $P$ is chosen randomly, it is almost everywhere diagonalizable on $\mathbb{C}$. According to Perron Frobenius theorem, the largest eigenvalue is 1 and the associated eigen-space is one-dimensional and equal to the vectorial space generated by $\mu_\text{stat}$. Without loss of generality, we can label the eigenvalues in decreasing order of their module: $$1=\lambda_1 > \lvert \lambda_2 \rvert \geq....\geq \lvert \lambda_m \rvert \geq 0$$ The convergence speed toward $\mu_\text{stat}$ is given by the second maximum module of the eigenvalues of $P$ [@boyd2004fastest], [@pierre1999markov]: $$\lambda(P)=\max_{i=2...m}{ |\lambda_i|}= \lvert \lambda_2 \rvert$$ The eigenvalues $\lambda_1=1,...,\lambda_m$ of $P$ and $P^t$ being equal, let us denote their associated eigenvectors $\mu_1=\mu_\text{stat},...,\mu_m$. For any initial probability density $\mu_0$, we find: $$\label{eqmu0} || (P^t)^n\mu_0 - \mu_\text{stat}|| \propto (\lambda(P))^n.$$ According to Eqs. (\[eqdn\]) and (\[eq:mix1\]), $\lambda(P)^{t(\epsilon)} \propto \epsilon$, i.e. $\lambda(P) \propto \epsilon^{1/t(\epsilon)}$. Hence, the smaller $\lambda(P)$ the shorter the mixing time (Fig. \[fig:KS1\]). $h_{KS}$ being a decreasing function of $t(\epsilon)$ and $\lambda(P)$ being an increasing function of $t(\epsilon)$, we deduce that $h_{KS}$ is a decreasing function of $\lambda(P)$. This link between maximum KSE and minimum mixing time actually also extends naturally to optimal diffusion coefficients. Such a notion has been introduced by Gomez-Gardenes and Latora [@gomez2008entropy] in networks represented by a Markov chain depending on a diffusion coefficient. Based on the observation that in such networks, KSE has a maximum as a function of the diffusion coefficient, they define an optimal diffusion coefficient as the value of the diffusion corresponding to this maximum. In the same spirit, one could compute an optimal diffusion coefficient with respect to the mixing time, corresponding to the value of the diffusion coefficient which minimizes the mixing time -or equivalently the smallest second largest eigenvalue $\lambda(P)$. This would roughly correspond to the diffusion model reaching the stationary time in the fastest time. To define such an optimal diffusion coefficient, we follow Gomez and Latora and vary the transition probability depending on the degree of the graph nodes. More accurately, if $k_i=\sum_j A(i,j)$ denotes the degree of node $i$, we set: $$\label{eq:diff1} p_{ij}=\frac{A_{ij}k_j^\alpha}{\sum_j A_{ij}k_j^\alpha}.$$ If $\alpha <0$ we favor transitions towards low degrees nodes, if $\alpha=0$ we find the typical random walk on network and if $\alpha>0$ we favor transitions towards high degrees nodes. We assume here that $A$ is symmetric. It may then be checked that the stationary probability density is equal to: $$\label{eq:diff2} \pi_{stat_i}=\frac{c_ik_i^\alpha}{\sum_j c_jk_j^\alpha},$$ where $c_i=\sum_j A_{ij}k_j^\alpha$, Using Eqs. (\[eq:diff1\]) and (\[eq:diff2\]), we check that the transition matrix is reversible and then has $m$ real eigenvalues. From this stationary probability density, we can thus compute both the KSE and the second largest eigenvalue $\lambda(P)$ as a function of $\alpha$. The result is provided in (Fig. \[fig:KS5\]). ![KSE (top) and $\lambda(P)$ (bottom) function of $\alpha$ for a network of size $m=400$ with a proportion of $0$ in $A$ equal to $1/3$. []{data-label="fig:KS5"}](mixtimeandhksfuncalpham400rho1div34.jpg){width="10cm"} We observe in (Fig. \[fig:KS5\]) that the KS entropy has a maximum at a value that we denote $\alpha_{KS}$, in agreement with the findings of [@gomez2008entropy]. Likewise, $\lambda(P)$ (i.e. the mixing time) presents a minimum for $\alpha=\alpha_{mix}$. Moreover, $\alpha_{KS}$ and $\alpha_{mix}$ are close. This means that the two optimal diffusion coefficients are close to each other. Furthermore, looking at the ends of the two curves, we can find two special Markov chains $P1$ and $P2$ such that $h_{KS}(P1) \leq h_{KS}(P2)$ and $t_1(\epsilon) \leq t_2(\epsilon)$, illustrating that the link between KSE and the minimum mixing time is only true in a general statistical sense. We have thus shown that, for a given transition matrix $P$ (or equivalently for given jump rules) the greater the KSE, the smaller the mixing time. We now investigate whether a similar property holds for dynamics, i.e. whether transition rules that maximise KSE are close to the ones minimizing the mixing time. For a given network, i.e. for a fixed $A$, there is a well known procedure to compute the transition matrix $P_{KS}$ which maximizes the KSE with the constraints $A(i,j)=0 \Rightarrow P_{KS}(i,j)=0$ [@burda2009localization]. It proceeds as follow: let us note $\lambda$ the greatest eigenvalue of $A$ and $\Psi$ the normalized eigenvector associated i.e $A\Psi=\lambda \Psi$ and $ \sum_i \Psi^2_i=1$. We define $P_{KS}$ such that: $$\label{eq:pks} P_{KS}(i,j)=\frac{A(i,j)}{\lambda}\frac{\Psi_j}{\Psi_i}.$$ We have $ \forall i$ $ \sum_j P_{KS}(i,j) =1$. Moreover, using the fact that $A$ is symmetric we find: $$\label{eq:sta} \sum_j P_{KS}(j,i)\Psi^2_j=\sum_j \frac{A(j,i)\Psi_i\Psi_j}{\lambda}=\Psi^2_i.$$ Hence, $P_{KS}^t \Psi^2=\Psi^2$ and the stationary density of $P_{KS}$ is $\pi_{stat}=\Psi^2$. Using Eqs. (\[eqhks\]) and (\[eq:pks\]), we have: $$\label{eq:pks2} h_{KS}=-\frac{1}{\lambda}\sum_{(i,j)} A(i,j) \Psi_i \Psi_j \log(\frac{A(i,j)}{\lambda}\frac{\Psi_i}{\Psi_j}).$$ Eq. (\[eq:pks2\]) can be split in two terms: $$\begin{aligned} \label{eq:pks3} h_{KS}&=&\frac{1}{\lambda}\sum_{(i,j)} A(i,j)\Psi_i\Psi_j\log(\lambda)\nonumber\\ &-&\frac{1}{\lambda}\sum_{(i,j)} A(i,j)\Psi_i\Psi_j\log(A(i,j)\frac{\Psi_j}{\Psi_i}).\end{aligned}$$ The first term is equal to $\log(\lambda)$ because $\Psi$ is an eigenvector of $A$ and the second term is equal to $0$ due to the symmetry of $A$. Thus: $$\label{eq:pks4} h_{KS}=\log(\lambda).$$ Moreover, for a Markov chain the number of trajectories of length $n$ is equal to $ N_n=\sum_{(i,j)} (A^n)(i,j)$. For a Markov chain the KSE can be seen as the time derivative of the path entropy leading that KSE is maximal when the paths are equiprobable. For an asymptotic long time the maximal KSE is: $$\label{eq:pks5} h_{KS_{max}}=\frac{\log(N_n)}{n} \rightarrow\log(\lambda),$$ by diagonalizing $A$. Using Eqs. (\[eq:pks4\]) and (\[eq:pks5\]) we find that $P_{KS}$ defined as in Eq. (\[eq:pks\]) maximises the KSE. Finally $P_{KS}$ verifies $\pi_{stat_i}P_{KS}(i,j)=\pi_{stat_j}P_{KS}(j,i)$ $\forall$ $(i,j)$ and thus $P_{KS}$ is reversible. In a similar way, we can search for a transition matrix $P_{mix}$ which minimizes the mixing time -or, equivalently the transition matrix minimizing its second eigenvalue $\lambda(P)$. This problem is much more difficult to solve than the first one, given that the eigenvalues of $P_{mix}$ can be complex. Nevertheless, two cases where the matrix $P_{mix}$ is diagonalizable on $\mathbb{R}$ can be solved [@boyd2004fastest]: the case where $P_{mix}$ is symmetric and the case where $P_{mix}$ is reversible for a given fixed stationary distribution. Let us first consider the case where $P$ is symmetric. The minimisation problem takes the following form: $$\label{eq:mixmix} \left\{ \begin{array}{rcr} \min\limits_{P \in S_n} \lambda(P) \\ P(i,j) \geq 0, P*\textbf{1}=\textbf{1} \\ A(i,j)=0 \Rightarrow P(i,j)=0\\ \end{array} \right.$$ given the strict convexity of $\lambda$ and the compactness of the stochastic matrices, this problem admits an unique solution. $P$ is symmetric thus $\textbf{1}$ is an eigenvector associated with the largest eigenvalue of $P$. Then the eigenvectors associated to $\lambda(P)$ are in the orthogonal of $ \textbf{1}$.The orthogonal projection on $\textbf{1}^{\perp}$ writes: $ I_d-\frac{1}{n}\textbf{1}\textbf{1}^t$ Moreover, if we take the matrix norm associated with the euclidiean norm i.e. for $M$ any matrix $|||M|||= \max \frac{ ||MX||_2}{||X||_2} \text{ } X \in \mathbb{R}^n \text{ } X\neq 0$ it is equal to the square root of the largest eigenvalue of $ MM^t$ and then if $M$ is symmetric it is equal to $\lambda(M)$. Then the minimization problem can be rewritten: $$\label{eq:mix2} \left\{ \begin{array}{rcr} \min\limits_{P \in S_n} ||| (I_d-\frac{1}{n}\textbf{1}\textbf{1}^t)P(I_d-\frac{1}{n}\textbf{1}\textbf{1}^t)|||=|||P-\frac{1}{n}\textbf{1}\textbf{1}^t|||\\ P(i,j) \geq 0, P*\textbf{1}=\textbf{1} \\ A(i,j)=0 \Rightarrow P(i,j)=0\\ \end{array} \right.$$ We solve this constrained optimization problem (Karush-Kuhn-Tucker) with Matlab and we denote $P_{mix}$ the matrix which minimizes this system. We remark that the mixing time of $P_{KS}$ is smaller than the mixing time of $P_{mix}$. This is coherent because in order to calculate $P_{KS}$ we take the minimum on all the matrix space whereas to calculate $P_{mix}$ we restrict us to the symmetric matrix space. Nevertheless, we can go a step further and calculate, the stationary distribution being fixed, the reversible matrix which minimizes the mixing time. If we note $\pi$ the stationary measure and $\Pi=diag(\pi)$. Then $P$ is reversible if and only if $\Pi P=\Pi^t P$. Then in particular $\Pi^{\frac{1}{2}}P\Pi^{-\frac{1}{2}}$ is symmetric and has the same eigenvalues as $\Pi$. Finally, $p=(\sqrt{\pi_1},...,\sqrt{\pi_n})$ is an eigenvector of $\Pi^{\frac{1}{2}}P\Pi^{-\frac{1}{2}}$ associated to the eigenvalue $1$. Then the minimization problem can be written as the following system: $$\label{eq:mix2} \left\{ \begin{array}{rcr} \min\limits_{P} ||| (I_d-\frac{1}{n}\textbf{q}\textbf{q}^t)\Pi^{\frac{1}{2}}P\Pi^{-\frac{1}{2}}(I_d-\frac{1}{n}\textbf{q}\textbf{q}^t)|||\\ =|||\Pi^{\frac{1}{2}}P\Pi^{-\frac{1}{2}}-\frac{1}{n}\textbf{q}\textbf{q}^t|||\\ P(i,j) \geq 0, P*\textbf{1}=\textbf{1}, \Pi P=\Pi^t P \\ A(i,j)=0 \Rightarrow P(i,j)=0\\ \end{array} \right.$$ When we implement this problem in Matlab with $\pi=\pi_{KS}$ we find a matrix $P_{mix}$ such that naturally $\lambda(P_{mix}) \leq \lambda(P_{KS})$. Moreover we can compare both dynamics by evaluating $|||P_{KS}-P_{mix}|||$ compared to $|||P_{KS}|||$ which is approximatively equal to $|||P_{mix}|||$. We remark that $|||P_{KS}-P_{mix}|||$ depends on the density $\rho$ of $0$ in the matrix $A$. For a density equal to $0$ the matrices $P_{KS}$ and $P_{mix}$ are equal and the quantity $|||P_{KS}-P_{mix}|||$ will increase continuously when $\rho$ increases. This is shown in (Fig. \[fig:KS2bis\]). ![$|||P_{KS}-P_{mix}|||/|||P_{KS}|||$ as a function of the density $\rho$ of $0$ present in $A$.[]{data-label="fig:KS2bis"}](normePksmoinsPmixfonctiondensitedeAm103.jpg){width="10cm"} From this, we conclude that the rules which maximize the KSE are close to those which minimize the mixing time. This becomes increasingly accurate as the fraction of removed links in $A$ is weaker. Since the calculation of $P_{mix}$ quickly becomes tedious for quite large values of $m$, we offer here a much cheaper alternative by computing $P_{KS}$ instead of $P_{mix}$. Moreover, maximizing the KSE appears today as a method to describe out of equilibrium complex systems [@monthus2011non], to find natural behaviors [@burda2009localization] or to define optimal diffusion coefficients in diffusion networks. This general observation however provides a possible rationale for selection of stationary states in out-of-equilibrium physics: it seems reasonable that in a physical system with two simultaneous equiprobable possible dynamics, the final stationary state will be closer to the stationary state corresponding to the fastest dynamics (smallest mixing time). Through the link found in this letter, this state will correspond to a state of maximal KSE. If this is true, this would provide a more satisfying rule for selecting stationary states in complex systems such as climate than the maximization of the entropy production, as already suggested in [@mihelich2014maximum]. [**Acknowledgments**]{} Martin Mihelich thanks IDEEX Paris-Saclay for financial support. Quentin Kral was supported by the French National Research Agency (ANR) through contract ANR-2010 BLAN-0505-01 (EXOZODI).

--- abstract: 'We study nonlinear waves in a nonrelativistic ideal and cold quark gluon plasma immersed in a strong uniform magnetic field. In the context of nonrelativistic hydrodynamics with an external magnetic field we derive a nonlinear wave equation for baryon density perturbations, which can be written as a reduced Ostrovsky equation. We find analytical solutions and identify the effects of the magnetic field.' address: 'Instituto de Física, Universidade de São Paulo, Rua do Matão Travessa R, 187, 05508-090 São Paulo, SP, Brazil' author: - 'D. A. Fogaça, S. M. Sanches Jr. and F. S. Navarra' title: Nonlinear waves in magnetized quark matter and the reduced Ostrovsky equation --- Introduction ============ There is a strong evidence that quark gluon plasma (QGP) has been observed in heavy ion collisions at RHIC and at LHC [@qgp1; @qgp2]. Deconfined quark matter may also exist in the core of compact stars [@qgp3]. Waves may be formed in the QGP [@w1; @w2]. In heavy ion collisions waves may be produced, for example, by fluctuations in baryon number, energy density or temperature caused by inhomogeneous initial conditions [@w2]. In order to study waves, it is very often assumed that they represent small perturbations in a fluid and hence one can linearize the equations of hydrodynamics and find their solutions, which are linear waves. Alternatively, instead of linearization we may use another procedure, called Reductive Perturbation Method (RPM) [@rpm], which preserves the nonlinearity of the original equations. This leads to nonlinear differential equations, whose solution describe nonlinear waves, such as solitons. In a series of works [@werev; @weset] we studied the existence and properties of nonlinear waves in hadronic matter and in a quark gluon plasma as well. The existence and effects of a magnetic field in quark stars has been studied since long time ago [@mag1] and became a hot topic in our days. In a different context, about ten years ago [@mag2] it was realized that a very strong magnetic field might be produced in relativistic heavy ion collisions and it might have some effect on the quark gluon plasma phase. A natural question is then: what it the effect of the magnetic field on the waves propagating through the QGP ? In a previous work [@we17] we studied the conditions for an ideal, cold and magnetized quark gluon plasma (QGP) to support stable and causal perturbations. These perturbations were considered in the linear approach and the QGP was treated with nonrelativistic hydrodynamics. We have derived the dispersion relation for density and velocity perturbations. The magnetic field was included both in the equation of state and in the equations of motion, where the term of the Lorentz force was considered. We have used three equations of state: a generic non-relativistic one, the MIT bag model EOS (for weak and strong magnetic field) and the mQCD EOS. The anisotropy effects caused by the B field were also manifest in the parallel and perpendicular sound speeds. We found that the existence of a strong magnetic field does not lead to instabilities in the velocity and density waves. Moreover, in most of the considered cases the propagation of these waves was found to respect causality. However causality might be violated in the strong field regime. The onset of causality violation might happen at very large densities and/or large values of the wave length (small values of the wavenumber $k$). The magnetic field changes the pressure, the energy density and the speed of sound. It also changes the equations of hydrodynamics. One of the conclusions of Ref. [@we17] is that the changes in hydrodynamics are by far more important than the changes in the equation of state. In the present study we extend our previous work to the case of nonlinear waves. We will investigate the effects of a strong and uniform magnetic field on nonlinear baryon density perturbations in an ideal and magnetized quark gluon plasma. We consider the magnetic field in the EOS and also in the Euler equation, which requires special attention in the RPM [@rpm] formalism. Our study could be applied to the deconfined cold quark matter in compact stars and to the cold quark gluon plasma formed in heavy ion collisions at intermediate energies at FAIR [@fair] or NICA [@nica]. We go beyond the linear approach used in [@we17] and improve the nonlinear treatment used in [@w2], now including the strong magnetic field effects. Some work along this line was already published in [@azam], where the authors concluded that increasing the magnetic field leads to a reduction in the amplitude of the nonlinear waves. More recently [@javi17], perturbations in a cold QGP were studied with nonrelativistic hydrodynamics with magnetic field effects in a nonlinear approach. Solitonic density waves were found as solutions of a modified nonlinear Schrodinger equation. The magnetic field was found to increase the phase speed of the soliton and to reduce its width. We will discuss below the differences between our study and the above mentioned works. Nonrelativistic hydrodynamics ============================= We start from the nonrelativistic Euler equation [@land] with an external uniform magnetic field. The same magnetic field affects the thermodynamical quantities appearing in the equation of state, as in [@we17]. The magnetic field of intensity $B$ is chosen to be in the $z-$direction and hence $\vec{B}=B \hat{z}$ . The three fermions species considered are the quarks: up ($u$), down ($d$) and strange ($s$) with the following respectively charges $Q_{u}= 2 \, Q_{e}/3$, $Q_{d}= - \, Q_{e}/3$ and $Q_{s}= - \, Q_{e}/3$, where $Q_{e}=0.08542$ is the absolute value of the electron charge in natural units [@glend]. Because of the external magnetic field, particles with different charges may assume different trajectories [@azam; @multif] and this justifies the use of the multi-fluid approach [@azam; @multif; @we17]. Throughout this work, we employ natural units ($\hbar=c=1$) and the metric used is $g^{\mu\nu}=\textrm{diag}(+,-,-,-)$. Starting from the hydrodynamics equations discussed in [@we17], and The Euler equation for the quark of flavor $f$ (f=u,d,s) reads: $${\rho_{m\,f}}\Bigg[{\frac{\partial \vec{v_f}}{\partial t}} + (\vec{v_f}\cdot \vec{\nabla}) \vec{v_f}\Bigg]= -\vec{\nabla}p +{\rho_{c\,f}}\Big(\vec{v_f} \times \vec{B} \Big) \label{nsgeralmag}$$ where ${\rho_{m\,f}}$ is the quark mass density. The charge density of the quark flavor $f$ is $\rho_{c\,f}$ [@azam] and the masses are: $m_{u}=2.2 \, MeV$, $m_{d}=4.7 \, MeV$, $m_{s}=96 \, MeV$ and $m_{e}=0.5 \, MeV$ [@pdg]. The continuity equation for the mass density $\rho_{m\,f}$ is[@land]: $${\frac{\partial \rho_{m\,f}}{\partial t}} + \vec{\nabla} \cdot (\rho_{m\,f} \, {\vec{v_f}})=0 \label{conteq}$$ The relationship between the mass density and the baryon density is ${\rho_{m}}_f=3m_{f} \,\, {\rho_{B}}_{f}$ [@we17]. The charge density for each quark is given by ${\rho_{c}}_{u}=2Q_{e}\,{\rho_{B}}_{u}$ , ${\rho_{c}}_{d}=-Q_{e}\,{\rho_{B}}_{d}$ and ${\rho_{c}}_{s}=-Q_{e}\,{\rho_{B}}_{s}$ . In general we write ${\rho_{c}}_{f}=3\,{Q_{f}} \, {\rho_{B}}_{f}$ for each quark $f$. Equation of state ================= In general, the equation of state (EOS) of the quark gluon plasma can be written as a relation between pressure $p$ and energy density $\epsilon$: $p = {c_s}^{2}\epsilon$, where $c_s$ is the speed of sound. As previously studied in [@we17; @we16; @soundes], when the fluid is immersed in an external uniform magnetic field, the pressure splits into a parallel (with respect to the direction of the external field), $p_{\parallel}$, and a perpendicular component, $p_{\perp}$. We have thus a parallel (${c_s}_{\parallel}$) and a perpendicular (${c_s}_{\perp}$) speed of sound, given by [@we17; @we16; @soundes]: $${({c_{s}}_{\parallel})}^{2}={\frac{\partial p_{\parallel}}{\partial \varepsilon}} \hspace{1.0cm} \textrm{and} \hspace{1.0cm} {({c_{s}}_{\perp})}^{2}={\frac{\partial p_{\perp}}{\partial \varepsilon}} \label{soundes}$$ and hence $p_{\parallel} \approx {({c_{s}}_{\parallel})}^{2} \, \varepsilon$ and $p_{\perp} \approx {({c_{s}}_{\perp})}^{2} \, \varepsilon$ . The pressure gradient can then be written as: $$\vec{\nabla}{p}=\Bigg({\frac{\partial {p}_{\perp}}{\partial x}}\,,\, {\frac{\partial {p}_{\perp}}{\partial y}}\,,\,{\frac{\partial {p}_{\parallel}}{\partial z}}\Bigg) \label{gradpresscartsforquarks}$$ The nonrelativistic equation of state ------------------------------------- As in [@we17], we take here the limit [@w2]: $\varepsilon \cong {\rho_{m}}$. Since ${\rho_{m}}=3m_{f}\,{\rho_{B}}_{f}$ and remembering that the pressure is anisotropic, the pressure gradient (\[gradpresscartsforquarks\]) for the quark of flavor f is given by [@we17]: $$\vec{\nabla} p = 3m_{f}\Bigg({({c_{s}}_{\perp})}^{2}\,{\frac{\partial{{\rho_{B}}_{f}}}{\partial x}}\,,\, {({c_{s}}_{\perp})}^{2}\,{\frac{\partial{{\rho_{B}}_{f}}}{\partial y}}\,,\, {({c_{s}}_{\parallel})}^{2}\,{\frac{\partial{{\rho_{B}}_{f}}}{\partial z}}\Bigg) \label{gradespresses}$$ The improved MIT equation of state ---------------------------------- The EOS which we call mQCD was derived in [@we11] and used in [@we16] and also in [@we17]. The energy density ($\varepsilon$), the parallel pressure ($p_{f\,\parallel}$) and the perpendicular pressure ($p_{f\,\perp}$), are given respectively by [@we16; @we17]: $$\varepsilon={\frac{27{g_{h}}^{2}}{16{m_{G}}^{2}}}\,({{\rho_{B}}})^{2} +{\mathcal{B}}_{QCD}+{\frac{B^{2}}{8\pi}} +\sum_{f=u}^{d,s}{\frac{|Q_{f}|B}{2\pi^{2}}}\sum_{n=0}^{n^{f}_{max}} 3(2-\delta_{n0}) \int_{0}^{k^{f}_{z,F}} dk_{z}\sqrt{m_{f}^{2}+k_{z}^{2}+2n|Q_{f}|B} \, , \label{epsilontempzeromagon}$$ $$p_{\parallel}={\frac{27{g_{h}}^{2}}{16{m_{G}}^{2}}}\,({{\rho_{B}}})^{2} -{\mathcal{B}}_{QCD}-{\frac{B^{2}}{8\pi}} +\sum_{f=u}^{d,s}{\frac{|Q_{f}|B}{2\pi^{2}}} \sum_{n=0}^{n^{f}_{max}} 3(2-\delta_{n0}) \int_{0}^{k^{f}_{z,F}} dk_{z} \,{\frac{{k_{z}}^{2}}{\sqrt{m_{f}^{2}+k_{z}^{2}+2n|Q_{f}|B}}} \label{parallelpressuremagon}$$ and $$p_{\perp}={\frac{27{g_{h}}^{2}}{16{m_{G}}^{2}}}\,({{\rho_{B}}})^{2} -{\mathcal{B}}_{QCD}+{\frac{B^{2}}{8\pi}} +\sum_{f=u}^{d,s}{\frac{|Q_{f}|^{2}B^{2}}{2\pi^{2}}} \sum_{n=0}^{n^{f}_{max}} 3(2-\delta_{n0}) n \int_{0}^{k^{f}_{z,F}} {\frac{dk_{z}}{\sqrt{m_{f}^{2}+k_{z}^{2}+2n|Q_{f}|B}}} \label{perppressuremagon}$$ The baryon density ($\rho_{B}$) is given by [@we16; @we17]: $$\rho_{B}=\sum_{f=u}^{d,s}\,{\frac{|Q_{f}|B}{2\pi^{2}}} \, \sum_{n=0}^{n^{f}_{max}}(2-\delta_{n0}) \, \sqrt{{\nu_{f}}^{2}-m_{f}^{2}-2n|Q_{f}|B} \hspace{0.7cm} \textrm{with} \hspace{0.5cm} n\leq n^{f}_{max}=int\Bigg[ {\frac{{\nu_{f}}^{2}-m_{f}^{2}}{2|Q_{f}|B}} \Bigg] \label{magbaryondens}$$ where $ {\it{int}}[a]$ denotes the integer part of $a$ and ${\nu_{f}}$ is the chemical potential for the quark $f$. As in [@we16] we define $\xi \equiv g_{h}/m_{G}$. Choosing $\xi=0$ we recover the MIT EOS. For a given magnetic field intensity, we choose the values for the chemical potentials ${\nu_{f}}$ which determine the density $\rho_{B}$. We also choose the other parameters: $\xi$ and ${\mathcal{B}}_{QCD}$. In this case the pressure gradient (\[gradpresscartsforquarks\]) becomes $$\vec{\nabla} p = \Bigg({\frac{27{g_{h}}^{2}}{8{m_{G}}^{2}}}\Bigg) \Bigg({\rho_{B}}_{f}\,{\frac{\partial{{\rho_{B}}_{f}}}{\partial x}}\,,\, {\rho_{B}}_{f}\,{\frac{\partial{{\rho_{B}}_{f}}}{\partial y}}\,,\, {\rho_{B}}_{f}\,{\frac{\partial{{\rho_{B}}_{f}}}{\partial z}}\Bigg) \label{mqcdgradespresses}$$ Nonlinear waves =============== Now we apply the Reductive Perturbation Method (RPM) [@rpm; @w2; @weset; @javi17; @azam] to the basic equations of hydrodynamics (\[nsgeralmag\]) and (\[conteq\]) to obtain the nonlinear wave equations that govern the baryon density perturbations. The RPM technique goes beyond the linearization approach and preserves nonlinear terms in the wave equations. The background density, upon which small perturbations occur, is defined by $\rho_{0}$, and it is usually given in terms of the ordinary nuclear matter density $\rho_{N}=0.17\, fm^{-3}$. According to the RPM technique we rewrite the equations (\[nsgeralmag\]) changing variables and going from the $(x,y,z,t)$ space to the $(X,Y,Z,T)$ space using the “stretched coordinates” defined by [@w2; @weset]: $X=\sigma^{1/2}(x-{c_{s\,\perp}}\,t)\hspace{0.1cm}, \hspace{0.2cm}Y=\sigma \, y\hspace{0.1cm}, \hspace{0.2cm} Z=\sigma \,z \hspace{0.5cm} \textrm{and} \hspace{0.5cm} T=\sigma^{3/2}\,t$. In our approach, following the RPM algebraic procedure [@rpm; @weset] we apply the following transformation to the magnetic field: $B=\sigma\, \tilde{B}$. In this way, we obtain the equations (\[nsgeralmag\]) and (\[conteq\]) in the $(X,Y,Z,T)$ space containing the (small) parameter $\sigma$, which is the expansion parameter of the dimensionless density and dimensionless velocities: $$\hat\rho_{B\,f}(x,y,z,t)={\frac{\rho_{B\,f}(x,y,z,t)}{\rho_{0}}}=1+\sigma {\rho_{f}}_{1}(x,y,z,t)+ \sigma^{2} {\rho_{f}}_{2}(x,y,z,t) + \sigma^{3} {\rho_{f}}_{3}(x,y,z,t)+ \dots \, , \label{roexpa}$$ $${\hat{v}_{f\,x}}(x,y,z,t)={\frac{{v}_{f\,x}(x,y,z,t)}{{c_{s\,\perp}}}}= \sigma {{v_{{f}\,x}}_{1}}(x,y,z,t)+ \sigma^{2} {{v_{{f}\,x}}_{2}}(x,y,z,t) + \sigma^{3} {{v_{{f}\,x}}_{3}}(x,y,z,t)+ \dots \, , \label{vxfexpa}$$ $${\hat{v}_{f\,y}}(x,y,z,t)={\frac{{v}_{f\,y}(x,y,z,t)}{{c_{s\,\perp}}}}= \sigma^{3/2} {{v_{{f}\,y}}_{1}}(x,y,z,t)+ \sigma^{2} {{v_{{f}\,y}}_{2}}(x,y,z,t) + \sigma^{5/2} {{v_{{f}\,y}}_{3}}(x,y,z,t)+ \dots \label{vyfexpa}$$ and $${\hat{v}_{f\,z}}(x,y,z,t)={\frac{{v}_{f\,z}(x,y,z,t)}{{c_{s\,\parallel}}}}= \sigma^{3/2} {{v_{{f}\,z}}_{1}}(x,y,z,t)+ \sigma^{2} {{v_{{f}\,z}}_{2}}(x,y,z,t) + \sigma^{5/2} {{v_{{f}\,z}}_{3}}(x,y,z,t)+ \dots \label{vzfexpa}$$ Next we use (\[roexpa\]) to (\[vzfexpa\]) to rewrite (\[nsgeralmag\]) and (\[conteq\]). We then neglect terms proportional to $\sigma^{n}$ for $n > 2$ and collect the remaining terms in a power series of $\sigma$, $\sigma^{3/2}$ and $\sigma^{2}$, solving them in order to obtain an equation in the $(X,Y,Z,T)$ space. This equation is finally written back in the usual $(x,y,z,t)$ space, yielding the nonlinear wave equation for the baryon density perturbation. The continuity equation (\[conteq\]) in the RPM gives: $$\sigma\Bigg\{-{\frac{\partial{{\rho_{f}}_{1}}}{\partial X}}+{\frac{\partial{{v_{f\,x}}_{1}}} {\partial X}}\Bigg\}+ \sigma^{2}\Bigg\{-{\frac{\partial{{\rho_{f}}_{2}}}{\partial X}}+{\frac{\partial{{v_{f\,x}}_{2}}} {\partial X}}+{\frac{1}{({c_{s\,\perp}})}}{\frac{\partial{{\rho_{f}}_{1}}}{\partial T}}+ {\rho_{f}}_{1}{\frac{\partial{{v_{f\,x}}_{1}}}{\partial X}}+ {v_{f\,x}}_{1}{\frac{\partial{{\rho_{f}}_{1}}}{\partial X}}$$ $$+{\frac{\partial{{v_{f\,y}}_{1}}}{\partial Y}}+\Bigg({\frac{{c_{s\,\parallel}}}{{c_{s\,\perp}}}}\Bigg) {\frac{\partial{{v_{f\,z}}_{1}}}{\partial Z}}\Bigg\}=0 \label{conteqcartes}$$ and the Euler equation (\[nsgeralmag\]) will be studied in following subsections. Nonrelativistic EOS ------------------- Applying the RPM procedure to Eq. (\[nsgeralmag\]) and using (\[gradespresses\]), we obtain the following set of equations in powers of the $\sigma$ parameter: $$\sigma\Bigg\{-{\frac{\partial{{v_{f\,x}}_{1}}}{\partial X}}+ {\frac{\partial{{\rho_{f}}_{1}}}{\partial X}}\Bigg\} $$ $$ +\sigma^{2}\Bigg\{-{\frac{\partial{{v_{{f}\,x}}_{2}}}{\partial X}} +{\frac{1}{({c_{s\,\perp}})}}{\frac{\partial{{v_{f\,x}}_{1}}}{\partial T}} +\,{v_{f\,x}}_{1}\,{\frac{\partial{{v_{f\,x}}_{1}}}{\partial X}} -{\rho_{{f}}}_{1}\,{\frac{\partial{{v_{f\,x}}_{1}}}{\partial X}} +{\frac{\partial{{\rho_{f}}_{2}}}{\partial X}} -{\frac{Q_{f}\,\tilde{B}}{m_{f}\,({c_{s\,\perp}})}}{v_{f\,y}}_{1}\Bigg\}=0 \, , \label{eulercartex}$$\ $$\sigma^{3/2}\Bigg\{-{\frac{\partial{{v_{f\,y}}_{1}}}{\partial X}}+ {\frac{\partial{{\rho_{f}}_{1}}}{\partial Y}}+ {\frac{Q_{f}\,\tilde{B}}{m_{f}\,({c_{s\,\perp}})}}{v_{f\,x}}_{1}\Bigg\} +\sigma^{2}\Bigg\{-{\frac{\partial{{v_{f\,y}}_{2}}}{\partial X}}\Bigg\}=0 \label{eulercartey}$$\ and $$\sigma^{3/2}\Bigg\{-{\frac{\partial{{v_{f\,z}}_{1}}}{\partial X}}+ \Bigg({\frac{{c_{s\,\parallel}}}{{c_{s\,\perp}}}}\Bigg){\frac{\partial{{\rho_{f}}_{1}}}{\partial Z}}\Bigg\} +\sigma^{2}\Bigg\{-{\frac{\partial{{v_{f\,z}}_{2}}}{\partial X}}\Bigg\}=0 \label{eulercartez}$$ Solving the equations (\[conteqcartes\]) to (\[eulercartez\]) we arrive at: $${\frac{\partial}{\partial X}}\Bigg[{\frac{\partial{{\rho_{f}}_{1}}}{\partial T}}+ ({c_{s\,\perp}})\,{\rho_{f}}_{1}{\frac{\partial{{\rho_{f}}_{1}}}{\partial X}}\Bigg] +{\frac{({c_{s\,\perp}})}{2}}\Bigg[ {\frac{\partial^{2}{{\rho_{f}}_{1}}}{\partial Y^{2}}} +\Bigg({\frac{{c_{s\,\parallel}}}{{c_{s\,\perp}}}}\Bigg)^{2}{\frac{\partial^{2} {{\rho_{f}}_{1}}}{\partial Z^{2}}} \Bigg]={\frac{(Q_{f}\,\tilde{B})^{2}}{2{m_{f}}^{2}\,({c_{s\,\perp}})}}{\rho_{f}}_{1} \label{roestretcart}$$ Writing (\[roestretcart\]) back in the cartesian space we obtain the following wave equation: $${\frac{\partial}{\partial x}}\Bigg[{\frac{\partial}{\partial t}}{\delta\rho_{B}}_{f} +({c_{s\,\perp}}){\frac{\partial}{\partial x}}{\delta\rho_{B}}_{f} +({c_{s\,\perp}}){\delta\rho_{B}}_{f}{\frac{\partial}{\partial x}}{\delta\rho_{B}}_{f}\Bigg] $$ $$ +{\frac{({c_{s\,\perp}})}{2}}\Bigg[ {\frac{\partial^{2}}{\partial y^{2}}}{\delta\rho_{B}}_{f} +\Bigg({\frac{{c_{s\,\parallel}}}{{c_{s\,\perp}}}}\Bigg)^{2} {\frac{\partial^{2}}{\partial z^{2}}}{\delta\rho_{B}}_{f} \Bigg]= {\frac{(Q_{f}\,B)^{2}}{2{m_{f}}^{2}\,({c_{s\,\perp}})}}{\delta\rho_{B}}_{f} \label{roecart}$$ where ${\delta\rho_{B}}_{f}\equiv\sigma{\rho_{f}}_{1}$ is the baryon density perturbation on the background $\rho_{0}$, as can be seen in (\[roexpa\]). Introducing the variable [@weset]: $$\xi=x+y+z \label{carttransf}$$ the equation (\[roecart\]) becomes: $${\frac{\partial}{\partial \xi}}\Bigg\{{\frac{\partial}{\partial t}}{\delta\rho_{B}}_{f} +\Bigg[{\frac{3}{2}}({c_{s\,\perp}})+{\frac{({c_{s\,\parallel}})^{2}}{2({c_{s\,\perp}})}}\Bigg] {\frac{\partial}{\partial \xi}}{\delta\rho_{B}}_{f} +({c_{s\,\perp}}){\delta\rho_{B}}_{f}{\frac{\partial}{\partial \xi}}{\delta\rho_{B}}_{f}\Bigg\}= {\frac{(Q_{f}\,B)^{2}}{2{m_{f}}^{2}\,({c_{s\,\perp}})}}{\delta\rho_{B}}_{f} \label{roecartxitau}$$ mQCD ---- Repeating the steps described in the last subsection and using (\[mqcdgradespresses\]), we obtain: $$\sigma\Bigg\{-{\frac{\partial{{v_{f\,x}}_{1}}}{\partial X}}+ \Bigg({\frac{9\,{g_{h}}^{2}\,{\rho_{0}}}{8\,m_{f}\,{m_{G}}^{2}\,({c_{s\,\perp}})^{2}}}\Bigg) {\frac{\partial{{\rho_{f}}_{1}}}{\partial X}}\Bigg\} +\sigma^{2}\Bigg\{-{\frac{\partial{{v_{{f}\,x}}_{2}}}{\partial X}} +{\frac{1}{({c_{s\,\perp}})}}{\frac{\partial{{v_{f\,x}}_{1}}}{\partial T}} +{v_{f\,x}}_{1}\,{\frac{\partial{{v_{f\,x}}_{1}}}{\partial X}} $$ $$ -{\rho_{{f}}}_{1}\,{\frac{\partial{{v_{f\,x}}_{1}}}{\partial X}} +{\rho_{{f}}}_{1}\,{\frac{\partial{{\rho_{f}}_{1}}}{\partial X}} +\Bigg({\frac{9\,{g_{h}}^{2}\,{\rho_{0}}}{8\,m_{f}\,{m_{G}}^{2}\, ({c_{s\,\perp}})^{2}}}\Bigg){\frac{\partial{{\rho_{f}}_{2}}}{\partial X}} -{\frac{Q_{f}\,\tilde{B}}{m_{f}\,({c_{s\,\perp}})}}{v_{f\,y}}_{1}\Bigg\}=0 \label{mqcdeulercartex} \, ,$$\ $$\sigma^{3/2}\Bigg\{-{\frac{\partial{{v_{f\,y}}_{1}}}{\partial X}}+ \Bigg({\frac{9\,{g_{h}}^{2}\,{\rho_{0}}}{8\,m_{f}\,{m_{G}}^{2}\, ({c_{s\,\perp}})^{2}}}\Bigg){\frac{\partial{{\rho_{f}}_{1}}}{\partial Y}}+ {\frac{Q_{f}\,\tilde{B}}{m_{f}\,({c_{s\,\perp}})}}{v_{f\,x}}_{1}\Bigg\} +\sigma^{2}\Bigg\{-{\frac{\partial{{v_{f\,y}}_{2}}}{\partial X}}\Bigg\}=0 \label{mqcdeulercartey}$$\ and $$\sigma^{3/2}\Bigg\{-\Bigg({\frac{{c_{s\,\parallel}}}{{c_{s\,\perp}}}}\Bigg) {\frac{\partial{{v_{f\,z}}_{1}}}{\partial X}}+ \Bigg({\frac{9\,{g_{h}}^{2}\,{\rho_{0}}}{8\,m_{f}\,{m_{G}}^{2}\, ({c_{s\,\perp}})^{2}}}\Bigg){\frac{\partial{{\rho_{f}}_{1}}}{\partial Z}}\Bigg\} +\sigma^{2}\Bigg\{-\Bigg({\frac{{c_{s\,\parallel}}}{{c_{s\,\perp}}}}\Bigg) {\frac{\partial{{v_{f\,z}}_{2}}}{\partial X}}\Bigg\}=0 \label{mqcdeulercartez}$$ Solving the set of equations (\[conteqcartes\]) and (\[mqcdeulercartex\]) to (\[mqcdeulercartez\]) we arrive at: $${\frac{\partial}{\partial X}}\Bigg[{\frac{\partial{{\rho_{f}}_{1}}}{\partial T}}+ {\frac{3}{2}}({c_{s\,\perp}})\,{\rho_{f}}_{1}{\frac{\partial{{\rho_{f}}_{1}}}{\partial X}}\Bigg] +{\frac{({c_{s\,\perp}})}{2}}\Bigg( {\frac{\partial^{2}{{\rho_{f}}_{1}}}{\partial Y^{2}}} +{\frac{\partial^{2}{{\rho_{f}}_{1}}}{\partial Z^{2}}} \Bigg)={\frac{(Q_{f}\,\tilde{B})^{2}}{2{m_{f}}^{2}\,({c_{s\,\perp}})}}{\rho_{f}}_{1} \label{mqcdroestretcart}$$ From the terms of order $\mathcal{O}(\sigma)$ we obtain the following constraint for the perpendicular speed of sound: $$({c_{s\,\perp}})^{2}= {\frac{9\,{g_{h}}^{2}\,{\rho_{0}}}{8\,m_{f}\,{m_{G}}^{2}}} \label{csordsigma1lin}$$ which coincides with the “effective sound speed” ${{\tilde{c}_{s}}}$ obtained in the linearization approach in [@we17]. Writing (\[mqcdroestretcart\]) back in cartesian coordinates, we find the following nonlinear wave equation: $${\frac{\partial}{\partial x}}\Bigg[{\frac{\partial}{\partial t}}{\delta\rho_{B}}_{f} +({c_{s\,\perp}}){\frac{\partial}{\partial x}}{\delta\rho_{B}}_{f} +{\frac{3}{2}}({c_{s\,\perp}}){\delta\rho_{B}}_{f}{\frac{\partial}{\partial x}}{\delta\rho_{B}}_{f}\Bigg] $$ $$ +{\frac{({c_{s\,\perp}})}{2}}\Bigg( {\frac{\partial^{2}}{\partial y^{2}}}{\delta\rho_{B}}_{f} +{\frac{\partial^{2}}{\partial z^{2}}}{\delta\rho_{B}}_{f} \Bigg)= {\frac{(Q_{f}\,B)^{2}}{2{m_{f}}^{2}\,({c_{s\,\perp}})}}{\delta\rho_{B}}_{f} \label{mqcdroecart}$$ where again, from (\[roexpa\]), we have ${\delta\rho_{B}}_{f}\equiv\sigma{\rho_{f}}_{1}$ . Using (\[carttransf\]) in (\[mqcdroecart\]) we find: $${\frac{\partial}{\partial \xi}}\Bigg[{\frac{\partial}{\partial t}}{\delta\rho_{B}}_{f} +2({c_{s\,\perp}}){\frac{\partial}{\partial \xi}}{\delta\rho_{B}}_{f} +{\frac{3}{2}}({c_{s\,\perp}}){\delta\rho_{B}}_{f}{\frac{\partial}{\partial \xi}}{\delta\rho_{B}}_{f}\Bigg]= {\frac{(Q_{f}\,B)^{2}}{2{m_{f}}^{2}\,({c_{s\,\perp}})}}{\delta\rho_{B}}_{f} \label{mqcdroecartxitau}$$ Reduced Ostrovsky Equation (ROE) ================================ In cartesian coordinates, we derived the “inhomogeneous three dimensional breaking wave equations” given by (\[roecart\]) and (\[mqcdroecart\]). By using (\[carttransf\]) we transformed these two equations into (\[roecartxitau\]) and (\[mqcdroecartxitau\]), respectively, where ${\delta\rho_{B}}_{f}(x,y,z,t) \rightarrow {\delta\rho_{B}}_{f}(\xi,t)$. The equations (\[roecartxitau\]) and (\[mqcdroecartxitau\]) can be put in the form: $${\frac{\partial}{\partial \xi}}\Bigg[{\frac{\partial}{\partial t}}{\delta\rho_{B}}_{f} +{\alpha}\,{\frac{\partial}{\partial \xi}}{\delta\rho_{B}}_{f} +\beta \,{\delta\rho_{B}}_{f}{\frac{\partial}{\partial \xi}}{\delta\rho_{B}}_{f}\Bigg]= \Gamma{\delta\rho_{B}}_{f} \label{roegeral}$$ with the nonlinear coefficient $\beta$ and the velocity of the dispersionless linear wave $\alpha$ defined in (\[roecartxitau\]) and (\[mqcdroecartxitau\]) for each case. The common dispersion coefficient $\Gamma$ for the two cases is given by $$\Gamma = {\frac{(Q_{f}\,B)^{2}}{2{m_{f}}^{2}\,({c_{s\,\perp}})}} \label{disperforced}$$ and it comes from the magnetic field term of the Euler equation (\[nsgeralmag\]) for each quark of flavor f. The magnetic field effects are also indirectly present in the coefficients of (\[roegeral\]), which come from the equation of state chosen for the magnetized medium. If the magnetic field were zero, (\[roegeral\]) would be converted into a breaking wave equation without soliton solutions. We can then say that the B field allows for localized solitonic solutions of Eq. (\[roegeral\]). Equation (\[roegeral\]) is known in the literature and it is called Reduced Ostrovsky equation (ROE) or Ostrovsky-Hunter equation (OHE) when $\Gamma>0$ [@roe], which is our case. The ROE is a particular case of the Ostrovsky equation [@ostrov] for a general function $f(\xi,t)$: $${\frac{\partial}{\partial \xi}}\Bigg[{\frac{\partial}{\partial t}}f + \alpha\,{\frac{\partial}{\partial \xi}}f +\beta \,f{\frac{\partial}{\partial \xi}}f +\Pi\,{\frac{\partial^{3}}{\partial \xi^{3}}}f\Bigg]= \Gamma \,f \label{ostrovsky}$$ when the high-frequency dispersion coefficient $\Pi$ vanishes. Equation (\[ostrovsky\]) describes internal waves and weakly nonlinear surface in a rotating ocean [@ostrov]. The equation (\[roegeral\]) can be solved analytically, as it is shown in the Appendix. The solution of (\[roegeral\]) reads: $${{\delta\rho_{B}}_{f}}(\xi,t)=-{\frac{6\gamma^{2}\lambda^{2}} {\beta \Gamma}}sech^{2} \Big[\lambda\Big(\Omega-\gamma t\Big) \Big] \label{roesolsexacta}$$ where $\lambda$ and $\gamma$ are integration constants. The latter is related to the propagation speed of the perturbation. Also $$\xi= x + y + z = \Omega + {\alpha} \, t +\xi_{0} +{\frac{6 \gamma \lambda}{\beta \Gamma}}\Big\{tanh\Big[\lambda \Big(\Omega-\gamma t\Big) \Big] -1\Big\} \label{introeagain}$$ For a given value of the coordinate $\xi$, we solve the above equation and find $\Omega$ which is then substituted in (\[roesolsexacta\]), which represents a traveling gaussian-looking pulse moving to the right and preserving its shape. As it can be seen in (\[roesolsexacta\]), the amplitude of the density wave is proportional to $1/\Gamma$ and hence increasing $B$ results in a decreasing amplitude. Similarly, waves of heavier flavor quarks have larger amplitudes. In (\[introeagain\]) we have $\alpha = 3({c_{s\,\perp}})/2 + ({c_{s\,\parallel}})^{2}/[2({c_{s\,\perp}})]$ and $\beta={c_{s\,\perp}}$ for the nonrelativistic EOS. For the mQCD EOS we have $\alpha = 2({c_{s\,\perp}})$ and $\beta = 3({c_{s\,\perp}})/2$. To illustrate the solitonic behavior of the rarefaction solution (\[roesolsexacta\]), we show in Figs. \[fig1\] and \[fig2\] the perturbation $|{{\delta\rho_{B}}_{f}}|$ as a function of $x$ for fixed values of $y=0$ and $z=0$ for two values of the time $t$. In both cases showed in Figs. \[fig1\] and \[fig2\], we consider the quark $up$ and three values of the magnetic field, that are chosen to satisfy $0<|{{\delta\rho_{B}}_{u}}|<1$ and respect (\[roexpa\]) (since ${\delta\rho_{B}}_{u}\equiv\sigma{\rho_{u}}_{1}$). For magnetic fields $\sim 10^{16} \, G$ or smaller, we obtain $|{{\delta\rho_{B}}_{u}}|>1$ . In Fig. \[fig1\] we show the results obtained with the nonrelativistic EOS for the parameters ${c_{s\,\perp}}={c_{s\,\parallel}}=0.3$, $\xi_{0}=20 \, fm$, $\lambda=1\, fm^{-1}$ and $\gamma=0.1$ . The propagation speed of the pulse is $\alpha+\gamma=0.7$ . In Fig. \[fig2\] we show the results obtained with the mQCD EOS for the parameters $B_{QCD}=70\,MeV/fm^{3}$, $g_{h}=0.05$, $m_{G}=300\, MeV$, $\xi_{0}=20 \, fm$, $\lambda=1\, fm^{-1}$ and $\gamma=0.1$ . The common chemical potential for all quarks is $\nu_{f}=300 \, MeV$ and for the chosen values of the magnetic field we have background densities $\rho_{0}=2\rho_{N} \sim 2.1\rho_{N}$ $(B=10^{17}\,G \sim 10^{19}\,G)$, which, with the use of (\[csordsigma1lin\]) lead to ${c_{s\,\perp}}\cong 0.2$ . The propagation speed of the pulse is $\alpha+\gamma=0.5$, which does not violate causality. Similar behavior is found when $|{{\delta\rho_{B}}_{u}}|$ is plotted as a function of the $y$ coordinate (perpendicular to the magnetic field) and of the $z$ coordinate (along the magnetic field). Conclusions =========== In this work we focused on nonlinear wave propagation in a cold and magnetized quark gluon plasma. Including the effects of a strong magnetic field both in the equation of state and in the basic equations of hydrodynamics, we derived from the latter a wave equation for a perturbation in the baryon density. This wave equation could be identified as the reduced Ostrovsky equation (ROE), which has a known analytical solution given by a rarefaction solitonic pulse of the baryon perturbation. The numerical analysis and a possible phenomenological application in the context of heavy ion collisions or in compact stars will be investigated in a future work. At a qualitative level we can observe that the most remarkable effect of the magnetic field, as can be seen in the coefficient $\Gamma$ by (\[disperforced\]), is to reduce the wave amplitude. We therefore corroborate and extend the conclusion found in [@azam]. Appendix ======== To establish the integrability of (\[roegeral\]), we employ the change of variables developed in [@roe]: $$\xi= \Omega + \beta \int_{-\infty}^{\eta} \psi(\Omega,\eta') d\eta' \,+\alpha\,\eta +\xi_{0} \hspace{0.4cm} \textrm{,} \hspace{0.6cm} t = \eta \hspace{0.6cm} \textrm{and} \hspace{0.6cm} {\delta\rho_{B}}_{f}(\xi,t)=\psi(\Omega,\eta) \label{introe}$$ where $\xi_{0}$ is an arbitrary constant. From (\[introe\]) we have the operators: $${\frac{\partial}{\partial \xi}} ={\frac{1}{h(\Omega,\eta)}}{\frac{\partial}{\partial \Omega}} \hspace{0.6cm} \textrm{and} \hspace{0.6cm} {\frac{\partial}{\partial t}}={\frac{\partial}{\partial \eta}}- {\frac{\beta}{h(\Omega,\eta)}}\, \psi {\frac{\partial}{\partial \Omega}} -{\frac{\alpha}{h(\Omega,\eta)}}\,{\frac{\partial}{\partial \Omega}} \label{introepers}$$ where the function $h(\Omega,\eta)$ is given by: $$h(\Omega,\eta)=1+\beta \int_{-\infty}^{\eta} \Big[{\frac{\partial} {\partial \Omega}} \psi(\Omega,\eta')\Big] \,\, d\eta' \label{introepersfuncg}$$ The equation (\[roegeral\]) rewritten in terms of (\[introepers\]) and (\[introepersfuncg\]) is: $$h(\Omega,\eta)={\frac{1}{\Gamma\,\psi}} {\frac{\partial}{\partial \Omega}}{\frac{\partial \psi}{\partial \eta}} \label{roealmostint}$$ From (\[introepersfuncg\]) we have: $${\frac{\partial h}{\partial \eta}}=\beta\,{\frac{\partial \psi}{\partial \Omega}} \label{delhdeleta}$$ Finally, inserting (\[roealmostint\]) in (\[delhdeleta\]) we arrive at the following equation: $$\psi\,{\frac{\partial^{2}}{\partial \eta^{2}}}\,{\frac{\partial \psi} {\partial \Omega}} -{\frac{\partial \psi}{\partial \eta}}{\frac{\partial} {\partial \Omega}}{\frac{\partial \psi}{\partial \eta}}-(\beta\Gamma) (\psi)^{2}\,{\frac{\partial \psi}{\partial \Omega}}=0 \label{roeintegravel}$$ which is the ROE equation (\[roegeral\]) rewritten in a integrable form. To solve (\[roeintegravel\]) we apply the hyperbolic tangent function method as described in [@w2; @weset; @tangh] and find the following exact solutions: $$\psi_{I}(\Omega,\eta)=-{\frac{6\gamma^{2}\lambda^{2}}{\beta\Gamma}}sech^{2} \Big[\lambda(\Omega-\gamma\eta) \Big] \hspace{0.7cm} \textrm{or} \hspace{0.7cm} \psi_{II}(\Omega,\eta)={\frac{4\gamma^{2}\lambda^{2}}{\beta\Gamma}} +\psi_{I}(\Omega,\eta) \label{roesols}$$ The parameters $\lambda$, which is the inverse of the width and $\gamma$, the speed, are integration constants and are free to be chosen. The negative sign in the $-sech^{2}[\dots]$ function in (\[roesols\]) describes a rarefaction pulse. Such negative sign is due the condition $(\alpha\Gamma)>0$. Considering $\psi_{I}$ from (\[roesols\]) in (\[introe\]) we obtain the following parametric solution of (\[roegeral\]): $${{\delta\rho_{B}}_{f}}(\xi,t)=-{\frac{6\gamma^{2}\lambda^{2}} {\beta\Gamma}}sech^{2} \Big[\lambda\Big(\Omega-\gamma t\Big) \Big] \label{roesolsexacta-ap}$$ with $$\xi=\Omega +\alpha\,t +\xi_{0} +{\frac{6\gamma\lambda}{\beta\Gamma}}\Big\{tanh\Big[\lambda \Big(\Omega-\gamma t\Big) \Big] -1\Big\} \label{introeagain-ap}$$ As previously mentioned, the last two expressions are (\[roesolsexacta\]) and (\[introeagain\]), respectively. We do not consider $\psi_{II}$ of (\[roesols\]) as solution of (\[roegeral\]). The reason is to avoid the divergence due the constant term of $\psi_{II}$ in the integral present in (\[introe\]): $$\beta \int_{-\infty}^{\eta}{\frac{4\gamma^{2}\lambda^{2}}{\beta\Gamma}} \, d\eta' \rightarrow \infty$$ 0.5cm This work was partially supported by the Brazilian funding agencies CAPES, CNPq and FAPESP (contract 2012/98445-4). 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--- abstract: | A filtration over a simplicial complex $K$ is an ordering of the simplices of $K$ such that all prefixes in the ordering are subcomplexes of $K$. Filtrations are at the core of Persistent Homology, a major tool in Topological Data Analysis. In order to represent the filtration of a simplicial complex, the entire filtration can be appended to any data structure that explicitly stores all the simplices of the complex such as the Hasse diagram or the recently introduced Simplex Tree \[Algorithmica ’14\]. However, with the popularity of various computational methods that need to handle simplicial complexes, and with the rapidly increasing size of the complexes, the task of finding a compact data structure that can still support efficient queries is of great interest.\ This direction has been recently pursued for the case of maintaining simplicial complexes. For instance, Boissonnat et al. \[SoCG ’15\] considered storing the simplices that are maximal for the inclusion and Attali et al. \[IJCGA ’12\] considered storing the simplices that block the expansion of the complex. Nevertheless, so far there has been no data structure that compactly stores the *filtration* of a simplicial complex, while also allowing the efficient implementation of basic operations on the complex.\ In this paper, we propose a new data structure called the Critical Simplex Diagram (CSD) which is a variant of the Simplex Array List (SAL) \[SoCG ’15\]. Our data structure allows to store in a compact way the filtration of a simplicial complex, and allows for the efficient implementation of a large range of basic operations. Moreover, we prove that our data structure is essentially optimal with respect to the requisite storage space. Next, we show that the CSD representation admits the following construction algorithms. - A new *edge-deletion* algorithm for the fast construction of Flag complexes, which only depends on the number of critical simplices and the number of vertices. - A new *matrix-parsing* algorithm to quickly construct relaxed Delaunay complexes, depending only on the number of witnesses and the dimension of the complex. author: - | Jean-Daniel Boissonnat[^1]\ INRIA Sophia Antipolis - Méditerranée, France.\ `Jean-Daniel.Boissonnat@inria.fr`. - | Karthik C. S.[^2]\ Department of Computer Science and Applied Mathematics,\ Weizmann Institute of Science, Israel.\ `karthik.srikanta@weizmann.ac.il`. bibliography: - 'References.bib' title: '**An Efficient Representation for Filtrations of Simplicial Complexes**' --- Introduction ============ Persistent homology is a method for computing the topological features of a space at different spatial resolutions [@EH10]. More persistent features are detected over a wide range of length and are deemed more likely to represent true features of the underlying space, rather than artifacts of sampling, noise, or particular choice of parameters. To find the persistent homology of a space [@BDM15; @BM14], the space is represented as a sequence of simplicial complexes called a filtration. The most popular filtrations are nested sequences of increasing simplicial complexes but more advanced types of filtrations have been studied where consecutive complexes are mapped using more general simplicial maps [@DFW14]. Persistent homology found applications in many areas ranging from image analysis [@CIDZ08; @PC14], to cancer research [@ABDMPP12], virology [@CCR13], and sensor networks [@DG07]. Thus, a central question in Computational Topology and Topological Data Analysis is to represent simplicial complexes and filtrations efficiently. The most common representation of simplicial complexes uses the Hasse diagram of the complex that has one node per simplex and an edge between any pair of incident simplices whose dimensions differ by one. A more compact data structure, called Simplex Tree (ST), was proposed recently by Boissonnat and Maria [@SimplexTree]. The nodes of both the Hasse diagram and ST are in bijection with the simplices (of all dimensions) of the simplicial complex. In this way, they explicitly store all the simplices of the complex and it is easy to attach information to each simplex (such as a filtration value). In particular, they allow to store in a natural way the filtration of complexes which are at the core of Persistent Homology and Topological Data Analysis. However, such data structures are redundant and typically very big, and they are not sensitive to the underlying structure of the complexes. This motivated the design of more compact data structures that represent only a sufficient subset of the simplices. A first idea is to store the 1-skeleton of the complex together with a set of blockers that prevent the expansion of the complex [@DataStructure3]. A dual idea is to store only the simplices that are maximal for the inclusion. Following this last idea, Boissonnat et al. [@BKT15] introduced a new data structure, called Simplex Array List, which was the first data structure whose size and query time were sensitive to the geometry of the simplicial complex. SAL was shown to outperform ST for a large class of simplicial complexes. Although very efficient, SAL, as well as data structures that do not explicitly store all the simplices of a complex, makes the representation of filtrations problematic, and in the case of SAL, impossible. In this paper, we introduce a new data structure called Critical Simplex Diagram (CSD) which has some similarity with SAL. CSD only stores the critical simplices, i.e., those simplices all of whose cofaces have a higher filtration value, and in this paper, we overcome the problems arising due to the implicit representation of simplicial complexes, by showing that the basic operations on simplicial complexes can be performed efficiently using CSD. In short, CSD compromises on the membership query (which is slightly worse than that for ST) in order to save storage and to perform insertion and removal efficiently. Our Contribution ---------------- At a high level, our main contribution through this paper is in developing a new perspective for the design of data structures representing simplicial complexes associated with a filtration. Previous data structures such as Hasse diagram and Simplex Tree interpreted a simplicial complex as a set of strings defined over the label set of its vertices and the filtration values as keys associated with each string. When a simplicial complex is perceived this way, a Trie is indeed a natural data structure to represent the complex. However, this way of representing simplicial complexes doesn’t make use of the fact that simplicial complexes are not arbitrary sets of strings but are constrained by a lot of combinatorial structure. In particular, simplicial complexes are closed under subsets and also (standard) filtrations are monotone functions. We exploit these constraints/structure by viewing a filtered simplicial complex with filtration range of size $t$ as a monotone function from $\{0,1\}^{|V|}$ to $\{0,1,\ldots,t\}$, where $V$ is the vertex set. We note that if a simplex is mapped to $t$ then, the simplex is understood to be not in the complex and if not, the mapping is taken to correspond to the filtration value of the simplex. In light of this viewpoint, we propose a data structure (CSD) which stores only the critical elements in the domain, i.e. those elements all of whose supersets (cofaces in the complex) are mapped to a strictly larger value. As a result, we are not only able to store data regarding a simplicial complex more efficiently but also explicitly utilize geometric regularity in the complex which would have been otherwise obscured. More concretely, we have the following result. \[main\] Let $K$ be a $d$-dimensional simplicial complex. Let $\kappa$ be the number of critical simplices in the complex. The data structure CSD representing $K$ admits the following properties: - The size of CSD is at most $\kappa d$. - The cost of basic operations (such as membership, insertion, removal, elementary collapse, etc.) through the CSD representation is $\tilde{\mathcal{O}}((\kappa \cdot d)^2)$. The proof of the above two items follows from the discussions in Section \[sizeofCSD\] and Section \[sec:operations\] respectively. We would like to point out here that while the cost of static operations such as membership is only $\tilde{\mathcal{O}}(d)$ for the Simplex Tree, to perform any dynamic operation such as insertion or removal, the Simplex Tree requires $\text{exp}(d)$ time. Moreover, as shown in Section \[sec:operations\], the cost of *most* basic operation using CSD is linear in $\kappa$. As a direct consequence of representing a simplicial complex only through the critical simplices, we note that the construction of any simplicial complex with filtration, will be very efficient through CSD, simply because we have to build a smaller data structure as compared to the existing data structures. More specifically, we propose a new *edge-deletion* algorithm for the construction of flag complexes on $n$ vertices with $\kappa$ critical simplices in time $\mathcal{O}\left(\kappa n^{2.38}\right)$. Additionally, we provide a *matrix-parsing* algorithm for building $d$-dimensional relaxed Delaunay complexes over the witness set $W$ in $\mathcal{O}(|W| d^2 \log |W|)$ time. In each of these cases, we show that the construction is more efficient when using CSD rather than ST, primarily because CSD is a compact representation. Organization of the paper ------------------------- In Section \[Preliminaries\], we introduce definitions which will be used throughout the paper and we provide new lower bounds on the space of data structures needed to store simplicial complexes. In Section \[CSD\], we introduce the new data structure CSD to represent simplicial complexes and their filtration. Further, we describe how the data structure can perform basic operations on the complex and also discuss about its efficiency. In Section \[Flag\], we give a new algorithm for the construction of Flag complexes. In Section \[Delaunay\], we give a new algorithm for the construction of relaxed Delaunay complexes. Finally, in Section \[Conclusion\], we conclude by highlighting some open directions for future research. Preliminaries {#Preliminaries} ============= A simplicial complex $K$ is defined over a (finite) vertex set $V$ whose elements are called the vertices of $K$ and is a set of non-empty subsets of $V$ that is required to satisfy the following two conditions: 1. $p\in V\Rightarrow \{p\}\in K$ 2. $\sigma\in K, \tau\subseteq\sigma\Rightarrow\tau\in K$ Each element $\sigma\in K$ is called a simplex or a face of $K$ and, if $\sigma\in K$ has precisely $s + 1$ elements $(s \ge -1)$, $\sigma$ is called an $s$-simplex and the dimension of $\sigma$, denoted by $d_\sigma$ is $s$. The dimension of the simplicial complex $K$ is the largest $d$ such that it contains a $d$-simplex. A *face* of a simplex $\sigma = \{p_0 ,..., p_s \}$ is a simplex whose vertices form a subset of $\{p_0 ,..., p_s \}$. A proper face is a face different from $\sigma$ and the facets of $\sigma$ are its proper faces of maximal dimension. A simplex $\tau\in K$ admitting $\sigma$ as a face is called a *coface* of $\sigma$. A maximal simplex of a simplicial complex is a simplex which has no cofaces. A simplicial complex is pure, if all its maximal simplices are of the same dimension. Also, a free pair is defined as a pair of simplices $(\tau,\sigma)$ in $K$ where $\tau$ is the only coface of $\sigma$. In this paper, the class of simplicial complexes of $n$ vertices in $d$ dimensions with $k$ maximal simplices out of the $m$ simplices in the complex is denoted by ${\cal K}(n,d,k,m)$, and $K$ denotes a simplicial complex in $ {\cal K} (n,d,k,m)$ . In Figure \[fig:SimplicialComplexExample\] we see a three dimensional simplicial complex on the vertex set $\{1,2,3,4,5,6\}$. This complex has two maximal simplices: the tetrahedron $[1234]$ and the triangle $[356]$. We use this complex as an example through out the paper. We adopt the following notation through out the paper: $[t]:=\{1,\ldots ,t\}$ and $\llbracket t\rrbracket=\{0,1,\ldots ,t\}$. A [*filtration*]{} over a simplicial complex $K$ is an ordering of the simplices of $K$ such that all prefixes in the ordering are subcomplexes of $K$. More concretely, a filtration of a complex is a function $f : K \to \mathbb{R}$ satisfying $f(\tau ) \le f(\sigma)$ whenever $\tau\subseteq\sigma$ [@EH10]. Moreover, we will assume that the filtration values range over $\llbracket t\rrbracket$. We say that a simplex $\sigma\in K$ is a **‘critical simplex’** if for all cofaces $\tau$ of $\sigma$ we have $f(\sigma)<f(\tau)$. For example, the critical simplices in the example described in Figure \[fig:SimplicialComplexExample\] are all the vertices, the edges $[56], [14], $ and $[24]$, the triangles $[356]$ and $[134]$, and the tetrahedron $[1234]$. Lower Bounds {#Lower Bounds} ------------ Boissonnat et al. proved the following lower bound on the space needed to represent simplicial complexes [@BKT15]. \[Lowerbound\] [@BKT15] Consider the class of all $d$-dimensional simplicial complexes with $n$ vertices containing $k$ maximal simplices, where $d\ge2$ and $k\ge n+1$, and consider any data structure that can represent the simplicial complexes of this class. Such a data structure requires $\log{\binom{\binom{n/2}{d+1}}{k-n}}$ bits to be stored. For any constant $\varepsilon\in (0,1)$ and for $\frac{2}{\varepsilon}n\le k\le n^{(1-\varepsilon)d}$ and $d\le n^{\varepsilon/3}$, the bound becomes $\Omega(kd\log n)$. We prove now a lower bound on the representation of filtrations of simplicial complexes. \[FiltrationLowerbound\] Let $\beta=\left\lfloor\frac{t+1}{d+1}\right\rfloor$ be greater than 1. For any simplicial complex $K$ of dimension $d$ containing $m$ simplices, the number of distinct filtrations $f:K\to\llbracket t\rrbracket$ is at least $\beta^{m}$. If $\beta>(d+1)^{\delta}$ for some constant $\delta>0$ then, any data structure that can represent filtrations of the class of all $d$-dimensional simplicial complexes containing $m$ simplices requires $\Omega(m\log t)$ bits to be stored. Let us fix a simplicial complex $K$ of dimension $d$ containing $m$ simplices. We will now build functions $f_i:K\to\llbracket t\rrbracket$. For every $i\in \left\llbracket{\beta}^m-1\right\rrbracket$, let $b(i)$ be the representation of $i$ as a $m$ digit number in base $\beta$ and let $b(i)_j$ be the $j^{\text{th}}$ digit of $b(i)$. Let $g$ be a bijection from $K$ to $[m]$. We define $f_i(\sigma)=d_{\sigma}\cdot\beta +b(i)_{g(\sigma)}+1$. We note that all the $f_i$s are distinct functions as for any two distinct numbers $i$ and $j$ in $\left\llbracket{\beta}^m-1\right\rrbracket$, we have that $b(i)\neq b(j)$. Finally, we note that each of the $f_i$s is a filtration of $K$. This is because, for any two simplices $\tau,\sigma\in K$, such that $\tau\subset\sigma$ (i.e., $d_\tau<d_\sigma$), and any $i\in \left\llbracket{\beta}^m-1\right\rrbracket$, we have that $f_i(\tau)\le (d_{\tau}+1)\cdot\beta < d_{\sigma}\cdot\beta+1\le f_i(\sigma)$. It follows that there are at least $\beta^{m}$ distinct filtrations of $K$. By the pigeonhole principle, we have that any data structure that can represent filtrations of $K$ requires $\log\left(\beta^{m}\right)$ bits. It follows that if $\beta>(d+1)^{\delta}$ for some constant $\delta>0$ then $\beta>\frac{(t+1)^\delta}{(\beta-1)^\delta}$. Thus, any data structure that can represent filtrations of $K$ requires at least $\frac{\delta}{1+\delta}\cdot m\log (t+1)=\Omega(m\log t)$ bits to be stored. Even if $\left\lfloor\frac{t}{d}\right\rfloor \le 1 $, we can show that any data structure that can represent $d$-dimensional simplicial complexes containing $m$ simplices with filtration ra nge $\llbracket t\rrbracket$ requires $\Omega(\frac{m\sqrt{t}}{d}\log t)$ bits. This can be shown by modifying the above proof as follows. Let $S_j$ be the set of all simplices of dimension $j-1$. We identify a subset $D$ of $[d]$ of size $\sqrt{t}$, such that $\sum_{j\in D}|S_{j}|$ is at least $\frac{m\sqrt{t}}{d}$. Therefore, for every set $S_j$, $j\in D$, we can associate $\sqrt{t}$ distinct filtration values, which leads to the lower bound. The lower bounds in Lemma \[FiltrationLowerbound\] is not sensitive to the number of critical simplices, and intuitively, any lower bound on the size of data structures storing complexes with filtrations needs to capture the number of critical simplices as a parameter. We adapt the proof of Theorem \[Lowerbound\] and combine it with the ideas from the proof of Lemma \[FiltrationLowerbound\], to obtain the following lower bound. \[CSDLowerBound\] Consider the class of all simplicial complexes on $n$ vertices of dimension $d$, associated with a filtration over the range of $\llbracket t\rrbracket$, such that the number of critical simplices is $\kappa$, where $d\ge2$ and $\kappa\ge n+1$, and consider any data structure that can represent the simplicial complexes of this class. Such a data structure requires $\log\left(\binom{\binom{n/2}{d+1}}{\kappa-n}t^{\kappa-n}\right)$ bits to be stored. For any constant $\varepsilon\in (0,1)$ and for $\frac{2}{\varepsilon}n\le \kappa\le n^{(1-\varepsilon)d}$ and $d\le n^{\varepsilon/3}$, the bound becomes $\Omega(\kappa (d\log n+\log t))$. The proof of the first statement is by contradiction. Let us define $h=\kappa-n\ge 1$ and suppose that there exists a data structure that can be stored using only $s<\log \alpha \stackrel{{\rm def}}{=}\log\left(\binom{\binom{n/2}{d+1}}{\kappa-n}t^{h}\right)$ bits. We will construct $\alpha$ simplicial complexes (associated with a filtration), all with the same set $P$ of $n$ vertices, the same dimension $d$, with exactly $\kappa$ maximal simplices, and with a filtration over the range of $\llbracket t \rrbracket$. By the pigeon hole principle, two different simplicial complexes[^3], say $K$ and $K^\prime$, are encoded by the same word. So any algorithm will give the same answer for $K$ and $K^\prime$. But, by the construction of these complexes, there is either a simplex which is in $K$ and not in $K^\prime$ or there is a simplex whose filtration value in $K$ is different from the simplex’s filtration value in $K^\prime$. This leads to a contradiction. The simplicial complexes and their associated filtration are constructed as follows. Let $P'\subset P$ be a subset of cardinality $n/2$, and consider the set of all possible simplicial complexes of dimension $d$ with vertices in $P'$ that contain $h$ critical simplices. We further assume that all critical simplices have dimension $d$ exactly. These complexes are $\beta=\binom{\binom{n/2}{d+1}}{h}$ in number, since the total number of maximal $d$ dimensional simplices is $\binom{n/2}{d+1}$ and we choose $h$ of them. Let us call them $\Gamma_1,\ldots,\Gamma_\beta$. We now extend each $\Gamma_i$ so as to obtain a simplicial complex whose vertex set is $P$ and has exactly $\kappa$ critical simplices. The critical simplices will consist of the $h$ maximal simplices of dimension $d$ already constructed (whose filtration value is set to one of the values in $[t]$) plus a number of maximal simplices of dimension 1 (whose filtration value is set to 0). The set of vertices of $\Gamma_i$, ${\rm vert}(\Gamma_i)$, may be a strict subset of $P'$. Let its cardinality be $\frac{n}{2}-r_i$ and observe that $0\leq r_i<\frac{n}{2}$. Consider now the complete graph on the $\frac{n}{2}+r_i$ vertices of $P\setminus {\rm vert}(\Gamma_i)$. Any spanning tree of this graph gives $\frac{n}{2}+r_i-1$ edges and we arbitrarily choose $\frac{n}{2}-r_i+1$ edges from the remaining edges of the graph to obtain $n$ distinct edges spanning over the vertices of $P\setminus {\rm vert}(\Gamma_i)$. We have thus constructed a 1–dimensional simplicial complex $K_i$ on the $\frac{n}{2}+r_i$ vertices of $P\setminus {\rm vert}(\Gamma_i)$ with exactly $n$ maximal simplices. Finally, we define the complex $\Lambda_i=\Gamma_i\cup K_i$ that has $P$ as its vertex set, dimension $d$, and $\kappa$ maximal simplices which are also the critical simplices. The filtration value of any simplex which is not maximal is defined to be the minimum of the filtration values of its cofaces in the complex. The set of $\Lambda_i$, $i=1, \cdots ,\beta$, where for each complex we associate $t^h$ different filtrations is the set of simplicial complexes (associated with a filtration) that we were looking for. The second statement in the theorem is proved through the following computation: $$\begin{aligned} \log\left(\dbinom{\dbinom{n/2}{d+1}}{\kappa-n}t^{\kappa-n}\right)\ge &\ \log{\left(\frac{n^{(d+1)(\kappa-n)}}{2^{(d+1)(\kappa-n)}(d+1)^{(d+1)(\kappa-n)}(\kappa-n)^{(\kappa-n)}}\right)}+(\kappa-n)\log t \\ = &\ (d+1)(\kappa-n)\log n -(d+1)(\kappa-n)- (d+1)(\kappa-n)\log (d+1)\\ &\ \phantom{{}(d+1)(\kappa-n)\log n -(d+1)} - (\kappa-n)\log (\kappa-n) +(\kappa-n)\log t\\ > &\ (d+1)(\kappa-n)\log n -3(d+1)(\kappa-n)-(d+1)(\kappa-n)\log d \\ &\ \phantom{{}(d+1)(\kappa-n)\log n -(d+1)\log n } - (\kappa-n)\log \kappa +(\kappa-n)\log t\\ \ge &\ (d+1)(\kappa-n)(\log n -3-\log d) - (\kappa-n)(1-\varepsilon)d\log n \\ &\ \phantom{{}(d+1)(\kappa-n)(\log n -3-\log d) - (\kappa-n)(1-\varepsilon)d} +(\kappa-n)\log t\\ \ge &\ d\varepsilon(\kappa-n)\log n + (\kappa-n)\log n -(d+1)(\kappa-n)\left(3+\frac{\varepsilon}{3}\log n\right)\\ &\ \phantom{{}(d+1)(\kappa-n)(\log n -3-\log d) - (\kappa-n)(1-\varepsilon)d} +(\kappa-n)\log t\\ \ge &\ \frac{2\varepsilon}{3}\left(1-\frac{\varepsilon}{2}\right)\kappa d\log n +\left(1-\frac{\varepsilon}{2}\right)\kappa\log t+\left(1-\frac{\varepsilon}{2}\right)\kappa\log n \\ &\ \phantom{{}(\kappa-n)\log n -(d+1)}-3d\left(1-\frac{\varepsilon}{2}\right)\kappa -\left(1-\frac{\varepsilon}{2}\right)\kappa\left(3+\frac{\varepsilon}{3}\log n\right) \\ = &\ \Omega(\kappa (d\log n+\log t))\end{aligned}$$ We note that in the above computation, the first inequality is obtained by applying the following bound on binomial coefficients: $\binom{n}{d}\ge\left(\frac{n}{d}\right)^d$. Simplex Tree ------------ Let $K\in {\cal K}(n,d,k,m)$ be a simplicial complex whose vertices are labeled from 1 to $n$ and ordered accordingly. We can thus associate to each simplex of $K$ a word on the alphabet set $[n]$. Specifically, a $j$-simplex of $K$ is uniquely represented as the word of length $j + 1$ consisting of the ordered set of the labels of its $j + 1$ vertices. Formally, let $\sigma = \{v_{\ell_0} , \ldots , v_{\ell_j} \}$ be a simplex, where $v_{\ell_i}$ are vertices of $K$ and $\ell_i \in [n]$ and $\ell_0 <\cdot\cdot\cdot < \ell_j$ . $\sigma$ is represented by the word $[\sigma] = [ \ell_0 , \cdots , \ell_j ]$. The simplicial complex $K$ can be defined as a collection of words on an alphabet of size $n$. To compactly represent the set of simplices of $K$, the corresponding words are stored in a tree and this data structure is called the Simplex Tree of $K$ and denoted by ST$(K)$ or simply ST when there is no ambiguity. It may be seen as a trie on the words representing the simplices of the complex. The depth of the root is 0 and the depth of a node is equal to the dimension of the simplex it represents plus one. We give a constructive definition of ST. Starting from an empty tree, insert the words representing the simplices of the complex in the following manner. When inserting the word $[\sigma] = [ \ell_0 ,\cdot\cdot\cdot, \ell_j ]$ start from the root, and follow the path containing successively all labels $\ell_0 , \cdot\cdot\cdot , \ell_i$, where $[ \ell_0 ,\cdot\cdot\cdot, \ell_i ]$ denotes the longest prefix of $[\sigma]$ already stored in the ST. Next, append to the node representing $[ \ell_0 ,\cdot\cdot\cdot, \ell_i ]$ a path consisting of the nodes storing labels $\ell_{i+1} ,\cdot\cdot\cdot, \ell_j$. The filtration value of $\sigma$ denoted by $f(\sigma)$ is stored inside the node containing the label $\ell_j$, in the above path. In Figure 2, we give ST for the simplicial complex shown in Figure \[fig:SimplicialComplexExample\]. If $K$ consists of $m$ simplices (including the empty face) then, the associated ST contains exactly $m$ nodes. Thus, we need $\Theta(m(\log n+\log t))$ space/bits to represent the nodes in ST (since each node stores a vertex which needs $\Theta(\log n)$ bits to be represented and also stores the filtration value of the simplex that the node corresponds to, which needs $\Theta(\log t)$ bits to be represented). We remark here that we don’t consider the space needed to maintain the edges of ST, as it is part of the data structure implementation of ST. We can compare the upper bound obtained to the lower bound of Lemma \[FiltrationLowerbound\]. In particular, ST matches the lower bound, when $t$ is not small. Now, we will briefly recapitulate the cost of doing some basic operations through ST on a simplicial complex. To check if a simplex $\sigma$ is in the complex, is equivalent to checking the existence of the corresponding path starting from the root in ST. This can be done very efficiently in time $\mathcal{O}(d_\sigma \log n)$ and therefore all operations which primarily determine on the membership query can be efficiently performed using ST. One such example, is querying the filtration value of a simplex. However, due to its explicit representation, insertion is a costly operation on ST (exponential in the dimension of the simplex to be inserted). Similarly, removal is also a costly operation on ST, since there is no efficient way to locate and remove all cofaces of a simplex. Consequently, topology preserving operations such as elementary collapse and edge contraction are also expensive for ST. These operation costs are summarized later in Table \[tab:OperationsonMSD\]. In the next section, we will introduce a new data structure which does a better job of balancing between static queries (e.g. membership) and dynamic queries (e.g. insertion and removal). Critical Simplex Diagram {#CSD} ======================== In this section, we introduce the [*Critical Simplex Diagram*]{}, ${\text{CSD}}(K)$, which is an evolved version of SAL [@BKT15]. CSD is a collection of $n$ arrays that correspond to the $n$ vertices of $K$. The elements of an array, referred to as [*nodes*]{} in the rest of the paper, correspond to copies of the vertex of $K$ associated to the array. Additionally, CSD has edges that join nodes of different arrays. Each connected component of edges in CSD represents a simplex of $K$. Not all simplices of $K$ are represented but only those simplices all of whose cofaces have a higher filtration value. We describe some notations used throughout the section below. Notations {#sec:notations} --------- Let $S_h$ be the set of simplices in the complex whose filtration value is $h$. Let $M_h$ be the maximal subset of $S_h$ containing all the critical simplices of $S_h$. For instance, in the complex of Figure \[fig:SimplicialComplexExample\], we have $M_0=\{1,2,3,4,5,6\}$, $M_1=\{[56]\}$, $M_2=\{[14],[356]\}$, $M_3=\{[134], [24]\}$, $M_4=\emptyset$, and $M_5=\{[1234]\}$. Moreover, we note that $M_0, M_1,...., M_t$ are all disjoint and denote by $M$ the union of all $M_h$. We denote by $\Psi_{\max}(i)$ the subset of simplices of $M_h$ that contain vertex $i$ for all possible $h\in \llbracket t \rrbracket$. In other words, Let $\Psi=\underset{i\in [n] }{\max}\ |\Psi_{\max}(i)|$. We denote by $\Gamma_j$ the largest number of maximal simplices of $K$ that a given $j$-simplex of $K$ may be contained in. We note the following bounds: $$k\ge \Gamma_0\ge\Gamma_1\ge \cdots \ge \Gamma_d = 1,$$ $$\Gamma_0\le \Psi \le m.$$ Moreover, when $t=0$, we have $\Psi=\Gamma_0$. We describe the construction of CSD below. Construction {#sec:construction} ------------ We initially have $n$ empty arrays $A_1,\ldots ,A_n$. The vertices of $K$ are associated to the arrays. Each array contains a set of nodes that are copies of the vertex associated to the array and are labelled by an ordered pair of integers (to be defined below). Nodes belonging to distinct arrays are joined by edges leading to a graph structure. The connected components of that graph represent the simplices in $M_0,M_1,\ldots,M_5$. All the nodes of such a simplex are labelled by a pair of integers. The first integer refers to the filtration value of the simplex and the second integer refers to a number used to index simplices that have the same filtration value. For instance, in Figure \[fig:CSDExample\] we have the CSD representation of the simplicial complex of Figure \[fig:SimplicialComplexExample\], and the triangle $[134]$ in $M_3$ is represented by 3 nodes, each with label $(3,1)$, that are connected by edges. Below we provide a more detailed treatment of the construction of CSD. Given $M_h$ for every $h\in\llbracket t\rrbracket$, we build the CSD by inserting the simplices in $M_h$ in decreasing ordering of $h$. For *every* simplex $\sigma=v_{\ell_0} \cdot\cdot\cdot v_{\ell_j}$ in $M_h$, we associate a unique key generated using a hash function $\mathcal{H}$, ${\cal H}(\sigma) \in \left[|M_h|\right]$, and insert the nodes with label $(h,\mathcal{H}(\sigma))$ in to the arrays $A_{\ell_0},\ldots ,A_{\ell_j}$. For every $j^\prime\in [j]$, we introduce an edge between $(h,\mathcal{H}(\sigma))$ in $A_{\ell_{0}}$ and $(h,\mathcal{H}(\sigma))$ in $A_{\ell_{j^\prime}}$. In other words, $\sigma$, a critical simplex is being represented in CSD by a connected component in the graph thus defined. Each connected component is a star graph on $d_{\sigma}+1$ nodes where $v_{\ell_0}$ is the center of the star. Furthermore, each node in CSD corresponds to a vertex in exactly one simplex. We denote by $A_i^\star$ the set of all nodes in $A_i$ of star graphs that correspond to maximal simplices in $K$ (the region with these nodes are shaded in Figure \[fig:CSDExample\]). Inside each $A_i$, we first sort nodes based on whether they are in $A_i^\star$ or not. Further, inside $A_i^\star$ and inside $A_i\setminus A_i^\star$, we sort the nodes according to the lexicographic order of their labels. We note that $A_i^\star$ is a contiguous subarray of $A_i$, i.e., all consecutive elements in $A_i^\star$ are also consecutive elements in $A_i$, as can be observed in Figure \[fig:CSDExample\]. We remark here that we use a hash function $\mathcal{H}$ to generate keys for simplices because it is an efficient way to reuse keys (in case of multiple insertions and removals). Size of the Critical Simplex Diagram {#sizeofCSD} ------------------------------------ The number of nodes in each $A_i$ is at most $\Psi$, and the total number of nodes in CSD, we denote by $|{\rm CSD}|$, is at most $\Psi\cdot n$. Note that the number of edges in CSD is also at most $\Psi\cdot n$ since CSD is essentially a collection of star graphs. Alternatively, we can bound the number of nodes by $|M|d$, where $M$ as defined in Section \[sec:notations\] is the union of all $M_h$. The actual relation between $\Psi$ and $|M|$ can be stated as follows: $$\sum_{i=1}^{n} |\Psi_{\max}(i)| = \sum_{\sigma\in M} (d_\sigma+1).$$ Further, in each node we store a filtration value (which requires $\log t$ bits) and a hashed value (which requires $\underset{h\in\llbracket t\rrbracket}{\max}\ \log |M_h|\le \log m$ bits). We can thus upper bound the space needed to store the nodes of CSD by $\Psi\cdot n(\log m+\log t)$ or by $|M|\cdot d(\log m+\log t)$. However, if we only store the filtration value and the hashed value at the center of the star graph then, we need only $|M|(d+\log m+\log t)$ space to store the nodes of CSD. In doing so, from any node of the star graph, we can still access/modify the filtration and hashed values in $\mathcal{O}(1)$ time. Thus, CSD matches the lower bound in Theorem \[CSDLowerBound\], up to constant factors (as $m=\mathcal{O}\left(n^d\right)$). In the case of ${\text{CSD}}$, we are interested in the value of $\Gamma_0$ and $\Psi$ which we use to estimate the worst-case cost of basic operations in ${\text{CSD}}$, in the following subsection. Operations on the Critical Simplex Diagram {#sec:operations} ------------------------------------------ Let us now analyze the cost of performing basic operations on ${\text{CSD}}$. First, we describe how to intersect arrays and update arrays in CSD as these are elementary operations on CSD which are required to perform basic operations on the simplicial complex that it represents. Next, we describe how to perform static queries such as the membership query. Following which we describe how to perform dynamic queries such as the insertion or removal of a simplex. Finally, we compare the efficiency of CSD with ST. We remark here that in order to perform the above operations efficiently, we will exploit the fact that the filtration value of a simplex that is not critical is equal to the minimum of the filtration values of its cofaces. ### Elementary Operations to maintain the Critical Simplex Diagram Below, we first discuss the implementation of the arrays in CSD using red-black trees, and thus we would have described how to search within an array and update an array of CSD. Next, we describe how to compute the intersection of the arrays in CSD, an operation needed to answer static queries. ### Implementation of the arrays {#implementation-of-the-arrays .unnumbered} We implement the arrays $A_i$ using a variant of the red-black trees, and this means we can search, insert, and remove an element inside $A_i$ in time $\mathcal{O}(\log |A_i|)$. Below we will discuss how to implement $A_i^\star$ and the same will hold for $A_i\setminus A_i^\star$ which we treat separately. Each subarray of $A_i^\star$ which have the same filtration value i.e., the same first coordinate, is implemented using a red-black tree. Now these subarrays described above partition $A_i^\star$ and we can label each partition with the common first coordinate value of its elements. We represent the set of these partitions using a red-black tree by storing the partitions label. Therefore, each $A_i$ is the union of two “doubly-composed” red-black trees. The way we search in $A_i$, is that we sequentially search in $A_i^\star$, followed by $A_i\setminus A_i^\star$. ### Intersecting arrays {#intersecting-arrays .unnumbered} Let $\sigma$ be a simplex of dimension $d_{\sigma}$ and denote its vertices by $v_{\ell_0}, \ldots ,v_{\ell_{d_\sigma}}$. We will need to compute $A_\sigma$, defined as the intersection of $A_{\ell_0},\dots ,A_{\ell_{d_\sigma}}$, and $A_\sigma^\star$, defined as the intersection of $A_{\ell_0}^\star,\dots ,A_{\ell_{d_\sigma}}^{\star}$. To compute $A_\sigma$, we first find out the array with fewest elements amongst $A_{\ell_0},\dots ,A_{\ell_{d_\sigma}}$. Then, for each element $x$ in that array, we search for $x$ in the other $d_\sigma$ arrays, which can be done in time $\mathcal{O}\left(d_\sigma\log \left(\underset{i}{\max}\ |A_{\ell_i}|\right)\right)$. Hence $A_\sigma$ can be computed in time $\mathcal{O}\left(\left(\underset{i}{\min}\ |A_{\ell_i}|\right)d_\sigma\log \left(\underset{i}{\max}\ |A_{\ell_i}|\right)\right)$. As we have seen before, $|A_{\ell_i}| \leq \Psi \le m$. We can compute $A_\sigma^\star$ in the same way as described above for $A_\sigma$ in time $\mathcal{O}\left(\left(\underset{i}{\min}\ |A_{\ell_i}^\star|\right)d_\sigma\log \left(\underset{i}{\max}\ |A_{\ell_i}^\star|\right)\right)$. As we have seen before, $|A_{\ell_i}^\star| \leq \Gamma_0 \le k$. ### Static Operations on the Critical Simplex Diagram {#staticoperations} The tree structure of ST provides an efficient representation to perform static operations. However, we show below that we are able to answer these static queries using CSD by only paying a multiplicative factor of $\Psi$ (in the worst case) over the cost of performing the same operation in ST. In the case of the membership query, the multiplicative factor is reduced to $\Gamma_0$. ### Membership of a Simplex {#membership-of-a-simplex .unnumbered} We first observe that $\sigma\in K$ if and only if $A_\sigma^\star\neq\emptyset$. This is because if $\sigma\in K$, then there exists a maximal simplex in $K$ which contains $\sigma$. The star graph associated to this maximal simplex has nodes in all the $A_{\ell_i}^\star$, and all those nodes have the same label. This implies that $A_\sigma^\star\neq\emptyset$, and the converse is also true. It follows that determining if $\sigma$ is in $K$ reduces to computing $A_\sigma^\star$ and checking whether it is non-empty. This procedure is very similar to the analogous procedure using SAL [@BKT15]. Therefore, membership of a simplex can be determined in time $\mathcal{O}(d_\sigma\Gamma_0\log k)$. Finally, we note that through the membership query, we are also able to decide if a simplex is maximal in the complex. We denote this new query by `is_maximal`, and will be used later for performing other operations. ### Access Filtration Value {#access-filtration-value .unnumbered} Given a simplex $\sigma$ of $K$ we want to access its filtration value $f(\sigma)$. We observe that $f(\sigma)$ is the minimal filtration value of the nodes in $A_\sigma$ since the filtration function is monotone w.r.t. inclusion. Hence, accessing the filtration value of $\sigma$ reduces to computing $A_\sigma$. Therefore, the filtration of a simplex can be determined in time $\mathcal{O}(d_\sigma\Psi\log \Psi)$. For example, consider the ${\text{CSD}}$ in Figure \[fig:CSDExample\]. We have to find the filtration value of $\sigma=[134]$ in the complex of Figure \[fig:SimplicialComplexExample\]. We see that $A_1\cap A_3 \cap A_4=\{(3,1),(5,1)\}$. This means that the filtration value of the triangle is $f([134])=\min\ (3,5)=3$. Finally, we note that through the filtration query, we are also able to decide if a simplex is critical in the complex. This new query, denoted by `is_critical`, will be used later for performing other operations. ### Computing Filtration Value of Facets {#computing-filtration-value-of-facets .unnumbered} Given a simplex $\sigma= v_{l_0},...,v_{l_{d_{\sigma}}}$, we could procure the filtration value of its $d_\sigma+1$ facets by the access filtration value query, and thus requiring a total running time of $\mathcal{O}(d_\sigma^2 \Psi\log \Psi)$. However, we will modify the access filtration value query to obtain filtration value of the $d_\sigma+1$ facets of $\sigma$ in running time of $\mathcal{O}(d_\sigma \Psi\log \Psi)$. Let $A_{\ell_r}=\underset{i}{\text{argmin}}\ \lvert A_{\ell_i}\rvert$, for some $r\in\llbracket d_\sigma\rrbracket$. Let $B$ be a subset of $A_{\ell_r}$ such that every element of $A_{\ell_r}$ which appears in exactly $d_{\sigma}-1$ of the sets in $A_{\ell_0},\ldots ,A_{\ell_{r-1}},A_{\ell_{r+1}},\ldots ,A_{\ell_{d_\sigma}}$ is in $B$. We can identify $B$ in time $\mathcal{O}(|A_{\ell_r}|d_\sigma\log\Psi )$. To each entry $(s,x)$ in $B$ we associate the index of the set $A_{\ell_i}$ which does not contain $(s,x)$ by a mapping $g$. We sort $B$ based on $g$, and in case of ties based on the first coordinate. The filtration value of the facet $\sigma^\prime=v_{\ell_0} \cdots v_{\ell_{j-1}}\cdot v_{\ell_{j+1}}\cdots v_{\ell_{d_\sigma}}$, is the minimal filtration value of the nodes in $B$ which are mapped to $j$ under $g$. If there are no nodes in $B$ mapped to $j$ under $g$ then, the filtration value of $\sigma^\prime$ is $f(\sigma)$. This computation for all the $d_\sigma+1$ facets requires a total time of $\mathcal{O}(|B|\log |B|+d_\sigma\log |B|)=\mathcal{O}((\Psi+d_\sigma)\log \Psi)$. Therefore the total running time is $\mathcal{O}(d_\sigma \Psi\log \Psi)$. ### Computing Filtration Value of Cofaces of codimension 1 {#computing-filtration-value-of-cofaces-of-codimension-1 .unnumbered} Given a simplex $\sigma$, we perform the access filtration value query to obtain all critical simplices that contain $\sigma$. We traverse the star graphs that contain the nodes in $A_\sigma$, to list these critical simplices in time $\mathcal{O}(d\Psi\log \Psi)$. From this list, we can compute the filtration value of all the cofaces of $\sigma$ of codimension 1 in $\mathcal{O}(d\Psi\log \Psi)$ time. ### Dynamic Operations on the Critical Simplex Diagram {#sec:dynamicoperations} Now, we will see how to perform dynamic operations on CSD. We note here that CSD is more suited to perform dynamic queries over ST because of its non-explicit representation, and this means that the amount of information to be modified is always less than ST. ### Lazy Insertion {#lazy-insertion .unnumbered} Lazy insertion is the operation of inserting $\sigma$ into the complex without checking if there are simplices in the complex which were previously critical but are now faces of $\sigma$ with the same filtration value as $f(\sigma)$. Additionally, in the case of lazy insertion, we assume that the information about $\sigma$ being a maximal simplex (or not) is known as part of the input. Lazy insertion will be extensively used in the later sections for preliminary construction of simplicial complexes. Lazy insertion in CSD requires $\mathcal{O}(d_\sigma\log\Psi)$ time (i.e., the cost of updating the arrays). ### Insertion {#insertion .unnumbered} Suppose we want to insert a simplex $\sigma$ with filtration value $f(\sigma)$ such that any coface of $\sigma$ in $K$ has a filtration value larger than $f(\sigma)$. The insertion operation consists of first checking if $\sigma$ is a maximal simplex in $K$ by the `is_maximal` query. If $\sigma$ is maximal, then we have to insert the simplex and remove or reallocate (based on filtration value) all simplices which were maximal simplices in $K$ but are now faces of $\sigma$. If $\sigma$ is not maximal, then we just have to lazy insert $\sigma$ into the complex. We remark here that we do not need to remove the faces of $\sigma$ which were previously critical simplices and had filtration value at least $f(\sigma)$ as their presence will not hinder any operation on CSD. Alternatively, we can think of performing a clean-up operation in parallel where such nodes are removed from CSD without affecting any other operation (this discussion is elaborated in the paragraph called ‘Robustness in Modification’ in Section \[Performance\]). Suppose $\sigma$ is a maximal simplex then, insert the star graph corresponding to $\sigma$ in $A_{\ell_0}^\star,\ldots ,A_{\ell_{d_\sigma}}^\star$. Updating the arrays $A_{\ell_i}$ takes time $\mathcal{O}(d_\sigma\log \Psi)$. Next, we have to check if there exist maximal simplices in $K$ which are now faces of $\sigma$, and either remove them if their filtration value is equal to $f(\sigma)$ or move them outside $A_{\ell_i}^\star$ if their filtration value is strictly less than that of $f(\sigma)$. We restrict our search for faces of $\sigma$ which were previously critical by looking for every vertex $v$ in $\sigma$, at the set of all maximal simplices which contain $v$, denoted by $Z_v$. We can compute $Z_v$ in time $\mathcal{O}( d_{\sigma}\Gamma_0 \log \Gamma_0)$. Then, we compute $\underset{v\in\sigma}{\cup}Z_v$ whose size is at most $(d_\sigma+1)\Gamma_0$ and check if any of these maximal simplices are faces in $\sigma$ (can be done in $\mathcal{O}(d_\sigma^2\Gamma_0)$ time). If such a face of $\sigma$ in $\underset{v\in\sigma}{\cup}Z_v$ has filtration value equal to $f(\sigma)$ then, we remove that connected component. To remove all such connected components takes time $\mathcal{O}(d_\sigma^2\Gamma_0\log \Gamma_0)$. On the other hand, if filtration value of the face is less than $f(\sigma)$ then, we will have to move the node outside $A_{\ell_i}^\star$ and place it appropriately to maintain the sorted structure of $A_{\ell_i}$. To reallocate all such connected components takes time $\mathcal{O}(d_\sigma^2\Gamma_0\log \Psi)$. Summarizing, to handle removal of face or reallocating the concerned faces of $\sigma$ which were previously maximal takes time at most $\mathcal{O}(d_\sigma^2\Gamma_0(\log \Gamma_0+\log \Psi))$. If $\sigma$ is not a maximal simplex then, we insert the star graph corresponding to $\sigma$ in $A_{\ell_0},\ldots ,A_{\ell_{d_\sigma}}$. Updating the arrays $A_{\ell_i}$ takes time $\mathcal{O}(d_\sigma\log \Psi)$. Therefore, the total running time in this case is $\mathcal{O}(d_\sigma \log \Psi)$. ### Removal {#removal .unnumbered} To remove a face $\sigma$, we first perform an access filtration value query of $\sigma$ (requires $\mathcal{O}(\Psi d_{\sigma}\log \Psi)$ time). We deal with the simplices in $ A_\sigma^\star$ and $ A_\sigma\setminus A_\sigma^\star$ separately. For every simplex $ \tau\in A_\sigma^\star$, i.e., for every coface $\tau$ of $\sigma$ in $K$ which is a maximal simplex, we remove its corresponding star graph from the CSD. Since there are at most $\Gamma_{d_{\sigma}}$ maximal simplices which contain $\sigma$, the above removal of star graphs can be done in $\mathcal{O}(\Gamma_{d_{\sigma}}d\log \Psi)$ time. Next, for each maximal simplex $\tau$ (containing $\sigma$) that we removed, and for every $i\in \llbracket d_\sigma\rrbracket $ we check if the facet of $\tau$ obtained by removing $v_{\ell_{i-1}}$ from $\tau$, is a maximal simplex. If yes, we lazy insert the facet as a maximal simplex with the same filtration value as $\tau$. If no, we lazy insert the facet as a non-maximal simplex with the same filtration value as $\tau$. Note that in order to check if the above mentioned $d_\sigma+1$ facets of $\tau$ are maximal, we do not have to make $d_{\sigma}+1$ `is_maximal` queries, but can do the same checking in $\mathcal{O}(\Gamma_{d_\sigma}d\log \Psi )$ time by using the same idea that is described in the ‘computing filtration value of facets’ paragraph in Section \[staticoperations\]. Additionally, we remark here that we can lazy insert the above selected facets of the the critical cofaces of $\sigma$, without checking if the facets themselves are critical because the argument made in the Insertion paragraph above for such cases apply here as well. Next, for every simplex $ \tau\in A_\sigma\setminus A_\sigma^\star$ i.e., for every coface $\tau$ of $\sigma$ in $K$ which is a critical (not maximal) simplex, we replace its corresponding star graph by star graphs of its $d_{\sigma}+1$ facets with the same filtration value, where the $i^{\text{th}}$ facet is obtained by removing $v_{\ell_{i-1}}$ from $\tau$. Introducing a star graph and updating the arrays $A_{\ell_i}$ takes time $\mathcal{O}(d_\tau\log \Psi)$. Further, if $\sigma$ is a critical simplex (can be checked by `is_critical` query) then, we know that there is a connected component representing $\sigma$. We replace this star graph by the star graph for all its facets which have the same filtration value. Therefore, the total running time is $\mathcal{O}(dd_\sigma\Psi\log \Psi)$. We note here that as before, we lazy insert the above selected facets of the the critical cofaces of $\sigma$, without checking if the facets themselves are critical. Therefore, the total time for insertion is $\mathcal{O}(\Psi dd_\sigma \log \Psi + \Gamma_{d_\sigma}d_\sigma d\log \Psi + \Gamma_{d_\sigma}^2d\log \Psi )=\mathcal{O}((\Psi d_\sigma + \Gamma_{d_\sigma}^2)d\log \Psi)$. ### Elementary Collapse {#elementary-collapse .unnumbered} A simplex $\tau$ is collapsible through one of its faces $\sigma$, if $\tau$ is the only coface of $\sigma$. Such a pair $(\sigma,\tau)$ is called a free pair, and removing both faces of a free pair is an elementary collapse. Given a pair of simplices $(\sigma,\tau)$, to check if it is a free pair is done by obtaining the list of all maximal simplices which contain $\sigma$, through the membership query (costs $\mathcal{O}(d_\sigma \Gamma_0\log\Gamma_0 )$ time) and then checking if $\tau$ is the only member in that list with codimension 1. If yes, then we remove $\tau$ from the CSD by just removing all the nodes in the corresponding arrays in time $\mathcal{O}(d_{\tau}\log \Psi)$. Next, for every facet $\sigma^\prime$ of $\tau$ other than $\sigma$, we check if $\sigma^\prime$ is a critical simplex (post removal of $\tau$) by asking the `is_critical` query. If yes, we lazy insert $\sigma^\prime$ in time $\mathcal{O}(d_\sigma\log \Psi)$. Finally, if $\sigma$ is a critical simplex then, we remove it in the same way we removed $\tau$ and for every facet of $\sigma$ we similarly check if it is a critical simplex (post removal of $\sigma$) by asking the `is_critical` query. If yes, we lazy insert that facet in time $\mathcal{O}(d_\sigma\log \Psi)$. Thus, the total running time is $\mathcal{O}(d_\sigma (d_\sigma\Psi\log \Psi+\Gamma_0\log\Gamma_0))=\mathcal{O}(d_\sigma^2 \Psi\log \Psi)$. ### Summary We summarize in Table \[tab:OperationsonMSD\] the asymptotic cost of basic operations discussed above and compare it with ST, through which the efficiency of ${\text{CSD}}$ is established. If the number of critical simplices is not large then $|{\text{CSD}}|$ is smaller than $|{\text{ST}}|$. The number of critical simplices is small unless we associate unique filtration values to a significant fraction of the simplices. For this subsection, we will assume that the number of critical simplices is small (this assumption will be justified in the next subsection). In this case, we have $\Psi$ to be small and thus the size of CSD is smaller than the size of ST. We observe that while performing static queries, we pay a factor of $\Psi$ or $\Gamma_0$ in the case of CSD over the cost of the same operation in ST. In the case of dynamic operations we observe that the dependence on the dimension is exponentially smaller in CSD than in ST. Therefore, even if the number of critical simplices is polynomial in the dimension then, there is an exponential gap between CSD and ST in both the storage and the efficiency of performing dynamic operations. Furthermore, in the case of insertion, CSD depends on $\Gamma_0$ and not $\Psi$ (recall that $\Gamma_0\le \Psi$). Thus, the efficient insertion operation in CSD allows for fast construction of simplicial complexes, as we will see in future sections. In short, CSD needs less storage than ST and performs dynamic operations more efficiently than ST while paying (mostly) a small multiplicative factor over ST in performing static queries. This is analogous to the trade-off between NFA (Non-deterministic Finite state Automaton) and DFA (Deterministic Finite state Automaton). Performance of CSD {#Performance} ------------------ CSD has been designed to store filtrations of simplicial complexes but it can be used to store simplicial complexes without a filtration. In this case, $|M|=k$ and CSD requires $\mathcal{O}(kd\log k)$ memory space, which matches the lower bound in Theorem \[Lowerbound\], when $k=n^{\mathcal{O}(1)}$. In this case, CSD is very similar to SAL [@BKT15]. Marc Glisse and Sivaprasad S. [@MarcSivaprasad] have performed experiments on SAL and concluded that it is not only smaller in size but also faster than the Simplex Tree in performing insertion, removal, and edge contraction. CSD is also a compact data structure to store filtrations, as its size matches (up to constant factors) the lower bound of $\Omega(|M|(d\log n+\log t))$ in Theorem \[CSDLowerBound\]. Moreover, if $\Psi$ is small, CSD is not only a compact data structure since $|{\text{CSD}}|$ is upper bounded by $\Psi n$, but, as shown in Table \[tab:OperationsonMSD\], CSD is also a very efficient data structure as all basic operations depend polynomially on $d$ (as opposed to ST for which some operations depend exponentially on $d$). As our analysis shows, we can express the complexity of CSD in terms of a parameter $\Psi$ that reflects some “local complexity” of the simplicial complex. In the worst-case, $\Psi=\Omega(m)$ as it can be observed in the complete complex with each simplex having a unique filtration value. However we conjecture that, even if $m$ is not small, $\Psi$ remains small for a large class of simplicial complexes of practical interest. This conjecture is supported by the following experiment. We considered a set of points obtained by sampling a Klein bottle in $\mathbb{R}^5$ and constructed its Rips filtration (see Section \[Flag\] for definition) using libraries provided by the GUDHI project [@GUDHI]. We computed $\Gamma_0$ and $\Psi$ for various values of $t$. The resulting simplicial complex on 10,000 vertices is 17 dimensional and has 10,508,486 simplices of which 27,286 are maximal. We record in Table \[tab:Rips\] below, the values of $|{\text{ST}}|$ and $|{\text{CSD}}|$ for the various filtration ranges of the Rips complex constructed above. In Figure \[fig:Rips\] is a graphical illustration of the data. \[dataset1\] We note from Table \[tab:Rips\] that $\Gamma_0$ is significantly smaller than $k(\approx 2.7 \times 10^4)$, and also that $\Gamma_0^{\text{avg}}$ is much smaller than $\Gamma_0$. Also, from Figure \[fig:Rips\], it is clear that there is an order of magnitude gap between $|{\text{CSD}}|$ and $|{\text{ST}}|$. Next, we note that $\Psi$ is remarkably smaller than $m$ (even notably smaller than $n$), and this implies efficient implementation of all operations. More importantly, we remark here that $\Psi^{\text{avg}}=|{\text{CSD}}|/n$ is at most $77.8$ in the above experiment. Finally, we observe that despite increasing $t$ at a rapid rate, $|{\text{CSD}}|$ grows very slowly after $t=100$. This is because the set of all possible filtration values of the Rips complex is small. Therefore, even for small values of $t$ the simplicial complex and its filtration is accurately captured by CSD. ### Local Sensitivity of the Critical Simplex Diagram {#local-sensitivity-of-the-critical-simplex-diagram .unnumbered} It is worth noting that while the cost of basic operations are bounded using $\Gamma_0$ and $\Psi$, the actual cost is bounded by parameters such as $\underset{i}{\min}\left(|A_{\ell_i}^\star|\right)$, $\underset{i}{\min}\left(|A_{\ell_i}|\right)$, and $Z_v$ (introduced in the Insertion paragraph in Section \[sec:dynamicoperations\]) to get a better estimate on the cost of these operations. These parameters are indeed local. To begin with, $\underset{i}{\min}\left(|A_{\ell_i}^\star|\right)$ captures the local information about a simplex $\sigma$ sharing a vertex with other maximal simplices of the complex. More precisely, it is the minimum, over all the vertices of $\sigma$, of the largest number of maximal simplices that contain the vertex. If $\sigma$ has a vertex which is contained in a few maximal simplices then, $\underset{i}{\min}\left(|A_{\ell_i}^\star|\right)$ is small. Similarly, $\underset{i}{\min}\left(|A_{\ell_i}|\right)$ is the minimum, over all the vertices of $\sigma$, of the largest number of critical simplices that contain the vertex. This value depends not only on the structure of the filtration function but also on the filtration range. Finally, $Z_v$ captures another local property of a simplex $\sigma$ – the set of all maximal simplices that contain the vertex $v$. Therefore, CSD is sensitive to the local structure of the complex. ### Robustness in Modification {#robustness-in-modification .unnumbered} We now demonstrate the robustness of CSD, i.e., its ability to perform queries *correctly* and *efficiently* even when it might have stored redundant data such as simplices which are not critical or multiple copies of the same simplex with different filtration values. Consider modifying the filtration value of some simplex $\sigma\in K$ from $f(\sigma)$ to $s_{\sigma}(<f(\sigma))$. In the case of ST, we will have to modify the filtration value inside the node containing $\sigma$ and additionally check (and modify if needed) its faces in decreasing order of dimension. This requires time $\Theta(2^{d_\sigma}d_\sigma(\log n+\log t))$. However, in the case of CSD, we can perform a lazy insertion of $\sigma$ into CSD in time $\mathcal{O}(d_\sigma\log \Psi)$, and the data structure is robust to such an insertion. This is because, all the operations can be performed correctly and with the same[^4] efficiency after the lazy insertion (even if some previously critical simplices need to be removed due to the lazy insertion of $\sigma$). For instance, consider the $\texttt{is\_critical}$ query on some simplex $\tau$. If $\tau$ was a face of $\sigma$ before modifying $f(\sigma)$ then, the *minimal* filtration value of the nodes in $A_{\tau}$ correctly gives the filtration value of $\tau$ as $s_\sigma$ will now be one of the entries in $A_\tau$. Otherwise, if $\tau$ was not a face of $\sigma$ then the filtration value of $\tau$ remains unchanged, as the lazy insertion of $\sigma$ has not introduced a new simplex, but only a new filtration value to an existing simplex. Therefore, we can think of using the data structure to manipulate simplicial complexes in very short time through a collection of lazy insertions and perform a clean-up operation at the end of the collection of lazy insertions, or even think of performing the clean-up operation in parallel to the lazy insertions. This idea has been applied in performing the insertion and removal operations as discussed in Section \[sec:dynamicoperations\]. We remark here that if we lazy insert $r$ simplices then in the worst case, $\Psi$ grows to $r+\Psi$. In other words, the presence of redundant simplices, implies that the efficiency dependence will now be on $r+\Psi$ instead of $\Psi$, but the redundancy will not affect the correctness of the operations. [A Sequence of Representations for Simplicial Complexes and their Filtrations]{} -------------------------------------------------------------------------------- Boissonnat et al. [@BKT15] in their paper on Simplex Array List described a sequence of data structures, each more powerful than the previous ones (but also bulkier). In that sequence of data structures $\langle \Lambda\rangle$, we had $\Lambda_{i}=i\mhyphen {\text{SAL}}$ (${\text{SAL}}$ referred to earlier in this paper is equal to $1\mhyphen{\text{SAL}}$). Further, they note that in the $i^{\textrm{th}}$ element of the sequence, every node which is not a leaf (sink) in the data structure corresponds to a unique $i$-simplex in the simplicial complex. Also for all $i\mhyphen{\text{SAL}}$, $i\in\mathbb{N}$, they state that it is a NFA recognizing all the simplices in the complex. As one moves along the sequence, the size of the data structure blows up by a factor of $d$ at each step. But in return, there is a gain in the efficiency of searching for simplices as the membership query depends on $\Gamma_i$ which decreases as $i$ increases. We note here that ${\text{CSD}}$ described in this paper is exactly the same as $0\mhyphen{\text{SAL}}$, when $t=0$ (we ignore the structure of the connected component, which is a path in SAL but a star in CSD). Therefore, ${\text{CSD}}$ supersedes $0\mhyphen{\text{SAL}}$. There is no change in representation of a simplex between SAL and CSD; instead we only store more simplices (i.e, all critical simplices) in CSD. Therefore, in the same vein as $\langle\Lambda\rangle$, we can define a sequence of data structures, each more powerful than the previous ones (but also bulkier). More formally, consider the sequence of data structures $\langle \Pi\rangle$, where $\Pi_{0}={\text{CSD}}$ and $\Pi_i = \underset{j=0}{\overset{t}{\bigcup}}i\mhyphen{\text{SAL}}(M_j)$, for all $i\in\mathbb{N}$. Further, for all $i\in\mathbb{N}$, we will refer to the data structure $\Pi_i$ by the name $i\mhyphen {\text{CSD}}$ (we will continue to refer to $0\mhyphen{\text{CSD}}$ as ${\text{CSD}}$). As we move along the sequence $\Pi$, the size of the data structure blows up by a factor of $d$ at each step. But in return, we gain efficiency in searching for simplices as the membership query depends on $\Gamma_i$ which decreases as $i$ increases. Additionally, we gain efficiency in accessing filtration value of a simplex as the complexity no longer depends on $\Psi=\Psi_0$ but on a smaller parameter, $\Psi_i$, which is the maximum number of critical cofaces that any $i$-simplex can have in the complex. Marc Glisse and Sivaprasad S. implemented SAL [@MarcSivaprasad] for Data Set mentioned in Section 3.4, and then performed insertion and removal of random simplices, and contracted randomly chosen edges. They observed that 1-SAL outperformed 0-SAL in low dimensions. However, 0-SAL performed better than 1-SAL in higher dimensions. Therefore, in similar vein, it would be worth exploring for which class of simplicial complexes, $i\mhyphen$CSD is the best data structure in the CSD family (for every $i\in\mathbb{N}$). Construction of Flag Complexes {#Flag} ============================== The flag complex of an undirected graph $G$ is defined as an abstract simplicial complex, whose simplices are the sets of vertices in the cliques of $G$. Let $(P, \|\cdot\|)$ be a metric space where $P$ is a discrete point-set. Given a positive real number $r > 0$, the Rips complex is the abstract simplicial complex $\mathcal{R}^r(P)$ where a simplex $\sigma\in\mathcal{R}^r(P)$ if and only if $\|p-q\|\le 2r$ for every pair of vertices of $\sigma$. Note that the Rips complex is a special case of a flag complex. Rips filtrations are widely used in Topological Data Analysis since they are easy to compute and they allow to robustly reconstruct the homology of a sample shape via the computation of its persistence diagram [@CCSGGO09]. We will describe a specific filtration for Flag complexes which is of significant interest as it includes the Rips filtration. The filtration value of a vertex is $0$. The filtration value of every edge in the complex is given as part of the input. The filtration value of a simplex of higher dimension is equal to the maximum of the filtration values of all the edges in the simplex. Edge-Deletion Algorithm for Construction of Flag Complexes ---------------------------------------------------------- Let $G$ be the (weighted) graph of the simplicial complex $K$. Let $\Delta$ denote the maximum degree of the vertices of $G$. To represent $K$ using ST, Boissonnat and Maria [@SimplexTree] propose computing and inserting the $\ell$-skeleton of $K$ into the ST and incrementally increasing $\ell$ from $1$ to $d$. Therefore, the time for construction of the ST representing the flag complex is $\mathcal{O}(mnd\log n)$. To represent $K$ using CSD, we propose an *edge-deletion* algorithm, which is significantly faster than the construction algorithm for ST. We recall that in Section \[sec:construction\] we defined $S_h$ to denote the set of simplices in the complex with filtration value $h$. #### Preprocessing Step. We first compute all maximal cliques in $G$ in time $\mathcal{O}(k\cdot n^\omega)$ [@MU04], where $\omega<2.38$ [@L14] is the matrix multiplication exponent, i.e., $n^\omega$ is the time needed to multiply two $n\times n $ matrices. We store these maximal simplices in a Prefix Tree (like MxST of [@BKT15]). The filtration value given to the edges provides a natural ordering to the edges of the complex. We consider edges in descending order of their filtration value. Let $e_i$ be the edge with the $i^{\text{th}}$ highest filtration value. Recall that all simplices containing $e_i$ are of filtration value $f(e_i)$ and are in $S_{f(e_i)}$. Fix $i=1$. #### Step 1. In this step, we would like to compute $M_{f(e_i)}$ in order to build CSD. A natural way to do that is by first computing $S_{f(e_i)}$, and then identifying the subset $M_{f(e_i)}$. Computing $S_{f(e_i)}$ requires time $\mathcal{O}\left(\left|S_{f(e_i)}\right|d\log n\right)$ and then computing $M_{f(e_i)}$ will require time $\widetilde{\mathcal{O}}\left(\left|S_{f(e_i)}\right|\cdot d\cdot \left|M_{f(e_i)}\right|\right)$ using the best known algorithms in literature [@Y92; @YJ93; @P97; @BP11]. However, we will not compute $M_{f(e_i)}$ from $S_{f(e_i)}$, but instead skip computing $S_{f(e_i)}$ and directly compute $M_{f(e_i)}$ to list all the maximal simplices only containing the edge $e_i$ in time $\mathcal{O}\left(\left|M_{f(e_i)}\right|\Delta^\omega\right)$ using the algorithm presented by Makino and Uno [@MU04] on a subgraph of $G$ in the following way. We build an induced subgraph $H$ of $G$ which contains the vertices of the edge $e_i$ and all the vertices which are adjacent to *both* the vertices of $e_i$. We note that every maximal clique in $H$ is a maximal clique in $G$ containing the edge $e_i$, and vice versa. Therefore, if we run Makino and Uno’s algorithm on $H$ (which contains at most $\Delta+1$ vertices), we obtain all the maximal cliques in $G$ containing the edge $e_i$. #### Step 2. Next, we recognize the maximal simplices of $K$ in $M_{f(e_i)}$ in time $\mathcal{O}\left(\left|M_{f(e_i)}\right|d\log n\right)$ by checking each simplex $\sigma$ in $M_{f(e_i)}$ with the Prefix tree built in the preprocessing step in time $\mathcal{O}(d_\sigma \log n)$ (per simplex). We remark here that all simplices in $M_{f(e_1)}$ are maximal simplices in $K$, since $e_1$ has the largest filtration value. #### Step 3. We perform lazy insertion of simplices in $M_{f(e_i)}$ into the CSD and since we have identified the maximal simplices in $M_{f(e_i)}$, we know whether to insert them in $A_j^\star$ or not, within each $A_j$. This takes time $\mathcal{O}\left(\left|M_{f(e_i)}\right|d\log \Psi\right)$. #### Step 4. Finally, we remove $e_i$ from $G$, increment $i$ by 1, and repeat the procedure from step 1 until $G$ has no edges left. This entire construction takes time $\mathcal{O}(|M|(\Delta^\omega+d\log(kd\Psi)))=\mathcal{O}\left(|M|n^{2.38}\right)$, which is significantly better than that of constructing a representation of $K$ by ST (which required time $\mathcal{O}(mnd\log n)$), as $|M|$ can be considerably (exponentially) smaller than $m$. Performance of CSD for Flag Complexes ------------------------------------- We would like to note here that the case when $k=\mathcal{O}(n)$, was argued to be of particular interest by Boissonnat et al. [@BKT15]. It can be observed in flag complexes, constructed from planar graphs and expanders [@ELS10], and in general, from nowhere dense graphs [@GKS13], and also from chordal graphs[@G80]. Generalizing, they noted that for all flag complexes constructed from graphs with degeneracy $\mathcal{O}(\log n)$ (degeneracy is the smallest integer $r$ such that every subgraph has a vertex of degree at most $r$), we have that $k=n^{\mathcal{O}(1)}$ [@ELS10]. We add to this list of observations by noting that the flag complexes of $K_{\ell}$-free graphs have at most $\max\{n,n\Delta^{\ell-2}/2^{\ell-2}\}$ maximal simplices [@P95], where $\Delta$ is the maximum degree of any vertex in the graph. Thus, when $\Delta$ and $\ell$ are constants, we have $k=\mathcal{O}(n)$. Finally, we note that the flag complexes of Helly circular-arc (respectively, circle) graphs [@G74; @D03], and boxicity-2 graphs [@S03] have $k=n^{\mathcal{O}(1)}$ from Corollary 4 of [@RS07]. This encompasses a large class of complexes encountered in practice and if the number of maximal simplices is small, ${\text{CSD}}$ is a very efficient data structure as $\Gamma_0\le k$. Adaptation to Simplicial Maps ----------------------------- A map $F:K\to K^\prime$ is simplicial if for every simplex $\sigma=\{v_0, v_1,\ldots , v_k\}$ in $K$, $F(\sigma)=\{F(v_0), F(v_1), \ldots , F(v_k)\}$ is a simplex in $K^\prime$. In this subsection, we discuss the adaptation of CSD to handle simplicial maps. For the purpose of demonstration, we consider Dey et al.’s [@DFW14] application of simplicial maps to topological data analysis over Rips complexes. They construct a sequence of Rips simplicial complexes, $\langle K_i^{\alpha_i}\rangle$ connected by simplicial maps, where $\alpha_i$ is the Rips parameter. The vertex set of $K_{i+1}^{\alpha_{i+1}}$ is obtained by extracting a subset (net) of the vertex set of $K_i^{\alpha_{i}}$. They define a map $\pi$ through which they map each vertex of $K_i^{\alpha_{i}}$ to its closest vertex in $K_{i+1}^{\alpha_{i+1}}$ . Next, they take $\alpha_{i+1}>\alpha_i$ so that the image of the edges of $K_i^{\alpha_{i}}$ are edges in $K_{i+1}^{\alpha_{i+1}}$. Since the complexes are Rips complexes, the image of all the simplices of $K_i^{\alpha_{i}}$ are in $K_{i+1}^{\alpha_{i+1}}$. To implement this procedure using CSD, we need to collapse the vertices of vert($K_i^{\alpha_{i}}$)$\setminus$vert($K_{i+1}^{\alpha_{i+1}}$) onto the vertices of $K_{i+1}^{\alpha_{i+1}}$ as given by $\pi$. First, we remark that this procedure is very expensive using ST, as vertex collapse is more expensive than the edge contraction operation, and Boissonnat and Maria [@SimplexTree] provide an essentially optimal algorithm running in $\mathcal{O}(md\log n)$ time for edge contraction. In the case of CSD, we will assume that $\left(|\text{vert}(K_i^{\alpha_{i}})|-|(\text{vert}(K_{i+1}^{\alpha_{i+1}})|\right)$ is small, as otherwise, we could reconstruct $K_{i+1}^{\alpha_{i+1}}$ entirely from scratch using the fast edge-deletion algorithm. For every vertex $v$ in vert($K_i^{\alpha_{i}}$)$\setminus$vert($K_{i+1}^{\alpha_{i+1}}$), we first build a set $T_v$ from the set of maximal simplices containing $v$ as follows. If $\sigma$ is a maximal simplex containing $v$, then we include the simplex $\tau=\sigma\cap\text{vert}(K_{i+1}^{\alpha_{i+1}})$ in $T_v$. Let $T=\cup\ T_v$. Next, for every vertex $v$ in vert($K_i^{\alpha_{i}}$)$\setminus$vert($K_{i+1}^{\alpha_{i+1}}$), we remove all the nodes in $A_v$ and its neighbors in other arrays. For every simplex in $T$, we perform the `is_critical` query. If the simplex was not critical in $K_i^{\alpha_{i}}$ but is critical in $K_{i+1}^{\alpha_{i+1}}$, then we lazy insert the simplex. On the other hand, if the simplex was critical in $K_i^{\alpha_{i}}$, then we check if it is maximal in $K_{i+1}^{\alpha_{i+1}}$ and reallocate appropriately. The total cost of performing the above procedure is $\mathcal{O}(|T|d\Psi\log\Psi)$, where $|T|=\mathcal{O}(\Gamma_0(|\text{vert}(K_i^{\alpha_{i}})|-|(\text{vert}(K_{i+1}^{\alpha_{i+1}})|))$. This running time is significantly better than that of implementing the simplicial map using ST, as $\Psi$, and consequently $\Gamma_0$, may be considerably (exponentially) smaller than $m$. Adding to the above argument the benefit of the considerably (exponentially) smaller size of CSD, it is clear that CSD better supports the implementation of simplicial maps than ST. Construction of Relaxed Delaunay Complexes {#Delaunay} ========================================== Let $Q$ be a finite subset of a metric space $(P, \|\cdot\|)$ where $P$ is a discrete point-set. Given a relaxation parameter $\rho\ge 0$, we define the notion of being ‘witnessed’ as follows. A simplex $\sigma = \{q_0,\ldots , q_{d_\sigma}\}\subseteq Q$ belongs to the relaxed Delaunay complex[^5] $\text{Del}^\rho(Q, P)$ [@D08; @BDG15] if and only if there exists $x\in P$ such that for all $q_i\in \sigma$, and for all $q\in Q$ the following holds: $$\begin{aligned} \|x-q_i\| \leq \|x-q\| + \rho\end{aligned}$$ The parameter $\rho$ defines a filtration on the relaxed Delaunay complexes, which has been used in topological data analysis. More explicitly, the filtration value of a simplex $\sigma$ in $\text{Del}^\rho(Q, P)$ is the smallest $\rho^\prime\le \rho$, such that $\sigma$ is in $\text{Del}^{\rho^\prime}(Q, P)$. For this entire section, we assume that the filtration range is $\llbracket t\rrbracket$. We define a matrix $D$ of size $|P|\times |Q|$ as follows. For every $x\in P$ and $\ell\in [|Q|]$ let $D(x,\ell)$ denote the $\ell^{\text{th}}$ nearest neighbor of $x$ in $Q$ (ties are broken arbitrarily). For every $x\in P$, $i\in\llbracket t\rrbracket$, let $\ell_x^i$ be the largest integer such that $\left|\|x-D(x,1)\|-\|x-D(x,\ell_x^i)\|\right|\le \rho i/t$. Let $\sigma_x^i=\{D(x,1),D(x,2),\ldots ,D(x,\ell_x^i)\}$ and let $W=\left\{\sigma_x^i\bigl\vert x\in P, i\in \llbracket t \rrbracket\right\}$. We note below that $M$ (recall notation from Section \[sec:notations\]), the set of critical simplices in the complex is contained in $W$. \[DelaunayStrong\] $M\subseteq W.$ Let $\tau=\{v_0,\ldots v_{d_\tau}\}$ be a critical simplex in $M$ with filtration value $f(\tau)$. By definition of $\text{Del}^\rho(Q, P)$, we have that there exists a point $x\in P$ which $\left(f(\tau)\rho/t\right)$-strongly witnesses $\tau$. Since $\tau$ is critical, we have that for every $q\in Q\setminus \tau$, the following holds. $$\forall i\in\llbracket d_\tau\rrbracket,\ \left|\|x-v_i\|-\|x-q\|\right|> \rho f(\tau)/t.$$ Therefore, we have that for every $i\in [d_\tau+1]$, $A(x,\ell_x^i)\in \tau$, or more precisely, $\sigma_x^{d_\tau+1}=\tau$. The above lemma provides a characterization of $\text{Del}^\rho(Q, P)$: it can have at most $|P|(d+1)$ critical simplices. We note here that typically $P$ is a relatively small set. For example, in the experiments performed by Boissonnat and Maria (Table 1 of [@SimplexTree]), we note that the cardinality of the witness set is about a few ten thousands while the number of simplices in the complex is over a hundred million. Therefore, this provides practical evidence of the compact representation of $\text{Del}^\rho(Q, P)$ through CSD. Under the assumption that for any $x,\ell$, $D(x,\ell)$ could be computed in $\mathcal{O}(1)$ time (i.e., $D$ is computed as part of the preprocessing), Boissonnat and Maria [@SimplexTree] described an algorithm to construct the ST representation of the relaxed witness complex. Their algorithm can be easily adapted to construct $\text{Del}_w^\rho(Q, P)$ in time $\mathcal{O}(tmd\log n)$. In the case of CSD, we propose a new *matrix-parsing* algorithm which builds $\text{Del}^\rho(Q, P)$ in time $\mathcal{O}(|P|d^2\log \Psi)$ (assuming an oracle to access $D$). It is easy to see that all the simplices in $W$ can be constructed in $\mathcal{O}(|W|d\log n)=\mathcal{O}(|P|d^2\log n)$ time by sequentially computing the simplices $\sigma_x^i$ for all the $x\in P$, i.e., by parsing the matrix $D$ one row at a time. From the discussions about the robustness of CSD discussed in Section \[Performance\], we know that we could lazy insert all the simplices in $W$ to the CSD and it would behave exactly like in the scenario wherein only the simplices in $M$ (which is a subset of $W$) are inserted. This lazy insertion of all the simplices in $W$ can be done in time $\mathcal{O}(|P|d^2\log \Psi)$. After the construction, we may perform a clean-up operation to remove the redundant simplices that were inserted. Conclusion {#Conclusion} ========== In this paper, we introduce a new data structure called the Critical Simplex Diagram (CSD) to represent filtrations of simplicial complexes. In this data structure, we store only those simplices which are critical with respect to the filtration value, i.e., we store a simplex if and only if all its cofaces are of a (strictly) higher filtration value than the filtration value of the simplex itself. We then show how to efficiently perform basic operations on simplicial complexes by only storing these (critical) simplices. This is summarized in Table \[tab:OperationsonMSD\]. Finally, we showed how to (quickly) construct the CSD representation of flag complexes and relaxed Delaunay complexes. As a future direction of research, we would like to obtain better bounds on $\Psi$ and $\Gamma_i$ for specific complexes such as the Rips complex or the relaxed Delaunay complex by assuming some notion of geometric regularity. Also, it would be interesting to obtain lower bounds on the various query times (such as membership, insertion/removal), by assuming an optimal storage of $\mathcal{O}(\kappa d\log n)$ ($\kappa=|M|$ is the number of critical simplices). From the standpoint of practice, we would like to find fast construction algorithms under the CSD representation for other simplicial complexes of interest such as the alpha complex and the relaxed witness complex. Finally, we would like to implement this data structure and check its performance versus the Simplex Tree in practice. Acknowledgements {#acknowledgements .unnumbered} ---------------- We would like to thank the anonymous reviewers whose comments helped us improve the presentation of the paper. [^1]: This work was partially supported by the Advanced Grant of the European Research Council GUDHI (Geometric Understanding in Higher Dimensions). [^2]: This work was partially supported by Irit Dinur’s ERC-StG grant number 239985. [^3]: i.e., two complexes which either differ on the simplices contained on the complex or on the filtration value of a simplex contained in both complexes. [^4]: up to constant additive factors in the worst case. [^5]: This complex was referred to as the $\rho$-relaxed strong Delaunay complex in [@D08] and as the $\rho$-relaxed Delaunay complex in [@BDG15].

--- abstract: 'We study the Dirichlet series $F_b(s)=\sum_{n=1}^\infty d_b(n)n^{-s}$, where $d_b(n)$ is the sum of the base-$b$ digits of the integer $n$, and $G_b(s)=\sum_{n=1}^\infty S_b(n)n^{-s}$, where $S_b(n)=\sum_{m=1}^{n-1}d_b(m)$ is the summatory function of $d_b(n)$. We show that $F_b(s)$ and $G_b(s)$ have continuations to the plane ${\mathbb{C}}$ as meromorphic functions of order at least 2, determine the locations of all poles, and give explicit formulas for the residues at the poles. We give a continuous interpolation of the sum-of-digits functions $d_b$ and $S_b$ to non-integer bases using a formula of Delange, and show that the associated Dirichlet series have a meromorphic continuation at least one unit left of their abscissa of absolute convergence.' author: - Corey Everlove bibliography: - 'sumofdigitsbibliography.bib' title: 'Dirichlet series associated to sum-of-digits functions' --- [^1] Introduction ============ There has been a great deal of study of properties of the radix expansions to an integer base $b\geq 2$ of integers $n$. For each integer base $b\geq 2$, every positive integer $n$ has a unique base-$b$ expansion $$n = \sum_{i \geq 0} \delta_{b,i}(n)b^i$$ with digits $\delta_{b,i}\in\{0,1,\dotsc,b-1\}$ given by $$\delta_{b,i} = \Bigl\lfloor \frac{n}{b^i} \Bigr\rfloor - b \Bigl\lfloor \frac{n}{b^{i+1}} \Bigr\rfloor.$$ This paper considers two summatory functions of base $b$ digits of $n$: 1. The *base-$b$ sum-of-digits function* $d_b(n)$ is $$d_b(n) = \sum_{i\geq 0} \delta_{b,i}(n).$$ 2. The *(base $b$) cumulative sum-of-digits function* $S_b(n)$ is $$S_b(n) = \sum_{m=1}^{n-1} d_b(m).$$ We follow here the convention of previous authors (including [@delange-75] and [@flajolet-94]), with the sum defining $S_b(n)$ running to $n-1$ instead of $n$. We associate to the functions $d_b(n)$ and $S_b(n)$ the Dirichlet series generating functions $$F_b(s) = \sum_{n=1}^\infty \frac{d_b(n)}{n^s}$$ and $$G_b(s) = \sum_{n=1}^\infty \frac{S_b(n)}{n^s}.$$ These Dirichlet series have abscissa of convergence $\operatorname{Re}(s)=1$ and $\operatorname{Re}(s)=2$, respectively. This paper studies the problem of the meromorphic continuation to ${\mathbb{C}}$ of Dirichlet series associated to the base-$b$ digit sums $d_b(n)$ and $S_b(n)$. Here we obtain the meromorphic continuation and determine its exact pole and residue structure. The pole structure contains half of a two-dimensional lattice and the residues involve Bernoulli numbers and values of the Riemann zeta function on the line $\operatorname{Re}(s)=0$. A meromorphic continuation of these functions was previously obtained in the thesis of Dumas [@dumas-thesis] by a different method, which specified a half-lattice containing all the poles but did not determine the residues; in fact infinitely many of the residues on his possible pole set vanish. The asymptotics of $S_b(n)$ have been extensively studied, see Section \[sec-previous-work\]. We mention particularly work of Delange [@delange-75], given below as Theorem \[thm-delange\], which gives an exact formula for $S_b(n)$ in terms of a continuous nondifferentiable function with Fourier coefficients involving values of the Riemann zeta function on the imaginary axis. Using an interpolation of Delange’s formula we formulate a continuous interpolation of $S_b(n)$ in the base parameter $b$, permitting definitions of $d_\beta(n)$ and $S_\beta(n)$ for a real parameter $\beta>1$. We obtain a meromorphic continuation of the associated Dirichlet series $F_{\beta}(s)$ and $G_{\beta}(s)$ to the half-planes $\operatorname{Re}(s)>0$ and $\operatorname{Re}(s)>1$, respectively. We note apparent fractal properties of $d_{\beta}(n)$ as $\beta$ is varied. Results ------- Our first results concern the meromorphic continuation of the functions $F_b(s)$ and $G_b(s)$ to the entire complex plane ${\mathbb{C}}$. \[thm-db\] For each integer base $b\geq 2$, the function $F_b(s) = \sum_{n=1}^\infty d_b(n)n^{-s}$ has a meromorphic continuation to ${\mathbb{C}}$. The poles of $F_b(s)$ consist of a double pole at $s=1$ with Laurent expansion beginning $$F_b(s)=\frac{b-1}{2\log b}(s-1)^{-2} + \biggl( \frac{b-1}{2\log b} \log(2\pi) - \frac{b+1}{4}\biggr)(s-1)^{-1} + O(1),$$ simple poles at each other point $s=1+2\pi i m / \log b$ with $m\in {\mathbb{Z}}$ ($m\neq 0$) with residue $$\operatorname*{Res}\biggl( F_b(s) , s=1+\frac{2\pi i m}{\log b} \biggr) = - \frac{b-1}{2\pi i m} \zeta\biggl(\frac{2\pi i m}{\log b}\biggr),$$ and simple poles at each point $s=1-k+2\pi i m /\log b$ with $k=1$ or $k\geq 2$ an even integer and with $m\in{\mathbb{Z}}$, with residue $$\operatorname*{Res}\biggl( F_b(s) , s=1-k+\frac{2\pi i m}{\log b} \biggr) = (-1)^{k+1}\frac{b-1}{\log b}\zeta\biggl(\frac{2\pi i m}{\log b}\biggr)\frac{B_k}{k!} \prod_{j=1}^{k-1} \biggl(\frac{2\pi i m}{\log b} - j\biggr)$$ where $B_k$ is the $k$th Bernoulli number. Theorem \[thm-db\] is proved by first considering the Dirichlet series $\sum \bigl(d_b(n)-d_b(n-1)\bigr) n^{-s}$ and then exploiting a relation between power series and Dirichlet series to recover $F_b(s)$. The proof is presented in Section \[sec-mero-cont\]. The meromorphic continuation of Dirichlet series attached to $b$-regular sequences, of which our Dirichlet series $F_b(s)$ is a particular example, was studied by Dumas in his 1993 thesis [@dumas-thesis]; this work also showed that the poles of $F_b(s)$ must be contained in a certain half-lattice, strictly larger than the half-lattice here. A similar method allows us to meromorphically continue the series $G_b(s)$ to the complex plane. \[thm-sb\] For each integer $b\geq 2$, the function $G_b(s)=\sum_{n=1}^\infty S_b(n)n^{-s}$ has a meromorphic continuation to ${\mathbb{C}}$. The poles of $G_b(s)$ consist of a double pole at $s=2$ with Laurent expansion $$G_b(s) = \frac{b-1}{2\log b}(s-2)^{-2} + \biggl(\frac{b-1}{2\log b}\bigl(\log(2\pi)-1\bigr)-\frac{b+1}{4}\biggr)(s-2)^{-1} + O(1),$$ a simple pole at $s=1$ with residue $$\operatorname*{Res}(G_b(s),s=1) = \frac{b+1}{12},$$ simple poles at $s=2 + 2\pi i m / \log b$ with $m\in{\mathbb{Z}}$ ($m\neq 0$) with residue $$\operatorname*{Res}\biggl( G_b(s) , s= 2 + \frac{2\pi i m}{\log b} \biggr) = - \frac{b-1}{2\pi i m}\biggl(1+\frac{2\pi i m}{\log b}\biggr)^{-1} \zeta\biggl(\frac{2\pi i m}{\log b}\biggr)$$ and simple poles at point $s=2-k + 2\pi i m / \log b$ with $k\geq 2$ an even integer and $m\in{\mathbb{Z}}$ with residue $$\operatorname*{Res}\biggl( G_b(s) , s= 2 - k + \frac{2\pi i m}{\log b} \biggr) = \frac{b-1}{\log b} \zeta\biggl(\frac{2\pi i m}{\log b}\biggr) \biggl(\frac{B_k}{k(k-2)!} \biggr) \prod_{j=1}^{k-2}\biggl(\frac{2\pi i m}{\log b} - j\biggr).$$ An interesting feature of the above theorems is the abundance of poles. Since each function $F_b(s)$ and $G_b(s)$ has $\asymp r^{2}$ poles in the disk $\lvert s \rvert<r$, we have the following corollary, which we discuss further in Section \[sec-meromorphic-functions\]. The functions $F_b(s)$ and $G_b(s)$ are meromorphic functions of order at least $2$ on ${\mathbb{C}}$. The Riemann zeta function, the Dirichlet L-functions, and the Dirichlet series generating functions of many important arithmetic functions (such as the Möbius function $\mu(n)$, the von Mangoldt function $\Lambda(n)$, the Euler totient function $\phi(n)$, and the sum-of-diviors functions $\sigma_\alpha(n)$) analytically continue as meromorphic functions of order $1$ on the complex plane. The Dirichlet series $F_b(s)$ and $G_b(s)$ thus have a different analytic character than many other Dirichlet series considered in number theory. In Section \[sec-non-int\], we use a formula of Delange [@delange-75] for $S_b(n)$ to define continuous real-valued interpolations of the functions $d_b(n)$ and $S_b(n)$ from integer bases $b\geq 2$ to a real parameter $\beta>1$. As before, we associate to these interpolated sum-of-digits functions the Dirichlet series $$F_\beta(s) = \sum_{n=1}^\infty \frac{d_\beta(n)}{n^s}$$ and $$G_\beta(s) = \sum_{n=1}^\infty \frac{S_\beta(n)}{n^s}.$$ We prove that these Dirichlet series each have a meromorphic continuation one unit to the left of their halfplane of absolute convergence. For the function $F_\beta(s)$ we have the following theorem. For each real $\beta>1$, the function $F_\beta(s)$ has a meromorphic continuation to the halfplane $\operatorname{Re}(s)>0$, with a double pole at $s=1$ with Laurent expansion $$F_\beta(s) = \frac{\beta - 1}{2\log \beta}(s-1)^{-2} + \biggl(\frac{\beta-1}{2\log \beta} \bigl(\log(2\pi)\bigr) - \frac{\beta+1}{4}\biggr) (s-1)^{-1} +O(1)$$ and simple poles at $s=1+2\pi i m / \log \beta$ for $m\in{\mathbb{Z}}$ with $m\neq 0$ with residue $$\operatorname*{Res}\biggl(F_\beta(s),s=1+\frac{2\pi i m}{\log \beta}\biggr) = -\frac{\beta-1}{2\pi i m}\zeta\biggl(\frac{2\pi i m}{\log \beta} \biggr).$$ For the function $G_\beta(s)$ we have the following theorem. For each real $\beta>1$, the function $G_\beta(s)$ is meromorphic in the region $\operatorname{Re}(s)>1$ with a double pole at $s=2$ with Laurent expansion $$G_\beta(s) = \frac{\beta-1}{2\log \beta}(s-2)^{-2} + \biggl(\frac{\beta-1}{2\log \beta} \bigl(\log(2\pi) - 1\bigr) - \frac{\beta+1}{4}\biggr)(s-2)^{-1} +O(1)$$ and simple poles at $s=2+2\pi i m / \log \beta$ for $m\in{\mathbb{Z}}$ with $m\neq 0$ with residue $$\operatorname*{Res}\biggl(G_b(s),s=2+\frac{2\pi i m}{\log \beta}\biggr) = -\frac{\beta-1}{2\pi i m}\biggl(1+\frac{2\pi i m}{\log \beta}\biggr)^{-1}\zeta\biggl(\frac{2\pi i m}{\log \beta} \biggr).$$ To prove these theorems, we start by obtaining the continuation of the series $G_\beta(s)$ by working directly with its Dirichlet series and then obtain the continuation of $F_\beta(s)$ by studying the relation between these two Dirichlet series. Previous work {#sec-previous-work} ------------- There has been much previous work studying the functions $d_b(n)$ and $S_b(n)$. The function $d_b(n)$ exhibits significant fluctuations as $n$ changes to $n+1$. It can only increase slowly, having $d_b(n+1) \leq d_b(n)+1$ but it can decrease by an arbitrarily large amount. The sequence $d_b(n)$ is a $b$-regular sequence in the sense of Allouche and Shallit [@allouche-shallit-92 Ex. 2, Sec. 7] and is a member of the $b$-th arithmetic fractal group $\Gamma_b({\mathbb{Z}})$ of Morton and Mourant [@morton-mourant-89 p. 256]. Chen et al. [@chen-hwang-zacharovas-14] survey results on the sum-of-digits function of random integers, and give many references. Concerning the cumulative sum-of-digits function, Mirsky [@mirsky-49] proved in 1949 that for any integer base $b\geq 2$, the function $S_b(n)$ has the asymptotic $$S_b(n) = \frac{b-1}{2\log b} n \log n + O(n).$$ In 1968 Trollope [@trollope-68] expressed the error term for the base-$2$ cumulative digit sum $S_2(n)$ in terms of a continuous everywhere nondifferentiable function, the Takagi function—see [@lagarias-12] for a survey of the properties of this function. In 1975 Delange [@delange-75] proved the following formula for $S_b(n)$, expressing the error term as a Fourier series with coefficients involving values of the Riemann zeta function on the imaginary axis. \[thm-delange\] The cumulative sum-of-digits function $S_b(n)$ satisfies $$\label{eq-delange-formula} S_b(n) = \frac{b-1}{2\log b} n \log n + h_b\biggl(\frac{\log n}{\log b} \biggr) n$$ where $h_b$ is a nowhere-differentiable function of period $1$. The function $h_b$ has a Fourier series $$h_b(x) = \sum_{k=-\infty}^\infty c_b(k)e^{2\pi i k x}$$ with coefficients $$c_b(k) = -\frac{b-1}{2\pi i k}\biggl(1+\frac{2\pi i k}{\log b}\biggr)^{-1}\zeta\biggl(\frac{2\pi i k}{\log b} \biggr)$$ for $k\neq 0$ and $$c_b(0) = \frac{b-1}{2\log b}\bigl(\log(2\pi) - 1\bigr) - \frac{b+1}{4}.$$ A complex-analytic proof of a summation formula for general $q$-additive functions, of which the base-$q$ sum-of-digits function is an example, was given by Mauclaire and Murata in 1983 [@mauclaire-murata-83a; @mauclaire-murata-83b; @murata-mauclaire-88]. A shorter complex-analytic proof of in the specific case of $S_2(n)$ was given by Flajolet, Grabner, Kirschenhofer, Prodinger, and Tichy in 1994 [@flajolet-94]. The method of Flajolet et al. is based on applying a variant of Perron’s formula to the Dirichlet series $$\sum_{n=1}^\infty \bigl(d_2(n)-d_2(n-1)\bigr) n^{-s}.$$ Grabner and Hwang [@grabner-hwang-05] study higher moments of the sum-of-digits function by similar complex-analytic methods. Our formulas for the residues of $F_b(s)$ and $G_b(s)$ involve the Bernoulli numbers. Kellner [@kellner-17] and Kellner and Sondow [@kellner-sondow-17] investigate another relation between sums of digits and Bernoulli numbers, proving that the least common multiple of the denominators of the coefficients of the polynomial $\sum_{i=0}^n n^k$, which can be written in terms of a Bernoulli polynomial, can be expressed as a certain product of primes satisfying $d_p(n+1)\geq p$. Sum-of-digits Dirichlet series ============================== First we consider basic properties of the Dirichlet series $$F_b(s) = \sum_{n=1}^\infty \frac{d_b(n)}{n^s}$$ attached to the base-$b$ digit sum of $n$ and the Dirichlet series $$G_b(s) = \sum_{n=1}^\infty \frac{S_b(n)}{n^s}$$ attached to the cumulative base-$b$ digit sum. For standard references on the basic analytic properties of Dirichlet series, see the books of Hardy and Riesz [@hardy-riesz] or Titchmarsh [@titchmarsh-tof Ch. IX]. Recall that each ordinary Dirichlet series $\sum a_n n^{-s}$ has an abscissa of conditional convergence $\sigma_c$ such that the Dirichlet series converges and defines a holomorphic function if $\operatorname{Re}(s)>\sigma_c$ and diverges if $\operatorname{Re}(s)<\sigma_c$. Each Dirichlet series also has an abscissa of absolute convergence $\sigma_a$ such that the Dirichlet series converges absolutely if $\operatorname{Re}(s)>\sigma_a$ and does not converge absolutely if $\operatorname{Re}(s)<\sigma_a$. For ordinary Dirichlet series, one always has $\sigma_a-1\leq \sigma_c \leq \sigma_a$, and $\sigma_c=\sigma_a$ if the coefficients $a_n$ are nonnegative reals. For each integer $b\geq 2$, the Dirichlet series $$\label{eq-fb-def-2} F_b(s) = \sum_{n=1}^\infty \frac{d_b(n)}{n^s}$$ converges and defines a holomorphic function for $\operatorname{Re}(s)>1$. A positive integer $n$ has $[\log n / \log b]+1$ digits when written in base $b$, each of which is at most $b-1$, so $$\label{eq-db-estimate} d_b(n) \leq (b-1) \biggl(\biggl\lfloor\frac{\log n}{\log b} \biggr\rfloor+1\biggr).$$ We then obtain the estimate $$\label{eq-sb-estimate} S_b(n) \ll n \log n$$ with an implied constant depending on $b$. This implies that the Dirichlet series has abscissa of absolute convergence at most 1 and therefore defines a holomorphic function for $\operatorname{Re}(s)>1$. For each integer $b\geq 2$, the Dirichlet series $$\label{eq-gb-def-2} G_b(s) = \sum_{n=1}^\infty \frac{S_b(n)}{n^s}$$ converges and defines a holomorphic function for $\operatorname{Re}(s)>2$. The estimate gives $$\sum_{m=1}^n S_b(m) \ll n^2 \log n,$$ which shows that the Dirichlet series converges for $\operatorname{Re}(s)>2$. It follows from Delange’s formula that $F_b(s)$ and $G_b(s)$ have abscissa of absolute convergence $\operatorname{Re}(s)=1$ and $\operatorname{Re}(s)=2$, respectively, and this can be proven directly using a more careful estimate of the function $d_b(n)$. We can also obtain the exact values of the abscissas of convergence as a corollary of our theorems on the meromorphic continuation of $F_b(s)$ and $G_b(s)$, since $F_b(s)$ has a pole at $s=1$ and $G_b(s)$ has a pole at $s=2$. As in previous work on Dirichlet series associated to $q$-additive sequences, it is advantageous to consider the Dirichlet series $$Z_b(s) =\sum_{n=1}^\infty \bigl(d_b(n)-d_b(n-1)\bigr) n^{-s}$$ obtained by differencing the coefficients of the series $F_b(s)$, setting $d_b(0)=0$. Identity in the following proposition appears in a more general form (for $q$-additive functions) in the work of Mauclaire and Murata [@mauclaire-murata-83a; @mauclaire-murata-83b; @murata-mauclaire-88] and is stated and proved explicitly for the sum-of-digits series by Allouche and Shallit [@allouche-shallit-90]. We give a more direct proof of this result. \[prop-differenced-zeta\] For each integer $b\geq 2$, the Dirichlet series $Z_b(s)$ has abscissa of absolute convergence $\sigma_a=1$, abscissa of conditional convergence $\sigma_c=0$, and has a meromorphic continuation to ${\mathbb{C}}$, satisfying $$\label{eq-zb-formula} Z_b(s) = \frac{b^s-b}{b^s-1}\zeta(s).$$ For bases $b\geq 3$, we have $\lvert d_b(n)-d_b(n-1) \rvert \geq 1$ for all $n$; if $b=2$, we have $\lvert d_b(n)-d_b(n-1)\rvert \geq 1$ for at least all odd $n$. Hence $\sigma_a\geq 1$. We also have $d_b(n)-d_b(n-1)\ll \log n$, so $\sigma_a\leq 1$. The abscissa of conditional convergence $\sigma_c=0$ follows from the bound $$\sum_{m\leq n} \bigl(d_b(m)-d_b(m-1)\bigr) = d_b(n) \ll \log n.$$ The effect of adding 1 on the digit sum in base-$b$ arithmetic depends on the divisibility of $n$ by $b$; in particular, we have $$d_b(n) - d_b(n-1) = 1 - k(b-1)$$ where $k$ is the largest integer such that $b^k \mid n$. We may also express this as $$d_b(n) - d_b(n-1) = \sum_{m\mid n} \alpha(m) \beta(m/n)$$ where $$\alpha(n) = \begin{cases} 1 &\text{if $n=b^k$ for some $k$}\\ 0 &\text{otherwise} \end{cases} \qquad \beta(n)= \begin{cases} 1-b &\text{if $b\mid n$} \\ 1 &\text{if $b \nmid n$} \end{cases}.$$ Then we have, for $\operatorname{Re}(s)>1$, $$Z_b(s) = \sum_{n=1}^\infty \bigl( d_b(n) - d_b(n-1) \bigr) n^{-s} = \sum_{n=1}^\infty \Bigl( \sum_{m\mid n} \alpha(m) \beta(m/n) \Bigr) n^{-s}$$ Writing the right side as a product of two Dirichlet series, we have $$Z_b(s) = \sum_{n=1}^\infty \alpha(n) n^{-s} \sum_{n=1}^\infty \beta(n) n^{-s} = \sum_{n=1}^\infty b^{-ns} \sum_{n=1}^\infty (1-b) (bn)^{-s}.$$ Summing the geometric series, we obtain $$Z_b(s) = \frac{1}{1-b^{-s}} \bigl( \zeta(s) - bb^{-s} \zeta(s)\bigr) = \frac{b^s-b}{b^s-1}\zeta(s)$$ as claimed. Equation then provides a meromorphic continuation of $Z_b(s)$ since the right side is meromorphic on ${\mathbb{C}}$. We will obtain information about the mermorphic continuation of $F_b(s)$ and $G_b(s)$ by considering the relation between these series and the series $Z_b(s)$. For future use, we list the poles of the function $Z_b(s)$. \[lem-poles-of-ndb-ds\] The function $Z_b(s)$ is meromorphic on ${\mathbb{C}}$, with simple poles at $s=2\pi i m / \log b$ for $m\in{\mathbb{Z}}$. The residue at each pole is $$\operatorname*{Res}\biggl(Z_b(s),s=\frac{2\pi i m}{\log b}\biggr) = - \frac{b-1}{\log b} \zeta\biggl(\frac{2\pi i m}{\log b}\biggr).$$ In particular, at $s=0$, the function $Z_b(s)$ has a Laurent expansion beginning $$Z_b(s) = \biggl(\frac{b-1}{2\log b}\biggr)s^{-1} + \biggl(- \frac{b+1}{4} + \frac{b-1}{2\log b} \log(2\pi) \biggr) + O(s).$$ The function $(b^s-b)/(b^s-1)$ has simple poles at $s=2\pi i m /\log b$ for each $m\in{\mathbb{Z}}$, with residue $$\operatorname*{Res}\biggl(\frac{b^s-b}{b^s-1},s=\frac{2\pi i m}{\log b}\biggr) = -\frac{b-1}{\log b}.$$ The Laurent expansion at $s=0$ follows from multiplying the expansions $$\frac{b^s-b}{b^s-1} = -\frac{b-1}{\log b}s^{-1} + \frac{b+1}{2} + O(s)$$ and $$\zeta(s) = -\frac{1}{2} -\frac{1}{2}\log(2\pi)s + O(s^2).$$ The function $\zeta(s)$ has only a simple pole at $s=1$, cancelled by a zero of $(b^s-b)$. Meromorphic continuation of $F_b(s)$ and $G_b(s)$ {#sec-mero-cont} ================================================= In this section, we show that for integers $b\geq 2$, the Dirichlet series $F_b(s)$ and $G_b(s)$ have a meromorphic continuation to ${\mathbb{C}}$ and determine the structure of the poles, proving Theorems \[thm-db\] and \[thm-sb\]. Bernoulli numbers ----------------- Our formulas for the meromorphic continuation of $F_b(s)$ and $G_b(s)$ involve Bernoulli numbers. For standard facts about the Bernoulli numbers and their basic properties, see [@abramowitz-stegun Ch.  23]. For a thorough reference on Bernoulli numbers, their history, and their relation to zeta functions, see [@arakawa-et-al]. The Bernoulli numbers $B_k$ are the sequence of rational numbers defined by the generating function $$\label{eq-bernoulli-gf} \frac{x}{e^x-1} = \sum_{k=0}^\infty \frac{B_k}{k!}x^k.$$ If $x$ is a complex variable, this series converges for $\lvert x \rvert < 2\pi$. Note that there are several competing notations for the Bernoulli numbers; with our definition, we have $B_0=1$, $B_1=-1/2$, and $B_2=1/6$. Because the function $$\frac{x}{e^x-1} + \frac{1}{2}x$$ is an even function, we find that $B_{2k+1}=0$ for all $k\geq 1$. Power series and Dirichlet series --------------------------------- To prove the meromorphic continuation of $F_b(s)$ and $G_b(s)$, we make use of the following classical relation between Dirichlet series and power series. \[prop-ps-ds-relation\] Let $\sigma_c$ be the abscissa of conditional convergence of the Dirichlet series $\sum_{n=1}^\infty a_n n^{-s}$. Then $$\Gamma(s) \sum_{n=1}^\infty a_n n^{-s} = \int_0^\infty \Bigl( \sum_{n=1}^\infty a_n e^{-nx} \Bigr) x^{s-1} \, dx$$ for $\operatorname{Re}(s) >\max(\sigma_c,0)$. See [@montgomery-vaughan eq. 5.23]. Proposition \[prop-ps-ds-relation\] allows us to translate the additive relations between the arithmetic functions $d_b(n)-d_b(n-1)$, $d_b(n)$, and $S_b(n)$, which are easily expressed in terms of power series generating functions, into relations between their associated Dirichlet series. Meromorphic continuation of $F_b(s)$ ------------------------------------ We now prove the meromorphic continuation of the Dirichlet series $F_b(s)$ by combining the relation between the Dirichlet series and power series generating functions of $d_b(n)$ with the relation between the power series generating functions of $d_b(n)$ and $d_b(n)-d_b(n-1)$. Let $$\label{eq-px-def} p(x) = \sum_{n=1}^\infty \bigl(d_b(n)-d_b(n-1)\bigr) x^n.$$ We note that $$\sum_{n=1}^\infty d_b(n)x^n = \frac{p(x)}{1-x}.$$ Then by Proposition \[prop-ps-ds-relation\], we have $$\Gamma(s) F_b(s) = \int_0^\infty \frac{1}{1-e^{-x}} p(e^{-x})x^{s-1}\, dx$$ for $\operatorname{Re}(s)>1$. The series expansion $$\frac{x}{1-e^{-x}} = \sum_{k=0}^\infty \frac{(-1)^kB_k}{k!} x^k,$$ which follows from , holds for $\lvert x \rvert<2\pi$. Since $$\Gamma(s)Z_b(s) = \int_0^\infty p(e^{-x})x^{s-1}\, dx,$$ for $\operatorname{Re}(s)>0$, we can write $$\label{eq-first-fb-expansion} F_b(s) = \sum_{k=0}^K \frac{(-1)^kB_k}{k!} \frac{\Gamma(s-1+k)}{\Gamma(s)} Z_b(s-1+k) + R_K(s)$$ with $$\label{eq-RK-def} R_K(s) = \frac{1}{\Gamma(s)}\int_0^\infty \Bigl(\frac{x}{1-e^{-x}} - \sum_{k=0}^K \frac{(-1)^k B_k}{k!} x^k \Bigr) p(e^{-x})x^{s-2} \, dx.$$ Note that $$\frac{\Gamma(s-1)}{\Gamma(s)} = \frac{1}{s-1}$$ and $$\frac{\Gamma(s-1+k)}{\Gamma(s)} = (s)(s+1)\dotsm (s+k-2).$$ Since $$\frac{x}{1-e^{-x}} - \sum_{k=0}^K \frac{(-1)^k B_k}{k!} x^k \ll x^{K+1}$$ as $x\rightarrow 0^+$, the integral in converges and defines a holomorphic function in the region $\operatorname{Re}(s)>1-K$. From Lemma \[lem-poles-of-ndb-ds\] we know that $Z_b(s)$ has simple poles at $s=2\pi ik/\log b$ for $k\in{\mathbb{Z}}$. The $k=0$ term $$\frac{1}{s-1}Z_b(s-1)$$ has a double pole at $s=1$ with Laurent expansion beginning $$\frac{1}{s-1}Z_b(s-1) = \frac{b-1}{2\log b}(s-1)^{-2} + \biggl( \frac{b-1}{2\log b} \log(2\pi) - \frac{b+1}{4}\biggr)(s-1)^{-1} + O(1),$$ and simple poles at each other point $s=1+2\pi i m /\log b$. Each term $$(-1)^k\frac{B_k}{k!}\prod_{j=0}^{k-2}(s+j) \cdot Z_b(s-1+k)$$ with $k=1$ or with $k$ an even integer with $k\geq 2$ has a simple pole at $s=1-k+2\pi i m/\log b$ for $m\in{\mathbb{Z}}$ with residue $$(-1)^{k}\frac{B_k}{k!} \prod_{j=1}^{k-1} \biggl(\frac{2\pi i m}{\log b} - j\biggr) \cdot \biggl(-\frac{b-1}{\log b}\biggr)\zeta\biggl(\frac{2\pi i m}{\log b}\biggr).$$ Since $K$ can be taken arbitrarily large, this proves the theorem. Meromorphic continuation of $G_b(s)$ ------------------------------------ We continue the function $G_b(s)$ to the plane in a similar fashion, using the fact that $S_b(n)$ is a double sum of the differences $d_b(n)-d_b(n-1)$ appearing in the series $Z_b(n)$. Define $p(x)$ by . We make use of the identity of power series $$\frac{x}{(1-x)^2} p(x) = \sum_{n=1}^\infty S_b(n)x^n.$$ Then by Proposition \[prop-ps-ds-relation\], we have $$\Gamma(s)G_b(s) = \int_0^\infty \frac{e^{-x}}{(1-e^{-x})^2} p(e^{-x}) x^{s-1} \, dx$$ for $\operatorname{Re}(s)>2$. From and noting that $$\frac{e^{-x}}{(1-e^{-x})^2}=-\frac{d}{dx}\biggl(\frac{1}{e^x-1}\biggr),$$ we have the power series expansion $$\frac{x^2e^x}{(e^x-1)^2} = 1-\sum_{k=2}^\infty \frac{B_k}{k(k-2)!}x^k.$$ Then for a fixed integer $K\geq 2$ can write $G_b(s)$ as $$\label{eq-gbs-expansion} G_b(s) = \frac{\Gamma(s-2)}{\Gamma(s)}Z_b(s-2) - \sum_{k=2}^K \frac{B_k}{k(k-2)!} \frac{\Gamma(s-2+k)}{\Gamma(s)} Z_b(s-2+k) + R_K(s)$$ with remainder $R_K$ given by $$R_K(s) = \frac{1}{\Gamma(s)}\int_0^\infty \biggl( \frac{x^2e^{-x}}{(1-e^{-x})^2} - 1 + \sum_{k=2}^K \frac{B_k}{k(k-2)!} x^k \biggr) p(e^{-x}) x^{s-3} \, dx.$$ The function $R_K(s)$ is holomorphic for $\operatorname{Re}(s)>2-K$. As before, we consider the poles of each term of . The first term $$\frac{1}{(s-1)(s-2)} Z_b(s-2)$$ has a double pole at $s=2$ with Laurent expansion $$\frac{b-1}{2\log b}(s-2)^{-2} + \biggl(\frac{b-1}{2\log b}\bigl(\log(2\pi)-1\bigr)-\frac{b+1}{4}\biggr)(s-2)^{-1}+ \dotsb,$$ a simple pole at each point $s=2+2\pi i m / \log b$ with $m\neq 0$, and a simple pole at $s=1$ with residue $(b+1)/12$. Each other term $$\frac{B_k}{k(k-2)!} \prod_{j=0}^{k-3} (s+j) \cdot Z_b(s-2+k).$$ has simple poles at $s=2-k+2\pi i m / \log b$ for all $m$. Instead of relating $G_b(s)$ to the series $Z_b(s)$ as we did in this proof, we could have also used the relation $$\Gamma(s) G_b(s) = \int_0^\infty \frac{e^{-x}}{1-e^{-x}} \biggl(\sum_{n=1}^\infty d_b(n) e^{-xn}\biggr) x^{s-1} \, dx,$$ following the proof of Theorem \[thm-db\] to write $G_b(s)$ in terms of $F_b(s)$. Order of $F_b(s)$ and $G_b(s)$ as meromorphic functions {#sec-meromorphic-functions} ------------------------------------------------------- The functions $F_b(s)$ and $G_b(s)$ are meromorphic functions on the complex plane with infinitely many poles on a left half-lattice. We now raise a further question about the analytic properties of these functions. Recall that the *order* of an entire function $f(z)$ on ${\mathbb{C}}$ is $$\rho = \inf \bigl\{ \rho\geq 0 \, \bigm\vert \, f(z) = O\bigl(\exp(\lvert z \rvert^{\rho+\varepsilon})\bigr)\text{ as $\lvert z \rvert\rightarrow\infty$} \bigr\}.$$ An entire function is of *finite order* if $\rho<\infty$. The order of a meromorphic function is defined as the order of growth of its associated Nevanlinna characteristic function. This definition is equivalent (by [@rubel] Lemma 15.6 and Theorem on p. 91) to the following. The *order* of a meromorphic function $f(z)$ is the infimum of $\rho\geq 0$ such that $f$ can be written as $f(z) = g(z)/h(z)$ for entire functions $g(z)$ and $h(z)$ of order $\rho$. Many of the common Dirichlet series of analytic number theory are meromorphic (or entire) functions of order $1$. Examples include the Riemann zeta function $\zeta(s)$, the Dirichlet $L$-functions $L(s,\chi)$, and their relatives; more generally, all Dirichlet series in the Selberg class (as introduced in Selberg [@selberg-92]) are meromorphic functions of order 1. By Proposition \[prop-differenced-zeta\], the function $Z_b(s)$ is meromorphic of order $1$. The functions $F_b(s)$ and $G_b(s)$, however, must have greater order by the following fact. Let $f(z)$ be a meromorphic function of order $\rho$ and let $n(r,a)$ be the number of zeros of $f(z)-a$ in the disc $\lvert z \rvert<r$. Then $n(r,a)=O(r^{\rho+\varepsilon})$. By Theorems \[thm-db\] and \[thm-sb\], the functions $F_b(s)$ and $G_b(s)$ each have $\gg r^2$ poles in the disc $\lvert z \rvert<r$. Hence we have the following corollary. The functions $F_b(s)$ and $G_b(s)$ are meromorphic functions of order at least $2$. A meromorphic function of order greater than 2 could still have only $O(r^2)$ poles in $\lvert z \rvert<r$, so without further information, we cannot deduce that $F_b(s)$ and $G_b(s)$ have order 2. Are the functions $F_b(s)$ and $G_b(s)$ mermorphic functions of order exactly $2$? Such a question has been answered positively in the related setting of strongly $q$-multiplicative functions: the Dirichlet series attached to such functions are entire of order exactly $2$ (see Alkauskas [@alkauskas-04]). Meromorphic continuation of Dirichlet series for non-integer bases {#sec-non-int} ================================================================== In this section, we consider the problem of extending the digit sums $d_b(n)$ and $S_b(n)$ from integer bases $b$ to real parameters $\beta>1$. There are a number of possible ways to do this. One natural approach concerns the notion of $\beta$-expansions introduced by Renyi [@renyi-57] and studied at length by Parry [@parry-60]. However, for non-integer values of $\beta$, the $\beta$-expansion of an integer generally has infinitely many digits, so the sum of the digits will generally be infinite. Digit sums related to a different digit expansion with respect to an irrational base were considered by Grabner and Tichy [@grabner-tichy-91]. The approach which we consider in this section is to use the formula of Delange to define the cumulative digit sum $S_\beta(n)$ for real parameters $\beta>1$, from which we can define a digit sum $d_\beta(n)$ by differencing. The resulting functions are continuous in the $\beta$-parameter. Extension to non-integer bases by Delange’s formula --------------------------------------------------- We begin by replacing the integer variable $b$ in Theorem \[thm-delange\], which gives a formula for $S_b(n)$ for integer bases $b\geq 2$, by a real parameter $\beta>1$. \[def-non-integer\] For $\beta\in{\mathbb{R}}$ with $\beta>1$, define a generalized cumulative sum-of-digits function $S_\beta(n)$ by $$\label{eq-sbeta-delange-def} S_\beta(n) := \frac{\beta-1}{2\log \beta} n \log n + h_\beta\biggl(\frac{\log n}{\log \beta} \biggr) n,$$ where the function $h_\beta(x)$ is defined by the Fourier series $$\label{eq-hbeta-fourier-series} h_\beta(x) = \sum_{k=-\infty}^\infty c_\beta(k)e^{2\pi i k x}$$ with coefficients $$c_\beta(k) = -\frac{\beta-1}{2\pi i k}\biggl(1+\frac{2\pi i k}{\log \beta}\biggr)^{-1}\zeta\biggl(\frac{2\pi i k}{\log \beta} \biggr)$$ for $k\neq 0$ and $$c_\beta(0) = \frac{\beta-1}{2\log \beta}(\log 2\pi - 1) - \frac{\beta+1}{4}.$$ Define the generalized sum-of-digits function $d_\beta(n)$ for real $\beta>1$ by $$d_\beta(n) := S_\beta(n+1) - S_\beta(n).$$ A plot of $S_\beta(10)$ as a function of $\beta$ for $1\leq\beta\leq 15$ is shown in Figure \[Sb10-plot\]. Note that $S_\beta(n)$ is approximately constant for $\beta\geq 10$. \[Sb10-plot\]  The function $h_\beta(x)$ ------------------------- In this section, we study properties of the function $h_\beta(x)$ appearing in Definition \[def-non-integer\] as a function of the variable $\beta>1$ and as a function of the variable $x$. When $\beta=b\in{\mathbb{N}}$, Delange showed that $h_b(x)$ is a continuous but everywhere non-differentiable real-valued function of $x$ with period 1. For each fixed $\beta>1$, the function $h_\beta(x)$ is a real-valued continuous function of $x$ on ${\mathbb{R}}$. The zeta function satisfies the bound $$\lvert\zeta(it)\rvert\ll t^{1/2+\varepsilon}$$ for $t\in{\mathbb{R}}$ (see for example [@titchmarsh-zeta eq. 5.1.3], so the Fourier coefficients of $h_\beta$ satisfy $$c_\beta(k)= -\frac{\beta-1}{2\pi i k}\biggl(1+\frac{2\pi i k}{\log \beta}\biggr)^{-1}\zeta\biggl(\frac{2\pi i k}{\log \beta} \biggr) \ll_\beta k^{-3/2+\varepsilon}.$$ This estimate shows that the Fourier series is absolutely and uniformly convergent for $x\in{\mathbb{R}}$, so gives a continuous function of $x$. The function $h_\beta(x)$ is real-valued for $x\in{\mathbb{R}}$ since the Fourier coefficients $c_\beta(k)$ satisfy $\overline{c_\beta(k)} = c_\beta(-k)$. \[fb2-plot\]  \[fbl2-plot\]  A plot of $h_\beta(2)$ as a function of the real parameter $\beta$ for $1\leq \beta \leq 8$ is shown in Figure \[fb2-plot\]. From the plot, it also appears that $h_\beta$ might be non-differentiable as a function of the real parameter $\beta$. For fixed $x\in{\mathbb{R}}$, is the function $h_\beta(x)$ everywhere non-differentiable as a function of the real variable $\beta$? Meromorphic continuation of $G_\beta(s)$ ---------------------------------------- Our proofs of the meromorphic continuation of $F_b(s)$ and $G_b(s)$ for integer bases relied on the identity $$Z_b(s) = \sum_{n=1}^\infty \bigl( d_b(n) - d_b(n-1) \bigr) n^{-s} = \frac{b^s-b}{b^s-1}\zeta(s).$$ If for non-integer $\beta>1$ we define $$Z_\beta(s) \coloneq \sum_{n=1}^\infty \bigl( d_\beta(n) - d_\beta(n-1) \bigr) n^{-s},$$ then $Z_\beta(s)$ is *not* equal to $$\label{eq-zbeta-wrong} \frac{\beta^s-\beta}{\beta^s-1}\zeta(s)$$ as is not an ordinary Dirichlet series. We must therefore take a different approach. We first consider the Dirichlet series generating function $$G_\beta(s) := \sum_{n=1}^\infty \frac{S_\beta(n)}{n^s}$$ for $\beta\in{\mathbb{R}}$ with $\beta>1$. Since the coefficients satisfy $$S_\beta(n) \asymp n \log n,$$ the Dirichlet series $G_\beta(s)$ has abscissa of absolute convergence $\sigma_a=2$. We show that the function $G_\beta(s)$ can be analytically continued to a larger halfplane. For each real $\beta>1$, the function $G_\beta(s)$ is meromorphic in the region $\operatorname{Re}(s)>1$ with a double pole at $s=2$ with Laurent expansion $$G_\beta(s) = \frac{\beta - 1}{2\log \beta}(s-2)^{-2} + c_\beta(0) (s-2)^{-1} +O(1)$$ and simple poles at $s=2+2\pi i k / \log \beta$ for $k\in{\mathbb{Z}}$ with $k\neq 0$ with residue $$\operatorname*{Res}\biggl(G_b(s),s=2+\frac{2\pi i k}{\log \beta}\biggr) = c_\beta(k),$$ where the numbers $c_\beta(k)$ are those in Definition \[def-non-integer\]. Using the definition of $S_\beta$, we have $$\begin{aligned} G_\beta(s)&=\sum_{n=1}^\infty \Biggl(\frac{\beta-1}{2\log \beta} n \log n + h_\beta\biggl(\frac{\log n}{\log \beta} \biggr) n\Biggr)n^{-s}\\ &= -\frac{\beta-1}{2\log \beta} \zeta'(s-1) + \sum_{n=1}^\infty h_\beta\biggl(\frac{\log n}{\log \beta} \biggr) n^{-(s-1)}.\end{aligned}$$ The function $\zeta'(s-1)$ is meromorphic on ${\mathbb{C}}$ with only singularity a double pole at $s=2$ with Laurent expansion $\zeta'(s-1)=-(s-1)^{-2} + O(1)$. Using the Fourier series for $h_\beta$, we have $$\begin{aligned} \sum_{n=1}^\infty h_\beta\biggl(\frac{\log n}{\log \beta} \biggr) n^{-(s-1)} &= \sum_{n=1}^\infty \sum_{k=-\infty}^\infty c_\beta(k) \exp\biggl(2 \pi i k \frac{\log n}{\log \beta}\biggr) n^{-(s-1)}\\ &= \sum_{n=1}^\infty \sum_{k=-\infty}^\infty c_\beta(k) n^{-(s-1 - 2\pi i k / \log \beta)}.\end{aligned}$$ This double sum is absolutely convergent, so we may exchange the sums, giving $$\sum_{n=1}^\infty h_\beta\biggl(\frac{\log n}{\log \beta} \biggr) n^{-(s-1)} = \sum_{k=-\infty}^\infty c_\beta(k) \zeta\biggl( s- 1 - \frac{2\pi i k}{\log \beta}\biggr).$$ If $\operatorname{Re}(s)>1$, then $$\zeta\biggl( s- 1 - \frac{2\pi i k}{\log \beta}\biggr) \ll k^{1/2+\varepsilon}$$ for any $\varepsilon>0$. On any compact set $K$ in the halfplane $\operatorname{Re}(s)>1$ not containing a point $s=2+2\pi i k / \log \beta)$ for any $k\in{\mathbb{Z}}$, the sum $$\sum_{k=-\infty}^\infty c_\beta(k) \zeta\biggl( s- 1 - \frac{2\pi i k}{\log \beta}\biggr)$$ is uniformly convergent on $K$; if the compact set $K$ contains a point of the form $s=2 + 2\pi i k_0 / \log \beta$, then one term of the sum has a simple pole with residue $c_\beta(k_0)$ while the remaining sum is uniformly convergent. When $\beta\geq 2$ is an integer, we know that the function $G_\beta(s)$ has a meromorphic continuation to the entire complex plane. For noninteger $\beta>1$, does the Dirichlet series $G_\beta(s)$ have a meromorphic continuation beyond $\operatorname{Re}(s)>1$? Meromorphic continuation of $F_\beta(s)$ ---------------------------------------- We now consider the Dirichlet series $$F_\beta(s) = \sum_{n=1}^\infty \frac{d_\beta(n)}{n^s}$$ for real $\beta>1$. We already know that this series has a meromorphic continuation to ${\mathbb{C}}$ when $\beta\geq 2$ is an integer. We show that for each real $\beta>1$, the Dirichlet series $F_\beta(s)$ has a meromorphic continuation to the halfplane $\operatorname{Re}(s)>0$. The function $F_\beta(s)$ has a meromorphic continuation to the halfplane $\operatorname{Re}(s)>0$, with a double pole at $s=1$ with Laurent expansion $$F_\beta(s) = \frac{\beta - 1}{2\log \beta}(s-1)^{-2} + \biggl(c_\beta(0) +\frac{\beta-1}{2\log \beta}\biggr) (s-1)^{-1} +O(1)$$ and simple poles at $s=1+2\pi i k / \log \beta$ for $k\in{\mathbb{Z}}$ with $k\neq 0$ with residue $$\operatorname*{Res}\biggl(F_\beta(s),s=1+\frac{2\pi i k}{\log \beta}\biggr) = \biggl(1+\frac{2\pi i k}{\log \beta}\biggr) c_\beta(k).$$ Let $$p(x) = \sum_{n=2}^\infty S_\beta(n)x^n,$$ so that $$\Gamma(s)\bigl(G_\beta(s)-S_\beta(1)\bigr) = \int_0^\infty p(e^{-x}) x^{s-1}\, dx.$$ By our definition of $d_\beta(n)$, we have $$\sum_{n=1}^\infty d_\beta(n) x^n + S_\beta(1)= (x^{-1}-1)p(x).$$ Hence by Proposition \[prop-ps-ds-relation\] we have $$\Gamma(s)\bigl(F_\beta(s)+S_\beta(1)\bigr) = \int_0^\infty (e^x-1)p(e^{-x})x^{s-1}\, dx$$ for $\operatorname{Re}(s)>1$. Using the power series expansion Then we write $$\label{eq-fbeta-formula-1} \Gamma(s)\bigl(F_\beta(s)+S_\beta(1)\bigr) = \Gamma(s+1)\bigl(G_\beta(s+1)-S_\beta(1)\bigr) + \int_0^\infty (e^x-1 - x) p(e^{-x})x^{s-1} \, dx.$$ Dividing by $\Gamma(s)$ and rearranging, we obtain $$\label{eq-fbeta-gbeta-relation} F_\beta(s) = -S_\beta(1)(s+1) + s G_\beta(s+1) + R(s)$$ where the remainder term $$R(s) = \frac{1}{\Gamma(s)} \int_0^\infty (e^x-1 - x) p(e^{-x})x^{s-1} \, dx$$ is holomorphic in $\operatorname{Re}(s)>0$ since $e^x-1-x\ll x^2$ as $x\rightarrow 0^+$. Since $G_\beta(s+1)$ is meromorphic in $\operatorname{Re}(s)>0$, we find that $F_\beta(s)$ is meromorphic in $\operatorname{Re}(s)>0$, with poles coming from the poles of $G_\beta(s+1)$. Since $$sG_\beta(s+1) = (s-1)G_\beta(s+1) + G_\beta(s+1),$$ we find that $F_\beta(s)$ has a double pole at $s=1$ with Laurent expansion as given in the theorem. At each other point $s=1+2\pi i m / \log \beta$, $F_\beta(s)$ has a simple pole. Meromorphic continuation of $F_\beta(s)$ to a larger halfplane would follow from continuation of $G_\beta(s)$ to a larger halfplane; in particular, by using more terms of the power series for $e^x$ in formula , we find that if $G_\beta(s)$ is meromorphic in $\operatorname{Re}(s)>c$ for some $c$, then $F_\beta(s)$ is meromorphic in $\operatorname{Re}(s)>c-1$. Acknowledgements ================ The author thanks Jeffrey Lagarias for many helpful comments. [^1]: Work partially supported by NSF grant DMS-1701576.

--- author: - | Wei-Gang Yuan, Xiao-Dong Zhang$^{\dagger}$\ [Department of Mathematics, and MOE-LSC]{}\ [Shanghai Jiao Tong University]{}\ [800 Dongchuan Road, Shanghai, 200240, P.R. China]{}\ title: 'The Second Zagreb Indices of Graphs with Given Degree Sequences [^1]' --- = -0.2 in = 0.25 in \[section\] \[theorem\][Corollary]{} \[theorem\][Definition]{} \[theorem\][Conjecture]{} \[theorem\][Question]{} \[theorem\][Lemma]{} \[theorem\][Proposition]{} \[theorem\][Example]{} \[theorem\][Problem]{} \[theorem\][Remark]{} Abstract The second Zagreb index of a graph G is denoted by $M_2(G)=\sum_{uv\in E(G)}d(u)d(v)$. In this paper, we investigate properties of the extremal graphs with the maximum second Zagreb indices with given graphic sequences, in particular graphic bicyclic sequences. Moreover, we obtain the relations of the second Zagreb indices among the extremal graphs with different degree sequences.\ [*Keywords:*]{} Second Zagreb index; graphic sequence; majorization; bicyclic graph.\ [[*MSC:*]{} 05C12, 05C07]{} 0.5cm Introduction ============ Throughout this paper, $G=(V,E)$ is a simple undirected graph with vertex set $V$ and edge set $E$. The distance between two vertices $u$ and $v$ which is denoted by $d(u,v)$ is the length of the shortest path that connects $u$ and $v$. For a vertex $v\in V$, $N(v)$ denotes the neighbor set of $v$ and $d(v)=|N(v)|$ denotes the degree of $v$. A vertex whose degree is one is called [*leaf.*]{} Moreover, $(d(v_1), \cdots, d(v_n))$ is called [*degree sequence*]{} of $G$. A nonnegative non-increased integer sequence $\pi=(d_1, d_2, \ldots , d_n)$ is called the [*graphic sequence*]{} if there exists a simple graph $G$ such that its degree sequence is exactly $\pi$. For convenience, we use $d^{(k)}$ to denote the $k$ same degrees $d$ in $\pi$. For example, $\pi=(4,4,2,2,1,1)$ is denoted by $(4^{(2)},2^{(2)},1^{(2)})$. Let $\pi$ be a given graphic sequence. Let $$\Gamma(\pi)=\{G |\ G{\mbox{ is a connected graph with degree sequences}~\pi}\}.$$ Without loss of generality, assume $d(v_i)=d_i$, for $1\le i\le n$, $v_i\in G\in \Gamma(\pi)$. The [*second Zagreb index*]{} [@Balaban1983] of a graph $G$ is definted by: $$M_2(G)=\sum_{uv\in E}d(u)d(v).$$ For a given graphic sequence $\pi$, let $$M_2(\pi)=max\{M_2(G): G\in \Gamma(\pi)\}.$$ A simple connected graph $G$ is called an [*optimal graph*]{} in $\Gamma(\pi)$ if $G\in\Gamma(\pi)$ and $M_2(G)=M_2(\pi)$. The second Zagreb index, whose origin may be dated back to [@Gutman2004] and [@Nikolic2003], plays an important role in total $\pi-$electron energy on molecular structure in chemical graph theory. There are two excellent surveys ([@Gutman2004],[@Nikolic2003]) on the Zagreb index, which summarize main properties and characterization of the topological index. Das et al. [@das2014] investigated the connections between the Zagreb index and the Wiener index. Estes and Wei [@estes2014] presented the sharp upper and lower bounds for the Zagreb indices of $k-$tree. For more information, the readers are referred to [@Balaban1983], [@Gutman2004], [@Gutman1975], [@Kier1976], [@Kier1986], [@Nikolic2003], [@Todeschini2000] and references therein. Recently, Liu and Liu [@Liu2012] characterized the all optimal trees in the set of trees with a given tree sequence. Further, they [@Liu2014] investigate some optimal unicycle graphs in the set of unicycle graphs with a given unicyclic graphic sequence. In this paper, we study properties of the optimal graphs in the set of all connected graphs with a given graphic sequence $\pi$ that satisfies some conditions, which generalize the main results in [@Liu2012] and [@Liu2014]. In addition, we present some optimal bicyclic graphs in the set of all bicyclic graphs with a given bicyclic graphic sequence and some relations of the maximum values of the second Zagreb indices with different bicyclic graphic sequences. The rest of this paper is organized as follows. In Section 2, some notations and the main results of this paper are presented . In Sections 3, 4 and 5, the proofs of the main results are presented, respectively. Preliminary and Main Results ============================ In order to present the main results of this paper, we introduce some more notations. Assume $G$ is a rooted graph with root $v_1$. Let $h(v)$ be the distance between $v$ and $v_1$ and $H_i(G)$ be the set of vertices with distance $i$ from vertex $v_1$. [@zhang2009] Let $G=(V,E)$ be a graph of root $v_1$. A well-ordering $\prec$ of the vertices is called breadth-first search ordering with non-increasing degrees (BFS-ordering for short) if the following holds for all vertices $u,v\in V$ : \(1) $u\prec v$ implies $h(u)\le h(v)$; \(2) $u\prec v$ implies $d(u)\ge d(v)$; \(3) if there are two edges $uu_1\in E(G)$ and $vv_1\in E(G)$ such that $u\prec v$, $h(u)=h(u_1)+1$ and $h(v)=h(v_1)+1$, then $u_1\prec v_1$. For a graphic sequence $\pi=(d_1,d_2,\ldots ,d_n)$ with $\sum_{i=1}^n d_i=2(n+c)$, $d_1\geq d_2\geq c+2$, c is an integer and $c\geq -1$. We may construct a graph $G_M^*(\pi)$ by following steps. Select $v_1$ as the root vertex and begin with $v_1$ of the zeroth layer. Select the vertices $v_2,v_3,v_4,\ldots,v_{d_1+1}$ as the first layer such that $N(v_1)=\{v_2,v_3,v_4,\ldots,v_{d_1+1}\}$; then, append $d_2-1$ vertices to $v_2$, $d_3-2$ vertices to $v_3$, $\cdots$, $d_{c+3}-2$ vertices to $v_{c+3}$ such that $N(v_2)=\{v_1,v_3,\ldots,v_{c+3},v_{d_1+2},v_{d_1+3},\ldots,v_{d_1+d_2-c-1}\}$, $N(v_3)=\{v_1,v_2,v_{d_1+d_2-c},\ldots,v_{d_1+d_2+d_3-c-3}\}$, $\cdots$, $N(v_{c+3})=\{v_1,v_2,v_{(\sum_{i=1}^{c+2}d_i)-3c},\ldots,\\v_{(\sum_{i=1}^{c+3}d_i)-3c-3}\}$. After that, append $d_{c+4}-1$ vertices to $v_{c+4}$ such that $N(v_{c+4})=\{v_1, v_{(\sum_{i=1}^{c+3}d_i)-3c-2},\ldots,v_{(\sum_{i=1}^{c+4}d_i)-3c-4}\}$; $\cdots$ . Note that $v_1v_2v_3$, $\ldots$, $v_1v_2v_{c_3}$ form $c+1$ triangles in $G_M^*(\pi)$. Obviously, $G_M^*({\pi})$ is a BFS-ordering graph. In particular, if $c=1,$ the graph $G_M^*({\pi})$ is denoted by $B_M^*(\pi)$. The first main result in this paper can be stated as follows. \[general\] Let $\pi=(d_1,d_2,\ldots ,d_n)$ be a graphic sequence. If it satisfies the following condition: $(i)$$\sum_{i=1}^n d_i=2(n+c)$, c is an integer and $c\geq -1$; $(ii)$$d_1\geq d_2\geq c+2$; $(iii)$$d_3\ge d_4=d_5=\cdots=d_{c+3}$, for $c\ge 0$; $(iv)$$d_n=1$; then $G_M^*({\pi})$ is an optimal graph in $\Gamma(\pi)$. In other words, for any graph $G\in \Gamma(\pi)$, $M_2(G)\le M_2(G_M^*(\pi))$. If $\pi$ is a tree degree sequence, then there exists only one tree with degree $\pi$ having a BFS order (for example, see [@Zhang2007]). Hence it follows from Theorem \[general\] that the main results in [@Liu2012] and [@Liu2014] hold for $c=-1$ and $c=0$, respectively. \[tree\]([@Liu2012]) Let $\pi$ be a tree degree sequence. The BFS-tree in $\Gamma(\pi)$ reaches the maximum second Zagreb index. \[unicyclic\]([@Liu2014]) Let $\pi=(d_1, \cdots, d_n)$ be a unicycle graphic sequence with $d_n=1$. Then there exists an optimal graph $G\in \Gamma(\pi)$ which has a BFS-ordering $\{v_1, \cdots, v_n\}$ with a triangle $v_1v_2v_3$. Moreover, condition $(iii)$ in Theorem \[general\] can not be deleted. For example, let $\pi=(4,4,3,3,2,1,1)$ which doesn’t satisfy condition$(iii)$. In Figure 1, $G$ is produced by the method in Theorem \[general\] and $G'$ is not isomorphic to $G$. It’s easy to see that $M_2(G')=M_2(G)+1$. (60,60) (25,52)[$v_{1}$]{} (25,50) (25,17)[$v_{2}$]{} (25,20) (30,37)[$v_{4}$]{} (25,50)[(0,-1)[30]{}]{}(25,50)[(1,-3)[5]{}]{}(30,35) (25,20)[(1,3)[5]{}]{} (16,37)[$v_{3}$]{} (20,35) (25,50)[(-1,-3)[5]{}]{}(25,20)[(-1,3)[5]{}]{} (56,37)[$v_{5}$]{} (55,35) (25,50)[(2,-1)[30]{}]{} (25,20)[(2,1)[30]{}]{} (6,35)[$v_{6}$]{} (10,35) (20,35)[(-1,0)[10]{}]{} (40,37)[$v_{7}$]{} (40,35) (30,35)[(1,0)[10]{}]{} (30,10)[$G$]{} (105,52)[$v_{1}$]{} (105,17)[$v_{2}$]{} (105,50)(105,50)[(0,-1)[30]{}]{}(105,50)[(1,-1)[15]{}]{}(120,35) (86,35)[$v_{3}$]{} (90,35)(90,35)[(1,1)[15]{}]{}(90,35)[(1,-1)[15]{}]{} (120,37)[$v_{4}$]{} (105,20)(105,20)[(1,1)[15]{}]{} (90,35)[(1,0)[30]{}]{} (120,52)[$v_{5}$]{} (120,50) (105,50)[(1,0)[15]{}]{} (120,22)[$v_{6}$]{} (120,20) (105,20)[(1,0)[15]{}]{} (135,52)[$v_{7}$]{} (135,50) (120,50)[(1,0)[15]{}]{} (108,10)[$G'$]{} (60,5)[**Figure 1**]{} In order to present the results of bicyclic graphs with given bicyclic graphic sequences, we introduce some more notations. A [*bicyclic graph*]{} is a connected graph with $n\geq 4$ vertices and $n+1$ edges. Let $\pi=(d_1, \cdots, d_n)$ be a graphic sequence. If $\pi$ is a degree sequence of some bicyclic graphs, $\pi$ is called a [*bicyclic graphic*]{}. For a given bicyclic graphic sequence $\pi$, let $${\mathcal{B}}_{\pi}=\{ G|\ G\ {\mbox{ is bicyclic graph with degree sequences}~\pi}\}$$ If $\pi$ is a bicyclic graphic sequence, then $\Sigma_{i=1}^n d_i=2n+2$. Denote by $B(p,q)$ a bicyclic graph of order $n$ obtained from two vertex-disjoint cycles $C_p$ and $C_q$ by identifying vertices $u$ of $C_p$ and $v$ of $C_q$ with $p+q-1=n$. Denote by $B(p,r,q)$ a bicyclic graph of order $n$ obtained from two vertex-disjoint cycles $C_p$ and $C_q$ by joining vertices $u$ of $C_p$ and $v$ of $C_q$ by a new path $uu_1u_2\cdots u_{r-1}v$ with length $r(r\geq1)$ with $p+q+r-1=n.$ Denote by $B(P_k,P_l,P_m)~(1\leq m\leq min\{k,l\})$ a bicyclic graph of order $n$ obtained from three pairwise internal disjoint paths $xv_1v_2\cdots v_{k-1}y$, $xu_1u_2\cdots u_{l-1}y$ and $xw_1w_2\cdots w_{m-1}y$ with $k+l+m-1=n$. Denote by $B(p,q;p_1,p_2,\ldots,p_s)$ a bicyclic graph of order $n$ obtained from $B(p,q)$ appending $s$ paths on the common vertex of the two cycles, where $p+q+p_1+\cdots+p_s-1=n$, $s$ is the number of leaves and $p_1,p_2,\ldots,p_s$ denote the lengths of the $s$ paths. The results of bicyclic graphic sequences can be stated as follows. \[main\] Let $\pi=(d_1, \cdots, d_n)$ be a bicyclic graphic sequence. Denote by $s$ the number of leaves in the graph of ${\mathcal{B}}_{\pi}$. (1). If $d_n=2$ and $d_2\ge 3$, then the optimal graphs in the set ${\mathcal{B}}_{\pi}$ are $B(p,1,q)$ or $B(P_k,P_l,P_1)$ with $p+q=n$ and $k+l=n$. In other words, for any $G\in {\mathcal{B}}_{\pi}$, $M_2(G)\le 4n+17$ with equality if and only if $G$ is $B(p,1,q)$ or $B(P_k,P_l,P_1)$ with $p+q=n$ and $k+l=n$. (2). If $d_n=2$ and $d_2= 2$, then the optimal graphs in the set ${\mathcal{B}}_{\pi}$ are $B(p,q)$ with $p+q=n$. In other words, for any $G\in {\mathcal{B}}_{\pi}$, $M_2(G)\le 4n+20$ with equality if and only if $G$ is $B(p,q)$ with $p+q=n$. (3). If $d_n=1$ and $d_2=2$ and $s\le \frac{n-5}{2}$, then the optimal graphs in the set ${\mathcal{B}}_{\pi}$ are $B(p,q;p_1,p_2,\ldots,p_s)$ with $p_i\geq 2$ for $1\leq i\leq s$. In other words, for any $G\in {\mathcal{B}}_{\pi}$, $M_2(G)\le 4n+2s^2+10s+20$ with equality if and only if $G$ is $B(p,q;p_1,p_2,\ldots,p_s)$ with $p_i\geq 2$ for $1\leq i\leq s$. (4). If $d_n=1$ and $d_2=2$ and $s> \frac{n-5}{2}$, then the optimal graphs in the set ${\mathcal{B}}_{\pi}$ are $B(3,3;2,\cdots,2,1,\cdots,1)$ with $p_1=\cdots=p_{n-s-5}=2$ and $p_{n-s-4}=\cdots=p_s=1$. In other words, for any $G\in {\mathcal{B}}_{\pi}$, $M_2(G)\le sn+6n+s+10$ with equality if and only if $G$ is $B(3,3;2,\cdots,2,1,\cdots,1)$ with $p_1=\cdots=p_{n-s-5}=2$ and $p_{n-s-4}=\cdots=p_s=1$. (5). If $d_n=1$ and $d_2\ge 3$, then $B_M^*(\pi)$ is an optimal graph in the set ${\mathcal{B}}_{\pi}$. $B_M^*(\pi)$ is not the unique optimal graph for $d_n=1$ and $d_2\ge 3$. For example, let $\pi=(4^{(5)},1^{(8)})$. Figure 2 shows two different optimal graphs. (60,58) (25,52)[$v_{1}$]{} (25,17)[$v_{2}$]{} (25,50)(25,50)[(0,-1)[30]{}]{}(25,50)[(1,-1)[15]{}]{}(40,35) (6,35)[$v_{3}$]{} (10,35)(10,35)[(1,1)[15]{}]{}(10,35)[(1,-1)[15]{}]{} (40,37)[$v_{4}$]{} (25,20)(25,20)[(1,1)[15]{}]{} (50,52)[$v_{5}$]{} (25,50)[(1,0)[25]{}]{}(50,50) (6,20)[$v_{6}$]{} (25,20)[(-1,0)[15]{}]{}(10,20) (4,52)[$v_{7}$]{}(14,52)[$v_{8}$]{} (10,35)[(1,3)[5]{}]{}(10,35)[(-1,3)[5]{}]{}(5,50)(15,50) (40,17)[$v_{9}$]{}(50,22)[$v_{10}$]{} (40,35)[(0,-1)[15]{}]{}(40,35)[(1,-1)[10]{}]{}(40,20)(50,25) (65,52)[$v_{11}$]{}(63,39)[$v_{12}$]{}(50,32)[$v_{13}$]{} (50,50)[(1,0)[15]{}]{}(50,50)[(1,-1)[11]{}]{}(50,50)[(0,-1)[15]{}]{} (65,50)(61,39)(50,35) (28,10)[$B_M^*({\pi})$]{} (115,52)[$v_{1}$]{} (115,17)[$v_{2}$]{} (115,50)(115,50)[(0,-1)[30]{}]{}(115,50)[(1,-1)[15]{}]{}(130,35) (130,37)[$v_{4}$]{} (115,20)(115,20)[(1,1)[15]{}]{} (100,45.5)[$v_{3}$]{} (115,50)[(-2,-1)[15]{}]{}(100,42.5) (100,24.5)[$v_{5}$]{} (100,42.5)[(0,-1)[15]{}]{}(115,20)[(-2,1)[15]{}]{}(100,27.5) (135,52)[$v_{6}$]{} (115,50)[(1,0)[20]{}]{}(135,50) (96,20)[$v_{7}$]{} (115,20)[(-1,0)[15]{}]{}(100,20) (81,47.5)[$v_{8}$]{}(81,37.5)[$v_{9}$]{} (100,42.5)[(-3,1)[15]{}]{}(100,42.5)[(-3,-1)[15]{}]{}(85,37.5)(85,47.5) (130,17)[$v_{10}$]{}(140,22)[$v_{11}$]{} (130,35)[(0,-1)[15]{}]{}(130,35)[(1,-1)[10]{}]{}(130,20)(140,25) (80,32.5)[$v_{12}$]{}(80,22.5)[$v_{13}$]{} (100,27.5)[(-3,1)[15]{}]{}(100,27.5)[(-3,-1)[15]{}]{}(85,22.5)(85,32.5) (118,10)[$G'$]{} (60,5)[**Figure 2**]{} For two different non-increasing graphic sequences $\pi=(d_1,d_2,\ldots,d_n)$ and $\pi'=(d_1',d_2',\ldots,d_n')$, we write $\pi\triangleleft\pi'$ if $\sum_{i=1}^nd_i=\sum_{i=1}^nd_i'$ and $\sum_{i=1}^jd_i\leq \sum_{i=1}^jd_i'$ for all $j=1,2,\ldots,n$. Such an ordering is called [*majorization*]{} [@Marshall1979]. \[differentdegree\] Let $\pi$ and $\pi'$ be two non-increasing bicyclic degree sequences. If $\pi\triangleleft\pi'$, then $M_2(\pi)\le M_2(\pi')$ with equality if and only if $\pi=\pi'$. Proof of Theorem \[general\] ============================ To prove the theorem, the following lemmas are needed. \[trans1\]([@Liu2012]) Let $G=(V,E)$ be a connected graph with $v_1u_1\in E, v_2u_2\in E$, $v_1v_2\notin E$ and $u_1u_2\notin E$. Let $G'=G-u_1v_1-u_2v_2+v_1v_2+u_1u_2$. If $d(v_1)\geq d(u_2)$ and $d(v_2)\geq d(u_1)$, then $M_2(G')\geq M_2(G)$. Moreover, $M_2(G')>M_2(G)$ if and only if both two inequalities are strict. \[trans2\]([@Liu2012]) Suppose $G\in \Gamma(\pi)$, and there exist three vertices u, v, w of a connected graph G such that $uv\in E(G), uw\notin E(G), d(v)<d(w)\leq d(u)$, and $d(u)>d(x)$ for all $x\in N(w)$. Then, there exists another connected graph $G'\in \Gamma(\pi)$ such that $M_2(G)<M_2(G')$. \[neighbor\]([@Liu2014]) For any graphic sequence $\pi$ with $n\geq 3$, there exists an optimal graph $G\in \Gamma(\pi)$ such that $\{v_2,v_3\}\subseteq N(v_1)$. \[i=3\] Let $\pi$ be a graphic sequence satisfying the conditions in Theorem \[general\]. Then there is an optimal graph $G\in\Gamma(\pi)$ such that $v_1v_2v_3$ forms a triangle. To prove Lemma \[i=3\], we need to prove following claims first. [**Claim 1.**]{} There is an optimal graph $G\in \Gamma(\pi)$ such that $\{v_2,v_3\}\subseteq N(v_1)$ and there exists a cycle $C_{t_1}\subseteq G$ such that $v_1\in C_{t_1}$. Assume that Claim 1 does not hold for any optimal graph $G\in \Gamma(\pi)$. By Lemma \[neighbor\], we may suppose that G is an optimal graph in $\Gamma(\pi)$ such that $\{v_2,v_3\}\subseteq N(v_1)$. So $v_1$ is not in any cycle of any optimal graph $G\in \Gamma(\pi)$. Since $d_1\geq c+2$, there exists a shortest path $P=u\cdots v_1 \cdots xy$ connecting $u$ and $y$ such that $v_1$ is on the path, where $u\in C_{t_1}$ and $d(y)=1$, $x\in P, x\in N(y)$. Suppose $w\in N(u)\bigcap V(C_{t_1})$. If $d(w)\leq d(x)$, let $G_1=G+ux+wy-wu-xy$. By Lemma \[trans1\] $M_2(G_1)\geq M_2(G)$. Note that $G_1\in \Gamma(\pi)$, $v_1$ is in some cycle of $G_1$ and ${v_2,v_3}\subseteq N(v_1)$, a contradiction. If $d(u)\leq d(x)$, let $G_2=G+wx+uy-wu-xy$. By Lemma \[trans1\] $M_2(G_2)\geq M_2(G)$. For the same reason, it’s a contradiction. Thus, $min\{d(u), d(w)\}>d(x)$. Then take $z\in (N(x)\bigcap V(P))\backslash \{y\}$. Similarly, $min\{d(u), d(w)\}>d(z)$. It can be proved that $min\{d(u), d(w)\}>d(v_1)$ by repeating this process, which is a contradiction. Thus, Claim 1 holds. [**Claim 2.**]{} There is an optimal graph $G\in \Gamma(\pi)$ such that there exists a cycle $C_{t_1}\subseteq G$ which contains $v_1v_2$ and $v_3\in N(v_1)$. Assume that Claim 2 does not hold for any optimal graph $G\in \Gamma(\pi)$. By Claim 1, there exists an optimal graph $G\in \Gamma(\pi)$ such that $v_1\in V(C_{t_1})$ and $\{v_2,v_3\}\subseteq N(v_1)$. Then $v_2\notin V(C_{t_1})$, and there are two cases for $v_1$ and $v_2$. Case 1. There is a shortest path $P=v_1v_2xy\cdots z$ connecting $v_1$ and $z$ such that $v_2$ is on the path P, where $d(z)=1$. Choose $\{u,v\}\subseteq V(C_{t_1})$ such that $uv\in E(C_{t_1})$ and suppose $max\{d(u),d(v)\}=d(u)$. If $d(u)\geq d(x)$, let $G'=G+uv_2+vx-uv-v_2x$. By Lemma \[trans1\], $M_2(G')\geq M_2(G)$ and note that $G'\in \Gamma(\pi)$ and Claim 2 holds for $G'$, a contradiction. Thus $max\{d(u),d(v)\}=d(u)<d(x)$. Repeating the above process, we can conclude $d(u)<d(z)=1$, a contradiction. So case 1 does not hold. Case 2. There is not any path connecting $v_1$ and $z$ such that $v_2$ is on the path, where $z$ is the arbitrary vertex in G and $d(z)=1$. So it is obvious that $v_2$ is in another cycle $C_{t_2}$ of $G$ and $v_1\notin C_{t_2}$. Let $u_1\in N(v_1)\bigcap V(C_{t_1})$ and $u_2\in N(v_2)\bigcap V(C_{t_2})$. By the definition of $v_1$, $v_2$, $d(v_1)\geq d(u_2)$, $d(v_2)\geq d(u_1)$. Let $G'=G-v_1u_1-v_2u_2+v_1v_2+u_1u_2$. By Lemma \[trans1\], $M_2(G')\geq M_2(G)$ and note that $G'\in \Gamma(\pi)$. $v_1v_2$ is in the same cycle of $G'$, a contradiction. So case 2 does not hold. Thus, Claim 2 holds. [**Claim 3.**]{} There is an optimal graph $G\in \Gamma(\pi)$ such that $\{v_1v_2, v_1v_3\}\subseteq E(C_{t_1})$. By Claim 2, there is an optimal graph $G\in \Gamma(\pi)$ such that there exists a cycle $C_{t_1}\subseteq G$ which contains $v_1v_2$ and $v_3\in N(v_1)$. If claim 3 does not hold, $v_3\notin V(C_{t_1})$, then $v_2v_3\notin E(G)$. Choose $u\in (V(C_{t_1})\bigcap N(v_2))\backslash \{v_1\}$ and $v\in N(v_3)\backslash \{v_1\}$. If $uv\in E(G)$, let $C_{t_1}=v_1v_2uvv_3v_1$ and $\{v_1v_2, v_1v_3\}\subseteq E(C_{t_1})$, a contradiction. So $uv\notin E(G)$. Let $G'=G+v_2v_3+uv-vv_3-uv_2$. By Lemma \[trans1\], $M_2(G')\geq M_2(G)$ and $G'\in \Gamma(\pi)$. Claim 3 holds for $G'$. Thus, by Claim 3, there is an optimal graph $G\in \Gamma(\pi)$ such that $\{v_1v_2, v_1v_3\}\subseteq E(C_{t_1})$. If $v_2v_3\notin E(G)$, choose $v\in (N(v_3)\bigcap V(C_{t_1}))\backslash \{v_1\}$. Because $d_2\geq 3$, there are two cases for the vertices in $N(v_2)$. Case 1. There is $u\in N(v_2)\backslash V(C_{t_1})$ such that $uv\notin E$. Let $G'=G+v_2v_3+uv-uv_2-vv_3$. By Lemma \[trans1\], $M_2(G')\geq M_2(G)$ and $G'\in \Gamma(\pi)$. Since $v_1v_2v_3$ forms a triangle in $G'$, Lemma \[i=3\] holds. Case 2. All vertices in $N(v_2)\backslash v_1$ connect with $v$. So $d(v)\ge 3$. Then $d(v_3)\ge 3$. We can choose $u\in N(v_2)\backslash V(C_{t_1})$ and $v'\in N(v_3)\backslash V(C_{t_1})$. Let $G'=G+v_2v_3+uv'-v_2u-v_3v'$. By Lemma \[trans1\], $M_2(G')\geq M_2(G)$ and $G'\in \Gamma(\pi)$. Since $v_1v_2v_3$ forms a triangle in $G'$, Lemma \[i=3\] is proved. \[v1vi\] Let $\pi$ be a graphic sequence satisfying the conditions in Theorem \[general\]. G is an optimal graph in $\Gamma(\pi)$. If $v_1v_2v_3$, $v_1v_2v_4$, $\ldots$, $v_1v_2v_{i-1}$ form $i-3$ triangles in G, where $4\le i\le c+2$, there is an optimal graph $G'$ (isomorphic or not isomorphic to G) in $\Gamma(\pi)$ such that $v_1'v_2'v_3'$, $v_1'v_2'v_4'$, $\ldots$, $v_1'v_2'v_{i-1}'$ form $i-3$ triangles in $G'$ and $v_1'v_i'\in E(G')$. If $v_1v_i\notin E(G)$, $\forall v\in N(v_1)\backslash \{v_2,\ldots,v_{i-1}\}, d(v)<d(v_i)$ otherwise we may exchange the label of $v$ and $v_i$. Then by Lemma \[trans2\] we may assume there exists $u\in N(v_i)$ such that $d(u)=d_1$. Suppose $u=v_j$, then $d_1=d_2=\cdots=d_j$. There are three cases for $u=v_j$: Case 1. $u=v_2$. The result holds after exchanging the label of $v_1$ and $v_2$. Case 2. $u\notin \{v_2,v_3,\ldots v_{i-1}\}$, i.e. $j>i$. Then $d_1=d_2=d_i=d_j$. Let $P$ be a shortest path from $v_1$ to $v_i$. If $\{v_2,\ldots,v_{i-1}\}\bigcap V(P)=\emptyset$, choose $x\in N(v_1)\bigcap V(P)$. Since $v_1\in N(x)\backslash v_i$ and $d_1=d_i\geq d(x)$, there must exist some vertex $y\in N(v_i)\backslash V(P)$ such that $y\notin N(x)$. Let $G'=G+v_1v_i+xy-v_1x-v_iy$. By Lemma \[trans1\] $M_2(G')\geq M_2(G)$. Note that $G'\in \Gamma(\pi)$ and $v_1v_i\in E(G')$, the result holds. If $\{v_2,\ldots,v_{i-1}\}\bigcap V(P)\neq\emptyset$, it can be proved similarly. Case 3. $u\in \{v_2,\ldots v_{i-1}\}$, i.e. $j<i$. Denote set $S=N(v_1)\backslash \{v_2,\ldots,v_{i-1},N(v_j)\}$, Case 3.1. $S\neq \emptyset$, choose $w\in S$. Note that $d(v_i)\geq d(w)$ and $d(v_1)\geq d(u)$. Let $G'=G+v_1v_i+v_jw-v_1w-v_iv_j$. Then $M_2(G')\geq M_2(G)$ by Lemma \[trans1\] and $G'\in \Gamma(\pi)$, $v_1v_i\in E(G')$. Case 3.2. $S=\emptyset$. Assume $U=\{v_3,v_4,\ldots,v_{j-1},v_{j+1},\ldots,v_{i-1}\}\backslash N(v_j)$ and $|U|=l>0$. Suppose $U=\{v_{i_1},v_{i_2},\ldots,v_{i_l}\}$. Note that $U\subseteq N(v_1)$. Since $d_1=d_j$, there exists not less than $l$ vertices in $N(v_j)\backslash N(v_1)$. Choose $l$ vertices $u_1,u_2,\ldots,u_l$ from $N(v_j)\backslash N(v_1)$. Let $G'=G+v_{i_1}v_j+\cdots+v_{i_l}v_j-v_{i_1}v_2-\cdots-v_{i_l}v_2+u_1v_2+\cdots+u_lv_2-u_1v_j-\cdots-u_lv_j$. It can be concluded that $M_2(G')\geq M_2(G)$ by using Lemma \[trans1\] $l$ times. Then relabel $v_j$ as $v_1$, $v_1$ as $v_2$ and $v_2$ as $v_j$ in $G'$. $v_1v_i\in E(G')$. If $|U|=0$, we can do the last step directly. Hence, $v_1v_i\in E(G')$. \[introduction\] Let $\pi$ be a graphic sequence satisfying the conditions in Theorem \[general\]. Then there is an optimal graph $G\in\Gamma(\pi)$ such that $v_1v_2v_3$, $\ldots$, $v_1v_2v_{c+3}$ form $c+1$ triangles. The lemma can be proved by induction. For $i=3,$ the result holds by Lemma \[i=3\]. Assume that for $i-1$, the assertion holds, i.e., there is an optimal graph $G\in \Gamma(\pi)$ in which $\{v_1,v_2,v_3\},\ldots,\{v_1,v_2,v_{i-1}\}$ form $i-3$ triangles. By Lemma \[v1vi\], we may assume $v_1v_i\in E(G)$. To finish the introduction, it suffices to prove the following claims. For convenience, let $C_j$ denote triangle $v_1v_2v_j$ for $3\le j\le i-1$. [**Claim 1.**]{} There is an optimal graph $G\in \Gamma(\pi)$ in which there exists a cycle $C_{t'}$ such that $v_1\in V(C_{t'})$, where $C_{t'}\neq C_j$ for $3\le j\le i-1$. If Claim 1 doesn’t hold for any optimal graph, $v_1\notin C$, $\forall C\neq C_j$ for $3\le j\le i-1$. Assume $C_{t'}$ is a cycle in G and $C_{t'}\neq C_j$ for $3\le j\le i-1$. Since $d_1\geq d_2\geq c+2$ and there are $c+1$ cycles, there exists two vertices $u,w\in V(C_{t'})$, $uw\in E(C_{t'})$ and a path $P=u\cdots v_1xy\cdots z$, where $x\notin\{v_2,v_3,\ldots,v_i\}$, $u\in C_{t'}$ and $d(z)=1$. Note that if $x=v_{j'}$, $2\le j'\le i$, we relabel the path by $u\cdots v_1v_{j'}xy\cdots z$ and start from $v_{j'}x$ instead of $v_1x$. Let $w\in N(u)\bigcap V(C_{t'})$. If $d(u)\geq d(x)$, let $G_1=G+uv_1+wx-v_1x-uw$. By Lemma \[trans1\] $M_2(G_1)\geq M_2(G)$. Note that $G_1\in \Gamma(\pi)$, $v_1$ is in some cycle not $C_j$ of $G_1$ for $3\le j\le i-1$, a contradiction. If $d(w)\geq d(x)$, let $G_2=G+wv_1+ux-v_1x-uw$. By Lemma \[trans1\] $M_2(G_2)\geq M_2(G)$. For the same reason, it’s a contradiction. Thus, $max\{d(u), d(w)\}<d(x)$. Then take $y\in (N(x)\bigcap V(P))\backslash \{v_1\}$. Similarly, $max\{d(u), d(w)\}<d(y)$. It can be proved that $max\{d(u), d(w)\}<d(z)=1$ by repeating this process, which is a contradiction. Thus, Claim 1 holds. [**Claim 2.**]{} There is an optimal graph $G\in \Gamma(\pi)$ in which there exists a cycle $C_{t'}$ such that $v_1v_2\in E(C_{t'})$, where $C_{t'}\neq C_j$ for $3\le j\le i-1$. If Claim 2 doesn’t hold for any optimal graph, by Claim 1, we may assume there is an optimal graph $G\in \Gamma(\pi)$ in which there exists a cycle $C_{t'}$ such that $v_1\in V(C_{t'})$ and $v_2\notin V(C_{t'})$, where $C_{t'}\neq C_j$ for $3\le j\le i-1$. Because $v_1\in V(C_{t'})$ and there remains $c+4-i$ cycles except $C_j$, $3\le j\le i-1$ and $d(v_2)-|N(v_2)\bigcap \{v_1,v_3,\ldots,v_{i-1}\}|=c+4-i$, there exists a vertex $z$ and a path $P=v_1v_2xy\cdots z$, where $d(z)=1$, $x\notin \{v_3,\ldots,v_{i-1}\}$ and P is the shortest path connecting $v_1$ and $z$ such that $v_2$ is on it. Choose $\{u,v\}\subseteq V(C_{t'})\backslash \{v_1\}$ such that $uv\in E(C_{t'})$. Note that if $v_1v_j\in E(C_{t'})$ for $2<j<i$, there is $C_{t''}\subseteq G$ such that $v_1v_2\in E(C_{t''})$. So $\{v_3,\ldots,v_{i-1}\}\bigcap\{u,v\}=\emptyset$. Suppose that $max\{d(u),d(v)\}=d(u)$. If $d(u)\geq d(x)$, let $G'=G+uv_2+vx-v_2x-uv$. By Lemma \[trans1\] $M_2(G')\geq M_2(G)$. Note that Claim 2 holds for $G'$ which is a contradiction. Thus, $max\{d(u),d(v)\}=d(u)<d(x)$. Similarly, $max\{d(u),d(v)\}<d(y)$. Repeating the above process, we will yield that $max\{d(u),d(v)\}<d(z)=1$, a contradiction. Thus, Claim 2 holds. [**Claim 3.**]{} There is an optimal graph $G\in \Gamma(\pi)$ in which there exists a cycle $C_{t'}$ such that $v_1v_2,v_1v_i\in E(C_{t'})$, where $C_{t'}\notin C_j$ for $3\le j\le i-1$. If Claim 3 doesn’t hold for any optimal graph, by Claim 2 and $v_1v_i\in E(G)$, we may assume there is an optimal graph $G\in \Gamma(\pi)$ in which there exists a cycle $C_{t'}$ such that $v_1v_2\in E(C_{t'})$ and $v_i\notin V(C_{t'})$, where $C_{t'}\neq C_j$ for $3\le j\le i-1$. Choose $u\in (N(v_2)\bigcap V(C_{t'}))\backslash \{v_1\}$ and $v\in N(v_i)\backslash \{v_1\}$. If $u=v$, Claim 3 holds. Thus, $u\neq v$. Then, Case 1. $u\notin \{v_3,\ldots,v_{i-1}\}$. Let $G'=G+v_2v_i+uv-v_2u-v_iv$. $M_2(G')\geq M_2(G)$ by Lemma \[trans1\]. Note that $G'\in \Gamma(\pi)$. So Claim 3 holds for $G'$. Case 2. $u\in \{v_3,\ldots,v_{i-1}\}$. Choose $w\in N(u)\bigcap V(C_{t'})\backslash \{v_2\}$. Case 2.1. $w\notin \{v_3,\ldots,v_{i-1}\}\backslash \{u\}$. Let $G'=G+uv_i+wv-uw-v_iv$. Because $d(v_i)\geq d(w)$ and $d(u)\geq d(v_i)$, $M_2(G')\geq M_2(G)$ by Lemma \[trans1\]. Note that $G'\in \Gamma(\pi)$. So Claim 3 holds for $G'$. Case 2.2. $w\in \{v_4,\ldots,v_{i-1}\}\backslash \{u\}$. Let $G'=G+uv_i+wv-uw-v_iv$. By condition $(iii)$, $d(v_i)=d(w)$. So $M_2(G')\geq M_2(G)$ by Lemma \[trans1\]. Note that $G'\in \Gamma(\pi)$. So Claim 3 holds for $G'$. Case 2.3. $w=v_3$. Then there is another cycle $C_{t''}=v_2v_3uv_1v_2$ in $G$ such that $v_1v_2\in E(C_{t''})$ and $v_i\notin V(C_{t''})$. Then by the same method using in Case 2.2, we can conclude that Claim 3 holds. [**Claim 4.**]{} There is an optimal graph $G\in \Gamma(\pi)$ such that $\{v_1,v_2,v_3\},\ldots,\{v_1,v_2,v_i\}$ form $i-2$ triangles in $G$. By Claim 3, there is an optimal graph $G\in \Gamma(\pi)$ in which there exists a cycle $C_{t'}$ such that $v_1v_2,v_1v_i\in E(C_{t'})$, where $C_{t'}\neq C_j$ for $3\le j\le i-1$. If Claim 4 doesn’t hold for any optimal graph, we may assume $v_2v_i\notin E(C_{t'})$. Choose $u\in (N(v_2)\bigcap V(C_{t'}))\backslash\{v_1\}$ and $v\in (N(v_i)\bigcap V(C_{t'}))\backslash \{v_1\}$. Note that $u$ and $v$ can be the same vertex. There are two cases for $u$. Case 1. $u\notin \{v_3,\ldots,v_{i-1}\}$ which implies $v\notin \{v_3,\ldots,v_{i-1}\}$. Case 1.1. $d_3\geq 3$. Choose $w\in N(v_3)\backslash \{v_1,v_2\}$. Let $G_1=G+v_3v_i+wv-v_3w-v_iv$ and $G_2=G_1+v_2v_i+v_3u-v_3v_i-v_2u$. By Lemma \[trans1\] $M_2(G_2)\geq M_2(G_1)\geq M_2(G)$. Note that $G_1,G_2\in \Gamma(\pi)$ and Claim 4 holds for $G_2$. Case 1.2. $d_3=2$. Then $d_1\geq c+3$ by condition $(iv)$. So we can choose a vertex $x\in N(v_1)\backslash \{v_2,\ldots,v_i\}$. Let $G'=G+v_2v_i+v_1u+xv-v_2u-v_iv-v_1x$. Note that $G'\in \Gamma(\pi)$ and $G'$ is connected and $d(v_i)=d(u)=d(v)=2\geq d(x)$. By elemental calculation, $M_2(G')-M_2(G)=(d_1-2)(2-d(x))\geq 0$. So $M_2(G')\geq M_2(G)$ and Claim 4 holds for $G'$. Case 2. $u\in \{v_3,\ldots,v_{i-1}\}$. Since $d_2\geq c+2$, we can choose $u'\in N(v_2)\backslash V(C_{t'})\backslash \\\{v_3,\ldots,v_{i-1}\}$. Let $G'=G+v_2v_i+u'v-v_2u'-v_iv$. By Lemma \[trans1\] $M_2(G')\geq M_2(G)$. It is easy to check that $G'\in \Gamma(\pi)$ and Claim 4 holds for $G'$. Thus, we can conclude by introduction that there is an optimal graph $G\in \Gamma(\pi)$ in which $\{v_1,v_2,v_3\},\{v_1,v_2,v_4\},\ldots,\{v_1,v_2,v_{c+3}\},$ form $c+1$ triangles. Now we are ready to prove Theorem \[general\]. The first part of the theorem have be proved by Lemma \[introduction\]. So we may assume $\{v_1,v_2,v_3\},\ldots,\{v_1,v_2,v_{c+3}\}$ form $c+1$ triangles in an optimal graph $G\in \Gamma(\pi)$. Then an ordering $\prec$ of $V(G)$ can be created by the breadth-first search as follows: firstly, let $v_1\prec v_2\prec\cdots\prec v_{c+3}$; secondly, append all neighbors $u_{c+4},\ldots,u_{d_1+1}$ of $N(v_1)\backslash \{v_2,\ldots,v_{c+3}\}$ to the order list, these neighbors are ordered such that $u\prec v$ whenever $d(u)>d(v)$ (in the remaining case the ordering can be arbitrary); thirdly, append all neighbors $u_{d_1+2},u_{d_1+3},\ldots,u_{d_1+d_2-2}$ of $N(v_2)\backslash \{v_1,v_3,\ldots,v_{c+3}\}$ to the ordered list, these neighbors are ordered such that $u\prec v$ whenever $d(u)>d(v)$ (in the remaining case the ordering can be arbitrary); with the same method we can append the vertices of $N(v_3)\backslash \{v_1,v_2\}, \cdots, N(v_{c+3})\backslash \{v_1,v_2\}$ to the ordered list. Then, append the vertices $N(x)\backslash \{v_1\}$ to the ordered list, where $d(x)=max\{d(y):y\in N(v_1)\backslash \{v_2,v_3,\ldots,v_{c+3}\}\}$. Repeat the last process recursively with all vertices $v_1,v_2,\ldots$, until all vertices of G are processed. Then $H_0=\{v_1\}$. By the construction of $\prec$, $u\prec v$ implies $h(u)\leq h(v)$. For $v\in H_i(G),i>0$, we call the unique vertex $u\in N(v)\bigcap H_{i-1}(G)$ the parent of v. So $u\prec v$, if $u$ is the parent of $v$. Moreover, because the vertices are appended to the ordered list recursively, if there are two edges $uu_1\in E(T)$ and $vv_1\in E(T)$ such that $u\prec v$, $h(u)=h(u_1)+1$ and $h(u)=h(v_1)+1$, then $u_1\prec v_1$. To prove the assertion, it suffices to show that $d(u)\geq d(v)$ holds for each two vertices $u,v\in V(G)$ and $u\prec v$. If the above proposition doesn’t hold, assume $v_i$ is the first vertex in the ordering of $\prec$ with the property $v_i\prec u$ and $d(v_i)<d(u)$ for some $u\in V(G)$. Clearly, $v_i\notin \{v_1,v_2,v_3,\ldots,v_{c+3}\}$ and if $v\prec v_i$, $d(v)\geq d(u)$ holds for each $u$ with $v\prec u$. Suppose $v_j$ is the first vertex in the ordering $\prec$ such that $v_i\prec v_j$ and $d(v_j)=max\{d(v_t):i+1\leq t\leq n\}$. By the choice of $v_i$, we can conclude that $v_i\prec v_j$, but $d(v_i)<d(v_j)$. Let $w_i$ and $w_j$ be the parents of $v_i$ and $v_j$, respectively. Note that $d(v_i)<d(v_j)$. Then $w_i\neq w_j$ and $w_i\prec w_j$ by the construction of $\prec$. It is obvious that $w_iv_j\notin E(G)$. Otherwise there is a cycle in $G$ such that $w_i, w_j, v_j$ are on it and $E(G)\geq n+c+1$ because $w_i\prec w_j $ and $v_j\notin \{v_1,v_2,v_3,\ldots,v_{c+3}\}$. Let’s consider the following two cases. Case 1. $w_iv_i$ is in the shortest path that connects $w_j$ and $v_1$. We can conclude that $w_i\prec v_i\prec w_j\prec v_j$ and $d(w_i)>d(v_j)>d(w_j)$ by the definition of $v_i$ and $v_j$. Now we shall prove the following Claim. [**Claim**]{}. There exists some $y\in N(v_j)\backslash \{w_j\}$ such that $d(w_i)=d(v_j)=d(y)$ and $v_iy\notin E(G)$. Because $v_i\notin \{v_1,v_2,v_3,\ldots,v_{c+3}\}$, $v_iy\notin E(G)$ holds for every $y\in N(v_j)\backslash \{w_j\}$ for the same reason of $w_iv_j\notin E(G)$. If $d(w_i)>d(y)$ holds for every $y\in N(v_j)\backslash \{w_j\}$, $d(w_i)>d(y)$ holds for all $y\in N(v_j)$ because $d(w_i)>d(w_j)$. So $d(w_i)\geq d(v_j)>d(v_i)$, $w_iv_i\in E(G)$ and $w_iv_j\notin E(G)$. By Lemma \[trans2\], there exists another graph $G'\in \Gamma(\pi)$ such that $M_2(G)<M_2(G')$, a contradiction. Thus, there exists some $y\in N(v_j)\backslash \{w_j\}$ such that $d(w_i)\leq d(y)$. On the other hand, by $w_i\prec v_i\prec w_j\prec v_j\prec y$ and the choice of $v_j$, we have $d(w_i)\geq d(v_j)\geq d(y)$. Hence, claim holds. Then there exists some $y\in N(v_j)\backslash \{w_j\}$ such that $d(w_i)=d(v_j)=d(y)>d(v_i)$. Let $G_1=G+w_iv_j+v_iy-w_iv_i-v_jy$. Clearly, $G_1\in \Gamma(\pi)$. By Lemma \[trans1\], $M_2(G)\leq M_2(G_1)$. Case 2. $w_iv_i$ is not in the shortest path that connects $w_j$ and $v_1$. Then $w_jv_i\notin E(G)$. Otherwise we can find a cycle in $G$ such that $v_1,v_i,w_j$ or $v_2,v_i,w_j$ are on it and $E(G)\geq n+c+1$, a contradiction. Let $G_1=G+w_iv_j+w_jv_i-w_iv_i-w_jv_j$. Then $G_1\in \Gamma(\pi)$. Because $w_i\prec v_i$ and $w_i\prec w_j$, $d(w_i)\geq d(w_j)$ by the choice of $v_i$. By Lemma \[trans1\], $M_2(G)\leq M_2(G_1)$. Note that $\{v_1,v_2,v_3\},\cdots,\{v_1,v_2,v_{c+3}\}$ still form $c+1$ triangles in $G_1$. After getting a new graph $G_1\in \Gamma(\pi)$ such that $M_2(G)\leq M_2(G_1)$ in the above two cases, we redefine the ordering $\prec$ to $V(G_1)$ as follows: Let $v_1\prec v_2 \prec \cdots \prec v_{i-1} \prec v_j$ be the first i vertices. Then, append the rest vertices by the same method which is used in the construction of $\prec$ of $V(G)$. In the redefined ordering, if $v\prec v_j$ or $v=v_j$, $d(v)\geq d(u)$ holds for all $v\prec u$. Moreover, by the construction of the redefined $\prec$, if there are two edges $uu_1\in E(T)$ and $vv_1\in E(T)$ such that $u\prec v$, $h(u)=h(u_1)+1$ and $h(u)=h(v_1)+1$, then $u_1\prec v_1$. We can also conclude $h(u)\leq h(v)$ if $u\prec v$. So repeating the above process at most $t(t\leq n-c-3)$ times, we can get an optimal graph $G_t\in \Gamma(\pi)$ such that $d(u)\geq d(v)$ holds for each two vertices $u,v\in V(G)$ and $u\prec v$. $G_t$ is isomorphic to the graph constructed in the theorem. Proof of Theorem \[main\] ========================= \[main1\] Let $\pi=(d_1, \cdots, d_n)$ be a bicyclic graphic degree sequence. (1). If $d_n=2$ and $d_2\ge 3$, then the optimal bicyclic graphs in the set ${\mathcal{B}}_{\pi}$ are $B(p,1,q)$ and $B(P_k,P_l,P_1)$ with $p+q=n$ and $k+l=n$. (2). If $d_n=2$ and $d_2=2$, then the optimal bicyclic graphs in the set ${\mathcal{B}}_{\pi}$ are $B(p,q)$ with $p+q=n$. If $d_n=2$ and $d_2\ge 3$, then the only possible degree sequence is $\pi= (3, 3, 2^{(n-2)})$ and $G$ is $B(p,r,q)$ or $B(P_k,P_l,P_m)$. It is easy to see that $M_2(B(p,1,q))=M_2(B(P_k,P_l,P_1))=4n+17 >M_2(B(p,r,q))=M_2(B(P_k,P_l,P_m))=4n+16$ for $r>1,m>1$. Hence (1) holds. If $d_n=2$ and $d_2= 2$, then $G$ is $B(p,q)$ with $p+q-1=n$. It is easy to see that $M_2(B(p, q))=4n+20$. Hence (2) holds. \[main2\] Let $\pi=(d_1, \cdots, d_n)$ be a bicyclic graphic sequence. Suppose the number of leaves in the graph of ${\mathcal{B}}_{\pi}$ is $s$. If $d_n=1$ and $d_2=2$, then the optimal bicyclic graphs in the set ${\mathcal{B}}_{\pi}$ are (1). $B(p,q;p_1,p_2,\ldots,p_s)$ with $p_i\geq 2$ for $1\leq i\leq s$ when $s\le\frac{n-5}{2}$. (2). $B(3,3;2,\cdots,2,1,\cdots,1)$ with $p_1=\cdots=p_{n-s-5}=2$ and $p_{n-s-4}=\cdots=p_s=1$ when $s>\frac{n-5}{2}$. We may write $\pi=(d_1,2^{(k)},1^{(s)})$, where $k=n-s-1$ and $d_1=2n-2k-s+2$. The lemma can be proved easily by exhaustion. (1). $s\le\frac{n-5}{2}$ i.e $k\geq s+4$ i.e. $n\leq 2k-3$. The optimal graphs are $B(3,3;k-s-2,2,2,\ldots,2),B(3,3;k-s-3,3,2,\ldots,2),\\ \cdots ,B(p,q;p_1,p_2,\ldots,p_s)$ whose second Zagreb indices are all equal to $2\times (n-k+3)(n-k+3)+2\times 2 \times (2k-n-1)+2\times 1 \times (n-k-1)=2n^2-4nk+2k^2+10n-6k+12=4n+2s^2+10s+20$, where $p_i\geq 2$ for $1\leq i\leq s$. (2). $s>\frac{n-5}{2}$ i.e. $4\leq k<s+4$ i.e. $n>2k-3$. The unique optimal graph of this case is $B(3,3;2,\cdots,2,1,\cdots,1)$ whose second Zagreb index is $2\times (n+3-k)k+1\times (n+3-k)(n-2k+3)+2\times 2 \times 2+1\times 2 \times (k-4)=n^2-nk+6n-k+9=sn+6n+s+10$, where $p_1=\cdots=p_{k-4}=2$ and $p_{k-3}=\cdots=p_s=1$. Now we are ready to prove Theorem \[main\]. It is easy to see that the assertion follows from Lemmas \[main1\], \[main2\] and Theorem \[general\]. Proof of Theorem \[differentdegree\] ==================================== In order to prove Theorem \[differentdegree\], we need some lemmas \[majorization\]([@Marshall1979]) Let $\pi$ and $\pi'$ be two different non-increasing graphic sequences. If $\pi\triangleleft\pi'$, then there exists a series of non-increasing graphic sequences $\pi_1,\pi_2,\ldots,\pi_k$ such that $\pi=\pi_0\triangleleft\pi_1\triangleleft\pi_2\triangleleft\ldots\triangleleft\pi_k\triangleleft\pi_{k+1}\triangleleft\pi'$, where $\pi_i$ and $\pi_{i+1}$ differ only in two positions and the differences are 1 for $0\leq i\leq k$. \[trans3\]([@Liu2012]) Let $u,v$ be two vertices of a connected graph G, and $w_1,w_2,\ldots,w_k$ $(1\leq k\leq d(v))$ be some vertices of $N(v)\backslash (N(u)\bigcup \{u\})$. Let $G'=G+w_1u+w_2u+\cdots+w_ku-w_1v-w_2v-\cdots-w_kv$. If $d(u)\geq d(v)$ and $\sum_{y\in N(u)}d(y)\geq \sum_{x\in N(v)}d(x)$, then $M_2(G')>M_2(G)$. \[d2=2\] Let $\pi=(d_1, \ldots, d_n)$ and $\pi'=(d_1', \ldots, d_n')$ be two bicyclic graphic degree sequence. Suppose that at most one following condition holds. (i)$d_2=3$ and $d_n=1$. (ii)$d_2'=3$ and $d_n'=1$. If there exist $1\le p<q\le n$ with $d_p=d_p'+1$, $d_q=d_q-1$ for $1\le p<q\le n$ and $d_i=d_i'$ for all $i\neq p, q$, then $M_2({\pi})<M_2({\pi'})$. This Lemma can be proved by exhaustion. Let $G_\pi$ be an optimal graph with degree sequence $\pi$. Then for each degree sequences $\pi$, the method to prove the lemma is to find all possible degree sequences $\pi_1,\pi_2$ such that $\pi_1\triangleleft\pi\triangleleft\pi_2$, where $\pi_1,\pi$ and $\pi,\pi_2$ differ only in two positions, where the difference are 1. After that, prove $M_2(G_{\pi_1})<M_2(G_{\pi})<M_2(G_{\pi_2})$. Without loss of generality, we may assume condition (i) doesn’t hold. There are four cases for $\pi$. Case 1. $\pi=(3,3,2^{(n-2)})$. It is easy to check that for any other bicyclic sequences $\pi'$ satisfying the conditions in Lemma \[d2=2\], $\pi\triangleleft\pi'$ holds and $M_2(\pi)=4n+17<M_2(\pi')$. Case 2. $\pi=(4,2^{(n-1)})$. The all possible sequences for $\pi_1$ and $\pi_2$ are $\pi_1=(3,3,2^{(n-2)})$ and $\pi_2=(5,2^{(n-2)},1)$, $\pi_2'=(4,3,2^{(n-3)},1)$. By the preceding proof and calculation, $M_2(G_{\pi_1})=4n+17$, $M_2(G_{\pi})=4n+20$, $M_2(G_{\pi_2})=2n^2-4nk+2k^2+10n-6k+12=4n+32~(k=n-2)$, $M_2(G_{\pi_2'})=M_2(B_M^*(\pi_2'))=4n+26$ and $M_2(G_{\pi_1})<M_2(G_{\pi})<M_2(G_{\pi_2'})<M_2(G_{\pi_2})$. Lemma \[d2=2\] holds for this case. Case 3. $\pi=(d_1,2^{(k)},1^{(s)})$,where $k\geq s+4$ i.e. $n\leq 2k-3$. The all possible sequences for $\pi_1$ and $\pi_2$ are $\pi_1=(d_1-1,2^{(k+1)},1^{(s-1)})$, $\pi_1'=(d_1-1,3,2^{(k-1)},1^{(s)})$ and $\pi_2=(d_1+1,2^{(k-1)},1^{(s+1)})$, $\pi_2'=(d_1,3,2^{(k-2)},1^{(s+1)})$. By the preceding proof and calculation, $M_2(G_{\pi_1})=2n^2-4nk+2k^2+6n-2k+8$, $M_2(G_{\pi_1}')=2n^2-4nk+2k^2+7n-3k+12$, $M_2(G_{\pi})=2n^2-4nk+2k^2+10n-6k+12$, $M_2(G_{\pi_2})=2n^2-4nk+2k^2+14n-10k+20~for~n\leq2k-5$; $M_2(G_{\pi_2})=n^2-n(k-1)+6n-(k-1)+9=n^2-nk+7n-k+10~for~n=2k-4,2k-3$, $M_2(G_{\pi_2'})=2n^2-4nk+2k^2+11n-7k+17~for~n\leq 2k-4$; $M_2(G_{\pi_2'})=2n^2-4nk+2k^2+14n-14k+28~for~n=2k-3$. So $M_2(G_{\pi_1})<M_2(G_{\pi_1'})<M_2(G_{\pi})<M_2(G_{\pi_2'}<M_2(G_{\pi_2})$ and Lemma \[d2=2\] holds for this case. Case 4. $\pi=(d_1,2^{(k)},1^{(s)})$, where $4\leq k<s+4$ i.e. $n>2k-3$. The all possible sequences for $\pi_1$ and $\pi_2$ are the same as the above case except that the $M_2$ is different. By the preceding proof and calculation, $M_2(G_{\pi_1})=n^2-nk+5n-k+8~for~n>2k-1$; $M_2(G_{\pi_1})=2n^2-4nk+2k^2+6n-2k+8~for~n=2k-2,2k-1$, $M_2(G_{\pi_1'})=n^2-nk+5n-k+12$, $M_2(G_{\pi})=n^2-nk+6n-k+9$, $M_2(G_{\pi_2})=n^2-nk+7n-k+10$, $M_2(G_{\pi_2'})=n^2-nk+6n-k+13$, Lemma \[d2=2\] also holds for this case. \[d2&gt;2\] Let $\pi=(d_1, \ldots, d_n)$ and $\pi'=(d_1', \ldots, d_n')$ be two bicyclic graphic degree sequence with $M_2({\pi})$ and $M_2({\pi'})$ being the maximum second Zagreb index in the set ${\mathcal{B}}_{\pi}$. Suppose that $d_2\ge 3$, $d_2'\ge 3$ and $d_n=1$, $d_n'=1$. If there exist $1\le p<q\le n$ with $d_p=d_p'+1$, $d_q=d_q-1$ for $1\le p<q\le n$ and $d_i=d_i'$ for all $i\neq p, q$, then $M_2({\pi})<M_2({\pi'})$. By Theorem \[main\], $M_2({\pi})=M_2(B_M^*({\pi}))$. So it suffice to show that $M_2(B_M^*({\pi}))<M_2(\pi')$. We have $v_p\prec v_q$ in the ordering of $V(B_M^*({\pi}))$ since $p<q$ and hence $d(v_p)\geq d(v_q)$. By the proof of the last part of Theorem \[general\], we have $\sum_{x\in N_{B_M^*({\pi})}(v_p)}d(x)\geq \sum_{y\in N_{B_M^*({\pi})}(v_q)}d(y)$. Let $P$ be the (one of) shortest path from $v_p$ to $v_q$ in $B_M^*({\pi})$. If $q=2$, then $d_q\geq 4$ because $d_q'=d_q-1\ge3$. If $3\leq q\leq 4$, then $d_q\geq 3$ because $d_q'=d_q-1=2$. If $q\geq 4$, then $d_q\geq 2$. In all these cases, there exists a vertex $v_k(k>q)$ such that $v_k\in N_{B_M^*({\pi})}(v_q)\backslash N_{B_M^*({\pi})}(v_p)$ and $v_k\notin V(P)$. Let $G=B_M^*({\pi})+v_pv_k-v_qv_k$. Note that $G\in \Gamma(\pi')$ and $d(v_p)\geq d(v_q)$. By Lemma \[trans3\], $M_2(B_M^*({\pi}))<M_2(G)\leq M_2(\pi')$. Now we are ready to prove Theorem \[differentdegree\]. Set $\pi=(d_1,d_2,\ldots,d_n)$ and $\pi'=(d_1',d_2',\ldots,d_n')$. Since $\pi\triangleleft\pi'$, by Lemma \[majorization\] we may suppose that $\pi$ and $\pi'$ differ only in two positions, where the difference are 1. So we may assume that $d_i=d_i'$ for $i\neq p,q$, and $d_p+1=d_p', d_q-1=d_q'$, $1\leq p<q\leq n$. The remaining parts of proofs follow from Lemmas \[d2=2\] and \[d2&gt;2\]. A. T. Balaban, I. Motoc, D. Bonchev and O. Mekenyan, Topological indices for structure-activity correlations, [*Topics Curr. Chem.*]{}, 114(1983) 21-55. K. Ch. Das, H. U. Jeon and N. Trinajstić, Comparison between the Wiener index and the Zagreb indices and the eccentric connectivity index for trees. [*Discrete Appl. Math.*]{}, 71(2014) 35-41. J. Estes and B. Wei, Sharp bounds of the Zagreb indices of $k-$trees. [*J. Comb. Optim.*]{}, 27(2014) 271-291. I. Gutman and K. C. Das, The first Zagreb index 30 years after, [*MATCH Commun. Math. Comput. Chem.*]{}, 50(2004) 83-92. I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. Total $\pi-$electron energy of alternant hydrocarbons, [*Chem. Phys. Lett.*]{}, 17(1972) 535-538. I. Gutman, B. Ruščić and C. F. Wilcox, Graph theory and molecular orbitals.12.Acyclic polyenes, [*J. Chem. Phys.*]{}, 62(1975) 3399-3405. A. Ilić and B. Zhou, On reformulated Zagreb indices, [*Discrete Appl. Math.,*]{} 160 (2012) 204-209. H. B. Hua and K. Ch. Das, The relationship between the eccentric connectivity index and Zagreb indices, [*Discrete Appl. Math.,*]{} 161 (2011) 2480-2491. L. B. Kier and L. H. Hall, [*Molecular Connectivity in Chemistry and Drug Research*]{}, Academic Press, San Francisco, 1976. L. B. Kier and L. H. Hall, [*Molecular Connectivity in Structure-Activity Analysis*]{}, Wiley, New York, 1986. M. H. Liu, B. L. Liu, The second Zagreb indices and Wiener polarity indices of trees with given degree sequences, [*MATCH Commun. Math. Comput. Chem.*]{}, 67(2012) 439-450. M. H. Liu, B. L. Liu, The second Zagreb indices of unicyclic graphs with given degree sequences, [*Discrete Appl. Math.*]{}, 167(2014) 217-221. A. W. Marshall, I. Olkin, [*Inequalities: Theory of Majorization and Its Applications*]{}, Academic Press, New York, 1979. S. Nikolić, G. Kovačević, A. Miličević and N. Trinajstić, The Zagreb indices 30 years after, [*Croat. Chem. Acta*]{}, 76 (2003) 113-124. R. Todeschini and V. Consonni, [*Handbook of Molecular Descriptors*]{}, Wiley VCH, Weinheim, 2000. X.-D. Zhang, The Laplacian spectral radii of trees with degree sequences, [*Discrete Math.*]{}, 308(2008) 3143-3150. X.-D. Zhang, The signless Laplacian spectral radius of graphs with given degree sequences, [*Discrete Appl. Math.*]{}, 157(2009) 2928-2937. [^1]: This work is supported by National Natural Science Foundation of China (No.11271256), Innovation Program of Shanghai Municipal Education Commission (No.14ZZ016) and Specialized Research Fund for the Doctoral Program of Higher Education (No.20130073110075).$^{\dagger}$Correspondent author: Xiao-Dong Zhang (Email: xiaodong@sjtu.edu.cn)

--- 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.

--- abstract: 'We examine the reliability of the merger trees generated for the Monte-Carlo modeling of galaxy formation. In particular we focus on the cold gas fraction predicted from merger trees with different assumptions on the progenitor distribution function, the timestep, and the mass resolution. We show that the cold gas fraction is sensitive to the accuracy of the merger trees at small-mass scales of progenitors at high redshifts. One can reproduce the Press–Schechter prediction to a reasonable degree by adopting a fairly large number of redshift bins, $N_{\rm step}\sim 1000$, in generating merger trees, which is a factor of ten larger than the canonical value used in previous literature.' author: - 'Mamoru <span style="font-variant:small-caps;">Shimizu</span>, Tetsu <span style="font-variant:small-caps;">Kitayama</span>, Shin <span style="font-variant:small-caps;">Sasaki</span>, and Yasushi [ Suto]{}' title: | Reliability of Merger Tree Realizations of Dark Halos\ in the Monte-Carlo Modeling of Galaxy Formation --- Introduction ============ Understanding the formation and evolution of galaxies is a fundamental step in linking the initial condition of the universe and the cosmological observational data. Recent systematic studies of high-redshift objects, such as quasars and Lyman-break galaxies, should provide important clues to the early universe, although their proper interpretation is often not so straightforward, mainly because those objects certainly do evolve in time. A theoretical study of galaxy evolution, especially its spectroscopic evolution, from a cosmological context, was begun by @tinsley80 and followed by many authors (e.g., [@Bruzual83]; [@AY86]; [@GR87]; [@CB91]; [@BC93]; [@KA97]). These studies are based on a so-called ‘one-zone’ model which assumes that a galaxy does not interact with other galaxies. It is now fairly established, however, that structures in the universe have built up hierarchically from small to large scales as in a cold dark matter (CDM) model. This means that a galaxy interacts and sometimes merges with other galaxies even if it was an isolated system at birth. The predictions in the one-zone model therefore may be significantly different from what happened to galaxies in a hierarchical universe. White and Frenk (1991) developed a detailed analytic formalism to describe the formation and evolution of galaxies while taking account of the hierarchical merging of dark-matter halos, gas cooling, star formation, and supernova feedback. Subsequent numerical approaches in modeling hierarchical merging of dark halos employ two somewhat different algorithms; one is called the ‘block model’ in which a random-Gaussian density fluctuation field is generated by dividing a hypothetical rectangular box recursively ([@CK88]; [@Cole91]; [@Cole94]). While this algorithm is simple and straightforward, the resulting halo masses are necessarily binned in discrete steps of a factor of two. The other generates a realization of halo merger trees according to a probability distribution function predicted by the extended Press–Schechter theory ([@Bower91]; [@Bond91]; [@KW93]; [@SK99]; [@SL99]). The latter is widely used in studying the cosmological evolution of galaxies in a hierarchical universe ([@KWG93]; [@baugh98]; [@SP99]; [@Cole00]; [@nagashima]). Throughout the present paper, we call the latter method the Monte-Carlo modeling of merger histories (simply, the Monte-Carlo modeling), while it is usually referred to as a semi-analytic model of galaxy formation (SAM). The most important ingredient in Monte-Carlo modeling is the conditional joint-probability distribution function of a set of *progenitor* halos of mass $M_2^{j}$ at a redshift of $z_2$, which is a part of a *parent* halo of mass $M_1$ at $z_1$, conceptually written as $$\begin{aligned} \label{eq:jointprob} {\rm Prob}(M_2^1, M_2^2, \cdots, M_2^N, z_2 | M_1, z_1) dM_2^1 dM_2^2 \cdots dM_2^N \cr \qquad (N=1, \cdots , \infty). \end{aligned}$$ Unfortunately only an analytical expression for the conditional one-point probability distribution function, Prob($M_2^i$, $z_2 |M_1$, $z_1$), is known based on the extended Press–Schechter theory (for the special case of the Poisson initial power spectra, see a different approach by [@SL99]); one thus needs to employ an additional *assumption* in generating realizations of merger trees of halos in general (e.g., [@KW93]; [@SK99]). Furthermore, any numerical procedure to generate them necessarily involves several *ad hoc* parameters due to the limitation of the available computation resources including the finite timestep of computation, the minimum mass of halos to be included in merger trees, and the maximum number of progenitors for each halo at each step. The purpose of the paper is to perform a systematic investigation of possible artificial effects of the above-mentioned problems on merger tree realizations, and to re-examine the validity of the Monte-Carlo modeling. In particular, we focus on the extent to which the resulting merger trees reproduce the conditional one-point probability distribution function predicted by the extended Press–Schechter theory, which directly changes the fraction of cold gas. Exactly for this reason, we adopt a conventional $\Lambda$CDM model with the cosmological parameters $\Omega_{0}=0.3$, $\lambda_{0}=0.7$, $h=0.7$, $\sigma_{8}=1.0$, and $\Omega_{\mathrm{B}}=0.015h^{-2}$ (e.g., Kitayama, Suto 1997; Kitayama et al. 1998), and neglect star formation and a feedback effect for definiteness. Merger Trees of Dark Matter Halos ================================= Constructing Merger Trees of Dark-Matter Halos \[subsec:construction\] ---------------------------------------------------------------------- Our model of merging histories of dark-matter halos is mainly based on that of Somerville and Kolatt (1999), which we adopt as our fiducial choice and slightly modify their original scheme as follows. We begin with a halo of mass of $M_1= M_{\mathrm{root}}$ at a redshift of $z_1=z_{\mathrm{min}}$, and consider its progenitors at a slightly earlier redshift of $z_2=z_1+\Delta z(z_1)$. Since the joint conditional probability for the progenitors \[equation (\[eq:jointprob\])\] is not known, we choose the $i$-th progenitor halo of mass $M_2^i$ according to the *one-point* conditional probability, Prob($M_2^i$, $z_2 |M_1$, $z_1$), as long as $M_2^i > M_{\rm res}$ and the total mass satisfies $$\begin{aligned} \label{eq:massconserve} \sum_{i=1}^N M_2^i < M_1 - \Delta M_{\rm acc}(<M_{\rm res}) ,\end{aligned}$$ where $$\begin{aligned} \Delta M_{\rm acc}(<M_{\rm res}) = \int_{0}^{M_{\rm res}}\!\!\!\!dM_2 M_2 \frac{dN}{dM_2}(M_2,z_2|M_1,z_1)\end{aligned}$$ is the expectation value of the total mass of halos smaller than the resolution mass ($M_{\rm res}$) with $dN/dM_2(M_2,z_2|M_1,z_1)$ being the appropriate conditional mass function \[equation (\[eq:eps-num\]) below\]. In other words, we distinguish the discrete merging and the continual accretion at mass $M_{\rm res}$, and do not resolve the halos below $M_{\rm res}$ in our merger trees. Once all relevant progenitor halos are selected, we repeat the above procedure recursively for each progenitor until the maximum redshift ($z_{\rm max}$). Unless otherwise stated, we set $z_{\mathrm{min}}=0$ and $z_{\mathrm{max}}=15$ in the present paper. For convenience, we list in table \[tab:parameters\] variables which are extensively discussed in the present paper. In the original method by Somerville and Kolatt (1999), one stops selecting progenitors when $M_1 - \sum_{i=1}^N M_2^i$ becomes less than $M_{\rm res}$, but without imposing the condition $M_2^i>M_{\rm res}$. They carefully tuned the timesteps depending on $M_{1}$ so that the resulting progenitor mass function becomes close to equation (\[eq:eps-num\]) below. Rather, we stop choosing the progenitor when $M_1 - \Delta M_{\rm acc}(<M_{\rm res}) - \sum_{i=1}^N M_2^i$ becomes negative, and the last selected progenitor $M_2^N$ is not included in the tree. In this case, the remaining mass $M_1 - \Delta M_{\rm acc}(<M_{\rm res}) - \sum_{i=1}^{N-1} M_2^i$ is not necessarily smaller than $M_{\rm res}$. We find that our method reproduces equation (\[eq:eps-num\]) even with the $M_1$-independent timesteps. Conditional Probability Distribution Function --------------------------------------------- The most important and subtle issue is the proper choice of the *one-point* conditional probability, Prob($M_2$, $z_2 |M_1$, $z_1$). Bower (1991) and Bond et al. (1991) derived the conditional probability of $M_2$ at $z_2$, which is a part of halo $M_1$ at $z_1$: $$\begin{aligned} \label{eq:eps-mass} \frac{dP}{dM_{2}}(M_{2},z_{2}|M_{1},z_{1}) &=& \frac{\delta_{\rm c,2}-\delta_{\rm c,1}}{\sqrt{2\pi(S_{2}-S_{1})^{3}}}\ \cr && \exp\!\left[ -\frac{(\delta_{\rm c,2}-\delta_{\rm c,1})^{2}}{2(S_{2}-S_{1})} \right] \left| \frac{dS_{2}}{dM_{2}} \right| ,\end{aligned}$$ where $\delta_{\mathrm{c},i} \sim 3 (12\pi)^{2/3} /20 D(z_i)$ (its useful approximate formula may be found in Kitayama, Suto 1996) is the critical over-density of the mass density field at a redshift of $z_{i}$, $D(z_i)$ is the linear growth rate, and $S_{i}\equiv\sigma^{2}(M_{i})$ is a mass variance of the density field top-hat smoothed over the mass scale $M_i$. Since equation (\[eq:eps-mass\]) is the *mass-weighted* probability for $M_2$, it is easily translated to the *number-weighted* probability that we need in the halo number counting: $$\label{eq:eps-num} \frac{dN}{dM_{2}}(M_{2},z_{2}|M_{1},z_{1}) = \frac{M_{1}}{M_{2}}\frac{dP}{dM_{2}}(M_{2},z_{2}|M_{1},z_{1}) .$$ (80mm,80mm)[Fig01OneStep.eps]{} Figures \[fig:OneStep\] and \[fig:100Step\] present how the progenitor number distribution of the merger tree realizations reproduces the theoretical prediction: $M_{\mathrm{root}}=1.3\times 10^{11}\;M_{\odot}$ (*Left*) and $1.3\times 10^{14}\;M_{\odot}$ (*Right*). In these plots, we adopt the logarithmically equal timestep in redshift: $$\begin{aligned} \label{eq:zbin} z_{\scriptscriptstyle(i)}=(1+z_{\mathrm{min}})\times \left( \frac{1+z_{\mathrm{max}}}{1+z_{\mathrm{min}}} \right)^{i/N_{\mathrm{step}}}-1 \cr \qquad (i=1, \cdots , N_{\rm step}),\end{aligned}$$ where $N_{\mathrm{step}}$ is the total number of the redshift bins. We defer the discussion concerning the choice of $N_{\mathrm{step}}$ to the next subsection (\[sec:timestep\]), and fix $N_{\mathrm{step}}=100$ throughout this subsection. The top panels in figure \[fig:OneStep\] show the result for the 1st timestep ($i=1$) corresponding to $z_{\scriptscriptstyle(1)}=0.028$ according to equation (\[eq:zbin\]). We call this model SK-n indicating the Somerville and Kolatt (1999) method with the number-weighted probability. The symbols indicate the average $(M_2/N_{\rm ens})(\Delta N_2/\Delta M_2)$ with the quoted error-bars being the corresponding one-sigma dispersion, where $\Delta N_2$ is the number of progenitors in the range of mass $M_2 \sim M_2 + \Delta M_2$, and we adopt $\Delta \log_{10}M_2 = 0.1$. In the SK-n model we generate the random numbers according to the number-weighted probability distribution function \[equation (\[eq:eps-num\])\]. Also we have to set the lower limit on the progenitor mass in adopting the SK-n model so as to avoid a divergent total probability. We adopted the lower limit of $10^{-3}M_{\rm res}$, and made sure that the mass function of progenitors of the mass range of our interest $M>M_{\mathrm{res}}$ is properly reproduced. The solid triangles show our result based on the algorithm outlined in the previous subsection. Somewhat surprisingly, they are completely different from the theoretical distribution that we use in generating the trees (solid curve). Note that figure \[fig:OneStep\] plots the number distribution multiplied by $M_2$, $M_2 \, dN/dM_2= M_1 \, dP/dM_2$ \[see equation (\[eq:eps-num\])\]. To understand the origin of the discrepancy, we generate the progenitors at the 1st timestep for $N_{\mathrm{ens}}$ realizations simultaneously as long as they satisfy $$\begin{aligned} \label{eq:nmassconserve} \sum_{i=1}^{N'} M_2^i < N_{\mathrm{ens}} \left[M_1 - \Delta M_{\rm acc}(<M_{\rm res}) \right] ,\end{aligned}$$ instead of the mass conservation \[equation (\[eq:massconserve\])\] for each individual parent halo. In the above, $N'$ is not the number of progenitors for a single halo at $z=z_{\rm min}$, but for an ensemble of $N_{\rm ens}$ halos with the same mass $M_{\rm root}$. The resulting distribution is plotted in open circles, and in fact shows good agreement with the theoretical curve. This is simply because we attempt to generate a joint distribution of progenitors with a repeated use of the conditional probability \[equation (\[eq:eps-num\])\] *incorrectly*; except for the first progenitor, the mass conservation for each halo \[equation (\[eq:massconserve\])\] introduces an additional cutoff at higher mass in the selection probability of progenitors. In fact the conditional probability \[equation (\[eq:eps-num\])\] for $(z_2-z_1)/z_1 \ll 1$ is sharply peaked at a mass scale $M_2$ just below the parent mass $M_1$, and thus even a small value of the first progenitor mass may effectively bias not to choose remaining progenitors in the peak. Thus, the resulting distribution is significantly biased toward low-mass objects, i.e., the number density of the low-mass objects exceeds the theoretical predictions by an order of magnitude (top panels in figure \[fig:OneStep\]). One way out of this problem is to generate many ($>100$) realizations simultaneously, as Kauffmann and White (1993) adopted. Even in this case, one needs to specify an additional assumption on how to plant a set of progenitors in a single merger tree *by hand*. Moreover, a practical implementation of this method requires one to discretize the halo mass, and thus becomes computationally demanding as both the mass and time resolutions increase. Another possibility is to artificially distort the input conditional probability so that the selected progenitors obey the distribution \[equation (\[eq:eps-num\])\]. While the required correction may be a fairly definite mathematical problem, we do not know the exact answer, and thus have to proceed in a phenomenological fashion. Basically this is the approach taken by Somerville and Kolatt (1999) and Sheth and Lemson (1999), who adopted the *mass-weighted* probability \[equation (\[eq:eps-mass\])\] as the theoretical input. The middle and bottom panels in figure \[fig:OneStep\] show the resulting distribution for SK-m ([@SK99]-mass weighted) and SL-m ([@SL99]-mass weighted), respectively. Clearly the resulting distributions (filled triangles) become much closer to equation (\[eq:eps-num\]) under the constraint \[equation (\[eq:massconserve\])\] although their original distributions \[i.e., without the constraint (\[eq:massconserve\])\] plotted in open circles are completely different. As this indicates, the input conditional probability for the current purpose should be small at lower mass scales of $M_2$ relative to equation (\[eq:eps-num\]). Thus, we also attempted to make the probability proportional to $(M_1/M_2)^\alpha (dP/dM_2)$. Note that the mass- and number-weighted probabilities \[equations (\[eq:eps-mass\]) and (\[eq:eps-num\])\] correspond to $\alpha=0$ and $1$. We were not able to obtain a similar degree of agreement for a value of $\alpha$ very different from 0, but did not find a significant change for $-0.2\lesssim \alpha \lesssim +0.2$. Thus, we decided to adopt $\alpha=0$ (the mass-weighted probability) as Somerville and Kolatt (1999). This choice has an advantage that a numerical routine to generate random numbers becomes easier than the cases of $\alpha \not= 0$. Define $x_2 = (\delta_{\rm c,2}-\delta_{\rm c,1})/\sqrt{S_{2}-S_{1}}$ that obeys Gaussian distribution of a unit variance. Equation (\[eq:eps-mass\]) implies that the desired distribution of $M_2$ can be simply given via $S_2 = S(M_2) = S_1 + (\delta_{\rm c,2}-\delta_{\rm c,1})^2/x_2^2$. Since Gaussian-distributed random numbers can be implemented easily, all we have to do is to supply the inverse function of the mass variance, $M_2 = S^{-1}(S_2)$. Our SK-m implementation seems to yield a larger discrepancy between theory and the merger tree realizations at small $M_2$ regimes than their original results. This may be due to the different choice of the timestep and the condition how to stop selecting progenitor halos. In any case, however, this discrepancy rapidly fades away in constructing the merger tree using many timesteps, as we show in figures \[fig:100Step\] and \[fig:1000Step\]. (80mm,80mm)[Fig02z01.eps]{} Figure \[fig:100Step\] plots a snapshot of the progenitor distribution at the 25th timestep ($z=z_{\scriptscriptstyle(25)}=1.0$), which makes sure that the mass-weighted probability reasonably works, even when we trace the merger tree by many steps. Also SK-m works a bit better than SL-m especially at small mass scales. Originally SL-m was proposed to correct for the halo exclusion effect, but does not work so efficiently at least in the range of parameters we surveyed. Figure \[fig:100Step\] also indicates that SK-n is substantially different from the analytical solution. This is due to the fact that SK-n tends to select relatively less massive progenitors preferentially (see figure \[fig:OneStep\]), and this tendency simply accumulates in many steps. On the contrary, the behavior of SK-m and SL-m becomes closer to the analytical solution than in the case of figure \[fig:OneStep\]. This is because the latter two models well approximate the probability distribution around $M_1$, which is the most important range when constructing real merger trees with many timesteps. Timestep \[sec:timestep\] ------------------------- The next question that we address is the appropriate choice of the timestep. While this is an equally important problem in the Monte-Carlo modeling, the previous authors did not discuss it in an explicit manner. (60mm,60mm)[Fig03Timescale.eps]{} Obviously, the timestep needs to be smaller than the dynamical timescale of halos just virialized at the redshift, $t_{\rm dyn, vir}(z)$, because they are the objects that serve as the initial condition for the Monte-Carlo modeling. Figure \[fig:timescale\] compares this timescale and our choice \[equation (\[eq:zbin\])\] for $N_{\rm step}=10$, 100, 1000. Also, we plot the cosmic time $t_{\rm cosm}(z)$ and the timestep corresponding to the linearly equal bin ($\Delta z=0.15$). Even this simple comparison indicates that the logarithmic time bin with $N_{\rm step} \gtrsim 100$ is required. While the important question is how small timescales one should resolve, it critically depends on the problem that one would like to address. Therefore, we rather ask how many timesteps we need to reproduce the progenitor distribution. (80mm,80mm)[Fig04Nstep1000.eps]{} To see explicitly how different values of $N_{\rm step}$ affect the realizations of merger trees, we plot in figure \[fig:1000Step\] the progenitor distribution with $N_{\rm step}=1000$ at redshifts of $z_{\scriptscriptstyle(1)}=0.003$, $z_{\scriptscriptstyle(146)}=0.499$, and $z_{\scriptscriptstyle(250)}=1.0$. A comparison of figures \[fig:100Step\] and \[fig:1000Step\] indicates that the average progenitor distribution is indeed slightly better reproduced by $N_{\rm step}=1000$ than $N_{\rm step}=100$, particularly at small mass scales. (80mm,80mm)[Fig05progmassfrac.eps]{} The difference between $N_{\mathrm{step}}=100$ and 1000 is more clearly illustrated when we plot the cumulative mass fraction of progenitors of mass exceeding a threshold value of $M_{\rm thre}$. More specifically, figure \[fig:massfrac\] compares the theoretical prediction, $$F_{\rm th}(>M_{\mathrm{thre}}; z) = \int^{M_{\mathrm{root}}}_{M_{\rm thre}}\!\!\!\!dM \frac{dP}{dM}(M,z|M_{\mathrm{root}},z_{\mathrm{min}}), \label{eq:fracpth}$$ with the average from the tree realizations, $$F_{\rm model}(>M_{\rm thre}; z) =\frac{1}{N_{\mathrm{ens}}M_{\mathrm{root}}} \sum_{M_{\mathrm{halo}}(z)\ge M_{\rm thre}}\mbox{\hspace{-7mm}}M_{\rm halo}(z),$$ for $M_{\rm root} = 1.3\times10^{11}\;M_{\odot}$ (*Left*) and $1.3\times10^{14}\;M_{\odot}$ (*Right*). In all panels shown here, the results with $N_{\rm step}=1000$ better reproduce the theoretical prediction, mainly because of the small-scale behavior (see figures \[fig:100Step\] and \[fig:1000Step\]). Since these small mass progenitors at earlier redshifts significantly contribute to radiative cooling and thereby subsequent star formation in the entire halo, this difference is indeed critical in the Monte-Carlo modeling of galaxy formation. We discuss the effect on gas cooling explicitly in the next section. (60mm,60mm)[Fig06ProgMassFrac.eps]{} We interpret the above as an empirical result due to the balance between the mass-weighted probability and the timesteps. If the use of the mass-weighted probability *were* strictly justified, the proper realizations with a smaller timestep would become more difficult numerically, and there is no reason why we could obtain better agreement with a larger $N_{\rm step}$. On the other hand, we understand that the mass-weighted probability is nothing but a phenomenological remedy of the problem, and with this choice $N_{\rm step}=1000$ seems to work better than $N_{\rm step}=100$ empirically. In fact, still larger values of $N_{\rm step}$ do not necessarily improve the result. Figure \[fig:mflargeNstep\] is a similar plot as figure \[fig:massfrac\] for $M_{\rm root}=1.3\times10^{14}\;M_\odot$, but with increasing $N_\mathrm{step}$. The cumulative mass fraction for $M>10^{11}\;M_\odot$ is almost unchanged, but the contribution from smaller mass progenitors steadily increases as $N_\mathrm{step}$ becomes larger. This reflects the fact that the empirical use of the mass-weighted conditional probability does not guarantee convergence of the result with respect to $N_\mathrm{step}$. We thus conclude that $N_{\mathrm{step}}\sim 1000$ is the optimal value to reproduce the Press–Schechter mass function in our method. Number of Progenitors --------------------- Finally, we briefly discuss how many progenitors, $N_{\rm prog}$, one should keep in order to properly reproduce the merger trees. Figure \[fig:Nprog\] displays the distribution functions at $z=1$ for the merger tree of $M_{\mathrm{root}}=1.3\times 10^{14}\;M_{\odot}$ at $z=0$. In this particular example, we use $N_{\rm step}=1000$ and the results are averaged over $N_{\rm ens}=100$ realizations for each $N_{\rm prog}$. Obviously a smaller value of $N_{\rm prog}$ does not properly link the merger tree back to higher redshifts, and the number of small-mass halos is systematically under-predicted compared with the extended Press–Schechter model (solid curve). While we do not set any upper limit on $N_{\mathrm{prog}}$, figure \[fig:Nprog\] indicates that $N_{\mathrm{prog}} \gtrsim 5$ is acceptable given the accuracy of the present scheme. Although some authors employ a binary merger tree in Monte-Carlo modeling, that scheme needs to be adjusted with a careful choice of the timestep and other parameters. (60mm,60mm)[Fig07Nprog.eps]{} In conclusion, we have found that a reasonable agreement between the theory and the merger tree realizations can be obtained by employing the mass-weighted conditional probability into the Somerville and Kolatt (1999) scheme with $N_{\mathrm{step}}\sim 1000$. Gas Cooling =========== So far, we have restricted our discussion to the gravitational aspect of the halo evolution. We next consider how the resulting merger trees affect the cold gas fraction in an individual halo. Of course, we must eventually discuss the effects on the efficiency of star formation, but we focus on gas cooling alone, since modeling star formation, the feedback from supernovae, the chemical evolution and so on necessarily introduce additional (numerical) parameters and the interpretation becomes more complicated. Thus, the principal aim of this section is to see if we can achieve a convergence of the cold gas fraction from different realizations of the merger trees. Description of Gas Cooling -------------------------- Our prescription of gas cooling in the merger trees goes as follows. First, we assume that the density profile of dark halos obeys the universal shape ([@nfw96]): $$\label{eq:nfw} \rho_{\rm halo}(r;M)= \left\{\begin{array}{cc} \displaystyle \frac{\bar{\rho}(z) \, \delta_\mathrm{c}}{ (r/r_\mathrm{s}) (1+r/r_\mathrm{s})^2} & (r< r_{\rm vir}) \\ \displaystyle 0 & (r>r_{\rm vir}), \end{array}\right.$$ where $\bar\rho(z) \equiv \Omega_0 \rho_{\rm c0} (1+z)^3$ is the mean density of the universe at $z$, $\rho_{\rm c0}$ is the present critical density, $\delta_\mathrm{c}(M)$ is the characteristic density excess, and $r_{\rm vir}(M)$ and $r_\mathrm{s}(M)$ indicate the virial radius and the scale radius of the halo, respectively. The virial radius is defined according to the spherical collapse model as $$r_{\rm vir}(M) \equiv \left(\frac{3M}{4\pi\bar{\rho} \Delta_{\rm nl}}\right)^{1/3} , \label{eq: r_vir}$$ and useful approximation for the critical over-density, $\Delta_{\rm nl}= \Delta_{\rm nl}(\Omega_0,\lambda_0)$, may be found in Kitayama and Suto (1996). The two parameters, $r_{\rm s}$ and $r_{\rm vir}$, are related in terms of the concentration parameter, $$\label{eq: concentration} c(M,z) \equiv \frac{r_{\rm vir}(M,z)}{r_\mathrm{s}(M,z)}.$$ We use an approximate fitting function from the simulation data of Bullock et al. (2001), $$c(M,z)=\frac{8.0}{1+z}\;\left(\frac{M}{10^{14}\;M_{\odot}}\right)^{-0.13}. \label{eq: c_Bullock}$$ The condition that the total mass inside $r_{\rm vir}$ be equal to $M$ relates $\delta_{\rm c}$ to $c$ as $$\delta_\mathrm{c} = {\Delta_{\rm nl} \over 3} {c^3 \over \ln(1+c) -c/(1+c)}.$$ @MSS97 showed that if the hot gas is isothermal and in hydrostatic equilibrium, the gas density profile is well approximated by the isothermal $\beta$-model, $$\rho_{\rm hot}(r) = \frac{\rho_{\rm hot, 0}}{[1+(r/r_\mathrm{c})^{2}]^{3\beta/2}}, \label{eq:rhogas}$$ where $r_\mathrm{c}\sim 0.22r_\mathrm{s}$. We fix $\beta=2/3$ for simplicity. The amplitude $\rho_{\rm hot, 0}$ is computed so as to reproduce the total hot gas in the halo when integrated up to $r=r_{\rm vir}$. The hot gas is gradually converted to cold gas according to the prescription below, but still the total baryon (hot + cold) fraction within the virial radius of each halo is set to the cosmic average, $\Omega_{\rm B}/\Omega_0$. Once the gas profile is specified, one can compute the cooling timescale at radius $r$ from the center of the halo, $$\label{eq:tcool} t_{\mathrm{cool}}=\frac{3}{2} \frac{\rho_{\mathrm{hot}}(r)}{\mu m_{\mathrm{p}}} \frac{k_{\mathrm{B}}T_{\mathrm{gas}}} {\Lambda(T_{\mathrm{gas}})n_{\mathrm{H}}^{2}(r)},$$ where $\mu$ is the mean molecular weight, $m_{\mathrm{p}}$ the proton mass, $k_{\mathrm{B}}$ the Boltzmann constant, $\Lambda(T_{\mathrm{gas}})$ the radiative cooling function for gas of temperature $T_{\mathrm{\mathrm{gas}}}$, and $n_{\mathrm{H}}(r)$ the number density of hydrogen (including both neutral and ionized). We assume that the gas has the primordial abundance of hydrogen and helium ($X=0.76$ and $Y=0.24$, and thus $n_{\rm H}=\rho_{\rm hot}/Xm_{\rm p}$), and compute the corresponding cooling function $\Lambda(T)$ (e.g., Sasaki and Takahara 1994). Since we neglect the molecular and metal cooling, $\Lambda(T)=0$ at $T\le 10^{4}\;\mathrm{K}$. (60mm,60mm)[Fig08tvirtom.eps]{} We further assume that the temperature of the hot gas, $T_{\mathrm{gas}}$, is equal to the virial temperature, $T_{\mathrm{vir}}$, of the halo. The relation between the virial temperature and the mass of a halo is plotted in figure \[fig:tvirtom\]. We can then solve equation (\[eq:tcool\]) for the cooling radius, $r_{\mathrm{cool}}$, within which the gas can cool within a given cooling timescale ($\tau_{\rm cool}$): $$\hspace*{-0.5cm} r_{\rm cool}(\tau_{\rm cool}) \equiv 0.22 \frac{r_{\rm vir}(M,z)}{c(M,z)} \sqrt{\frac{2\mu \Lambda(T_{\mathrm{vir}}) \rho_{\rm hot,0}\tau_{\rm cool}} {3m_\mathrm{p} k_{\rm B}T_{\rm vir}} - 1 } .$$ It now remains to define the origin of $\tau_{\rm cool}$. Actually, this is fairly arbitrary in a sense, and we adopt the following simple picture. When a halo of mass $M_{\rm f}$ *forms* at the formation redshift, $z_{\rm f}$, its hot gas is supposed to reach the profile \[equation (\[eq:rhogas\])\] instantaneously. This is defined to be the origin of $\tau_{\rm cool}$ for the halo. In the subsequent timesteps, we neglect the change in the hot gas profile even if the halo mass ($M$) grows due to mergers and the cooling radius is computed with $\tau_{\rm cool}$ set to the elapsed cosmic time since $z_{\rm f}$. When $M$ exceeds $2M_{\rm f}$, the halo is replaced by a *newly* formed massive halo, and the hot gas profile is reset to the profile \[equation (\[eq:rhogas\])\] corresponding to the new mass and the virial temperature, and we reset the origin of $\tau_{\rm cool}$ as the new formation epoch. Incidentally, we made sure that the value of $1.5M_{\rm f}$ instead of $2M_{\rm f}$ does not change the result, which is consistent with the finding of Cole et al.(2000). We apply this procedure for all halos, and the cold gas in each progenitor halo is simply accumulated (without reheated) according to the merger trees. Cold Gas Fraction in the Monte-Carlo Realization of Merging Histories --------------------------------------------------------------------- In a practical implementation of the merger tree algorithm, one has to stop tracing the progenitors of halos of mass below the resolution mass, $M_{\rm res}$. We discuss the relevance of our choice of $M_{\rm res}$ by looking at the cold gas fraction. Previous authors often apply the cutoff at a fixed mass or a circular velocity of halos; for instance, @Cole00 consider halos with $M > 5\times 10^{9}h^{-1}\;M_{\odot}$ in their merger trees, while Somerville and Primack (1999) take account of halos with the circular velocity exceeding $40\;\mathrm{km\ s}^{-1}$ (corresponding to the virial temperature $T_{\rm vir}\sim 6\times10^{4}\;\mathrm{K}$). The latter condition comes from an estimate of the smallest scale of halos which can cool in the presence of the UV background (e.g., [@TW96]; [@KI00]). In our present analysis, the cooling function, $\Lambda(T)$, vanishes below $T=10^4\;\mathrm{K}$, since we neglect both the UV heating and the metal/molecular cooling. Thus, we set the resolution mass, $M_{\rm res}(z)$, as $M(T_{\rm vir}=10^4\;{\rm K})$. As figure \[fig:tvirtom\] shows, this scale increases rapidly with time, resulting in a significant improvement of the computing time. On the other hand, this might systematically underestimate the cold gas fraction, since halos of mass below $M_{\rm res}(z_1)$ may have progenitors of mass larger than $M_{\rm res}(z_2)$ at the earlier epoch ($z_2>z_1$). Fortunately, this is not an important effect, as we show below. (60mm,60mm)[Fig09Gasmres.eps]{} To see this in detail, we plot in figure \[fig:convmres\] the cold gas fraction averaged over all progenitors at $z$ of a root halo of mass $M_{\rm root}$ at $z=0$, $$\begin{aligned} \label{eq:coldgasfrac} f_{\rm cold}(z;M_{\rm root}) \equiv \frac{\Omega_0}{\Omega_{\rm B}M_{\rm root}} \sum_{M_{\rm prog}>M_{\rm res}} M_{\rm cold}(M_{\rm prog}) ,\end{aligned}$$ for a merger tree with $N_{\rm step}=1000$. If we adopt a constant value for the resolution mass, $M_{\rm res} < 10^8\;M_{\odot}$ yields the convergent result for the cold gas fraction. Exactly the same convergence is obtained for the time-dependent $M_{\rm res}(z)$ when the value is set to $M(T_{\rm vir}=10^4\;{\rm K})$, but not if we use $M(T_{\rm vir}=5\times 10^4\;{\rm K})$, for instance. This critical value is expected to vary depending on the thermal history of the universe, but the appropriate value for $T_{\rm vir}$ is straightforwardly read off from the relevant cooling function. Actually, $M_{\rm res}(z)$ increases in this case and exceeds $10^9\;M_{\odot}$ (see figure \[fig:tvirtom\]). Thus, the required merging tree is less demanding from a computational point of view than that for $M_{\rm res}=10^{8}\;M_{\odot}$, for instance, as illustrated in table \[tab:cputime\]. Thus we decide to choose $M_{\rm res}(z) = M(T_{\rm vir}=10^4\;{\rm K})$ for gas with the primordial abundance. Finally, we show the convergence with respect to $N_{\rm step}$. Figure \[fig:convnsamp\] plots the cold gas fraction for $M_{\rm root}=1.3\times 10^{11}\;M_{\odot}$ (*Upper*) and $1.3\times 10^{14}\;M_{\odot}$ (*Lower*) for merging trees with different $N_{\rm step}$. The results are fairly in agreement for small $M_{\rm root}$, but are very different for large $M_{\rm root}$. This is because the merger tree at small mass scales, especially at $M(T_{\rm vir}=10^4 {\rm K}) < M < 10^{10}M_\odot$, is well reproduced only when we use $N_{\rm step}=1000$ (see figure \[fig:massfrac\]). When we repeat the same calculation sampling every 10 steps from the $N_{\rm step}=1000$ tree, the result is almost indistinguishable. Thus, we conclude that the progenitor distribution at small scales is quite essential in the estimate of the cold mass fraction of large halos. Incidentally the use of the timestep much smaller than $t_{\rm dyn, vir}(z)$ enables one to describe the collapse and gas cooling more realistically than the instantaneous approximation. While we do not attempt this in the present paper, this would improve the estimate of the cold gas fraction quantitatively. (60mm,60mm)[Fig10Sampling.eps]{} Conclusions and Discussion ========================== We attempted several convergence tests of the merger trees generated with the Monte-Carlo method. While this method provides a useful tool for modeling galaxy formation in a complementary manner to more intensive cosmological simulations with ad hoc recipes of galaxy formation (e.g., Cen, Ostriker 1992; Weinberg et al. 1997; Yoshikawa et al. 2001), the lack of an explicit expression for the joint distribution function of progenitors \[equation (\[eq:jointprob\])\] requires one to put an additional assumption in practice. We confirmed that a repeated use of the *mass-weighted* conditional probability \[equation (\[eq:eps-num\])\] reasonably reproduces the progenitor distribution predicted in the extended Press–Schechter theory if one adopts fairly small timesteps in redshift, $N_{\rm step} \sim 1000$, a factor of ten larger than a typical value used in previous work. We note, however, that one can alternatively achieve a similar result by fine-tuning the timestep as a function of $M_{1}$ (e.g., [@SK99]) instead of equation (\[eq:zbin\]), as we adopted here. One may avoid the above problem also by using merger trees generated via $N$-body simulations ([@gif99a]; Somerville et al. 2001). In fact, they claim that the agreement between the $N$-body simulations and the Monte-Carlo method is good. Benson et al. (2001) compared the SPH simulations and the Monte-Carlo modeling, and concluded that both agree with each other on the cold gas mass fraction and mass function of the halos. While this comparison is encouraging, it is not yet clear if the lack of the joint distribution function of progenitors \[equation (\[eq:jointprob\])\] in the Monte-Carlo modeling may not be essential. Thus, further detailed studies are definitely important to test the reliability of [*both*]{} $N$-body and the Monte-Carlo modeling in generating merger tree realizations. We thank Kazuhiro Shimasaku and Tomonori Totani for discussions and suggestions in the early phase of this work. This research was supported in part by the Grant-in-Aid from Monbu-Kagakusho, Japan (07CE2002, 12304009, 12640231). T.K. gratefully acknowledges support from Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists (7202). Arimoto, N., & Yoshii, Y. 1986, , 164, 260 Baugh, C. M., Cole, S., Frenk, C. S., & Lacey, C. G. 1998, , 498, 504 Benson, A. J., Pearce, F. R., Frenk, C. S., Baugh, C. M., & Jenkins, A. 2001, , 320, 261 Bond, J., Cole, S., Efstathiou, G., & Kaiser, N. 1991, , 379, 440 Bower, R. 1991, , 248, 332 Bruzual, A. G. 1983, , 273, 105 Bruzual, A. G., & Charlot, S. 1993, , 405, 538 Bullock, J. S., Kolatt, T. S., Sigad, Y., Somerville, R. S., Kravtsov, A. V., Klypin, A. A., Primack, J. R., & Dekel, A. 2001, , 321, 559 Charlot, S., & Bruzual, A. G. 1991, , 367, 126 Cen, R., & Ostriker, J. P. 1992, , 399, L113 Cole, S. 1991, , 367, 45 Cole, S., Aragón-Salamanca, A., Frenk, C. S., Navarro, J. F., & Zepf, S. 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P., & Suto, Y. 2001, , 558, 520 Symbol Adopted value Physical meaning ------------------- -------------------------------- ---------------------------------------------------------- $M_{\rm root}$ — mass of halo at $z=z_{\mathrm{min}}$ $T_{\rm vir}$ — virial temperature of halo $M_{\rm res}$ $M(T_{\rm vir}=10^4\;{\rm K})$ minimal mass of progenitors resolved in each merger tree $\tau_{\rm cool}$ halo mass doubling time cooling time scale for gas in hosting halos $N_{\rm step}$ 1000 number of redshift bins (logarithmically equal interval) $N_{\rm ens}$ — number of realizations of merger trees $z_{\rm min}$ 0 minimum redshift of merger trees $z_{\rm max}$ 15 maximum redshift of merger trees : Summary of the variables used in building merger trees and in gas cooling. \[tab:parameters\] --------------------- ------ -------------------------------------- ---------- ------ ------ $1.3\times 10^{11}$ 100 $10^8\;M_{\odot}$ 3876 3.4 0.55 $1.3\times 10^{11}$ 100 $M(T_\mathrm{vir}=10^4\;\mathrm{K})$ 1687 2.9 0.43 $1.3\times 10^{11}$ 1000 $10^8\;M_{\odot}$ 56057 15.1 4.4 $1.3\times 10^{11}$ 1000 $M(T_\mathrm{vir}=10^4\;\mathrm{K})$ 20556 6.8 2.3 $1.3\times 10^{14}$ 100 $10^8\;M_{\odot}$ 1122420 244 75.6 $1.3\times 10^{14}$ 100 $M(T_\mathrm{vir}=10^4\;\mathrm{K})$ 640409 137 46.5 $1.3\times 10^{14}$ 1000 $10^8\;M_{\odot}$ 29725672 6514 1993 $1.3\times 10^{14}$ 1000 $M(T_\mathrm{vir}=10^4\;\mathrm{K})$ 18875982 4140 1400 --------------------- ------ -------------------------------------- ---------- ------ ------ : CPU timing of the Monte-Carlo modeling for one merger tree on a 21264 alpha $600\;\mathrm{MHz}$ machine. \[tab:cputime\]

--- abstract: 'A fuzzy expert system (FES) for the prediction of prostate cancer (PC) is prescribed in this article. Age, prostate-specific antigen (PSA), prostate volume (PV) and $\%$ Free PSA ($\%$FPSA) are fed as inputs into the FES and prostate cancer risk (PCR) is obtained as the output. Using knowledge based rules in Mamdani type inference method the output is calculated. If PCR $\ge 50\%$, then the patient shall be advised to go for a biopsy test for confirmation. The efficacy of the designed FES is tested against a clinical data set. The true prediction for all the patients turns out to be $68.91\%$ whereas only for positive biopsy cases it rises to $73.77\%$. This simple yet effective FES can be used as supportive tool for decision making in medical diagnosis.' author: - 'Juthika Mahanta$^{1,}$[^1]' - 'Subhasis Panda$^{2,}$[^2]' bibliography: - 'biblio.bib' title: Fuzzy expert system for prediction of prostate cancer --- Introduction {#intro} ============ Artificial intelligence (AI) is defined as the intelligence processed by machines. With the advancement in the computer system, machines exhibit tasks which normally need human intelligence. Development of AI techniques has revolutionized many areas like robotics, transportation, education, marketing etc. including medical diagnosis and health care. Medical diagnosis deals with the analysis of complex medical data. The primary job in medical diagnosis is to reach to a decision using expert’s logical reasoning. Handling large complex data and many uncertainties make this job very difficult. AI appears to be very handy for this job. AI in medical diagnosis has added expert human reasoning in simulation of computer-aided diagnosis process. There are different AI methods used in medical diagnosis, fuzzy logic is one of the most popular one. AI is the technique of mimicking human intelligence by the help of advance computer systems. Human brain takes natural languages as inputs which aren’t feasible to be represented by Boolean logic (either true or false). So, there is a requirement of representation outside these two possibilities. Fuzzy logic exactly does that. Fuzzy logic appears closer to the way human brain works. Therefore, in AI, fuzzy logic shows the sign of being a natural choice.\ Uncertainties and imprecision are connected with every aspect of our day to day life activities. Specifically in medical diagnosis domain, one encounters a lot of uncertainty and vagueness. It becomes very difficult to identify a particular disease from the said symptoms of the patients, as it contains lot of approximate and inaccurate information. On the other hand, a particular symptom can possibly lead to many different disjoint diseases, whereas for the same disease, the symptom may manifest itself in completely different ways from person to person. There are inherent uncertainties in the process of decision making in medical diagnosis, even for an expert too. To tackle these inexact, linguistic inputs, fuzzy logic based expert system is turned out to be very useful. The concepts of fuzzy set and fuzzy logic were introduced by Prof. L. A. Zadeh [@zadeh]. In contrast to binary logic, fuzzy logic deals with multi-valued logic which is a mathematical tool to represent the real world effectively. Due to its usefulness and simplicity, fuzzy logic has drawn huge attention of interdisciplinary researchers round the globe.\ Fuzzy logic based expert systems are widely used in many areas of medical diagnosis and decision-making process. Particularly, in the area of prostate cancer, very few literature are available [@saritas; @benecchi; @yuksel; @seker; @kar; @castanho; @abbod_review; @lorenz]. These literature address the problem in different angles and also use fuzzy logic in disjoint ways. Some researchers have used hybrid system to treat this problem. We focus our attention to fuzzy logic based expert system to predict the prostate cancer risk. From careful study of the literature, we found that $\%$FPSA is a very crucial parameter, along with age, PSA and PV for early detection of PC. Therefore, we formulate a fuzzy expert system by taking care of all these inputs.\ The paper is organized as follows. Section \[fes\] describes the fuzzy expert system, where all the inputs and the output are discussed. In section \[res\], we apply our FES to a medical data set and discuss our findings. Finally we summarize and conclude in section \[con\]. Fuzzy Expert System (FES) {#fes} ========================= An expert system which uses fuzzy logic instead of Boolean logic is called fuzzy expert system (FES). A FES is a form of artificial intelligence which deals with membership functions and some prescribed rule base to evaluate a set of data. Fuzziness is introduced to the crisp inputs of a FES by means of suitable membership functions. Once membership functions are defined for all input variables, then they are fed to a particular inference method for further action. Here, we have used Mamdani (max-min) inference method which is most popular in literature. Rules of the FES developed here are of IF-THEN form. Mamdani type inference method results in fuzzy sets as output. For a given set of input values, some relevant rules will be fired to produce a fuzzy output in Mamdani type inference method. Fuzzy output is defuzzified using different techniques to obtain crisp output. Centroid method is used for defuzzification in our FES. General structure of a FES is shown in below figure \[fig:0\]. ![General architecture of a FES.[]{data-label="fig:0"}](fes_block.eps){width="\textwidth"} We have used medical data of the patients as given in reference [@saritas]. Depending on the data set, the range of different inputs are determined. In the following two subsections, we discuss about different inputs and the output of the FES. Input variables --------------- ### Age Age of man is an important parameter to calculate the risk factor for prostate cancer. For a man having no family history of PC, the chance of getting it increases after the age of 50. This number changes from race to race. However, two out of three PCs are diagnosed in men at the age of 65 or above. The input variable “age" is represented by four fuzzy sets, namely, “very young", “young", “middle age" and “old". First and fourth fuzzy sets are represented by trapezoidal membership functions whereas for second and third we have used triangular membership functions. Table \[tab:1\] lists crisp sets and the corresponding fuzzy sets for the input “age". The membership functions for the same are plotted in figure \[fig:1\]. [lll]{} Input variable & Crisp set & Fuzzy set\ Age (year) & 0-30 & Very young\ & 20-50 & Young\ & 30-60 & Middle age\ & 40-100 & Old\ ![Membership functions for “age".[]{data-label="fig:1"}](age.eps){width="\textwidth"} ### Prostate-specific Antigen (PSA) PSA has altered drastically the management of prostate cancer in men. The PSA test for blood can provide early stage detection of PC [@brawer99]. PSA is a protein secreted by the prostate gland which helps to keep the semen in liquid form. Some parts of this protein will pass into blood which give rise to the increase in normal PSA level. Elevation in PSA level in blood depends up on the health of prostate gland and the age of the person. A healthy prostate will release less PSA in blood compared to a cancerous gland. So, a rise in PSA level over normal range could be a possible indicator of PC. Although, elevated PSA level may be caused due to other factors like acute bacterial prostatitis, enlargement of prostate and other urinary retention. The measurement of PSA is expressed as nanograms per milliliter of blood. The normal range of PSA number can be age specific and also race specific. The input variable “PSA" is represented by five fuzzy sets, e.g., “very low", “low", “middle", “high" and “very high". For the first and fifth sets we have used trapezoidal membership functions while for the rest, triangular membership functions are used. In table \[tab:2\], we have shown the crisp sets and the corresponding fuzzy sets. The plot of the membership functions for the input “PSA" is displayed in figure \[fig:2\]. [lll]{} Input variable & Crisp set & Fuzzy set\ PSA (ng/ml) & 0-4 & Very low\ & 2-8 & Low\ & 4-12 & Middle\ & 8-16 & High\ & 12-50 & Very high\ ![Membership functions for “PSA".[]{data-label="fig:2"}](psa.eps){width="\textwidth"} ### Prostate Volume (PV) A healthy human male’s prostate is marginally larger than a walnut. It is a crucial parameter for early detection of PC. There is a characteristic pattern in the growth of prostate with age. That pattern can change from race to race. With the increase in the prostate volume there is a possibility of sampling error in systematic sextant needle biopsy. It is wise to use prostate volume as a factor while determining the necessity to repeat biopsy with initial negative result [@brawer_pv]. Prostate is mainly divided in four zones in pathological terminology and total prostate volume as well as transition zone volume are measured in ultrasound. According to transrectal ultrasound (TRUS) guidance [@zhang], prostate width ($W$) (maximal transverse diameter) is estimated on an axial image. Prostate length ($L$) (longitudinal diameter, the distance between proximal external sphincter and urinary bladder) and height ($H$) (maximal antero-posterior diameter) are measured on a mid-sagittal image [@bangma]. The total prostate volume (TPV) is calculated using the prolate elliptical formula, TPV = $\frac{\pi} {6} \times W \times L \times H$. The transition zone volume can be calculated using the same formula by measuring the required dimensions from the ultrasound. The length, width and height changes with the age. The rate of change for the length is significant compared to other two dimensions as the man approaches the age 60. So, based on the above information we have divided the input “prostate volume" in four fuzzy sets such as “small", “middle", “big" and “very big". Trapezoidal and triangular membership functions are used to represent them. Crisp sets, fuzzy sets and the membership functions are listed in table \[tab:3\] and figure \[fig:3\] respectively for the input variable “prostate volume". [lll]{} Input variable & Crisp set & Fuzzy set\ PV (ml) & 0-60 & Small\ & 30-120 & Middle\ & 60-200 & Big\ & 180-300 & Very big\ ![Membership functions for “PV".[]{data-label="fig:3"}](pv.eps){width="\textwidth"} ### Percentage of Free PSA ($\%$FPSA) PSA is a protein which exists in different forms in serum. PSA circulates through body via bound to some other proteins or in unbound form. Free PSA test measure the ratio of the unbound PSA to bound PSA whereas normal PSA test measure the total PSA (both bound and unbound) level in blood [@labmed; @ito]. $\%$FPSA is calculated as $\frac{{\rm Free ~ PSA}}{{\rm Total PSA}} \times 100\%$. As we already discussed that PSA level may rise not only due to cancerous prostate but because of many other reasons. Therefore, for the early detection of the PC, which is potentially in its curable stage, requires a lower cutoff for PSA level, gives rise to avoidable biopsies. It is also observed that the men with PC are likely to have $\%$FPSA lower than those of benign disease [@labmed; @catalona]. So, along with an elevated PSA level, $\%$FPSA cutoff will be a good indicator for the early stage detection. Based on this we have categorized the input variable “$\%$FPSA" in three fuzzy sets, namely, “low", “middle" and “high". For their representation we have used trapezoidal and triangular membership functions. Crisp sets, fuzzy sets and the membership functions are shown in table \[tab:4\] and figure \[fig:4\]. [lll]{} Input variable & Crisp set & Fuzzy set\ $\%$FPSA & 0-11 & Low\ & 9-21 & Middle\ & 18-100 & High\ ![Membership functions for “$\%$FPSA".[]{data-label="fig:4"}](fpsa.eps){width="\textwidth"} Output Variable --------------- ### Prostate Cancer Risk (PCR) The output of the FES is PCR. Numerical value of it will help us to identify the benignant one or the malignant one by considering PSA, age, PV and $\%$FPSA of the patient as input variables for the FES. If the PCR $\%$ value is $\ge 50$, then we can anticipate that the patient has a high chance of having PC. Therefore, he will be recommended for biopsy test for confirmation. The positive biopsy result confirms our prediction whereas negative result contradicts. The case of PCR $\%$ value $<50$ can be understood in similar fashion. The output variable, PCR is categorized into three fuzzy sets, namely, “low", “middle" and “high". Trapezoidal and triangular membership functions are used to represent them. The table \[tab:5\] and the figure \[fig:5\] are displaying these fuzzy sets and their membership functions respectively. [lll]{} Output variable & Crisp set & Fuzzy set\ Prostate Cancer Risk ($\%$) & 0-30 & Low\ & 10-50 & Middle\ & 45-100 & High\ ![Membership functions for “PCR".[]{data-label="fig:5"}](pcr.eps){width="\textwidth"} Fuzzy Rule Base --------------- In this FES, four input variables are age, PSA, PV and $\%$FPSA which are represented by four, five, four and three membership functions respectively. So, our FES has total $4\times5\times4\times3=240$ rules to estimate PCR as the output, which is also characterized by three membership functions. Some of the selected rules are displayed below. ``` {fontsize="\footnotesize"} 1. If (Age is vy) and (PSA is vl) and (PV is small) and (%FPSA is low) then (PCR is low) (1) 8. If (Age is vy) and (PSA is vl) and (PV is big) and (%FPSA is mid) then (PCR is low) (1) 65. If (Age is yo) and (PSA is vl) and (PV is middle) and (%FPSA is mid) then (PCR is low) (1) 97. If (Age is yo) and (PSA is hi) and (PV is small) and (%FPSA is low) then (PCR is high) (1) 130. If (Age is ma) and (PSA is vl) and (PV is verybig) and (%FPSA is low) then (PCR is low) (1) 172. If (Age is ma) and (PSA is vh) and (PV is middle) and (%FPSA is low) then (PCR is high) (1) 196. If (Age is ol) and (PSA is lo) and (PV is middle) and (%FPSA is low) then (PCR is low) (1) 240. If (Age is ol) and (PSA is vh) and (PV is verybig) and (%FPSA is high) then (PCR is mid) (1) ``` For example, rule $1$ can be elucidated as, if age of the patient is very young, PSA is very low, PV is small and $\%$FPSA is low then PCR for that patient is low. The wieght of this rule is maximum which is specified by number $1$ in the parentheses after the rule. Other rules can be spelled out in similar manner. Surface plot gives a visualization of the rules applied in FES. Since, our FES has four inputs and one output, it is not possible to plot the dependence of the output against all inputs. So, in surface plot we have plotted the output against any two input variables by keeping some reference values for the rest two inputs. Fig \[fig:7\] displays two such surface plots where relevant parameters are mentioned in the caption. For example, PCR is high (very high) for $\%$FPSA value less than $20\%$ ($10\%$) irrespective of the patient’s age. ![Surface plot of four-input FES (left) by taking two variables as inputs and PCR as output while keeping reference values for other two inputs. For the figure on the left, age & $\%$FPSA are inputs and reference values for PV & PSA are $100$ and $6$ respectively. The figure on the right, we have taken PSA & age as inputs while fixing reference values for PV and $\%$FPSA at 100 and $15$ respectively. []{data-label="fig:7"}](surf1.eps "fig:"){width="63.00000%"} ![Surface plot of four-input FES (left) by taking two variables as inputs and PCR as output while keeping reference values for other two inputs. For the figure on the left, age & $\%$FPSA are inputs and reference values for PV & PSA are $100$ and $6$ respectively. The figure on the right, we have taken PSA & age as inputs while fixing reference values for PV and $\%$FPSA at 100 and $15$ respectively. []{data-label="fig:7"}](surf2.eps "fig:"){width="60.00000%"} Defuzzification & Mamdani Inference Engine ------------------------------------------ Defuzzification is the process of obtaining a precise quantity from a fuzzy set. In this FES, we have employed centroid method to defuzzify which is given by $${\bar z}=\frac{\int \mu_C(z)\cdot z~ dz}{\int \mu_C(z)~dz}.$$ For a given set of input variables, viz. age=$56$, PSA=$9.05$, PV=$39$ and $\%$FPSA=$8.51$, we calculate the corresponding degrees of membership by identifying the proper fuzzy sets for each input variable. This particular choice of inputs will fire, say $k$ number of rules. Truth degree ($\alpha_i$) of the $i^{\rm th}$ firing rule is determined by taking min of corresponding membership values of each input variable. Thereafter, max of all $\alpha_i$ will be the membership value of PCR. This is how Mamdani max-min inference technique is used. Then applying the centroid method, we obtain crisp value of PCR. Steps are shown below for the above mentioned input data. 1. Age = $56$, $\mu_{\rm ma}(56)=0.2$ and $\mu_{\rm ol}(56)=0.533$. 2. PSA = $9.05$, $\mu_{\rm mi}(9.05)=0.7375$ and $\mu_{\rm hi}(9.05)=0.2625$. 3. PV = $39$, $\mu_{\rm sl}(39)=0.7$ and $\mu_{\rm mi}(39)=0.3$. 4. $\%$FPSA = $8.51$, $\mu_{\rm lo}(8.51)=0.83$. For the above set of input data, eight rules will come into action and yield 1. $\alpha_{145}={\rm min}(0.2,0.7375,0.7,0.83)=0.2$, 2. $\alpha_{148}={\rm min}(0.2,0.7375,0.3,0.83)=0.2$, 3. $\alpha_{157}={\rm min}(0.2,0.2625,0.7,0.83)=0.2$, 4. $\alpha_{160}={\rm min}(0.2,0.2625,0.3,0.83)=0.2$, 5. $\alpha_{205}={\rm min}(0.533,0.7375,0.7,0.83)=0.533$, 6. $\alpha_{208}={\rm min}(0.533,0.7375,0.3,0.83)=0.3$, 7. $\alpha_{217}={\rm min}(0.533,0.2625,0.7,0.83)=0.2625$, 8. $\alpha_{220}={\rm min}(0.533,0.2625,0.3,0.83)=0.2625$, which further provides &= [max]{}(\_[145]{},\_[148]{},\_[157]{},\_[160]{},\_[205]{},\_[208]{},\_[217]{},\_[220]{}),\ &= [max]{}(0.2,0.2,0.2,0.2,0.533,0.3,0.2625,0.2625)= 0.533. Using $\alpha = 0.533$, and applying centroid method we get the PCR is $74.01\%$, which is much greater than our cutoff value of $50\%$. Therefore, in this case, the patient should be advised to go for a biopsy to confirm. We have used fuzzy logic toolbox of Matlab software for calculation of the entire set of medical data of 119 patients. Block diagram of FES is portrayed in the figure \[fig:6\]. ![FES structure[]{data-label="fig:6"}](mod_block.eps){width="120.00000%"} Results and Discussion {#res} ====================== We have applied our proposed FES to analyze the data (given in reference [@saritas]) presented in table \[tab:6\]. Here, the range of different inputs are chosen depending upon the minimum and the maximum values of the respective input variables. For example, for the input “PV", the maximum value is $235$ ml and the minimum value is $15$ ml. Therefore, we have chosen the range for “PV" to be $0-300$ ml. Feeding the four inputs into the FES, we obtain PCR as the output. If the value of PCR is greater or equal to $50\%$, then the patient should be advised to go for a biopsy test to confirm whether the prostate problem is of benignant or malignant type. We have compared outcomes of our FES with the results of biopsy. Out of total $119$ patients, $61$ patients had positive biopsy results, while the rest $58$ patients had negative results. The true prediction by our FES is $68.91\%$. The proposed FES has correctly predicted the positive biopsy results for $45$ patients ($73.77\%$) and the negative biopsy results for $37$ patients ($63.79\%$). True prediction percentage of our FES is better than that of the existing systems for prediction in literature. Saritas [*et. al*]{} [@saritas] has claimed $64.71\%$ true prediction for the same set of data, while online calculator for PC prediction and FPSA/PSA ratio provide only $62.18\%$ and $60.50\%$ respectively. Inclusion of one parameter has definitely increased the number of rules, but the prediction of the designed FES has become more accurate. Summary and Conclusions {#con} ======================= We have developed a fuzzy rule based expert system to predict the chances of having prostate cancer (PC) from a given set of input parameters. For this FES, we used age, PSA, PV and $\%$FPSA as the inputs to calculate prostate cancer risk (PCR) as the output. Existing literature on FES for PC, has mainly dealt with first three of the above mentioned inputs. The cutoff of $\%$FPSA is also a very crucial parameter along the PSA value, as it can reduce avoidable biopsy tests significantly. Most important advantages, apart from the cost of biopsy, the patient who do not need biopsy will be free from uneasiness and worries of the procedure, as well as the possible aftermath medical complications [@labmed]. Therefore, inclusion of this parameter in our FES was essential and it has altered the outcome appreciably. For a case study of $119$ patients, our FES has correctly predicted for $82$ patients ($68.91\%$). It is to be noted that, true prediction percentage for positive biopsy cases is excellent ($73.77\%$) whereas for the negative biopsy cases, it deviates slightly. The salient feature of our FES is that, the true prediction for positive biopsy is much higher compared to negative biopsy cases. This may cause an unnecessary biopsy test (along the with the pain associated with it) but it will certainly save a life. One should keep in mind that the FES designed here is not to confirm whether a person is having PC or not, but to assist a doctor to take a decision whether to go for biopsy or not. Since, four inputs are considered for this FES, number of rules has increased which makes it little complicated to start with. Once the burden of deciding these rules based on experts’ opinion are done, it does the prediction efficaciously. On the other hand, the behaviour of different bio-markers varies drastically from race to race, from demographic region to region, family history, life style, food habits and also depends on many more hidden variables. So, modification in cutoff values has to be done accordingly to use the FES. Hybridizing our FES with other AI techniques and incorporating more experts’ opinions may lead to an improved result. Acknowledgements {#acknowledgements .unnumbered} ================ Authors would like to thank Dr. Bidesh Karmakar and Dr. Aniruddha Dewri for the useful discussions made with them during the manuscript preparation. [\*[12]{}c]{} \ Age & PSA & PV & & Biopsy & & Age & PSA & PV & & Biopsy &\ (year) & (ng/ml) & (ml) & & result & & (year) & (ng/ml) & (ml) & & result &\ [[**  – continued from previous page**]{}]{}\ Age & PSA & PV & & Biopsy & & Age & PSA & PV & & Biopsy &\ (year) & (ng/ml) & (ml) & & result & & (year) & (ng/ml) & (ml) & & result &\ \ 44 & 7.6 & 38 & 10.53 & Negative & 45.42 & 67 & 15.93 & 69 & 6.09 & Positive & 74.82\ 51 & 6.76 & 15 & 4.14 & Positive & 57.78 & 67 & 28 & 47 & 15.00 & Positive & 74.15\ 51 & 44 & 83 & 31.82 & Positive & 30.00 & 68 & 5.09 & 47 & 2.36 & Negative & 42.96\ 53 & 4.5 & 39 & 18.89 & Negative & 19.96 & 68 & 5.51 & 45 & 11.25 & Negative & 21.28\ 53 & 5.83 & 25 & 6.86 & Negative & 53.32 & 68 & 7.2 & 33 & 3.61 & Positive & 67.21\ 53 & 8.34 & 25 & 7.43 & Negative & 73.83 & 68 & 9.25 & 91 & 3.57 & Positive & 74.03\ 54 & 5.62 & 28 & 14.95 & Negative & 21.96 & 68 & 12.1 & 61 & 16.12 & Negative & 74.25\ 54 & 17.3 & 90 & 27.46 & Negative & 30.00 & 68 & 23.7 & 109 & 10.04 & Positive & 73.55\ 54 & 17.3 & 45 & 8.90 & Positive & 73.91 & 68 & 140 & 117 & 14.29 & Positive & 74.80\ 55 & 10.51 & 54 & 22.45 & Negative & 23.57 & 68 & 140 & 54 & 3.29 & Positive & 74.70\ 56 & 8.9 & 26 & 34.16 & Negative & 18.80 & 69 & 8.8 & 34 & 8.98 & Positive & 74.40\ 56 & 9.05 & 39 & 8.51 & Positive & 74.07 & 69 & 11.06 & 38 & 29.84 & Negative & 26.64\ 56 & 16 & 146 & 8.44 & Negative & 74.07 & 69 & 15.31 & 74 & 30.57 & Positive & 30.00\ 57 & 12.56 & 52 & 65.84 & Negative & 30.00 & 69 & 61 & 46 & 9.93 & Negative & 73.65\ 58 & 4.48 & 67.5 & 16.07 & Negative & 16.09 & 69 & 70.56 & 45 & 6.02 & Positive & 73.98\ 58 & 4.62 & 48 & 11.04 & Negative & 17.62 & 69 & 146 & 29 & 7.33 & Positive & 75.10\ 58 & 5.2 & 58 & 23.46 & Negative & 12.80 & 70 & 5.39 & 120 & 19.11 & Negative & 23.58\ 58 & 16.39 & 27 & 92.07 & Negative & 30.00 & 70 & 5.39 & 42 & 12.80 & Negative & 19.91\ 59 & 0.28 & 168 & 42.86 & Negative & 13.30 & 70 & 13 & 40 & 15.46 & Negative & 74.39\ 59 & 8.36 & 55 & 7.54 & Positive & 74.31 & 70 & 13.95 & 119 & 13.76 & Negative & 74.02\ 59 & 18.2 & 77 & 17.75 & Negative & 73.76 & 70 & 19.2 & 44 & 10.10 & Positive & 73.49\ 59 & 19.48 & 79 & 25.00 & Positive & 30.00 & 70 & 21.94 & 29 & 7.11 & Positive & 75.17\ 59 & 22.51 & 42 & 7.02 & Negative & 74.22 & 70 & 27.7 & 63 & 8.99 & Negative & 74.40\ 59 & 22.65 & 66 & 10.82 & Negative & 73.88 & 71 & 6.08 & 48 & 21.38 & Positive & 13.52\ 60 & 6.58 & 65 & 14.74 & Negative & 24.82 & 71 & 12.64 & 50 & 7.99 & Positive & 74.39\ 60 & 10.6 & 30 & 16.79 & Positive & 61.27 & 71 & 22 & 57 & 12.00 & Positive & 74.59\ 60 & 11.45 & 46 & 19.48 & Negative & 53.47 & 72 & 6.64 & 32 & 27.41 & Negative & 12.43\ 60 & 14.79 & 38 & 6.90 & Positive & 74.39 & 72 & 13.31 & 33 & 3.83 & Positive & 74.40\ 60 & 15.51 & 35 & 21.02 & Negative & 30.00 & 72 & 13.31 & 33 & 3.76 & Positive & 74.40\ 61 & 4.6 & 37 & 10.87 & Negative & 25.02 & 72 & 20 & 48 & 7.90 & Positive & 74.22\ 61 & 10.33 & 62 & 25.36 & Negative & 23.78 & 72 & 46 & 36 & 10.70 & Positive & 73.81\ 61 & 10.36 & 35 & 19.79 & Negative & 47.88 & 72 & 77 & 48 & 8.31 & Positive & 74.22\ 61 & 10.59 & 56 & 17.00 & Positive & 60.72 & 73 & 4.65 & 41 & 41.94 & Negative & 12.52\ 61 & 18.3 & 62 & 6.99 & Positive & 74.46 & 73 & 7.25 & 19 & 5.52 & Negative & 67.73\ 62 & 6.12 & 52 & 24.18 & Negative & 12.86 & 73 & 7.6 & 74 & 31.32 & Positive & 12.09\ 62 & 6.2 & 25 & 4.35 & Positive & 56.05 & 73 & 19 & 90 & 6.84 & Positive & 73.98\ 62 & 8.37 & 43 & 11.23 & Negative & 40.23 & 73 & 29.52 & 91 & 9.82 & Negative & 73.73\ 62 & 8.79 & 45 & 10.92 & Positive & 48.39 & 73 & 47.4 & 87 & 15.89 & Positive & 74.11\ 62 & 20 & 53 & 5.20 & Positive & 74.55 & 74 & 12.52 & 27 & 11.82 & Negative & 74.47\ 62 & 51.74 & 29 & 6.80 & Positive & 74.55 & 74 & 150 & 54 & 16.67 & Positive & 74.09\ 63 & 8.8 & 31 & 22.50 & Positive & 19.22 & 75 & 4.61 & 16 & 17.57 & Positive & 18.39\ 64 & 5.7 & 36 & 29.82 & Negative & 12.71 & 75 & 10 & 34 & 7.60 & Positive & 73.98\ 64 & 6.96 & 45 & 9.20 & Negative & 60.28 & 76 & 9.81 & 56 & 37.41 & Negative & 21.46\ 64 & 8 & 40 & 7.50 & Positive & 74.39 & 76 & 13.61 & 61 & 19.91 & Positive & 52.54\ 64 & 11.08 & 26 & 10.11 & Negative & 59.05 & 76 & 13.83 & 54 & 19.96 & Positive & 51.76\ 64 & 16.28 & 21 & 6.94 & Positive & 74.70 & 76 & 21 & 86 & 5.43 & Positive & 74.15\ 65 & 4.39 & 30 & 21.64 & Negative & 13.42 & 77 & 10 & 60 & 6.00 & Positive & 73.98\ 65 & 5.15 & 47 & 15.73 & Negative & 19.46 & 77 & 12.05 & 28 & 27.05 & Positive & 30.00\ 65 & 7.61 & 23 & 5.78 & Positive & 70.95 & 77 & 56 & 51 & 7.34 & Positive & 74.46\ 65 & 7.82 & 75 & 22.76 & Negative & 13.04 & 78 & 4.5 & 180 & 20.44 & Negative & 18.05\ 65 & 8.33 & 32 & 14.53 & Positive & 38.08 & 78 & 26.1 & 46 & 8.62 & Negative & 74.07\ 66 & 4.38 & 33 & 23.52 & Negative & 12.78 & 78 & 26.13 & 235 & 8.27 & Negative & 74.96\ 66 & 6.72 & 61 & 13.84 & Positive & 25.38 & 78 & 31.6 & 57 & 8.86 & Negative & 74.50\ 66 & 7.65 & 89 & 23.66 & Negative & 12.90 & 79 & 17.1 & 41 & 7.60 & Negative & 74.31\ 66 & 9 & 74 & 18.89 & Positive & 53.84 & 80 & 69.51 & 28 & 28.77 & Positive & 30.00\ 66 & 9.86 & 49 & 23.83 & Negative & 21.69 & 81 & 4.5 & 28 & 21.56 & Positive & 13.45\ 67 & 4.39 & 28 & 0.91 & Negative & 23.99 & 81 & 68.36 & 52 & 35.27 & Positive & 30.00\ 67 & 5.65 & 24 & 10.27 & Positive & 46.65 & 88 & 10.4 & 32 & 7.50 & Positive & 74.22\ 67 & 6.24 & 65 & 21.96 & Negative & 13.31 & & & & & &\ 67 & 8.2 & 36 & 20.37 & Positive & 31.78 & & & & & &\ 67 & 9.68 & 41 & 7.44 & Positive & 74.18 & & & & & &\ [^1]: juthika@math.nits.ac.in [^2]: subhasis@phy.nits.ac.in

--- author: - Wensheng Cheng - Yan Zhang - Xu Lei - Wen Yang - Guisong Xia bibliography: - 'segmentation.bib' - 'change\_detection.bib' title: Semantic Change Pattern Analysis ---

--- author: - 'R. K. Zamanov' - 'K. A. Stoyanov' - 'J. Martí' - 'G. Y. Latev' - 'Y. M. Nikolov' - 'M. F. Bode' - 'P. L. Luque-Escamilla' date: 'Received April 20, 2016; accepted July 8, 2016' title: 'Optical spectroscopy of Be/gamma-ray binaries' --- Introduction ============ The rapid and sustained progress of high energy and very high energy astrophysics in recent years enabled the identification of a new group of binary stars emitting at TeV energies (e.g. Paredes et al. 2013). These objects, called $\gamma$-ray binaries, are high-mass X-ray binaries that consist of a compact object (neutron star or black hole) orbiting an optical companion that is an OB star. There are five confirmed $\gamma$-ray binaries so far: PSR B1259-63/LS 2883 (Aharonian et al. 2005), LS 5039/V479 Sct (Aharonian et al. 2006),  (Albert et al., 2009), HESS J0632+057/MWC 148 (Aharonian et al. 2007), and 1FGL J1018.6-5856 (H.E.S.S. Collaboration et al. 2015). Their most distinctive fingerprint is a spectral energy distribution dominated by non-thermal photons with energies up to the TeV domain. Recently, Eger et al. (2016) proposed a binary nature for the $\gamma$-ray source HESS J1832-093/2MASS J18324516-092154 and this object probably belongs to the family of the $\gamma$-ray binaries as a sixth member. The binary system, PSR B1259-63 is unique, since it is the only one where the compact object has been identified as a radio pulsar (Johnston et al. 1992, 1994). The nature of the compact object is known in PSR B1259-63 as a neutron star, and in AGL J2241+4454/MWC 656 as a black hole (Casares et al. 2014). Although not included in the confirmed list, MWC 656 was selected as a target here despite not having shown all the observational properties of a canonical $\gamma$-ray binary yet. It was only occasionally detected by the AGILE observatory at GeV energies and not yet detected in the TeV domain (see Aleksi[ć]{} et al. 2015). Nevertheless, the fact that the black hole nature of the compact companion is almost certain renders it very similar to the typical $\gamma$-ray binaries. In the other systems the nature of the compact object remains unclear (e.g. Dubus 2013). In addition to these objects, there are several other binary systems ($\eta$ Car, Cyg X-1, Cyg X-3, Cen X-3, and SS 433) that are detected as GeV sources, but not as TeV sources so far. Here we report high-resolution spectral observations of , MWC 148, and MWC 656, and discuss circumstellar disc size, disc truncation, interstellar extinction, and rotation of their mass donors. The mass donors (primaries) of these three targets are emission-line Be stars. The Be stars are non-supergiant, fast-rotating B-type and luminosity class III-V stars which, at some point in their lives, have shown spectral lines in emission (Porter & Rivinius 2003). The material expelled from the equatorial belt of a rapidly rotating Be star forms an outwardly diffusing gaseous, dust-free Keplerian disc (Rivinius et al. 2013). In the optical/infrared band, the two most significant observational characteristics of Be stars and their excretion discs are the emission lines and the infrared excess. Moving along the orbit, the compact object passes close to this disc, and sometimes may even go through it causing significant perturbations in its structure. This circumstellar disc feeds the accretion disc around the compact object and/or interacts with its relativistic wind. Observations ============ [c c c c l l l ]{} Date-obs & exp-time & S/N & Orb. phase &\ yyyymmdd...hhmm & & $H\alpha$ & &\ \ [** ** ]{} &\ 20140217...1923 & 60 min & 20 & 0.455 &\ 20140314...1746 & 60 min & 42 & 0.396 &\ 20150805...0009 & 60 min & 45 & 0.579 &\ \ [**MWC 148** ]{} &\ 20140113...1857 & 60 min & 56 & 0.758 &\ 20140217...2031 & 60 min & 44 & 0.870 &\ 20140218...1826 & 60 min & 62 & 0.872 &\ 20140313...2002 & 60 min & 54 & 0.946 &\ 20140314...1855 & 60 min & 81 & 0.949 &\ 20140315...1833 & 60 min & 46 & 0.952 &\ \ [**MWC 656** ]{} &\ 20150705...2259 & 30 min & 55 & 0.691 &\ 20150804...0017 & 30 min & 45 & 0.173 &\ 20150804...2229 & 30 min & 56 & 0.188 &\ \ High-resolution optical spectra of the three northern Be/$\gamma$-ray binaries were secured with the fibre-fed Echelle spectrograph [*ESpeRo*]{} attached to the 2.0 m telescope of the National Astronomical Observatory Rozhen, located in Rhodope mountains, Bulgaria. The spectrograph uses R2 grating with 37.5 grooves/mm, Andor CCD camera 2048 x 2048 px, 13.5x13.5 $\mu m$ px$^{-1}$ (Bonev et al. 2016). The spectrograph provides a dispersion of 0.06 Åpx$^{-1}$ at 6560 Å  and 0.04 Åpx$^{-1}$ at 4800 Å. The spectra were reduced in the standard way including bias removal, flat-field correction, and wavelength calibration. Pre-processing of data and parameter measurements are performed using various routines provided in IRAF. The journal of observations is presented in Table \[tab.J\], where the date, start of the exposure, exposure time, and signal-to-noise ratio at about $\lambda 6600$ Å are given. The orbital phases are calculated using $HJD_0= 2443366.775$, $HJD_0= 2454857.5,$ and $HJD_0= 2453243.7$ for , MWC 148, and MWC 656, respectively, and orbital periods given in Sect. \[sect.2\]. Emission line profiles of , MWC 148, and MWC 656 are plotted on Fig.\[f1.examp\]. Spectral line parameters equivalent width (W) and distance between the peaks ($\Delta V$) for the prominent lines ($H\alpha$, H$\beta$, H$\gamma$, $HeI \lambda 5876,$ and $FeII \lambda 5316$) are given in Table \[tab.2\]. The typical error on the equivalent width is below $\pm 10$ % for lines with $W > 1$ Å  and up to $\pm 20$% for lines with $W \lesssim 1$ Å. The typical error on $\Delta V$ is $\pm 10$ . It is worth noting that [**(1)**]{} in   FeII lines are not detectable; [**(2)**]{} In MWC 656 on spectrum 20150705 the HeI $\lambda5876$ line is not visible (probably emission fills up the absorption). In addition to the Rozhen data we use 98 spectra of MWC 148 and 68 spectra of MWC 656 (analysed in Casares et al. 2012) from the archive of the 2.0 m Liverpool Telescope[^1] (Steele et al. 2004). These spectra were obtained using the Fibre-fed RObotic Dual-beam Optical Spectrograph (FRODOSpec; Morales-Rueda et al. 2004). The spectrograph is fed by a fibre bundle array consisting of $12\times12$ lenslets of 0.82 arcsec each, which is reformatted as a slit. The spectrograph was operated in a high-resolution mode, providing a dispersion of 0.8 Åpx$^{-1}$ at 6500 Å, 0.35 Åpx$^{-1}$ at 4800 Å, and typical $S/N \gtrsim 100$. FRODOSpec spectra were processed using the fully automated data reduction pipeline of Barnsley et al. (2012). The typical error on the equivalent width is $\pm 10$ % and on $\Delta V$ is $\pm 20$ . [cccccccccccccccccccclll]{} & & & & & &\ date-obs & $W_\alpha$ & $\Delta V_\alpha$ & $W_\beta$ & $\Delta V_\beta$ & $W_\gamma$ & $\Delta V_\gamma$ & $W_{HeI5876}$ & $\Delta V_{HeI5876}$ & $W_{FeII5316}$ & $\Delta V_{FeII}$ &\ yyyymmdd.hhmm & Å &   & Å & & Å &   & Å  &   & Å  &   &\ \ [** ** ]{} &\ 20140217.1923 & -8.6 & 316 & -0.71 & 416 & +0.5 & ... & +0.39 & 455 & ... & ... &\ 20140314.1746 & -8.1 & 309 & -1.12 & 411 & +0.7 & ... & +0.17 & 456 & ... & ... &\ 20150805.0009 & -8.2 & 337 & -1.16 & 421 & +0.8 & ... & +0.40 & 416 & ... & ... &\ \ [**MWC 148** ]{} &\ 20140113.1857 & -29.5 & 105 & -4.19 & 182 & -1.38 & 188 & -0.49 & 231 & -0.49 & 210 &\ 20140217.2031 & -30.9 & 92 & -4.27 & 162 & -1.24 & 178 & -0.37 & 243 & -0.37 & 177 &\ 20140218.1826 & -29.3 & 87 & -4.12 & 161 & -1.12 & 171 & -0.39 & 244 & -0.39 & 182 &\ 20140313.2002 & -29.0 & ... & -3.41 & 165 & -0.77 & 149 & -0.39 & 246 & -0.39 & 186 &\ 20140314.1855 & -28.5 & ... & -3.79 & 170 & -1.04 & 159 & -0.37 & 251 & -0.37 & 185 &\ 20140315.1833 & -26.8 & ... & -3.84 & 168 & -1.01 & 175 & -0.51 & 264 & -0.51 & 186 &\ \ [**MWC 656** ]{} &\ 20150705.2259 & -23.3 & ... & -2.26 & 246 & -0.42 & 301 & +0.0 & ... & -0.40 & 275 &\ 20150804.0017 & -21.9 & ... & -2.12 & 244 & -0.23 & 289 & +0.22 & ... & -0.48 & 240 &\ 20150804.2229 & -21.2 & ... & -1.98 & 246 & -0.34 & 311 & +0.19 & ... & -0.42 & 227 &\ \ \[tab.2\] Objects: System parameters {#sect.2} ==========================   (V615 Cas) was identified as a $\gamma$-ray source with the $COS B$ satellite 35 years ago (Swanenburg et al. 1981). For the orbital period of , we adopt $P_{orb}=26.4960 \pm 0.0028$ d, which was derived with Bayesian analysis of radio observations (Gregory 2002) and an orbital eccentricity $e=0.537$, which was obtained on the basis of the radial velocity of the primary (Casares et al. 2005; Aragona et al. 2009). For the primary, Grundstrom et al. (2007) suggested a B0V star with radius $R_1=6.7 \pm 0.9$ . A B0V star is expected to have on average $M_1 \approx 15$  (Hohle et al. 2010). We adopt $v \sin i = 349 \pm 6$   for the projected rotational velocity of the mass donor (Hutchings & Crampton 1981, Zamanov et al. 2013). MWC 148 (HD 259440) was identified as the counterpart of the variable TeV source HESS J0632+057 (Aharonian et al. 2007). We adopt $P_{orb} = 315 ^{+6}_{-4}$ d derived from the X-ray data (Aliu et al. 2014), which is consistent with the previous result of $321 \pm 5$ days (Bongiorno et al. 2011). For this object Aragona et al. (2010) derived $T_{eff} = 27500 - 30000$ K, $\log g = 3.75 - 4.00$, $M_1 = 13.2 - 19.0$ , and $R_1 = 7.8 \pm 1.8 $ . For the calculations in Sect.\[Disc.size\], we adopt $e=0.83$, periastron at phase 0.967 (Casares et al. 2012), and $v \sin i = 230 - 240$   (Moritani et al. 2015). MWC 656 (HD 215227) is the emission-line Be star that lies within the positional error circle of the $AGILE$ $\gamma$-ray source AGL J2241+4454 (Lucarelli et al. 2010). It is the first and until now the only detected binary composed of a Be star and a black hole (Casares et al. 2014). For the orbital period, we adopt $P_{orb}=60.37 \pm 0.04$ d obtained with optical photometry (Williams et al. 2010), $e=0.10 \pm 0.04$ estimated on the basis of the radial velocity measurements and $v \sin i = 330 \pm 30$  (Casares et al. 2014). For the primary, Williams et al. (2010) estimated $T_{eff} = 19000 \pm 3000$ K, $\log g = 3.7 \pm 0.2 $, $M_1 = 7.7 \pm 2.0$ , $R_1 = 6.6 \pm 1.9$ . Casares et al. (2014) considered that the mass donor is a giant (B1.5-2 III) and give a mass range $M_1 = 10 - 16$ . On average a B1.5-2 III star is expected to have about $R_1 \approx 8.3 - 8.8$   (Straizys & Kuriliene 1981). From newer values of the luminosity (Hohle et al. 2010), such a star is expected to have $M_1 \approx 8.0 - 10.0$  and radius $R_1 \approx 9.5 - 10$ . We adopt $R_1 \approx 10$   for the calculations in Sect.\[Disc.size\]. [cccc|cccccllllll]{} Date-obs & $R_{disc}(H\alpha)$ & $R_{disc}(H\alpha)$ & $R_{disc}(H\alpha)$ & $R_{disc}(H\beta)$ & $R_{disc}(H\gamma)$ & $R_{disc}(HeI5876)$ & $R_{disc}(FeII)$ &\ yyyymmdd.hhmm &   &   &   &   &   &   &   &\ & () & () & () & & & & &\ \ [** ** ]{} &\ 20140217.1923 & 33 & 32 & 36 & 19 & ... & 16 & ... &\ 20140314.1746 & 34 & 33 & 33 & 19 & ... & 16 & ... &\ 20150805.0009 & 29 & 31 & 34 & 18 & ... & 19 & ... &\ \ [**MWC 148** ]{} &\ 20140113.1857 & 156 & 165 & 180 & 52 & 49 & 32 & 39 & &\ 20140217.2031 & 205 & 208 & 190 & 66 & 54 & 29 & 55 & &\ 20140218.1826 & 226 & 211 & 178 & 66 & 59 & 29 & 52 & &\ 20140313.2002 & ... & 201 & 176 & 63 & 52 & 29 & 50 & &\ 20140314.1855 & ... & 189 & 172 & 60 & 56 & 27 & 50 & &\ 20140315.1833 & ... & 193 & 160 & 60 & 50 & 25 & 50 & &\ \ [**MWC 656** ]{} &\ 20150705.2259 & ... & 213 & 174 & 63 & 42 & ... & 51 & &\ 20150804.0017 & ... & 216 & 162 & 64 & 46 & ... & 66 & &\ 20150804.2229 & ... & 213 & 156 & 63 & 40 & ... & 74 & &\ \ \[tab.D\] Circumstellar disc ================== Peak separation in different lines {#peak.sep} ---------------------------------- For the Be stars, the peak separations in different lines follow approximately the relations (Hanuschik et al. 1988) $$\begin{aligned} \Delta V_\beta \approx 1.8 \Delta V_\alpha \label{H3.1} \\ \Delta V_\gamma \approx 1.2 \Delta V_\beta \approx 2.2 \Delta V_\alpha \label{H3.2} \\ \Delta V_{\rm FeII} \approx 2.0 \Delta V_\alpha \label{H3.3} \\ \Delta V_{\rm FeII} \approx 1.1 \Delta V_\beta \label{H3.4} ,\end{aligned}$$ where Eq. \[H3.4\] is derived from Eqs. \[H3.1\] and \[H3.3\]. For  using the measurements in Table \[tab.2\], we obtain $\Delta V_\beta = 1.30 \pm 0.04 \, \Delta V_\alpha$ and $\Delta V_{HeI5876}= 1.38 \pm 0.13 \, \Delta V_\alpha $. The ratio $\Delta V_\beta / \Delta V_\alpha$ is considerably below the average value for the Be stars (see Eq.\[H3.1\]). We obtain $\Delta V_\beta = 1.78 \pm 0.06 \, \Delta V_\alpha$, $\Delta V_\gamma = 1.07 \pm 0.03 \, \Delta V_\beta$, and $\Delta V_{FeII5316} = 1.12 \pm 0.03 \, \Delta V_\beta$, $\Delta V_{HeI5876} = 1.47 \pm 0.10 \, \Delta V_\beta$ for MWC 148. We use only three spectra for $H\alpha$ (20140113, 20140217, and 20140218) when two peaks in  are visible. The value of $\Delta V_\beta / \Delta V_\alpha \approx 1.78$ is very similar to 1.8 in Be stars, the ratio $\Delta V_{FeII5316} / \Delta V_\beta \approx 1.07 $ is similar to 1.1 in Be stars, and the value of $\Delta V_\gamma / \Delta V_\beta \approx 1.07$ is again similar to the value 1.2 for Be stars. We estimate $\Delta V_\beta = 1.72 \pm 0.18 \, \Delta V_\alpha$ for MWC 656 (using six spectra from the Liverpool Telescope FRODOSpec, where two peaks are visible in both $H\alpha$ and $H\beta$), $\Delta V_\gamma = 1.22 \pm 0.04 \, \Delta V_\beta$, $\Delta V_{FeII5316} = 1.01 \pm 0.10 \, \Delta V_\beta$, where all three ratios are similar to the corresponding values (Eq. \[H3.1\], \[H3.2\], \[H3.4\]) in Be stars. We do not see two peaks on high-resolution Rozhen spectra of this object. However two peaks are clearly distinguishable on a few of the LT spectra. In every case of detection/non-detection of the double peak structure, $W\alpha$ is very similar $19 < W\alpha < 25$ Å. The comparison of the peak separation of different emission lines shows that MWC 148 and MWC 656 have circumstellar disc that is similar to that of the normal Be stars. At this stage considerable deviation from the behaviour of the Be stars is only detected in . In this star the $H\alpha$-emitting disc is only 1.7 times larger than the H$\beta$-emitting disc, while in normal Be stars it is 3.3 times larger. This probably is one more indication that outer parts of the disc are truncated as a result of the relatively short orbital period. Disc size {#Disc.size} --------- For rotationally dominated profiles, the peak separation can be regarded as a measure of the outer radius ($R_{disc}$) of the emitting disc (Huang 1972) $$\left( \frac{\Delta V} {2\,v\,\sin{i}} \;\right) = \; \left( \frac {R_{disc}}{R_1}\;\right)^{-j} , \label{Huang}$$ where $j=0.5$ for Keplerian rotation, $j=1$ for angular momentum conservation, $R_1$ is the radius of the primary, and $v\,\sin{i}$ is its projected rotational velocity. Eq. \[Huang\] relies on the assumptions that (1) the Be star is rotating critically, and (2) that the line profile shape is dominated by kinematics and radiative transfer does not play a role. When the two peaks are visible in the emission lines, we can estimate the disc radius using Eq. \[Huang\]. The calculated disc size for different emission lines are given in Table \[tab.D\]. In the $H\alpha$ emission line of   two peaks are clearly visible on all of our spectra. However in MWC 656 the $H\alpha$ emission line seems to exhibit three peaks (see Fig. \[f1.examp\]). Two peaks in $H\alpha$ emission of MWC 148 are clearly detectable on January-February 2014 observations. Two peaks are not distinguishable on the spectra obtained in March 2014 (when the companion is at periastron), which probably indicates perturbations in the outer parts of the disc caused by the orbital motion of the compact object. In the $H\beta$ line two peaks are visible on all the Rozhen spectra. We take this opportunity to obtain an estimation of the $R_{disc}(H\alpha)$ using $\Delta V_\beta$; the ratios $\Delta V_\beta / \Delta V_\alpha$ (as obtained in Sect. \[peak.sep\]), and Eq.\[Huang\]. The $R_{disc}(H\alpha)$ values calculated in this way are given in Table \[tab.D\] and indicated with $(^b)$. Disc size and $W_\alpha$ {#RdWa} ------------------------ The disc size and $W_\alpha$ are connected (see also Hanuschik 1989; Grundstrom & Gies 2006). This simply expresses the fact that $R_{disc}$ grows as $W_\alpha$ becomes larger. In Fig. \[f3.EW.dV\], we plot $\log \Delta V_\alpha / 2 $  versus $\log W_\alpha$. In this figure 138 data points are plotted for Be stars taken from Andrillat (1983), Hanuschik (1986), Hanuschik et al. (1988), Dachs et al. (1992), Slettebak et al. (1992), and Catanzaro (2013). In this figure, the data for Be/$\gamma$-ray binaries are also plotted. The three Be/$\gamma$-ray binaries are inside the distribution of normal Be stars. There is a moderate to strong correlation between the variables with Pearson correlation coefficient -0.63, Spearman’s (rho) rank correlation 0.64, and $p$-$value \sim 10^{-15}$. The dependence is of the form $$\; \; \log \; (\Delta V_{\alpha}/2 v \sin i ) = -a \; \log W_\alpha + b, \label{Han2}$$ and the slope is shallower for stars with $ W_\alpha < 3$ Å  as noted by Hanuschik et al. (1988). For 120 data points in the interval $3 \le W_\alpha \le 50$ Å, using a least-squares approximation we calculate $a=0.592 \pm 0.030$ and $b=0.165 \pm 0.036$. This fit as well as the correlation coefficients are calculated using only normal Be stars. Using Eq. \[Huang\] and Eq. \[Han2\] we then obtain $$\left( \frac {R_{disc}}{R_1}\;\right)^{-j} = 1.462 \; W_\alpha^{-0.592}. \label{Rd.W1}$$ Having in mind that (1) the discs of the Be stars are near Keplerian (Porter & Rivinius 2003, Meilland et al. 2012); (2) the Be stars rotate at rates below the critical rate (e.g. Chauville et al. 2001), and (3) at higher optical depths the emission line peaks are shifted towards lower velocities (Hummel & Dachs 1992), we calculate the disc radius with the following formula: $$\frac {R_{disc}}{R_1} = \; \epsilon \; 0.467 \; \; W_\alpha^{1.184}, \label{Rd.W2}$$ where $\epsilon$ is a dimensionless parameter (see also Zamanov et al. 2013), for which we adopt $\epsilon = 0.9 \pm 0.1$. The disc sizes calculated with Eq. \[Rd.W2\] are given in Table \[tab.D\] and are denoted with $(^c)$. As can be seen, the values agree with those obtained with the conventional method. We estimate average values of the dimensionless quantity $R_{disc} / \epsilon R_1 = 8.7 \pm 1.9$ (for ), $R_{disc} / \epsilon R_1 = 43 \pm 5$ (for MWC 148), and $R_{disc} / \epsilon R_1 = 18.0 \pm 1.1$ (for MWC 656), respectively. Disc truncation {#disc.trunc} --------------- The orbit of the compact object, the average size of $H\alpha$ disc, the average size of $H\beta$ disc, and the Be star are plotted in Fig. 3. The coordinates X and Y are in solar radii. The histograms of $H\alpha$ disc size, calculated using Eq. \[Rd.W2\], are plotted in Fig. 4. For , we use our new data and published data from Paredes et al. (1994), Steele et al. (1996), Liu & Yan (2005), Grundstrom et al. (2007), McSwain et al. (2010), and Zamanov et al. (1999, 2013). We use Rozhen and Liverpool Telescope spectra for MWC 148 and MWC 656. In all three stars the distribution of $R_{disc}$ values has a very well pronounced peak. The tendency for the disc emission fluxes to cluster at specified levels is related to the truncation of the disc at specific disc radii by the orbiting compact object (e.g. Coe et al. 2006). Okazaki & Negueruela (2001) proposed that these limiting radii are defined by the closest approach of the companion in the high-eccentricity systems and by resonances between the orbital period and the disc gas rotational periods in the low-eccentricity systems. The resonance radii are given by $${R_{n:m}^{3/2}} = \frac{m \; (G \: M_1)^{1/2}}{2 \: \pi} \: \frac{P_{orb}}{n}, \label{eq.resona}$$ where $G$ is the gravitational constant, $n$ is the integer number of disc gas rotational periods, and $m$ is the integer number of orbital periods. The important resonances are not only those with $n:1$, but can also be $n:m$ in general. For   (assuming $M_1 \approx 15$ , $M_2 \approx 1.4$ , $e \approx 0.537$), we estimate the distances between the components $a(1-e) \approx 44$  and $a(1+e)\approx 146$  for the periastron and apastron, respectively. As can be seen from Fig. \[f5.hist\], the disc size is $R_{disc} \sim a(1-e)$ and it never goes close to $a(1+e)$. The resonances that correspond to disc size are between 5:1 and 1:1, and the peak on the histogram corresponds to the 2:1 resonance. For MWC 148, with the currently available data ($M_1 \approx 15$ , $M_2 \approx 4$ , $e \approx 0.83$), we estimate $a(1-e) \approx 88$   and $a(1+e)\approx 951$   for the periastron and apastron, respectively. In Fig. \[f5.hist\], it is apparent that $a(1-e) < R_{disc} < a(1+e)$. The 2:1 resonance is the closest to the peak of the distribution. We note in passing that the disc size in this star could have a bi-modal distribution (a second peak with lower intensity seems to emerge close to 4:1 resonance radius). For MWC 656 (assuming $M_1 \approx 9$ , $M_2 \approx 4$ , $e \approx 0.1$), we estimate $a(1-e) \approx 137$  and $a(1+e)\approx 167$  for the periastron and apastron, respectively. In Fig. \[f5.hist\], it is seen that the 1:1 resonance is very close to the peak of the distribution and $a(1-e) \lesssim R_{disc} \lesssim a(1+e)$. The disc size rarely goes above $a(1+e)$. Interstellar extinction: Estimates of $E(B-V)$ from interstellar lines ====================================================================== There is a strong correlation between equivalent width of the diffuse interstellar bands (DIBs) and reddening (Herbig 1975; Puspitarini et al. 2013). There is also a calibrated relation of reddening with the equivalent width of the interstellar line $KI \lambda7699$Å  (Munari & Zwitter 1997). Aiming to estimate the interstellar extinction towards our objects, we measure equivalent widths of $KI \lambda7699$Å and DIBs $\lambda6613, \lambda5780, \lambda5797$. [**LSI+61$^0$303:** ]{} For this object, Hunt et al. (1994) use $E(B-V)=0.93$ (Hutchings & Crampton 1981). Howarth (1983) using the 2200 Å extinction bump obtained $E(B-V)=0.75 \pm 0.1$. Steele et al. (1998) estimated $E(B-V)=0.70 \pm 0.40$ from Na I D$_2$ and $E(B-V)= 0.65 \pm 0.25$ from diffuse interstellar bands. For , we measure $0.17 \le W(KI \lambda7699) \le 0.19$ Å, $0.17 < W(DIB \lambda6613) < 0.19$ Å, $0.34 < W(DIB \lambda5780) < 0.41$ Å, $0.09 < W(DIB \lambda5797) < 0.16$ Å, which following Munari & Zwitter (1997) and Puspitarini et al. (2013) calibrations corresponds to $E(B-V)=0.84 \pm 0.08$. [**MWC 148:** ]{} For this star, Friedemann (1992) estimated E(B-V)=0.85 from the 217 nm band. We measure $0.14 < W(KI \lambda7699) < 0.20$ Å, $0.13 < W(DIB \lambda6613) < 0.18$ Å, $0.28 < W(DIB \lambda5780) < 0.34$ Å, $0.13 < W(DIB \lambda5797) < 0.15$ Å, which corresponds to a slightly lower value $E(B-V)=0.77 \pm 0.06$. [**MWC 656:** ]{} For this star, Williams et al. (2010) gave a low value E(B-V)=0.02. Casares et al. (2014) estimated E(B-V)=0.24. We measure $0.04 < W(DIB \lambda6613) < 0.06$ Å, $0.09 \le W(DIB \lambda5780) \le 0.10$ Å, $0.03 < W(DIB\lambda5797) < 0.05$ Å, and estimate $E(B-V)=0.25 \pm 0.02$. Rotational period of the mass donors ==================================== In close binary systems the rotation of the companions of compact objects is accelerated by mass transfer and tidal forces (e.g. Ablimit & Lü 2012). Adopting the parameters given in Sect. \[sect.2\], we estimate the rotational periods of the mass donors $P_{rot} \approx 0.92$ d (for ), $P_{rot} \approx 0.91$ d (MWC 148), and $P_{rot} \approx 0.86$ d (MWC 656). For the $\gamma$-ray binary LS 5039, the mass donor is a O6.5V((f)) star with $R_{1} = 9.3 \pm 0.6 $ , inclination $i \approx 24.9^0$, $= 113 \pm 8$   (Casares et al. 2005). We calculate $P_{rot} = 1.764$ d. Because of the short orbital period $P_{orb} = 3.91$ d, the rotation of the mass donor could be pseudo-synchronized with the orbital motion (Casares et al. 2005). Radio observations of the pulsar in PSR B1259-63/LS 2883 allow the orbital parameters to be precisely established: $P_{orb} = 1236.72$ d and eccentricity of e = 0.87 (Wang et al. 2004; Shannon et al. 2014). For the primary, Negueruela et al. (2011) estimated $R_1 = 9.0 \pm 1.5$ ,  $ = 260 \pm 15$ , and $i_{orb} \approx 23^0$, which give $P_{rot} = 0.689$ d. In Fig. \[f2.Prot\] we plot $P_{rot}$ versus $P_{orb}$ for a number of high-mass X-ray binaries (see also Stoyanov & Zamanov 2009). High-mass X-ray binaries with a giant/supergiant component, Be/X-ray binaries, and Be/$\gamma$-ray binaries are plotted with different symbols in this figure. The solid line represents synchronization ($P_{rot} = P_{orb}$). Among the five $\gamma$-ray binaries with known rotational velocity of the mass donor, the rotation of the mass donor is close to synchronization with the orbital motion in only one (LS 5039). The four Be/$\gamma$-ray binaries are not close to the line of synchronization. They occupy the same region as the Be/X-ray binaries; in other words regarding the rotation of the mass donor the Be/$\gamma$-ray binaries are similar to the Be/X-ray binaries. In these, the tidal force of the compact object acts to decelerate the rotation of the mass donor. The spin-down of the Be stars due to angular momentum transport from star to disc (Porter 1998) is another source of deceleration. In the well-known Be star $\gamma$ Cas, Robinson & Smith (2000) found that the X-ray flux varied with a period P = 1.1 d, which they interpreted as the rotational period of the mass donor. A similar period is detected in photometric observations (Harmanec et al. 2000; Henry & Smith 2012). This periodicity is probably due to the interaction between magnetic field of the Be star and its circumstellar disc or the presence of some physical feature, such as a spot or cloud, co-rotating with the star. The optical emission lines of MWC 148 are practically identical to those of $\gamma$ Cas. All detected lines in the observed spectral range (Balmer lines, HeI lines and FeII lines) have similar equivalent widths, intensities, profiles, and even a so-called wine-bottle structure noticeable in the $H\alpha$ line (see Fig. 1 in Zamanov et al. 2016). Bearing in mind the above estimations and the curious similarities between: (i) the mean 20-60 keV X-ray luminosity of $\gamma$ Cas and  (Shrader et al. 2015) and (ii) the optical emission lines of $\gamma$ Cas and MWC 148, we consider that periodicity $\sim\!1$ day could be detectable in X-ray/optical bands in the Be/$\gamma$-ray binaries and could provide a direct measurement of the rotational period of the mass donor. Discussion ========== The three Be/$\gamma$-ray binaries discussed here have non-zero eccentricities and misalignment between the spin axis of the star and the spin axis of the binary orbit could be possible (Martin et al. 2009). The inclination of the primary star in  to the line of sight is probably $ i_{Be} \sim 70^0$ (Zamanov et al. 2013). Aragona et al. (2014) derived $a_1 \sin i_{orb} = 8.64 \pm 0.52$. Assuming $M_1 = 15$, $M_2 = 1.4$ , we estimate $ i_{orb} \sim 67^0 - 73^0$. It appears that there is no significant deviation of the orbital plane from the equatorial plane of the Be star. The emission lines of MWC 148 are very similar to those of $\gamma$ Cas. The emission lines are most sensitive to the footpoint density and inclination angle (Hummel 1994). It means that in MWC 148 the Be star inclination should be similar to that of $\gamma$ Cas, for which the inclination is in the range $40^0 - 50^0$ (Clarke 1990, Quirrenbach et al. 1997). For MWC 148, Casares et al. (2012) estimated $a_1 \sin i_{orb} = 77.6 \pm 25.9$, which for $M_1 = 15$   and a 4  black hole gives $ 45^0 \lesssim i_{orb} \lesssim 65^0$. For MWC 656, Casares et al. (2014) give $M_1 \sin ^3 i_{orb} = 5.83 \pm 0.70$. Bearing in mind the range of $8$  $\le M_1 \le 10$ , this gives $53^0 \lesssim i_{orb} \lesssim 59^0$. Inclination of the Be star can be evaluated from the full width at zero intensity of the FeII lines $FWZI/2 \sin i = (G M_1/ R_1)^{1/2}$. From FWZI of FeII lines (Casares et al. 2012) and using $R_1= 9.5 - 10.0$   we estimate $i_{Be} \approx 53 - 61^0$. It appears that both planes are almost complanar. There are no signs of considerable deviation between the two planes. The opening half-angle of the Be stars’ circumstellar disc are $\sim \! 10^0$ (Tycner et al. 2006; Cyr et al. 2015) and it means that the compact object is practically orbiting in the plane of the circumstellar disc. The comparison between the orbit and circumstellar disc size (see Fig. \[f5.orbit\] and Sect. \[disc.trunc\]) shows that in these three objects we have three different situations: - In   the neutron star passes through the outer parts of the circumstellar disc at periastron, but it does not enter deeply in the disc; - In MWC 148 the compact object goes into the innermost parts of the disc (passes through the innermost parts and penetrates deeply in the disc) during the periastron passage; - In MWC 656 the black hole is constantly accreting from the outer edge of the circumstellar disc. In MWC 656, this means that the compact object (black hole) is at the disc border at all times and as a consequence it will have a higher and stable mass accretion rate along the entire orbit. It seems to be a very clear case of disc truncation in which the circumstellar disc is cut almost exactly at the black hole orbit. Because the $H\alpha$ peaks are connected with the outermost parts of the disc, the above three items probably explain the observational findings: 1\. In  the $H\alpha$ emission line has a two-peak profile at all times (in all our spectra in the period 1987 - 2015),because the circumstellar disc size is relatively small and the neutron star passes only through the outermost parts of the circumstellar disc at periastron; 2. In MWC 148, the big jumps in the $H\alpha$ parameters, $W_\alpha$, full width at half maximum and radial velocity (see Fig. 4 of Casares 2012) occur because the compact object enters (reaches) the inner parts of the disc during the periastron passage. 3\. In MWC 656 the double-peak profile is not often visible because the black hole is at the outer edge at all times and makes perturbations exactly in the places where the $H\alpha$ peaks are formed. It is worth noting that when the compact object causes large-scale perturbations, distorted profiles, such as those observed in 1A 535+262 (Moritani et al. 2011, 2013), will appear. If only a small portion of the outer disc is perturbed then it will appear in the central part of the emission line profile (e.g. in between the peaks or even filling the central dip) because the outer parts of the disc produce the central part of the $H\alpha$ emission line profile. Similar additional emission is already detected in the $H\alpha$ spectra of  (Paredes et al. 1990; Liu et al. 2000; Zamanov & Marti 2000). Gamma-ray emission has been repeatedly observed to be periodic in the system LSI+61303 (Albert et al. 2009; Saha et al. 2016) and also very likely in MWC148, where its periodic X-ray flares are highly correlated with TeV emission (Aliu et al. 2014). A similar situation occurs in the case of LS 5039 (Aharonian et al. 2006), 1FGL J1018.6-20135856 (H. E. S. S. Collaboration et al. 2015), and PSR B1259-63 (H.E.S.S. Collaboration et al. 2013), which has the longest orbital period observed in a $\gamma$-ray binary (about 3 yr). This is likely related to the very different physical conditions sampled by the compact companion as it revolves around the primary star in an eccentric orbit. Non-thermal emission (e.g. Dubus 2013) is produced from particles accelerated at the shock between the wind of the pulsar and matter flowing out from the primary star (the polar wind as well as the disc). The X-ray and $\gamma$-ray light curve of MWC 148 shows two maxima at orbital phases 0.35 and 0.75 and minimum at apastron passage (Acciari et al. 2009; Aliu et al. 2014). X-ray and $\gamma$-ray fluxes are correlated as mentioned above, in agreement with leptonic emission models, where relativistic electrons lose energy by synchrotron emission and inverse Compton emission (Maier & for the VERITAS Collaboration 2015). The highly eccentric orbital geometry sketched in the central panel of Fig. \[f5.orbit\] is also in good agreement with a periodic flaring system. In the case of MWC 656, $\gamma$-rays have been detected occasionally (Williams et al. 2010), but so far never reaching the TeV energy domain. The main differences from previous sources are the fact that the compact object has been shown to be a black hole and its orbit is only moderately eccentric (Casares et al. 2014). Based on our spectroscopic observations, the size of the excretion disc is such that the black hole is accreting matter only from its lower density outer edges. As mentioned before, this implies that the accretion rate is stable but at the same time low. Indeed, the quiescent X-ray emission level of the system is as weak as $\sim 10^{-8}$ Eddington luminosity (Munar-Adrover et al. 2014). Therefore, episodic $\gamma$-ray flares such as those detected by AGILE, likely require some enhancement of mass loss from the primary Be star or clumps in its circumstellar disc. We speculate here that the physical mechanism responsible for $\gamma$-ray emission in the MWC 656 black hole context could be related to the alternative microquasar-jet scenario also proposed for $\gamma$-ray binaries (see e.g. Romero et al. 2007). The fact that MWC 656 seems to adhere to the low-luminosity end of the X-ray/radio correlation for hard state compact jets also points in this direction (Dzib et al. 2015). Conclusions =========== From the spectroscopic observations of the three Be/$\gamma$-ray binaries we deduce that in  the neutron star crosses the outer parts of the circumstellar disc at periastron, in MWC 148 the compact object passes deeply through the disc during the periastron passage, and in MWC 656 the black hole is accreting from the outer parts of the circumstellar disc during the entire orbital cycle. The histograms in all three stars show that the disc size clusters at specific levels, indicating the circumstellar disc is truncated by the orbiting compact object. We estimate the interstellar extinction towards , MWC 148, and MWC 656. The rotation of the mass donors is similar to that of the Be/X-ray binaries. We suggest that the three stars deserve to be searched for a periodicity of about $1.0$ day. The authors are grateful to an anonymous referee for valuable comments and suggestions. This work was partially supported by grant AYA2013-47447-C3-3-P from the Spanish Ministerio de Economía y Competitividad (MINECO), and by the Consejería de Economía, Innovación, Ciencia y Empleo of Junta de Andalucía as research group FQM-322, as well as FEDER funds. Ablimit, I., Lü, G L., 2012, Sci. Chi.: Phys.Mech. 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--- abstract: 'We study the evolution of LTB Universe models possessing a varying cosmological term and a material fluid.' author: - 'LUIS P. CHIMENTO' - DIEGO PAVÓN title: INHOMOGENEOUS EXACT SOLUTIONS WITH VARYING COSMOLOGICAL TERM --- Introduction ============ There is at present an increasing feeling in the astrophysic community that the cosmological constant is not zero but should contribute substantially to the mass-energy of the Universe -see Weinberg [*et al.*]{} [@DHW] and references therein. This may be so if the energy of the quantum vacuum spontaneously decayed into matter and radiation, hence reducing the cosmological term to a value compatible with astronomical constraints -see for instance Overduin [*et al.*]{} [@JMO] and references therein. On the other hand, recently it has been pointed out that because of sources evolution it may well happen that the Universe is in reality inhomogeneous and describable by the Lamaître- Tolmann-Bondi (LTB) metric [@MHE]. Further motivations conductive to use inhomogeneous metrics can be found in Krasinski [@KRS]. Metric and models ================= We consider a spatially flat LTB metric $$ds^{2} = - dt^{2} + Y^{'2} \, dr^{2} + Y^{2}\, (d\theta^{2} + sin^{2} \theta \, d\phi^{2}), \; \; (Y = Y(r, t)) \label{1}$$ whose source is a perfect fluid, with equation of state $P = (\gamma - 1) \rho$, plus a varying cosmological term $\Lambda(t)$. The non-trivial Einstein equations are $$\rho + \Lambda = \frac{1}{Y^{2} \, Y^{'}} (\dot{Y}^{2} \, Y)^{'} \, , \label{2}$$ $$P - \Lambda = - \frac{1}{Y^{2} \dot{Y}} (\dot{Y}^{2}\, Y)^{.} \, , \label{3}$$ $$\frac{\ddot{Y}}{Y} + \left( \frac{\dot{Y}}{Y}\right)^{2} - \frac{\ddot{Y}^{'}}{Y^{'}}-\frac{\dot{Y}}{Y}\frac{\dot{Y}^{'}}{Y^{'}}=0 \qquad (8 \pi G = 1). \label{4}$$ In general the solutions can be expressed as $Y(r, t) = R(r)^{2/3} Z(t)^{2/3\gamma} $. Next we summarize different secenarios of interest -see Chimento and Pavón [@LD] for details. 1. For $\gamma$ and $\Lambda$ constants one obtains $$Y_{1} = R^{2/3} (r) \, C_{1}^{2/3\gamma} \, cosh^{2/3\gamma} \left( \frac{\sqrt{3 \gamma \Lambda}}{2} \; t + \varphi_{1} \right) \, , \\ \label{5}$$ $$Y_{2} = R^{2/3} (r) \, C_{2}^{2/3\gamma} \, sinh^{2/3\gamma} \left( \frac{\sqrt{3 \gamma \Lambda}}{2} \; t + \varphi_{2} \right). \label{6}$$ Obviously both sets of solutions have a final inflationary stage. 2. When $\gamma = \mbox{constant}$ and $$\Lambda (t) = \frac{\lambda_{0}^{2}}{t^{2}} \quad (\lambda_{0}^{2} = \mbox{constant}) \, , \label{7}$$ it follows that $$Z(t) =C_{1} \, t^{m_{+}} + C_{2} \, t^{m_{-}} \label{8}$$  \ where $ m_{\pm} = \left(1 \, \pm \, \sqrt{1 + 3 \gamma \lambda_{0}^{2}}\right)/2 $. Inflationary solutions may occur for large enough $\lambda_{0}^{2}$. 3. For $\gamma = \mbox{constant}$ and $$\Lambda = \lambda_{0}^{2} \, t^{n-2} \; \; \; (n \neq 0, \, 2) \, , \label{9}$$ the solution can be expressed as a combination of Bessel functions $$Z = C_{1} \, t^{1/2} \, J_{1/n}\left(\frac{\lambda_{0}}{n} \sqrt{- 3\gamma} \, t^{n/2} \right)$$ $$+ C_{2} \, t^{1/2} \, J_{-1/n}\left(\frac{\lambda_{0}}{n} \sqrt{- 3\gamma} \, t^{n/2} \right). \label{10}$$ The behavior at the asymptotic limits depends on $n$. For $0 < n <2$ one has the following: (i) When $ t \rightarrow 0$ one obtains $Z \sim C_{1} \, t + C_{2}$ -one can choose $C_{2} = 0$ to have the initial singularity at $t = 0$. (ii) When $t \rightarrow \infty$ there follows $Z \sim t^{\frac{1}{2}-\frac{n}{4}} \; cos \, t^{n/2}$.\ Likewise for $ n < 0 $: (i) when $ t \rightarrow 0 \; $ one obtains $Z \sim t^{\frac{1}{2}-\frac{n}{4}} \; cos \, (t^{n/2} + \varphi)\;. $ (ii) When $ t \to \infty \; $ one obtains $ Z \sim t \, .$ 4. For $\gamma = \mbox{constant}$ and $$\Lambda (t) =\lambda_{0}^{2}+c\mbox{e}^{-\alpha t} \qquad (c < 0)\, , \label{11}$$ where $\lambda_{0}^{2}$, $\alpha$ and $c$ are constants, again the general solution is a combination of Bessel functions $$Z = C_{1} \,J_{\frac{\lambda_{0}}{\alpha}\sqrt{3\gamma}} \left(\frac{\sqrt{- 3\gamma c}}{\alpha}\,\mbox{e}^{\frac{-\alpha}{2} t}\right)$$ $$+ \, C_{2} \,J_{-\frac{\lambda_{0}}{\alpha}\sqrt{3\gamma}} \left(\frac{\sqrt{- 3\gamma c}} {\alpha}\,\mbox{e}^{\frac{-\alpha}{2} t}\right) \, , \label{12}$$ with $$C_2=-\frac{J_{\frac{\lambda_{0}}{\alpha}\sqrt{- 3\gamma}} \left(\frac{\sqrt{- 3\gamma c}}{\alpha}\right)} {J_{-\frac{\lambda_{0}}{\alpha}\sqrt{- 3\gamma}} \left(\frac{\sqrt{- 3\gamma c}}{\alpha}\right)}\,C_1 \label{13}$$ in order to fix the initial singularity at $t = 0$. When $t \rightarrow 0$ one has $Z \sim t$. At the final stage, when $t \rightarrow \infty$ and $\Lambda \rightarrow \lambda_{0}^{2}$, one obtains the following asymptotic behavior $$Y\approx R^{2/3}(r)\,\mbox{e}^{\frac{\lambda_{0}} {\sqrt{-3\gamma c}}\,t} \, . \label{14}$$ For the particular case $\lambda_{0}^{2} = 0 $ and in the limit $t \rightarrow \infty$, there is a solution whose final behavior is $$Y\approx R^{2/3}(r)\,t^{2/3\gamma} \, . \label{15}$$ 5. For $\gamma = \gamma (t)$ and $\Lambda = \Lambda (t)$ it can be found expressions for both quantities, $$\Lambda(t)=\frac{4C^2(t-t_0)^{2n}}{3\gamma_0 n^2(n+1)^2} \left[1+\frac{(t-t_0)^{n+1}}{C}\right]^{\frac{2-n}{n}}, \label{16}$$ $$\gamma(t)=\gamma_0\left[1+\frac{(t-t_0)^{n+1}}{C}\right]^{-\frac{2+n}{n}}, \label{17}$$ as well as an asymptotic solution for $Y(t,r)$ $$Y\approx R^{2/3} (r) \, T_{0}^{2/3\gamma_0} \, \left[\frac{(n+1)(n+2)}{n}(t-t_0)\right]^{2/3\gamma_0} \, , \label{18}$$ where $T_{0}, \gamma_{0}, t_{0}, \mbox{and} C $ are constants. It is worthy of note that, for $t\gg t_0$ we have both $\gamma \rightarrow \gamma_0$ and $\Lambda \rightarrow 0$. To examine the singular structure of the plane LTB metric (1) we have calcuated the curvature scalar and evaluated it at the points where the coefficients of the metric $Y^{'2}$ and/or $Y^{2}$ vanish. All the solutions we have found for $\gamma = \mbox{constant}$ except (\[5\]) have a singularity at $t = 0$, i.e. the big-bang singularity. Conclusions =========== We have found the coefficients of the LTB metric assuming that the early Universe possessed a time varying cosmological term, and that the adiabatic index of the material fluid were either constant or not.\ (a) All the solutions we have derived contain an arbitrary function of the radial coordinate.\ (b) For $\gamma = $ constant all the solutions, except (\[5\]) have a singularity at $t = 0$, i.e. the big-bang singularity.\ (c) Constant as well as varying cosmological terms give rise asymptotically to exponential inflation -see (\[5\]), (\[6\]) and (\[14\]).\ (d) For $\Lambda(t) \propto e^{-\alpha t}$ there exist solutions which behave as though the Universe were asymptotically matter dominated at late times when $\gamma = 1$, i.e. $Y \propto t^{2/3}$. Acknowledgements {#acknowledgements .unnumbered} ================ This work was partially supported by the Spanish Ministry of Education under Grant PB94-0718, and the University of Buenos Aires under Grant EX-260. References {#references .unnumbered} ========== [99]{} D.H. Weinberg, R.A.C Croft, Lars Hernquist, N. Katz and M. Pettini; [*Report*]{} astro-ph/9810011. J.M. Overduin and F.I. Cooperstock, [*Phys. Rev.*]{} D [**58**]{}, 43506 (1998). N. Mustapha, C. Hellaby and G.F.R. Ellis, [*Mon. Not. R. Astron. Soc.*]{} [**292**]{}, 817 (1997); gr-qc/9808079. A. Krasinski, [*Inhomogeneous Cosmological Models*]{} (Cambridge University Press, Cambridge, 1997). L.P. Chimento and D. Pavón, [*Gen. Relativ. Grav.*]{} [**30**]{}, 643 (1998).

--- abstract: 'We show that the homology of ordered configuration spaces of finite trees with loops is torsion free. We introduce configuration spaces with sinks, which allow for taking quotients of the base space. Furthermore, we give a concrete generating set for all homology groups of configuration spaces of trees with loops and the first homology group of configuration spaces of general finite graphs. An important technique in the paper is the identification of the $E^1$-page and differentials of Mayer-Vietoris spectral sequences for configuration spaces.' address: - 'Department of Mathematics, Reed College, Oregon, USA' - 'Institut für Mathematik, Freie Universität Berlin, Germany' author: - Safia Chettih - Daniel Lütgehetmann bibliography: - 'conf-graph.bib' title: The Homology of Configuration Spaces of Trees with Loops --- Introduction ============ For a topological space $X$ and a finite set $S$ we define the *configuration space of $X$ with particles labeled by $S$* as $$\operatorname{Conf}_S(X) {\mathrel{\mathop:}=}{\mathopen{}\mathclose\bgroup\originalleft}\{f\colon S\to X \text{ injective} {\aftergroup\egroup\originalright}\} \subset \operatorname{map}(S, X).$$ For $n\in{\mathbb{N}}$ we write $\mathbf{n}{\mathrel{\mathop:}=}\{1, 2, \ldots, n\}$ and $\operatorname{Conf}_n(X){\mathrel{\mathop:}=}\operatorname{Conf}_{\mathbf{n}}(X)$. This is usually called the $n$-th ordered configuration space of $X$. Let $G$ be a finite connected graph (i.e. a connected 1-dimensional CW complex with finitely many cells). We are interested in the homology of configurations of $n$ ordered particles in $G$, that is, $H_*(\operatorname{Conf}_n(G))$. A main ingredient in proving results about configurations in graphs is the existence of combinatorial models for the configuration spaces. In [@Abrams00], Abrams introduced a discretized model for the configuration space of $n$ points in a graph which is a cubical complex, allowing the spaces to be studied using techniques from discrete Morse theory and connecting them with right-angled Artin groups (see [@Farley05], [@Crisp04]). A similar discretized model for non-$k$-equal configuration spaces in a graph, where up to $k-1$ points are allowed to collide, was constructed in [@Chettih16], providing inspiration for the configuration with sinks introduced in this paper. Not long after the introduction of Abrams’ model, [Ś]{}witkowski introduced a cubical complex which is a deformation retract of the space of unordered configurations of $n$ points in a graph (see [@Swiat01]). In this model, instead of the points moving discrete distances along the graph, the points move from an edge to a vertex of valence at least two or vice versa. This gives a sharper bound for the homological dimension of these configuration spaces as the dimension of the complex is bounded from above by the number of vertices in the graph (see [@Ghrist01], [@Farley05] for proofs that this bound also holds for Abrams’ model). An analogous model holds for ordered configurations (see [@Luetgehetmann14]), by keeping track of the order of points on an edge. The combinatorial model for configurations with sinks has structure similar to the latter models. In order to describe the homology of $\operatorname{Conf}_n(G)$ we will compare it to a modified version of configuration spaces: we add “sinks” to our graphs. Sinks are special vertices in the graph where we allow particles to collide. For ordinary configuration spaces, if we collapse a subgraph $H$ of $G$ then this does *not* induce a map $$\operatorname{Conf}_n(G)\dashrightarrow\operatorname{Conf}_n(G/H)$$ because some of the particles could be mapped to the same point in $G/H$. If, however, we turn the image of $H$ under $G\to G/H$ into a sink, there is now an induced map on configuration spaces. Our first theorem shows that in the *ordered* case, there is no torsion and a geometric generating system for a large class of finite graphs. A finite connected graph $G$ is called a *tree with loops* if it can be constructed as an iterated wedge of star graphs and copies of $S^1$. A homology class $\sigma\in H_q(\operatorname{Conf}_n(G))$ is called the *product of classes $\sigma_1\in H_{q_1}(\operatorname{Conf}_{T_1}(G_1))$ and $\sigma_2\in H_{q_2}(\operatorname{Conf}_{T_2}(G_2))$* for $q_1+q_2=q$ if it is the image of $\sigma_1\otimes \sigma_2$ under the map $$H_q(\operatorname{Conf}_n(G_1\sqcup G_2)) \to H_q(\operatorname{Conf}_n(G))$$ induced by an embedding $G_1\sqcup G_2\hookrightarrow G$. Analogously, iterated products are induced by embeddings $G_1\sqcup G_2 \sqcup \ldots \sqcup G_n \hookrightarrow G$. For $k\ge 3$ let $\operatorname{Star}_k$ be the star graph with $k$ leaves, $\operatorname{H}$ the tree with two vertices of valence three and $S^1$ a circle with one vertex of valence 2. We call a class $\sigma\in H_q{\mathopen{}\mathclose\bgroup\originalleft}( \operatorname{Conf}_n(G) {\aftergroup\egroup\originalright})$ a *product of basic classes* if $\sigma$ is an iterated product of classes in groups of the form $H_j(\operatorname{Conf}_{n_i}(G_i))$ where $j$ equals 0 or 1 and $G_i$ is a star graph, the $\operatorname{H}$-graph, the circle $S^1$ or the interval $I$. \[thm:trees\] Let $G$ be a tree with loops and let $n$ be a natural number. Then the integral homology $H_q{\mathopen{}\mathclose\bgroup\originalleft}( \operatorname{Conf}_n(G); {\mathbb{Z}}{\aftergroup\egroup\originalright})$ is torsion-free and generated by products of basic classes for each $q\ge0$ A 1-class in $S^1$ moves all particles around the circle, a 1-class in a star graph uses the essential vertex to shuffle around the particles, and a 1-class in the $\operatorname{H}$-graph uses one of the vertices to reorder the particle and then undoes this reordering using the other vertex. The proof of will show that 2-classes in an $\operatorname{H}$-graph are given by sums of products of 1-classes in the two stars, and there are no higher dimensional classes in these three types of graphs. The proof of rests on an inductive argument on the number of essential vertices of a graph. We construct a basis for the configuration space of a star graph with loops such that the $E^1$-page of the Mayer-Vietoris spectral sequence induced by our gluing splits over that basis. We can identify a part of the homology of the $E^1$-page with configuration spaces where some of the points have been forgotten, and the rest of the homology with a configuration space where the star graph has been collapsed to a sink (see for the definition of sink configuration spaces). The gluing process does not create torsion, so torsion-freeness follows from explicit calculations of the homology of ordered configurations in star graphs with loops. An explicit generating set of homology classes with known relations is essential to our proof. A basis for the homology of ordered configurations of two points in a tree was first constructed in [@Chettih16], which highlighted the role of basic classes of the $\operatorname{H}$ graph in the configuration space of wedges of graphs. See also [@BF09] and [@FH10] for descriptions of product structure in configurations of two points on planar and non-planar graphs. The Mayer-Vietoris principle was previously used to compute the homology of (unordered) configuration spaces of graphs in [@MaSa16]. For more general graphs, the analogous theorems do not hold: \[thm:non-product-general-graph\] If $G$ is any finite graph and $n$ a natural number, then the *first* homology group $H_1(\operatorname{Conf}_n(G))$ is generated by basic classes. However, for each $i\ge2$ there exists a finite graph $G$ and a number $n$ such that $H_i(\operatorname{Conf}_n(G))$ is not generated by products of 1-classes. We provide explicit examples for the second statement. Abrams and Ghrist were aware of the second part of this result in 2002 ([@AbramsGhrist02]), but their example does not generalize to arbitrary dimensions. More specifically, they showed that $\operatorname{Conf}_2(K_5)$ and $\operatorname{Conf}_2(K_{3,3})$ are homotopic to surfaces of genus $6$ and $4$ respectively, where $K_5$ is the complete graph on five vertices and $K_{3,3}$ is the complete bipartite graph on $3+3$ vertices. Both theorems above can be generalized to the case where arbitrary subsets of the vertices are turned into sinks. In between versions of this paper, Ramos considered configurations where all the vertices of a graph are sinks, approaching them through the lens of representation stability ([@Ramos17]). His theorems concerning torsion-freeness and bounds on homological dimension are special cases of the theorems above. In an earlier version of this paper, we asserted torsion-freeness for arbitrary finite graphs. However, our proof relied on a basis which we discovered does not split the Mayer-Vietoris spectral sequence in the way we described it. Our investigation of obstructions to constructing an appropriate basis led us to the counterexamples in . Such a basis may still exist, and we believe the following: \[conj:torsion-free\] Let $G$ be a finite graph and $n$ a natural number. Then the integral homology $H_q(\operatorname{Conf}_n(G);{\mathbb{Z}})$ is torsion-free for each $q \geq 0$. To answer this question for general graphs, more work is needed on relations in the homology of configuration spaces of graphs with many cycles. The paper proceeds as follows: we introduce a combinatorial model for configurations with “sinks” in order to calculate the homology of a few specific examples in . After the Mayer-Vietoris spectral sequence is established in , we construct our desired basis and argue inductively by gluing on stars with loops in . The case of the first homology in an arbitrary graph comprises , with counterexamples for higher homology. Our techniques in this section are substantially different since we no longer have bases which split the spectral sequence. Acknowledgements ---------------- The second author was supported by the Berlin Mathematical School and the SFB 647 “Space – Time – Matter” in Berlin. The authors want to thank Elmar Vogt and Dev Sinha for helpful discussions, and the referee, whose comments helped us make the paper more readable. Quotient and Mayer-Vietoris constructions {#sec:conf-with-sinks} ========================================= In order to describe the homology of $\operatorname{Conf}_n(G)$ we will compare it to a modified version of configuration spaces: we add “sinks” to our graphs. Sinks are special vertices in the graph where we allow particles to collide, and they enable us to collapse subgraphs and get an induced map on configuration spaces. This does not work for ordinary configuration spaces: if we collapse a subgraph $H$ of $G$ then this does *not* induce a map $$\operatorname{Conf}_n(G)\dashrightarrow\operatorname{Conf}_n(G/H)$$ because some of the particles could be mapped to the same point in $G/H$. For a number $n\in{\mathbb{N}}$, a graph $G$ and a subset $W$ of $G$’s vertices we define the following configuration space with sinks: $${\operatorname{Conf}_{n}^\mathrm{sink}}(G,W) = {\mathopen{}\mathclose\bgroup\originalleft}\{(x_1,\ldots,x_n)\,|\, \text{for $i\neq j$ either $x_i\neq x_j$ or $x_i=x_j\in W$}{\aftergroup\egroup\originalright}\}\subset G^n.$$ Looking at the collapse map $G\to G/H$ again, there is now an induced map on configuration spaces if we turn the image of $H$ under $G\to G/H$ into a sink: $$\operatorname{Conf}_n(G) \to {\operatorname{Conf}_{n}^\mathrm{sink}}(G/H, H/H).$$ A combinatorial model --------------------- We can extend the techniques of [@Swiat01] and [@Luetgehetmann14] to obtain a cube complex model of configuration spaces with sinks. More precisely we will define a deformation retraction $r\colon {\operatorname{Conf}_{n}^\mathrm{sink}}(G,W)\to{\operatorname{Conf}_{n}^\mathrm{sink}}(G,W)$ such that the image of $r$ has the structure of a finite cube complex. Each axis of such a cube will correspond to the combinatorial movement of one particle. A combinatorial movement here is either given by the movement from an essential non-sink vertex onto an edge or along a single edge from one sink to the other. Each vertex and each such edge can only be involved in one of those combinatorial movements at a time, so the dimension of this cube complex will be restricted by the number of essential non-sink vertices and the edges connecting two sinks. A cube complex $K$ is the quotient of a disjoint union of cubes $X=\bigsqcup_{\lambda\in\Lambda}[0,1]^{k_\lambda}$ by an equivalence relation $\sim$ such that the quotient map $p\colon X\to X/\!\!\sim\ = K$ maps each cube injectively into $K$ and we only identify faces of the same dimensions by an isometric homeomorphism. The definition above differs slightly from the original definition by Bridson and Häfliger, in that it allows two cubes to be identified along more than one face. This is a necessary property for the complex we wish to describe. \[prop:combinatorial-model-sinks\] Let $G$ be a finite graph, $W$ a subset of the vertices and $n\in{\mathbb{N}}$. Then ${\operatorname{Conf}_{n}^\mathrm{sink}}(G,W)$ deformation retracts to a finite cube complex of dimension $\min\{n, |V_{\ge2}| + |E_W| \}$, where $V_{\ge 2}$ is the set of non-sink vertices of $G$ of valence at least two and $E_W$ is the set of edges incident to two sinks. The naive approach would be to retract particles in the interior of an edge to positions equidistant throughout the edge. However, this fails to be continuous as the number of particles in the interior changes, such as when a particle moves off a vertex. To fix this, we construct an additional parameter which controls the distance of the outermost particles on an edge from the vertices. Give $G$ the path metric such that every edge has length 1. For this proof, we define half edges in $G$: every edge consists of two distinct half edges $h^\iota_e$ and $h^\tau_e$. For each half edge $h$ we denote by $v(h)$ the vertex incident to $h$ and by $e(h)$ the edge corresponding to $h$ *with the orientation determined by the half edge*. If $h$ is a half edge, then $\overline{h}$ is the other half of $e(h)$, and $e(h) = - e{\mathopen{}\mathclose\bgroup\originalleft}(\overline{h}{\aftergroup\egroup\originalright})$. The general idea is now the following: the retraction $r$ only changes the position of particles *inside* (closed) edges of the graph. We move as many particles of a given configuration $\mathbf{x}=(x_1,\ldots,x_n)$ as possible into the sinks, so that $r(\mathbf{x})$ has at most one particle in the interior of any edge incident to a sink. Furthermore, the particles of $r(\mathbf{x})$ on each single edge will be equidistant, except for the outermost particles, which may be closer to the vertices, see . The main difficulty will be to define for each configuration $\mathbf{x}$ and each half edge $h$ the parameter $t_h\in[0,1]$ determining the distance of the particles from the corresponding vertex. Decreasing $t_h$ to zero represents moving the particle on the edge that is nearest to the vertex $v(h)$ towards that vertex. To avoid multiple particles approaching the same vertex, we therefore require that for any pair of half edges $h\neq h'$ with $v(h)=v(h')$ only one of the two values $t_h$ and $t_{h'}$ can be strictly smaller than 1. \[circle dotted/.style=[dash pattern=on .05mm off 1mm, line cap=round]{}, line width = .7pt\] (qdots) at (1,0) [$\cdots$]{}; (-4,0) edge\[|-\] (qdots.west) (qdots.east) edge\[-|\] (4,0); (0,0) circle (0.1); ; (0,0) circle (0.1); ; (0,0) circle (0.1); ; (0,0) circle (0.1); ; (0,0) circle (0.1); ; (-4,0) – node\[below = .8em, pos=0.5\] [$t_{h^\iota_e}\cdot c_e$]{}(-3.4,0); (-3.4,0) – node\[above = .8em, pos=0.5\] [$c_e$]{}(-1.4,0); (1.6,0) – node\[above = .8em, pos=0.5\] [$c_e$]{}(3.6,0); (3.6,0) – node\[below = .8em, pos=0.5\] [$t_{h^\tau_e}\cdot c_e$]{}(4,0); For fixed $(x_1, \ldots, x_n)\in{\operatorname{Conf}_{n}^\mathrm{sink}}(G,W)$ we now define the image $r(\mathbf{x})$. The first step is to construct the parameter $t_h$. Let $v\in V(G)$ be a vertex and denote by $H_v$ the set of half edges $h$ with $v(h)=v$. If $H_v$ has only one element, then we set $t_h=1$ because we do not want to move particles towards a vertex of valence 1. Also, if $v$ is occupied by one of the particles $x_i$, then we set $t_h=1$ for all $h\in H_v$ because we do not want to move particles towards an occupied vertex. Now assume that the valence of $v$ is at least two and it is not occupied by a particle. If for a half edge $h\in H_v$ the edge $e(h)$ contains no particles or $v(h)$ is a sink, set $t_h=1$. Otherwise, the particles on $e(h)$ cut the edge into segments, and we order these segments according to the orientation of $e(h)$ given by $h$. Let $\ell_h$ be the quotient of the length of the first segment by the length of the second segment, capped to the interval $[0,1]$, unless $v{\mathopen{}\mathclose\bgroup\originalleft}(\overline{h}{\aftergroup\egroup\originalright})$ is a sink. If it is a sink, let $\ell_h$ be the length of the first segment. We treat these cases differently because particles on edges incident to sinks move from vertex to vertex instead of from edge to vertex. We now define $$t_h {\mathrel{\mathop:}=}\min{\mathopen{}\mathclose\bgroup\originalleft}\{1, \frac{\ell_h}{\displaystyle\min_{h'\in H_v-\{h\}} \ell_{h'}}{\aftergroup\egroup\originalright}\}.$$ Notice: - if $\ell_h=\ell_{h'}$, then $t_h=t_{h'}=1$, - if only one of the $\ell_h$ goes to zero, then also $t_h$ goes to zero, and - at most one of the $t_h$ for $h\in H_v$ is strictly smaller than 1. Given these parameters $t_h$ for all half edges $h$ we now construct the configuration $r((x_1, \ldots, x_n))$. The particles on the vertices are not moved by the retraction, so it remains to describe the change of position for the particles in the interior of an edge $e$. We will not change the order of the particles but only their position within the edge, and to make the description more concise we choose once and for all an isometric identification of each edge $e$ with $[0,1]$ such that $v(h^\iota_e)=0$. **If $\mathbf{e}$ is not incident to a sink vertex** the new position of the $j$-th vertex on $e$ will be given by $(t_{h^\iota_e} + j-1)\cdot c_e$, where $k_e\ge 1$ is the number of particles in the interior of $e$ and $c_e{\mathrel{\mathop:}=}(t_{h^\iota_e} + k_e - 1 + t_{h^\tau_e})^{-1}$ will be the distance between the particles on that edge. This gives all particles on the edge the same distance and only modifies the distances from the vertices, see . It remains to be shown that the positions of the particles on the edge vary continuously as $t_h$ goes to 0. This is true when $t_{h^\iota_e}>0$, and notice that for $t_{h^\iota_e}=0$ the images of the particles will be the same as if we considered the first particle to be on $v(h^\iota_e)$ and $t_{h^\iota_e}=1$: this would change $t_{h^\iota_e}$ from $0$ to $1$ and reduce $k_e$ by one, so that $c_e$ will be exactly the same. The analogous result also holds for $h^\tau_e$. This shows that the position of the particles on this closed edge after applying $r$ is continuous in the original configuration. **If $\mathbf{e}$ is incident to precisely one sink vertex** then we can assume that this sink vertex corresponds to $0\in[0,1]$. All particles on $e$ except the last one are then moved to $0$, the last particle is moved to $1-t_{h^\tau_e}\in[0,1]$. **If both vertices incident to $\mathbf{e}$ are sinks** we slide all particles away from $1/2\in[0,1]$ with speed given by their distance from $1/2$ until at most one particle is left in the interior $(0,1)$ of the interval. This gives a configuration having one particle on $e$ and the rest on the sinks. The map described above is continuous and a retraction, i.e. satisfies $r^2=r$. In the description we only changed the positions of particles on individual edges, so there is an obvious homotopy from the identity to $r$ by just adjusting the positions of the particles on each edge individually. The image of $r$ has the structure of a cube complex: the 0-cells are configurations where all particles in the interior of each interval cut the interval into pieces of equal length, and additionally no particle is in the interior of an edge with one or two sinks. A $k$-cube is given by choosing such a $0$-cell, $k$ distinct particles which are either outmost on their edge or on a sink and move them to an adjacent vertex. Such a choice of $k$ movements determines a $k$-cube if and only if we can realize the movements independently, namely if - no two particles move along the same edge, - no two particles move towards the same *non-sink* vertex and - no particle moves towards an occupied *non-sink* vertex. Each direction of the cube corresponds to the movement of one of the particles. By the description of the choices involved for finding $k$-cubes we immediately get the restriction on the dimension. For more details about the general construction of the cube complex (without sinks), see [@Luetgehetmann14]. It will be useful for subsequent proofs to have a notion for pushing in new particles from the boundary of the graph. Let $G$ be a graph and $e$ be a leaf. For a finite set $S$ and an element $s\in S$, define the map $$\iota_{e,s}\colon{\operatorname{Conf}_{S-\{s\}}^\mathrm{sink}}(G,W)\hookrightarrow{\operatorname{Conf}_{S}^\mathrm{sink}}(G,W)$$ by slightly pushing in the particles on $e$ and putting $s$ onto the univalent vertex of $e$. Let $G$ be a graph. For finite sets $S'\subset S$, define the map $$\pi_{S'}\colon {\operatorname{Conf}_{S}^\mathrm{sink}}(G,W)\to{\operatorname{Conf}_{S'}^\mathrm{sink}}(G,W)$$ by forgetting the particles $S-S'$. If $S'=\{s\}$ then we write instead $\pi_s{\mathrel{\mathop:}=}\pi_{\{s\}}$. Notice that the composition $\pi_{S-\{s\}}\circ \iota_{e,s}$ is homotopic to the identity. Let $X=\Sigma_i \alpha_iX_i$ be a cellular chain in the combinatorial model of ${\operatorname{Conf}_{S}^\mathrm{sink}}(G, W)$ . The particle $s$ is called a *fixed particle of $X$* if there exists a cell $c$ of the graph $G$ such that $\pi_s(X_i)$ is contained in the interior of $c$ for all $X_i$. Here, the interior of a vertex is the vertex itself. Notice that fixed particles may still move inside their edge to preserve equidistance, but they never leave their edge or vertex. The homology for small graphs ----------------------------- For later use we calculate the homology of some of these configuration spaces with sinks. \[prop:homology-sinks\] $$\begin{aligned} H_i{\mathopen{}\mathclose\bgroup\originalleft}( {\operatorname{Conf}_{n}^\mathrm{sink}}(I, \varnothing) {\aftergroup\egroup\originalright}) &= \begin{cases} {\mathbb{Z}}\Sigma_n & i = 0\\ 0 & \text{else} \end{cases}\\ H_i{\mathopen{}\mathclose\bgroup\originalleft}( {\operatorname{Conf}_{n}^\mathrm{sink}}(S^1, \varnothing) {\aftergroup\egroup\originalright}) &= \begin{cases} {\mathbb{Z}}{\mathopen{}\mathclose\bgroup\originalleft}(\Sigma_n/\mathrm{shift}{\aftergroup\egroup\originalright})\cong {\mathbb{Z}}^{(n-1)!} & i = 0,1\\ 0 & \text{else} \end{cases}\\ H_i{\mathopen{}\mathclose\bgroup\originalleft}( {\operatorname{Conf}_{n}^\mathrm{sink}}(I, \{0\}) {\aftergroup\egroup\originalright}) &= \begin{cases} {\mathbb{Z}}& i = 0\\ 0 & \text{else} \end{cases}\\ H_i{\mathopen{}\mathclose\bgroup\originalleft}( {\operatorname{Conf}_{n}^\mathrm{sink}}(I, \{0, 1\}) {\aftergroup\egroup\originalright}) &= \begin{cases} {\mathbb{Z}}& i = 0\\ {\mathbb{Z}}^{(n-2)2^{n-1}+1} & i = 1\\ 0 & \text{else} \end{cases}\\ H_i{\mathopen{}\mathclose\bgroup\originalleft}( {\operatorname{Conf}_{n}^\mathrm{sink}}(S^1, \{0\}) {\aftergroup\egroup\originalright}) &= \begin{cases} {\mathbb{Z}}& i = 0\\ {\mathbb{Z}}^{n} & i = 1\\ 0 & \text{else} \end{cases} \end{aligned}$$ The first two are clear. The interval with one sink has contractible configuration space: we can just gradually pull all particles into the sink. For the last two cases, note that the spaces are obviously connected by pulling all particles onto one of the sinks. Furthermore, by they are homotopic to 1-dimensional cube complexes. Computing the Euler characteristic gives the described ranks: There is a zero cube for every distribution of particles onto the two sinks, which means that there are $2^n$ of them. We have a 1-cell for each choice of one moving particle and every distribution of the remaining ones onto the two sinks, so there are $n2^{n-1}$ many 1-cells. Notice that this is the 1-skeleton of the $n$-dimensional cube. Thus, the Euler characteristic is $(2-n)2^{n-1}$, which determines the rank of the first homology group. There is precisely one zero cell, namely the one where all particles are on the sink. There is one 1-cell for each choice of one particle moving along the edge, giving $n$ 1-cells and therefore the Euler characteristic $1-n$. Notice that this is a bouquet of circles. \[rem:sink-cycles-as-H-cycles\] Cycles in $H_1({\operatorname{Conf}_{n}^\mathrm{sink}}(I, \{0, 1\}))$ can be regarded as cycles in the ordinary configuration space of the H-graph $\operatorname{Conf}_n(\operatorname{H})$, see . Replace both spaces by their combinatorial models and define a continuous map as follows: take a 0-cell of the configuration space with sinks and replace particles sitting on a sink vertex with them sitting on the corresponding lower leaf of the H-graph in their canonical ascending order. Moving a particle $x$ from one sink vertex to the other is then given by moving all particles blocking $x$’s path to the vertex to the upper leaf, moving $x$ onto the horizontal edge, moving the particles on the upper leaf back to the lower leaf and repeating the same game on the other side in reverse. This determines a continuous map between combinatorial models and thus induces a map on cellular 1-cycles. (-1,0) – (1,0); (-1,0) circle (.1cm); (1,0) circle (.1cm); at (0,0) [3]{}; at (0,0) [5]{}; at (0,0) [6]{}; at (0,0) [2]{}; at (0,0) [4]{}; at (0,0) [1]{}; (.15, .1) – (.4, .1); (.15, -.1) – (.4, -.1); at (2.3,0) [$\leftrightsquigarrow$]{}; (-1, 1) – (0,0); (-1, -1) – (0,0); (0, 0) – (2,0); (2, 0) – (3,1); (2, 0) – (3,-1); at (0,0) [6]{}; ; at (0,0) [5]{}; ; at (0,0) [3]{}; ; at (0,0) [4]{}; ; at (0,0) [2]{}; ; at (0,0) [1]{}; ; (.75, .1) – (1, .1); (.75, -.1) – (1, -.1); This map is injective in homology: composing the map with the map collapsing the two pairs of leaves to sinks gives a map that is homotopic to the identity, showing that the homology of ${\operatorname{Conf}_{n}^\mathrm{sink}}(I,\{0,1\})$ is a direct summand of the homology of $\operatorname{Conf}_n(\operatorname{H})$. A Mayer-Vietoris spectral sequence for configuration spaces {#sec:mv-spectral-sequence} ----------------------------------------------------------- To compute the homology of the configuration space of a space $X$ we can decompose $X$ into smaller spaces and patch together local results. A structured way to do this is by using the Mayer-Vietoris spectral sequence associated with a countable open cover. Let $J$ be a countable ordered index set and $\{V_j\}_{j\in J}$ an open cover of $X$, then we define the following countable open cover $\mathcal{U}(\{V_j\})$ of $\operatorname{Conf}_n(X)$: for each $\phi\colon \mathbf{n}\to J$ we define $U_\phi$ to be the set of all those configurations where each particle $i$ is in $V_{\phi(i)}$, i.e. $$U_\phi {\mathrel{\mathop:}=}\bigcap_{i\in\mathbf{n}} \pi_i^{-1}{\mathopen{}\mathclose\bgroup\originalleft}( V_{\phi(i)} {\aftergroup\egroup\originalright}).$$ These sets are open and cover the whole space, so they define a spectral sequence $$E^1_{p,q} = \bigoplus_{\{\phi_0,\ldots,\phi_p\}} H_q{\mathopen{}\mathclose\bgroup\originalleft}( U_{\phi_0}\cap\cdots\cap U_{\phi_p} {\aftergroup\egroup\originalright}) \Rightarrow H_*{\mathopen{}\mathclose\bgroup\originalleft}( \operatorname{Conf}_n(X) {\aftergroup\egroup\originalright})$$ converging to the homology of the whole space. For a proof of the convergence of this spectral sequence, see [@Chettih16 Proposition 2.1.9, p. 13]. Notice that $$U_{\phi_0}\cap\cdots\cap U_{\phi_p} = \bigcap_{i\in\mathbf{n}} \bigcap_{0\le j\le p} \pi_i^{-1}{\mathopen{}\mathclose\bgroup\originalleft}( V_{\phi_j(i)} {\aftergroup\egroup\originalright}).$$ For brevity, we will also write $$U_{\phi_0\cdots\phi_p} {\mathrel{\mathop:}=}U_{\phi_0} \cap \cdots \cap U_{\phi_p}.$$ The boundary map $d_1$ is given by the alternating sum of the face maps induced by $$U_{\phi_0}\cap\cdots\cap U_{\phi_p} \hookrightarrow U_{\phi_0}\cap\cdots\cap \widehat{U_{\phi_i}}\cap\cdots\cap U_{\phi_p}$$ forgetting the $i$-th open set from the intersection. Of course, this construction generalizes to configuration spaces with sinks. Configurations of particles in trees with loops {#sec:trees} =============================================== We will more generally prove for all graphs as in the statement of the theorems with any (possibly empty) subset of the vertices of valence one turned into sinks. The proof will proceed by induction over the number of essential vertices (i.e. vertices of valence at least three). We first prove the base case: \[prop:tree-base-case\] Let $G$ be a finite connected graph with precisely one essential vertex and $W$ a subset of the vertices of valence 1. Then $H_1({\operatorname{Conf}_{n}^\mathrm{sink}}(G,W))$ is free and generated by basic classes. Notice that if we talk of $\operatorname{H}$-classes in a graph *with sinks* $(G,W)$ then we allow some of the leaves of $\operatorname{H}$ to be collapsed to a sink under the map $\operatorname{H}\to G$. In the proof, we will need the following definition: \[def:configurations-of-tuples\] For finite sets $T\subset S$, a finite graph $G$, a subset $K\subset G$, and sinks $W\subset V(G)$ write $\Gamma=(G,K)$ and define $${\operatorname{Conf}_{S,T}^\mathrm{sink}}( \Gamma, W) = \{ f\colon S\to G\,|\, f(T)\subset K \}\subset {\operatorname{Conf}_{S}^\mathrm{sink}}(G,W).$$ As a consequence of the definition, we get $${\operatorname{Conf}_{S, \emptyset}^\mathrm{sink}}(\Gamma, W) = {\operatorname{Conf}_{S}^\mathrm{sink}}(G, W)$$ and $${\operatorname{Conf}_{S, S}^\mathrm{sink}}(\Gamma, W) = {\operatorname{Conf}_{S}^\mathrm{sink}}(K, W\cap K).$$ By , ${\operatorname{Conf}_{n}^\mathrm{sink}}(G,W)$ is homotopy equivalent to a graph, so the first homology is free. To see that it is generated by basic classes, we inductively use a Mayer-Vietoris long exact sequence. For a sink $w\in W$ let $\Gamma_w = (G,G-\{w\})$. Notice that $${\operatorname{Conf}_{S,\emptyset}^\mathrm{sink}}(\Gamma_w,W) = {\operatorname{Conf}_{S}^\mathrm{sink}}(G,W).$$ and $$\begin{aligned} {\operatorname{Conf}_{S, S}^\mathrm{sink}}(\Gamma_w,W) &= {\operatorname{Conf}_{S}^\mathrm{sink}}(G-\{w\}, W-\{w\}) \\ &\simeq {\operatorname{Conf}_{S}^\mathrm{sink}}(G, W-\{w\}), \end{aligned}$$ where the last homotopy equivalence follows because $w$ has valence 1. For two sinks $w_0\neq w_1$ we therefore have $${\operatorname{Conf}_{S, S}^\mathrm{sink}}(\Gamma_{w_0},W) \simeq {\operatorname{Conf}_{S,\emptyset}^\mathrm{sink}}(\Gamma_{w_1}, W-\{w_0\}).$$ Moving elements from $S-T$ to $T$ and using the above identifications, we will show by induction on $|S-T|$ and the number of sinks $|W|$ that the first homology of all spaces ${\operatorname{Conf}_{S,T}^\mathrm{sink}}(\Gamma,W)$ is generated by basic classes. In the base case, we have $T=\emptyset$ and $W=\emptyset$, so the space we are investigating is the ordinary configuration space $\operatorname{Conf}_S(G)$, which is generated by basic classes by (this is not a circular argument, the proposition is only stated and proven later since it is the main step to compute the first homology of configuration spaces of arbitrary finite graphs). For the induction step, choose an arbitrary $s\in S-T$ and take the open covering $\{V_1, V_2\}$ of ${\operatorname{Conf}_{S,T}^\mathrm{sink}}(\Gamma_{w_0}, W)$ given by the subsets $$V_1 {\mathrel{\mathop:}=}\pi_s^{-1}(G-\{w_0\}) \quad \text{ and } \quad V_2 {\mathrel{\mathop:}=}\pi_s^{-1}(\{ x\in G\,|\, d_G(x,w_0) < 1\}).$$ (-1,0) – (0.8,0); (-1,0) – (-2.5, 1); (-1,0) – (-2.8, 0); (-1,0) – (-2.5,-1); plot \[smooth\] coordinates [(-1, 0) (-1.3, 1) (-1, 1.3) (-0.7, 1) (-1, 0)]{}; (0.8,0) circle (.1cm); at (0.8, -.4) [$w_0$]{}; (-2.5,-1) circle (.1cm); plot \[smooth cycle\] coordinates [(-4.5, 0) (-2, -2) (-1, -2) (0.6, 0) (-1, 2) (-2, 2)]{}; at (-3.7,0) [$U_1$]{}; plot \[smooth cycle\] coordinates [(1.5, 0) (0.5, -1) (-0.7, 0) (0.5, 1)]{}; at (1.9,0) [$U_2$]{}; The interesting part of the Mayer-Vietoris long exact sequence is the following: $$\begin{aligned} H_1(V_1)\oplus H_1(V_2) &\to H_1({\operatorname{Conf}_{S,T}^\mathrm{sink}}(\Gamma_{w_0},W)) \to H_0(V_1\cap V_2) \\ &\to H_0(V_1)\oplus H_0(V_2). \end{aligned}$$ We have $V_1\simeq {\operatorname{Conf}_{S, T\sqcup\{s\}}^\mathrm{sink}}(\Gamma_{w_0},W)$, and $V_2$ is homotopy equivalent to a disjoint union of the space ${\operatorname{Conf}_{S-\{s\}, T}^\mathrm{sink}}(\Gamma_{w_0},W)$ and several copies of ${\operatorname{Conf}_{S'}^\mathrm{sink}}(G, W-\{w_0\})$ for different finite sets $S'\subset S$. Those latter components of $V_2$ arise if particles of $T$ sit between $s$ and $w_0$, preventing $s$ to move to the sink. The set $S'$ is then given by the set of all particles on the other side of $s$. The first component is identified by moving $s$ to the sink and forgetting it. The first homology of both of these spaces is by induction generated by basic classes. Therefore, it remains to show that the classes coming from the kernel $H_0(V_1\cap V_2)\to H_0(V_1)\oplus H_0(V_2)$ are generated by basic classes. In $V_1\cap V_2$ the particle $s$ is trapped on the edge $e$ between $w_0$ and the central vertex. We can represent each connected component by a configuration where all particles sit on $e$. The remaining particles are then distributed to both sides of $s$. Restricted to the connected components where there is a particle of $T$ on the $w_0$-side of $s$, the map $$V_1\cap V_2\hookrightarrow V_2$$ is a homeomorphism onto the corresponding connected components of $V_2$ because those particles in $T$ prevent $s$ from moving to the sink $w_0$. The image of that restricted inclusion is disjoint from the image of the remaining components, so to find elements in the kernel of $$H_0(V_1\cap V_2)\to H_0(V_1)\oplus H_0(V_2)$$ we can restrict ourselves to the union $X$ of components where no element of $T$ is on the $w_0$-side of $s$. The inclusions $X\to V_1$ and $X\to V_2$ map all these connected components to the same component of $V_1$ and $V_2$, respectively, because we can use either the sink or the essential vertex to reorder the particles. Therefore, the kernel of the map to $H_0(V_1)\oplus H_0(V_2)$ is generated by differences of distinct ways of putting particles in $S-T$ to the two sides of $s$, and the lifting process turns these differences into $\operatorname{H}$-classes involving $w_0$ and the central vertex, proving the claim. A basis for configurations in graphs with one essential vertex {#sec:basis-conf-stars} -------------------------------------------------------------- The key to proving the induction step is choosing for each leaf $e$ a particular system of bases for all first homology groups $H_1({\operatorname{Conf}_{\bullet}^\mathrm{sink}}(G,W))$ with the following property: if a representative of a basis element has fixed particles on the leaf $e$ then changing the order of these particles should give another basis element, and all these basis elements should be distinct. Furthermore, adding and forgetting fixed particles of representatives of basis elements should again give elements in the chosen system of bases. For the description of such a system of bases, fix the graph $G$, the set of sinks $W$ and the leaf $e$. For all finite sets $S$ we will choose a system of spanning trees $T_S$ in the combinatorial model of ${\operatorname{Conf}_{S}^\mathrm{sink}}(G,W)$. As constructed in , this model is a graph. For each edge $\xi$ in the combinatorial model, the system $T_\bullet$ will have the following properties: - The edge $\xi$ determines a set $F_\xi$ of fixed particles on the leaf $e$. The symmetric group $\Sigma_{F_\xi}\le \Sigma_n$ acts on the combinatorial model by precomposition, and we want that the orbit $\Sigma_{F_\xi}\cdot \xi$ is completely contained in either $T_S$ or $G-T_S$. - Given $s\not\in S$ we have a map ${\operatorname{Conf}_{S}^\mathrm{sink}}(G,W)\to{\operatorname{Conf}_{S\sqcup\{s\}}^\mathrm{sink}}(G,W)$ by adding the particle $s$ to the end of the leaf $e$. Then $\xi$ should be in $T_S$ if and only if the image of $\xi$ under that map is contained in $T_{S\sqcup \{s\}}$. We now inductively choose the system of spanning trees $T_S$. For $S=\emptyset$, we define $T_\emptyset = \emptyset$. Given a non-empty set $S$, complete the forest $$\bigsqcup_{s\in S} \iota_{e,s}{\mathopen{}\mathclose\bgroup\originalleft}( T_{S-\{s\}} {\aftergroup\egroup\originalright})$$ to a spanning tree $T_S$ in an arbitrary way. If $S'\subset S$ then $T_{S'}$ appears as subtrees of $T_S$ by adding the particles $S-S'$ to the leaf $e$ in all different orders. While completing this forest we only add edges that have no fixed particles on $e$, otherwise, one of the trees $T_{S-\{s\}}$ was not maximal in ${\operatorname{Conf}_{S-\{s\}}^\mathrm{sink}}(G,W)$. This yields a spanning tree $T_S$ of ${\operatorname{Conf}_{S}^\mathrm{sink}}(G,W)$, inductively describing spanning trees for all finite sets $S$ with the properties listed above. This defines a system of bases $\mathcal{B}_\bullet$ of $H_1({\operatorname{Conf}_{\bullet}^\mathrm{sink}}(G,W))$ with the following properties: - for $\sigma\in\mathcal{B}_S$ the class $\sigma^\eta$ given by adding a set of particles $T$ in some order $\eta$ to the end of the leaf $e$ is an element of $\mathcal{B}_{S\sqcup T}$, - for $\sigma\in\mathcal{B}_S$ the classes $\sigma^\eta$ and $\sigma^{\eta'}$ for two orderings $\eta\neq\eta'$ of $T$ are distinct, - every $\sigma\in\mathcal{B}_S$ has precisely one *minimal representative* $\sigma_{\min}\in\mathcal{B}_{S'}$ for $S'\subset S$ such that $(\sigma_{\min})^\eta=\sigma$ for some ordering $\eta$ of $S-S'$ (meaning that the set $S'$ is minimal with respect to this property) and - we always have ${\mathopen{}\mathclose\bgroup\originalleft}( \sigma^\eta {\aftergroup\egroup\originalright})_{\min} = \sigma_{\min}$. Given $\sigma\in\mathcal{B}_S$ and the corresponding minimal cycle $C$, define $S'$ to be the set of fixed particles of $C$ which are on $e$. Then $\pi_{S-S'}(\sigma)$ defines the minimal representative $\sigma_{\min}\in\mathcal{B}_{S-S'}$. With this definition it is straightforward to check the four properties described above. The spectral sequence for the induction step -------------------------------------------- Let $(G,W)$ be a tree with loops with any subset of the vertices *of valence one* turned into sinks, and $v$ an essential vertex which is connected to precisely one other essential vertex $w$ via an edge $e$. Define the following two open subspaces of $G$: $$L {\mathrel{\mathop:}=}{\mathopen{}\mathclose\bgroup\originalleft}\{ x\in G \,|\, \text{$d_G(x,v) < 1$} {\aftergroup\egroup\originalright}\}$$ and $$K {\mathrel{\mathop:}=}{\mathopen{}\mathclose\bgroup\originalleft}\{ x\in G \,|\, d_G(x, G-L) < 1 {\aftergroup\egroup\originalright}\},$$ where $d_G$ is the path metric giving every internal edge of $G$ length 1 and every leaf length $1/2$. In other words, $K$ is the connected component of $G-\{v\}$ containing $w$, see . (-1,0) – (1,0); (-1,0) – (-2.5, 1); (-2.6,.666667\*1.6) – (-2.8, .666667\*1.8); (-2,.66667) – (-2.5, .66667 - .5\*0.73334); (-2.6, .66667 - .6\*0.73334) – (-2.8,.66667 - .8\*0.73334); plot \[smooth\] coordinates [(-2, .66667) (-1.9, 1) (-1.75, 1) (-1.7, .85) (-2, .66667)]{}; (-1.5,.33333) – (-2, .33333 - .5\*.666667); (-2.1,.33333 - .6\*.6666667) – (-2.3,.33333 - .8\*.6666667); (-1,0) – (-2,-1); (-2.1,-1.1) – (-2.3,-1.3); (-1.5,-.5) – (-1,-1); (-.9,-1.1) – (-0.7,-1.3); at (-1.6, -1.5) [$K$]{}; at (0, 0.25) [$e$]{}; (1,0) – (2.5, 1); (1,0) – (2.8, 0); (1,0) – (2.5,-1); plot \[smooth\] coordinates [(1, 0) (1.3, 1) (1, 1.3) (0.7, 1) (1, 0)]{}; at (-1, -.3) [$w$]{}; at (1, -.3) [$v$]{}; at (1.6, -1.5) [$L$]{}; plot \[smooth cycle\] coordinates [(-4, 0) (-2, -2) (-1, -2) (0.8, 0) (-1, 2) (-2, 2)]{}; plot \[smooth cycle\] coordinates [(3.5, 0) (2, -2) (1, -2) (-0.8, 0) (1, 2) (2, 2)]{}; The intersection $L\cap K$ is the interior of the edge $e$. The graph $K$ has strictly fewer essential vertices than $G$, so by induction we can assume that its configuration spaces (with sinks) of any number of particles are torsion-free and generated by products of basic classes. As described in , construct the open cover $\mathcal{U}(\{ K, L \})$ of ${\operatorname{Conf}_{n}^\mathrm{sink}}(G,W)$ and look at the corresponding Mayer-Vietoris spectral sequence $E^*_{\bullet,\bullet}$. The open cover has one open set for each map $\phi\colon\mathbf{n}\to \{K, L\}$, restricting particle $i$ to the open set $\phi(i)$. We have $$\begin{aligned} U_{\phi_0\cdots\phi_p} &= U_{\phi_0}\cap\cdots\cap U_{\phi_p}\\ &= \bigcap_{i\in\mathbf{n}} \bigcap_{0\le j\le p} \pi_i^{-1}{\mathopen{}\mathclose\bgroup\originalleft}( V_{\phi_j(i)} {\aftergroup\egroup\originalright})\\ &\simeq \coprod_{j\in J} {\operatorname{Conf}_{S^j_L}^\mathrm{sink}}(L, W_L)\times {\operatorname{Conf}_{T^j_K}^\mathrm{sink}}(K,W_K),\end{aligned}$$ where $J$ is a finite index set, $S^j_L\sqcup T^j_K\subset\mathbf{n}$ and $W_L$ and $W_K$ are the sinks of $L$ and $K$, respectively. To see this, notice that each connected component of such an intersection has three types of particles: - particles which can move everywhere in $L$, - particles which can move everywhere in $K$, - particles which are restricted to the intersection $L\cap K$. A particle $x$ of the last type either has $\{\phi_0(x), \ldots, \phi_p(x)\}=\{K,L\}$ or is trapped by another particle. Since each connected component of the configuration space of particles in the interval $L\cap K$ is contractible, we get an identification as described above simply by forgetting the particles restricted to the intersection. The order of the particles on this intersection will be important for the face maps given by going from $(p+1)$-fold intersections to $p$-fold intersections by forgetting one of the open sets. The $E^1$-page consists at position $(p,q)$ of the $q$-th homology of all $(p+1)$-fold intersections of the open sets $U_\phi$. By the identification above and the Künneth theorem, each $E^1_{p,q}$ is given as $$E^1_{p,q}\cong \bigoplus_{j\in J'}\bigoplus_{q_L+q_K=q} H_{q_L}({\operatorname{Conf}_{S^j_L}^\mathrm{sink}}(L, W_L)) \otimes H_{q_K}({\operatorname{Conf}_{S^j_K}^\mathrm{sink}}(K, W_K)),$$ where $J'$ is some finite indexing set. Here we used that we know that the configuration spaces of $L$ have free homology. Recall that attached to each of those summands there is an ordering of the particles $\mathbf{n}-S^j_L-S^j_K$, which are sitting on the interior of $e$. The face maps forgetting one of the open sets from a $(p+1)$-fold intersection yielding a $p$-fold intersection only affect the particles restricted to the intersection $L\cap K$: for some (but possibly none) of them the restriction is removed, allowing them to move in all of either $L$ or $K$. Under the identification above, these particles are added to the sets $S^j_L$ or $S^j_K$ and put to the edge $e$ of $L$ or $K$, respectively, in the order determined by their order on $L\cap K$. Since the configuration space of $L$ is 1-dimensional by these summands of $E^1_{p,q}$ are only non-trivial for $q_L\in\{0,1\}$. The horizontal boundary map $d_1$ preserves $q_L$, so the $E^1$-page splits into two parts $(^0\!E^1, ^0\!d_1)$ and $(^1\!E^1, ^1\!d_1)$ consisting of all direct summands with $q_L = 0$ and $q_L=1$, respectively. The key point is now that $^1\!E^2$ is concentrated in the zeroth column, we understand $^0\!E^\infty$, and the two spectral sequences don’t interact. The homology of $^1\!E^1$ ------------------------- As described in , choose a system of bases $\mathcal{B}_\bullet$ for $H_1({\operatorname{Conf}_{\bullet}^\mathrm{sink}}(L, W_L))$ for the edge of $L$ corresponding to $e$. This determines a direct sum decomposition of the direct summands of every module $^1\!E^1_{p,q}$ as follows: $$\begin{aligned} H_1({\operatorname{Conf}_{S^j_L}^\mathrm{sink}}(L, W_L)) &\otimes H_{q-1}({\operatorname{Conf}_{S^j_K}^\mathrm{sink}}(K, W_K)) \\&\cong \bigoplus_{\sigma\in\mathcal{B}_{S^j_L}} {\mathbb{Z}}_\sigma\otimes H_{q-1}({\operatorname{Conf}_{S^j_K}^\mathrm{sink}}(K, W_K)).\end{aligned}$$ Here, ${\mathbb{Z}}_\sigma$ is the free abelian group on the single generator $\sigma$. By the description of the face maps above and the properties of the system of bases, the boundary map $^1\!d_1$ does not change the *minimal* representative of the first tensor factor. Grouping these summands by their corresponding minimal representative $\sigma_0$ yields a decomposition of each row $^1\!E^1_{\bullet,q}$ into summands denoted by $(E^1[\sigma_0], d^{\sigma_0}_1)$, which is a decomposition *as chain complexes*. We now compute the homology of one of these chain complexes $E_{\bullet,q}^1[\sigma_0]$ for fixed $\sigma_0$ and $q\ge0$. Let a minimal $\sigma_0\in\mathcal{B}_S$ for some $S\subset{\mathbf{n}}$ be given (i.e.$(\sigma_0)_{\min}=\sigma_0$), then every $\sigma\in\mathcal{B}_{S'}$ appearing in one of the second tensor factors of the modules in the chain complex $E^1_{\bullet,q}[\sigma_0]$ is given by adding fixed particles $S'-S$ to $\sigma_0$, putting them in some ordering to the end of $e$ (away from $v$). Since there are no relations between the different orderings of the particles $S'-S$, we can forget the particles $S$ and replace $L$ by an interval: Let $^KE^*_{\bullet,\bullet}$ be the Mayer-Vietoris spectral sequence for ${\operatorname{Conf}_{{\mathbf{n}}-S}^\mathrm{sink}}(K, W_K)$ corresponding to the cover $\{K,L\}$ pulled back by the inclusion $K\hookrightarrow G$. The chain complex $E^1_{\bullet,q}[\sigma_0]$ is isomorphic to the chain complex $^KE^1_{\bullet,q}$ by forgetting the particles $S$ involved in $\sigma_0$ and looking at cycles of the remaining particles. The open cover of $K$ is very special: one of the open sets is the whole space itself. We will now show that because of that, the $E^2$-page is concentrated in the zeroth column. The open cover of ${\operatorname{Conf}_{{\mathbf{n}}-S}^\mathrm{sink}}(K,W_K)$ is indexed by maps $\psi\colon{\mathbf{n}}-S\to \{K, L\cap K \}$. For the map $\psi_\mathrm{all}$ sending everything to $K$, we have $U_{\psi_\mathrm{all}}={\operatorname{Conf}_{{\mathbf{n}}-S}^\mathrm{sink}}(K,W_K)$. Hence, for each tuple $(\psi_0,\ldots,\psi_p)$ with $\psi_i\neq\psi_\mathrm{all}$ for all $i$ the inclusion $$U_{\psi_0}\cap\cdots\cap U_{\psi_p}\cap U_{\psi_\mathrm{all}} \to U_{\psi_0}\cap\cdots\cap U_{\psi_p}$$ and therefore the face maps $$H_q(U_{\psi_0}\cap\cdots\cap U_{\psi_p}\cap U_{\psi_\mathrm{all}}) \to H_q(U_{\psi_0}\cap\cdots\cap U_{\psi_p})$$ are the identity. Notice that precisely one of the $p+2$ face maps with that source lands in an intersection without $U_{\psi_\mathrm{all}}$. By adding $^K\!d_1$ boundaries we can thus assume that every homology class of the chain complex $(^KE^1_{p,q}, {^K}d_1)$ has a representative which is trivial in all direct summands $H_q(U_{\psi_0\cdots\psi_p})$ where none of the $\psi_i$ is $\psi_\mathrm{all}$. The composition of maps $$\bigoplus_{\substack{\psi_0<\cdots<\psi_p\\ \exists i: \psi_i=\psi_\mathrm{all}}} \!\!\!\!\!H_q(U_{\psi_0\cdots\psi_p}) \xrightarrow{^K\!d_1} \bigoplus_{\psi_0<\cdots<\psi_{p-1}} \!\!\!\!\!\!\!H_q(U_{\psi_0\cdots\psi_{p-1}}) \twoheadrightarrow \bigoplus_{\substack{\psi_0<\cdots<\psi_{p-1}\\ \not\exists i: \psi_i=\psi_\mathrm{all}}} \!\!\!\!\!\!\!H_q(U_{\psi_0\cdots\psi_{p-1}}),$$ where the second map collapses all direct summands with one of the $\psi_i$ equal to $\psi_\mathrm{all}$, is injective by the observation above (actually the images of the direct summands intersect trivially, and restricted to one such summand the map onto its image is given by either the identity or multiplication by $-1$). In particular, the map $^K\!d_1$ restricted to the intersections including $U_{\psi_\mathrm{all}}$ is injective (unless we are in the zeroth degree), and the homology is trivial. Therefore, the homology of $E^1_{\bullet,q}[\sigma_0]$ is zero in degrees $i\neq 0$ and given by $${\mathbb{Z}}_{\sigma_0}\otimes H_{q-1}{\mathopen{}\mathclose\bgroup\originalleft}({\operatorname{Conf}_{{\mathbf{n}}-S}^\mathrm{sink}}(K,W_K) {\aftergroup\egroup\originalright})$$ for $i=0$, which by induction is free and generated by products of basic classes. In conclusion, the homology of $^1\!E^1$ is free, concentrated in the zeroth column and generated by products of basic classes. Denote this bigraded module by $E^\infty[K]$. The homology of $^0\!E^1$ and the $E^\infty$-page ------------------------------------------------- The other part, $^0\!E^1$, is actually the first page of the Mayer-Vietoris spectral sequence $E_{\bullet,\bullet}^*[G/L]$ of $G$ with $L-e$ collapsed to a sink with respect to the image of the open cover $\mathcal{U}(\{K,L\})$. By induction, this spectral sequence $E_{\bullet,\bullet}^*[G/L]$ converges to a free infinity page, and the corresponding homology is generated by products of basic classes. The $E^2$-page of our original spectral sequence is hence given by the direct sum of the two bigraded modules $E^2[G/L]$ and $E^\infty[K]$, which differs from $E^2[G/L]$ only in the zeroth column. We will now show that for each $2\le\ell\le\infty$ the $E^\ell$-page is the direct sum of $E^\ell[G/L]$ and $E^\infty[K]$. For $p>0$ and $q\ge 0$ look at the map $d_2$ starting in $E^2_{p,q}$. This map is constructed by representing each class in $E^2_{p,q}$ on the chain level (i.e. on the $E^0$-page), mapping it via the horizontal boundary map to $E^0_{p-1,q}$, lifting it to $E^0_{p-1,q+1}$ and applying the horizontal map again, landing in $E^0_{p-2,q+1}$. The element of $E^2_{p-2,q+1}$ represented by this cycle is the image of the class we started with under $d_2$. The lifting of the particles in $L$ always connects pairs of distinct orderings of particles on $e$ via a path through the central vertex of $L$. The end result does not depend on the choice of such a lift, so we always take the following one: choose (once and for all) two leaves $e_1, e_2$ of $L$ that are different from $e$, then connecting two orderings $\nu\neq\nu'$ of a $S=\{s_1,\ldots,s_m\}$ is given by starting with the configuration $\nu$ on $e$, sliding all particles between $s_1$ and the central vertex to $e_2$, moving $s_1$ to $e_1$, moving the other particles back to $e$ and repeating this for all particles $s_2,\ldots, s_m$. Repeating the same for $\nu'$ we get two paths which glued together give a path $\gamma[\nu,\nu']$ between the two configurations. By construction it is clear that $\gamma[\nu,\nu'] + \gamma[\nu',\nu''] = \gamma[\nu,\nu'']$, so the only closed loop arising in such a way is the trivial path. The construction of the image of a class under $d_2$ as described above produces segments $\gamma[\nu,\nu']$ adding up to a cycle, which hence must be trivial. This shows that $d_2$ maps to zero in $E^\infty[K]$ and hence that $E^3 \cong E^3[G/L] \oplus {E^\infty[K]}$. By the same reasoning, this is true for all pages, proving that $$E^\infty \cong E^\infty[G/L] \oplus {E^\infty[K]}.$$ In conclusion, the $E^\infty$-page is torsion-free and the corresponding homology is generated by products of basic classes. For graphs with precisely one vertex of valence at least three and any subset of the vertices of valence 1 turned into sinks the theorems follow from . By induction on the number of essential vertices, we then use the calculation of the spectral sequence above to prove this for any graph as in the statement of the two theorems with any subset of the vertices of valence 1 turned into sinks. In particular, this proves the statement for the case where none of the vertices are sinks. Configurations of particles in general finite graphs {#sec:general-graph} ==================================================== In this section we prove that the *first* homology of configuration spaces of graphs with rank at least one is generated by basic classes. In contrast to the case of trees with loops, we prove that in general the higher homology groups are *not* generated by products of 1-classes. The first homology of configurations in general graphs {#sec:first-homology-general-graph} ------------------------------------------------------ For a graph $G$, we choose distinct edges $e_1, \ldots, e_\ell$ such that cutting those edges in the middle yields a tree. Fix identifications of $[0,1]$ with each of the $e_i$ and denote for $x\in[0,1]$ by $x_{e_i}$ the corresponding point on the edge $e_i$. Then, define the tree $K$ as $$K = G - \bigcup_{1\le i\le \ell} [1/3, 2/3]_{e_i},$$ where $[1/3,2/3]_{e_i} = \{ x_{e_i} \,|\, x \in [1/3, 2/3] \}$. The idea is now to start with the configuration space of $K$ embedded into the configuration space of $G$ and to release the particles into the bigger graph $G$ one at a time. For $\Gamma = (G,K)$ recall the definition of ${\operatorname{Conf}_{S,T}^\mathrm{sink}}(\Gamma,W)$ (). We will prove that $H_1({\operatorname{Conf}_{S,T}^\mathrm{sink}}(\Gamma,W))$ is always generated by basic classes. The second part of will be proven in the next section. We will again proceed by constructing an open cover and investigating the Mayer-Vietoris spectral sequence. Let ${\operatorname{Conf}_{S,T}^\mathrm{sink}}(\Gamma, W)$ with $S-T$ non-empty be given, then choose an arbitrary element $s\in S-T$ and construct the following open cover: for each $i$, define two open subsets $U_{+e_i}$ and $U_{-e_i}$ of ${\operatorname{Conf}_{S,T}^\mathrm{sink}}(\Gamma,W)$ by $$\begin{aligned} U_{+e_i} &= {\mathopen{}\mathclose\bgroup\originalleft}\{ f\colon S\to G \,|\, \text{$f(s)\not\in [1/3, 2/3]_{e_j}$ for $j\neq i$ and $f(s)\neq 2/3_{e_i}$} {\aftergroup\egroup\originalright}\}\\ U_{-e_i} &= {\mathopen{}\mathclose\bgroup\originalleft}\{ f\colon S\to G \,|\, \text{$f(s)\not\in [1/3, 2/3]_{e_j}$ for $j\neq i$ and $f(s)\neq 1/3_{e_i}$} {\aftergroup\egroup\originalright}\}.\end{aligned}$$ plot \[smooth\] coordinates [(2, 0) (2.6, .2) (2.8, 0) (2.6, -.2) (2, 0)]{}; plot \[smooth cycle\] coordinates [ (-4, 0) (-3, 1) (0.7, 1.2) (1, 0.2) (1.4, 1.1) (2.6, 0) (1.4, -1.1) (0, -.3) (-1.4, -1.1) (-3, -1) ]{}; (-2,0) – (2,0); plot \[smooth\] coordinates [(-2, 0) (-1, .6) (0, .8) (1, .6) (2, 0)]{}; plot \[smooth\] coordinates [(-2, 0) (-1, -.6) (0, -.8) (1, -.6) (2, 0)]{}; (-2,0) – (-3,0); (-2,0) – (-2.7,0.7); (-2,0) – (-2.7,-0.7); at (0, 1) [$e_1$]{}; at (0, -1) [$e_2$]{}; at (3.1, 0) [$e_3$]{}; Let $T' = T\sqcup \{s\}$ and $\Gamma'=(G-[1/3,2/3]_{e_i},K)$. \[prop:identification-intersection-general-graph\] The intersections of those open sets can be identified as follows: $$\begin{aligned} U_{\pm e_i} &\simeq {\operatorname{Conf}_{S,T'}^\mathrm{sink}}(\Gamma, W)\\ U_{-e_i}\cap U_{+e_i} &\simeq {\operatorname{Conf}_{S,T'}^\mathrm{sink}}(\Gamma,W) \sqcup {\operatorname{Conf}_{S-\{s\}, T}^\mathrm{sink}}(\Gamma',W)\\ U_{\pm e_i}\cap U_{\pm e_j} &\simeq {\operatorname{Conf}_{S,T'}^\mathrm{sink}}(\Gamma, W). \end{aligned}$$ Any intersection of at least three of those open sets is again homotopy equivalent to ${\operatorname{Conf}_{S,T'}^\mathrm{sink}}(\Gamma,W)$. The inclusions induced by going from $p$-fold intersections to $(p-1)$-fold intersections are homotopic to the identity on the components ${\operatorname{Conf}_{S,T'}^\mathrm{sink}}(\Gamma,W)$ and given by adding the particle $s$ to $1/2_{e_i}$ for the configurations in each component ${\operatorname{Conf}_{S-\{s\},T}^\mathrm{sink}}(\Gamma',W)$. These latter components are not hit by any such inclusion. If the intersection of any number of these open sets contains open sets $U_{\pm e_i}$ and $U_{\pm e_j}$ for $i\neq j$ then the particle $s$ is restricted from entering all $[1/3,2/3]_{e_i}$, so this intersection is actually precisely the same as ${\operatorname{Conf}_{S,T'}^\mathrm{sink}}(\Gamma,W)$. Since every intersection of $\ge 3$ of those sets contains two such open sets, there are only two cases remaining, namely 1-fold intersections and the intersection $U_{-e_i}\cap U_{+e_i}$. The space $U_{+e_i}$ is almost the same as ${\operatorname{Conf}_{S,T'}^\mathrm{sink}}(\Gamma,W)$, the only difference is that the particle $s$ is also allowed in the segment $[1/3, 2/3)_{e_i}$. By sliding $s$ back into the interval $[0, 1/3)_{e_i}$ whenever necessary and moving all particles between $0_{e_i}$ and $s$ accordingly, we see that this space is homotopy equivalent to ${\operatorname{Conf}_{S,T'}^\mathrm{sink}}(\Gamma,W)$. The analogous reasoning identifies $U_{-e_i}$. The intersection $U_{-e_i}\cap U_{+e_i}$ has two connected components: the component where $s$ is in $(1/3,2/3)_{e_i}$ and the one where it is in $K$. The second component is again on the nose equal to ${\operatorname{Conf}_{S,T'}^\mathrm{sink}}(\Gamma,W)$. Modify the first component by a homotopy moving $s$ to $1/2_{e_i}$ and sliding all other particles on $e_i$ away from $s$ into the intervals $[0, 1/3)_{e_i}$ and $(2/3, 1]_{e_i}$, then forgetting the particle $s$ gives an identification with ${\operatorname{Conf}_{S-\{s\},T}^\mathrm{sink}}(\Gamma',W)$, proving the first claim. By our identification above the description of the inclusion maps given by forgetting one of the intersecting open sets is easily deduced. If one of these inclusions would hit a component ${\operatorname{Conf}_{S-\{s\},T}^\mathrm{sink}}(\Gamma',W)$, then the particle $s$ would need to be on the interval $(1/3, 2/3)_{e_i}$, which it never is for any triple intersection. This allows us to describe generators for the first homology of the configuration space of any finite graph. We formulate this as a separate proposition in order to use it for the case where $K$ is a graph with precisely one essential vertex since this case is needed to prove . \[prop:1cycles-general-graph-induction\] Let $G$ be a connected finite graph, $K\subset G$ a tree defined as above and $W$ a subset of the vertices. If $H_1({\operatorname{Conf}_{S}^\mathrm{sink}}(K, W))$ is generated by basic classes for all finite sets $S$ then also $H_1({\operatorname{Conf}_{S,T}^\mathrm{sink}}(\Gamma,W))$ is generated by basic classes for all pairs of finite sets $T\subset S$, where $\Gamma=(G,K)$. We prove this by looking at the spectral sequence constructed from the open cover described above. To prove the statement we only need to show that moving one element out of $T$ preserves the property that the homology is generated by basic classes. We can assume that the configuration space of $K$ is connected since the only case where this is not true is if $G$ is $S^1$ without sinks, and this case is true by definition. We will now argue by induction on the number of elements in $S-T$. The induction start $S=T$ is precisely that $H_1({\operatorname{Conf}_{S}^\mathrm{sink}}(K,W))$ is generated by basic classes, so we only need to check the induction step. In the induction step, we only get 1-classes at $E^\infty_{0,1}$ and $E^\infty_{1,0}$. The module $E^\infty_{0,1}$ is a quotient of $E^1_{0,1}$, which is generated by 1-classes of $U_{\pm e_i}\simeq{\operatorname{Conf}_{S,T'}^\mathrm{sink}}(\Gamma,W)$, so by induction by classes of the required form. The chain complex $E^1_{\bullet,0}$ is given by the chain complex of the nerve of the cover (which is a simplex) and one additional copy of ${\mathbb{Z}}$ for each intersection $U_{+e_i}\cap U_{-e_i}$. Restricted to $H_0(U_{-e_i}\cap U_{+e_i})\cong {\mathbb{Z}}\oplus{\mathbb{Z}}$ the face maps $${\mathbb{Z}}\oplus{\mathbb{Z}}\cong H_0(U_{-e_i}\cap U_{+e_i}) \to H_0(U_{\pm e_i}) \cong {\mathbb{Z}}$$ are given by $(x,y)\mapsto \pm(x+y)$. Therefore, all elements $(x, -x)$ are in the kernel of $d_1$. These elements correspond to $S^1$ movements of $s$ along the edge $e_i$: by mapping $U_{-e_i}\cap U_{+e_i}\hookrightarrow U_{-e_i}$ the particle $s$ is allowed to leave $(1/3,2/3)_{e_i}$ via one of the sides, connecting it to a configuration where $s$ is on the tree $K$. The other inclusion allows $s$ to leave via the other side, connecting it to that same configuration with $s$ on $K$. Mapping this to ${\operatorname{Conf}_{S,T}^\mathrm{sink}}(\Gamma,W)$ yields a cycle where $s$ moves along $K$ and $e_i$. We can choose a representative such that all other particles are fixed and that this movement follows an embedded circle in $G$. Subtracting such kernel elements, we can modify every cycle of $(E^1_{\bullet,0}, d_1)$ such that it is zero in all copies of $H_0({\operatorname{Conf}_{S-\{s\},T}^\mathrm{sink}}(\Gamma',W))$. Since the remaining part of the chain complex is the chain complex of a simplex, there are no other 1-classes, concluding the argument. By , the homology group $H_1(\operatorname{Conf}_{S}(K))$ is generated by basic classes for any finite tree $K$, so the theorem follows from . Non-product generators ---------------------- In this section, we describe an example of a homology class of the configuration space of a graph that cannot be written as a sum of product classes. The easiest example we were able to find so far is a 2-class of $\operatorname{Conf}_3{\mathopen{}\mathclose\bgroup\originalleft}( B_3 {\aftergroup\egroup\originalright})$, where $B_3$ is the banana graph of rank three, i.e. two vertices $v,w$ connected via four edges, see . To construct the class, we first construct classes in $\operatorname{Conf}_2(\operatorname{Star}_4)$. Let $S\subset\mathbf{3}$ be a set of two particles, then the first homology group of $\operatorname{Conf}_S(\operatorname{Star}_3)$ is one-dimensional, a generator can be represented by a sum of twelve edges, each with coefficient +1: start with both particles on different edges, then in turns move the particles to the free edge until the initial configuration is restored. plot \[smooth\] coordinates [(-2, 0) (-1, 0.7) (0, 1) (1, 0.7) (2, 0)]{}; plot \[smooth\] coordinates [(-2, 0) (-1, 0.25) (0, 0.35) (1, 0.25) (2, 0)]{}; plot \[smooth\] coordinates [(-2, 0) (-1, -0.25) (0, -0.35) (1, -0.25) (2, 0)]{}; plot \[smooth\] coordinates [(-2, 0) (-1, -0.7) (0, -1) (1, -0.7) (2, 0)]{}; at (-2.3, 0) [$v$]{}; at (2.3, 0) [$w$]{}; plot \[smooth\] coordinates [(-1.98, 0) (-1, 0.7)]{}; plot \[smooth\] coordinates [(-2.008, 0) (-1, 0.25)]{}; plot \[smooth\] coordinates [(-1.97, 0) (-1, -0.7)]{}; Now choose a bijection of $\mathbf{3}$ with the leaves of $\operatorname{Star}_3$ and $\mathbf{4}$ with the leaves of $\operatorname{Star}_4$. This defines four 1-cycles in $\operatorname{Conf}_S(\operatorname{Star}_4)$ by including $\operatorname{Star}_3$ into $\operatorname{Star}_4$ in all order-preserving ways (with respect to these identifications). Now we add those four cycles together with the following signs: each inclusion of $\operatorname{Star}_3$ is determined by the edge $i\in\mathbf{4}$ that is missed. The 1-cycle corresponding to this $i$ gets the sign $(-1)^i$. This sum is actually equal to zero: The 1-cells of these cycles are given by one particle moving from one edge to the central vertex and the other particle sitting on another edge. Each such cell appears precisely twice, once for each way of choosing a third edge from the remaining two leaves. If these two remaining leaves are cyclically consecutive in $\mathbf{4}$ the corresponding cycles have different signs, otherwise, these two cells inside the 1-cycles appear with different signs, so in both cases, they add up to zero. Including $\operatorname{Star}_4$ into $B_3$ (mapping the central vertex to $v$) gives a sum of four 1-cycles coming from embedding $\operatorname{Star}_3$ into $B_3$ in different ways (see ), which evaluates to zero. Now let $t$ be the third particle, i.e. $S\sqcup \{t\} = \mathbf{3}$, then take for each of those four 1-cycles in $\operatorname{Conf}_S(B_3)$ the product of the cycle with the 1-cell moving particle $t$ from the remaining one of the four edges to the vertex $v$. Doing this construction for all three choices of $S$ gives a sum of 144 2-cells, and the claim is that this is, in fact, a 2-cycle in the combinatorial model of the configuration space. We can think of this cycle as 12 cylinders of a 1-cycle in the star of $v$ multiplied with another particle moving to the other vertex $w$, whose boundary 1-cells get identified in a certain way, see . Let $t\in\mathbf{3}$, then one part of the boundary of four of those cylinders is given by the 1-cycles of the particles $\mathbf{3}-\{t\}$ with $t$ sitting on $w$. By construction, those four 1-cycles add up to zero. It remains to investigate the parts where the third particle is in the middle of the edge. These 1-cells are precisely given by two particles sitting in the middle of two edges and a third particle moving from another edge to $v$. Each such cell appears precisely twice: once for every choice of which one of the fixed particles moves to $w$ and which one belongs to the star movement. By analogous reasoning, these two occurrences have opposite signs, so the total contribution is zero. plot \[smooth\] coordinates [(-2, 0) (-1, 0.7) (0, 1) (1, 0.7) (2, 0)]{}; plot \[smooth\] coordinates [(-2, 0) (-1, 0.25) (0, 0.35) (1, 0.25) (2, 0)]{}; plot \[smooth\] coordinates [(-2, 0) (-1, -0.25) (0, -0.35) (1, -0.25) (2, 0)]{}; plot \[smooth\] coordinates [(-2, 0) (-1, -0.7) (0, -1) (1, -0.7) (2, 0)]{}; (-1.3, .7) – (-1.7, 0.4); (1.25, .3) – (1.6, 0.2); at (-2.3, 0) [$v$]{}; at (2.3, 0) [$w$]{}; at (0,0) [1]{}; at (0,0) [3]{}; at (0,0) [2]{}; Thus, the boundary cells of the twelve cylinders add up to zero, yielding a non-trivial cycle. By the dimension of our combinatorial model, there are no three-cells, so this does not represent the zero class. Notice that there are no product classes since every $S^1$ generator uses both vertices and there are too few particles for two $\operatorname{H}$-classes or star classes. By looking at the identifications and calculating the Euler characteristic, one sees that the resulting cycle is, in fact, a closed surface of genus 13 embedded into the combinatorial model of the configuration space. In fact, by pushing in 2-cells where strictly less than three edges are involved (starting with those involving only one edge, followed by those involving precisely two edges) and afterward pushing in the 1-dimensional intervals where particles move to an occupied edge it is straightforward to show the following: $\operatorname{Conf}_3(B_3)$ is homotopy equivalent (equivariantly with respect to the action of the symmetric group $\Sigma_3$) to a closed surface of genus 13. In between versions of this paper, Wiltshire-Gordon independently showed this homotopy equivalence using explicit computer calculations of the groups $H_*(\operatorname{Conf}_3(B_3))$, see [@WiGo17 Example 2.1, p. 4]. We now prove the rest of , whose first part was proven in . A counterexample for the second homology group was described above, all that remains is to describe how to use this to construct counterexamples for higher homology groups. By adding $k$ disjoint $S^1$ graphs, connecting each of them to $v$ via a single edge and adding $k$ particles we can take the product of this non-product cycle with the $k$-cycle given by the product of the $k$ particles moving inside the $S^1$’s. This gives a class in the $(k+2)$-nd homology group of the configuration space of $k+3$ particles in this graph, which by analogous reasoning cannot be written as a sum of product classes. This shows that this phenomenon appears in every homology degree except for the zeroth and first.

--- abstract: 'In this paper we address the problem of information-constrained optimal control for an interconnected system subject to one-step communication delays and power constraints. The goal is to minimize a finite-horizon quadratic cost by optimally choosing the control inputs for the subsystems, accounting for power constraints in the overall system and different information available at the decision makers. To this purpose, due to the quadratic nature of the power constraints, the LQG problem is reformulated as a linear problem in the covariance of state-input aggregated vector. The zero-duality gap allows us to equivalently consider the dual problem, and decompose it into several sub-problems according to the information structure present in the system. Finally, the optimal control inputs are found in a form that allows for offline computation of the control gains.' author: - 'V. Causevic$^{\dagger}$, P. Ugo Abara$^{\dagger}$ and S. Hirche$ $[^1][^2][^3]' title: '**Information-Constrained Optimal Control of Distributed Systems with Power Constraints** ' --- INTRODUCTION ============ Technological advances in computation and communication, and societal needs have revived the research interest in control of interconnected systems [@networkflows]. Some examples include smart grids, communication networks, and transportation systems. Traditionally, arguments in favor of distributed control (compared to centralized) are geographically distributed sensors, limited local computational power at the plant side, robustness against single-node failure and information privacy.\ In general, the design of distributed control is difficult because it imposes information constraints on individual decision makers. Such constraints arise due to either partial information exchange between decision makers or communication delay. In the problem we address herein, decision makers are able to communicate the full information they receive - either due to own measurements or from other decision makers, however, with delay. In other words, information constraints are due to communication delays between decision makers. The information constraints, sometimes referred to as information structure, play a key role in determining the optimal control and decide on its computational tractability. Indeed, in [@witsenhausen1968] a linear quadratic Gaussian team problem is constructed with a non-classical information pattern and it is shown that a linear controller is not necessarily optimal. This problem is addressed in [@ho-chi1972] where it is shown that the so-called partially nested information structure guarantees existence of optimal control laws that are linear in the associated information. Finally, a strong result characterizing the class of all information-constrained problems which may be cast as a convex program is given in [@rotkowitz2006tac].\ Inspiration for our approach is given by the work in [@nayyar2013tac] which suggests that the information hierarchy existing between the decision makers can be exploited to obtain the optimal solution. First explicit solutions to linear quadratic Gaussian team problems that adopt similar approach are given in [@lamperski2012cdc]. The authors however, consider a typical unconstrained linear quadratic team problem. But in reality, e.g. actuation capabilities are limited and thus must be accounted for in the design procedure.\ The main contribution of this paper is a method to compute optimal control laws, for a power-constrained system with given information structure. We assume the latter to be induced by a one-step communication delays between the decision makers. To this end, the problem is reformulated in its dual Lagrangian form, where the covariance of the state-input aggregated vector is defined as decision variable. The information structure is then exploited to split the optimization problem into simpler sub-problems that have alike structure. Indeed, in-network control [@InNetwork] is seen as the decomposition of a complex task into smaller sub-tasks resulting in computationally inexpensive local control actions. From an application point of view, the goal is to implement and analyze the developed approach within a network infrastructure, exploiting the possibility of existing (but limited) in-network processing, in order to improve control performance.\ The remainder of the paper is outlined as follows. We start with problem setup in \[sec: problem statement\]. The method to decouple problem into several sub-problems via covariance decomposition is presented in section \[sec:info dec\]. In section \[sec:dual\] we provide structural characterization of the solution to the problem and finally conclusions are given in \[sec:conclude\]. Problem setting {#sec: problem statement} =============== Consider a large-scale interconnected dynamical system composed of $N$ physically-coupled linear time-invariant (LTI) subsystems. Formally, the physical interconnections are described through a graph $\mathcal{G} = \left(\mathcal{V}, \mathcal{E}\right)$. We will refer to it as the physical interconnection graph. Each node $i \in \mathcal{V}$ corresponds to one of the subsystems $i\in\{1, \ldots, N\}$. An edge $(j,i) \in \mathcal{E}$ if dynamics of node $i$ is directly affected by node $j$. We assume that $\mathcal{G} $ is connected and undirected, i.e., $(i,j) \in \mathcal{E}$ if and only if $(j,i) \in \mathcal{E}$. The set of direct neighbors of decision maker $i$ is defined as $\mathcal{N}_i = \{j \,\vert (j,i) \in \mathcal{E}\}$. The length of the shortest path between nodes $i$ and $j$ will be denoted by $d_{ij}$. Clearly, if $j \in \mathcal{N}_i$ then $d_{ij} = 1$. The dynamics of the $i$-th subsystem is given by a first order stochastic difference equation $$\begin{aligned} &x_i (k+1)= A_i x_i (k)+ B_i u_i (k) + \sum_{j \in \mathcal{N}_{i}} A_{ij} x_j (k)+ w_i (k), \label{eq: main NCS} {\addtocounter{equation}{1}\tag{\theequation}}\end{aligned}$$ where $A_i \in \mathbb{R}^{n_i \times n_i}$, $A_{ij} \in \mathbb{R}^{n_i \times n_j}$, $B_i \in \mathbb{R}^{n_i \times m_i}$, $ x_i (k) \in \mathbb{R}^{n_i}$ is the state and $u_i (k) \in \mathbb{R}^{m_i}$ is the control signal of the $i$-th subsystem. The noise process $w_i (k) \in \mathbb{R}^{n_i}$ is zero-mean i.i.d. Gaussian noise with covariance matrix $\Sigma_{w} $. The initial state $x_i (0)$ is a random variable with zero-mean and finite covariance $\Sigma_{x} $. Moreover, $x_i (0)$ and $w_i (k)$ are assumed to be pair-wise independent at each time instant $k$ and every $i$. For a more compact notation, equation can be rewritten as $$x (k+1) = A x(k) + B u(k) + w(k) \label{eq: global system}$$ where the stacked vectors are $x (k) = ({x_1 ^\top (k)}, \ldots , {x_N ^\top (k)})^\top \in \mathbb{R}^n$, $ w(k) = ({w_1 ^\top (k)}, \ldots, {w_N ^\top (k)})^\top \in \mathbb{R}^n$, $ u(k) = ({u_1 ^\top (k)}, \ldots, {u_N ^\top (k)})^\top \in \mathbb{R}^m$, $n=\sum_{i=1}^{N} n_i$ and $m=\sum_{i=1}^{N} m_i$. The admissible control policies at time instant $k$ are measurable functions of the information available to each decision maker $i$ (sometimes also referred to as player $i$) $$u_i (k) = \gamma_k^i(\mathcal{I}_k^i) \label{eq:gama}$$ where $\mathcal{I}_k^i, \ k=0,\ldots,T-1,$ is defined as $$\begin{aligned} &\mathcal{I}_k^i = \{\mathcal{I}_{k-1}^i, x^i_{k}, u_{k-1}^i\} \underset{j \in \mathcal{N}_i}\bigcup \{\mathcal{I}^j_{{k-1}}\}, \quad k>0, {\addtocounter{equation}{1}\tag{\theequation}}\label{eq:information set}\end{aligned}$$ and $\mathcal{I}_0^i=\lbrace x_0 ^i \rbrace$. In other words, the information set of each decision maker $i$ is updated at time instant $k$ by the current state and the one-step delayed information from the direct neighbors $\mathcal{N}_i$. The objective is to minimize the following global control cost $$\begin{aligned} {\addtocounter{equation}{1}\tag{\theequation}}\label{eq: quadratic cost 1} J_{\mathcal{C}} = {\rm E}\left[ \sum_{k=0}^{T-1} { \begin{bmatrix} x(k) \\ u(k) \end{bmatrix}}^\top Q \begin{bmatrix} x(k) \\ u(k) \end{bmatrix} + x(T)^\top Q_{T} x(T) \right]\end{aligned}$$ where the matrix $Q$ is partitioned according to the vector $z (k) $ = $\left[{x(k)}^\top {u(k)}^\top\right]^\top$ i.e. $${\addtocounter{equation}{1}\tag{\theequation}}\label{eq:matrixQ} Q= \begin{bmatrix}Q_{xx} & Q_{xu} \\ Q_{ux} & Q_{uu} \end{bmatrix}.$$ The matrix $Q_{uu}$ is assumed to be positive-definite matrix, while $Q$ and $Q_T$ are assumed to be semi-definite positive. We also assume controllability of pair (A,B) as well as detectability of $(Q^{\frac{1}{2}},A)$. Moreover, it is assumed that each decision maker knows the parameters of the overall system.\ The cost is to be minimized under power constraints, which are defined as $$\begin{aligned} {\addtocounter{equation}{1}\tag{\theequation}}& {\rm E}\left[ {z(k)}^\top W_i \, z (k) \right] \leq p_k^i, \quad \forall i= 1, \ldots, M \label{eq: power constraints}\end{aligned}$$ where $W_i \in \mathbb{R}^{(n+m) \times (n+m)}$, $i=1,\ldots,M$, is a positive semi-definite weighting matrix. By appropriate choice of $W_i$, the set of constraints in captures either constraints present in the power of the overall system, or those related to the individual subystems. Ultimately, the problem is formally stated as $$\begin{aligned} {\addtocounter{equation}{1}\tag{\theequation}}\label{eq: problem1} \min_{\gamma_{0:T-1}} \qquad & {\rm E}\left[ \sum_{k=0}^{T-1} { \begin{bmatrix} x(k) \\ u(k) \end{bmatrix}}^\top Q \begin{bmatrix} x(k) \\ u(k) \end{bmatrix} + x(T)^\top Q_{T} \, x(T) \right]\\ \text{s.t.} \qquad & \eqref{eq: global system}, \eqref{eq:gama}, \eqref{eq: power constraints}\end{aligned}$$ where $\gamma_k = [ \gamma^{1}_{k},\ldots,\gamma^{N}_{k}]$ is composed of all players control policies. Before stating the main result of this section we define the notion of partial nestedness [@PN]. The information structure $\mathcal{I}_{k} = \left\lbrace \mathcal{I}_{k} ^1, \ldots, \mathcal{I}_{k} ^N \right\rbrace$ is partially nested if, for every admissible policy , whenever $u_i(\tau)$ affects $\mathcal{I}_{k} ^j$, then $\mathcal{I}_{\tau} ^i \subset \mathcal{I}_{k} ^j$. The information structure defined by is partially nested. \[lemma: partially nested\] Let $d_{ji}$ be the length of shortest path $j \rightarrow i$ in the physical interconnection graph. Considering , the information set $\mathcal{I}^i_k$ is influenced by measurement $x_j (k-d_{ji})$, or equivalently by $u_j (k-d_{ji}-1)$. Thus, to check if information structure is partially nested, one should verify the condition: $\mathcal{I}_{k-d_{ji}-1}^j \subset \mathcal{I}_k^i$. Recalling the assumption that graph $\mathcal{G}$ is connected and undirected, the information sets of decision makers $i$ and $j$ are explicitly written as $$\begin{aligned} \mathcal{I}_k^i= &\underset{n = 1,\ldots, N}\bigcup \left\lbrace x_n (0:k-d_{ni}) \right\rbrace, \\ \mathcal{I}_{k-d_{ji}-1}^j =& \underset{n = 1,\ldots, N}\bigcup \left\lbrace x_n (0:k-d_{nj}-d_{ji}-1)\right\rbrace, \end{aligned}$$ which reduces the partial nestedness condition to: $d_{nj}+d_{ji}+1 \geq d_{ni} $. Since $d_{ni}$ is the length of the shortest path between nodes $n$ and $i$ in $\mathcal{G}$, one can write: $d_{ni} \leq d_{nj} + d_{ji} < d_{nj}+d_{ji}+1 $ which concludes the proof. \ Taking into consideration that problem is subject to power constraints, it is convenient to reformulate it in terms of covariance as the new decision variable $$V(k)= {\rm E} \left[ z(k) z(k)^\top \right] ={\rm E} \left[ {\begin{bmatrix} x(k) \\ u(k) \end{bmatrix}} \begin{bmatrix} x(k) \\ u(k) \end{bmatrix}^\top\right]$$ With the additional constraint given by , problem is posed as a covariance selection problem $$\begin{aligned} {\addtocounter{equation}{1}\tag{\theequation}}\label{eq: problem2} \min_{V(0:T-1)\succeq 0} \quad & tr (Q_{T} V_{xx} (T)) +\sum_{k=0}^{T-1} tr (Q V(k)) \\ \text{s.t.} \quad & F V(0) F^{\top} = \Sigma_x \\ \quad & \begin{bmatrix} A & B \end{bmatrix} V(k) \begin{bmatrix} A & B \end{bmatrix}^\top + \Sigma_w = F V(k+1) F^\top \\ & tr(W_i V(k)) \le p_k ^{i} , \quad \forall i= 1, \ldots, M \end{aligned}$$ where $F= \begin{bmatrix} I & 0 \end{bmatrix}$. Part of the result above is derived from the fact that, for a generic matrix $\Theta$ the following identity holds $$\begin{aligned} {\rm E}\left[ z(k)^\top \Theta z(k) \right] = tr\left( \Theta V(k) \right).\end{aligned}$$ Additionally, rewriting the system dynamics equation in terms of a covariance variable $V$ $$\begin{aligned} & F {V}(k+1)F^\top = V_{xx} (k+1)={\rm E}\left[ x(k+1) x(k+1)^\top \right] \\ & = \begin{bmatrix} A & B \end{bmatrix} {V}(k)\begin{bmatrix} A & B \end{bmatrix}^\top + \Sigma_w \end{aligned}$$ and translating the initial condition ${\rm E}\left[ x(0) x(0)^\top \right] = \Sigma_x $ into $$\begin{aligned} V_{xx} (0)&= {\rm E}\left[ x(0) x(0)^\top \right] = F {V}(0)F^\top = \Sigma_x ,\end{aligned}$$ the form in is obtained. Information decomposition {#sec:info dec} ========================= Covariance Decomposition {#par: covariance decompostion} ------------------------ For the sake of simplicity of derivation we demonstrate the method on a two-player system. Considering the state equation , each decision maker at each time instant $k$ is able to compute the estimate of the state $x$ based on the common information $\mathcal{I}_k^0$ the two players have at time instant $k$, i.e. $$\begin{aligned} {\addtocounter{equation}{1}\tag{\theequation}}\label{eq:common information set} \mathcal{I}_k^0 = \mathcal{I}_k^1 \cap \mathcal{I}_k^2 = \{x({0:k-1}), u({0:k-1})\},\end{aligned}$$ later referred to as the coordinator’s information set. The estimate is given by $$\begin{aligned} {\addtocounter{equation}{1}\tag{\theequation}}\label{eq:estimator} & \hat{x} (k) ={\rm E}\left[x(k) |\mathcal{I}_k^0 \right] = A x (k-1) + B u (k-1), \end{aligned}$$ since ${\rm E}\left[w(k-1) |\mathcal{I}_k^0 \right] = 0$. Locally, after measuring its own state $x_i$ each decision maker can compute the local noise value at the previous time step as $$\begin{aligned} {\addtocounter{equation}{1}\tag{\theequation}}& \omega_{i} (k) = w_{i} (k-1) = x_i (k) - M_i^\top\hat{x} (k) \label{eq:noise1}\end{aligned}$$ where $M_1^\top = \begin{bmatrix} I & 0 \end{bmatrix}^\top$, $M_2^\top = \begin{bmatrix} 0 & I \end{bmatrix}^\top$. The quantities $\hat{x} , \omega_{1}, \omega_{2}$ form a pair-wise independent components of state. Due to linearity of the state decomposition given by , and partial nestedness of the information structure one can represent the optimal control input in the form $$\begin{aligned} {\addtocounter{equation}{1}\tag{\theequation}}\label{eq:input decomposition} & u(k) = \hat{\phi} (k)+\begin{bmatrix} \phi_1 (k) \\ \phi_2 (k) \end{bmatrix} \end{aligned}$$ where $\hat{\phi} (k) = -{L}_0 (k) \hat{x} (k)$, $\phi_1 (k) = - L_1 (k) \omega_1 (k)$ and $\phi_2 (k) = - L_2 (k) \omega_2 (k)$, for some gains $L_0, L_1, L_2$. Aiming for the decomposition of problem , we define a vector $\bar {z}$ of state components $\hat{x},\omega_{1},\omega_{2}$ and input components $\hat{\phi},\phi_{1},\phi_{2}$ $${\addtocounter{equation}{1}\tag{\theequation}}\label{eq:zbar} \bar {z}(k) = \begin{bmatrix} \hat {x}(k)^\top & \hat {\phi}(k)^\top \vert \omega_1(k)^\top & \phi_1(k)^\top \vert \omega_2(k) ^\top & \phi_2(k) ^\top \end{bmatrix} ^\top$$ whose blocks are independent. [Additionally, denoting the state decomposition - and input decomposition in as $$\begin{aligned} (x^{0} (k) ,x^{1} (k) , x^{2} (k))=(\hat{x}(k),\omega_1 (k), \omega_2 (k)),\\ (u^{0} (k) ,u^{1} (k) , u^{2} (k))=(\hat{\phi}(k),\phi_1 (k), \phi_2 (k)).\end{aligned}$$ the covariance matrix of $\bar{z}(k)$ is given by $$\label{eq: augmented state variance} \bar{V}(k) = {\rm E} \left[\bar{z}(k)\bar{z}(k)^\top\right] = \left[ \begin{array}{ccc} {V}^0(k) & 0 & 0 \\ 0 & V^1(k) & 0 \\ 0 & 0 & V^2(k) \end{array}\right]$$ where covariance matrices $V^l, l \in \{0,1,2\},$ of the individual blocks are $$\begin{aligned} & {V}^l (k) = {\rm E}\left[ \begin{bmatrix} {x}^l(k) \\ u^l (k) \end{bmatrix}\begin{bmatrix} x(k) \\ u(k) \end{bmatrix}^\top \right] = \begin{bmatrix} V_{{x}^l{x}^l} (k) & V_{x^lu^l} (k)\\ V_{u^lx^l} (k) & V_{u^lu^l} (k) \end{bmatrix}.\end{aligned}$$ The sparsity of $\bar{V}$ is due to block-independency of the vector $\bar{z}$ and due to presence of zero-mean Gaussian noise.\ Finally, recalling , for the sake of compactness, $A$ and $B$ are partitioned as $$\begin{aligned} A = \begin{bmatrix} A_1 \vert A_2 \end{bmatrix}, \quad &B = \begin{bmatrix} B_1 \vert B_2 \end{bmatrix}. \end{aligned}$$ where $A_1 \in \mathbb{R}^{n \times n_1}$, $A_2 \in \mathbb{R}^{n \times n_2}$, $B_1 \in \mathbb{R}^{n \times m_1}$ and $B_2 \in \mathbb{R}^{n \times m_2}$. Similarly, referring to , matrix $Q$ is partitioned as $$\begin{aligned} Q = [Q_{x_1} \vert Q_{x_2} \vert Q_{u_1} \vert Q_{u_2}] \end{aligned}$$ where $Q_{x_1} \in \mathbb{R}^{(n + m)\times n_1}$, $Q_{x_2} \in \mathbb{R}^{(n + m)\times n_2}$, $Q_{u_1} \in \mathbb{R}^{(n + m)\times m_1}$, and $Q_{u_2} \in \mathbb{R}^{(n + m)\times m_2}$. Furthermore, we define the following two matrices $$\begin{aligned} Q^1 = [Q_{x_1} \vert Q_{u_1} ],\quad Q^2 = [Q_{x_2} \vert Q_{u_2} ].\end{aligned}$$ ]{} Equivalent Problem Formulation ------------------------------ In order to rewrite the constraints appearing in equation as a function of $\bar{V}$, vectors $x(k)$, $u(k)$, $z(k)$ are obtained pre-multiplying the new variable $\bar {z}(k)$ according to $$\begin{aligned} {\addtocounter{equation}{1}\tag{\theequation}}\label{eq: extraction of state} x(k) = C_x \bar{z}(k), &\qquad u(k) = C_u\bar{z}(k), & \quad z(k) = C \bar{z}(k).\end{aligned}$$ where $$\begin{aligned} C = \begin{bmatrix} C_x\\ C_u \end{bmatrix} = \left[ \begin{array}{c|c|c} \begin{aligned} I \quad 0 \end{aligned} & \begin{aligned} I \quad 0 \\ 0 \quad 0 \end{aligned} & \begin{aligned} 0 \quad 0 \\ I \quad 0 \end{aligned}\\ \hline \begin{aligned} 0 \quad I \end{aligned} & \begin{aligned} 0 \quad I \\ 0 \quad 0 \end{aligned} & \begin{aligned} 0 \quad 0 \\ 0 \quad I \end{aligned} \end{array}\right]. \end{aligned}$$ The evolution of the original state $x(k)$ expressed as a function of $\bar{z}(k)$ is now $$x(k+1) = \begin{bmatrix} A & B \end{bmatrix} C \,\bar{z}(k) + w(k). \label{eq: fake evolution}$$ Combining the expressions in equations , and the variance of the state $x$ can be written as $$\begin{aligned} {\addtocounter{equation}{1}\tag{\theequation}}\label{eq: variance xx} V_{xx} (k)&= {\rm E}\left[ x(k) x(k)^\top \right] \\ & = {\rm E}\left[ (C_x \bar{z}(k))(C_x \bar{z}(k))^\top \right] = C_x \bar{V}(k)C_x^\top\end{aligned}$$ In the same way the variance of input $u(k)$ equals $$\begin{aligned} {\addtocounter{equation}{1}\tag{\theequation}}\label{eq: variance uu} V_{uu} (k)&= {\rm E}\left[ u(k) u(k)^\top \right]= C_u \bar{V}(k)C_u^\top\end{aligned}$$ Finally, from and , the evolution of the system’s state imposes the following recursive covariance equation $$\begin{aligned} {\addtocounter{equation}{1}\tag{\theequation}}\label{eq: variance deco} & C_x \bar{V}(k+1)C_x^\top = V_{xx} (k+1)={\rm E}\left[ x(k+1) x(k+1)^\top \right] \\ & = \begin{bmatrix} A & B \end{bmatrix} C \, {\rm E} \left[\bar{z}(k)\bar{z}(k)^\top\right] C^\top\begin{bmatrix} A & B \end{bmatrix}^\top + {\rm E} \left[{w}(k)w(k)^\top\right] \\ & = \begin{bmatrix} A & B \end{bmatrix} C \, \bar{V}(k)C^\top\begin{bmatrix} A & B \end{bmatrix}^\top + \Sigma_w. \end{aligned}$$ Similarly from the assumption on the state initial condition, the equivalent condition for the covariance is written as $$\begin{aligned} {\addtocounter{equation}{1}\tag{\theequation}}\label{eq: variance xx initial condition} V_{xx} (0)&= {\rm E}\left[ x(0) x(0)^\top \right] = C_x \bar{V}(0)C_x^\top = \Sigma_x.\end{aligned}$$ We then have the following proposition which is the main achievement of this subsection. \[eq:proposition\] Let $\bar{V}$ be the covariance of the extended vector $\bar{z}$. Problem is equivalent to $$\begin{aligned} {\addtocounter{equation}{1}\tag{\theequation}}\label{eq: quadratic cost variafnce} \min_{\footnotesize \bar{V}(0:T) \succeq 0}\quad & tr(C_{x}^\top Q_{T} C_{x} \bar{V}(T)) + \sum_{k=0}^{T-1} tr({C^\top Q C \bar{V}(k)}) \\ \text{s.t.} \quad & C_{x} \bar{V}(0) C_{x}^\top =\Sigma_x \\ & C_{x} \bar{V}(k+1) C_{x}^\top = \begin{bmatrix} A & B \end{bmatrix} C \bar{V}(k) C^\top \begin{bmatrix} A & B \end{bmatrix}^\top + \Sigma_w \\ & tr(C^\top W_i C \bar{V}(k)) \le p^{i}_k \end{aligned}$$ The proof follows from problem in and equations and . \[rem\] Although the methodology is presented for the case of 2-player system, it can be extended to a system of $N$ players using an algorithmic approach for state decomposition [@lamperski2015]. Information-oriented Optimization via Dual Decomposition {#sec:dual} ======================================================== In this section we proceed to define the dual problem to , which allows to transform the original constrained problem into an unconstrained one. To this end, we introduce dual variables $S(k) \in \mathbb{R}^{n \times n}, k=0,\ldots,T,$ to account for constraints on the evolution of $\bar{V} (k)$, as defined in and . Additionally, dual scalar variables $\tau_i (k) \in \mathbb{R}^+\, , {k=0,\ldots,T-1},$ are defined to account for power constraints in the overall system. Computation of Dual Variables ----------------------------- Introducing the Langrange multipliers $S(0), \ldots, S(T)$ and $\tau_i (0), \ldots, \tau_i (T-1)$ the primal problem is equivalent to $$\begin{aligned} \max_{\footnotesize S(0:T), \tau_i (0:T-1)} & \min_{\footnotesize \bar{V}(0:T)}\quad tr \left (S(0)(\Sigma_x - C_x \bar{V}(0) C_{x}^\top) \right) \\ & + tr(C_{x}^\top Q_{T} C_{x} \bar{V}(T)) + \sum_{k=0}^{T-1} tr({Q C \bar{V}(k) C^\top})\\ & +\sum_{k=0}^{T-1} tr \left( S(k+1) \begin{bmatrix} A & B \end{bmatrix} C \bar{V}(k) C^\top \begin{bmatrix} A & B \end{bmatrix}^\top \right) \\ & +\sum_{k=0}^{T-1} tr \left( S(k+1) (\Sigma_w - C_x \bar{V}(k+1) C_x^\top ) \right) \\ & +\sum_{k=0}^{T-1} \sum_{i=1}^{M} tr \left( \tau_i(k) (C^\top W_i C \bar{V}(k) - p_k^i \right) {\addtocounter{equation}{1}\tag{\theequation}}\label{eq: quadratic cost variance}\end{aligned}$$ where the constraints in now appear as part of the cost in form of linear operators on covariance matrix $\bar{V}(k)$. Defining the Hamiltonian of the system $$\begin{aligned} &H(T) = tr \{ C_x^T \left( Q_{T} - S(T) \right)C_x \bar{V}(T)\}\\ &H(k) = tr \{C^\top (Q + \begin{bmatrix} A & B \end{bmatrix}^\top S(k+1) \begin{bmatrix} A & B \end{bmatrix} + \\ & - \begin{bmatrix} S(k) & 0 \\ 0 & 0 \end{bmatrix} + \sum_{i=1}^{M} \tau_i(k) W_i)C \bar{V}(k)\}\quad \text{for}\ k=0,\ldots,T-1\end{aligned}$$ the dual problem in is rewritten as $$\begin{aligned} \max_{\footnotesize S(0:T), \tau_i (0:T-1)} \min_{\footnotesize \bar{V}(0:T)}\quad & H(T) + \sum_{k=0}^{T-1} \{ H(k) + \Sigma_w \, tr S(k+1)\}+ \\ & + \Sigma_x \, tr S(0) -\sum_{k=0}^{T-1} \sum_{i=1}^{M} \tau_i(k) p_k^i. \label{eq: quadiance}\end{aligned}$$ With the boundary condition on the Hamiltonian it follows $H(T)=0$, hence $S(T)=Q_{T}$. The dual function is finite if and only if $${\addtocounter{equation}{1}\tag{\theequation}}\label{eq:feasabil} Q + \begin{bmatrix} A & B \end{bmatrix}^\top S(k+1) \begin{bmatrix} A & B \end{bmatrix} -\begin{bmatrix} S(k) & 0 \\ 0 & 0 \end{bmatrix} + \sum_{i=1}^{M} \tau_i(k) W_i \succeq 0.$$ Since the primal problem is convex and constraints are affine, Slater’s condition can be relaxed. Indeed, the constraints in are composed of linear equalities and inequalities and domain of the defined cost function is open, the Slater’s condition reduces to feasibility. To this end, it is easy to verify that the set of constraints in defines a non-empty region. Hence, the dual problem is equivalent to the primal and is stated as $$\begin{aligned} {\addtocounter{equation}{1}\tag{\theequation}}\label{eq:eriafnce} \max_{\footnotesize S(0:T), \tau_i (0:T-1)} \quad & tr(S(0)) \Sigma_x + \Sigma_w \sum_{k=1}^{T} trS(k) - \sum_{k=0}^{T-1}\sum_{i=1}^{M} \tau_i(k) p^i_k \\ \text{s.t.} \quad & Q(k)\\ & + \begin{bmatrix} A^\top S(k+1) A - S(k) & A^\top S(k+1) B \\ B^\top S(k+1) A & B^\top S(k+1) B \end{bmatrix} \succeq 0\\ & S(T+1) = 0\end{aligned}$$ where the constraint in is obtained from by defining $$\begin{aligned} Q (k) = \left \lbrace \begin{aligned} &Q + \sum_{i=1}^{M} \tau_i (k) W_i, \quad & k=1,\ldots,T-1\\ &\begin{bmatrix} Q_T & 0 \\ 0 & 0 \end{bmatrix}, & k=T. \end{aligned} \right.\end{aligned}$$ With fixed values of $\tau_i$, the previous equation is maximized for every time-instant $k$ with $$\begin{aligned} {\addtocounter{equation}{1}\tag{\theequation}}\label{eq:recursive} S(k) &= A^\top S(k+1) A + Q_{xx}(k) - L(k)^\top Y(k) L(k) \\ Y(k) &= (B^\top S(k+1) B + Q_{uu}(k)) \\ L(k) &= Y(k)^{-1} (B^\top S(k+1)A + Q_{xu}^\top (k))\end{aligned}$$ which can be proved by analogously to [@Gattami2010]. Indeed, the choice of $S(k)$ should be made such that $tr S(k)$ is maximized and at the same time constraint in is satisfied, under the condition that the optimal value of $S(k+1)$ is known. To this end, since any choice of $S(k)$ with trace greater than the trace of violates the constraint in , the choice in is optimal. The variables $\tau_i$ have to be computed numerically from accounting for . Optimal Information-constrained Control --------------------------------------- In this subsection we show how to obtain the solution via information decomposition. In paragraph \[par: covariance decompostion\] we introduced state, input and covariance decomposition. In the 2-player’s case, we obtain three information sets: $\mathcal{I}_0,\mathcal{I}_1,\mathcal{I}_2$, that are defined by , and referred herein as the coordinator, first subsystem and second subsystem respectively. Moreover, the coordinator is assumed to have the following information about the overall system $$\begin{aligned} \left(A_0, B_0, Q^0, x^0 (k) \right) \triangleq \left(A, B, Q, \hat{x}(k) \right). \end{aligned}$$ Before stating the main result of this paper, we define the expression for $J^l, \ l=0,1,2$ as $$\begin{aligned} {\addtocounter{equation}{1}\tag{\theequation}}\label{eq: expression of J} {J}^l ({V}^l,S,\tau) &= tr \left( Q_{T} {F}_l^\top {V}^l(T) {F}_l\right) + \sum_{k=0}^{T-1} tr\left(Q^l {V}^l (k)\right) \\ &+ tr\left\lbrace S(k+1) \left(\begin{bmatrix} A_l \vert B_l \end{bmatrix} {V}^l(k) \begin{bmatrix} A_l \vert B_l \end{bmatrix} ^\top \right) \right\rbrace\\ &- tr\left\lbrace S(k+1) \left( {F}_l^\top {V}^l (k+1) {F} + \frac{\Sigma_w }{3}\right) \right\rbrace\\ &+tr \left\lbrace S(0) \left({F}_l^\top {V}^l(0) {F}_l - \frac{\Sigma_x }{3}\right) \right\rbrace \\ &+ \sum_{k=0}^{T-1} \sum_{i=1}^{M} tr \left( \tau_i(k) W_i {V}^l (k) - q_k^i \right) \end{aligned}$$ where ${F}_0$, $F_1$ and $F_2$ are such that $$\begin{aligned} {\addtocounter{equation}{1}\tag{\theequation}}\label{eq: expression of F} {F}_0 {V}^0 (k) {F_0}^\top &= V_{\hat{x}\hat{x}}(k),\\ {F}_1 {V}^1 (k) {F_1}^\top &= \begin{bmatrix} V_{\omega_1\omega_1}(k) & 0\\ 0 & 0 \end{bmatrix},\\ {F}_2 {V}^2 (k) {F_2}^\top &=\begin{bmatrix} 0 & 0\\ 0 & V_{\omega_2 \omega_2}(k) \end{bmatrix}. \end{aligned}$$ Moreover, the definition of $q_k ^i$ is given by identity: ${p_k ^i = 3 q_k ^i}$. We can now state the main result of this paper. \[theorem\] Let the system dynamics be given by equation . Considering the optimization problem defined in and denoting by $S(k)$ and $\tau_{i} (k)$ the optimal values of the dual variables introduced in we state the following. 1. The problem is decoupled into the sum of independent sub-problems that are linear in the respective decision variables, i.e., it is equivalent to $$\begin{aligned} {\addtocounter{equation}{1}\tag{\theequation}}\label{eq: variance hhb} & \sum_{l = 0}^{2}\quad \min_{\footnotesize {V}^l (0:T)} {J}^l ({V}^l (0:T),S (0:T),\tau_{1:M} (0:T-1)) \end{aligned}$$ where ${J}^l$ is defined in and $V^l$, $l=0,1,2$ are defined in . 2. The optimal covariances $V^l$, $l=0,1,2$ of are computed according to $$\begin{aligned} {\addtocounter{equation}{1}\tag{\theequation}}\label{eq: optimal Vs} &V^l(k) = \begin{bmatrix} V^l_{xx}(k) & V^l_{xu}(k) \\ V^l_{ux}(k) & V^l_{uu}(k) \end{bmatrix},\\ &{V}^l_{xx} (0) = \frac{\Sigma_x}{3}, \\ &{V}^l_{ux} (k) = - {L}_l (k) {V}^l_{xx} (k), \\ &{V}^l_{uu} (k) = {V}^l_{ux} (k) \left({V}^l_{xx}(k)\right)^{-1} {V}^l_{xu} (k), \\ &{V}^l_{xx} (k+1) = \begin{bmatrix} {A}_l & {B}_l \end{bmatrix} {V}^l (k) \begin{bmatrix} {A}_l & {B}_l \end{bmatrix}^\top + \Sigma_w . \end{aligned}$$ where ${L}_l (k)$ is $$\begin{aligned} {\addtocounter{equation}{1}\tag{\theequation}}\label{eq:Ll} {L}_l (k) = \left({B}_l^\top S(k+1) {B}_l + {Q}^l_{uu}\right)^{-1}\left ({A}_l^\top S(k+1) {B}_l + {Q}^l_{xu}\right)^\top. \end{aligned}$$ From Proposition , problem  and are equivalent. Furthermore, from equations , and accounting for the specific structure of matrix $C_x$ one gets $$\begin{aligned} C_x \bar{V} (k) C_x ^\top &= V_{xx} (k) = V_{\hat{x}\hat{x}}(k) + \begin{bmatrix} V_{\omega_1\omega_1}(k) & 0\\ 0 & V_{\omega_2 \omega_2}(k) \end{bmatrix}\\ &= {F}_0 {V}^0 (k) {F_0}^\top + F_1 V^1 (k) F_1^\top + F_2 V^2 (k) F_2^\top \end{aligned}$$ where ${F_0}$, $F_1$ and $F_2$ are extraction matrices since $V_{\hat{x}\hat{x}}(k)$, $V_{\omega_1\omega_1}(k)$ and $V_{\omega_2 \omega_2}(k)$ are square submatrices of ${V}^0 (k), V^1 (k)$ and $V^2 (k)$ respectively. On the other hand, from the block-diagonal structure of $\bar{V}(k)$ and sparsity of $C$, one obtains $$\begin{aligned} tr({Q C \bar{V}(k) C^\top}) &= tr \left( Q {V}^0(k)\right) + tr \left( Q^1 {V}^1(k)\right) + tr \left( Q^2 {V}^2(k)\right) \end{aligned}$$ Analogously, we obtain $$\begin{aligned} \begin{bmatrix} A & B \end{bmatrix} C \bar{V} (k) C^\top \begin{bmatrix} A & B \end{bmatrix}^\top &= \begin{bmatrix} A & B \end{bmatrix} {V}^0 (k) \begin{bmatrix} A & B \end{bmatrix} ^\top\\ & + \begin{bmatrix} A_1 & B_1 \end{bmatrix} {V}^1 (k) \begin{bmatrix} A_1 & B_1 \end{bmatrix}^\top \\ & + \begin{bmatrix} A_2 & B_2 \end{bmatrix} {V}^2 (k) \begin{bmatrix} A_2 & B_2 \end{bmatrix}^\top\end{aligned}$$ With algebraic reordering the proof of the first part is completed. The second and fifth equation of stated follow respectively from the condition on the variance of the initial state and equation . To prove the second and third equation, observe that the decoupled problems in have a similar structure. Therefore, with the optimal values of $S(k)$ and $\tau_i (k)$, each problem in equation is written as $$\begin{aligned} \min_{\footnotesize {V}^l (0:T-1) \succeq 0} \sum_{k=0}^{T-1} tr({Z}^l (k) {V}^l (k)) + \sum_{k=0}^{T} tr(S(k)) - \sum_{k=0}^{T-1} \sum_{i=1}^{M} \tau_i (k) q^{i}_k \end{aligned}$$ where ${Z}^l (k) $ is given by $$\begin{aligned} {Z}^l (k) = \begin{bmatrix} {X}_l {Y_l}^{-1} {X_l}^\top & {X_l} \\ {X_l}^\top & {Y_l} \end{bmatrix}\end{aligned}$$ and the values of matrices $ {X}_l $ and $ {Y}_l $ are computed recursively $$\begin{aligned} & {X_l} = {A_l}^\top S(k+1) {B_l} + Q^l_{xu} \\ & {Y_l} = {B_l}^\top S(k+1) {B_l} + Q^l_{uu}.\end{aligned}$$ To conclude the proof, exploiting the linearity of the subproblems, in order to compute the optimal covariances $V_l$ it is sufficient to verify if the condition $ tr(Z^l (k) V^l (k)) = 0$ is satisfied for a certain choice of the covariance matrix $V_l$. Indeed $$\begin{aligned} tr(Z^l (k) V^l (k))&=tr \begin{bmatrix} {X}_l {Y_l}^{-1} {X_l}^\top V_{xx} ^l + X_l V_{ux} ^l & * \\ * & X_l ^ \top V_{xu} ^l + Y_l V_{uu} ^l \end{bmatrix} \end{aligned}$$ By imposing to the diagonal elements in latter equation to be zero and recalling the assumption on positive-definitness (and thus invertibility) of $Q_{uu} ^l$ it follows: $$\begin{aligned} V_{ux} ^l &= - Y_l ^{-1} X_l ^\top V_{xx} ^l = - L_l V_{xx} ^l \\ V_{uu} ^l &= - Y_l ^{-1} X_l ^T V_{xu} ^l = V_{ux} ^l (V_{xx} ^l)^{-1} V_{xu} ^l\end{aligned}$$ which concludes the proof. Consider the system and the optimization problem defined in . For the 2-player system, the optimal control law is given by $$\begin{aligned} {\addtocounter{equation}{1}\tag{\theequation}}\label{eq: optimal control law} &u(k) = u^0 (k)+\begin{bmatrix} u^1 (k) \\ u^2 (k) \end{bmatrix} \end{aligned}$$ where ${u}^l (k), l=0,1,2$ is computed as $$\begin{aligned} u^l (k) = - L_l (k) x^{l} (k) \end{aligned}$$ and $L_l$ is defined by . According to Proposition \[eq:proposition\], the problem defined in is equivalent to the covariance selection problem in . Since the latter is decomposed in Theorem \[theorem\] and optimal values of covariances are provided by , the optimal control law follows in straightforward manner. Indeed, the control inputs referring to the coordinator and two subsystems are given by $$\begin{aligned} {\addtocounter{equation}{1}\tag{\theequation}}\label{eq:a123} & {u}_l (k) = -{V}^l_{ux} (k) {{V}^l_{xx}}^{-1} (k) {x}^{l} (k) = - L^l (k) x^l (k) \end{aligned}$$ where $$\begin{aligned} &{L}_{0} (k) = (B^\top S(k+1) B + Q_{uu}) ^{-1} (A^\top S(k+1) B + Q_{xu})^\top\\ &L_{1} (k) =(B_1^\top S(k+1) B_1 + Q^1 _{uu}) ^{-1} (A_1^\top S(k+1) B_1 + Q^1 _{xu})^\top\\ &L_{2} (k) = (B_2^\top S(k+1) B_2 + Q^2 _{uu}) ^{-1} (A_2^\top S(k+1) B_2 + Q^2 _{xu})^\top.\end{aligned}$$ Interpretation of Control Input Structure ----------------------------------------- Consider a 2-player network with one-step communication delay as depicted in Fig. \[fig:2-player\]. It can be transformed into an equivalent network by introducing a dummy node, herein referred to as coordinator ${\mathcal{C}}$ (this is illustrated in Fig. \[fig:2-player with coordinator at time t\]). The colocated control units of subsystems $\mathcal{S}_1$ and $\mathcal{S}_2$ are of limited computational power (e.g. they might be routers, switches etc.) and limited memory. The coordinating unit ${\mathcal{C}}$ is assumed to be able to perform more complex computations. However, as it can be noted, it also has access to limited information about the overall system - more precisely, it knows a one-step delayed information about both subsystems. In our approach ${\mathcal{C}}$ computes and stores the sequences of $S(0:T)$ and ${\tau_i (0:T-1)}$ offline, based on equations and . During the system execution, at time-instant $k$, the coordinator ${\mathcal{C}}$ sends the matrix $S(k+1)$ to the local units. Then, using equations , the calculation of the gains $L_1(k)$ and $L_2(k)$ is computed locally at the control units of $\mathcal{S}_1$ and $\mathcal{S}_2$ by matrix multiplications, thus avoiding additional memory requirements. Moreover, the coordinator ${\mathcal{C}}$, computes the estimate of the overall state based on delayed knowledge, and passes the command to units $\mathcal{S}_1$ and $\mathcal{S}_2$. Hence, the corresponding inputs to be applied to the plants are computed using local measurements and the control signal from the coordinator. \(a) at (0,0) [$\mathcal{S}_1$]{}; (a1) at (3,0) [$\mathcal{S}_2$]{}; \(a) to\[bend left\] node [$1$]{} (a1) ; (a1) to\[bend left\] node [$1$]{} (a); \[ht!\] \(a) at (0,0) [$\mathcal{S}_1$]{}; (a1) at (3.7,0) [${\mathcal{C}}$]{}; (a2) at (7.4,0) [$\mathcal{S}_2$]{}; ; \(a) to\[bend left\] node [$1$]{} (a1) ; (a1) to\[bend left\] node (CC2) [$0$]{} (a); (a1) to\[bend right\] node (CC1) [$0$]{} (a2); (a2) to\[bend right\] node [$1$]{} (a1); ; ; $$\begin{aligned} &\phi_1 (k) = - {L}_1 (k) {x}^1 (k) \quad & \quad \phi_2 (k) = - {L}_2 (k) {x}^2 (k)\\ &u_1(k) = \phi_1 (k) + \begin{bmatrix}I \vert 0 \end{bmatrix}\hat{u} (k) \quad & \quad u_2(k) = \phi_2 (k) + \begin{bmatrix}0 \vert I \end{bmatrix}\hat{u} (k) \end{aligned}$$ CONCLUSIONS {#sec:conclude} =========== In this paper a framework for power-constrained optimization based on information decomposition is introduced. The linear quadratic control problem with power constraints is decomposed accordingly through covariance decomposition and Lagrangian dual reformulation. As presented, the obtained equivalent problem is linear in the new decision variables and the control gains are computed offline. The approach adopted can be extended to a network of several players. [99]{} Boyd, Vandenberghe, and Faybusovich, *Convex Optimization*, vol. 51(11), 2006 A. Giridhar and P. R. Kumar, *Toward a theory of in-network computation in wireless sensor networks*, in IEEE Communications Magazine, vol. 44, no. 4, pp. 98-107, April 2006 A. Gattami, *Generalized Linear Quadratic Control*, IEEE Transactions on Automatic Control, 55(1) 131-136, 2010 A. Nayyar, A. Mahajan, and D. Teneketzis, *Decentralized Stochastic Control with Partial History Sharing: A Common Information Approach*, IEEE Transactions on Automatic Control, vol. 58, no. 7, 2013 Ravindra K. Ahuja, Thomas L. Magnanti, and James B. Orlin, *Network flows: theory, algorithms and applications*, Prentice Hall, 1993 H. S. Witsenhausen, *A counterexample in stochastic optimum control*, SIAM J. Control, vol. 6, no. 1, pp. 131–147, 1968 Y.-C. Ho and K’ai-Ching Chu, *Team decision theory and information structures in optimal control problems-Part I*, IEEE Transactions on Automatic Control, vol. 17, no. 1, pp. 15–22, 1972 A. Lamperski, L. Lessard, *Optimal Decentralized State-Feedback Control with Sparsity and Delays*, Automatica, 2015 M. Rotkowitz and S. Lall, *Characterization of Convex Problems in Decentralized Control*, IEEE Transactions on Automatic Control, vol. 51, no. 2, 2006 A. Lamperski and J. Doyle, *Dynamic Programming Solutions for Decentralized State-Feedback LQG Problems with Communication Delays*, American Control Conference, 2012 N. Matni, A. Lamperski, J. Doyle, *Optimal Two Player LQR State Feedback With Varying Delay*, In IFAC Proceedings Volumes, 2014 [^1]: This project has received funding from European Union’s Horizon 2020 Framework Programme for Research and Innovation under grant agreement No 674875 and the German Research Foundation (DFG) within the Priority Program SPP 1914 ”Cyber-Physical Networking". [^2]: $^{\dagger}$ Both authors contributed equally to this work. [^3]: $^{1}$ V. Causevic, P. Ugo Abara and S. Hirche are with the Chair of Information-oriented Control, Technical University of Munich, Germany; [http://www.itr.ei.tum.de, {vedad.causevic, ugoabara, hirche}@tum.de]{}

--- abstract: 'We report the discovery of two Mira variable stars (Miras) toward the Sextans dwarf spheroidal (dSph) galaxy. We performed optical long-term monitoring observations for two red stars in the Sextans dSph. The light curves of both stars in the $I_{\rm c}$ band show large-amplitude (3.7 and 0.9 mag) and long-period ($326\pm 15$ and $122\pm 5$ days) variations, suggesting that they are Miras. We combine our own infrared data with previously published data to estimate the mean infrared magnitudes. The distances obtained from the period-luminosity relation of the Miras ($75.3^{+12.8}_{-10.9}$ and $79.8^{+11.5}_{-9.9}$ kpc , respectively), together with the radial velocities available, support memberships of the Sextans dSph ($90.0\pm 10.0$ kpc). These are the first Miras found in a stellar system with a metallicity as low as ${\rm [Fe/H]\sim -1.9}$, than any other known system with Miras.' author: - 'Sakamoto, Tsuyoshi' - 'Matsunaga, Noriyuki' - 'Hasegawa, Takashi' - 'Nakada, Yoshikazu' title: 'Discovery of Mira variable stars in the metal-poor Sextans dwarf spheroidal galaxy' --- Introduction ============ Miras are pulsating stars with initial masses between 0.8 and 8 solar masses in the Asymptotic Giant Branch (AGB) phase, and eject material via stellar winds into the interstellar medium (e.g., Habing 1996). The ejected material contain chemical elements that have been dredged up from the interior (e.g., carbon and s-process elements). A large amount of dust forms in the ejecta of the Miras, and then such dust grains regulate the cooling of the interstellar medium and the fragmentation of collapsing molecular clouds into stars. Thus, Miras play an important role in providing the heavy elements and dust grains from the early Universe to the present day. The lack of appropriate theoretical models of Miras currently prevents us from determining their fundamental properties, such as metallicities, based on observational data for Miras. Therefore, Miras in stellar systems with known metallicity and/or age distribution can be important tracers to study the evolution of Miras and their impacts on chemical enrichment. For example, most of the Galactic globular clusters are old stellar systems with a single metallicity or a narrow metallicity distribution, offering an important sample of low-mass and metal-poor Miras (Frogel & Whitelock 1998; Feast et al. 2002). We note that Miras are found only in the clusters with ${\rm [Fe/H]>-1}$. Another interesting sample is low- to intermediate-mass Miras found in the Magellanic Clouds. A formidable amount of literature exists on those objects (e.g. Ita et al. 2004ab; Fraser et al. 2008; Soszyński et al. 2009; Groenewegen et al. 2009). Galactic dSphs provide us with a sample of even lower metallicity objects than the globular clusters. It is known that the fainter galaxy tends to have the lower mean metallicity (Norris et al. 2010). Therefore, the faint dSphs are excellent places to study metal-poor Miras if any Mira is found. Recent monitoring surveys in Galactic dSphs have discovered several Miras (Fornax dSph, Whitelock et al. 2009; Leo I dSph, Menzies et al. 2010; Sagittarius dSph, Lagadec et al. 2009; Sculptor dSph, Menzies et al. 2011). Among the dSphs with previously known Miras, the Sculptor dSph is the most metal deficient one, which has the metallicity distribution with a peak at ${\rm [Fe/H]}=-1.56$ and a dispersion of 0.48 (Kirby et al. 2009). Sloan et al. (2009) reported the evidence of circumstellar dust around one of the Miras in the Sculptor dSph (Menzies et al. 2011), based on the [*Spitzer Space Telescope*]{} spectroscopy. This suggests that AGB stars in low-metallicity environments can make a significant contribution to dust formation in the early Universe. Our target galaxy, the Sextans dSph, shows a metallicity distribution with a peak at ${\rm [Fe/H]}=-1.9$ and is one of the most metal-poor dSphs in the Galaxy (Battaglia et al. 2011). So far, two monitoring surveys have been conducted for the center of this galaxy, revealing dozens of short-period variable stars (Mateo et al. 1995; Lee et al. 2003). No Mira has been found previously.In this paper, we report the discovery of two Miras toward the Sextans dSph. In Section 2, we describe optical and infrared photometric observations, and discuss their membership to the Sextans dSph. In Section 3, we discuss the chemical properties and their impacts. Observations and Results ======================== Our targets ----------- In order to explore Miras in the Sextans dSph, we selected two target stars from the photometric catalogs presently available. These are listed in Table \[tab:Object\]. The first target \#1, SDSS J101525.93$-$020431.8, was selected using the color criteria, $J-H>0.7$, $H-K_{\rm s}>0.3$ on the 2MASS catalog (Skrutskie et al. 2006) and $g-r>0.8$, $r-i>0.3$ on the SDSS catalog (Adelman-McCarthy et al. 2008). These criteria are also used for our monitoring survey of Miras in the Galactic halo (Sakamoto et al., in preparation). The target \#1 is carbon-rich, showing a spectrum with the strong CN absorption band at 7900${\rm \AA}$ (Mauron et al. 2004; Cruz et al. 2007). The second target \#2, SDSS J101234.29$-$013440.8, was later added because of its variation detected in QUEST1 (QUasar Equatorial Survey Team, Phase 1) variability survey (Rengstorf et al. 2009). The $R$-band light curve of \#2 over 2 years in the QUEST1 showed a variation with a large amplitude ($\Delta R \geq 1.2$ mag) and a long period (over 100 days), although their time sampling was not good enough to estimate the period. The target \#2 is oxygen-rich, showing a spectrum with clear TiO molecular absorption lines (Suntzeff et al. 1993). $I_c$-band photometry --------------------- We conducted photometric monitoring observations of the two selected targets in the direction of the Sextans using the 2KCCD camera attached to the 105-cm f/3.0 Schmidt telescope at Kiso Observatory (Itoh et al. 2001). The observations started in December 2008 and February 2010 for the targets \#1 and \#2, respectively, and were repeated until February 2012. Time series $I_c$-band images were obtained.The data were reduced following standard procedures with IRAF, including bias subtraction (both the level of the overscan region in each image and the bias pattern taken on each night) and the flat-field correction with $I_c$-band dome-flat images. Instrumental magnitudes of the targets and comparison stars were measured with aperture photometry using the IRAF/APPHOT package. The comparison stars were selected from the SDSS database (Adelman-McCarthy et al. 2008), and their $I_c$ magnitudes were calculated by using the transformation of Jordi et al. (2006), $$I_c=i'+(-0.386\pm 0.004)(i'-z')-(0.397\pm 0.001).$$ Using these $I_c$ magnitudes for calibrating the magnitude scale, we obtained the magnitudes of our target stars as listed in Tables 2 and 3. Fig. \[fig:LC\] plots the $I_c$ variations against the Modified Julian Date (MJD). Both stars show the long-period and large-amplitude variation characteristic of either Miras or semi-regular variables. The peak-to-valley amplitudes ($\Delta I_c$) are 3.72 mag for the target \#1 and 0.94 mag for the target \#2. Miras are generally considered to have the $I_c$ amplitude greater than 0.9-1.0 mag (Ita et al. 2004ab; Matsunaga et al. 2005). The target \#1 is clearly a Mira, whereas the target \#2 falls between Miras and semi-regulars. The light curve of the target \#1 shows a clear modulation over the entire observation run indicating a long-term variation which is often observed for carbon-rich Miras (Whitelock et al. 2003). We subtract this long-term trend which is fitted by a sine curve with a period of 1500 days. The residual light curve shows a regular periodic variation as expected. Then, we applied the Phase Dispersion Minimization (PDM, Stellingwerf 1978) to the residual curve to obtain a period 326 days as the best estimate. It should be noted that due to the insufficient coverage of the light curve, especially around the expected minima, we cannot exclude the possibility of the star having a period roughly half of the 326 days based only on our $I_c$ band data. Nevertheless, the shorter period is unlikely, considering that the star is so red, $(H-K_{\rm s}) \gtrsim 1.0$, and all of such red LMC Miras in Ita et al. (2004) have periods of 300$-$500 days. We examined various assumptions on the long-term trend (e.g., sine curves with different periods and/or amplitudes), and the estimated period stays within 15 days from the above value. Thus, the period of \#1 is estimated to be $326\pm 15$ days. In contrast, the light curve of the target \#2 appears to show no long-term trend, and the period of 122 days is obtained with the PDM. Sine curves with periods of 117$-$127 days seems consistent with the photometric data. Thus, the periods of \#2 is estimated to be $122\pm 5$ days. Near-infrared photometry ------------------------ For the target \#1, we obtained near infrared images in the $J$, $H$, and $K_{\rm s}$-bands using the IRSF 1.4-m telescope at South African Astronomical Observatory (Nagayama et al. 2003). Our near-infrared observations were carried out on June 14, 2011 and April 28, 2012, and the magnitudes of the target were calibrated based on 2MASS magnitudes of the neighboring stars. In addition, the near-infrared magnitudes for our targets were collected from a few large-scale catalogues (2MASS, Skrutskie et al. 2006; UKIRT DR6, Lawrence et al. 2007; DENIS, Epchtein et al. 1994). Table \[tab:Kband\] lists the magnitudes available from these sources, where the magnitudes were transformed into those in the IRSF photometric system (Kato et al. 2007). The magnitudes in the DENIS and UKIRT DR6 photometric systems were transformed into the 2MASS system by using the equations in Carpenter (2001) and Hewett et al. (2006), respectively, and then further transformed into the IRSF system by using Kato et al. (2007). The $J$, $H$, and $K_{\rm s}$ magnitudes list in Table \[tab:Kband\] and show significant variations. The mean magnitudes are obtained by taking averages of four or three independent data in Table 4. Here we check the uncertainty in the transformation between the different catalogs by comparing the magnitudes of stars around our Miras after the transformation into the IRSF magnitude. The infrared magnitudes well match between the 2MASS, UKIRT, and IRSF photometry measurements within their uncertainties, whereas the $K_{\rm s}$ magnitudes obtained from the DENIS are different from those in the others by $\sim 0.3$. We added a shift to the above DENIS magnitudes to fit them onto the 2MASS magnitude scale (the magnitudes in Table 4 are on this corrected scale). We estimate the uncertainties in the random-phased mean magnitudes in the $K_{\rm s}$-band as follows. Carbon-rich Miras with periods of 300$-$350 days in the solar neighborhood and nearby dwarf galaxies appear to have $K_{\rm s}$-band amplitudes of 0.5–1.0 mag (Whitelock et al. 2003). O-rich Miras with periods of 100$-$150 days have the $K_{\rm s}$-band amplitude of 0.14$-$0.88 mag in the sample of Whitelock et al. (2000). The amplitudes of \#1 and \#2 are assumed to be 0.75 mag and 0.51 mag in the $K_s$-band, respectively, suggesting that the observed mean magnitudes are located within $\pm$ 0.38 mag and $\pm$ 0.26 mag around their true mean magnitudes, respectively. Our infrared datasets consist of four and three independent data for \#1 and \#2, and thus the uncertainties in the mean magnitudes are approximately 0.19 mag and 0.15 mag. Distances and radial velocities ------------------------------- To discuss whether or not our target stars are associated with the Sextans dSph, we estimate the distances on the basis of the period-luminosity (PL) relation of Miras. Miras, as well as semi-regulars with a relatively regular variation and an amplitude close to 1 mag, like \#2, are generally found to lie on the Mira PL relation. Thus, we can apply this relation to both of our targets (Ita et al. 2004ab). Adopting the distance modulus of the LMC to be $18.50\pm 0.02$ (Alves 2004), the PL relation for the Miras (Ita & Matsunaga 2011) is expressed as $$M_K=(-3.675\pm 0.076)\log P+(1.456\pm 0.173).$$ Ita & Matsunaga (2011) also found that the redder Miras tend to show large offsets from the PL relation. They suggested that such offsets are caused by the dust shells around red Miras ($J-K_{\rm s}\gtrsim 2$), and we discuss the implication for our Miras in Section 3. For \#1, the offset in the $K_{\rm s}$ band is estimated to be $0.12\pm 0.09$ mag (Ita & Matsunaga 2011). In contrast, the target \#2 is so blue that such an offset is negligible, if any. The interstellar extinctions in the Galaxy in the $K_{\rm s}$ band are calculated to be $0.02$ mag from the map in Schlegel et al. (1998). After correcting the above offset and the interstellar extinction, we obtain distances of $75.3^{+12.8}_{-10.9}$ kpc and $79.8^{+11.5}_{-9.9}$ kpc for the targets \#1 and \#2. The uncertainties in the distances include those in periods, period-luminosity relation, and mean $K_{\rm s}$-band magnitudes. The estimated distances are consistent with that of the Sextans dSph, $90.0\pm 10.0$ kpc (Lee et al. 2003, 2009). Here we discuss the effect of the long-term trend on the distance esitmate for \#1. Long-term trends have been detected from optical and infrared wavelengths, however the relationship among different wavelengths has not been established, in particular when our data points are poor. In addition, the observation dates for the 2MASS and the UKIRT were MJD 51174 and 53726, respectively, whereas our $I_c$ observations started on MJD=54836. The long interval between the infrared and the $I_c$ observations makes the estimation of the long-term trend difficult. However, the long-term trend is suggested to be partly related with absorption by the circumstellar dust (Winters et al. 1994). The circumstellar extinction is also suggested to be correlated with the deviation from the PL relation (Ita & Matsunaga 2011), and its deviation is corrected using the ($H-K_{\rm s}$) color in this work. Thus, the above correction considering the ($H-K_{\rm s}$) color works, at least partly, as the correction of the long-term trend. In fact, the UKIRT photometry indicates that the \#1 was both fainter and redder at its epoch than at the epochs of other photometry. Even if we exclude the UKIRT magnitudes from our calculation, with the proper correction applied, we get the same result within the uncertainty. We check the reliablity of the distance for \#2. The Ic-band PL relation in Ita & Matsunaga (2011) is relatively tight near the period of our Mira \#2 (P=122 days), although it has a higher dispersion than the Ks-band PL relation. The distance based on the Ic-band PL relation is estimated to be $86.6^{+23.1}_{-18.2}$  kpc, consistent with the distance by the Ks-band PL relation ($79.8^{+11.5}_{-9.9}$ kpc). The systematic radial velocity of the Sextans dSph is 226.0 ${\rm km~s^{-1}}$ with a dispersion of 8.4 ${\rm km~s^{-1}}$ (Battaglia et al. 2011). For the targets \#1 and \#2, the radial velocities at a single epoch are $202\pm 12$ km s$^{-1}$ (Mauron et al. 2004) and $228.2\pm 2$ km s$^{-1}$ (Suntzeff et al. 1993). The radial velocity of a Mira measured with the optical spectra shows a time variation due to the pulsation by 10–25 ${\rm km~s^{-1}}$ (Alvarez et al. 2001). The radial velocities of foreground stars in the direction of the Sextans dSph show a peak around 0–50 ${\rm km~s^{-1}}$ and a weak tail toward a large velocity. Probabilities of foreground stars with the velocities of our targets are low, and this clearly support the memberships to the Sextans dSph. Another possible system that the targets might be associated with is a stellar stream in the Galactic halo. The SDSS reveals the presence of a 60-degrees-long stream extending from the Ursa Major to the Sextans (e.g., Grillmair 2006). However, its heliocentric distance is about 20 kpc, much smaller than those of our targets. The model of Law et al. (2005) predicts the presence of the Sagittarius stream at a heliocentric distance of 20-60 kpc toward the Sextans dSph in the oblate dark-matter halo of the Galaxy, although the stream has not been identified in its direction. Furthermore, the predicted radial velocities are 250–290 km s$^{-1}$, which are larger than the measured values of the targets. Thus, both of our targets are most likely associated with the Sextans dSph. Discussion ========== As mentioned in Section 1, our targets in the Sextans dSph are expected to be metal-poor. Further constraints can be inferred from their locations in the galaxy. The target \#1 is located within the tidal elliptical radius of the Sextans dSph (160 arcmin, ellipticity of 0.35), but in its outer part. Unfortunately, the metallicities for a large sample of stars around the target \#1 are not yet available. Nevertheless, Battaglia et al. (2011) reported that the Sextans dSph has a clear metallicity gradient: most stars beyond an elliptical radius of 48 arcmin (ellipticity 0.35) in projection appear to have \[Fe/H\]$<-$2.2, while metal-rich stars are concentrated toward the center. The target \#1 is located beyond the elliptical radius of 48 arcmin, suggesting that its metallicity is very low. As shown in Section 2.4, the target \#1 is very red, in particular at the UKIRT epoch, $J-K_{\rm s}=2.96$. Ita & Matsunaga (2011) suggested the existence of the dust shells around such red Miras from their magnitudes fainter than the PL relation. The deviation of the $K_{\rm s}$ magnitude in each catalog from that of the period-luminosity relation appears to follow a function of the ($J-K_{\rm s}$) and ($H-K_{\rm s}$) in Ita & Matsunaga (2011).Sloan et al. (2009, 2012) and Matsuura et al. (2007) detected the SiC dust excess for the spectra of the carbon-rich AGB stars in nearby metal-poor dSphs. The color and period of target \#1 are similar to those in the carbon-rich Miras in Sloan et al. (2012). Thus, we suggest a possible presence of the circumstellar dust around carbon-rich Miras with the metallicity $\sim$ 100 times lower than the solar. The Sextans dSph has a lower mean metallicity than any other dSph in Sloan et al. (2012), and the target \#1 can impose an important limit on the metallicity dependence of the dust content around carbon-rich Miras. The target \#2 is located in the inner part of the Sextans dSph. In contrast to the outer part, relatively metal-rich stars (\[Fe/H\]$>-1.0$) coexist with the metal-poor stars at the inner part, and thus the limit on the metallicity of \#2 is not so strong. The short-period and oxygen-rich chemistry suggest that the target \#2 is similar to low-mass Miras found in Galactic globular clusters. Summary ======= We discovered two Miras ($P=326$ and 122 days, respectively) toward the metal-poor Sextans dSph by performing photometric monitoring for red stars over periods of 3-4 years, although the shorter-period one may be a semi-regular. The distances and radial velocities of the objects are consistent with those of the Sextans dSph, which suggests that the two objects belong to the Sextans dSph. Thus, these objects are found in the lowest metallicity dSph with Miras, \[Fe/H\]$<-2$. 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J., Gauger, A., Sedlmayr, E. 1994, , 290, 623 [llllllllll]{} \#1 (SDSS J101525.93$-$020431.8) &10:15:25.93&$-$02:04:31.8&$326\pm 15$ &17.0&3.72 &$75.3^{+12.8}_{-10.9}$ &$202\pm 12$ &Carbon-rich\ \#2 (SDSS J101234.29$-$013440.8)&10:12:34.29&$-$01:34:40.8&$122\pm 5$ &15.7&0.94 &$79.8^{+11.5}_{-9.9}$ &$228.2\pm 2.0$&Oxygen-rich\ Sextans dSph &10:13:03&$-$01:36.9&—&—&—&$90.0^{+10.0}_{-10.0}$&$226\pm 8.4$&—\ [cccccc]{}\ MJD & $I_c$ (mag) & $e_{I_c}$ (mag) &MJD & $I_c$ (mag) & $e_{I_c}$ (mag)\ 54836.77&17.30&0.08&55646.49&15.53&0.01\ 54875.69&17.20&0.06&55662.53&15.59&0.02\ 54904.57&17.04&0.06&55663.49&15.64&0.03\ 55174.78&18.50&0.13&55694.50&16.08&0.06\ 55221.64&17.99&0.09&55694.50&16.03&0.06\ 55249.61&17.24&0.05&55718.48&16.10&0.06\ 55288.48&16.81&0.03&55879.84&15.03&0.02\ 55314.58&16.87&0.09&55880.82&15.10&0.02\ 55342.49&17.31&0.10&55899.85&14.78&0.01\ 55580.79&15.66&0.07&55899.85&14.78&0.02\ 55580.79&15.65&0.08&55942.73&14.79&0.03\ 55611.56&15.38&0.03&55970.55&14.84&0.03\ [ccccccccc]{}\ MJD & $I_c$ (mag) & $e_{I_c}$ (mag) &MJD & $I_c$ (mag) & $e_{I_c}$ (mag) & MJD & $I_c$ (mag) & $e_{I_c}$ (mag)\ 55249.60&15.54&0.01&55663.49&15.57&0.03&55898.73&15.24&0.04\ 55289.42&15.27&0.03&55694.50&16.11&0.07&55898.73&15.30&0.03\ 55316.47&15.71&0.02&55718.47&15.95&0.08&55939.77&15.81&0.12\ 55344.47&15.87&0.03&55876.77&15.24&0.02&55940.76&15.98&0.05\ 55580.78&15.68&0.08&55877.78&15.18&0.03&55942.72&15.82&0.03\ 55580.78&15.78&0.09&55879.83&15.17&0.05&55967.70&15.70&0.12\ 55611.55&15.55&0.03&55879.84&15.23&0.03&55967.70&15.76&0.10\ 55611.55&15.51&0.05&55880.81&15.24&0.05&55967.70&15.70&0.06\ 55646.65&15.33&0.01&55880.81&15.27&0.04&&&\ 55662.53&15.52&0.03&55898.72&15.28&0.04&&&\ --------- ---------- ------------------- ------------------- ------------------- ---------- ------------------- ------------------- ------------------- Object Filter MJD $J$ $H$ $K_{\rm s}$ MJD $J$ $H$ $K_{\rm s}$ 2MASS 51174.35 $14.017\pm 0.026$ $12.908\pm 0.025$ $11.981\pm 0.026$ 51174.31 $14.128\pm 0.022$ $13.567\pm 0.029$ $13.391\pm 0.036$ UKIRT 53726.68 $15.142\pm 0.005$ $13.596\pm 0.004$ $12.179\pm 0.004$ 53726.68 $13.853\pm 0.002$ $13.303\pm 0.002$ $13.139\pm 0.003$ IRSF1 55726.76 $13.830\pm 0.005$ $12.470\pm 0.005$ $11.430\pm 0.005$ — — — — IRSF2 56045.76 $13.420\pm 0.005$ $12.470\pm 0.005$ $11.430\pm 0.005$ — — — — DENIS — — — — 50480.25 — — $13.34\pm 0.17$ Average — 14.10 12.86 11.76 — 13.99 13.43 13.34 --------- ---------- ------------------- ------------------- ------------------- ---------- ------------------- ------------------- ------------------- : $J$, $H$, and $K_{\rm s}$ magnitudes of our targets available in our photometry with the IRSF/SIRIUS and other near-infrared catalogs.The magnitudes in other photometric systems were transformed into the IRSF system. \[tab:Kband\]

--- abstract: 'A novel soft-photon amplitude is proposed to replace the conventional Low soft-photon amplitude for nucleon-nucleon bremsstrahlung. Its derivation is guided by the standard meson-exchange model of the nucleon-nucleon interaction. This new amplitude provides a superior description of $pp\gamma$ data. The predictions of this new amplitude are in close agreement with potential-model calculations, which implies that, contrary to conclusions drawn by others, off-shell effects are essentially insignificant below pion-production threshold.' address: - | Department of Physics and Institute for Nuclear Theory,\ Brooklyn College of CUNY, Brooklyn, NY 11210 - 'Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545' author: - 'M.K. Liou' - 'R. Timmermans and B.F. Gibson' title: | Novel soft-photon analysis of pp below\ pion-production threshold --- Bremsstrahlung processes have been used as a tool to investigate electromagnetic properties of resonances, details of reaction mechanisms, and off-shell properties of scattering amplitudes. The most succesful example in the first case is the determination of the magnetic moments of the $\Delta^{++}$ ($\Delta^0$) from $\pi^+p\gamma$ ($\pi^-p\gamma$) data in the energy region of the $\Delta$(1232) resonance [@Lin91]. In the case of reaction mechanisms, a well-known example is the extraction of nuclear time delays from the $p ^{12}C\gamma$ data near the 1.7-MeV resonance [@Mar76]. The time delay distinguishes between direct and compound nuclear reactions. The initial goal of nucleon-nucleon bremsstrahlung investigations was to distinguish among various phenomenological potential models of the fundamental two-nucleon interaction. Most measured $pp\gamma$ cross sections could, in fact, be reasonably described by potential-model calculations, but the difference between predictions from any two realistic potentials appears to be too small to be distinguished by the data. For more than 30 years, the conventional Low soft-photon amplitude [@Low58] has been widely used for studying nuclear and particle bremsstrahlung processes. It seemingly provides a good description of the data for some processes. For instance, Nyman [@Nym68] and Fearing [@Fea80] used this amplitude to calculate $pp\gamma$ cross sections which were in reasonable agreement with several measurements and potential-model calculations. However, it was recently pointed out by Workman and Fearing [@Wor86] that the results from this conventional Low amplitude differ significantly from the potential-model calculations for the TRIUMF data at 280 MeV [@Mic90]. This difference was interpreted as evidence for “off-shell effects” in the $pp\gamma$ process. The main purpose of this Letter is to propose a novel soft-photon amplitude to replace the conventional Low prescription. This new amplitude, the derivation of which is guided by the structure of the standard meson-exchange model of the two-nucleon interaction, is relativistic, manifestly gauge invariant, and consistent with the soft-photon theorem. It belongs to one of the two general classes of recently derived soft-photon amplitudes [@Lio93]. We demonstrate that the $pp\gamma$ data from low energies to energies near the pion-production threshold can be consistently described by the new amplitude. Most importantly, we point out here that our amplitude essentially eliminates the discrepancy between the soft-photon approximation and the potential-model calculations. That is, we demonstrate that “off-shell effects” are essentially negligible. Finally, we explore why the conventional Low amplitude works for some cases but fails for others. In order to elucidate these points, let us consider photon emission accompanying the scattering of two spin-1/2 particles $A$ and $B$, $$A(q_i^\mu) + B(p_i^\mu) \rightarrow A(q_f^\mu) + B(p_f^\mu) + \gamma(K^\mu) \ . \label{eq:ppg}$$ Here, $q_i^\mu$ ($q_f^\mu$) and $p_i^\mu$ ($p_f^\mu$) are the initial (final) four-momenta for particles $A$ and $B$, respectively, and $K^\mu$ is the four-momentum for the emitted photon with polarization $\varepsilon^\mu$. Particle $A$ ($B$) is assumed to have mass $m_A$ ($m_B$), charge $Q_A$ ($Q_B$), and anomalous magnetic moment $\kappa_A$ ($\kappa_B$). For process (\[eq:ppg\]), we can define the following Mandelstam variables: $s_i=(q_i+p_i)^2$, $s_f=(q_f+p_f)^2$, $t_q=(q_f-q_i)^2$, $t_p=(p_f-p_i)^2$, $u_1=(p_f-q_i)^2$, and $u_2=(q_f-p_i)^2$. Since a soft-photon amplitude depends only on either ($s$,$t$) or ($u$,$t$), chosen from the above set, we can derive two distinct classes of soft-photon amplitudes: $M^{(1)}_\mu(s,t)$ and $M^{(2)}_\mu(u,t)$ [@Lio93]. The general amplitude from the first class is the two-$s$–two-$t$ special amplitude $M^{TsTts}_\mu(s_i,s_f;t_q,t_p)$; that from the second class is the two-$u$–two-$t$ special amplitude $M^{TuTts}_\mu(u_1,u_2;t_q,t_p)$. The distinguishing characteristics of these amplitudes come from the fact that they are evaluated at different elastic-scattering or on-shell points (energy and angle). The soft-photon theorem does not specify how these on-shell points are to be selected. The modified procedure for deriving these soft-photon amplitudes is described in detail in Ref. [@Lio93]. In this procedure, the fundamental tree diagrams of the underlying elastic scattering process play an important role in deriving the two general amplitudes. Thus, we argue that $M^{TsTts}_\mu$ should be used to describe those processes which are resonance dominated \[such as $p ^{12}C\gamma$ near 1.7 MeV and $\pi^\pm p\gamma$ in the $\Delta$(1232) region\], whereas $M^{TuTts}_\mu$ should be used to describe those processes which are exchange-current dominated (such as the $np\gamma$ process). For the $pp\gamma$ process, which exhibits neither strong resonance effects nor significant $u$-channel exchange-current effects, both amplitudes can be used in theory, although this has never been tested in conjunction with experimental data. We provide here the results of such an analysis. We emphasize that the general amplitude $M^{TuTts}_\mu$ (not $M^{TsTts}_\mu$) arises naturally for nucleon-nucleon bremsstrahlung [*if*]{} the derivation is guided by the standard meson-exchange model of the two-nucleon interaction. The amplitude $M^{TuTts}_\mu$ for the $pp\gamma$ process can be written in terms of five invariant amplitudes $F^e_\alpha$ ($\alpha=1,\ldots,5$) as $$M^{TuTts}_\mu = \sum_{\alpha=1}^5 \left[ Q_A\overline{u}(q_f)X_{\alpha\mu}u(q_i)\overline{u}(p_f)g^\alpha u(p_i) + Q_B\overline{u}(q_f)g_\alpha u(q_i)\overline{u}(p_f)Y^\alpha_\mu u(p_i) \right] \ , \label{eq:MTuTt}$$ where $$\begin{aligned} X_{\alpha\mu} &=& F^e_\alpha(u_1,t_p)\left[ \frac{q_{f\mu}+R^{q_f}_\mu}{q_f\cdot K}-\frac{(p_i-q_f)_\mu}{(p_i-q_f)\cdot K} \right] g_\alpha \nonumber \\ && - F^e_\alpha(u_2,t_p) g_\alpha \left[ \frac{q_{i\mu}+R^{q_i}_\mu}{q_i\cdot K}-\frac{(q_i-p_f)_\mu}{(q_i-p_f)\cdot K} \right] \ , \label{eq:Xamu} \\ Y^\alpha_\mu &=& F^e_\alpha(u_2,t_q)\left[ \frac{p_{f\mu}+R^{p_f}_\mu}{p_f\cdot K}-\frac{(q_i-p_f)_\mu}{(q_i-p_f)\cdot K} \right] g^\alpha \nonumber \\ &&- F^e_\alpha(u_1,t_q) g^\alpha \left[ \frac{p_{i\mu}+R^{p_i}_\mu}{p_i\cdot K}-\frac{(p_i-q_f)_\mu}{(p_i-q_f)\cdot K} \right] \ . \label{eq:Yamu}\end{aligned}$$ In Eqs. (\[eq:MTuTt\]-\[eq:Yamu\]), we have defined $$\begin{aligned} (g_1,g_2,g_3,g_4,g_5) & \equiv & (1,\sigma_{\mu\nu}/\sqrt{2},i\gamma_5\gamma_\mu,\gamma_\mu,\gamma_5) \ , \nonumber \\ (g^1,g^2,g^3,g^4,g^5) & \equiv & (1,\sigma^{\mu\nu}/\sqrt{2},i\gamma_5\gamma^\mu,\gamma^\mu,\gamma_5) \ , \nonumber\end{aligned}$$ and the factors $R^Q_\mu$ ($Q=q_f,q_i,p_f,p_i$) can be expressed as $$R^Q_\mu = \frac{1}{4}\left[\gamma_\mu,\slash\hspace{-.28cm}K\right] + \frac{\kappa}{8m}\left\{\left[\gamma_\mu,\slash\hspace{-.28cm}K\right], \slash\hspace{-.25cm}Q\right\} \ . \label{eq:RQmu}$$ In Eq. (\[eq:RQmu\]), $m$ ($=m_A=m_B$) and $\kappa$ ($=\kappa_A=\kappa_B$) are the mass and the anomalous magnetic moment of the proton, $\slash\hspace{-.25cm}Q=Q^\mu\gamma_\mu$, and we have used $\left[F,G\right] \equiv FG-GF$ and $\left\{F,G\right\} \equiv FG+GF$. As one can see from Eqs. (\[eq:Xamu\]) and (\[eq:Yamu\]), the invariant amplitudes $F^e_\alpha$ depend on $u$ and $t$. The same amplitudes but as functions of $s$ and $t$ can be obtained if we use the condition $s+t+u=4m^2$. For example, $F^e_\alpha(u_1,t_p)=F^e_\alpha(s_{1p},t_p)$ where $s_{1p}+t_p+u_1=4m^2$. Since $F^e_\alpha(s_{1p},t_p)$ ($\alpha=1,\ldots,5$) are invariant amplitudes for the $pp$ elastic process, the Feynman amplitude $F(s_{1p},t_p)$ defined by Goldberger [*et al*]{}. [@Gol60] can be written in terms of the five Fermi covariants ($S,T,A,V,P$) as $$F(s_{1p},t_p) = F^e_1(s_{1p},t_p)S+F^e_2(s_{1p},t_p)T+F^e_3(s_{1p},t_p)A +F^e_4(s_{1p},t_p)V+F^e_5(s_{1p},t_p)P \ .$$ The amplitude $M^{TsTts}_\mu(s_i,s_f;t_q,t_p)$ can be formally obtained from the amplitude $M^{TuTts}_\mu(u_1,u_2;t_q,t_p)$ given by Eqs. (\[eq:MTuTt\]), (\[eq:Xamu\]), and (\[eq:Yamu\]) by making the following substitutions: ($i$) $Q_B\rightarrow -Q_B$ and ($ii$) $p_i^\mu \leftrightarrow -p_f^\mu$ and $g^\alpha R^{p_i}_\mu \leftrightarrow -R^{p_f}_\mu g^\alpha$, keeping $R^{q_i}_\mu$, $R^{q_f}_\mu$, and the spinors $\overline{u}$ and $u$ unchanged. However, we emphasize that the two are not the same numerically. If all $F^e_\alpha(s_x,t_y)$ ($\alpha=1,\ldots,5$, $x=i,f$, and $y=q,p$) in $M^{TsTts}_\mu$ are expanded about average $s$, $\overline{s}$, and average $t$, $\overline{t}$, then the first two terms of the expansion give the conventional Low amplitude $M^{{\rm Low}(s,t)}_\mu(\overline{s},\overline{t})$. This particular choice ($\overline{s},\overline{t}$) for the on-shell point at which the Low amplitude is evaluated is just an [*ad hoc*]{} prescription, although it provided a reasonable description of $pp\gamma$ data until the TRIUMF measurements at 280 MeV. We have studied the amplitudes $M^{TuTts}_\mu$, $M^{TsTts}_\mu$, and $M^{{\rm Low}(s,t)}_\mu$ and have applied them to calculate $pp\gamma$ cross sections at various energies, using state-of-the-art phase shifts from the latest Nijmegen $pp$ partial-wave analysis [@Ber90]. Anecdotal results are shown in Figs. \[fig1\], \[fig2\], and \[fig3\]. At 42 MeV for $\theta_q=\theta_p=26^\circ$ (see Fig. \[fig1\]) the coplanar cross sections calculated from $M^{TsTts}_\mu$ are much larger than the Manitoba data [@Jov71]. The amplitudes $M^{TuTts}_\mu$ and $M^{{\rm Low}(s,t)}_\mu$, on the other hand, give similar results which agree well with both the data (within the experimental error) and the representative Hamada-Johnston-potential calculation [@Lio72]. The results calculated using $M^{{\rm Low}(s,t)}_\mu$ are close to those obtained by Nyman and Fearing. In Fig. \[fig2\] our coplanar cross sections calculated from $M^{TuTts}_\mu$ and $M^{{\rm Low}(s,t)}_\mu$ at 157 MeV for $\theta_q=\theta_p=35^\circ$ are compared with the Harvard data [@Got67] and a Paris-potential calculation [@Jet93]. (Other potential-model calculations [@Wor86; @Her92; @Bro92; @Kat93] which include relativistic spin-corrections etc.are similar.) Cross sections calculated using the amplitude $M^{TsTts}_\mu$ are missing from Figs. \[fig2\] and \[fig3\], because they are factors larger than those plotted. Again the amplitudes $M^{TuTts}_\mu$ and $M^{{\rm Low}(s,t)}_\mu$ give very similar results at this energy and agree reasonably with both the potential-model curve and the Harvard data. However, at an energy near the pion-production threshold and far from the on-shell point, the two amplitudes $M^{TuTts}_\mu$ and $M^{{\rm Low}(s,t)}_\mu$ predict quite different results. This is demonstrated in Fig. \[fig3\]. At 280 MeV for $\theta_q=12.4^\circ$ and $\theta_p=12^\circ$, the curve calculated from $M^{TuTts}_\mu$ agrees well with the published TRIUMF data [@Mic90] and with the curves calculated using the Paris potential and the Bonn potential [@Mic90]. The amplitude $M^{{\rm Low}(s,t)}_\mu$, on the other hand, predicts cross sections which are too small for forward ($\theta_\gamma \leq 30^\circ$) and backward ($\theta_\gamma \geq 150^\circ$) photon angles. That $M^{{\rm Low}(s,t)}_\mu$ can describe most of the older $pp\gamma$ data but fails to fit the new TRIUMF data has already been pointed out by Fearing. What is emphasized here is that the new amplitude $M^{TuTts}_\mu$ describes data where the conventional Low amplitude $M^{{\rm Low}(s,t)}_\mu$ fails. In other words, the correct soft-photon amplitude which describes the $pp\gamma$ data consistently is $M^{TuTts}_\mu$. How can we understand the failure of the conventional Low amplitude $M^{{\rm Low}(s,t)}_\mu$? Consider the expressions given in Eqs. (\[eq:MTuTt\]-\[eq:Yamu\]). If we impose the on-shell condition, $s+t+u=4m^2$, we can write $F^e_\alpha(u_1,t_p)=F^e_\alpha(s_{1p},t_p)$, $F^e_\alpha(u_2,t_p)=F^e_\alpha(s_{2p},t_p)$, $F^e_\alpha(u_1,t_q)=F^e_\alpha(s_{1q},t_q)$, and $F^e_\alpha(u_2,t_q)=F^e_\alpha(s_{2q},t_q)$, where $s_{1p}=s_i-2q_f\cdot K$, $s_{2p}=s_i-2p_i\cdot K$, $s_{2q}=s_i-2p_f\cdot K$, and $s_{1q}=s_i-2q_i\cdot K$. This shows that $F^e_\alpha$ will be evaluated at four different energies and four different angles in constructing $M^{TuTts}_\mu$. (Potential-model calculations also use four-energy-four-angle amplitudes.) In contrast, $M^{TsTts}_\mu$ is evaluated at two energies and four angles, while $M^{{\rm Low}(s,t)}_\mu$ is evaluated at just one energy and one angle. To be specific, at 100 MeV for $\theta_q=\theta_p=\theta_\gamma=30^\circ$, we have $s_{1p}=3.648$ GeV$^2$, $s_{2q}=3.640$ GeV$^2$, $s_{2p}=3.632$ GeV$^2$, and $s_{1q}=3.655$ GeV$^2$, whereas $s_i=3.709$ GeV$^2$ and $s_f=3.578$ GeV$^2$, and finally $\overline{s}=3.644$ GeV$^2$. These quantities are the dominant factors determining the calculated cross sections. Since $s_{1p} \simeq s_{2p} \simeq s_{2q} \simeq s_{1q} \simeq \overline{s}$ (the differences in c.m. energy between $s_{1p}$, $s_{2p}$, $s_{2q}$, and $s_{1q}$ on one hand, and $\overline{s}$ on the other hand, are less than about 3 MeV), $M^{{\rm Low}(s,t)}_\mu$ and $M^{TuTts}_\mu$ predict similar results at energies lower than 100 MeV and for large proton angles. However, the value of $s_i$ is much larger than the value of $s_f$. This is equivalent to a c.m. energy difference of some 34 MeV. This large difference between $s_i$ and $s_f$ is the primary reason for the huge cross sections predicted by $M^{TsTts}_\mu$. As the incident energy increases (or the proton angles decrease), the values of the four energies, $s_{1p}$, $s_{2p}$, $s_{1q}$, and $s_{2q}$, will no longer be close to one another, and they differ significantly from $\overline{s}$, as well as $s_i$ and $s_f$. Thus, the cross sections calculated using the amplitude $M^{TuTts}_\mu$ will differ from those calculated using either $M^{{\rm Low}(s,t)}_\mu$ or $M^{TsTts}_\mu$. A more systematic analysis, including other relevant factors, will be given elsewhere. In conclusion, we have demonstrated that the amplitude $M^{TuTts}_\mu$, not the conventional Low amplitude $M^{{\rm Low}(s,t)}_\mu$ nor the amplitude $M^{TsTts}_\mu$, is the correct soft-photon amplitude to be used in describing the nucleon-nucleon bremsstrahlung processes. Furthermore, below the pion-production threshold this novel amplitude $M^{TuTts}_\mu$ provides a description of the $pp\gamma$ data that is the equal of contemporary potential-model calculations, implying off-shell effects are insignificant in the kinematic range measured to date. We would like to thank M. Rentmeester for providing the phase shifts from the latest Nijmegen proton-proton partial-wave analysis. The work of M.K.L. was supported in part by the City University of New York Professional Staff Congress-Board of Higher Education Faculty Research Award Program, while the work of R.T. and B.F.G. was performed under the auspices of the Department of Energy. Dahang Lin, M.K. Liou, and Z.M. Ding, Phys. Rev. C [**44**]{}, 1819 (1991), and references therein; M.K. Liou and Dahang Lin (unpublished). C. Maroni, I. Massa, and G. Vannini, Nucl. Phys. [**A273**]{}, 429 (1976); C.K. Liu, M.K. Liou, C.C. Trail, and P.M.S. Lesser, Phys. Rev. C [**26**]{}, 723 (1982); H. Taketani, M. 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--- address: | The Johns Hopkins University\ Dept. of Physics and Astronomy\ 3400 N. Charles Street\ Baltimore, MD 21218 author: - 'Edward M. Murphy[@byline]' date: 'September 21, 1998' title: A Prosaic Explanation for the Anomalous Accelerations Seen in Distant Spacecraft --- ‘=11 2 bylinecite thanks 2maketitle2thanksauthoraddresstitledateabstract pacs \#1 abstract =-1000pt 0=-0 by17.5pt width 0pt height0 [  ]{}\#1 \#1 pacs =-1000pt 0=-0 by20pt PACS numbers: \#1 maketitle2 preprint title =-1000pt authoraddress date [=0.10753==-1pc]{} abstract pacs ‘=12 \#1[[$\backslash$\#1]{}]{} 2 Introduction ============ Anderson, et al. [@anderson98] have recently modeled the accelerations acting on the Pioneer 10, Pioneer 11, Ulysses, and Galileo spacecraft. They find an anomalous, excess acceleration of $\rm 8.5\times10^{-8}\;cm\;s^{-2}$ directed towards the Sun. They have ruled out excess gravitational forces due to the Galaxy and unidentified planetesimals, errors in the orbital and rotational parameters of the Earth, spacecraft gas leaks, and errors in the planetary ephemeris as explanations for the acceleration. In addition, the authors rule out radiation pressure from thermal radiation generated by the spacecraft radioisotope thermoelectric generators (RTGs). Anderson et al. assume that the thermal radiation generated by the RTGs is isotropically radiated and results in no net force on the spacecraft. However, I believe that this assumption overlooks the fact that the electrical energy produced by the RTGs is dissipated in a non-isotropic manner. Spacecraft designed to travel beyond the inner solar system cannot rely on currently available solar cells to provide power, as the size of the solar arrays would be prohibitively large. Therefore, missions to the outer solar system have used RTGs to provide power. RTGs rely on the thermal energy generated by the radioactive decay of Pu$^{238}$ to heat a semiconductor junction which generates an electrical current. RTGs have an electrical conversion efficiency of a few percent [@piscane94]. For example, the Ulysses RTGs generate 4500 W of thermal power and produce 280 W of electrical power (at the beginning of the mission). The available power decreases with time due to degradation in the semiconductor junction and, to a much lesser degree, the decay of the Pu$^{238}$ [@piscane94]. The excess thermal power generated by the RTGs is dumped radiatively by cooling fins located on the outer surface of the cylindrical RTG structure (this is true of all the spacecraft considered here). The geometry of the fins is complex and the thermal radiation dissipated from the surface will not be isotropic. However, a cursory examination of the Pioneer and Ulysses RTG designs shows that they are cylindrically symmetric. Even though it is not isotropic, the escaping radiation will not impart a net force on the spacecraft since it is dissipated symmetrically. The same is not true of the electrical energy created by the RTGs. The electrical energy is transported to a main power bus from where it is distributed to the individual subsystems of the satellite to provide power for operating the electronics. These electronics are typically found in a single large electronics bay (which contains most of the essential systems) and science instruments distributed throughout and around the spacecraft. To prevent the electronics from overheating, the waste heat dissipated by the electronics is radiated from the spacecraft by surface radiators. In most cases, radiators are located on the anti-solar side of the spacecraft to prevent the panels from being heated by solar radiation. Because the radiator panels are preferentially located on the anti-solar side of the spacecraft, their radiation will cause an acceleration of the spacecraft toward the Sun. From conservation of momentum arguments, it is easy to show that the acceleration, $a_P$, produced by an amount of radiated power, $P$, is $a_{P}=P\:(m\:c)^{-1}$ where $m$ is the mass of the spacecraft and $c$ is the speed of light. This assumes that the radiated power is tightly collimated (i.e. it carries all the momentum in a single direction). In fact, however, the radiation from a flat plate is spread over 2$\pi$ steradians. In the case of a flat Lambertian source (i.e. one in which the intensity is independent of viewing angle [@boyd83]), the momentum carried away perpendicular to the plate surface will be 2/3 of the total. Ulysses ======= The Ulysses spacecraft is spin stabilized with the rotation axis pointing approximately toward the Earth (and the Sun when the spacecraft is near aphelion). The anti-Earth (anti-solar) side is always in the spacecraft shadow. The majority of the electrical components in the Ulysses spacecraft are located in a single thermal enclosure [@standley98]. The waste heat from the electronics is radiated through a large, flat radiator panel on the anti-solar side of the spacecraft. The interior electronics radiate their heat to the panel, which in turn radiates the heat into space. In addition, the traveling wave tube amplifier (which dissipates 43 W alone) is directly thermally coupled to the surface radiator. Except for the anti-solar side, the spacecraft is covered in multi-layer insulation (MLI) blankets. A large 1.65 meter diameter antenna covers most of the Earth facing side. A power budget for the Ulysses spacecraft for January 1998 [@standley98] indicates that Ulysses’ systems are drawing $231\pm3$ W of electrical power. Of this, I calculate that $27\pm10$ W is dissipated by scientific instruments and heaters outside the main thermal enclosure and another 20 W is radiated by the transmitter. Therefore, in a steady state, the main thermal enclosure must radiate $184\pm13$ W of power. Because the error estimates are systematic, rather than statistical, I have added them directly rather than in quadrature. Some fraction of this power will escape through the MLI thermal blankets and the remainder will be radiated through the large surface radiator on the anti-solar side of the spacecraft. MLI blankets typically radiate 8 W m$^{-2}$ [@piscane94] into space. Only the blankets on the side of the spacecraft will radiate internal heat because the solar facing blankets of Ulysses allow a net input of heat into the thermal enclosure due to solar heating, though this input power is small ($\sim$ 2 W) compared to electrical power when the spacecraft is at aphelion. When near perihelion, Ulysses compensates for the excess input solar heating by dumping excess electrical energy into resistors on the outside of the spacecraft. About 50 W of excess electrical power is dumped when the spacecraft is near perihelion [@standley98]. I calculate that there are $3.0\pm1.0\:{\rm m}^{2}$ of MLI blankets on the sides of the main thermal enclosure of Ulysses resulting in $24\pm8$ W escaping through the MLI blankets. This implies that the total power radiated through the spacecraft radiator on the anti-solar side of Ulysses is ($160\pm21$) W. The acceleration produced by this power is $a_{P}=(10.3\pm1.3)\times 10^{-8}\:{\rm cm}\:{\rm s}^{-2}$ assuming a spacecraft mass of 345 kg and that the radiator is a Lambertian source (2/3 of the momentum is carried away perpendicular to the radiator). Since the radiator faces away from the Sun, the direction of this acceleration is toward the Sun. This matches, to within the errors, the anomalous acceleration reported by Anderson et al. [@anderson98] for Ulysses of $a_{P}=(12\pm3)\times10^{-8}\: {\rm cm}\:{\rm s}^{-2}$. Pioneer 10 and 11 ================= Toward the ends of their missions, the Pioneer 10/11 spacecraft were drawing 80 W of electrical power [@anderson98] from their RTGs, which was sufficient to power the essential spacecraft systems and possibly one or two scientific instruments. Of this, 9 W is transmitted as RF power [@anderson98]. The essential electrical systems are located in a cylindrical hub beneath the high-gain antenna. The waste heat generated by the electronics is radiated from a series of fins on the anti-solar side of the spacecraft. Since the majority of the science instruments have been turned off, essentially all of the 71 W of internally dissipated electrical power is radiated from the fins. Solar heating is negligible at the Pioneers’ distance from the Sun. Assuming that the current mass of Pioneer 10/11 is 250 kg, the radiated power generates an $a_{P}=6.3\times10^{-8}\: {\rm cm}\:{\rm s}^{-2}$ again assuming a Lambertian source. However, the radiator fins of the back of the Pioneer spacecraft are highly non-Lambertian sources. In fact, the fins are likely to collimate the outgoing radiation to a significant degree. If the radiation were fully collimated, the resulting acceleration would be $a_{P}=9.5\times10^{-8}\: {\rm cm}\:{\rm s}^{-2}$. The actual value is likely to lie between these extremes. These estimates of the acceleration due to radiative cooling closely match the $a_{P}=8.5\times10^{-8}\;{\rm cm}\;{\rm s}^{-2}$ reported by Anderson, et al. [@anderson98] for the Pioneer 10 and 11 spacecraft. The essential electrical systems must remain powered at all times. Although the thermal power output of the RTGs is expected to decrease with time, the power drawn by the essential spacecraft electronics is nearly constant and, therefore, the acceleration imparted by the thermal radiation from the spacecraft radiators should also be constant with time. As the missions have progressed, various science instruments have been turned on and off. Most of these instruments are located on the periphery of the spacecraft, or on long booms reaching 3 meters or more from the spacecraft. These instruments are, typically, small enough that they radiate their power isotropically and do not employ a radiator system. Therefore, as the the science instruments are cycled on and off, there should be little effect on the net acceleration. Conclusions =========== I have shown that the most likely explanation for the anomalous accelerations found by Anderson et al. in the Pioneer 10/11 and Ulysses spacecraft is radiation pressure from spacecraft radiators which prevent the buildup of heat in the electronics systems. A more detailed analysis of the thermal designs of the spacecraft, including an examination of their power requirements with time, would allow us to reduce the associated errors and would make it possible to search for additional accelerations in the Anderson et al. data at a level of $a_{P}=1-2\times10^{-8}\;{\rm cm}\;{\rm s}^{-2}$. I wish to thank Shaun Standley of the Ulysses Project at JPL for very informative conversations concerning the Ulysses thermal design and power budget. I also wish to thank Scott Friedman and Alexandra Cha for helpful conversations. This work was supported by NASA contract NAS5-32985 (FUSE). Electronic address: emurphy@pha.jhu.edu. J. D. Anderson, P. A. Laing, E. L. Lau, A. S. Liu, M. M. Nieto, and S. G. Turyshev, Phys. Rev. Lett., in press (1998), gr-qc/9808081. V. L. Piscane and R. C. Moore, [*Fundamentals of Space Systems*]{}, (Oxford University Press, New York, 1994). R. W. Boyd, [*Radiometry and the Detection of Optical Radiation*]{}, (John Wiley & Sons, New York, 1983). S. Standley, private communication (1998).

--- abstract: 'It was recently observed that sand flowing down a vertical tube sometimes forms a traveling density pattern in which a number of regions with high density are separated from each other by regions of low density. In this work, we consider this behavior from the point of view of kinetic wave theory. Similar density patterns are found in molecular dynamic simulations of the system, and a well defined relationship is observed between local flux and local density – a strong indicator of the presence of kinetic waves. The equations of motion for this system are also presented, and they allow kinetic wave solutions. Finally, the pattern formation process is investigated using a simple model of interacting kinetic waves.' address: 'HLRZ-KFA Jülich, Postfach 1913, W-5170 Jülich, Germany' author: - Jysoo Lee and Michael Leibig title: 'Density Waves in Granular Flow: A Kinetic Wave Approach' --- Systems of granular particles (e.g. sand) exhibit many interesting phenomena, such as segregation under vibration or shear, density waves in the outflow through tubes and hoppers, and probably most strikingly, the formation of heap and convection cell under vibration [@s84; @c90; @jn92; @m92]. In granular flows through a narrow vertical tube, Pöschel found [@p92] that the particles do not flow uniformly, but form high density regions which travel as coherent structures with a velocity different from the center of mass velocity. He also reproduced these density waves using molecular dynamics (MD) simulations [@p92]. However, the motion of these high density regions and the mechanism which is responsible for their formation are not fully understood. In this paper, we present numerical and theoretical evidence that these density waves are of a kinetic nature [@lw55]. Using MD simulations, we measure the dependence of the particle flux on the density. We find a well-defined flux-density relation – an indication that a kinetic wave theory describes the behavior. A direct measurement of the velocity of these high density regions shows a dependence on the mean density which is in good agreement with the predictions from kinetic wave theory. On the theoretical side, we consider one dimensional equations of motion for the density and the velocity fields in the tube. These equations, together with Bagnold’s law for friction [@b54], allow kinetic density wave solutions. In order to understand the formation of these high density regions, we consider the general problem of interacting kinetic waves. We first show numerically that a system with an initially random density field evolves to a configuration in which neighboring regions have a high density contrast. At the early stage of development, we can show analytically that the density contrast between nearby regions increases linearly with time. We first discuss the MD simulations of the system, and begin with a brief description of the interparticle force laws that were used in our calculations. The particles interact with each other (or with a wall) only if they are in contact. The force that acts on particle $i$ due to particle $j$ can be divided into two components. The first, $F^{n}_{j \to i}$, is parallel to the vector $\vec{r} \equiv \vec{R_i} - \vec{R_j}$, where $\vec{R_i}$ and $\vec{R_j}$ are the coordinates of the centers of particles $i$ and $j$ respectively. We refer to this as the normal component. The second component, orthogonal to $\vec{r}$, is the shear component $F^{s}_{j \to i}$. The normal component is given by \[eq:f\] $$\label{eq:fn} F^{n}_{j \to i} = k_n (a_i + a_j - |r|)^{3/2} - \gamma _{n} m_e {\vec{v} \cdot \vec{r} \over |r|},$$ where $a_i (a_j)$ is the radius, and $m_i$ $(m_j)$ the mass of particle $i$ $(j)$. Also, $m_e$ is the effective mass $m_i m_j / (m_i + m_j)$, and $\vec{v} \equiv d\vec{r}/dt$. The first term in Eq.(\[eq:fn\]) is the Hertzian elastic force, where $k_{n}$ is a material dependent elastic constant. The second term is a velocity dependent friction term, where $\gamma _{n}$ is a normal damping coefficient. The shear component is given as $$\label{eq:fs} F^{s}_{j \to i} = -\gamma _{s} m_e {\vec{v} \cdot \vec{s} \over |s|},$$ where $\vec{s}$ is defined by rotating $\vec{r}$ clockwise by $\pi/2$. The shear force, Eq. (\[eq:fs\]), is simply a velocity dependent friction term similar to the second term in the normal component. Finally, we must specify the interaction between a particle and a wall. The force on particle $i$, in contact with a wall, is given by Eqs. (\[eq:f\]) with $a_{j} = \infty$ and $m_{j} = \infty$. The choice of the interactions defined by Eqs. (\[eq:f\]) is rather typical in the MD simulations of granular material [@mdgranule]. A detailed explanation of the interaction is given elsewhere [@lh93]. For simplicity, we study granular flows in $2$ dimensions and use a fifth order predictor-corrector scheme to integrate the equations of motion, calculating both the positions and velocity of each particle at all times. The tube is modeled by two vertical sidewalls of length $L$ with a separation $W$, and we apply a periodic boundary condition in the vertical direction. Between the sidewalls, particles of radii $0.1$ are initially filled with a uniform density of $\rho _o$ (throughout this paper, numerical values are given in CGS unit). The particles begin to move under the influence of gravity, and soon reach a steady state, where the gravitational force is balanced by the frictional force from the interactions with the sidewalls. In Fig. \[fig:mdtube\], we show the time evolution of the density and the velocity fields for $L = 15$ and $W = 1$, measured at every $5$ ms. At a given time, we divide the tube into $15$ vertical regions of equal length, and measure the density and the average velocity in each region. These fields are displayed as a vertical strip of square boxes, where each box corresponds to a region in the tube. The grayscale of the box is proportional to the value of the field in that region. The parameters we used in this simulation are $k_{n} = 1.0 \times 10^{6}, \gamma _{n} = \gamma _{s} = 5.0 \times 10^{2}$, with the time step $5.0 \times 10^{-5}$. The initial density $\rho _{o}$ is 25 particles per unit area. In the figure, we find (1) a region of large density fluctuations is formed out of the initially uniform system, (2) the fluctuations seem to travel with almost constant velocity (different from the center of mass velocity), and (3) there seems to be strong correlation between the density and the velocity fields. These findings remain true for the simulations we have performed with different values of $\gamma$, $k_{n}$ and $\rho _{o}$, except when $\rho _o$ is very small, where a steady state is not reached. These traveling density patterns were first observed in the simulations by Pöschel [@p92]. In order to quantitatively study the correlation between the density and the velocity fields, we measure the local particle flux as a function of the local density in the following manner. Once the system has reached a steady state, we measure the mean velocity $v_i$ and the density $\rho _i$ in region $i$. The flux $j(\rho )$ for a given density $\rho$ is then taken to be $\rho \cdot \langle v (\rho) \rangle$, where $\langle\rangle$ is a time average over all regions which had a particular density $\rho$. The flux-density curve, obtained by averaging over $10{,}000$ iterations, are shown in Fig.\[fig:jrho\]. Here, the parameters are the same as those of Fig.\[fig:mdtube\]. The fact that a well-defined flux-density curve exists suggest that the density waves (traveling density fluctuations) are kinetic in nature. Furthermore, the flux-density curve for the granular flow resembles that of a traffic flow, which is considered as a prime example of the systems which shows kinetic waves [@lw55]. One additional piece of evidence that the density waves are of a kinetic nature is their dependence on the initial density $\rho _{o}$. The theory of kinetic waves predicts [@lw55] that small density fluctuations in a uniform density background $\rho _{o}$ travel with a velocity $$\label{eq:kvel} U(\rho _{o}) = {dj(\rho ) \over d\rho } \mid _{\rho = \rho _{o}},$$ which is the slope of the flux-density curve at the mean density. We thus expect a large negative velocity for small $\rho_{o}$, a decrease to zero velocity at $\rho_{o} \approx 15$, with an increasingly large positive velocity as $\rho_{o}$ is increased further. To check this, we measure the wave velocities for several values of $\rho _{o}$ (keeping all other parameters fixed as above). Writing the mean density $\rho _o$ and the corresponding velocity $U(\rho _o)$ as $(\rho _o, U(\rho _o))$, we find $(10.0, -41 \pm 2),\ (15.0, 5 \pm 9),\ (18.7, 12 \pm 11)$ and $(22.5, 113.0 \pm 4)$, which are all consistent with the above prediction. We now consider the theoretical aspect of the density waves. Consider the equations of motion which govern the time evolution of the density $\rho (x,t)$ and the velocity $v(x,t)$ fields for a granular flow in a vertical tube. The first equation is that of mass conservation \[eq:motion\] $${\partial \over \partial t}\rho + {\partial \over \partial x} (\rho v) = 0,$$ and the second is a momentum conservation equation $$\rho {\partial \over \partial t}v + \rho v {\partial \over \partial x} v = F(x,t),$$ where $F(x,t)dx$ is the total amount of force acting on the particles in a region $[x,x+dx]$. The force $F(x,t)$ has three contributions—gravity, internal pressure, and friction from the sidewalls. The exact form of the internal pressure and the friction is not known. Here, we use Bagnold’s law, which is believed to be valid in the grain inertia regime [@b54]. Therefore, the force $F(x,t)$ is $$\label{eq:bagnold} F(x,t) = -\rho g W - \text{sign}(v) \rho _{B} f_{xy}(p) v^{2}- D {\partial \over \partial x}[\rho _{B} f_{xx}(p) v^{2}].$$ Here, $g$ is gravitational acceleration, $\rho _{B}$ the density of the material which forms the particles, $p$ the packing fraction ($\rho = \rho _{B} p$) and $D$ is the diameter of the particles. We assume the thickness of the shear layer to be of order of $D$. Also, $f_{xx}$ and $f_{xy}$ are geometry dependent functions, which contain the information about the density dependence of the forces. The uniform density solution of Eq. (\[eq:motion\]) with the force given by Eq. (\[eq:bagnold\]) is $$\begin{aligned} \label{eq:uniform} \rho (x,t) = & \rho _{B} p_{o} \cr v (x,t) = & - \sqrt{p_{o}gW / f_{xy}(p_{o})}\end{aligned}$$ If we add small density fluctuation $p = p_{o} + dp$ in the uniform density flow, it can be shown [@l93b] that the fluctuation travels with a velocity $$\label{eq:kvel2} U(p_{o}) \simeq - {1 \over 2} \sqrt{p_{o}gW / f_{xy}(p_{o})} \cdot {3 f_{xy}(p_{o}) - p_{o}df_{xy}(p_{o})/dp \over f_{xy}(p_{o})}.$$ Equations (\[eq:uniform\]) and (\[eq:kvel2\]) are exactly what one expects if the kinetic wave theory is to apply – uniform flow is a solution to the equations of motion, and density fluctuations travel with a density dependent velocity. Thus, it is clear then that the motion of the density pattern can be understood by applying the ideas of kinetic wave theory. However, this basic formalism only describes the motion of a pre-existing density pattern. It does not explain the observation that regions with large density contrasts are being formed out of the uniform background. Our simulations show that the large scale density pattern begins as a collection of small fluctuations in the density. These small fluctuations grow in time and a pattern emerges in which large density contrasts exist between neighboring regions. The evolution to such a state can be understood by considering the system as set of interacting kinetic waves. A detailed treatment of the general problem of interacting kinetic waves can be found elsewhere [@l93]; in this paper, we present only the results from a simple model for the pattern evolution process in sand flowing down a tube. Consider the early stages of the flow in which the density of sand is nearly uniform at $\rho \approx \rzero$. Because of the roughness of the grains, the roughness of the walls or from the stochastic nature of the inelastic collisions, small density fluctuations appear in the system. In the interacting density wave approach, we treat the fluctuations as a set of distinct density regions with interfaces whose velocities are determined by a discrete form of Eq.(\[eq:kvel\]). In this case, the interface separating a region of density $\rho_1$ from a region with density $\rho_2$ moves with a velocity, $U(1,2)$, given by \[eq:discrete\] $$U(1,2) = {j(\rho_1) - j(\rho_2) \over \rho_1 - \rho_2},$$ which is the kinetic wave theory result for interfacial velocities involving finite density differences [@lw55]. The evolution of the system is determined by the motion of the interfaces and, as shall be shown, the nature of their interactions leads to a final state in which large density contrasts occur. In the computational and analytic results that follow, we choose a specific form for the density fluctuations in the system. In our model, the initial positions of the interfaces are taken as a set of $\nz$ points placed randomly on the interval $[0, L]$, with regions between successive interfaces being assigned a density randomly in the range $[\rzero - W, \rzero + W]$. The principal virtues of this model are its simplicity and the fact that there are no correlations in the initial state which might influence the final structure. A more realistic model for the fluctuations of the system would require a microscopic understanding of each specific source of noise. It is also necessary to choose a form for the flux curve $\flxdens$. We have taken the parabolic form $$\flxdens = J_o{\rho\over R} \left(1 - {\rho\over R}\right),$$ where $R$ is the density at which no flow occurs, and $J_o$ is one quarter of the maximum flux of the system. This curve was chosen for several reason. The first is that its simplicity eases some of the hardships of analytical calculations. The second reason is that for density fluctuations over a sufficiently small range, the true flux response can be approximated by this form (with $R$ and $J_o$ being fitting parameters). And finally, it is a first approximation to the form observed for the $\flxdens$ observed in Fig. \[fig:jrho\]. Numerical simulation of this system is a very straight-forward exercise. The values of the densities in two adjacent regions determine the velocity of the associated interface. Consider three successive density regions A, B and C. During the course of the simulation, the interface A-B may encounter the interface B-C. This indicates that all of the mass that was inside region B has been completely “swallowed up” by the regions A and C. In this case, the interfaces A-B and B-C are replaced by a single A-C interface. The velocity of this interface can be calculated from the densities in regions A and C. Thus, it is a matter of tracking all of the interfaces, checking for collisions, and when they occur, replacing the two old interface with a single new one. Therefore, this technique does not allow for any density values other than those initially present, and the number of interfaces is always decreasing. For convenience, the simulations were done using periodic boundary condition. The first set of results shown below are from a simulation in which there are 400 interfaces initially placed randomly in the interval $[0, 1]$ (i.e. $L = 1$). We also choose the values $J_o = 1$ and $R = 1$. The densities are chosen at random from the interval $[0.3, 0.8]$ (i.e. $\rzero = .55R$, $W = .25R$). Figure \[fig:kevol\](a) shows the initial density configuration, while figure \[fig:kevol\](b) shows the system after a time $t = .486$ (where time is measured in the units of $RL/J_o$), and there are only 33 interfaces which remain along the interval. The system has evolved to a state in which the density contrast is very high between neighboring regions, and this behavior was observed for all values of $\rho_o$ and $W$. This increase in the density contrast can be characterized quantitatively in the following way. Let the density of each region be $\rho_i$, with $i$ indexing the different regions, and $N(t)$ be the number of regions at time $t$. Define the quantity $$\Mt \equiv {1\over \nt} \sum_{i = 1}^{\nt} |\rho_i - \rho_{i + 1}|,$$ where $\rho_{N(t)+1} \equiv \rho_1$. The larger the value of $\Mt$ the larger the density contrast between neighboring regions. Figure \[fig:mt\] shows the quantity $\Mt - \Mz$ averaged over 10 simulations with $\nz = 10{,}000$ interfaces. At early times, there is a linear increase in $\Mt$ with a crossover to a nearly constant value at late times. At early times, it is possible to calculate $\Mt$ analytically and the results are shown as the dotted line in Figure \[fig:mt\]. In this regime, the changes in $\Mt$ are dominated by the interaction of interfaces whose movements are determined by the initial configuration of the system. The calculation averages over all possible configurations of the initial random densities and interfaces, determines the time at which each interface collision occurs and how much that collision changes the value of $\Mt$. In this regime, the agreement with the simulation is good. It is also possible to show exactly that $\Mz = 2W/3$. At later times, after there have been many collisions between interfaces, the structure of the system depends on the nature of the earlier evolution. Thus, this long time behavior is much more difficult to calculate. The results from the calculation described above break down in this regime because the distribution of density regions is no longer that of the initial random distribution. At long times $\Mt \approx 2W$. Thus, the density contrast at long times is, on average, as large as the largest density contrasts present in the initial configuration. It turns out that the interacting kinetic waves do not create large contrasts. Rather, the interfaces from the initial distribution which survive are those that have a very large density contrast [@l93]. Thus, while the noise in the system may provide a variety of such contrasts, the interacting kinetic waves will destroy all but the very largest. This paper outlines a kinetic wave approach to understanding the density patterns observed in sand flow along a vertical tube (many of the details omitted here can be found in references [@l93b] and [@l93]). However, these ideas certainly do not constitute a complete theory for the patterns observed in the experimental system. The role that the flow of air plays in this process [@air], as well as the sources of noise in the system, are certainly not well understood. Further experimental investigation of these issues would be most enlightening. From a theoretical point of view, it is not clear whether the frictional force at the wall and the internal pressure obey Bagnold’s law. While this form has been observed in the sheer cell geometry [@b54; @shear], there has been no direct measurement of the frictional force for gravity driven flow. Finally, it is known that the interface between two regions of differing densities may not be a stable structure [@lw55], and that diffusive effects may strongly influence the long time behavior of a system of interacting kinetic waves. The authors would like to thank the members of the HLRZ Many Body Group for stimulating discussion throughout this work. S. B. Savage, Adv. Appl. Mech. [**24**]{}, 289 (1984); S. B. Savage, [*Disorder and Granular Media*]{} ed. D. Bideau, North-Holland, Amsterdam (1992). C. S. Campbell, Annu. Rev. Fluid Mech. [**22**]{}, 57 (1990). H. M. Jaeger and S. R. Nagel, Science [**255**]{}, 1523 (1992). A. Mehta, Physica A [**186**]{}, 121 (1992). T. Pöschel, HLRZ preprint 89/92 (1992). M. J. Lighthill and G. B. Whitham, Proc. Roy. Soc. A [**229**]{}, 281 and 317 (1955). R. A. Bagnold, Proc. R. Soc. London A [**225**]{}, 49 (1954). For example, P. A. Cundall and O. D. L. Strack, Géotechnique [**29**]{}, 47 (1979); P. K. Haff and B. T. Werner, Powder Technol. [**48**]{}, 239 (1986); Y. M. Bashir and J. D. Goddard, J. Rheol. [**35**]{}, 849 (1991); P. A. Thompson and G. S. Grest, Phys. Rev. Lett. [**67**]{}, 1751 (1991); G. Ristow, J. Physique I [**2**]{}, 649 (1992); Y-h. Taguchi, Phys. Rev. Lett. [**69**]{}, 1371 (1992); J. A. C. Gallas, H. J. Herrmann and S. Sokołowski, Phys. Rev. Lett. [**69**]{}, 1375 (1992); D. C. Hong and J. A. McLennan, Physica A [**187**]{}, 159 (1992). J. Lee and H. J. Herrmann, J. Phys. A [**26**]{}, 373 (1993). J. Lee, HLRZ preprint 44/93 (1993). M. Leibig, HLRZ preprint 42/93 (1993). D. Bideau, private communication; A. Hansen, private communication. For example, S. Savage and M. Sayed, J. Fluid Mech. [**142**]{}, 391 (1984); C. Campbell, J. Fluid Mech. [**203**]{}, 449 (1989).

--- abstract: 'The dielectric function for electron gas with parabolic energy bands is derived in a fractional dimensional space. The static response function shows a good dimensional dependence. The plasma frequencies are obtained from the roots of the dielectric functions. The plasma dispersion shows strong dimensional dependence. It is found that the plasma frequencies in the low dimensional systems are strongly dependent on the wave vector. It is weakly dependent in the three dimensional system and has a finite value at zero wave vector.' author: - 'K. M. Mohapatra' - 'Dr. B. K. Panda' title: 'Plasma dispersion in fractional-dimensional space' --- Introduction ============ When the well width of a quantum well (QW) is extremely narrow and its barrier height that causes the in-plane confinement is infinite, the QW shows two-dimensional (2D) electronic and optical properties. The infinitely wide QW exhibits the three-dimensional (3D) bulk properties of the well material[@Matos]. The electronic and optical properties in a QW with finite barrier height and narrow well width show 3D behavior of the barrier material. It happens since the envelope functions for electrons and holes spread into the barrier region partially restoring the 3D characteristics of the system. On the other hand, the electronic and optical properties in a finite QW with sufficiently wide well width show 3D characteristics of the well material. Consequently the QW with finite well width and barrier height shows the fractional dimensional behavior which is somewhere in between 2D and 3D. This has been demonstrated by Ishida[@Ishida] in the calculation of plasma dispersion in a superlattice. The same behavior has also been demonstrated in the calculation of exciton[@Jai] and polaron[@Smondyrev] ground state properties. The anisotropic interactions in an anisotropic solid are treated as ones in an isotropic fractional dimensional space, where the dimension is determined by the degree of anisotropy[@He]. Thus only a single parameter known as the degree of dimensionality $(\alpha)$ is needed to describe the system. In the quantum well structures the width of the QW can also serve for determining $\alpha$ of the system. The fractional dimensional $\alpha D$ space is not Euclidean space, it is spectroscopic dimension which is observed[@Bak]. The $(\alpha$D) space is not a vector space and the coordinates in this space are termed as [*pseudo coordinates*]{}[@Stillinger]. The advantage of the $\alpha$D space approach over the conventional method for calculating different electronic and optical properties in the low dimensional systems is that it is easier to apply this method. For example, the $\alpha D$ space approach has been successfully employed to calculate exciton binding energy in QWs in an analytic method[@He1; @Matos1], while the conventional method needs involved numerical calculations[@Jai]. Similarly the polaron properties in the $\alpha D$ space have been studied in a simple method[@Polaron] whereas the conventional method needs quite a bit of computational effort[@Smondyrev]. The technique has also been used to study biexcitons[@Biex1; @Biex2; @Biex3], magnetoexciton[@Magnet1; @Magnet2], exciton-exciton interaction[@Exex], exciton-phonon interaction[@Exph], Stark shift of exciton complexes in weak electric field[@Stark], refractive index[@Tanguy], impurity and donor states[@Imp1; @Harrison; @Imp2], Pauli blocking effect[@Pauli], exciton-phonon interaction[@Exphn], exciton-polaron interaction[@Expol] and magnetopolaron[@Magpol]. The absorption spectra in a quantum wire shows the fractional dimensional space behavior with the dimension of the system lies between 1 and 3 depending on the size of the system[@Karlsson]. Several properties of the charged boson system have been studied in the $\alpha D$ space using the Singwi-Tosi-Land-Sjölander (STLS) method[@Boson]. The Luttinger liquid[@Castelliani] and the breakdown of Fermi liquid due to long range interaction[@Wirefd] in the fractional dimensional space with the dimension between 1 and 2 have been studied. In the fractional dimensional space the plasma frequencies in the long-wavelength limit have been derived from the real part of the dielectric function both in the quantum and classical limits[@Panda]. However, the full treatment of the dielectric function in the $\alpha D$ space for finding plasma frequency has not been carried out for Fermi gas. The present paper aims to fill up this gap and study the correlation energy. Dielectric function =================== In the $\alpha D$ space, the dielectric function with the wave vector $q$ and frequency $\omega$ of the external charge is defined as[@Vignale] $$\epsilon_{\alpha D}(q,\omega)=1-v_{\alpha D}(q)\chi^{0}(q,\omega),$$ where $v_{\alpha D}(q)$ is the Fourier transform of Coulomb potential $e^{2}/\epsilon_{\infty}r$ in $\alpha$D space and $\chi^{0}_{\alpha D}(q,\omega)$ is the irreducible polarizability function. The expression for $v_{\alpha D}(q)$ is given as[@Stillinger], $$v_{\alpha D}(q)=\frac{(4\pi)^{\frac{\alpha-1}{2}} \Gamma\biggl(\frac{\alpha-1}{2}\biggr)e^{2}} {q^{\alpha-1}}\label{eq:cq},$$ where $\Gamma(x)$ is the Euler gamma function. The irreducible polarizability function is defined as[@Vignale] $$\chi^{0}_{\alpha D}(q,\omega)=\frac{2}{V_{\alpha D}} \sum_{\bf k}\frac{f({\bf k})- f({\bf k}+{\bf q})} {E_{\bf k}-E_{{\bf k}+{\bf q}}+\hbar\omega+i\epsilon}\label{eq:pol1},$$ where $f({\bf k})$ is the Fermi-Dirac distribution function, $V_{\alpha D}$ is the volume in $\alpha$D space and $\epsilon\rightarrow 0^{+}$. Rearranging Eq.(\[eq:pol1\]), we find $$\chi^{0}_{\alpha D}(q,\omega)=-\frac{2}{V_{\alpha D}}\sum_{\bf k}f({\bf k}) \left[\frac{1}{E_{{\bf k}+{\bf q}}- E_{\bf k}-\hbar\omega-i\epsilon} +\frac{1}{E_{\bf k-{\bf q}}-E_{{\bf k}}+ \hbar\omega+i\epsilon}\right]\label{eq:pol}$$ We consider the zero-temperature limit and the parabolic energy dispersion, $E_{\bf k}=\hbar^2k^2/2m^{\ast}$ where $m^{\ast}$ is the effective mass of electron. The summation over [**k**]{} in the $\alpha D$ space approach is transferred into integration over $k$ and $\theta$ as $$\sum_{\bf k}=\frac{V_{\alpha D}}{(2\pi)^{\alpha}} \frac{2\pi^{\frac{\alpha-1}{2}}}{\Gamma\biggl(\frac{\alpha-1}{2}\biggr)} \int^{k_F}_{0}k^{\alpha-1}dk\int^{\pi}_{0} \sin^{\alpha-2}\theta d\theta\label{eq:int}$$ In the $\alpha D$ space, the Fermi momentum $k_{F}$ is related to $r_{s}$ as $$k_{F}r_{s}a_{B}=\beta_{\alpha}\label{eq:rs},$$ where $a_{B}$ is the Bohr’s radius and $\beta_{\alpha}=[2^{d-1}\{\Gamma(1+d/2)\}^2]^{1/\alpha}$. $$\begin{aligned} \chi^{0}_{\alpha D}(q,\omega)&=&- \frac{2^{2-\alpha}} {\pi^{\frac{\alpha+1}{2}}\Gamma\biggl(\frac{\alpha-1}{2}\biggr)} \int^{k_F}_{0}k^{\alpha-1}dk \int^{\pi}_{0} \sin^{\alpha-2}\theta d\theta\nonumber\\ & & \left[\frac{1}{E_{q}+\hbar^{2}kq\cos\theta/m^{\ast}- \hbar\omega-i\epsilon}+ \frac{1}{E_{q}-\hbar^{2}kq\cos\theta/m^{\ast}+ \hbar\omega+i\epsilon}\right]\label{eq:pol2}\end{aligned}$$ We have the identity $$\frac{1}{x\pm i\epsilon}=P\biggl[\frac{1}{x}\biggr] \mp i\pi\delta(x)\label{eq:identity},$$ where $P[1/x]$ is principal part of $1/x$ and $\delta(x)$ is the Dirac delta function. Real part of the dielectric function ------------------------------------ Using Eq.(\[eq:identity\]) in Eq.(\[eq:pol2\]), we find $$\begin{aligned} Re[\chi^{0}_{\alpha D}(q,\omega)]&&=- \frac{2^{2-\alpha}} {\pi^{\frac{\alpha+1}{2}}\Gamma\biggl(\frac{\alpha-1}{2}\biggr)} \int^{k_F}_{0}k^{\alpha-1}dk \int^{\pi}_{0} \sin^{\alpha-2}\theta d\theta\nonumber\\ && P\Biggl[\frac{1}{E_{q}+\hbar^{2}kq\cos\theta/m^{\ast}- \hbar\omega}+ \frac{1}{E_{q}-\hbar^{2}kq\cos\theta/m^{\ast}+ \hbar\omega}\Biggr]\label{eq:pol}\end{aligned}$$ The first and second integrals in Eq.(\[eq:pol\]) diverge when $\theta=\theta_{-}$ and $\theta=\theta_{+}$, respectively where $\theta_{\pm}=\arccos[m^{\ast}(\hbar\omega\pm E_{q})/\hbar^2kq$. $$\begin{aligned} Re[\chi^{0}_{\alpha D}(q,\omega)]&&=- \frac{2^{2-\alpha}} {\pi^{\frac{\alpha+1}{2}}\Gamma\biggl(\frac{\alpha-1}{2}\biggr)} \int^{k_F}_{0}k^{\alpha-1}dk\nonumber\\ && \left[\int^{\pi}_{0}d\theta \frac{\left(\sin^{\alpha-2}\theta-\sin^{\alpha-2}\theta_-\right)} {E_q-\hbar\omega+\hbar^2kq\cos\theta/m^{\ast}} + \int^{\pi}_{0}d\theta \frac{\left(\sin^{\alpha-2}\theta-\sin^{\alpha-2}\theta_+\right)} {E_q+\hbar\omega-\hbar^2kq\cos\theta)/m^{\ast}}\right]\end{aligned}$$ Carrying out the $\theta$ integration analytically, we find $$\begin{aligned} Re[\chi^{0}_{\alpha D}(q,\omega)]&=&- \frac{2^{2-\alpha}}{\pi^{\frac{\alpha}{2}} \Gamma\biggl(\frac{\alpha}{2}\biggr)} \int^{k_{F}}_{0}k^{\alpha-1}dk\nonumber\\ & & \Biggl[\frac{{_2F_1}\biggl(1,\frac{\alpha-1}{2},\alpha-1; \frac{2\hbar^2kq/m^{\ast}}{(E_q-\hbar\omega) -\hbar^2kq/m^{\ast}}\biggr)}{(E_q-\hbar\omega) -\hbar^2kq/m^{\ast}} +\frac{\pi\sin^{\alpha-2}\theta_-} {(E_q-\hbar\omega)^2-(\hbar^2kq/m^{\ast})^2}\nonumber \\ && +\frac{{_2F_1}\biggl(1,\frac{\alpha-1}{2},\alpha-1; \frac{2\hbar^2kq/m^{\ast}}{(E_q+\hbar\omega) +\hbar^2kq/m^{\ast}}\biggr)}{(E_q+\hbar\omega) +\hbar^2kq/m^{\ast}} +\frac{\pi\sin^{\alpha-2}\theta_+} {(E_q+\hbar\omega)^2-(\hbar^2kq/m^{\ast})^2}\Biggr],\end{aligned}$$ where ${_2F_1}$ is the Gauss Hypergeometric function. The real part of the susceptibility $\chi^{0}(q,\omega)$ does not contain any divergent part in the range $\hbar v_{F}q-E_{q}<\hbar\omega<\hbar v_{F}q+E_{q}$. In this range the real part of the susceptibility is obtained as $$\begin{aligned} Re[\chi^{0}_{\alpha D}(q,\omega)]&&=- \frac{2^{2-\alpha}k^{\alpha}_{F}}{\pi^{\frac{\alpha}{2}} \Gamma\biggl(\frac{\alpha-1}{2}\biggr)} \sum^{\infty}_{m=0}\frac{1}{(\alpha+2m)} (\hbar v_{F}q)^{2m}\nonumber \\ && \times\Biggl[\frac{1}{E_{q}-\hbar\omega)^{2m+1}} +\frac{1}{(E_{q}+\hbar\omega)^{2m+1}}\Biggr] \sum^{m}_{l=0}(-1)^{l}{m\choose l} \frac{\Gamma\biggl(\frac{\alpha+2l-2}{2}\biggr)} {\Gamma\biggl(\frac{\alpha+2l}{2}\biggr)}\label{eq:ch}\end{aligned}$$ Imaginary part of the dielectric function ----------------------------------------- The imaginary part of the irreducible polarizability function can be derived from Eq.(\[eq:pol2\]) by using Eq.(\[eq:identity\]) as $$\begin{aligned} Im[\chi^{0}_{\alpha D}(q,\omega)]&=& -\frac{2^{2-\alpha}}{\pi^{\frac{\alpha-1}{2}} \Gamma\biggl(\frac{\alpha-1}{2}\biggr)} \int^{k_{F}}_{0}k^{\alpha-1}dk \int^{\pi}_{0}\sin^{\alpha-2}\theta d\theta\nonumber \\ &&\times\biggl[\delta\biggl(E_{q}+ \hbar^2kq\cos\theta/m^{\ast}-\hbar\omega\biggr)- \delta\biggl(E_{q}-\hbar^2kq\cos\theta/m^{\ast}+ \hbar\omega\biggr)\biggr]\label{Imag2}\end{aligned}$$ Integrating over $\theta$, we find $$\begin{aligned} Im[\chi^{0}_{\alpha D}(q,\omega)]= -\frac{2^{2-\alpha}m^{\ast}}{\pi^{\frac{\alpha-1}{2}} \Gamma\biggl(\frac{\alpha-1}{2}\biggr)} && \biggl[\Theta(k_{F}-k_{-})\int^{k_{F}}_{k_{-}}k^{\alpha-2}dk \biggl(1-\frac{k^{2}_{-}}{k^{2}}\biggr)^{\frac{\alpha-3}{2}}\nonumber \\ &&\times -\Theta(k_{F}-k_{+})\int^{k_{F}}_{k_{+}}k^{\alpha-2}dk \biggl(1-\frac{k^{2}_{+}}{k^{2}}\biggr)^{\frac{\alpha-3}{2}} \biggr]\label{eq:Imag3}\end{aligned}$$ where $k_{\pm}=\frac{m^{\ast}}{\hbar^{2}q}|\hbar\omega\pm E_{q}|$. Performing the integration over $k$-space gives the result $$Im[\chi^{0}_{\alpha D}(q,\omega)]= -\frac{2^{2-\alpha}m^{\ast}}{(\alpha-1)\pi^{\frac{\alpha-1}{2}} \Gamma\biggl(\frac{\alpha-1}{2}\biggr)\hbar^{2}q} \biggl[\Theta(k_{F}-k_{-})(k^{2}_{F}-k^{2}_{-})^{\frac{\alpha-1}{2}}- \Theta(k_{F}-k_{+})(k^{2}_{F}-k^{2}_{+})^{\frac{\alpha-1}{2}}\biggr]\label{eq:imga4}$$ Static response function and Plasma dispersion ---------------------------------------------- The scaled static response function $F_{\alpha D}(q)= \chi^{(0)}_{\alpha D}(q,\omega\rightarrow 0)/ \chi^{(0)}_{\alpha D}(0,\omega\rightarrow 0)$ is a measure of the number of excited states available to the system for vanishing excitation energy. Therefore $F_{\alpha D}(q,0)$ vanishes in the systems where there is a gap in the excitation spectrum. The calculated $F_{\alpha D}(q,0)$ for $\alpha=$1, 1.5, 2, 2.5 and 3 at $k_{F}=$0.5 a.u. and $m^{\ast}=m_{0}$ are shown in Fig.1. According to Eq.(\[eq:rs\]), the $r_{s}$ values corresponding to $k_{F}=$0.5 a.u. are $r_{s}$=1.57, 2.25, 2.8, 3.34 and 3.83 for $\alpha$=1, 1.5, 2, 2.5 and 3, respectively. The static response functions in integer dimensions 1D, 2D and 3D have been previously reported[@Vignale]. There are singularities at $q=2k_{F}$ in all dimensions. In 1D there is a logarithm divergence. This singular behavior is responsible for Peirls instability which is the spontaneous formation of density wave at $q=2k_{F}$. In 1.5D the response function is weaker and there is a weak kink at $q=2k_{F}$. In 2D the kink at $q=2k_{F}$ is quite significant. As the dimension is increased, the kink decreases and the derivatives of the response functions for $2\le\alpha\le 3$ diverge at $q=2k_{F}$. The divergence in response function at $q=2k_{F}$ results in oscillations with periodicity 2$k_{F}$ in the Fourier transformation of $F_{\alpha D}(q,0)$. These are Friedel oscillations which are direct consequence of the existence of Fermi surface. The plasmon frequency $\omega_{p}(q)$ can be obtained from the roots of $\epsilon_{\alpha D}(q,\omega)=0$ which gives the condition, $$1-v_{\alpha D}(q)Re[\chi^{0}_{\alpha D}(q,\Omega_{\alpha D}(q))]= 0\label{eq:root}$$ Since the analytic solution of this equation does not exist, we find $\Omega_{\alpha D}(q)$ by numerical method. The plasmon frequencies at $k_{F}$=0.5 a.u. for different $\alpha$ values and are shown in Fig.2 . The plasma frequency in 3D[@Lindhard] has got a finite value at $q=0$. The plasma frequencies for 2D agree with those of Stern[@Stern]. The plasma frequency in 1D has been calculated following Das Sarma and Hwang[@DasSarma]. For other dimensions the plasma frequency plasma frequency vanishes at $q=0$. The condition for the existence of undamped plasma oscillations is the the plasma frequency $\Omega_{\alpha D}(q)$ needs to be higher than $\omega_{+}(q)=\hbar(2k_{F}q+q^2)/2m^{\ast}$ which is the boundary frequency of the single particle regime. The plasma line and e-h line never intersect, but are tangential at $q_{c}$. For $q>q_{c}$, the dielectric function has no root and it is called Landau damping. The plasma frequency touches the boundary frequency of the single particle regime at $\omega_{+}=\hbar(2k_{F}q+q^2)/2m^{\ast}$ In order to understand this effect, we evaluate the plasma frequency in the long-wavelength limit. In the long-wavelength limit $q\rightarrow 0$, $Im[\chi^{0}_{\alpha D}(q,\omega)]=0$. Taking $m=0$ and 1 in the $m$ summation in Eq.(\[eq:ch\]), we find $$Re[\chi^{0}_{\alpha D}(q,\omega)]=- \frac{2^{1-\alpha}k^{\alpha}_{F}}{\pi^{\frac{\alpha}{2}} \Gamma\biggl(1+\frac{\alpha}{2}\biggr)} \Biggl[\frac{2E_{q}}{E^{2}_{q}-\hbar^{2}\omega^{2}} +\frac{2\hbar^{2}v^{2}_{F}q^{2}}{\alpha+2} \biggl(\frac{E^{3}_{q}+3E_{q}\hbar^{2}\omega^{2}} {(E^{2}_{q}-\hbar^{2}\omega^{2})^{3}} \biggr)\Biggr]\label{eq:lwl1}$$ For $E_{q}<<\hbar\omega$, $(E^{2}_{q}-\hbar^{2}\omega^{2})^{-1}= -(1+E^{2}_{q}/\hbar^2\omega^2)/\hbar^2\omega^{2}$ and $(E^{3}_{q}+3E_{q}\hbar^{2}\omega^{2})/(E^2_{q}-\hbar^{2}\omega^2)^3= -3E_{q}/\hbar^{4}\omega^{4}$. Substituting these values in Eq.(\[eq:lwl1\]), we find $$Re[\chi^{0}_{\alpha D}(q,\omega)]= \frac{2^{1-\alpha}q^{2}k^{\alpha}_{F}}{\pi^{\frac{\alpha}{2}}m^{\ast} \Gamma\biggl(1+\frac{\alpha}{2}\biggr)\omega^2} \left[1+\frac{E^{2}_{q}+ 3\hbar^2v^{2}_{F}q^2/(\alpha+2)}{\hbar^2\omega^2}\right]\label{eq:lwl2}$$ Substituting Eq.(\[eq:lwl2\]) in Eq.(\[eq:root\]), the long-wavelength plasma frequency $\Omega^{lw}_{\alpha D}(q)$ is obtained as $$\Omega^{lw}_{\alpha D}(q)=\omega_{\alpha D}(q) \left[1+\frac{3v^{2}_{F}q^2} {2(\alpha+2)\omega^{2}_{\alpha D}}+ \frac{\hbar^2q^4}{8m^{\ast}\omega^{2}_{\alpha D}}\right],$$ where the classical plasma frequency $\omega_{\alpha D}(q)$ is given by[@eqn] $$\omega_{\alpha D}= \sqrt{\frac{2^{2\alpha-2}\Gamma(1+\frac{\alpha}{2}) \Gamma(\frac{\alpha-1}{2})e^{2}q^{3-\alpha}} {\sqrt{\pi}\epsilon_{\infty}m^{\ast}r^{\alpha}_{s}}}.\label{eq:clas}$$ The dimensionless density parameter is related to the electron density $n_{\alpha D}$ as[@eqn] $$r_{s}=\Biggl[\frac{\Gamma(1+\frac{\alpha}{2})} {\pi^{\frac{\alpha}{2}}n_{\alpha D}}\Biggr]^{\frac{1}{\alpha}}.$$ We can easily understand the long-wavelength $q$-dependence in plasma frequency in Fig.1 by inspecting Eq.(\[eq:clas\]). The plasma frequency in 3D system is nonzero at $q=0$ as it independent of $q$ vector. For $\alpha<3$, the plasma frequency vanishes at $q=0$. Conclusion ========== The dielectric function for electron gas in the fractional dimensional space has been derived in the RPA. Using the irreducible susceptibilities the static response functions for different $\alpha$ values have been calculated. The static response functions show derivative divergence for all dimensions except for 1D system where there is a logarithm singularity. However, the response function for 1.5D electron gas is weak. The plasma dispersion has been found from the root of the dielectric function. The plasma frequency for low dimensional systems vanishes at $q=0$. It gradually approaches towards bulk value when $\alpha$ increases. Ericsson[@Plasma2D] experimentally verified the plasma frequencies in a wide QW. 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B [**44**]{}, 1936 (1996) Standard expression in literature is $\omega_{\alpha D}= \sqrt{(4\pi)^{{{\alpha-1}\over 2}}\Gamma({{\alpha-1}\over 2})e^2n_{\alpha D} q^{3-\alpha}/\epsilon_{\infty}m^{\ast}}$, where high-frequency dielectric constant $\epsilon_{\infty}$ is taken unity. M. A. Ericsson, A. Pinczuk, B. S. Dennis, C. F. Hirjibehedin, S. H. Simon, L. N. Pfeiffer and K. W. West, Physica E [**6**]{}, 165 (2000), C. F. Hirjibehedin, A. Pinczuk, B. S. Dennis, L. N. Pfeiffer and K. W. West, Phys. Rev. B. [**65**]{}, 161309 (2002) ![Static response functions in fractional dimensions as a function of wave vector. Here $k_{F}$=0.5 a.u. and $m^{\ast}=m_{0}$. Different color lines are denoted as $\alpha=$ 1 (red line), 1.5 (green line), 2 (blue line), 2.5 (violet line) and 3 (cyan line).[]{data-label="fig:spin1"}](response.eps){width="100.00000%"} ![Plasma dispersion in fractional dimensions in several dimensions. Here $k_{F}$=0.5 a.u. and $m^{\ast}=m_{0}$. The black dot-dashed line represents the boundary of the single particle regime. Different color lines correspond to different dimensions as in Fig.1. Plasma frequencies are denoted by solid lines while plasma frequencies at long-wavelength limit are denoted by dashed lines.[]{data-label="fig:spin2"}](plasfrac.eps){width="100.00000%"}

--- abstract: | Thermal inflation usually requires an inflationary potential with nonrenormalizable operators (NROs). We demonstrate how O’Raifeartaigh models with or without NROs can provide thermal inflation and a solution to the moduli problem, as well as provide SUSY breaking. We then discuss a scenario where generalized O’Raifeartaigh potentials (with NROs) are included in a SUGRA where the supergravity and O’Raifeartaigh potentials provide negative and a positive contributions to the cosmological constant respectively. Tuning these contributions to nearly cancel can provide the present value of the dark energy. PACS numbers: 98.80.Cq, 12.60.Jv, 95.35.+d address: | [*$^{1}$School of Physics, University of Edinburgh, Edinburgh EH9 3JZ, Great Britain*]{}\ [*$^{2}$Department of Physics and Astronomy, Vanderbilt University, Nashville TN 37235, USA*]{} author: - 'Arjun Berera$^{1}$ [^1] and Thomas W. Kephart$^{1,2}$ [^2]' title: 'Inflation and Generalized O’Raifeartaigh SUSY models' --- In Press Physics Letters B 2003 There is considerable belief that the fundamental model of particle physics respects local and/or global supersymmetry at high energy. Inflationary cosmology appears to provide further support to this expectation. Due to the ability of supersymmetry to protect against radiative corrections, such models provide powerful means to realize ultra-flat potentials, which are necessary from inflation density perturbation constraints. However, alongside this benefit, cosmological implementations of supergravity and SUSY models generally lead to undesired particles, such as the spin 3/2 gravitino in supergravity models [@gravitino] and various spin zero particles of mass $\sim 10^{2-3} {\rm GeV}$ [@moduli]. In particular for cosmological inflation, whether supercooled [@si] or warm [@wi], which end at conventional high temperature scales, $T \stackrel{>}{\sim} 10^{10} {\rm GeV}$, overabundances of unwanted SUSY particles is a real problem, sometimes termed the moduli problem [@moduli; @Lyth:1995hj]. SUSY must not survive at low energy scales, where physics clearly is not supersymmetric, with current limits set by particle physics experiments indicating SUSY must break above the electroweak scale $\sim 10^3 {\rm GeV}$. It is reasonable to expect that symmetry breaking and more specifically SUSY breaking has cosmological implications. For example, one scenario termed thermal inflation [@Lyth:1995hj; @Lyth:1995ka; @Lazarides:ja; @Barreiro:1996dx; @Asaka:1999xd] uses symmetry breaking to overcome the problem of overabundance of unwanted particles created by SUSY at high temperature. A second problem related to SUSY, for which cosmologists are universally and anxiously awaiting an explanation, is the presentday cosmological constant $\rho_{\Lambda}$. Observation of type IA supernova data have indicated an accelerating universe [@supernova], which could be explained by a cosmological constant of $70 \%$ of the critical density, which implies a vacuum energy component $\rho_{\Lambda} \sim 10^{-10} {\rm eV}^4$. Recently the first year WMAP data has independently verified the presence of a cosmological constant, finding $\Omega_{\Lambda} = 0.73 \pm 0.04$ [@wmap]. In this Letter, we will demonstrate that generalized O’Raifeartaigh models [@O'Raifeartaigh:pr] can realize thermal inflation and solve the presentday cosmological constant problem. Recall that spontaneous global SUSY breaking can be accomplished by the O’Raifeartaigh mechanism that requires at least three chiral supermultiplets. The minimal model has a superpotential of the form $$W(\phi ,\chi ,\eta )=a \chi \left[ \phi ^{2}-M^{2}\right] +m\eta \phi .$$ SUSY is broken since the requirement $\frac{\partial W}{\partial \phi_{i}}=0$, with $\phi_{i}={\phi,\chi,\eta}$, cannot be satisfied for all three fields. In other words the three conditions, $$\begin{aligned} \phi ^{2}-M^{2}=0, \phi =0, 2a\chi \phi +m\eta =0 ,\end{aligned}$$ cannot be simultaneously satisfied. Our purpose is to demonstrate that within their compact structure, these models contain nontrivial cosmological implications. We will begin with a review of thermal inflation, to understand the relevant scales necessary for such scenarios. Generalizations of the O’Raifeartaigh models are then presented and solutions are derived for thermal inflation and the presentday cosmological constant. We then briefly discuss embedding O’Raifeartaigh models in supergravity (SUGRA) and other fundamental theories, as well as particle physics implications of such models. The thermal inflation scenario is comprised of two phases of inflation. The first phase is the normal one, typically motivated by GUT physics and pictured to end, after reheating, at a high temperature $T \stackrel{>}{\sim} 10^{10} {\rm GeV}$. In this phase, the large scale physics is determined, such as density fluctuations. The key new feature that underlies thermal inflation is that it requires the presence of a scalar field $\phi$, often called the flaton, which has a symmetry breaking potential with the properties that at high temperature, $T > V_0^{1/4}$ symmetry is unbroken with $\phi=0$ where the scale of the potential is $V_0^{1/4} \approx 10^9 {\rm GeV}$. On the other hand, at $T=0$ symmetry is broken with the minimum now at $\phi \approx 10^9 {\rm GeV}$ and with the scalar particles acquiring a mass $m_{\phi} \sim 10^{2-3} {\rm GeV}$. Given such a potential, a second phase of inflation, termed thermal inflation, commences. In this picture, for $T > V_0^{1/4}$ the scalar field finite temperature effective potential locks the flaton field at $\phi = 0$ and the universe is in a hot big bang regime. Once $T < V_0^{1/4}$, the potential energy of this field dominates the energy density of the universe, thereby driving inflation, which to a good approximation is assumed to be an isentropic expansion. Due to the high temperature corrections to the effective potential, in the initial phase of thermal inflation, the scalar field remains locked at its high temperature point, $\phi = 0$. However, since inflationary expansion is rapidly cooling the universe, it implies the effective potential is evolving to its zero temperature form. Eventually, in what is estimated to be $\stackrel{<}{\sim} 15$ e-folds, the scalar field VEV no longer is locked at zero, and is able to roll down to its new minimum. The effect of the second phase of inflation is to lower the temperature of the universe from $T \sim 10^9 {\rm GeV}$ to $T \sim 10^3 {\rm GeV}$. This alone does not solve any overabundance problems since the abundance ratios $n/s$ for all species remain constant. However, subsequent to thermal inflation the scalar field oscillates, thereby producing scalar particles of mass $m_{\phi} \sim 10^{2-3} {\rm GeV}$ and lighter. These particles eventually decay, producing a huge increase in entropy, thereby adequately diluting the abundances of unwanted relics. Finally, in order not to affect the success of hot big bang nucleosynthesis, the temperature after decay of scalar particles is constrained to be above $\sim 10 {\rm MeV}$. Note, the desired features of thermal inflation could also occur for a continuous phase transition and a nonisentropic, warm-inflationary type expansion, which dampens the flaton’s motion during its evolution to its new minimum [@wi; @br]. The details of the thermal inflation scenario outlined above can be found in [@Lyth:1995hj; @Lyth:1995ka; @Lazarides:ja; @Barreiro:1996dx; @Asaka:1999xd]. The key point demonstrated in these papers is that all the desired features of this scenario follow, provided a potential with the properties described above is present. Considerable work on thermal inflation studies the consequences of such potentials, but many fewer works attempt to find explicit models of such potentials. Thermal inflation is typically carried out with potentials containing higher ($4$) dimension operators that are suppressed by powers of the Planck mass. In most studies of thermal inflation [@Lyth:1995hj; @Lyth:1995ka; @Lazarides:ja; @Barreiro:1996dx; @Asaka:1999xd], SUSY breaking is handled separately, for example, through nonperturbative means, such as the Affleck-Dine mechanism. Here we observe that a generalization of the O’Raifeartaigh potential, with one term replaced by a higher dimension operator can provide SUSY breaking, thermal inflation, and potentially, the presentday cosmological constant. Aside from the compactness of this solution, another advantage is that SUSY breaking terms are calculable at the tree level in the renormalizable O’Raifeartaigh model, so one has more control in model building. For the generalized O’Raifeartaigh model, loop level calculations would diverge. However, the basic motivation of the higher dimension operators is string theory which would serve to cut off all divergences and still leaves the model with some degree of control. To treat the cosmological moduli and cosmological constant problems, consider the generalization of the O’Raifeartaigh model superpotential, $$W(\phi ,\chi ,\eta )=a\chi \left[ \phi^2 -M^{2}\right] +\lambda\eta \frac{\phi ^{n+1}}{m_{Pl}^{n-1}} . \label{genor}$$ The $\frac{\partial W}{\partial \phi _{i}}=0$ conditions now become $$\begin{aligned} \frac{\partial W}{\partial \chi }=a[\phi^2 -M^{2}]=0 \nonumber \\ \frac{\partial W}{\partial \eta }=\lambda\frac{\phi ^{n+1}}{m_{Pl}^{n-1}} =0 \nonumber \\ \frac{\partial W}{\partial \phi }=2a\chi \phi +(n+1)\lambda\eta \frac{\phi ^{n}}{m_{Pl}^{n-1}} =0 ,\end{aligned}$$ and since these cannot be simultaneously satisfied, SUSY is broken. To carry out the calculation of thermal inflation and the cosmological constant, we need the Higgs potential $V=\left( \frac{\partial W}{\partial \phi _{i}}\right) ^{*}\left( \frac{\partial W}{\partial \phi _{i}}\right) $, which is $$\begin{aligned} V & = & a^{2}|\phi^2-M^{2}|^2+\lambda^2m_{Pl}^{4} \frac{|\phi|^{2(n+1)}}{m_{Pl}^{2(n+1)}} \nonumber \\ & + & | 2a\phi \chi +(n+1)\lambda\eta\frac{\phi^n}{m_{Pl}^{n-1}}|^2 . \label{genorpot}\end{aligned}$$ The first objective is to show at zero temperature this potential has the correct features and scales for thermal inflation and the presentday cosmological constant. For this the minimum of $V$ is required. There is a single family parameter of minima with $\langle \phi \rangle \neq 0$, given by setting the third term in the potential to zero. This gives the condition $\langle \eta \rangle = x \langle \chi \rangle$, with $x= -2am_{pl}^{n-1}/(n+1)\lambda \langle \phi \rangle^{n-1}$. A number of possibilities exist for this direction. First if the flat direction is uncorrected by the full theory, then there will be a massless boson $b = \chi+ x \eta$. If this boson couples sufficiently weakly to standard model fields, it does not upset the cosmology. In particular, although it will not thermalize, it still redshifts away. For a more strongly coupled $b$, particle physics familon limits apply [@Ammar:2001gi]. If the full theory corrects the O’Raifeartaigh potential, the mass generated for $b$ will allow it to contribute to the dark matter density or $b$ could generate an additional moduli problem at lower scale. A further implication is that corrections from outside the O’Raifeartaigh potential could allow the overconstrained set of conditions on the VEVs to be relaxed in a way that spoils the O’Raifeartaigh mechanism and could restore SUSY. We assume this does not happen or if it did we would have to modify the O’Raifeartaigh potential to again make the system overconstrained and thus break SUSY. This model also will have a goldstone fermion (goldstino) once global $N=1$ SUSY is broken. Nevertheless, should such particles be produced, they will redshift away like radiation. However, the other fermions generally will have mass and this leads to an interesting possibility. These fermionic components could be identified with right-handed neutrinos, for example if the U(1) symmetry of the generalized O’Raifeartaigh model was identified with B $-$ L. In this case a leptonic asymmetry can be produced, which can lead to baryongenesis based on the scenario of [@Fukugita:1986hr]. Taking the minimum of the Higgs potential gives $$\begin{aligned} \frac{dV}{d\phi} = 0 = -4a^2M^2 \phi + 4a^2 \phi^3 +2(n+1) \lambda^2m_{pl}^4 \frac{\phi^{2n+1}}{m_{pl}^{2n+2}}. \label{dv}\end{aligned}$$ Defining $a \equiv M/m_{pl}$, we are interested in the regime $a,\lambda \ll 1$, for which the solution to Eq. (\[dv\]) is $$\langle \phi^2_{\rm min} \rangle \approx M^2[1 - \frac{(n+1)}{2} \lambda^2 a^{2n-4}].$$ At this minima $$V_{\rm min} \equiv V(\langle \phi^2_{\rm min} \rangle) \approx \lambda^2 a^{2(n+1)} m_{Pl}^4 ,$$ $$m_{\phi}^2 \equiv V''(\phi_{\rm min}) \approx 8 a^2 M^2 ,$$ and $$V_0 \equiv V(\phi=0) = a^2 M^4 .$$ Choosing the scale $M \sim 10^{10-11} {\rm GeV}$ leads to $$\begin{aligned} V_0^{1/4} & \approx & 10^{5-8} {\rm GeV} \nonumber \\ m_{\phi} & \approx & 10^{2-4} {\rm GeV} , \label{scales}\end{aligned}$$ which are the desired properties for the thermal inflation zero temperature potential. Moreover, $\lambda$ remains a free parameter along with a choice for the index $n$ of the higher dimensional operator. This implies the value of $V_{\rm min}$ remains at our discretion, and it can be chosen to give the desired scale of the presentday cosmological constant. In particular, for $ V_{\rm min} = \rho_{\Lambda} \approx 10^{-10} eV^4$ it implies the condition $\lambda \approx 10^{-53 + 8n}$. So, for example, for $n=2$ it requires $\lambda \approx 10^{-37}$ whereas for $n=6$, $\lambda \approx 10^{-5}$. It is interesting that both the moduli and cosmological constant problems can be solved by this model, but parametrically neither of these two examples are particularly desirable. $\lambda$ must be highly fine tuned in the $n=2$ case, and although $\lambda$ is a typical coupling for the lepton sector of the standard model when $n=6$, the relevant term in the O’Raifeartaigh model Higgs potential is of order $|\phi|^{14}$. Another undesirable feature of the model in its present form is, since it only respects global SUSY, after symmetry breaking for $\phi$, since $V_{\rm min} \approx 0$, SUSY remains only very weakly broken [@witten] and so uninteresting for particle physics. Later we will propose a scenario where incorporating this model within a local supersymmetric theory can overcome all these problems, yet preserve those features attractive for solving cosmological problems. At low-temperature the O’Raifeartaigh potential has the shape and scales necessary for thermal inflation and at $T=0$ its minima can be chosen to give the scale of the presentday cosmological constant. For thermal inflation, it still must be confirmed that at high temperature, $T > a^{1/2} M$, thermal corrections to the effective potential stabilize $\phi$ at zero. Lowest order finite temperature corrections to SUSY models shift the mass as shown in [@de97], and this argument can be modified to the generalized O’Raifeartaigh models. Since $\lambda$ is tiny, the dominant high temperature corrections will come in Eq. (\[genorpot\]) from the terms $a^2 \phi^4$ and $4a^2 \phi^2 \chi^2$, which lead to high-T terms $\sim a^2T^2 \phi^2$ and $\sim a^2 T^2 \chi^2$. Thus, for $T> a^{1/2}M$ the minimum of the effective potential will be as desired at $<\phi>_T = < \chi>_T = <\eta>_T = 0$. It is interesting to note that independent of the thermal inflation problem, for an appropriate choice of scales, the potential Eq. (\[genorpot\]) can be implemented just to address the cosmological constant problem. In particular, the minimum scale necessary to obtain adequate vacuum energy is $M \stackrel{>}{\sim} \Lambda_{QCD}$. An interesting case is when $M \sim 10^3 {\rm GeV}$, the electroweak scale, where for the simplest nonrenormalizable potential $n=2$, $$V(\phi_{min} = M) = \lambda^2 10^{16} {\rm eV}^4 ,$$ which is at the scale of $\rho_{\Lambda}$ for $\lambda \sim 10^{-13}$. Moreover independent of $\lambda$, at the minimum $m_{\phi} \equiv \sqrt{V''(\phi_{min})} \approx 10^{-(3-5)} {\rm eV}$. This is an interesting scale as it is in the neighborhood of the neutrino mass mixing parameters. After thermal inflation, once the flaton $\phi$ is near its minimum, it will oscillate and thereby enter a reheating phase similar to that after supercooled inflation. The particle production history that develops has the same range of possibilities and outcomes as studied in other thermal inflation works [@Lyth:1995hj; @Lyth:1995ka; @Lazarides:ja; @Barreiro:1996dx; @Asaka:1999xd]. For example, if $\phi$ is a gauge singlet as in Eq. (\[genor\]), then reheating will create $\phi$-bosons. Also, $\phi$ can couple to charged scalars which can mediate decay into gauge particles. On the other hand, it is possible to easily generalize our O’Raifeartaigh models by letting $\phi$ be in the adjoint representation of some gauge group G, while keeping $\chi$ and $\eta$ as singlets of G. Thus, making the replacement $\phi^{p} \rightarrow {\rm Tr} (A^{p})$, a nonvanishing VEV for $A$ can break G to a set of degenerate minima, although gravity will lift the degeneracy (see below). For example for G = SU(N), $\langle A \rangle$ can be diagonalized by a SU(N) transformation so G may break to subgroups of the form $$H= \prod_i SU(N_i) \times U^p(1) ,$$ where $\sum_i(N_i-1) + p = N-1$, i.e., H has the same rank as G. This form of O’Raifeartaigh models has more possibilities of dissipating the vacuum energy. The O’Raifeartaigh type models we have been discussing up to now have global SUSY. In this case, the symmetry breaking considered above does not lead to SUSY breaking at scales of interest to particle physics, since the vacuum energy at the minimum is essentially zero. The full theory is expected to start off locally supersymmetric, thus be a supergravity theory. It is well known that global SUSY models can have many degenerate minima as long as SUSY is unbroken. SUGRA lifts this degeneracy [@Weinberg:id] and only one minimum can have zero energy, with the others having negative energy. These results discussed in [@Weinberg:id] were explicitly stated to exclude models of the O’Raifeartaigh [@O'Raifeartaigh:pr] and Fayet-Iliopoulos [@fi] type. If O’Raifeartaigh potentials are included, they break SUSY and make positive contributions to vacuum energy. It is thus quite possible that one of the negative vacuum energy SUGRA minima receives an additional positive vacuum energy contribution from the O’Raifeartaigh sector. Thus, while both the (+) and ($-$) contributions are large, the [*residual vacuum energy*]{} can be small and positive. This could be the true vacuum energy of the universe, and so explain the observed cosmological constant. For example consider the parameters necessary for near balancing vacuum contributions at a scale relevant to particle physics SUSY symmetry breaking, $\sim 10^{3} {\rm GeV}$. For the scale considered in Eq. (\[scales\]), $M \sim 10^{10-11} {\rm GeV}$, for $n=2$ to obtain $V_{\rm min}^{1/4} \stackrel{<}{\sim} 10^3 {\rm GeV}$, it requires $\lambda \stackrel{<}{\sim} 10^{-5}$, which is a realistic value. It remains a model building challenge to realize this effect through a natural mechanism. It appears the inclusion of O’Raifeartaigh superpotentials in the full SUGRA has interest for both particle physics and cosmology. While breaking SUSY adequately to generate potentially interesting phenomenological particle spectra, the O’Raifeartaigh potential can also shift the vacuum to a small positive value, generating the cosmological constant and from our above treatment, the same model can solve the moduli problem by permitting realization of thermal inflation. This scenario is promising and it seems worth further developing toward a realistic model. An initial step is to understand the origin of such models from fundamental theories. It is known that various compactifications of string theory have a number of light scalar singlets in their spectrum. For instance many models obtained from type IIB strings via orbifolding AdS$_5 \times$ S$^5$ lead to such scalars. The form of the superpotential is certainly model dependent. For example, it will depend on the initial string theory, or more generally the initial region of parameter space in M-theory, and details of the compactification. However, the occurrence of scalars are generic, so O’Raifeartaigh potentials can naturally arise in SUGRA and thus lead to our cosmology. To summarize, in this Letter we have shown that generalized O’Raifeartaigh models can have powerful implications both for cosmology and uniting cosmology with particle physics. Within the compact structure of these models, we have shown that they can solve the moduli problem and potentially lead to a solution of the cosmological constant problem. In an attempt to unify the symmetry breaking necessary to solve these cosmological problems with that necessary to break SUSY in particle physics models, a new interpretation of the presentday accelerating universe emerges providing dark matter and a “balanced” residual vacuum energy. This is an intriguing coincidence of solutions, given that O’Raifeartaigh type models may arise generically from fundamental theories. While we believe our scenario is provocative, more work needs to be done if it is to be developed into a completely satisfying model. Some way to avoid fine tuning of positive and negative contributions to the vacuum energy, or at least the renormalization of the fine tuning parameters (This is already provided for in the superpotential above the SUSY scale.) would be a important step. Another avenue to follow would be to develop a similar scenario for Fayet-Iliopoulos [@fi] D-term SUSY breaking. We thank Thomas Binoth and Michael Kr[ä]{}mer for a useful discussion. 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--- abstract: 'We use metastable NaCl-structure Ti$_{0.5}$Al$_{0.5}$N alloys to probe effects of configurational disorder on adatom surface diffusion dynamics which control phase stability and nanostructural evolution during film growth. First-principles calculations were employed to obtain potential energy maps of Ti and Al adsorption on an ordered TiN(001) reference surface and a disordered Ti$_{0.5}$Al$_{0.5}$N(001) solid-solution surface. The energetics of adatom migration on these surfaces are determined and compared in order to isolate effects of configurational disorder. The results show that alloy surface disorder dramatically reduces Ti adatom mobilities. Al adatoms, in sharp contrast, experience only small disorder-induced differences in migration dynamics.' author: - 'B. Alling' - 'P. Steneteg' - 'C. Tholander' - 'F. Tasnádi' - 'I. Petrov' - 'J. E. Greene' - 'L. Hultman' title: 'Effects of configurational disorder on adatom mobilities on Ti$_{1-x}$Al$_{x}$N(001) surfaces' --- Thin film growth is a complex physical phenomenon controlled by the interplay of thermodynamics and kinetics. This complexity facilitates the synthesis of metastable phases, such as Ti$_{1-x}$Al$_{x}$N alloys, which are not possible to obtain under equilibrium conditions and broaden the range of available physical properties in materials design. Fundamental understanding of elementary growth processes, such as adatom diffusion, governing nanostructural and surface morphological evolution during thin film growth can only be developed by detailed studies of their dynamics at the atomic scale. Research has mostly been carried out using elemental metals, as reviewed in refs. [@Jeong1999; @Antczak2007]. Much less is known about the atomic-scale dynamics of compound surfaces, and particularly little about complex, configurationally disordered, pseudobinary alloys which are presently replacing elemental and compound phases in several commercial applications. Kodambaka et al. [@Kodambaka2000; @*Kodambaka2002s; @*Kodambaka2003; @*Kodambaka2002b; @*Kodambaka2006] and Wall et al. [@Wall2004; @Wall2005] used scanning tunneling microscopy to determine surface diffusion activation energies $E_s$ on both TiN(001) and TiN(111). However, due to the vast difference between experimental and adatom hopping time scales, determining diffusion pathways requires theoretical approaches via first-principles methods that are capable of providing clear atomistic representation on the ps time scale. Gall et al. [@Gall2003surf] employed first-principles calculations to show that $E_{s}$ for Ti adatom diffusion on TiN is much lower on the (001) than the (111) surface and used this diffusional anisotropy to explain the evolution of (111) preferred orientation during growth of essentially strain-free polycrystalline films. The correspondingly large differences in chemical potentials result in Ti adatoms having higher residence times on (111) than on (001) grains. Here, we use cubic Ti$_{1-x}$Al$_{x}$N(001), a metastable NaCl-structure pseudobinary alloy, as a model system to probe the role of short-range disorder on cation diffusivities which control phase stability, surface morphology, and nanostructural evolution during growth. Ti$_{1-x}$Al$_{x}$N alloys with x $\sim 0.5$, synthesized by physical vapor deposition (PVD) far from thermodynamic equilibrium [@Hakansson1987; @*Adibi1991; @*Greczynski2011], are commercially important for high-temperature oxidation [@McIntyre1990] and wear-resistant applications [@Prengel1997; @*PalDey2003; @*Mayrhofer2003; @Horling2005]. Alloying TiN with AlN has also been shown to alter surface reaction pathways controlling film texture and nanostructure [@Horling2005; @Beckers2005; @Petrov1993; @Adibi1993b]. Unfortunately, atomic-scale understanding of the growth of these important, and more complex, materials systems is presently rudimentary as best. Surface diffusion on a metal alloy, the CuSn system in ordered configurations and in the dilute limit [@Chen2010PRL], has only recently been considered using first-principles. However, it is well known that *configurational disorder* can have large effects on the physical properties of solid solutions [@Ruban2008REV]. We employ first-principles calculations using the projector augmented wave method [@Blochl1994] as implemented in the Vienna Ab-Initio Simulation Package (VASP) [@Kresse1993], to determine the energetics of cation adsorption and diffusion on ordered TiN(001) and conÞgurationally-disordered Ti$_{0.5}$Al$_{0.5}$N(001) surfaces. Electronic exchange correlation effects are modeled using the generalized gradient approximation [@Perdew1996]. The plane wave energy cut-off is set to 400 eV. We sample the Brillouin zone with a grid of $3\times3\times1$ k-points. TiN(001), for reference, and Ti$_{0.5}$Al$_{0.5}$N(001) surfaces are modeled using slabs with four layers of $3\times3$ in-plane conventional cells with 36 atoms per layer. Calculated equilibrium lattice parameters, $a_0$, of bulk TiN, 4.255 Å, and Ti$_{0.5}$Al$_{0.5}$N, 4.179 Å, are employed. The vacuum layer above the surfaces corresponds to $5.5 a_0$. The adatoms are spin polarized, which is found to be important for Ti adatoms with its partially filled 3d-shell, but not for Al. To investigate diffusion on a configurationally-disordered surface, the Ti$_{0.5}$Al$_{0.5}$N(001) slab is modeled using the special quasirandom structure (SQS) method [@Zunger1990]. We impose a homogenous layer concentration profile and minimize the correlation functions on the first six nearest-neighbor shells for the slab as a whole. {width="90.00000%"} Convergence of diffusion barriers is tested with respect to the geometrical and numerical details of the calculations. $E_s$ results are within 0.04 eV of the converged value, partly due to error cancelation between the effects of treating Ti semicore states as core and the limited number of layers; both are of the order of 0.08 eV, but with opposite signs. Our primary focus is the observed differences in cation dynamics on the two surfaces. We begin by calculating the adsorption energy $E^{Al,Ti}_{ads}(x,y)$ for Ti and Al adatoms as a function of positions x and y on both ordered TiN(001) and disordered Ti$_{0.5}$Al$_{0.5}$N(001) surfaces, $$E_{ads}^{Al,Ti}(x,y)=E_{slab+ad}^{Al,Ti}(x,y)-E_{slab}-E_{atom}^{Al,Ti}.$$ $E^{Al,Ti}_{slab+ad}$is the energy of the slab with an adatom at $(x, y)$, $E_{slab}$ is the energy of the pure slab with no adatoms, and $E^{Al,Ti}_{atom}$ is the energy of an isolated Al or Ti atom in vacuum. We use a fine grid of sampling points, $\Delta x = \Delta y = 0.05a_0$. In each calculation, the adatom is fixed within the plane and relaxed out of plane. The upper two layers of the slab are fully relaxed, while the lower two layers are stationary. A periodic polynomial interpolation between the calculated points is used to obtain a smooth energy surface. Adsorption-energy profiles for Al and Ti atoms on TiN(001) and Ti$_{0.5}$Al$_{0.5}$N(001) surfaces are shown in Figs. 1(a)-1(d). The most favorable sites for Al adatoms on both surfaces are directly above N atoms at bulk cation positions. For Al on TiN(001), Fig. 1(a), $E^{Al}_{ads}$ is -2.54 eV. On Ti$_{0.5}$Al$_{0.5}$N(001), Fig. 1(b), $E^{Al}_{ads}$ varies from -2.39 to -1.52 eV on the bulk cation sites depending on their local environment. Ti adatoms have two stable adsorption sites: fourfold hollows, surrounded by two N and two metal atoms, and the bulk site on-top N. For TiN(001), Fig. 1(c), $E^{Ti}_{ads}$ = -3.50 eV in the hollow site and -3.27 eV above N. On the alloy surface, Fig. 1(d), $E^{Ti}_{ads}$ varies from -3.42 to -2.58 eV in the hollow sites and -3.23 to -2.67 eV in on-top sites. Al-rich environments are much less favorable for both Al and Ti adatoms as can be seen in the lower right regions of Figs. 1(b) and 1(d). The overall preferred sites for Ti on Ti$_{0.5}$Al$_{0.5}$N(001) are fourfold hollow positions with one Ti and one Al nearest metal neighbors; not two Ti atoms as might have been expected. ![ (Color online) Ti adatom diffusion paths from the center to the edge of (001) surfaces of (a) TiN, and (b) disordered Ti$_{0.5}$Al$_{0.5}$N. (c) The probability as a function of time, that Al and Ti adatoms placed at random positions in the center of a circular grain of radius $8.5 a_{0}$ have not yet reached the grain boundary on TiN(001) and Ti$_{0.5}$Al$_{0.5}$N(001).[]{data-label="fig:flowfig"}](composite4){width="45.00000%"} In order to quantify the impact of disorder on diffusion, transition state theory within a kinetic Monte Carlo approach is used to determine the mobilities of independent adatoms. The probability at each time step for a Ti or Al adatom at site $i$ to jump to site $j$ is calculated as $$\label{eq:prob} \Gamma_{ij}=\nu_0~ \mathrm{exp} \left( \frac{-\Delta E_{ij}}{k_B T} \right)$$ where $\Delta E_{ij} = \left( E_{ij} - E_i \right)$ is the difference between the adsorption energy in the local minima $i$ and at the saddle point defining the barrier height $E_{ij}$ between sites $i$ and $j$. The temperature $T$ is 800 K, a representative value for PVD growth of transition-metal nitride thin films. For convenience, we choose the attempt frequency $\nu_0$ to be the same for Ti and Al on both TiN(001) and Ti$_{0.5}$Al$_{0.5}$N(001) surfaces, but note that Al adatoms should have a slightly higher attempt frequency than Ti due to their lower mass. We determine $n_i(t)$, the probability density of finding adatoms on a given site $i$, at time $t$, corresponding to an ensemble average of a large number of individual cases. The most probable Al and Ti diffusion paths are identified by imposing a constant probability density of adatoms at the centers of circular grains with radii $8.5 a_0$, and then propagating the probability density using Eq. \[eq:prob\]. Adatoms crossing a grain boundary are not allowed to cross back. Thus, we obtain an adatom probability flow between sites i and j from the center of the grain outward, $$F_{ij}= n_i\Gamma_{ij}-n_j\Gamma_{ji}.$$ Steady-state results are plotted for Ti adatoms in Fig. 2 as grayscale intensity proportional to $F_{ij}$. Panel 2(a) shows that the flow of Ti adatoms across the ordered TiN(001) surface is symmetric and utilizes all $[110]$ paths. However, the flow of Ti atoms across Ti$_{0.5}$Al$_{0.5}$N(001), panel 2(b), simulated using periodically repeated SQSs, is almost completely absent in the energetically least favorable regions; most diffusion takes place along special paths. Such paths are indicated in Fig. 1(d) (Fig. 1(b) for Al adatoms) by white solid and dashed lines corresponding approximately to connections among the most favorable local energy minima. Next, we determine the timescales of adatom diffusion on the two surfaces. Fig. 2(c) is a plot of the probability as a function of time that adatoms, individually placed at a randomly chosen site close to the center of a circular grain, have not yet reached the grain boundary. On the pure TiN(001) surface, Al and Ti adatoms show similar behavior as the somewhat higher barriers for Al diffusion are compensated by Ti adatoms having three times as many local minima positions for a constant grain size. The striking result, however, is that Ti adatoms diffuse much slower on Ti$_{0.5}$Al$_{0.5}$N(001) than on TiN(001), while the rates for Al adatoms on the two surfaces are nearly equal. Since both Ti and Al adatoms diffuse predominantly along preferential paths on the disordered TiAlN(001) surface, the mobility differences are, in large part, explained by differences in energy profiles along these paths. ![(Color online) Adsorption energies of Al (upper graph) and Ti (lower graph) adatoms along favorable diffusion paths on ordered TiN(001) and disordered Ti$_{0.5}$Al$_{0.5}$N(001) surfaces. For the disordered alloy surface, energy profiles are plotted for both the solid and dashed paths across the SQS shown in Fig. \[fig:Esurf\].[]{data-label="fig:bar2D"}](8barriers2D){width="70.00000%"} Fig. 3 contains plots of $E^{Al,Ti}_{ads}$ relative to the most favorable adsorption site, along the preferred diffusion paths on Ti$_{0.5}$Al$_{0.5}$N indicated in Figs. 1(b) and 1(d). The arrows in Fig. 1 define the starting position for the energy-path plots in Fig. 3. Corresponding $E^{Al,Ti}_{ads}$ plots on TiN(001) are included for comparison. The calculated Al adatom diffusion activation energy on TiN(001) is $E_s = 0.47$ eV. Both the solid and dashed low-energy paths for Al on Ti$_{0.5}$Al$_{0.5}$N(001) exhibit the signature of configurational disorder with alternating deep and less-deep energy minima. However, the individual barrier heights are, in most cases, considerably lower on the disordered surface with the maximum barrier height just $1.2\times$ larger than on TiN(001). $E_s$ for Ti adatoms on TiN(001) is 0.40 eV and the smaller barrier for jumping out of the minima atop N is 0.17 eV. The individual barriers for Ti on Ti$_{0.5}$Al$_{0.5}$N(001) are similar, but a series of less favorable energy minima, combined with asymmetric jump probabilities, creates additional migration obstacles approximately 2/3 along the outlined paths. The maximum obstacles are $2.0\times$ and $1.6\times$ higher than $E_s$ on TiN(001) for the dashed and solid diffusion paths, respectively, explaining the dramatic reduction of mobility in this case. The mass difference between Al and Ti atoms (which we ignored in these calculations) affects $\nu_0$ and will further increase the mobility difference. These results illustrate the complex effects that configurational disorder can induce on surface diffusion. They also help to understand aspects of the growth behavior of Ti$_{1-x}$Al$_{x}$N thin films. Our observed increase in the residence time of Ti adatoms on Ti$_{0.5}$Al$_{0.5}$N(001) vs. TiN(001) is consistent with the experimentally reported transition in texture for polycrystalline TiN films, grown at relatively low temperatures with little or no ion irradiation, from (111) [@Greene1995] toward (001) upon alloying with AlN [@Horling2005]. In addition, the higher mobility of Al, with respect to Ti, adatoms on Ti$_{1-x}$Al$_{x}$N explains the results of Beckers et al. showing AlN enrichment in (111) and Al depletion in (001) oriented grains [@Beckers2005]. In conclusion, we have compared the adsorption-energy landscape and the migration mobilities of Ti and Al adatoms on ordered TiN(001) and disordered Ti$_{1-x}$Al$_{x}$N(001) surfaces. The configurational disorder on the alloy surface results in the formation of deep trap sites for Ti adatoms which, together with an asymmetric adsorption energy map, dramatically decreases the Ti adatom mobility. In contrast, Al adatom mobilities are nearly the same on TiN(001) and disordered Ti$_{1-x}$Al$_{x}$N(001) surfaces due to a much smaller disorder-induced spread in energy minima values and more symmetric diffusion probability distributions along the most favorable paths on the alloy surface. These results explain observed differences in preferred orientation and nanostuctural evolution during growth of polycrystalline TiN and Ti$_{1-x}$Al$_{x}$N films. We acknowledge financial support by the Swedish Foundation for Strategic Research (SSF), the Swedish Research Council (VR), and the European Research Council (ERC). The simulations were carried out using supercomputer resources provided by the Swedish national infrastructure for computing (SNIC). 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--- abstract: 'Given a Kählerian holomorphic fiber bundle $F {\hookrightarrow}M {\rightarrow}X$, whose fiber $F$ is a compact homogeneous Kähler manifold, we describe the perturbed Hermitian–Einstein equations relative to certain holomorphic vector bundles ${{\mathcal}E} {\rightarrow}M$ . With respect to special metrics on ${\mathcal}E$, there is a dimensional reduction procedure which reduces this equation to a system of equations on $X$ known as the twisted coupled vortex equations.' address: - | Department of Mathematics\ University of Illinois at Urbana–Champaign\ Urbana IL 61801 USA - | Department of Mathematics\ Eastern Illinois University\ Charleston IL 61920 USA\ and Department of Mathematics\ University of Illinois at Urbana–Champaign\ Urbana IL 61801 USA - | Department of Mathematics\ University of Illinois at Urbana–Champaign\ Urbana IL 61801 USA author: - 'Steven B. Bradlow${}^1$' - 'James F. Glazebrook' - 'Franz W. Kamber${}^2$' title: | Dimensional reduction of\ the perturbed Hermitian–Einstein equation --- Proc. in Global Anaysis, Diff. Geometry and Lie Algebras, Balkan Geometry Press, Bucharest 2001, pp. 43–62. [^1] [^2] Introduction ============ Dimensional reduction techniques are applicable to studying special solutions to partial differential equations particularly in the presence of a group action where invariant solutions are of interest. The invariant solutions may be interpreted as solutions to an associated set of equations on a lower dimensional space of orbits of the group action. However, one may ask if there is no group action, is it still possible to dimensionally–reduce the original system? A positive answer points to the study of the Hermitian–Einstein (HE) equation with respect to special metrics on holomorphic bundles together with some extra data. The purpose of this paper is to outline a construction leading to dimensional reduction of a class of equations which we call the [*perturbed Hermitian–Einstein equations*]{} (briefly, the PHE equations) on a Hermitian holomorphic vector bundle ${{\mathcal}E} {\longrightarrow}M$ where $M$ is a compact Kähler manifold. We stress that the term perturbed has here a delicate interpretation as will be apparent from the text. In fact, the PHE equations are actually more general than the HE equations because they possess an extra perturbation term. This extra term arises from the fact that in this case, $M$ is the total space of a holomorphic fiber bundle $F {\hookrightarrow}M {\longrightarrow}X$, where $X$ is a compact Kähler manifold and the fiber $F$ is a compact Kählerian homogeneous space. Now ${\mathcal E}$ as a holomorphic vector bundle is obtained via a holomorphic extension of certain holomorphic vector bundles on $M$ and is equipped with an invariant hermitian metric. This metric together with the Kobayashi form of the extension and some natural conditions on $F$, imply that the PHE equation is equivalent to a system of equations on $X$, namely the [*twisted coupled vortex equations*]{}. The overall construction, on which there are several variations, relies on results relating to the representation theory of complex semisimple Lie groups and the Bott–Borel–Weil theorem. In addition, the PHE equation can be obtained as a moment map equation. Here we will outline the general construction of [@BGKfour] leading to the twisted coupled vortex equations (cf [@BGKone] [@BGKtwo] [@GProne]). The existence theory of the solutions of such twisted coupled equations is discussed via the Hitchin–Kobayashi correspondence in [@BGKthree]. References [@AGone] [@AGtwo] contain an independent study of several aspects of this theory and focus on other questions. Some preliminaries ================== The Kähler manifold $M$ ----------------------- Let us commence by describing the compact homogeneous Kähler manifold $F$, that is, for connected complex Lie groups $G$ and $P$ with $G$ semisimple and $P \subset G$ parabolic, we set $F = G / P \cong U / K$ where $$\label{symmetric} G = {\operatorname{Hol}}(F)_e~, ~U = {\operatorname{Hol}_{{\operatorname{Iso}}}}(F)_e~, ~K = U \cap P~.$$ Furthermore, $F$ is a simply connected algebraic manifold, the groups $U$ and $K$ are connected compact Lie groups, with $U$ semisimple and $K$ the centralizer of a torus (hence $K \subset U$ has maximal rank) and any $G$–invariant hermitian metric on $F$ is a Kähler (for further details see [@BE] [@Botttwo] [@Kobtwo]). The equivariant holomorphic vector bundles on $G / P$ are homogeneous vector bundles [@Botttwo] given by representations $(\rho, V_\rho)$ of the parabolic subgroup $P$ $$\label{homogen} \rho {\mapsto}{{\mathcal}V}_\rho = G \times_P V_\rho~.$$ Let $X$ be a compact Kähler manifold and $P_G {\rightarrow}X$ a holomorphic principal $G$–bundle. The homogeneous vector bundle ${{\mathcal}V}_\rho$ extends to a holomorphic vector bundle on the associated holomorphic fiber bundle $M = P_G \times_G F = P_G / P$ by the formula $$\label{ext2} {\widetilde}{{\mathcal}V}_\rho \cong P_G \times_P V_\rho \to M = P_G / P~.$$ We call ${\widetilde}{{\mathcal}V}_\rho$ the [*canonical extension*]{} of ${{\mathcal}V}_\rho$ . With regards to the fundamental group ${\Gamma}= \pi_1(X)$, we suppose that $M$ has the structure of a generalized flat bundle [@KTone] $$\label{flatbundle} F {\hookrightarrow}M = {\widetilde}X \times_\Gamma F {\overset {\pi}{{\rightarrow}}} X ~,$$ with holonomy ${\alpha}: \Gamma \to U$ . Letting ${\omega}_F$ and ${\omega}_X$ denote the Kähler forms of $F$ and $X$ respectively, the extension to $M$ of the (invariant) Kähler form ${\omega}_F$ , is given by ${\widetilde}{{\omega}}_F = p^*{\omega}_F/{\alpha}$ , where $p : {\widetilde}X \times F {\rightarrow}F$ , is the natural projection. Then by [@BGKtwo] (Proposition $8.1$), there exists a family of Kähler metrics on $M$ with corresponding weighted Kähler forms $$\label{kform} {\omega}_{{\sigma}} = \pi^* {\omega}_X + {\sigma}{\widetilde}{{\omega}}_F ~,$$ where ${\sigma}> 0$ is a constant parameter. The bundle types of the extension on $M$ ---------------------------------------- Let ${{\mathcal}V}_{\rho_i} = U \times_K V_{\rho_i} \to F = U/K$ be homogeneous holomorphic vector bundles with canonical extensions ${\widetilde}{{\mathcal}V}_{\rho_i} {\rightarrow}M$ for $i=1, 2$ . Further, let ${\mathcal}W_i {\rightarrow}X$ be holomorphic vector bundles and set ${\mathcal}E_i = {{\pi^* {{\mathcal}W}_i} {\otimes}_{\mathbb C} {{\widetilde}{{\mathcal}V}_{\rho_i}}}$ . We consider the class of holomorphic vector bundles ${{\mathcal}E} {\rightarrow}M$ given by proper holomorphic extensions of the form $$\label{ext} {\mathbb E}~:~{0 {\rightarrow}{{\mathcal}E}_1 {\longrightarrow}{{\mathcal}E} {\longrightarrow}{{\mathcal}E}_2 {\rightarrow}0}~.$$ Such extensions are classified by the ${\operatorname{Ext}}^1$–functor (see e.g. [@Hart]) which in our case is of the form $$\label{extension5} {\operatorname{Ext}}^1_{{{\mathcal}O}_M}({{\mathcal}E}_2 , {{\mathcal}E}_1) \cong H^{0,1}(M, {{{\mathcal}H}om_{\mathbb C}( {{{\mathcal}E}_2} , {{{\mathcal}E}_1} )} ) \cong H^{0,1} (M, {{\pi^* {{\mathcal}W}} {\otimes}_{\mathbb C} {{\widetilde}{{{\mathcal}V}_{\rho}}}} )~,$$ where we set ${{\mathcal}W} = {{{\mathcal}H}om_{\mathbb C}( {{{\mathcal}W}_2} , {{{\mathcal}W}_1} )}$ and ${{\mathcal}V}_\rho = {{{\mathcal}H}om_{\mathbb C}( {{{\mathcal}V}_{\rho_2}} , {{{\mathcal}V}_{\rho_1}} )}$ . Note that in the latter case we have $\rho = \rho_1 \otimes \rho_2^*$ . For any holomorphic vector bundle ${{\mathcal}W} \to X$ and any homogeneous vector bundle ${{\mathcal}V} \to F$, there is an exact sequence derived from the Borel–Leray spectral sequence [@BGKtwo] [@HZ] : $$\label{edge} \begin{aligned} 0 &{\rightarrow}H^{0,1} (X, {{{{\mathcal}W}} {\otimes}_{\mathbb C} {{{\mathcal}H}^0 (F, {{\mathcal}V}}}) ) {\overset {\pi^*}{{\longrightarrow}}} H^{0,1} (M, {{\pi^* {{\mathcal}W}} {\otimes}_{\mathbb C} {{\widetilde}{{\mathcal}V}}}) {\overset {\Phi}{{\longrightarrow}}} H^0 (X, {{{{\mathcal}W}} {\otimes}_{\mathbb C} {{{\mathcal}H}^{0,1} (F, {{\mathcal}V}}})) {\rightarrow}\\ &{\overset {d_2}{{\longrightarrow}}} H^{0,2} (X, {{{\mathcal}W} {\otimes}_{\mathbb C} {{{\mathcal}H}^{0}(F,{{\mathcal}V})}}) {\overset {\pi^*}{{\longrightarrow}}} H^{0,2}(M, {{\pi^* {{\mathcal}W}} {\otimes}_{\mathbb C} {{\widetilde}{{\mathcal}V}}})~, \end{aligned}$$ where $\pi^*~, ~\Phi$ are the edge homomorphisms. In the flat case we can say more about the edge map $\Phi$  [@BGKfour]. Here we use the notation ${\mathbf H}^q (F,{{\mathcal}V})$ to indicate the fact that the fiber cohomologies ${{\mathcal}H}^q (F,{{\mathcal}V})$ are flat holomorphic bundles. \[edge2\]  Suppose that the fiber bundle $F {\hookrightarrow}M \to X$ is flat, with holonomy ${\alpha}: \Gamma \to U$ . For any holomorphic vector bundle ${{\mathcal}W} \to X$ and any equivariant vector bundle ${{\mathcal}V} \to F$, we have a short exact sequence $$\label{edge3} \begin{aligned} 0 &{\rightarrow}H^{0,1} (X, {{{\mathcal}W} {\otimes}_{\mathbb C} {{\mathbf H}^0 (F, {{\mathcal}V}}}) ) {\overset {\pi^*}{{\longrightarrow}}} H^{0,1} (M, {{\pi^* {{\mathcal}W}} {\otimes}_{\mathbb C} {{\widetilde}{{\mathcal}V}}}) \\ &{\overset {\Phi}{{\longrightarrow}}} H^0 (X, {{{{\mathcal}W}} {\otimes}_{\mathbb C} {{\mathbf H}^{0,1}(F, {{\mathcal}V}}})) {\rightarrow}0~. \end{aligned}$$ This has the following consequence (cf [@BGKtwo] Proposition $7.1$). \[ExtProp\] Suppose that ${{\mathcal}V}$ satisfies the vanishing condition $$H^0 (F, {{\mathcal}V}) = 0~.$$ Then the following hold$~:$ - The holomorphic extensions of the form ${\eqref}{ext}$ are classified by $${\operatorname{Ext}}^1_{{{\mathcal}O}_M} ({{\mathcal}E}_2, {{\mathcal}E}_1) \cong H^{0,1}(M, {{\pi^* {{\mathcal}W}} {\otimes}_{\mathbb C} {{\widetilde}{{\mathcal}V}}} ) {\overset {\Phi} {\cong}} H^0 (X, {{{\mathcal}W} {\otimes}_{\mathbb C} {{\mathbf H}^{0,1} (F, {{\mathcal}V})}})~,$$ and we have $$H^0(M, {{\pi^* {{\mathcal}W}} {\otimes}_{\mathbb C} {{\widetilde}{{\mathcal}V}}} ) = 0~,$$ for any holomorphic vector bundle ${\mathcal}W$ on $X$ . - If $\Gamma = \pi_1 (X)$ acts trivially on $H^{0,1} (F, {{\mathcal}V})$, then the bundle ${{\mathcal}H}^{0,1} (F, {{\mathcal}V})$ of fiber cohomologies is holomorphically trivial and we have the Kunneth formula $$H^{0,1}(M, {{\pi^* {{\mathcal}W}} {\otimes}_{\mathbb C} {{\widetilde}{{\mathcal}V}}} ) {\overset {\Phi} {\cong}} {{H^0 (X, {{\mathcal}W})} {\otimes}_{\mathbb C} {H^{0,1} (F, {{\mathcal}V})}}~.$$ The Kobayashi form of the extension ----------------------------------- In order to describe the representative of the extension class one needs to construct a right inverse to the edge homomorphism $\Phi$ in Proposition [ \[edge2\]]{}. This is done in [@BGKfour] and we summarize the necessary results in the proposition below. It will be useful to keep in mind the following diagram of holomorphic maps $$\label{maps} \begin{CD} {\widetilde}X \times F @>{{\tilde}\pi}>> {\widetilde}X \\ @VVqV @VV{q_0}V \\ M = {\widetilde}X \times_{\Gamma} F @>{\pi}>> X \\ \end{CD}$$ and the relevant cohomology groups as determined by the diagram $$\label{automorphic} \begin{CD} H^{0,1} ({\widetilde}X \times F, {{{\tilde}q^* {{\mathcal}W}} {\otimes}_{\mathbb C} {p^* {{\mathcal}V}_\rho}})^{\Gamma} @>{{\widetilde}\Phi}>> {{\operatorname{Hom}}_{\Gamma}( {H^{0,1}(F, {{\mathcal}V}_\rho)^*} , {H^0 ({\widetilde}X, q_0^*{{\mathcal}W})} )} \\ @A{\cong}A{q^*}A @A{\cong}A{q_0^*}A \\ H^{0,1}(M, {{\pi^* {{\mathcal}W}} {\otimes}_{\mathbb C} {{\widetilde}{{\mathcal}V}_\rho}}) @>{\Phi}>> H^0 (X, {{{\mathcal}W} {\otimes}_{\mathbb C} {{\mathbf H}^{0,1}(F, {{\mathcal}V}_\rho)}})~. \\ \end{CD}$$ \[kobform\]With regards to the edge homomorphism $\Phi$ in [edge3]{}, we have the following : - For a given ${\beta}_0 \in H^0 (X, {{{\mathcal}W} {\otimes}_{\mathbb C} {{\mathbf H}^{0,1}(F, {{\mathcal}V}_\rho) }})$ there exists a canonical class $[\bar{\beta}] \in H^{0,1}(M, {{\pi^* {{\mathcal}W}} {\otimes}_{\mathbb C} {{\widetilde}{{\mathcal}V}_\rho}})$ such that $\Phi ([\bar{\beta}]) = {\beta}_0$ . - The Kobayashi form ${\beta}\in H^{0,1}(M, {{{\mathcal}H}om_{\mathbb C}( {{{\mathcal}E}_2} , {{{\mathcal}E}_1} )}) \cong H^{0,1} (M, {{\pi^* {{\mathcal}W}} {\otimes}_{\mathbb C} {{\widetilde}{{\mathcal}V}_\rho}})$ of the holomorphic extension [ext]{} $${\mathbb E}~:~ 0 {\rightarrow}{{\pi^* {{\mathcal}W}_1} {\otimes}_{\mathbb C} {{\widetilde}{{\mathcal}V}_{\rho_1}}} {\longrightarrow}{{\mathcal}E} {\longrightarrow}{{\pi^* {{\mathcal}W}_2} {\otimes}_{\mathbb C} {{\widetilde}{{\mathcal}V}_{\rho_2}}} {\rightarrow}0~,$$ decomposes as $$[{\beta}] = \pi^* [{\beta}_X] + [\bar {\beta}]~,$$ where $[{\beta}_X] \in H^{0,1} (X, {{{{\mathcal}W}} {\otimes}_{\mathbb C} {{\mathbf H}^0 (F, {{\mathcal}V}_\rho)}})$ and $\bar {\beta}$ is the right inverse of ${\beta}_0$ . Thus we have $\Phi ([{\beta}]) = \Phi ([\bar {\beta}]) = {\beta}_0$ . Our discussion of extension classes on $M$ leads naturally to the following definitions of holomorphic objects on $X$ : - A [*holomorphic quadruple*]{}  $Q = ({{\mathcal}W}_1, {{\mathcal}W}_2, [{\beta}_X], {\beta}_0)$ is given by two holomorphic vector bundles ${{\mathcal}W}_i \to X$, together with cohomology classes $$[{\beta}_X] \in H^{0,1} (X, {{{{\mathcal}W}} {\otimes}_{\mathbb C} {{\mathbf H}^0 (F, {{\mathcal}V}_\rho )}}) ~\qquad~,~\qquad~ {\beta}_0 \in H^0 (X, {{{{\mathcal}W}} {\otimes}_{\mathbb C} {{\mathbf H}^{0,1} (F, {{\mathcal}V}_\rho )}})~.$$ A holomorphic quadruple of the form $Q = ({{\mathcal}W}_1, {{\mathcal}W}_2, 0, 0)$, that is $[{\beta}_X] = 0~, ~{\beta}_0 = 0$ is called [*degenerate*]{} (see [@BGKfour]). - A [*twisted holomorphic triple*]{}  $T_0 = ({{\mathcal}W}_1, {{\mathcal}W}_2, {\beta}_0)$ is given by two holomorphic vector bundles ${{\mathcal}W}_i \to X$, together with a holomorphic homomorphism $${\beta}_0 ~:~ {\mathbf H}^{0,1}(F, {{\mathcal}V}_\rho)^* {\longrightarrow}{{\mathcal}W} = {{{\mathcal}H}om_{\mathbb C}( {{{\mathcal}W}_2} , {{{\mathcal}W}_1} )}~.$$ If a basis $\{ {\hat \eta}_j \}$ of $H^{0,1}(F,{{\mathcal}V}_\rho)$ is specified, we denote by $T_0$ also the $k$–triple $T_0 = ({{\mathcal}W}_1, {{\mathcal}W}_2, {\tilde}\phi)$, where ${\tilde}\phi = ({{\tilde}\phi}_j)_{j = 1, \ldots, k}$ are the coefficients in the expansion of ${\beta}_0$ (holomorphic triples are considered in [@BGPone] [@BGKthree]). - A [*twisted $1$–cohomology triple*]{}  $T_1 = ({{\mathcal}W}_1, {{\mathcal}W}_2, [{\beta}_X])$ is given by two holomorphic vector bundles ${{\mathcal}W}_i \to X$, together with a cohomology class $$[{\beta}_X] \in H^{0,1} (X, {{{\mathcal}W} {\otimes}_{\mathbb C} {{\mathbf H}^0 (F, {{\mathcal}V}_\rho)}})~,$$ classifying a holomorphic extension on $X$ of the form $$\label{extension6} {\mathbb W}~:~ 0 {\rightarrow}{{{{\mathcal}W}_1} {\otimes}_{\mathbb C} {{\mathbf H}^0 (F, {{\mathcal}V}_\rho)}} {\longrightarrow}{\widetilde}{{\mathcal}W} {\longrightarrow}{{\mathcal}W}_2 {\rightarrow}0~$$ ($1$–cohomology triples are considered in [@BGPtwo]  [@DUW] ). Each of the above classes plays a significant role in the dimensional reduction theory [@BGKfour]. Here we will restrict attention mainly to holomorphic triples and proceed to state a result which makes use of Corollary [ \[ExtProp\]]{} and provides the explicit form of the extension class $\beta$ . \[BetaLemma2\][@BGKfour]  Suppose that the homogeneous vector bundle ${{\mathcal}V}_\rho$ satisfies the vanishing condition in Corollary ${~\ref{ExtProp}}$. - Relative to a basis ${\hat \eta}_j = [\eta_j]$ of $H^{0,1} (F, {{\mathcal}V}_\rho)$, the holomorphic triples $T_0 = ({{\mathcal}W}_1, {{\mathcal}W}_2, {\beta}_0)$ are of the form $$q_0^* ~{\beta}_0 = {\sum_{j=1}^{k} ~{{\tilde}\phi_j {\otimes}\hat \eta_j}}~,$$ where ${\tilde}\phi = ({\tilde}\phi_j)_{j = 1, \ldots, k} \in H^0 ({\widetilde}X, ~q_0^*{{\mathcal}W})^k$ is a $k$–tuple of holomorphic sections. - There is a one–one–correspondence between $k$–tuples ${\tilde}\phi = ({\tilde}\phi_j)_{j = 1, \ldots, k}$ of holomorphic sections and extension classes $$[{\beta}] \in {\operatorname{Ext}}^1_{{{\mathcal}O}_M}({{\mathcal}E}_2 , {{\mathcal}E}_1) \cong H^{0,1}(M, {{\pi^* {{\mathcal}W}} {\otimes}_{\mathbb C} {{\widetilde}{{\mathcal}V}}} ) \cong H^0 (X, {{{\mathcal}W} {\otimes}_{\mathbb C} {{\mathbf H}^{0,1} (F, {{\mathcal}V}_\rho)}})~,$$ given by $$q^* {\beta}= {\sum_{j=1}^{k} ~{{{{\tilde}\pi^* {\tilde}\phi_j} {\otimes}{p^* {\eta}_j}}}}~.$$ The perturbation terms associated to a holomorphic extension ============================================================ So far we have described how extensions ${\mathbb E}$ in [ext]{} are classified by $[{\beta}] \in H^{0,1}(M, {{\pi^* {{\mathcal}W}} {\otimes}_{\mathbb C} {{\widetilde}{{\mathcal}V}_\rho}})$ and thanks to Lemma [ \[BetaLemma2\]]{} we have an explicit form of $\beta$ which will be instrumental in the reduction procedure. Owing to the generality of our construction, certain technical features which did not arise in [@BGKone] [@BGKtwo] now become apparent and lead to the formulation of the PHE equation. Henceforth we assume some familiarity with the differential geometry of operators on Kähler manifolds (references are [@Kobthree] [@Weil]). In particular, ${{\Lambda_{\sigma}}}$ will denote the operator of contraction with respect to the Kähler form $\omega_{{\sigma}}$ in [kform]{}. Integration over the fiber -------------------------- We define integration over the fiber in the flat fiber bundle $M \to X$ , $$\pi_* = {\int_F ~{\operatorname{Tr}}}~:~ A^0 (M, ~{{{{{\mathcal}E}nd_{\mathbb C}( {\pi^* {{\mathcal}W}} )}} {\otimes}_{\mathbb C} {{{{\mathcal}E}nd_{\mathbb C}( {{\widetilde}{{\mathcal}V}_\rho} )}}} ) {\longrightarrow}{{\operatorname{End}}_{\mathbb C}({X, {{\mathcal}W}})}~,$$ at the level of $\Gamma$–invariant sections $$\pi_* = {\int_F ~{\operatorname{Tr}}}~:~ A^0 ({\widetilde}X \times F,~ {{{{{\mathcal}E}nd_{\mathbb C}( {{\tilde}q^* {{\mathcal}W}} )}} {\otimes}_{\mathbb C} {{{{\mathcal}E}nd_{\mathbb C}( {p^* {{\mathcal}V}_\rho} )}}} )^{\Gamma} {\longrightarrow}{{\operatorname{End}}_{\mathbb C}({{\widetilde}X, ~q_0^* {{\mathcal}W}})}^{\Gamma}~$$ by the formula $$\label{fibint1} {\int_F ~{\operatorname{Tr}}}~{{{\tilde}\pi^* {\varphi}} {\otimes}{p^* \psi}} = { \frac{1}{{\operatorname{Vol}}(F)} } ~{\tilde}{\varphi}~{\int_F ~{\operatorname{Tr}}}(\psi) ~{\operatorname{dvol}}_F = \frac{1}{\ell! ~{\operatorname{Vol}}(F)} ~{\tilde}{\varphi}~{\int_F ~{\operatorname{Tr}}}(\psi) ~{\omega}_F^\ell~.$$ This is well–defined, since the volume form ${\operatorname{dvol}}_F = { \frac{{\omega}_F}{\ell!} }$ is $U$–invariant. Here ${\operatorname{Tr}}$ is induced by the normalized trace on the fiber, that is the trace on the bundle ${{{\mathcal}E}nd_{\mathbb C}( {{{\mathcal}V}_\rho} )}$ . We remark that the flatness of the fiber bundle is not necessary in order to define integration over the fiber. Observe that the ‘basic’ terms ${\beta}_X$ and ${\beta}_0$ both involve data on the fiber, holomorphic homomorphisms in the case of ${\beta}_X$ and holomorphic extensions in the case of ${\beta}_0$ . There are particular curvature terms ${{\Lambda_{\sigma}}}({\beta}\wedge {\beta}^*)$ and ${{\Lambda_{\sigma}}}({\beta}^* \wedge {\beta})$ which depend on hermitian metrics $h_i$ on ${{\mathcal}W}_i$ and the fixed invariant hermitian metrics $k_i$ on the homogeneous bundles ${{\mathcal}V}_{\rho_i}$ . \[betadecomp\]  - The endomorphisms  $-\iota {\int_F ~{\operatorname{Tr}}}~{{\Lambda_{\sigma}}}({\beta}\wedge {\beta}^*) \in {{\operatorname{End}}_{\mathbb C}({{{\mathcal}W}_1})}$  and  $\iota {\int_F ~{\operatorname{Tr}}}~{{\Lambda_{\sigma}}}({\beta}^* \wedge {\beta}) \in {{\operatorname{End}}_{\mathbb C}({{{\mathcal}W}_2})} $ are non–negative hermitian endomorphisms of ${{\mathcal}W}_i~.$ - If ${\int_F ~{\operatorname{Tr}}}~{{\Lambda_{\sigma}}}({\beta}\wedge {\beta}^*) = 0$ or ${\int_F ~{\operatorname{Tr}}}~{{\Lambda_{\sigma}}}({\beta}^* \wedge {\beta}) = 0$ , then ${\beta}= 0$ . - For ${\beta}= \pi^* {\beta}_X + \bar{\beta}$ as in Proposition ${~\ref{kobform}}$, we have $$\begin{aligned} {{\Lambda_{\sigma}}}({\beta}\wedge {\beta}^*) &= {{\Lambda_{\sigma}}}(\pi^* {\beta}_X \wedge \pi^* {\beta}_X^*) + {{\Lambda_{\sigma}}}(\bar{\beta}\wedge \bar{\beta}^*)~, \\ {{\Lambda_{\sigma}}}({\beta}^* \wedge {\beta}) &= {{\Lambda_{\sigma}}}(\pi^* {\beta}_X^* \wedge \pi^* {\beta}_X) + {{\Lambda_{\sigma}}}(\bar{\beta}^* \wedge \bar{\beta})~. \end{aligned}$$ \[vortobstr\] The [*perturbation terms*]{} ${{\mathfrak}d}_i ({\beta}, {\sigma})$ associated to ${\beta}$ are defined by : $$\begin{aligned} {{\mathfrak}d}_1 ({\beta}, {\sigma}) &= {{\Lambda_{\sigma}}}({\beta}\wedge {\beta}^*) - {{\pi^* ~{\int_F ~{\operatorname{Tr}}}~{{\Lambda_{\sigma}}}({\beta}\wedge {\beta}^*)} {\otimes}{{\widetilde}{\mathbf I}_{1}}}~, \\ {{\mathfrak}d}_2 ({\beta}, {\sigma}) &= {{\Lambda_{\sigma}}}({\beta}^* \wedge {\beta}) - {{\pi^* ~{\int_F ~{\operatorname{Tr}}}~{{\Lambda_{\sigma}}}({\beta}^* \wedge {\beta})} {\otimes}{{\widetilde}{\mathbf I}_{2}}}~. \end{aligned}$$ The following properties are derived directly from the definition : - ${\int_F ~{\operatorname{Tr}}}~{{\mathfrak}d}_i ({\beta}, {\sigma}) = 0~,$ that is the perturbation terms vanish under integration over the fiber. - For ${\beta}= \pi^* {\beta}_X + \bar{\beta}$ as above, we have ${{\mathfrak}d}_i ({\beta}, {\sigma}) = {{\mathfrak}d}_i (\pi^* {\beta}_X, {\sigma}) + {{\mathfrak}d}_i (\bar{\beta}, {\sigma})~,~ i = 1, 2~.$ The linear maps $\lambda_i$ --------------------------- Relative to an orthonormal basis $\{ \hat \eta_j \}$ of $H^{0,1}(F, {{\mathcal}V}_\rho)$, we define linear homomorphisms $$\lambda_i~:~ {{\operatorname{End}}_{\mathbb C}({H^{0,1}(F, {{\mathcal}V}_\rho)})} \to {{\operatorname{End}}_{\mathbb C}({ {{\mathcal}V}_{\rho_i} })}~,$$ by the formulas $$\label{lambdaij} \lambda_1 (\eta_{ij}) = \frac{1}{\iota} ~{\Lambda_{F}} (\eta_i \wedge \eta_j^*)~, ~\qquad~ \lambda_2 (\eta_{ij}) = \iota ~{\Lambda_{F}} (\eta_i^* \wedge \eta_j)~,$$ where $\eta_{ij}$ is the standard basis of ${{\operatorname{End}}_{\mathbb C}({H^{0,1}(F, {{\mathcal}V})})} \cong {\operatorname{\mathfrak {gl}}}(k, {\mathbb C})$ . Since $\eta_{ij}^* = \eta_{ji}$ and ${\Lambda_{F}} (\eta_i \wedge \eta_j^*)^* = - ~{\Lambda_{F}} (\eta_j \wedge \eta_i^*)$ , we have $\lambda_i (\xi)^* = \lambda_i (\xi^*)$ . There are induced maps on sections $$\begin{aligned} {\lambda_i}_*~&:~A^0 (X, ~{{{{{\mathcal}E}nd_{\mathbb C}( {{{\mathcal}W}_i} )}} {\otimes}_{\mathbb C} {\mathbf{End}_{{\mathbb C}}(H^{0,1}(F, {{\mathcal}V}_\rho) )}} ) {\longrightarrow}A^0 (X, ~{{{{{\mathcal}E}nd_{\mathbb C}( {{{\mathcal}W}_i} )}} {\otimes}_{\mathbb C} {\mathbf{End}_{{\mathbb C}}({{\mathcal}V}_{\rho_i})}} )~, \\ \pi^*~&:~A^0 (X, ~{{{{{\mathcal}E}nd_{\mathbb C}( {{{\mathcal}W}_i} )}} {\otimes}_{\mathbb C} {\mathbf{End}_{{\mathbb C}}({{\mathcal}V}_{\rho_i})}}) {\longrightarrow}A^0 (M, ~{{{{{\mathcal}E}nd_{\mathbb C}( {\pi^* {{\mathcal}W}_i} )}} {\otimes}_{\mathbb C} {{{{\mathcal}E}nd_{\mathbb C}( {{\widetilde}{{\mathcal}V}_{\rho_i}} )}}} )~. \end{aligned}$$ The main advantage of the maps $\lambda_i$ consists in the fact that they allow us to express forms like ${{\Lambda_{\sigma}}}(\bar{\beta}\wedge \bar{\beta}^*)$ in terms of basic data. \[reduction\]  The forms ${{\Lambda_{\sigma}}}(\bar{\beta}\wedge \bar{\beta}^*)$ and ${{\Lambda_{\sigma}}}(\bar{\beta}^* \wedge \bar{\beta})$ are determined by the formulas $$\label{reduction2} \begin{aligned} {{\Lambda_{\sigma}}}(\bar{\beta}\wedge \bar{\beta}^*) &= \frac{\iota}{{\sigma}} ~\pi^* ~{\lambda_1}_* ({\beta}_0 \wedge {\beta}_0^*)~, \\ {{\Lambda_{\sigma}}}(\bar{\beta}^* \wedge \bar{\beta}) &= - ~\frac{\iota}{{\sigma}} ~\pi^* ~{\lambda_2}_* ({\beta}_0^* \wedge {\beta}_0)~. \end{aligned}$$ For ${\beta}= \pi^* {\beta}_X + \bar{\beta}~, ~{\beta}_0 = \Phi (\bar{\beta})$ the perturbation terms ${{\mathfrak}d}_i ({\beta}, {\sigma})$ are now seen to be of the form $${{\mathfrak}d}_i ({\beta}, {\sigma}) = \pi^* {{\mathfrak}d}_i ({\beta}_X) + \frac{1}{{\sigma}} ~\pi^* {{\mathfrak}d}_i ({\beta}_0)~.$$ Observe in particular that the scaling parameter ${\sigma}$ for the fiber metric appears as a parameter in the perturbation terms and this is the reason for the terminology. The perturbed Hermitian–Einstein equation ========================================= Recalling the holomorphic vector bundle ${\mathcal}E {\rightarrow}M$ in [ext]{}, our next step is to describe an integrable unitary (metric) connection on ${\mathcal}E$ and compute its curvature. Then we will combine this with the background material so far established and proceed to our intended system of equations. The connection on ${\mathcal}E {\rightarrow}M$ and its curvature ---------------------------------------------------------------- The holomorphic vector bundle ${\mathcal}E$ admits a smooth decomposition ${E} = {E}_1 \oplus {E}_2$ . Relative to this decomposition, we denote by $\mathbf h$ an invariant hermitian metric on ${\mathcal}E$ of the form $$\label{special} \mathbf h = {\mathbf h}_1 \oplus {\mathbf h}_2~,$$ where ${\mathbf h}_i = h_i' {\otimes}{\tilde}k_i$ on ${{\mathcal}E}_i = {{\pi^* {{\mathcal}W}_i} {\otimes}_{\mathbb C} {{\widetilde}{{\mathcal}V}_{\rho_i}}}$ is given by invariant $(basic)$ hermitian metrics $h_i'$ on $\pi^* {{\mathcal}W}_i$ and the extension ${\tilde}k_i$ of $U$–invariant Hermitian–Einstein metrics $k_i$ on ${{\mathcal}V}_{\rho_i}~.$ Relative to the smooth decomposition of ${\mathcal}E$ and the hermitian metric $\mathbf h$ , the unitary integrable connection $\mathbf A$ on $({{\mathcal}E}, \mathbf h)$ is given by $$\label{connext1} \mathbf A = \bmatrix ~\mathbf A_1 & {\beta}\\ -{\beta}^* & \mathbf A_2~ \endbmatrix ~,$$ where ${\mathbf A}_i$ are the Chern connections of $({{\mathcal}E}_i, \mathbf h_i)~,$ and ${\beta}\in A^{0,1}(M, {{{\mathcal}H}om_{\mathbb C}( {{{\mathcal}E}_2} , {{{\mathcal}E}_1} )})$ is the representative of the extension class in ${\operatorname{Ext}}^1_{{{\mathcal}O}_M}({{\mathcal}E}_2 , {{\mathcal}E}_1)$ relative to [ext]{}. A routine calculation (cf e.g. [@Kobthree]) shows that the curvature of $\mathbf A$ has the form $$\label{curvext1} F_{\mathbf h} = F_{\mathbf A} = \begin{bmatrix} ~F_{\mathbf h_1} - {\beta}\wedge {\beta}^* & D' {\beta}\\ - D'' {\beta}^* & F_{\mathbf h_2} - {\beta}^* \wedge {\beta}~ \end{bmatrix}~,$$ where $$\label{dconnect} D: A^1 (M, {{{\mathcal}H}om_{\mathbb C}( {{{\mathcal}E}_1} , {{{\mathcal}E}_2} )}) {\rightarrow}A^2 (M, {{{\mathcal}H}om_{\mathbb C}( {{{\mathcal}E}_1} , {{{\mathcal}E}_2} )})~,$$ is constructed from $\mathbf{A}_1$ and $\mathbf{A}_2$ in the standard way. Further, we let $D'$ and $D''$ denote the $(1,0)$ and $(0,1)$ components of $D$ respectively, so that $D = D' \oplus D''$ . Now for the integrable unitary connection $A_i$ on $({{\mathcal}W}_i, h_i)$ , and the Hermitian–Einstein metric connection ${{\widetilde}A}_i$ on $({\widetilde}{{\mathcal}V}_{\rho_i}, {\tilde}k_i)$, we have $$\mathbf A_i = \pi^*A_i {\otimes}{\widetilde}{\mathbf I}_{i} + {\mathbf I}_{i} {\otimes}{{\widetilde}A}_i~,$$ and the corresponding curvature form of type $(1,1)$ can be expressed as $$\label{curvext2} F_{\mathbf h_{i}} = F_{\pi^* h_{i}} {\otimes}{\widetilde}{\mathbf I}_{i} + {\mathbf I}_{i} {\otimes}F_{{\tilde}k_i}~,$$ where ${\mathbf I}_i = \pi^* \mathbf I_{{{\mathcal}W}_i}$ and ${\widetilde}{\mathbf I}_{i} = {\mathbf I}_{{\widetilde}{{\mathcal}V}_{\rho_i}}$ . Next, we define the slope of ${\mathcal}E$ relative to $\omega_{{\sigma}}$ for ${\sigma}>0$, by $$\label{muslope} \lambda = {{\mu_{{\mathbb E}} ({\sigma})}} = {{\mu_{{\mathcal}E} ({\sigma})}} = \frac{{{\deg_{\sigma} ({\mathcal}E)}}}{{\operatorname{rank}}({\mathcal}E)}~. $$ We recall that for a compact Kähler manifold $(M, \omega_M)$ and a holomorphic vector bundle ${\mathcal}E {\rightarrow}M$, the bundle ${\mathcal}E$ is [*stable*]{} ([*semistable*]{}) with respect to $\omega_M$, if for any proper holomorphic subbundle ${\mathcal}E' \subset {\mathcal}E$ for which $0 < {\operatorname{rank}}({\mathcal}E') < {\operatorname{rank}}({\mathcal}E)$, we have $\mu_{{\mathcal}E'} < \mu_{{\mathcal}E}$ (respectively, $\mu_{{\mathcal}E'} \leq \mu_{{\mathcal}E}$). The bundle ${\mathcal}E$ is said to be [*polystable*]{} if ${\mathcal}E$ is a direct sum of stable bundles of equal slope (see e.g. [@Kobthree]). \[PHE\] Relative to $(M, {\omega}_{\sigma})$ and ${\mathcal}E {\rightarrow}M$ as above, the [*perturbed Hermitian–Einstein equation*]{} (the PHE equation) is defined to be $$\label{defhes} \iota ~( {{\Lambda_{\sigma}}}{F_{\mathbf h}} + {{\mathfrak}d} ({\beta}, {\sigma}) ) = 2 \pi ~\lambda ~{\mathbf I}_{{{\mathcal}E}}~.$$ If ${{\mathfrak}d} ({\beta}, {\sigma}) = 0$, then [defhes]{} reduces to the usual Hermitian–Einstein equation [@Kobthree] : $$\iota ~{{\Lambda_{\sigma}}}{F_{\mathbf h}} = 2 \pi ~\lambda ~{\mathbf I}_{{{\mathcal}E}}~.$$ The term ‘perturbed’ signifies the inclusion of the perturbation term ${{\mathfrak}d} ({\beta}, {\sigma})$ in [defhes]{} which to some extent accounts for the fact that the usual Hermitian–Einstein equation does not in general admit a nice decomposition with respect to the holomorphic fiber bundle $M {\rightarrow}X$, even in the product case. The PHE equation necessitates working with a corresponding restricted type of (poly)stability. A dimensional reduction theorem =============================== The calibration condition on the fiber $F$ ------------------------------------------ We now fix the fiber data as follows. Recall that on $F$ we are given homogeneous holomorphic bundles ${{\mathcal}V}_{\rho_i} {\rightarrow}F$ associated to complex representations $(\rho_i~, V_{\rho_i}) \in R(K)$ as in [homogen]{}. The degree ${\deg_{F} ({{\mathcal}V}_{\rho_i})}$ and hence the slope $\mu_{\rho_i} = {{{\mu_{{{\mathcal}V}_{\rho_i}}}}}$ may be computed in terms of the weights of the representations $(\rho_i~, V_{\rho_i}) \in R(K)$ by the methods of [@BH]. Further, the representations $(\rho_i~, V_{\rho_i}) \in R(K)$ are assumed to be sums of irreducible representations of equal slope. By [@Kobthree] IV, Prop. 6.1, the $K$–invariant hermitian metrics on the irreducible components of the representation spaces $V_{\rho_i}$ are unique up to homothety and determine $U$–invariant Hermitian–Einstein structures on the corresponding homogeneous bundles. It follows that the homogeneous bundles ${{\mathcal}V}_{\rho_i} {\rightarrow}F$ have $U$–invariant Hermitian–Einstein structures and are therefore polystable. If the representations $(\rho_i~, V_{\rho_i})$ are irreducible, then the bundles ${{\mathcal}V}_{\rho_i}$ are stable (simple) by [@Kobthree] IV, Prop. 6.4 (cf also [@Kobtwo] [@Ramanan]). Conversely, if the bundles ${{\mathcal}V}_{\rho_i}$ are stable, the representations $(\rho_i~, V_{\rho_i})$ must be irreducible. In this case, two homogeneous bundles ${{\mathcal}V}_{\rho_i}$ are isomorphic as holomorphic vector bundles, if and only if the representations $(\rho_i, V_{\rho_i})$ are equivalent [@Ramanan]. As a consequence, the canonical extensions ${\widetilde}{{\mathcal}V}_{\rho_i} \to M$ of the homogeneous bundles ${{\mathcal}V}_{\rho_i} \to F$ admit Hermitian–Einstein structures $$\label{hefiber1} \iota ~{{\Lambda_{\sigma}}}F_{{\tilde}k_i} = {2 \pi} ~{\tilde}\mu_{\rho_i}~ {\widetilde}{\mathbf I}_{i} ~,$$ with constant given by $$\label{hefiber2} {\tilde}\mu_{\rho_i} = \mu_{{\widetilde}{{\mathcal}V}_{i}} = {\frac{\mu_{\rho_i}}{{\sigma}}}~.$$ It follows that if ${{\mathcal}V}_{\rho_i} \to F$ is stable $(\text{simple})$, then ${\widetilde}{{\mathcal}V}_{\rho_i} \to M$ is stable $(\text{simple})$. It is reasonable to require certain [*calibration conditions*]{} for the homogeneous vector bundles ${{\mathcal}V}_{\rho_i}$ . In view of the Bott–Borel–Weil theorem [@Botttwo], the representation theory suggests several possibilities. Here we will assume a slope condition of the form  $$\label{muslope1} \mu_{\rho} = \mu_{\rho_1} - \mu_{\rho_2} < 0~.$$ It follows that $H^0 (F, {{\mathcal}V}_\rho) = H^0 (F, {{{\mathcal}H}om_{\mathbb C}( {{{\mathcal}V}_{\rho_2}} , {{{\mathcal}V}_{\rho_1}} )}) = 0$ and $V_\rho^K = {{\operatorname{Hom}}_{K}( {V_{\rho_2}} , {V_{\rho_1}} )} = 0$ . Therefore we are in the situation of Corollary [ \[ExtProp\]]{}. Since the cohomology of the irreducible components must now occur in positive degrees, the Bott–Borel–Weil theorem implies that the corresponding dominant weights must have Bott index $\geq 1$ . In fact, there are other possible calibration conditions which could be assumed for other choices of the Bott index and consequently a different system of equations on $X$ can be realized [@BGKfour]. The reduction of the PHE equation to the twisted coupled vortex equations ------------------------------------------------------------------------- \[Main1\]  Let $F {\hookrightarrow}M = {\widetilde}X \times_\Gamma F {\overset {\pi}{{\rightarrow}}} X$ be a flat holomorphic fiber bundle of compact Kähler manifolds. Suppose that the homogeneous holomorphic bundles ${{\mathcal}V}_{\rho_i}$ on $F$ satisfy the calibration condition ${\eqref}{muslope1}~,$ that is ${\mu_{\rho}} = {\mu_{\rho_1}} - {\mu_{\rho_2}} < 0~,$ and therefore $H^0 (F, {{\mathcal}V}_\rho) = 0~.$ Consider the proper holomorphic extension $$\label{extension4a} {\mathbb E}~:~ 0 {\rightarrow}{{\pi^* {{\mathcal}W}_1} {\otimes}_{\mathbb C} {{\widetilde}{{\mathcal}V}_{\rho_1}}} {\overset {i}{{\longrightarrow}}} {{\mathcal}E} {\overset {p}{{\longrightarrow}}} {{\pi^* {{\mathcal}W}_2} {\otimes}_{\mathbb C} {{\widetilde}{{\mathcal}V}_{\rho_2}}} {\rightarrow}0~$$ where ${\mathbb E}$ corresponds to the holomorphic triple $T_0 = ({{\mathcal}W}_1, {{\mathcal}W}_2, {\beta}_0)~.$ For ${\sigma}>0$, let $$\lambda = {{\mu_{{\mathbb E}} ({\sigma})}} = {{\mu_{{\mathcal}E} ({\sigma})}} = \frac{{{\deg_{\sigma} ({\mathcal}E)}}}{{\operatorname{rank}}({\mathcal}E)}~, $$ and define the vortex parameters $\tau_i$ by $$\tau_i = \tau_i ({\sigma}) = {{\mu_{{\mathcal}E} ({\sigma})}} - {\frac{\mu_{\rho_i}}{{\sigma}}}~.$$ Then the following statements are equivalent$~:$ - There exist invariant hermitian metrics of the form $\mathbf h$ on the extension bundle ${\mathcal}E$ which satisfy the perturbed Hermitian–Einstein equation $$\label{defhes1} \iota ~( {{\Lambda_{\sigma}}}{F_{\mathbf h}} + \frac{1}{{\sigma}} ~\pi^* {{\mathfrak}d} ({\beta}_0) ) = 2 \pi ~\lambda ~{\mathbf I}_{{{\mathcal}E}}~,$$ relative to $(M, {\omega}_{\sigma})$ . - There exist hermitian metrics $h_i$ on ${{\mathcal}W}_i$ which satisfy the twisted coupled vortex equations $$\label{genvortex1} \begin{aligned} \iota ~{\Lambda_{X}}{F_{h_1}} + ~\frac{1}{{\sigma}} ~{\int_F ~{\operatorname{Tr}}}~{\lambda_1}_* ({\beta}_0 \wedge {\beta}_0^*) &= 2 \pi ~\tau_1 ~\mathbf I_{{{\mathcal}W}_1}~, \\ \iota ~{\Lambda_{X}}{F_{h_2}} - ~\frac{1}{{\sigma}} ~{\int_F ~{\operatorname{Tr}}}~{\lambda_2}_* ({\beta}_0^* \wedge {\beta}_0) &= 2 \pi ~\tau_2 ~\mathbf I_{{{\mathcal}W}_2}~. \end{aligned}$$ - There exist hermitian metrics $h_i$ on ${{\mathcal}W}_i$ which satisfy the twisted coupled multivortex equations $$\label{vortex1} \begin{aligned} \iota ~{\Lambda_{X}}{F_{h_1}} + \frac{1}{{\sigma}} ~{\Phi_1} &= 2 \pi ~\tau_1 ~\mathbf I_{{{\mathcal}W}_1}~, \\ \iota ~{\Lambda_{X}}{F_{h_2}} - \frac{1}{{\sigma}} ~{\Phi_2} &= 2 \pi ~\tau_2 ~\mathbf I_{{{\mathcal}W}_2}~, \end{aligned}$$ where $\Phi_i \in {{\operatorname{End}}_{\mathbb C}({{{\mathcal}W}_i})}$ are non–negative hermitian endomorphisms satisfying $$q_0^* \Phi_1 = {\sum_{j=1}^{k} ~{{\tilde}\phi_j \circ {\tilde}\phi_j^*}} ~\qquad~ ~,~ ~\qquad~ q_0^* \Phi_2 = {\sum_{j=1}^{k} ~{{\tilde}\phi_j^* \circ {\tilde}\phi_j}}~.$$ Here $k = {\dim_{{\mathbb C}}}H^{0,1} (F, {{\mathcal}V}_\rho)$, and the adjoints $\phi_j^*$ are taken with respect to the metrics $h_i$ . There is a one–to–one correspondence between solutions in $(1)$ and $(2)$, $(3)$ given by the assignment $h_i {\mapsto}h_i' = \pi^* h_i$ . The data in the equations [genvortex1]{} and [vortex1]{} depend only on the associated holomorphic triple $T_0 = ({{\mathcal}W}_1, {{\mathcal}W}_2, {\beta}_0)$ . \[split\]  If the extension ${\mathbb E}$ is holomorphically split, that is $[{\beta}] = 0~,$ the solutions of the PHE on ${{\mathcal}E}$ relative to $(M, {\omega}_{{\sigma}})~,$ respectively the corresponding solutions $(h_1, h_2)$ of the twisted coupled vortex equations ${\eqref}{vortex1}$, degenerate to solutions of the uncoupled Hermitian–Einstein equations on each ${{\mathcal}W}_i~.$ Outline of the proof -------------------- The proof of the theorem follows from some technical lemmas which reflect in part upon the flat structure of [flatbundle]{}. We will outline several of the steps involved following [@BGKfour] extending the special cases of [@BGKtwo] [@GPrfour]. First of all, we have  : - ${{\Lambda_{\sigma}}}F_{\pi^* h_i}= \pi^* {\Lambda_{X}} F_{h_i}~;$ - ${{\Lambda_{\sigma}}}F_{{\tilde}k_i}= \frac{1}{{\sigma}} ~ {\widetilde}{{\Lambda_{F}} F_{k_i}}~.$ Next, using [curvext1]{}, [curvext2]{} and [hefiber1]{}, the PHE equation is equivalent to the equation $$\begin{aligned} &\begin{bmatrix} ~{{\pi^* ( \iota ~{\Lambda_{X}} F_{h_1} + {2 \pi} ~( {\tilde}\mu_{\rho_1} - \lambda ) ~{\mathbf I}_{{{\mathcal}W}_1} )} {\otimes}{{\widetilde}{\mathbf I}_{\rho_1}}} & \iota ~{{\Lambda_{\sigma}}}D' {\beta}\\ - \iota ~{{\Lambda_{\sigma}}}D'' {\beta}^* & {{\pi^* ( \iota ~{\Lambda_{X}} F_{h_2} + {2 \pi} ~( {\tilde}\mu_{\rho_2} - \lambda ) ~{\mathbf I}_{{{\mathcal}W}_2} )} {\otimes}{{\widetilde}{\mathbf I}_{\rho_2}}}~ \end{bmatrix} \label{curvext3} \\ &= ~\iota ~ \begin{bmatrix} ~{{\Lambda_{\sigma}}}( {\beta}\wedge {\beta}^* ) - {{\mathfrak}d}_1 ({\beta}, {\sigma}) & 0 \\ 0 & {{\Lambda_{\sigma}}}( {\beta}^* \wedge {\beta}) - {{\mathfrak}d}_2 ({\beta}, {\sigma})~ \end{bmatrix} ~. \notag\end{aligned}$$ By the definition of the perturbation terms, this last expression equals $$\iota ~ \begin{bmatrix} ~{{\pi^* ~{\int_F ~{\operatorname{Tr}}}~{{\Lambda_{\sigma}}}({\beta}\wedge {\beta}^*)} {\otimes}{{\widetilde}{\mathbf I}_{\rho_1}}} & 0 \\ 0 & {{\pi^* ~{\int_F ~{\operatorname{Tr}}}~{{\Lambda_{\sigma}}}({\beta}^* \wedge {\beta})} {\otimes}{{\widetilde}{\mathbf I}_{\rho_2}}}~ \end{bmatrix} ~.$$ Hence we obtain the equivalent system of equations : $$\label{lambdabeta} {{\Lambda_{\sigma}}}D'{\beta}= 0 ~\qquad~ ~,~ ~\qquad~ {{\Lambda_{\sigma}}}D''{\beta}^* = 0~,$$ and $$\label{genbasic2} \begin{aligned} \iota ~{\Lambda_{X}} F_{h_1} - {2 \pi} ~\tau_1 ({\sigma}) ~{\mathbf I}_{{{\mathcal}W}_1} &= \iota ~{\int_F ~{\operatorname{Tr}}}~{{\Lambda_{\sigma}}}( {\beta}\wedge {\beta}^* )~, \\ \iota ~{\Lambda_{X}} F_{h_2} - {2 \pi} ~\tau_2 ({\sigma}) ~{\mathbf I}_{{{\mathcal}W}_2} &= \iota ~{\int_F ~{\operatorname{Tr}}}~{{\Lambda_{\sigma}}}( {\beta}^* \wedge {\beta})~. \end{aligned}$$ The remainder of the proof deals with some analysis of ${\beta}$ and showing that the off–diagonal terms in [curvext3]{} are zero. It follows from Lemma [ \[BetaLemma2\]]{} that we may choose the smooth decomposition of ${{\mathcal}E}$ , such that $q^* {\beta}$ is of the form $$q^* {\beta}= {\sum_{j=1}^{k} ~{{{{\tilde}\pi^* {\tilde}\phi_j} {\otimes}{p^* \eta_j}}}}~,$$ where $\eta_j \in A^{0,1} (F, {{\mathcal}V}_\rho)$ are $\Delta_{{\bar {\partial}}}$–harmonic $(0,1)$–forms representing an orthonormal basis of $H^{0,1}(F, {{\mathcal}V}_\rho)$ . Combining this with the Hodge formulas $$\label{Lambdaformula} \begin{aligned} {{\Lambda_{\sigma}}}D' {\beta}- D' {{\Lambda_{\sigma}}}{\beta}&= \iota ~{\bar {\partial}}^* {\beta}~, \\ {{\Lambda_{\sigma}}}{\bar {\partial}}{\beta}- {\bar {\partial}}{{\Lambda_{\sigma}}}{\beta}&= - ~\iota ~{D'}^* {\beta}~. \end{aligned}$$ in [@Kobthree] , we obtain the following equivalent properties for the metrics $h_i$ on ${{\mathcal}W}_i$ (cf [@BGKone]). \[lambdalemma4\]  Suppose that $\Delta_{{\bar {\partial}}} \eta_j = 0$, that is the forms $\eta_j \in A^{0,1} (F, {{\mathcal}V}_\rho)$ are harmonic. Then we have - ${{\Lambda_{\sigma}}}D' {\beta}= 0~;$ - ${{\Lambda_{\sigma}}}D'' {\beta}^* = 0~;$ - ${\bar {\partial}}^* {\beta}= 0~;$ - $\Delta_{{\bar {\partial}}} {\beta}= 0$, that is the form ${\beta}\in A^{0,1} (M, {{\pi^* {\mathcal}W} {\otimes}_{\mathbb C} {{\widetilde}{{\mathcal}V}_\rho}})$ is harmonic. From the calibration condition [muslope1]{}, Corollary [ \[ExtProp\]]{} and Lemma [ \[reduction2\]]{} we deduce that $$\label{fibintbeta1} \begin{aligned} \frac{1}{\iota} ~{\int_F ~{\operatorname{Tr}}}~{{\Lambda_{\sigma}}}({\beta}\wedge {\beta}^*) &= \frac{1}{{\sigma}} ~{\int_F ~{\operatorname{Tr}}}~{\lambda_1}_* ({\beta}_0 \wedge {\beta}_0^*) = \frac{1}{{\sigma}} ~\Phi_1~, \\ \iota ~{\int_F ~{\operatorname{Tr}}}~{{\Lambda_{\sigma}}}({\beta}^* \wedge {\beta}) &= \frac{1}{{\sigma}} ~{\int_F ~{\operatorname{Tr}}}~{\lambda_2}_* ({\beta}_0^* \wedge {\beta}_0) = \frac{1}{{\sigma}} ~\Phi_2~. \end{aligned}$$ Using again the above expression from Lemma [ \[BetaLemma2\]]{}, the non–negative hermitian endomorphisms $\Phi_i$ of ${{\mathcal}W}_i$ admit the expansion $$\label{fibintbeta2} q_0^* \Phi_1 = {\sum_{i,j}^{} ~{{\tilde}\phi_i \circ {\tilde}\phi_j^* ~\langle \eta_i, \eta_j \rangle}} = {\sum_{j=1}^{k} ~{{\tilde}\phi_j \circ {\tilde}\phi_j^*}} ~\qquad~,~\qquad ~ q_0^* \Phi_2 = {\sum_{i,j}^{} ~{{\tilde}\phi_i^* \circ {\tilde}\phi_j ~\langle \eta_i^*, \eta_j^* \rangle}} = {\sum_{j=1}^{k} ~{{\tilde}\phi_j^* \circ {\tilde}\phi_j}}$$ The Theorem now follows essentially from [genbasic2]{}, [fibintbeta1]{} and [fibintbeta2]{}. The Reduction Theorem for invariant extensions ---------------------------------------------- Here we assume the following stronger conditions on the data on the fiber. - The representations $(\rho_i, V_{\rho_i}) \in R(K)$ are irreducible. - $\mu_{\rho} = \mu_{\rho_1} - \mu_{\rho_2} < 0$ . - $H^{0,1} (F, {{\mathcal}V}_\rho)^G \neq 0$ . It follows from the Bott–Borel–Weil theorem [@Botttwo] and the multiplicity formulas in [@PRV] that the multiplicity of the trivial representation in $H^{0,1} (F, {{\mathcal}V}_\rho)$ is at most $1$ , that is we have $H^{0,1} (F, {{\mathcal}V}_\rho)^G \cong H^1 ({\mathfrak p}, {\mathfrak k}_{{\mathbb C}} ; V_\rho) \cong {\mathbb C}$ . Under the above assumptions, the terms ${{\mathfrak}d}_i ({\beta}_0)$ vanish and Theorem [ \[Main1\]]{} takes on a more familiar form. In fact, we are now essentially in the situation of [@BGKtwo Theorem $8.9$]. \[Main3\] Suppose that the extension ${\mathbb E}$ is invariant, that is the Kobayashi form $[{\beta}]$ of ${\mathbb E}$ satisfies $${[\beta]} \in {{\operatorname{Ext}}^1_{{{\mathcal}O}_{M}}({{\mathcal}E}_{2} , {{\mathcal}E}_{1})}_{\mathbf 1} \cong {{H^0 (X, {{\mathcal}W})} {\otimes}_{\mathbb C} {H^{0,1}(F, {{\mathcal}V}_\rho)^U}}~.$$ Then the following statements are equivalent: - The invariant metric $\mathbf h$ satisfies the Hermitian–Einstein equation $$\iota ~{{\Lambda_{\sigma}}}{F_{\mathbf h}} = 2 \pi ~\lambda ~{\mathbf I}_{{{\mathcal}E}}~.$$ - The metrics $h_1$ and $h_2$ satisfy the coupled vortex equations$~:$ $$\label{vortex2} \begin{aligned} \iota {\Lambda_{X}}{F_{h_1}} + \frac{1}{\sigma} ~\phi \circ \phi^* &= 2 \pi ~\tau_1 ~\mathbf I_{{{\mathcal}W}_1}~, \\ \iota ~{\Lambda_{X}}{F_{h_2}} - \frac{1}{\sigma} ~\phi^* \circ \phi &= 2 \pi ~\tau_2 ~\mathbf I_{{{\mathcal}W}_2}~. \end{aligned}$$ Examples -------- Bott’s generalization of the Borel–Weil theorem [@Botttwo] states that for an irreducible $P$–module $(\rho, V_{\rho})$ , the induced cohomology $H^{0,*} (G/P, {{\mathcal}V}_\rho)$ is either equal to zero or it is an irreducible $G$–module. The theory underlying the Bott–Borel–Weil (BBW) theorem can be used to compute examples of irreducible $P$–modules ${{\mathcal}V}_{\rho}$ which satisfy the calibration conditions for both of the reduction theorems as stated above. In principle it seems that a plentiful supply of such examples can be computed for many types of the Kähler homogeneous space $F= G/P$, in particular for the case of invariant extensions. The reference [@BE] (Chapters $1$–$5$) outlines a technology for doing this by means of computational rules. It is based on enumerating the theory of affine actions of the Weyl group of $\mathfrak g$ and the Bott–Kostant induction (cf [@BGKfour]). The procedure starts by considering the Dynkin diagram for a given $\mathfrak g$ where one or more nodes $\bullet$ are replaced by a crossed node $\times$ when there is a non–parabolic simple root. In this way the Dynkin diagram for $F = G/P$ is obtained. For instance, a maximal parabolic $\mathfrak p$ subalgebra (as in the case of the compact irreducible Hermitian symmetric spaces) admits a single crossed node and a Borel subalgebra $\mathfrak b$ has crosses through every node as is the case for the full flag manifold $G/B$ over $\mathbb C^{\ell +1}$ . The weights of the representations are exhibited by such diagrams (see below) by placing (integer) coefficients over each node in accordance with certain rules. If the aim is to obtain a $1$–dimensional irreducible $\mathfrak g$–module, we would select suitable node coefficients for the diagram such that on taking a single affine Weyl group reflection over the appropriate crossed node leads to zeros over each node in the diagram and hence this selection corresponds to the trivial $\mathfrak g$–module provided by the BBW theorem. Let $F = \mathbb CP^4$ and take ${\mathcal}V_\rho$ to be the irreducible $P$–module ${\Omega}_F^1$ . Starting from the corresponding Dynkin diagram  $\ {\begin{picture}(75,12) \put(0,2){\line(5,0){20}} \put(20,2){\line(5,0){20}} \put(45,2){\line(5,0){20}} \put(-4,-1){$\times$} \put(20,2){\circle*{4}} \put(45,2){\circle*{4}} \put(65,2){\circle*{4}} \put(-2,6){\shortstack{\small -2}} \put(17,6){\shortstack{\small 1}} \put(43,6){\shortstack{\small 0}} \put(62,6){\shortstack{\small 0}} \end{picture}}$  and then taking a single affine reflection, the trivial module is obtained. Thus $H^{1}(F, {{\mathcal}V}_\rho)\cong \mathbb C$ , and the cohomology in all other degrees is zero by the BBW Theorem. The dual module ${{\mathcal}V}_\rho^* \cong {\mathcal}T^{1,0}_F$ has the corresponding diagram  $\ {\begin{picture}(75,12) \put(0,2){\line(5,0){20}} \put(20,2){\line(5,0){20}} \put(45,2){\line(5,0){20}} \put(-4,-1){$\times$} \put(20,2){\circle*{4}} \put(45,2){\circle*{4}} \put(65,2){\circle*{4}} \put(-2,6){\shortstack{\small 1}} \put(17,6){\shortstack{\small 0}} \put(43,6){\shortstack{\small 0}} \put(62,6){\shortstack{\small 1}} \end{picture}}$ . That for the canonical line bundle ${\mathcal}K_F = {\Omega}_F^4$ is  $\ {\begin{picture}(75,12) \put(0,2){\line(5,0){20}} \put(20,2){\line(5,0){20}} \put(45,2){\line(5,0){20}} \put(-4,-1){$\times$} \put(20,2){\circle*{4}} \put(45,2){\circle*{4}} \put(65,2){\circle*{4}} \put(-2,6){\shortstack{\small -5}} \put(17,6){\shortstack{\small 0}} \put(43,6){\shortstack{\small 0}} \put(62,6){\shortstack{\small 0}} \end{picture}}$  from which ${{\mathcal}V}_\rho {\otimes}{{\mathcal}K}_F$ is represented by  $\ {\begin{picture}(75,12) \put(0,2){\line(5,0){20}} \put(20,2){\line(5,0){20}} \put(45,2){\line(5,0){20}} \put(-4,-1){$\times$} \put(20,2){\circle*{4}} \put(45,2){\circle*{4}} \put(65,2){\circle*{4}} \put(-2,6){\shortstack{\small -7}} \put(17,6){\shortstack{\small 1}} \put(43,6){\shortstack{\small 0}} \put(62,6){\shortstack{\small 0}} \end{picture}}$ . On taking four affine reflections on the latter we obtain   $\ {\begin{picture}(75,12) \put(0,2){\line(5,0){20}} \put(20,2){\line(5,0){20}} \put(45,2){\line(5,0){20}} \put(-4,-1){$\times$} \put(20,2){\circle*{4}} \put(45,2){\circle*{4}} \put(65,2){\circle*{4}} \put(-2,6){\shortstack{\small 1}} \put(17,6){\shortstack{\small 0}} \put(43,6){\shortstack{\small 0}} \put(62,6){\shortstack{\small 1}} \end{picture}}$ . Thus Serre–duality and the BBW theorem imply that the irreducible $G$–module $$H^0 (F, {{\mathcal}V}_\rho^*) \cong H^4 (F, {{\mathcal}V}_\rho {\otimes}{{\mathcal}K}_F) \cong {{\mathfrak}g}~:~~ {\begin{picture}(75,12) \put(0,2){\line(5,0){20}} \put(20,2){\line(5,0){20}} \put(45,2){\line(5,0){20}} \put(-4,-1){$\times$} \put(20,2){\circle*{4}} \put(45,2){\circle*{4}} \put(65,2){\circle*{4}} \put(-2,6){\shortstack{\small 1}} \put(17,6){\shortstack{\small 0}} \put(43,6){\shortstack{\small 0}} \put(62,6){\shortstack{\small 1}} \end{picture}}$$ and the cohomology in all other degrees is zero, so we have $H^0(F, {{\mathcal}V}_\rho^*)^U = H^1 (F, {{\mathcal}V}_\rho^*)^U = 0$ . Here we take $F$ to be the $9$–dimensional partial flag manifold over ${\mathbb C}^5$ whose compact representation is $$F \cong SU (5)/ S (U (1) \times U (2) \times U (1) \times U (1))~.$$ It is an example of a homogeneous Kähler manifold which is not a symmetric space. Consider the irreducible $P$–module ${{\mathcal}V}_\rho$ as represented by   $\ {\begin{picture}(75,12) \put(0,2){\line(5,0){20}} \put(20,2){\line(5,0){20}} \put(45,2){\line(5,0){20}} \put(-4,-1){$\times$} \put(20,2){\circle*{4}} \put(41,-1){$\times$} \put(61,-1){$\times$} \put(-2,6){\shortstack{\small -2}} \put(17,6){\shortstack{\small 1}} \put(43,6){\shortstack{\small 0}} \put(62,6){\shortstack{\small 0}} \end{picture}}$ . A single affine reflection leads to $H^{1}(F, {{\mathcal}V}_\rho) \cong \mathbb C$ and zero cohomology in all other degrees. As for the dual module ${\mathcal}V_\rho^*$, the diagram is   $\ {\begin{picture}(75,12) \put(0,2){\line(5,0){20}} \put(20,2){\line(5,0){20}} \put(45,2){\line(5,0){20}} \put(-4,-1){$\times$} \put(20,2){\circle*{4}} \put(41,-1){$\times$} \put(61,-1){$\times$} \put(-2,6){\shortstack{\small 1}} \put(17,6){\shortstack{\small 1}} \put(43,6){\shortstack{\small -1}} \put(62,6){\shortstack{\small 0}} \end{picture}}$  which can be seen to correspond to a singular weight and hence $H^q(F, {{\mathcal}V}_\rho^*) = 0$, for all $q \geq 0$ . We remark that other types of examples can be formulated following e.g. [@MS] Theorem B. The moment map and the PHE equation =================================== Given a hermitian metric $h$ on $E$ , the space ${{\mathcal}C} (E)$ of integrable ${\bar {\partial}}$–operators, for which ${\bar {\partial}}^2 = 0$ , corresponds bijectively to the space ${{\mathcal}A} (E, h)$ of unitary integrable connections whose curvature satisfies $F^{0,2}_{h} = 0$ . Here $E$ denotes the underlying smooth vector bundle of ${\mathcal}E$ . Thus each element ${\bar {\partial}}_E \in {\mathcal C}(E)$ defines a unique holomorphic structure ${{\mathcal}E} = (E, {\bar {\partial}}_E)$ on $E$ , for which it is the canonical ${\bar {\partial}}$–operator. The complex gauge group ${\operatorname{Aut}}(E)$ acts on ${\mathcal C}(E)$ via the action $g({\bar {\partial}}) = g \circ {\bar {\partial}}\circ g^{-1}$ , for $g \in {\operatorname{Aut}}(E)$ . For our purpose, we restrict attention to the [*unitary*]{} gauge (sub)group denoted by ${\mathcal G}$ . The quotient ${\mathcal C}(E)/{{\mathcal G}}$ is the space of equivalence classes of integrable holomorphic structures on $E$ up to unitary equivalence. The Lie algebra of ${\mathcal G}$ is given by ${\operatorname{Lie}}({\mathcal G}) \cong {\operatorname{End}}_s (E)$, where ${\operatorname{End}}_s (E)$ is the Lie algebra of global skew–hermitian endomorphisms of $E$ . Background references to this section are [@AB] [@Donone] [@Kobthree]. The restricted gauge group and the moment map --------------------------------------------- Since $M$ is Kähler, the inner product $$\langle {\alpha}_1,{\alpha}_2 \rangle = \frac{1}{\iota (m-1)!{\operatorname{Vol}}(M)} \int_M {\operatorname{Tr}}~({\alpha}_1 \wedge {\alpha}_2^*) \wedge {\omega}_{\sigma}^{m-1}~,$$ for ${\alpha}_1,{\alpha}_2 \in T_{{\bar {\partial}}}~{\mathcal C}(E) \cong A^{0,1}(M, {\operatorname{End}}_s (E))$, induces a Kähler structure on ${\mathcal C}(E)$ where the Kähler form ${\omega}$ is defined by $ {\omega}({\alpha}_1, {\alpha}_2) = {\operatorname{im}}~\langle {\alpha}_1,{\alpha}_2 \rangle$ . The standard action of ${\mathcal G}$ on ${\mathcal C}(E)$ preserves ${\omega}$ and induces an associated equivariant moment map $$\label{moment0} \begin{aligned} \nu = \nu ({\mathcal G})~: ~ {\mathcal C}(E) &{\longrightarrow}{\operatorname{Lie}}({\mathcal G}) \subset {\operatorname{Lie}}({\mathcal G})^* \cong L^2({{\operatorname{Lie}}({\mathcal G})}) \\ {\bar {\partial}}&\mapsto {{\Lambda_{\sigma}}}{F_h} ~, \end{aligned}$$ where ${\bar {\partial}}$ corresponds to the unitary integrable connection $(A, h)$ . This moment map is determined up to a constant in the center of the Lie algebra and may also be written as $$\label{einsteinmoment} \nu ({\bar {\partial}}) = {{\Lambda_{\sigma}}}{F_{h}} ~+~ 2 \pi \iota ~\lambda ~{\mathbf I}_{{\mathcal}E}~.$$ Note that $\nu^{-1} (0)$ is empty unless $\lambda = \mu_{E}$ , the slope of $E$ . In this section we assume that the representations $(\rho_i, V_{\rho_i})$ are irreducible. We consider the subspace ${{\mathcal}A} ({\mathbb E}, {\mathbf h}) \subset {{\mathcal}A} (E, {\mathbf h})$ of unitary integrable connections ${\mathbf A}$ of the form $({\mathbf A}_1, {\mathbf A}_2, {\beta})$ , as in [connext1]{}, where $\mathbf h$ is a (fixed) special metric $\mathbf h$ as in [special]{}. Let ${\mathcal C}({\mathbb E}) \subset {\mathcal C}(E)$ be the subspace of holomorphic structures determined by ${{\mathcal}A} ({\mathbb E}, {\mathbf h})$ . The elements ${\bar {\partial}}_E \in {\mathcal C}({\mathbb E})$ determine a holomorphic structure on the extension $\mathbb E$ in [extension4a]{}, that is ${\bar {\partial}}_E$ is of the form $$\label{holomoment} {\bar {\partial}}_E = \begin{bmatrix} ~{\bar {\partial}}_{\pi^* W_1} \otimes {\widetilde}{\mathbf{I}}_1 + \mathbf{I}_1 {\otimes}{\bar {\partial}}_{{\widetilde}{{\mathcal}V}_1} & \beta \\ 0 & {\bar {\partial}}_{\pi^* W_2} {\otimes}{\widetilde}{\mathbf{I}}_2 + \mathbf{I}_2 {\otimes}{\bar {\partial}}_{{\widetilde}{{\mathcal}V}_2}~ \end{bmatrix}~.$$ We further consider the subspaces ${\mathcal C}_0 ({\mathbb E}) \cong {{\mathcal}A}_0 ({\mathbb E}, {\mathbf h})$ , consisting of the elements in ${\mathcal C}({\mathbb E}) \cong {{\mathcal}A} ({\mathbb E}, {\mathbf h})$ such that ${\beta}$ is $\Delta_{{\bar {\partial}}}$–harmonic, that is ${\bar {\partial}}^* {\beta}= 0~.$ Then ${{\mathcal}A}_0 ({\mathbb E}, {\mathbf h})$ admits a mapping $$\label{fiber} {{\mathcal}H}_{{\bar {\partial}}}^{0,1} (M, {{{\mathcal}H}om_{\mathbb C}( {{{\mathcal}E}_2} , {{{\mathcal}E}_1} )}) {\longrightarrow}{\mathcal A}_0 ({\mathbb E}, \mathbf h) {\overset {\Pi}{{\longrightarrow}}} {\mathcal A}(W_1, h_1) \times {\mathcal A}(W_2, h_2)~,$$ where the dimension ${\dim_{{\mathbb C}}}{{\mathcal}H}_{{\bar {\partial}}}^{0,1} (M, {{{\mathcal}H}om_{\mathbb C}( {{{\mathcal}E}_2} , {{{\mathcal}E}_1} )})$ is upper–semicontinuous as a function on the base. We specify a subgroup ${\mathcal G}_0 \subset {\mathcal G}$ which acts symplectically on ${\mathcal C}(E)$ and on ${\mathcal C}_0({\mathbb E})$ via restriction to the latter. For $u_i \in {\mathcal G}_{W_i}$, the subgroup ${\mathcal G}_0$ is defined by $$\label{gaugetwo} {\mathcal G}_0 = \{ \begin{bmatrix} ~ \pi^* u_1 {\otimes}{\widetilde}{\mathbf{I}}_{1} & 0 \\ 0 & \pi^* u_2 \otimes {\widetilde}{\mathbf{I}}_{2}~ \end{bmatrix} \} \cong {\mathcal G}_{W_1} \times {\mathcal G}_{W_2} \subset {\mathcal G}~.$$ The subgroup ${\mathcal G}_0$ leaves ${\mathcal C}_0({\mathbb E})$ invariant and fixes the holomorphic structures on the fiber. In fact, ${\mathcal G}_0$ is the maximal subgroup of ${\mathcal G}$ with this property, since by the irreducibility of the $V_{\rho_i}$ there are no non–trivial $U$–equivariant gauge transformations on the homogeneous bundles ${{\mathcal}V}_{\rho_i}$ , that is ${{\operatorname{End}}_{U}( {{{\mathcal}V}_{\rho_i}} )} \cong {{\operatorname{End}}_{\mathbb C}({V_{\rho_i}})}^K \cong {{\mathbb C}} \cdot {\operatorname{Id}}$ . On smooth elements (relative to $\mathbf h = \mathbf h_1 \oplus \mathbf h_2$ as above), we have a commutative diagram $$\label{orthomoment2} \CD {\operatorname{Lie}}({\mathcal G}) @>\subset>> {\operatorname{Lie}}({\mathcal G})^* \cong L^2({{\operatorname{Lie}}({\mathcal G})}) \\ @AA\pi^*A @VP_0VV \\ {\operatorname{Lie}}({\mathcal G}_0) @>\subset>> {\operatorname{Lie}}({\mathcal G}_0)^* \cong L^2({{\operatorname{Lie}}({\mathcal G}_0)}) \\ \endCD$$ where $P_0$ denotes orthogonal projection and for $a \in {\operatorname{Lie}}({\mathcal G}_0)$ the following relationship is satisfied : $$\langle P_0({{\Lambda_{\sigma}}}{F_{\mathbf h}}) ~,~ a \rangle = \frac{\iota}{n!{\operatorname{Vol}}(X)} \int_X {\operatorname{Tr}}~ (P_0({{\Lambda_{\sigma}}}F_{\mathbf h}) \circ a^*)~{\omega}_{X}^n~.$$ Observing that the projection $P_0$ is essentially given by integration over the fiber, we obtain the main result concerning the moment map interpretation of the PHE equation. \[mainmoment\][@BGKfour]  With regards to the inclusion $j : {\mathcal C}_0 ({\mathbb E}) \hookrightarrow {\mathcal C}(E)$ , consider the map $$\nu_0 : {\mathcal C}_0 ({\mathbb E}) {\longrightarrow}{\operatorname{Lie}}({\mathcal G}_0)^* ~,$$ as defined by $\nu_0 = P_0 \circ \nu \circ j$ . Then the following hold  : - The map $\nu_0$ is a moment map for the action of ${\mathcal G}_0$ on ${\mathcal C}_0 ({\mathbb E})$ . - The following diagram commutes with respect to the inclusion of smooth elements$~:$ $$\begin{CD} {\mathcal C}(E) @>\nu >> {\operatorname{Lie}}({\mathcal G}) @>\subset >> {\operatorname{Lie}}({\mathcal G})^* \cong L^2({{\operatorname{Lie}}({\mathcal G})}) \\ @AAjA @AA\pi^*A @VP_0VV \\ {\mathcal C}_0 ({\mathbb E}) @>\nu_0 >> {\operatorname{Lie}}({\mathcal G}_0) @>\subset>> {\operatorname{Lie}}({\mathcal G}_0)^* \cong L^2({{\operatorname{Lie}}({\mathcal G}_0)}) \\ \end{CD}$$ - The PHE equation $$\iota ({{\Lambda_{\sigma}}}F_{\mathbf h} ~+~ {{\mathfrak}d} ({\beta}, {\sigma})) = 2 \pi ~\lambda ~\mathbf I_{{\mathcal}E}~,$$ is equivalent to the ${\mathcal G}_0$–moment map equation $\nu_0({\bar {\partial}})=0$ . 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Bradlow, S.B., Daskalopoulos, G., García-Prada, O. and Wentworth, R.A., *Stable augmented bundles over Riemann surfaces,* in Vector Bundles in Algebraic Geometry (Eds. Hitchin et al.), Cambridge Univ. Press 1995, pp. 15–67. Bradlow, S.B. and García-Prada, O., *Stable triples, equivariant bundles and dimensional reduction*, Math. Ann. **304** (1995), 225–252. Bradlow, S.B. and García-Prada, O., *Higher cohomology triples and holomorphic extensions*, Comm. Anal. Geom. **3** (1995), 421–463. Bradlow, S.B., Glazebrook, J.F. and Kamber, F.W., *A new look at the vortex equations and dimensional reduction*, in Proc. of the First Brazil–USA Workshop on Geometry, Topology and Physics 1996, Verlag Walter de Gruyter & Co, Berlin 1997, pp. 85–106. Bradlow, S.B., Glazebrook, J.F. and Kamber, F.W., *Reduction of the Hermitian–Einstein equation on Kählerian fiber bundles*, Tohoku Math. Jour. **51** (1999), 81–124. 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Hirzebruch, F., *Topological Methods in Algebraic Geometry*, Grundlehren der Math. Wiss. **131** (3rd. Ed.), Springer Verlag, Berlin–Heidelberg–New York 1966. Kamber, F.W. and Tondeur, Ph., *Foliated Bundles and Characteristic Classes*, Lect. Notes in Math., **493**, Springer Verlag, Berlin–Heidelberg–New York, 1975. Kobayashi, S., *Homogeneous vector bundles and stability*, Nagoya Math. Jour. **101** (1986), 37–54. Kobayashi, S., *Differential Geometry of Complex Vector Bundles*, Princeton University Press, Princeton 1987. Merkulov, S. and Schwachhöfer, L., *Classification of irreducible holonomies of torsion–free affine connections*, Ann. Math. **150** (1999), 77–149. Parthasarathy, K. R., Ranga Rao, R., Varadarajan, V.S., *Representations of complex semi–simple Lie groups and Lie algebras*, Ann. Math. [**8**5]{} (1967), 383–429. Ramanan, S., *Holomorphic vector bundles on homogeneous spaces*, Topology **5** (1966), 159–177. Weil, A., *Variétés Kähleriennes*, Act. Sci. et Ind. **1267**, Hermann, Paris 1958. [^1]: ${}^1$ Supported in part by The National Science Foundation under Grant DMS-9703869. [^2]: ${}^2$Supported in part by The National Science Foundation under Grant DMS-9504084.

--- author: - '[](mailto:xcolor@ukern.de)' date: ' () [^1]' title: 'Color extensions with the package — various examples' --- \[2007/01/21 v2.11 Color logging test (UK)\] The purpose of this file is to demonstrate a variety of capabilities including the logging facilities of the package. By playing around with different values of ``, one can observe the different behavior in the `log` file. Predefined colors ================= 1 Color definition and application ================================ Comma-separated and space-separated definitions: = = = = = = . Test with named colors: Test: ; Test: ; Test: ; Test: . [Test with ``]{} Current color application: and , and . Current color test with ``: TestTest TestTest TestTest Color in tables =============== ------------------- ----- test row test row test row test row test row test row test row test row [&gt;l]{}[test]{} row ------------------- ----- Color information ================= Type test: namedef[@foo1]{}[foo1]{}namedef[@foo2]{}[@[foo2]{}]{}namedef[@foo3]{}[@[foo3]{}]{}namedef[@foo4]{}[@[foo4]{}]{} [^1]: This file (`.tex`) is part of the distribution which can be downloaded from the CTAN mirrors `CTAN/macros/latex/contrib/xcolor/` or the homepage `www.ukern.de/tex/xcolor.html`. Please send error reports and suggestions for improvements to `xcolor@ukern.de`.

--- abstract: | Observations of radio halos and relics in galaxy clusters indicate efficient electron acceleration. Protons should likewise be accelerated and, on account of weak energy losses, can accumulate, suggesting that clusters may also be sources of very high-energy (VHE; $E>100$ GeV) gamma-ray emission. We report here on VHE gamma-ray observations of the Coma galaxy cluster with the VERITAS array of imaging Cherenkov telescopes, with complementing -LAT observations at GeV energies. No significant gamma-ray emission from the Coma cluster was detected. Integral flux upper limits at the 99% confidence level were measured to be on the order of $(2-5)\times 10^{-8}\ {\rm ph.\,m^{-2}\,s^{-1}}$ (VERITAS, $>220\ {\rm GeV}$) and $\sim 2\times 10^{-6}\ {\rm ph.\,m^{-2}\, s^{-1}}$ (, $1-3\ {\rm GeV}$), respectively. We use the gamma-ray upper limits to constrain CRs and magnetic fields in Coma. Using an analytical approach, the CR-to-thermal pressure ratio is constrained to be $< 16\%$ from VERITAS data and $< 1.7\%$ from data (averaged within the virial radius). [These upper limits are starting to constrain the CR physics in self-consistent cosmological cluster simulations and cap the maximum CR acceleration efficiency at structure formation shocks to be $<50\%$. Alternatively, this may argue for non-negligible CR transport processes such as CR streaming and diffusion into the outer cluster regions. ]{} Assuming that the radio-emitting electrons of the Coma halo result from hadronic CR interactions, the observations imply a lower limit on the central magnetic field in Coma of $\sim (2 - 5.5)\,\mu{\rm G}$, depending on the radial magnetic-field profile and on the gamma-ray spectral index. Since these values are below those inferred by Faraday rotation measurements in Coma (for most of the parameter space), this [renders]{} the hadronic model a very plausible explanation of the Coma radio halo. Finally, since galaxy clusters are dark-matter (DM) dominated, the VERITAS upper limits have been used to place constraints on the thermally-averaged product of the total self-annihilation cross section and the relative velocity of the DM particles, ${\left\langle \sigma v \right\rangle}$. author: - 'T. Arlen, T. Aune, M. Beilicke, W. Benbow, A. Bouvier, J. H. Buckley, V. Bugaev, K. Byrum, A. Cannon, A. Cesarini, L. Ciupik, E. Collins-Hughes, M. P. Connolly, W. Cui, R. Dickherber, J. Dumm, A. Falcone, S. Federici, Q. Feng, J. P. Finley, G. Finnegan, L. Fortson, A. Furniss, N. Galante, D. Gall, S. Godambe, S. Griffin, J. Grube, G. Gyuk, J. Holder, H. Huan, G. Hughes, T. B. Humensky, A. Imran, P. Kaaret, N. Karlsson, M. Kertzman, Y. Khassen, D. Kieda, H. Krawczynski, F. Krennrich, K. Lee, A. S Madhavan, G. Maier, P. Majumdar, S. McArthur, A. McCann, P. Moriarty, R. Mukherjee, T. Nelson, A. O’Faoláin de Bhróithe, R. A. Ong, M. Orr, A. N. Otte, N. Park, J. S. Perkins, M. Pohl H. Prokoph, J. Quinn, K. Ragan, L. C. Reyes, P. T. Reynolds, E. Roache, J. Ruppel, D. B. Saxon, M. Schroedter, G. H. Sembroski, C. Skole, A. W. Smith, I. Telezhinsky, G. Tešić, M. Theiling, S. Thibadeau, K. Tsurusaki, A. Varlotta, M. Vivier, S. P. Wakely, J. E. Ward, A. Weinstein, R. Welsing, D. A. Williams, B. Zitzer' - 'C. Pfrommer, A. Pinzke' bibliography: - 'refs.bib' title: 'Constraints on Cosmic Rays, Magnetic Fields, and Dark Matter from Gamma-Ray Observations of the Coma Cluster of Galaxies with VERITAS and ' --- Introduction ============ Clusters of galaxies are the largest virialized objects in the Universe, with typical sizes of a few Mpc and masses on the order of $10^{14}$ to $10^{15} M_{\odot}$. According to the currently favored hierarchical model of cosmic structure formation, larger objects formed through successive mergers of smaller objects with galaxy clusters sitting on top of this mass hierarchy [see @article:Voit:2005 for a review]. Most of the mass ($\sim$80%) in a cluster is dark matter (DM), as indicated by galaxy dynamics and gravitational lensing [@article:DiaferioSchindlerDolag:2008]. Baryonic gas making up the intra-cluster medium (ICM) contributes about 15% of the total cluster mass and individual galaxies account for the remainder (about 5%). The ICM gas mass also comprises a significant fraction of the observable (baryonic) matter in the Universe. The ICM is a hot ($T\sim 10^{8}$ K) plasma emitting thermal bremsstrahlung in the soft X-ray regime [see, e.g., @article:Petrosian:2001]. This plasma has been heated primarily through collisionless structure-formation shocks that form as a result of the hierarchical merging and accretion processes. Such shocks and turbulence in the ICM gas in combination with intra-cluster magnetic fields also provide a means to accelerate particles efficiently [see, e.g., @article:ColafrancescoBlasi:1998; @article:Ryu_etal:2003]. Many clusters feature megaparsec scale halos of nonthermal radio emission, indicative of a population of relativistic electrons and magnetic fields permeating the ICM [@article:Cassano_etal:2010]. There are two competing theories to explain radio halos. In the “hadronic model”, the radio-emitting electrons and positrons are produced in inelastic collisions of cosmic-ray (CR) ions with the thermal gas of the ICM [@article:Dennison:1980; @article:EnsslinPfrommerMiniatiSubramanian:2011]. In the “re-acceleration model”, a long-lived pool of 100-MeV electrons—previously accelerated by formation shocks, galactic winds, or jets of active galactic nuclei (AGN)—interacts with plasma waves that are excited during states of strong ICM turbulence, e.g., after a cluster merger. This may result in second order Fermi acceleration and may produce energetic electrons ($\sim 10$ GeV) sufficient to explain the observable radio emission [@article:SchlickeiserSieversThiemann:1987; @article:BrunettiLazarian:2010]. Observations of possibly nonthermal emission from clusters in the extreme ultraviolet [EUV; @article:SarazinLieu:1998] and hard X-rays [@article:RephaeliGruber:2002; @article:Fusco-Femiano_etal:2004; @article:Eckert_etal:2007] may provide further indication of relativistic particle populations in clusters, although the interpretation of these observations as nonthermal diffuse emission has been disputed on the basis of more sensitive observations [see, e.g., @article:Ajello_etal:2009; @article:Ajello_etal:2010; @article:Wik_etal:2009]. Galaxy clusters have, for many years, been proposed as sources of gamma rays. If shock acceleration in the ICM is an efficient process, a population of highly relativistic CR protons and heavy ions is to be expected in the ICM. The main energy-loss mechanism for CR hadrons at high energies is pion production through the interaction of CRs with nuclei in the ICM. Pions are short lived and decay. The decay of neutral pions produces gamma rays and the decay of charged pions produces muons, which then decay to electrons and positrons. Due to the low density of the ICM ($n_{\mathrm{ICM}}\sim 10^{-3}$ cm$^{-3}$), the large size and the volume-filling magnetic fields in the ICM, CR hadrons will be confined in the cluster on timescales comparable to, or longer than, the Hubble time [@article:Volk_etal:1996; @article:Berezinsky_etal:1997] and they can therefore accumulate. For a given CR distribution function, the hadronically induced gamma-ray flux is directly proportional to the CR-to-thermal pressure fraction, $X_{\mathrm{CR}}={\left\langle P_{{\mathrm{CR}}} \right\rangle}/ {\left\langle P_{\mathrm{th}} \right\rangle}$ [see, e.g., @article:EnsslinPfrommerSpringelJubelgas:2007], where the brackets indicates volume averages. A very modest $X_{{\mathrm{CR}}}$ of a few percent implies an observable flux of gamma rays [e.g., @article:PfrommerEnsslin:2004b]. Hydrostatic estimates of cluster masses, which are determined by balancing the thermal pressure force and the gravitational force, are biased low by the presence of any substantial nonthermal pressure component, including a CR pressure contribution. Similarly, a substantial CR pressure can bias the temperature decrement of the cosmic microwave background (CMB) due to the Sunyaev-Zel’dovich effect in the direction of a galaxy cluster. This could then severely jeopardize the use of clusters to determine cosmological parameters. Comparing X-ray and optical potential profiles in the centers of galaxy clusters yields an upper limit of 20-30% of nonthermal pressure (that can be composed of CRs, magnetic fields or turbulence) relative to the thermal gas pressure [@article:Churazov_etal:2008; @article:Churazov_etal:2010]. An analysis that compares spatially resolved weak gravitational lensing and hydrostatic X-ray masses for a sample of 18 galaxy clusters detects a deficit of the hydrostatic mass estimate compared to the lensing mass of $20\%$ at $R_{500}$ – the radius within which the mean density is 500 times the critical density of the Universe – suggesting again a substantial nonthermal pressure contribution on large scales [@article:Mahdavi_etal:2008]. Observing gamma-ray emission is a complementary method of constraining the pressure contribution of CRs that is most sensitive to the cluster core region. However, it assumes that the CR component is fully mixed with the ICM and may not allow for a detection of a two-phase structure of CRs and the thermal ICM. An $X_{\mathrm{CR}}$ of only a few percent is required in order to produce a gamma-ray flux observable with the current generation of gamma-ray telescopes, rendering this technique at least as sensitive as the dynamical and hydrostatic methods (which are more general in that they are sensitive to any nonthermal pressure component). Gamma-ray emission can also be produced by Compton up-scattering of ambient photons, for example CMB photons, on ultra-relativistic electrons. Those electrons can either be secondaries from the CR interactions mentioned above, or injected into the ICM by powerful cluster members and further accelerated by diffusive shock acceleration or turbulent reacceleration processes [@article:SchlickeiserSieversThiemann:1987 and references therein]. A third mechanism for gamma-ray production in a galaxy cluster could be self-annihilation of a DM particle, e.g., a weakly interacting massive particle (WIMP). As already mentioned, about 80% of the cluster mass is in the form of dark matter, which makes galaxy clusters interesting targets for DM searches [@article:EvansFerrerSarkar:2004; @article:BergstromHooper:2006; @article:PinzkePfrommerBergstrom2009; @article:Cuesta_etal:2011] despite their large distances compared to other common targets for DM searches, such as dwarf spheroidal galaxies [@article:Strigari_etal:2007; @article:Acciari_etal:2010; @article:Aliu_etal:2009] or the Galactic Center [@article:Kosack_etal:2004; @article:Aharonian_etal:2006; @article:Aharonian_etal:2009b; @article:Abramowski_etal:2011]. While several observations of clusters of galaxies have been made with satellite-borne and ground-based gamma-ray telescopes, a detection of gamma-ray emission from a cluster has yet to be made. Observations with EGRET [@article:Sreekumar_etal:1996; @article:Reimer_etal:2003] and the Large Area Telescope (LAT) on board the Gamma-ray Space Telescope [@article:Ackermann_etal:2010] have provided upper limits on the gamma-ray fluxes (typically $\sim10^{-9}$ ph cm$^{2}$ s$^{-1}$ for -LAT observations) for several galaxy clusters in the MeV to GeV band. Upper limits on the very-high-energy (VHE) gamma-ray flux from a small sample of clusters, including the Coma cluster, have been provided by observations with ground-based imaging atmospheric Cherenkov telescopes [IACTs; @article:Perkins_etal:2006; @inproc:Perkins_etal:2008; @article:Aharonian_etal:2009a; @article:Aleksic_etal:2010; @article:Aleksic_etal:2012]. The Coma cluster of galaxies (ACO 1656) is one of the most thoroughly studied clusters across all wavelengths [@article:Voges_etal:1999]. Located at a distance of about 100 Mpc [$z=0.023$; @article:StrubleRood:1999], it is one of the closest massive clusters [$M \sim10^{15}M_{\odot}$; @article:Smith:1983; @article:Kubo_etal:2008]. It hosts both a giant radio halo [@article:Giovannini_etal:1993; @article:Thierbach_etal:2003] and peripheral radio relic, which appears connected to the radio halo with a “diffuse” bridge [see discussion in @article:BrownRudnick:2010]. It has been suggested [@article:Ensslin_etal:1998] and successively demonstrated by cosmological simulations which model the nonthermal emission processes [@article:PfrommerEnsslinSpringel:2008; @article:Pfrommer:2008; @article:Battaglia_etal:2009; @article:Skillman_etal:2011], that the relic could well be an infall shock. Extended soft thermal X-ray (SXR) emission is evident from the ROSAT all-sky survey in the 0.1 to 2.4 keV band [@article:BrielHenryBohringer:1992]. Observations with XMM-Newton [@article:Briel_etal:2001] revealed substructure in the X-ray halo supported by substantial turbulent pressure of at least $\sim 10 \%$ of the total pressure [@article:Schuecker_etal:2004]. The Coma cluster is a natural candidate for gamma-ray observations. In this article, results from the VERITAS observations of the Coma cluster of galaxies are reported, with complementing analysis of available data from the Large Area Telescope (LAT) on board the Gamma-ray Space Telescope. The VERITAS and -LAT data have been used to place constraints on cosmic-ray particle populations, magnetic fields, and dark matter in the cluster. Throughout the analyses, a present day Hubble constant of $H_{0} = 100h$ km s$^{-1}$ Mpc$^{-1}$ with $h=0.7$ has been used. VERITAS Observations, Analysis, and Results =========================================== The VERITAS gamma-ray detector [@article:Weekes_etal:2002] is an array of four 12 m-diameter imaging atmospheric Cherenkov telescopes [@article:Holder_etal:2006] located at an altitude of $\sim$1250 m a.s.l. at the Fred Lawrence Whipple Observatory in southern Arizona (31$^{\circ}$ 40 30 N, 110$^{\circ}$ 57 07 W). Each of the telescopes is equipped with a 499-pixel camera covering a 3.5$^{\circ}$ field of view. The array, completed in the fall of 2007, is designed to detect gamma-ray emission from astrophysical objects in the energy range from 100 GeV to more than 30 TeV. Depending on the zenith angle and quality selection criteria imposed during the data analysis, the effective energy range may be narrower than that. The energy resolution is $\sim 15$% and the angular resolution (68% containment) is $\sim 0.1^{\circ}$ per event at 1 TeV and slightly larger at low energy. At the time of the Coma cluster observations, the sensitivity of the array allowed for detection of a point source with a flux of 1% of the steady Crab Nebula flux above 300 GeV at the confidence level of five standard deviations ($5\sigma$) in under 45 hours.[^1] The Coma cluster was observed with VERITAS between March and May in 2008 with all four telescopes fully operational. The total exposure amounts to 18.6 hours of quality-selected live time, i.e., time periods of astronomical darkness with clear sky conditions and no technical problems with the array. The center of the cluster was tracked in *wobble* mode, where the expected source location is offset from the center of the field of view by 0.5 degrees, to allow for simultaneous background estimation [@article:Fomin_etal:1994]. All of the observations were made in a small range with average zenith angle $\sim 21^{\circ}$. The data analysis was performed following the standard VERITAS procedures described in @inproc:Cogan_etal:2007 and @inproc:Daniel_etal:2007. Prior to event reconstruction and selection, all shower images are calibrated and cleaned. Showers are then reconstructed for events with at least two telescopes contributing images that pass the following quality selection criteria: more than four participating pixels in the camera, number of photoelectrons in the image larger than 75, and the distance from the image centroid to the center of the camera less than $1.43^{\circ}$. These quality selection criteria impose an energy threshold[^2] of about 220 GeV. In addition, events for which only images from the two closest-spaced telescopes[^3] survive quality selection are rejected, as they introduce an irreducible high background rate due to local muons, degrading the instrument sensitivity [@article:MaierKnapp:2007]. Gamma-ray-like events are separated from the CR background by imposing selection criteria (cuts) on the mean-scaled length and width parameters [@article:Aharonian_etal:1997; @article:Krawczynski_etal:2006] calculated from a parametrized moment analysis of the shower images [@inproc:Hillas:1985]. These parameters are averages over the four telescopes weighted with the total amplitude of the images, that measure the image moments width and length scaled with values expected for gamma rays. In this analysis, events with a mean-scaled length in the range 0.05-1.19 and a mean-scaled width in the range 0.05-1.08 are selected as gamma-ray-like events. These ranges for the gamma-hadron separation cuts were optimized [*a priori*]{} for a weak point source (3% Crab Nebula flux level) and a differential spectral index of 2.4, using data taken on the Crab Nebula during the same epoch. Because the VHE gamma-ray spectrum for the Coma cluster is expected to be a power-law function with an index of about 2.3 [@article:PinzkePfrommer:2010], these cuts are suitable for the analysis of the Coma cluster data set. It is noted that slightly varying the spectral index ($\pm$ 0.2) does not significantly impact the cuts used for quality selection and gamma-hadron separation in this work. The Coma cluster is a very rich cluster of galaxies with many plausible sites for gamma-ray emission: the core region, the peripheral radio relic, and individual powerful cluster member galaxies. VERITAS has a large enough field of view to allow investigation of several of these scenarios. In this work, the focus has been on the core region and three cluster members. The core region is treated as either a point source or a mildly extended source, a uniform disk with intrinsic radius $0.2^{\circ}$ or $0.4^{\circ}$, similar to the extension of the thermal soft X-ray emission from the core. There is evidence of a recent merger event between the two central galaxies NGC 4889 and NGC 4874 [@article:Tribble:1993]. There is also evidence for an excess of nonthermal X-ray emission from these galaxies as well as from the galaxy NGC 4921 [@article:Neumann_etal:2003]. Therefore, searches for point-like VHE gamma-ray emission have been conducted at the locations of these galaxies. The regions of interest considered in this work are summarized in Table \[table:roi\]. The ring-background model [@article:Aharonian_etal:2001] is used to estimate the background due to CRs misinterpreted as gamma rays (the cuts described above reject more than 99% of all CRs). The total number of events in a given region of interest is then compared to the estimated background from the off-source region scaled by the ratio of the solid angles to produce a final excess or deficit. The VHE gamma-ray significance is then calculated according to Formula 17 in @article:LiMa:1983. Significance skymaps over the VERITAS field of view produced with a $0.2^{\circ}$ integration radius are shown in Figure \[fig:skymaps\] with overlaid X-ray and radio contours from the ROSAT all-sky survey [@article:BrielHenryBohringer:1992] and GBT 1.4 GHz observations [@article:BrownRudnick:2010] respectively. Depending on the assumed extent of the source and the point-spread function, we can define an ON region, into which a defined fraction of the source photons should fall. No significant excess of VHE gamma rays from the Coma cluster was detected with VERITAS, as illustrated by the $\theta^{2}$ distribution shown in Figure \[fig:thetasq\], in which source events would pile up at small values of $\theta^2$ for a point source and fall into a somewhat wider range of $\theta^2$ values for an extended source. The $\theta^{2}$ distribution is a plot of event density versus the square of the angular separation from a given location. It permits a comparison of the ON-source event distribution with that of other locations, in this case a ring-shaped region, into which only background events should fall, the so-called OFF-source region. The $\theta^{2}$ distribution extends out to 0.42 square degrees to cover both the case of point-like and extended emission from the core of the Coma cluster. The $\theta^{2}$ distributions for the member galaxies also considered in this work are very similar to that in Figure \[fig:thetasq\] and show no excess of gamma rays. A 99% confidence level upper limit is calculated for each region of interest using events from the ON-source and OFF-source regions and the method described by @article:Rolke_etal:2005 assuming a Gaussian-distributed background. A lower bound of zero is imposed on the gamma-ray flux from the Coma cluster, which prevents artificially low flux upper limits in the case that the best-fit source flux is formally negative. Figure \[fig:sigdist\] shows the distribution of significances over the VERITAS skymap, which is well fit by a Gaussian with a mean close to zero and a standard deviation within a few percent of unity. Table \[table:results\] lists the upper limits for the selected regions of interest shown in Table \[table:roi\]. These upper-limit calculations depend on the gamma-ray spectrum, which in this work is assumed to be a power law in energy, $dN/dE\propto E^{-\alpha}$, where the spectral index $\alpha$ was allowed to have a value of 2.1, 2.3, or 2.5. -LAT Analysis and Results ========================= LAT on board has observed the Coma cluster in all-sky survey mode since its launch in June 2008. -LAT is sensitive to gamma rays in the 20 MeV to $\sim300$ GeV energy range and is complementary to the VERITAS observations. @article:Ackermann_etal:2010 reported on the search for gamma-ray emission from thirty-three galaxy clusters in the data from the first 18 months, including the Coma cluster, for which an upper limit of $4.58\times10^{-9}$ ph cm$^{-2}$ s$^{-1}$ in the 0.2 to 100 GeV energy band was reported. This limit is expected to improve as the exposure is increased. In this work an updated analysis is presented as a complement to the VERITAS results which includes data taken between August 5, 2008 and April 17, 2012. The LAT-data analysis of this work follows the same procedure as described in detail in @2012ApJS..199...31N and was performed with the Fermi Science Tools version 9.23.1. To only include events with high probability of being photons, the P7SOURCE class and the corresponding P7SOURCE\_V6 instrument-response functions were used throughout this work. A zenith-angle cut of 100$^\circ$ was applied to eliminate albedo gamma rays from the Earth’s limb, excluding time intervals during which any part of the region of interest (ROI) was outside the field of view. In addition, time intervals were removed during which the observatory was transiting the Southern Atlantic Anomaly or the rocking angle exceeded 52$^\circ$. The ROI is defined to be a square region of the sky measuring $14^{\circ}$ on a side and centered on $\alpha_{J2000}=194.953$ and $\delta_{J2000}=27.9806$, the nominal position of the Coma cluster. Only photons with reconstructed energy greater than 1 GeV are considered, for which the 68%-containment radius of the point-spread function (PSF) is narrower than $\sim0.8^{\circ}$. The Fermi-LAT collaboration estimates the systematic uncertainties on the effective area at 10 GeV to be around 10% [^4]. The background emission in the ROI was modeled using fourteen point sources listed in the second LAT source catalog [@2012ApJS..199...31N], the LAT standard Galactic diffuse emission component (`gal_2yearp7v6_v0.fit`), and the corresponding isotropic template (`iso_p7v6source.txt`) that accounts for extragalactic emission and residual cosmic-ray contamination. Due to the large tails of the PSF at low energy, further fourteen point sources, lying $\sim4^\circ$ outside the ROI, were included in the source model. The energy spectra of twenty-four sources are described by a power law. The remaining four sources[^5], being bright sources, are modeled with additional degrees of freedom using the log-normal representation, which is typically used for modeling Blazar spectra. The analysis is performed in three energy bins: 1–3 GeV, 3–10 GeV, and 10–30 GeV. To find the best fit spectral parameters, a binned maximum-likelihood analysis [@1996ApJ...461..396M] is performed for each energy bin on a map with $0.1^{\circ}$ pixel size in gnomonic (TAN) projection, covering the entire ROI. To determine the significance of the sources, and in particular that of the Coma cluster, the analysis tool uses the likelihood-ratio test statistic [@1996ApJ...461..396M] defined as, $${\rm TS}=-2\left(\ln L_0-\ln L\right),$$ where $L_0$ is the maximum likelihood value for the null hypothesis and $L$ is the maximum likelihood with the additional source at a given position on the sky. In the likelihood analysis the spatial parameters of the sources were kept fixed at the values given in the catalog, whereas the spectral parameters of the point sources in the ROI, along with the normalization of the diffuse components, were allowed to freely vary. We analyzed three cases in which the gamma-ray emission from the Coma cluster was assumed to follow a power-law spectrum with a photon index $\alpha=2.1$, 2.3, and 2.5. The spectral indices of all point sources were permitted to freely vary between $\alpha=0$ and $\alpha=5$. We considered the emission as being caused both by a point-like and a spatially extended source (a uniform disk) with radius $r=0.2^\circ$ or $r=0.4^\circ$, as in the VERITAS analysis. No significant gamma-ray signal was detected. For one free parameter, the flux from the Coma cluster, the detection significance is computed as the square root of the test statistic (TS follows a $\chi_1^2$ distribution). The highest test statistic was obtained for the high-energy band, where TS $\sim0.8$ for the point source model, TS $\sim0.7$ for the disk model with $r=0.2^{\circ}$, and TS $\sim2$ for the disk model with $r=0.4^{\circ}$. We therefore used the profile likelihood method [@article:Rolke_etal:2005] to derive flux upper limits at the 99% confidence level in the energy range 1–30 GeV, assuming both an unresolved, point-like or spatially extended emission, as shown in Table \[table:fermi\]. Gamma Ray Emission from Cosmic Rays =================================== We decided to adopt a multifaceted approach to constrain the CR-to-thermal pressure distribution in the Coma cluster using the upper limits derived from the VERITAS and -LAT data in this work. This approach includes (1) a simplified multi-frequency analytical model that assumes a constant CR-to-thermal energy density and a power-law spectrum in momentum, (2) an analytic model derived from cosmological hydrodynamical simulations of the formation of galaxy clusters, and (3) a model that uses the observed intensity profile of the giant radio halo in Coma to place a lower limit on the expected gamma-ray flux in the hadronic model – where the radio-emitting electrons are secondaries from CR interactions and which is independent of the magnetic field distribution. This last approach translates into a minimum CR pressure which, if challenged by tight gamma-ray limits/detections, permits scrutiny of the hadronic interaction model of the formation of giant radio halos. Alternatively, realizing a spatial CR distribution that is consistent with the flux upper limits, and requiring the model to match the observed radio data, enables us to derive a lower limit on the magnetic field distribution. We stress again that this approach assumes the validity of the hadronic interaction model. Modeling the CR distribution through different techniques enables us to bracket our lack of understanding about the underlying plasma physics that shapes the CR distribution hence to reflect the Bayesian priors that are imposed on the modeling [see @article:PinzkePfrommerBergstrom for a discussion]. Simplified analytical model {#sec:simple} --------------------------- We start by adopting a simplified analytical model that assumes a power-law CR spectrum and a constant CR-to-thermal pressure ratio, i.e., we adopt the isobaric model of CRs following the approach of @article:PfrommerEnsslin:2004b. To be independent of additional assumptions and in line with earlier work in the literature, we do not impose a low-momentum cutoff, $q$, on the CR distribution function, i.e., we adopt $q=0$. Since, [*a priori*]{}, the CR spectral index is unconstrained,[^6] we vary it in the range $2.1<\alpha<2.5$, which is compatible with the radio spectral index of the giant radio halo of the Coma cluster after accounting for the spectral steepening at frequencies $\nu\sim5~{\mathrm}{GHz}$ due to the Sunyaev-Zel’dovich effect [@article:Ensslin:2002; @article:PfrommerEnsslin:2004b].[^7] To model the thermal pressure, we adopt the electron density profile for the Coma cluster that has been inferred from ROSAT X-ray observations [@article:BrielHenryBohringer:1992] and use a constant temperature of $kT= 8.25$ keV throughout the virial region. Table \[table:constraints\_simple\] shows the resulting constraints on the CR-to-thermal pressure ratio, $X_{{\mathrm{CR}}} = {\left\langle P_{{\mathrm{CR}}} \right\rangle}/{\left\langle P_{\mathrm}{th} \right\rangle}$, averaged within the virial radius, $R_{\mathrm}{vir}=2.2$ Mpc, that we define as the radius of a sphere enclosing a mean density that is 200 times the critical density of the Universe. Constraints on $X_{\mathrm{CR}}$ with VERITAS flux upper limits (99% CL) strongly depend on $\alpha$. This is due to the comparably large energy range from GeV energies (that dominate the CR pressure, provided $\alpha>2$ and the CR population has a nonrelativistic low-momentum cutoff, i.e., $q<m_{p}c$, where $m_{p}$ is the proton mass) to energies at 220 GeV, where our quality selection criteria imposed the energy threshold. These gamma-ray energies correspond to 1.6 TeV CRs – an energy ratio of more than 3 orders of magnitude, which explains the sensitivity to small changes in $\alpha$. The flux measurements within 0.2$^{\circ}$ are the most constraining due to a competition between the integrated signal and the background as the integration radius increases. This yields limits on $X_{\mathrm{CR}}$ between 0.048 and 0.43 (for $\alpha$ varying between 2.1 and 2.5), with a constraint of $X_{\mathrm{CR}}<0.1$ for $\alpha=2.3$ (close to the spectral index predicted by the simulations of @article:PinzkePfrommer:2010 around 220 GeV). Constraints on $X_{\mathrm{CR}}$ with -LAT limits (99% CL) depend only weakly on $\alpha$ because GeV-band gamma rays are produced by CRs with energies near the relativistic transition, that dominantly contribute to the CR pressure. $X_{\mathrm{CR}}$-constraints with -LAT limits are most constraining for an aperture of 0.4$^{\circ}$; despite the slightly weaker flux upper limits in comparison to the smaller radii of integration, we expect a considerably larger gamma-ray luminosity due to the increasing volume in this model. The best limit of $X_{\mathrm{CR}}< 0.012$ is achieved for $\alpha=2.3$, while the limit for $\alpha=2.1$ is only slightly worse $(X_{\mathrm{CR}}<0.017)$. Simulation-based approach {#sec:simulation} ------------------------- We complement the simplified analytical analysis with a more realistic and predictive approach derived from cosmological hydrodynamical simulations. We adopt the universal spectral and spatial gamma-ray model developed by @article:PinzkePfrommer:2010 to estimate the emission from decaying neutral pions which in clusters dominates over the inverse-Compton (IC) emission above 100 MeV. Given a density profile as, e.g., inferred by cosmological simulations or X-ray observations, the analytic approach models the CR distribution and the associated radiative emission processes from radio to the gamma-ray band. This formalism was derived from high-resolution simulations of clusters of galaxies that included radiative hydrodynamics, star formation and supernova feedback, and it followed the CR physics by tracing the most important injection and loss processes self-consistently while accounting for the CR pressure in the equation of motion [@article:PfrommerSpringelEnsslinJubelgas; @article:EnsslinPfrommerSpringelJubelgas:2007; @article:JubelgasSpringelEnsslinPfrommer:2008]. The results are in line with earlier numerical results on some of the overall characteristics of the CR distribution and the associated radiative emission processes [@article:DolagEnsslin:2000; @article:MiniatiRyuKangJones:2001; @article:Miniati:2003; @article:Pfrommer_etal:2007; @article:PfrommerEnsslinSpringel:2008; @article:Pfrommer:2008]. The overall normalization of the CR and gamma-ray distribution scales nonlinearly with the acceleration efficiency at structure formation shocks. Following recent observations of supernova remnants [@article:Helder_etal:2009] as well as theoretical studies [@article:KangJones:2005], we adopt an optimistic but nevertheless realistic value of this parameter and assume that 50% of the dissipated energy at strong shocks is injected into CRs, with this efficiency decreasing rapidly for weaker shocks. Since the vast majority of internal formation shocks (merger and flow shocks) are weak shocks with Mach numbers $M\lesssim3$ [e.g., @article:Ryu_etal:2003], they do not contribute significantly to the CR population in clusters. Instead, strong shocks during the formation epoch of clusters and strong accretion shocks at the present time (at the boundary of voids and filaments/supercluster regions) dominate the acceleration of CRs which are adiabatically transported through the cluster. Hence, the model provides a plausible upper limit for the CR contribution from structure formation shocks in galaxy clusters which can be scaled with the effective acceleration efficiency. Other possible CR sources, such as AGN and starburst-driven galactic winds have been neglected for simplicity but could in principle increase the expected gamma-ray yield. These cosmological simulations only consider advective transport of CRs by [bulk gas flows that inject a turbulent cascade, leading to]{} centrally-enhanced density profiles. However, other means of CR transport such as diffusion and streaming may flatten the CR radial profiles. [The CRs stream along magnetic field lines in the opposite direction of the CR number density gradient (at any energy). In the stratified cluster atmosphere, this implies a net flux of CRs towards larger radii, equalizing the CR number density with time if not counteracted by advective transport. It has been suggested that advection velocities only dominate over the CR streaming velocities for periods with trans- and supersonic cluster turbulence during a cluster merger and drop below the CR streaming velocities for relaxing clusters. As a consequence, a bimodality of the CR spatial distribution is expected to result; with merging (relaxed) clusters showing a centrally concentrated (flat) CR energy density profile [@article:EnsslinPfrommerMiniatiSubramanian:2011]. This translates into a bimodality of the expected diffuse radio and gamma-ray emission of clusters, since more centrally concentrated CRs will find higher target densities for hadronic CR proton interactions. As a result of this, relaxed clusters could have a reduced gamma-ray luminosity by up to a factor of five [@article:EnsslinPfrommerMiniatiSubramanian:2011].]{} Hence, tight upper limits on the gamma-ray emission can constrain a combination of acceleration physics and transport properties of CRs. We adopt the density profile of thermal electrons as discussed in §\[sec:simple\] and model the temperature profile of the Coma cluster with a constant central temperature of $kT= 8.25$ keV and a characteristic decline toward the cluster periphery in accordance with a fit to the universal profile derived from cosmological cluster simulations [@article:PinzkePfrommer:2010; @article:Pfrommer_etal:2007] and the behavior of a nearby sample of deep [*Chandra*]{} cluster data [@article:Vikhlinin_etal:2005]. This enables us to adopt the spatial and spectral distribution of CRs according to the model by @article:PinzkePfrommer:2010 that neglects the contribution of supernova remnants, AGN, and cluster galaxies. Figure \[fig:spectrum\] shows the expected integral spectral energy distribution of Coma within the virial radius (dotted line). This suggests a spectral index of $\alpha=2.1$ in the energy interval 1-3 GeV and $\alpha=2.3$ for energies probed by VERITAS ($>220$ GeV). Also shown are integrals of the differential spectrum for finite energy intervals across the angular apertures tested in this study ([dashed]{} lines). These model fluxes (summarized in Table \[table:constraints\_simple\]) are compared to and VERITAS flux upper limits for the same energy intervals. Constraints on $X_{\mathrm{CR}}$ with the gamma-ray flux limit of in the energy interval 1-3 GeV ($<0.4^\circ$) are most constraining, [since that combination of a specific energy interval and aperture minimizes the ratio of the upper limit to the expected model flux. In particular, this upper limit is 24% below]{} the model predictions that assume an optimistically large shock-acceleration efficiency and CR transport parameters as laid out above. [Hence this enables us to constrain a combination of maximum shock acceleration efficiency and CR transport parameters.]{} In our further analysis, we use the most constraining -LAT flux limits in the energy interval 1-3 GeV as well as the gamma-ray flux limits of VERITAS in the energy range above 220 GeV. Figure \[fig:xcr\] shows the CR-to-thermal pressure ratio, $X_{{\mathrm{CR}}} = {\left\langle P_{{\mathrm{CR}}} \right\rangle}/{\left\langle P_{\mathrm}{th} \right\rangle}$, as a function of radial distance, $R$, from the Coma cluster center and contained within $R$. All radii are shown in units of the virial radius, $R_{\mathrm}{vir}=2.2$ Mpc. To compute the CR pressure, we assume a low-momentum cutoff of the CR distribution at $q = 0.8\,m_{p}c$, where $m_{p}$ is the proton mass. [This is suggested by cosmological cluster simulations and reflects]{} the high Coulomb cooling rates at low CR energies. The CR-to-thermal pressure ratio rises toward the outer regions on account of the higher efficiency of CR acceleration at the peripheral accretion shocks compared to the weak central flow shocks. Adiabatic compression of a mixture of CRs and thermal gas disfavors the CR pressure relative to the thermal pressure on account of the softer equation of state of CRs. The weak increase of $X_{{\mathrm{CR}}}$ toward the core is due to the comparably fast thermal cooling of gas. In the case of VERITAS, for the most constraining regions tested (within an aperture of radius 0.2$^{\circ}$), the predicted CR pressure is a factor of 7.2 below the inferred upper limits of VERITAS (see Table \[table:results\] and assuming a spectral index of $\alpha=2.3$ which matches the simulated one at energies $E_\gamma=200$ GeV). To first order, we can scale the averaged CR-to-thermal pressure ratio of our model by that factor, keep the spatial behavior and obtain an integrated limit of the CR-to-thermal pressure ratio of $X_{{\mathrm{CR}}}<0.112$ within 0.2$^{\circ}$ that translates to a limit within the cluster virial radius of $X_{\mathrm{CR}}<0.162$ (solid lines of Figure \[fig:xcr\]). This limit is less constraining by 50% in comparison to the simplified analytical model, which gives $X_{\mathrm{CR}}<0.1$. This difference is explained by the concavity of the simulated spectrum which therefore carries more pressure at GeV energies than a pure power-law spectrum with $\alpha=2.3$. [As already aluded to,]{} the most constraining -LAT upper limit in the energy interval 1-3 GeV ($<0.4^\circ$) is a factor of [0.76 smaller]{} than our model predictions (assuming $\alpha=2.1$ which is very close to the simulated spectral index for the energy range 1-3 GeV). Scaling our integrated CR-to-thermal pressure profile yields a constraint of $X_{{\mathrm{CR}}}<0.012$ within 0.4$^{\circ}$ that translates to a limit within the cluster virial radius of $X_{\mathrm{CR}}<0.017$ (dashed lines of Figure \[fig:xcr\]). The $X_{\mathrm{CR}}$ constraint evaluated within the cluster virial radius is comparable to the constraint of $X_{\mathrm{CR}}<0.017$ in our simplified model. Naturally, with the -LAT limits we probe the region around GeV energies that dominate the CR pressure, and we do not expect any differences to the simplified power-law model in comparison to the universal CR spectrum with its concave CR spectrum found in the simulations. Minimum gamma-ray flux {#sec:Fmin} ---------------------- For clusters that host radio halos, we can derive a minimum gamma-ray flux in the hadronic model of radio halos – where the radio-emitting electrons are secondaries from CR interactions. Hadronic interactions channel about the same power into secondary electrons and $\pi^{0}$-decay gamma rays. A stationary distribution of CR electrons loses all its energy to synchrotron radiation for strong magnetic fields ($B \gg B_{\mathrm}{CMB} \simeq (1+z)^2\,3.2 \mu{\rm G}$, where $B_{\mathrm}{CMB}$ is the equivalent magnetic field strength of the CMB so that $B_{\mathrm}{CMB}^2/8\pi$ equals the CMB energy density). Thus the ratio of gamma-ray to synchrotron flux becomes independent of the spatial distribution of CRs and thermal gas [@article:Voelk:1989; @article:Pohl:1994; @article:Pfrommer:2008], in particular with $\alpha_{\nu}\simeq 1$ as the observed synchrotron spectral index. Hence we can derive a minimum gamma-ray flux in the hadronic model $$\label{eq:Fmin} F_{\gamma,{\mathrm}{min}} = \frac{{\displaystyle}A_{\gamma}}{{\displaystyle}A_{\nu}}\frac{{\displaystyle}L_{\nu}}{{\displaystyle}4\pi D_{{\mathrm}{lum}}^{2}},$$ where $L_{\nu}$ is the observed luminosity of the radio mini-halo, $D_{{\mathrm}{lum}}$ denotes the luminosity distance to the respective cluster, and $A_\gamma$ and $A_\nu$ are dimensional constants that depend on the hadronic physics of the interaction [@article:Pfrommer:2008; @Pfrommer_etal:2008]. Lowering the magnetic field would require an increase in the energy density of CR electrons to reproduce the observed synchrotron luminosity and thus increase the associated gamma-ray flux. To derive a minimum gamma-ray flux that can be compared to the upper limits, we need to determine the radio flux within the corresponding angular regions. To this end, we fit the point-source-subtracted, azimuthally-averaged radio-halo profile at 1.38 GHz [@article:Deiss_etal:1997] with a $\beta$-model, $$\label{beta} S_{\nu} (r_{\bot})= S_{0} \left[ 1 + \left( \frac{r_{\bot}}{r_{{\mathrm}{c}}}\right)^{2}\right]^{-3\beta + 1/2},$$ where $S_{0} = 1.1 \times 10^{-3}\,{\mathrm}{Jy\,arcmin}^{-2}$, $r_{{\mathrm}{c}} = 450$ kpc, and $\beta = 0.78$. Within the error bars, this profile is consistent with 326-MHz data taken by @article:Govoni_etal:2001 when scaled with a radio spectral index of 1.15. The results for the minimum gamma-ray flux $F_{\gamma,{\mathrm}{min}}(>220~{\mathrm}{GeV})$ and the minimum CR-to-thermal pressure ratio $X_{{\mathrm{CR}},\,{\mathrm}{min}} = X_{\mathrm{CR}}F_{\gamma,{\mathrm}{min}}/F_{\gamma,{\mathrm}{iso}}$ are shown in Table \[table:constraints\], where $F_{\gamma,{\mathrm}{iso}}$ is the gamma-ray flux in the simplified model introduced in §\[sec:simple\]. [Even in the most constraining cases, and assuming $\alpha\leq 2.3$, these are a factor of $\sim 60$ below the VERITAS upper limits (for $\alpha=2.1$, $<0.2^{\circ}$) and a factor of $\sim 20$ below the -LAT upper limits (for $\alpha=2.3$, $<0.4^{\circ}$)]{}. Note that these minimum gamma-ray fluxes are sensitive to the variation of the CR proton spectral index with energy as a result of, for example, momentum-dependent diffusion. Assuming a plausible value for the central magnetic field of Coma of 5 $\mu$G [@article:Bonafede_etal:2010], the radio-halo emission at GHz frequencies is dominated by electrons with energy $E_{\mathrm}{e} \sim 2.5$ GeV (which corresponds to proton energies $E_{\mathrm}{p} \sim 40$ GeV). Gamma rays with an energy of $200$ GeV are produced by CR protons with an energy of $E_{\mathrm}{p} \sim 1.6$ TeV – a factor of 50 higher than those probed by radio halo observations. A steepening of the CR proton spectral index of 0.2 between 40 GeV and 1.6 TeV would imply a decrease in the minimum gamma-ray flux by a factor of two. Constraining the Magnetic Field {#sec:B} ------------------------------- In the previous section, we have obtained an absolute lower limit on the gamma-ray emission in the hadronic model by assuming high magnetic fields, $B\gg B_{\mathrm}{CMB}$. We can turn the argument around and use our upper limit on the gamma-ray emission (and by extension on the CR pressure) to infer a lower limit on the magnetic field needed to explain the observed radio emission. This, again, assumes the validity of the hadronic model of radio halos, in which the radio-emitting electrons are secondaries from CR interactions. A stronger gamma-ray constraint will tighten the magnetic-field limit. In case of a conflict with magnetic field measurements by other methods, e.g., Faraday rotation measure (RM),[^8] the hadronic model of radio halos would be challenged. The method we use to constrain the magnetic field inherits a dependence on the assumed radial scaling which we parametrize as $$\label{eq:B} B(r) = B_{0} \,\left(\frac{n_{{\mathrm}{e}}(r)}{n_{{\mathrm}{e}}(0)}\right)^{\alpha_B},$$ as suggested by Faraday RM studies and numerical magnetohydrodynamical (MHD) simulations [@article:Bonafede_etal:2010; @article:Bonafede_etal:2011 and references therein]. Here $n_{{\mathrm}{e}}$ denotes the Coma electron density profile [@article:BrielHenryBohringer:1992]. In fact, the magnetic field in the Coma cluster is among the best constrained, because its proximity permits RM observations of seven radio sources located at projected distances of 50 to 1500 kpc from the cluster center. The best-fit model yields $B_{0} = 4.7^{+0.7}_{-0.8}\,\mu$G and $\alpha_{B} = 0.5^{+0.2}_{-0.1}$ [@article:Bonafede_etal:2010]. We aim to constrain the central field strength, $B_{0}$, and we permit the magnetic decline, $\alpha_{B}$, to vary within a reasonable range of $\Delta\alpha_{B}=0.2$ as suggested by [those]{} Faraday RM studies. We proceed as follows: 1. Given a model for the magnetic field with $\alpha_B$ and an initial guess for $B_0$, we determine the profile of the CR-to-thermal pressure ratio, $X_{{\mathrm{CR}}}(r)$, by matching the hadronically-produced synchrotron emission to the observed radio-halo emission over the entire extent. To this end, we deproject the fit to the surface-brightness profile of Eq. \[beta\] (using an Abel integral equation, see Appendix of @article:PfrommerEnsslin:2004b) yielding the radio emissivity, $$\label{eq:Coma:radio} j_{\nu} (r) = \frac{S_{0}}{2\pi\, r_{{\mathrm}{c}}}\, \frac{6\beta - 1}{\left(1 + r^{2}/r_{{\mathrm}{c}}^{2}\right)^{3 \beta}}\, \mathcal{B}\left(\frac{1}{2}, 3\beta\right) = j_{\nu,0} \left(1 + r^2/r_{{\mathrm}{c}}^{2}\right)^{-3 \beta},$$ where $\mathcal{B}$ denotes the beta function. It is generically true for weak magnetic fields ($B<B_{{\mathrm}{CMB}}$) in the outer parts of the Coma halo that the product $X_{{\mathrm{CR}}}(r)X_{B}(r)$ (where $X_B$ denotes the magnetic-to-thermal energy density ratio) has to increase by a factor of about 100 toward the radio-halo periphery to account for the observed extent. If we were to adopt a steeper magnetic decline such as $\alpha_{B}=0.5$ which produces a flat $X_{B}(r)$, the CR-to-thermal pressure ratio would have to rise accordingly by a factor of 100. 2. Given this realization for $X_{{\mathrm{CR}}}$, we compute the pion-decay gamma-ray surface-brightness profile, integrate the flux within a radius of $(0.13, 0.2, 0.4)$ degree, and scale the CR profile in order to match the corresponding VERITAS/flux upper limits. This scaling factor, $X_{{\mathrm{CR}},0}$, depends on the CR spectral index, $\alpha$, (assuming a power-law CR population for simplicity), the radial decline of the magnetic field, $\alpha_{B}$, and our initial guess for $B_{0}$. 3. We then solve for $B_{0}$ while matching the observed synchrotron profile and fixing the profile of $X_{{\mathrm{CR}}}(r)$ as determined through the previous two steps. Note that for $B_{0} \gg B_{{\mathrm}{CMB}}$ and a radio spectral index of $\alpha_{\nu}=1$, the solution would be degenerate since the luminosity of the radio halo scales as $$\label{eq:Lnu} L_{\nu} \propto \int dV Q(E)\,\frac{B^{1+\alpha_\nu}}{B^2 + B_{{\mathrm}{CMB}}^2} \to \int dV Q(E),$$ where $Q(E)$ denotes the electron source function. 4. Inverse-Compton cooling of CR electrons on CMB photons introduces a characteristic scale of $B_{{\mathrm}{CMB}}\simeq 3.2\,\mu$G which imprints as a nonlinearity on the synchrotron emissivity as a function of magnetic field strength (see Eq. (\[eq:Lnu\])). Hence we have to iterate through the previous steps until our solution for the minimum magnetic field $B_{0}$ converges. Table \[table:Bmin\] shows the resulting lower limit of the central magnetic field ranging from $B_{0} = 0.5$ to $1.4\,\mu$G in case the of VERITAS and from $B_{0} = 1.4$ to $5.5\,\mu$G in the case of -LAT.[^9] Since these lower limits on $B_{0}$ are below the values favored by Faraday RM for [most of the]{} parameter space spanned by $\alpha_{B}$ and $\alpha$ [(and never exceed the values for the phenomenological Faraday RM-inferred $B$-model)]{}, the hadronic model is a viable explanation of the Coma radio halo. [In fact, the -LAT upper limits start to rule out the parameter combination of $\alpha_{B}\gtrsim 0.7$ and $\alpha \gtrsim 2.5$ for the hadronic model of the Coma radio halo.]{} Future gamma-ray observations of the Coma cluster may put more stringent constraints on the parameters of the hadronic model. A few remarks are in order. (1) For the VERITAS limits, the hardest CR spectral indices correspond to the tightest limits on $B_{0}$, because the CR flux is constrained around 1 TeV and a comparably small fraction of CRs at 100 GeV would be available to produce radio-emitting electrons. A high magnetic field would be required to match the observed synchrotron emission. The opposite is true for the upper limits at 1 GeV, which probe CRs around a pivot point of 8 GeV: a soft CR spectral index implies a comparably small fraction of CRs at 100 GeV and hence a strong magnetic field is needed to match the observed synchrotron flux. (2) For a steeper magnetic decline (larger $\alpha_{B}$), the CR number density needs to be larger to match the observed radio-emission profiles, which would yield a higher gamma-ray flux so that the upper limits are more constraining. This implies tighter lower limits for $B_{0}$. (3) Interestingly, in all cases, the 0.4$^{\circ}$-aperture limits are the most constraining. For a given magnetic realization, a substantially increasing CR-to-thermal pressure profile is needed to match the observed radio profiles, and therefore that CR realization produces a larger flux within 0.4$^{\circ}$ in comparison with the simplified CR model ($X_{{\mathrm{CR}}} = {\mathrm}{const.}$), for which the 0.2$^{\circ}$-aperture limits are more constraining in the case of VERITAS. Physically, the large CR pressure in the cluster periphery may arise from CR streaming into the large available phase space in the outer regions. As a final word, in Table \[table:Bmin\] we show the corresponding values for the CR-to-thermal pressure ratio (at the largest emission radius at 1 Mpc) such that the model reproduces the observed radio surface-brightness profile.[^10] They should be interpreted as upper limits since they are derived from flux upper limits. For the -LAT upper limits, they range from 0.08 to 0.27; hence the $X_{{\mathrm{CR}}}$ profiles always obey the energy condition, i.e., $P_{{\mathrm{CR}}} < P_{\mathrm{th}}$, over the entire range of the radio-halo emission ($< 1$ Mpc).[^11] The corresponding values for $X_{\mathrm{CR}}$ in the cluster center are smaller than 0.01 for the entire parameter space probed in this study. We conclude that the hadronic model is not challenged by current Faraday RM data and is a perfectly viable possibility in explaining the Coma radio-halo emission. Emission from Dark Matter Annihilations ======================================= As already mentioned in the introduction, most of the mass in a galaxy cluster is in the form of DM. While the nature of DM remains unknown, a compelling theoretical candidate is a WIMP. The self-annihilation of WIMPs can produce either monoenergetic gamma-ray lines or a continuum of secondary gamma rays that deviates significantly from the power-law spectra observed from most conventional astrophysical sources, with a sharp cut-off at the WIMP mass. These spectral features together with the expected difference in the intensity distribution compared to conventional astrophysical sources allow a clear, indirect detection of DM. The expected gamma-ray flux due to self-annihilation of WIMPs in a dark-matter halo is given by $$\frac{d\Phi_{\gamma}(\Delta\Omega,E)}{dE}= \frac{{\left\langle \sigma v \right\rangle}}{8\pi m_{\chi}^{2}}\,\frac{dN_{\gamma}}{dE}\, J(\Delta\Omega), \label{eqn:WIMPflux}$$ where ${\left\langle \sigma v \right\rangle}$ is the thermally-averaged product of the total self-annihilation cross section and the relative WIMP velocity, $m_{\chi}$ is the WIMP mass, $\frac{dN_{\gamma}}{dE}$ is the differential gamma-ray yield per annihilation[^12], $\Delta\Omega$ is the observed solid angle, and $J$ is the so-called astrophysical factor – a factor which determines the DM annihilation rate and depends on the DM distribution. Given the upper limit on the observed gamma-ray rate, defined as the ratio of the event number detected within the observing time $T_{\mathrm{obs}}$, $R_{\gamma}(99\%\ \mathrm{CL}) = N_{\gamma}(99\%\ \mathrm{CL}) / T_{\mathrm{obs}}$, we can place constraints on the WIMP parameter space $(m_{\chi}, {\left\langle \sigma v \right\rangle})$. Integrating Eq. (\[eqn:WIMPflux\]) over energy we find $${\left\langle \sigma v \right\rangle}(99\%\ \mathrm{CL}) < R_{\gamma}(99\%\ \mathrm{CL})\, \frac{8\pi m_{\chi}^{2}}{J(\Delta\Omega)}\, \left[\int^{m_{\chi}}_{0} dE\ A_{\mathrm{eff}}\,\frac{dN_\gamma (E)}{dE}\right]^{-1},$$ where $A_{\mathrm{eff}}$ is the effective area of the gamma-ray detector. Because the self-annihilation of a WIMP is a two-body process, the astrophysical factor $J(\Delta\Omega)$ is the line-of-sight integral of the DM density squared $$J(\Delta\Omega)=\int_{\Delta\Omega}d\Omega\int d\lambda\ \rho_{\chi}^{2}(\lambda,\Omega),$$ where $\lambda$ represents the line of sight. In this work, we have modeled the Coma DM distribution with a Navarro, Frenk and White (NFW) profile [@article:NavarroFrenkWhite:1997], $$\rho_{\chi}(r)=\rho_{s}\left(\frac{r}{r_{s}}\right)^{-1}\left(1+\frac{r}{r_{s}}\right)^{-2},$$ where $r_{s}$ is the scale radius and $\rho_{s}$ is the scale density. Using weak-lensing measurements of the virial mass in the Coma cluster and the DM halo mass-concentration relation derived from $N$-body simulation of structure formation [@article:Bullock_etal:2001], @article:Gavazzi_etal:2009 find, and list in their Table 1 (note that they define $R_{\rm vir}=R_{\rm 100})$, $M_{\rm vir}=M_{\rm 200} =9.7(+6.1/-3.5)\cdot 10^{14}\, h^{-1}\ {\rm M_\odot}$ and $C_{\rm vir}=C_{\rm 200}=3.5(+1.1/-0.9)$, which we translate into the density-profile parameters $r_{s}=0.654$ Mpc and $\rho_{s}=4.4\times 10^{14}$ M$_{\odot}$/Mpc$^{3}$. Note that the uncertainties are not necessarily distributed as a Gaussian, and also arise from the choice of dark-matter profile. According to the latest high-resolution DM-only simulations of nine rich galaxy clusters, the inner regions of the smooth density profiles are quite well approximated by the NFW formula [@article:Gao_etal:2012]. However, gravitational interactions of DM with baryons may modify these predictions. This could give rise to either an increasing inner density slope due to adiabatic contraction of the DM component in response to cooling baryons in the central regions or a decreasing density slope due to violent baryonic feedback processes pushing gas out of the center by, e.g., energy injection through AGNs. However, on scales $r\gtrsim 0.45$ Mpc or more than 20% of $R_{\rm vir}$ (which are of relevance for the present work), different assumptions about the inner slope of the smooth DM density profile translate to uncertainties in the resulting astrophysical factor. Table \[table:astrofactor\] lists the astrophysical factors calculated for the different VERITAS apertures considered in this work. Table \[table:astrofactor\] also lists the astrophysical factor calculated for the background region, which is used to estimate the gamma-ray contamination from DM annihilation in the background region. As long as the DM contribution to the event number in the background region is negligible, the upper limits derived here directly scale with the astrophysical factor, ${\rm UL} (<\sigma v>)\propto J^{-1}$. The analysis uses a ring region to estimate the background in a ON region. We have to compute the expected level of gamma-ray emission from DM annihilation in the ring region in order to check that it is negligible with respect to the level of gamma-ray emission from DM annihilation in the ON region. This is equivalent to compute the astrophysical factor of the ON and OFF source region since this quantity is related to the rate of DM annihilations. The resulting exclusion curves in the $({\left\langle \sigma v \right\rangle}, m_{\chi})$ parameter space are shown in Figure \[fig:dm\] for three different DM self-annihilation channels, W$^{+}$W$^{-}$, b$\bar{\mathrm{b}}$, and $\tau^{+}\tau^{-}$. Depending on the DM annihilation channel, the limits are on the order of $10^{-20}$ to $10^{-21}$ cm$^{3}$ s$^{-1}$. The minimum for each exclusion curve and corresponding DM particle mass is listed in Table \[table:DMlimits\]. We stress that these limits are derived with conservative estimates of the astrophysical factor $J$. They do not include any boost to the annihilation rate possibly due to DM substructures populating the Coma halo, which could scale down the present limits by a factor $O(1000)$ in the most optimistic cases [@article:PinzkePfrommerBergstrom; @article:Gao_etal:2012]. We also note that when the size of the integration region is increased, the limits on ${\left\langle \sigma v \right\rangle}$ result from a competition between the gain in the astrophysical factor ${\left\langle J \right\rangle}$ and the integrated background. For integration regions larger than 0.2$^{\circ}$ in radius, the astrophysical factors no longer compensate for the increased number of background events, and the signal-to-noise ratio deteriorates. Discussion and Conclusions ========================== We have reported on the observations of the Coma cluster of galaxies in VHE gamma rays with VERITAS and complementary observations with the *Fermi*-LAT. VERITAS observed the Coma cluster of galaxies for a total of 18.6 hours of high-quality live time between March and May in 2008. No significant excess of gamma rays was detected above an energy threshold of $\sim 220$ GeV. The *Fermi*-LAT has observed the Coma cluster in all-sky survey mode since its launch in June 2008. We have used all data available from launch to April 2012 for an updated analysis compared to published results [@article:Ackermann_etal:2010]. Again, no significant excess of gamma rays was detected. We have used the VERITAS and *Fermi*-LAT data to calculate flux upper limits at the 99% confidence level for the cluster core (considered as both a point-like source and a spatially-extended emission region) and for three member galaxies. The flux upper limits obtained were then used to constrain properties of the cluster. We have employed various approaches to constrain the CR population and magnetic field distribution that are complementary in their assumptions and hence well suited to assessment of the underlying Bayesian priors in the models. (1) We used a simplified “isobaric CR model” that is characterized by a constant CR-to-thermal pressure fraction and has a power-law momentum spectrum. While this model is not physically justified [*a priori*]{}, it is simple and widely used in the literature and captures some aspects of more elaborate models such as (2) the simulation-based analytical approach of @article:PinzkePfrommer:2010. The latter is a “first-principle approach” that predicts the CR distribution spectrally and spatially for a given set of assumptions. It is powerful since it only requires the density profile as input due to the approximate universality of the CR distribution (when neglecting CR diffusion and streaming). Note, however, that inclusion of these CR transport processes may be necessary to explain the radio-halo bimodality. (3) Finally, we used a pragmatic approach which models the CR and magnetic distributions in order to reproduce the observed emission profile of the Coma radio halo. While this approach is also not physically justified, it is powerful because it shows what the “correct” model has to achieve and can point in the direction of the relevant physics. Within this pragmatic approach, we employ two different methods. Firstly, adopting a high magnetic field everywhere in the cluster ($B\gg B_{\mathrm}{CMB}$) yields the minimum gamma-ray flux in the hadronic model of radio halos which we find to be a factor of 20 (60) below the most constraining flux upper limits of -LAT (VERITAS). Secondly, by matching the radio-emission profile (i.e., fixing the radial CR profile for a given magnetic field model) and by requiring the pion-decay gamma-ray flux to match the flux upper limits (i.e., fixing the normalization of the CR distribution), we obtain lower limits on the magnetic field distribution under consideration. Our limits for the central magnetic field range from $B_{0} = 0.5$ to $1.4\,\mu$G (for VERITAS flux limits) and from $B_{0} = 1.4$ to $5.5\,\mu$G (for -LAT flux limits). Since all [*(but one) of*]{} these lower limits on $B_0$ are below the values favored by Faraday RM, $B_{0} = 4.7^{+0.7}_{-0.8}\,\mu$G [@article:Bonafede_etal:2010], the hadronic model is a very attractive explanation of the Coma radio halo. [The -LAT upper limits start to rule out the parameter combination of $\alpha_{B}\gtrsim 0.7$ and $\alpha \gtrsim 2.5$ for the hadronic model of the Coma radio halo.]{} Applying our simplified “isobaric CR model” to the most constraining VERITAS limits, we can constrain the CR-to-thermal pressure ratio, $X_{\mathrm{CR}}$, to be below 0.048–0.43 (for a CR or gamma-ray spectral index, $\alpha$, varying between 2.1 and 2.5). We obtain a constraint of $X_{\mathrm{CR}}<0.1$ for $\alpha=2.3$, the spectral index predicted by simulations at energies around 220 GeV. This limit is more constraining by a factor of 1.6 than that of the simulation-based model which gives $X_{\mathrm{CR}}<0.16$. This difference is due to the concave form of the simulated spectrum which provides more pressure at GeV energies in comparison to a pure power-law spectrum of $\alpha=2.3$. The -LAT flux limits constrain $X_{\mathrm{CR}}$ to be below 0.012–0.017 (for $\alpha$ varying between 2.3 and 2.1), only weakly depending on the assumed CR spectral index. Assuming $\alpha=2.1$, which is very close to the simulated spectral index for the energy range of 1–3 GeV, we obtain a constraint which is identical to that from our simulation-based model within the virial radius of $X_{\mathrm{CR}}<0.017$. That constraint improves to $X_{{\mathrm{CR}}}<0.012$ for an aperture of 0.4$^\circ$ corresponding to a physical scale of $R \simeq R_{200}/3 \simeq 660$ kpc. [These upper limits are now starting to constrain the CR physics in self-consistent cosmological cluster simulations and cap the maximum CR acceleration efficiency at structure formation shocks to be $<50\%$. Alternatively, this may argue for non-negligible CR transport processes such as CR streaming and diffusion into the outer cluster regions [@article:Aleksic_etal:2012].]{} These are encouraging results in that we constrain the CR pressure (of a phase that is fully mixed with the ICM) to be at most a small fraction ($<0.017$) of the overall pressure. As a result, hydrostatic cluster masses and the total Comptonization parameter due to the Sunyaev-Zel’dovich effect suffer at most a very small bias due to CRs. We have also used the flux upper limits obtained with VERITAS to constrain the thermally-averaged product of the total self-annihilation cross section and the relative velocity of DM particles. Modeling the Coma cluster DM halo with a NFW profile we derived limits on ${\left\langle \sigma v \right\rangle}$ to be on the order of $10^{-20}$ to $10^{-21}$ cm$^{-3}$ s$^{-1}$ depending on the chosen aperture. These limits are based on conservative estimates of the astrophysical factor, where a possible boost to the annihilation rate due to DM substructures in the cluster halo has been neglected. Including such a boost could scale down the present limits by a factor $O(1000)$ in the most optimistic cases. This research is supported by grants from the U.S. Department of Energy Office of Science, the U.S. National Science Foundation and the Smithsonian Institution, by NSERC in Canada, by Science Foundation Ireland (SFI 10/RFP/AST2748) and by STFC in the U.K. We acknowledge the excellent work of the technical support staff at the Fred Lawrence Whipple Observatory and at the collaborating institutions in the construction and operation of the instrument. The *Fermi* LAT Collaboration acknowledges generous ongoing support from a number of agencies and institutes that have supported both the development and the operation of the LAT as well as scientific data analysis. These include the National Aeronautics and Space Administration and the Department of Energy in the United States, the Commissariat à l’Energie Atomique and the Centre National de la Recherche Scientifique / Institut National de Physique Nucléaire et de Physique des Particules in France, the Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in Italy, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), High Energy Accelerator Research Organization (KEK) and Japan Aerospace Exploration Agency (JAXA) in Japan, and the K. A. Wallenberg Foundation, the Swedish Research Council and the Swedish National Space Board in Sweden. Additional support for science analysis during the operations phase is gratefully acknowledged from the Istituto Nazionale di Astrofisica in Italy and the Centre National d’Études Spatiales in France. C.P. gratefully acknowledges financial support of the Klaus Tschira Foundation. A.P. acknowledges NSF grant AST-0908480 for support. [lcc]{} Core & [12$^{\mathrm{h}}$59$^{\mathrm{m}}$48.7$^{\mathrm{s}}$]{} & [+27$^{\circ}$5850.0]{}\ NGC 4889 & [13$^{\mathrm{h}}$00$^{\mathrm{m}}$08.13$^{\mathrm{s}}$]{} & [+27$^{\circ}$5837.03]{}\ NGC 4874 & [12$^{\mathrm{h}}$59$^{\mathrm{m}}$35.71$^{\mathrm{s}}$]{} & [+27$^{\circ}$5733.37]{}\ NGC 4921 & [13$^{\mathrm{h}}$01$^{\mathrm{m}}$26.12$^{\mathrm{s}}$]{} & [+27$^{\circ}$5309.59]{}\ \[table:roi\] [lccccccccc]{} Core & 0 & 17 & 0.84 & 2.59 & (0.78%) & 2.78 & (0.83%) & 2.97 & (0.89%)\ & 0.2 & -41 & -1.0 & 1.96 & (0.59%) & 2.09 & (0.63%) & 2.21 & (0.66%)\ & 0.4 & -26 & -0.30 & 4.44 & (1.3%) & 4.74 & (1.4%) & 5.02 & (1.5%)\ NGC 4889 & 0 & 3 & 0.14 & - & - & 1.85 & (0.55%) & - & -\ NGC 4874 & 0 & -14 & -0.71 & - & - & 1.51 & (0.45%) & - & -\ NGC 4921 & 0 & -4 & -0.23 & - & - & 2.41 & (0.72%) & - & - \[table:results\] [lccc]{} & 1.882 (0.000) & 0.759 (0.000) & 0.671 (0.830)\ & 2.109 (0.152) & 0.899 (0.000) & 0.719 (0.740)\ & 2.438 (0.201) & 1.232 (0.619) & 0.875 (1.387)\ & 1.946 (0.000) & 0.788 (0.000) & 0.667 (0.874)\ & 2.180 (0.169) & 0.941 (0.000) & 0.725 (0.828)\ & 2.524 (0.246) & 1.275 (0.742) & 0.869 (1.390)\ & 2.008 (0.000) & 0.816 (0.000) & 0.663 (0.915)\ & 2.246 (0.189) & 0.979 (0.020) & 0.720 (0.864)\ & 2.606 (0.291) & 1.313 (0.856) & 0.861 (1.387)\ \[table:fermi\] [cccccc]{}\ 0 & 0.1 & 0.23 & 0.97 & 1.9 & 14.8\ 0.2 & 0.048 & 0.10 & 0.43 & 2.9 & 7.2\ 0.4 & 0.067 & 0.15 & 0.62 & 4.4 & 10.8\ 0 & 0.035 & 0.024 & 0.033 & 1.4 & 1.34\ 0.2 & 0.024 & 0.017 & 0.022 & 2.1 & 1.00\ 0.4 & 0.017 & 0.012 & 0.016 & 3.2 & 0.76\ \[table:constraints\_simple\] [ccccccc]{}\ 0 & 1.6 & 0.7 & 0.3 & 6.7 & 6.1 & 11\ 0.2 & 3.1 & 1.4 & 0.6 & 7.8 & 7.2 & 13\ 0.4 & 6.3 & 2.8 & 1.3 & 9.8 & 9.0 & 16\ 0 &   3.5 &   4.8 &   6.4 & 6.7 & 6.1 & 11\ 0.2 &   6.8 &   9.3 & 12.5 & 7.8 & 7.2 & 13\ 0.4 & 13.5 & 18.6 & 25.0 & 9.8 & 9.0 & 16\ \[table:constraints\] [ccccccc]{} 0.3 & 0.69 & 0.57 & 0.48 & 1.38 & 1.95 & 2.68\ 0.5 & 0.97 & 0.80 & 0.68 & 1.94 & 2.74 & 3.78\ 0.7 & 1.40 & 1.17 & 0.99 & 2.80 & 3.97 & 5.50\ &\ 0.3 & 0.46 & 1.05 &   4.55 & 0.11 & 0.08 & 0.11\ 0.5 & 0.74 & 1.70 &   7.47 & 0.18 & 0.13 & 0.17\ 0.7 & 1.09 & 2.59 & 11.55 & 0.27 & 0.19 & 0.26\ \[table:Bmin\] [ccc]{} 0 & $5.7\times 10^{16}$ & $1.3\times 10^{14}$ (negligible)\ 0.2 & $8.1\times 10^{16}$ & $4.4\times 10^{14}$ ($<0.01{\left\langle J \right\rangle}_{{\mathrm}{signal}}$, negligible)\ 0.4 & $9.4\times 10^{16}$ & $1.3\times 10^{15}$ ($\simeq0.01{\left\langle J \right\rangle}_{{\mathrm}{signal}}$, negligible) \[table:astrofactor\] [lccc]{} W$^{+}$W$^{-}$ & 0 & 2000 & $1.1\times 10^{-20}$\ & 0.2 & 1900 & $4.3\times 10^{-21}$\ & 0.4 & 1900 & $8.4\times 10^{-21}$\ $b\bar{b}$ & 0 & 3500 & $1.2\times 10^{-20}$\ & 0.2 & 3400 & $4.4\times 10^{-21}$\ & 0.4 & 3500 & $8.7\times 10^{-21}$\ $\tau^{+}\tau^{-}$ & 0 & 670 & $2.4\times 10^{-21}$\ & 0.2 & 650 & $9.1\times 10^{-22}$\ & 0.4 & 660 & $1.8\times 10^{-21}$ \[table:DMlimits\] [^1]: The integral flux sensitivity above 300 GeV was improved by about 30% with the relocation of one telescope in the summer of 2009. [^2]: The energy threshold is defined as the energy corresponding to the maximum of the product function of the observed spectrum and the collection area. It does not vary significantly for the different source scenarios and assumed spectral indices reported in this work. [^3]: In the array configuration prior to summer 2009, two telescopes had a separation of only 35 m. [^4]: http://fermi.gsfc.nasa.gov/ssc/data/analysis/LAT\_caveats.html [^5]: 2FGLJ1303.1+2435, 2FGLJ1310.6+3222, 2FGLJ1226.0+2953, and 2FGLJ1224.9+2122 [^6]: The hadronic interaction physics guarantees that the CR spectral index coincides with that of the resulting pion-decay gamma-ray emission at energies $E\gg 1$ GeV that are well above the pion bump [see discussion in @article:PfrommerEnsslin:2004b]. [^7]: Assuming a magnetic field of 1 $\mu$G, the CR protons responsible for the GHz radio emitting electrons have an energy of $\sim100$ GeV and are $\sim$ 20 times less energetic than those CR protons responsible for 200-GeV gamma-ray emission. [^8]: Generally, Faraday RM analyses of the magnetic field strength by, e.g., background sources observed through clusters, are degenerate with the magnetic coherence scale and may be biased by the unknown correlation between magnetic and density fluctuations. [^9]: Note that a central magnetic field of $3\,\mu$G corresponds in the Coma cluster to a magnetic-to-thermal energy density ratio of $X_B=0.005$. [^10]: Note that in this section, we determine the radial behavior of $X_{\mathrm{CR}}$ by adopting a specific model for the magnetic field and requiring the modeled synchrotron surface-brightness profile to match the observed data of the Coma radio halo. This is in contrast to the simplified analytical CR model where $X_{\mathrm{CR}}$ is constant (§ \[sec:simple\]) and to the simulation-based model where $X_{\mathrm{CR}}(r)$ is derived from cosmological cluster simulations (§ \[sec:simulation\]). [^11]: See Figure 3 in @article:PfrommerEnsslin:2004a for the entire parameter range assuming minimum energy conditions, and @article:PfrommerEnsslin:2004b, Figure 7 for a parametrization as adopted in this study. We caution, however, that the minimum-energy condition is violated at the outer radio-halo boundary for the range of minimum magnetic-field values inferred by this study. [^12]: In this work, we have calculated the differential gamma-ray yield per annihilation using the Pythia Monte Carlo simulator.

--- abstract: 'The Auger Engineering Radio Array (AERA) aims at the detection of air showers induced by high-energy cosmic rays. As an extension of the Pierre Auger Observatory, it measures complementary information to the particle detectors, fluorescence telescopes and to the muon scintillators of the Auger Muons and Infill for the Ground Array (AMIGA). AERA is sensitive to all fundamental parameters of an extensive air shower such as the arrival direction, energy and depth of shower maximum. Since the radio emission is induced purely by the electromagnetic component of the shower, in combination with the AMIGA muon counters, AERA is perfect for separate measurements of the electrons and muons in the shower, if combined with a muon counting detector like AMIGA. In addition to the depth of the shower maximum, the ratio of the electron and muon number serves as a measure of the primary particle mass.' address: - '^1^ Institute of Nuclear Physics, Karlsruhe Institute of Technology, Karlsruhe, Germany' - '^2^ Observatorio Pierre Auger, Av. San Martín Norte 304, 5613 Malargüe, Argentina' - 'Full author list: <http://www.auger.org/archive/authors_2015_09.html>' author: - E M Holt^1^ for the Pierre Auger Collaboration^2^ title: 'The Auger Engineering Radio Array and multi-hybrid cosmic ray detection' --- Introduction ============ The Auger Engineering Radio Array (AERA) is a modern radio experiment for the detection of cosmic ray air showers. It is the radio extension of the Pierre Auger Observatory, located in the Province of Mendoza in Argentina. With its area of 17km$^2$ AERA is the world’s largest experiment in the field of cosmic ray radio detection. The coincident measurement with the other low energy extensions of the Pierre Auger Observatory [@auger] enables the simultaneous measurement of a number of air shower properties. The Pierre Auger Observatory is an experiment for ultra-high-energy cosmic rays [@auger]. It combines different detection techniques to obtain complementary information about extensive cosmic ray air showers. 1660 water-Cherenkov stations form the surface detector (SD) and are distributed with a spacing of 1500m over an area of 3000km$^2$. They measure the particles of the showers at ground. Extensive air showers induce radiation in the ultraviolet range due to excitation of atmospheric nitrogen by the shower particles. On moonless clear nights, this radiation is detected by 27 fluorescence telescopes at four sites, overlooking the SD area and enabling the hybrid detection of showers. Several enhancements, aiming at lower energies to 10$^{17}$eV, were installed in one part of the Observatory. The Auger Muons and Infill for the Ground Array (AMIGA) [@AMIGA] covers an area of about 20km$^2$. In AMIGA the spacing between the water-Cherenkov stations is reduced to 750m, yielding full efficiency for air showers with a primary energy down to $10^{17.5}$eV. Buried muon scintillators, at 2.3m depth, accompany the water-Cherenkov stations for a better separation of electrons and muons in a shower. Seven stations with muon detectors, forming the “Unitary Cell”, have been taking data since 2013. Three high-elevation air fluorescence telescopes (High Elevation Auger Telescope – HEAT), observe low energy showers higher in the atmosphere. AERA is located inside the region of the dense stations, enabling a combined detection of showers and cross calibration of the detectors. In addition, AERA uses the particle detectors as an external trigger. The radio emission of air showers is mainly induced by two mechanisms: a) the geomagnetic effect due to deflection of the charged particles of the shower in the geomagnetic field [@geomagn], and b) the Askaryan effect due to positron annihilation and ionization of atmospheric molecules by the shower particles, which cause an excess of negative charges in the shower front [@askaryan]. Hence, the radio emission contains information on the development of the shower, in particular the position of the shower maximum X$_{\textrm{max}}$. In first order, the radio emission is caused purely by the electromagnetic part of the shower. Data taking is possible for almost 100% of the time. Hence, in comparison with the fluorescence detector (FD), an X$_{\textrm{max}}$ measurement is possible around the clock. Furthermore, radio detection – in contrast to particle detection – becomes more efficient for more inclined showers due to a larger footprint of the emission at ground [@inclined]. The Auger Engineering Radio Array - detector description ======================================================== AERA was built in three phases. Starting in September 2010, AERA24 was deployed with 24 radio detection stations (RDS) featuring logarithmic periodic dipole antennas (LPDA) [@LPDA] with a spacing of 144m between the stations. AERA24 was used to investigate the radio emission itself and to develop techniques for radio detection of cosmic rays. For the second phase, AERA124, another 100 RDS were installed in May 2013. These RDS feature a different antenna type, the so called butterfly antenna [@butterfly], and improved hardware. They are distributed with spacings of 250m and 375m, and together with the first 24 antennas, cover an area of about 6km$^2$. With this configuration AERA measures several thousand cosmic ray events per year from a primary energy of about 10$^{17}$eV up to the highest energies. In March 2015, 25 additional butterfly antenna stations were installed on a grid with a 750m spacing, aiming mainly on the detection of horizontal air showers (&gt;55$^{\circ}$ zenith angle). With some additional prototype stations, AERA153 now consists of 153 RDS on an array of about 17km$^2$. A map of AERA with the different phases and the other detectors of the Pierre Auger Observatory is shown in figure \[fig:AERAmap\]. An AERA butterfly antenna together with a water-Cherenkov station and buried muon scintillator of AMIGA is sketched in figure \[fig:detectors\]. All stations of AERA are equipped with a solar panel, battery and signal processing hardware. They work completely autonomous in the field and send the collected data upon request to a central data facility via a WiFi link. The antennas are aligned along the magnetic north-south and east-west direction. They are triggered externally by the particle and fluorescence detectors and from internal triggers. The signals are bandpass-filtered to the range of 30 – 80MHz. Results from AERA ================= AERA was built for several different purposes: to improve the understanding of the radio emission mechanisms, for cosmic ray physics in the transition region between galactic and extragalactic cosmic rays, and to test the feasibility of a large-scale radio array for the highest energies. Probing the theory of radio emission ------------------------------------ The different radio-emission mechanisms produce differently polarized radio emission – linearly polarized from the geomagnetic effect and radial polarized towards the shower axis from the Askaryan effect. By measuring the polarization of the radio emission, it is possible to study the contributions from the different effects. Since the contribution from the geomagnetic effect depends on the angle to the geomagnetic field, this contribution is different for different detector sites. Polarization measurements in AERA revealed a radial component with a mean contribution of 14% [@polarization], aside from the linear polarization. This measurements agree well with the theory of the geomagnetic and Askaryan emission processes. Properties of the primary cosmic ray particle --------------------------------------------- The main properties of a cosmic ray are its direction, energy and mass. Radio measurements of AERA are sensitive to all of these properties. The reconstruction of these properties is done by the Auger Offline software framework [@Offline], which includes all detector systems for combined analysis. [*Arrival direction of the primary cosmic ray particle:*]{} The shower axis corresponds to the arrival direction of the primary particle. This axis is reconstructed from the timing information of the radio pulses in the radio stations using a wavefront model for the radio emission. For this, a plane wave is used as first-order approximation for the wavefront shape. The reconstructed direction is in good agreement with the direction from the SD. [*Energy of the primary cosmic ray particle:*]{} The energy contained in the radio emission yields information about the primary cosmic ray energy. To calculate the radiation energy, the measured electric-field strength of the 30 – 80MHz radiation at the station positions is converted to the energy density. We use a two-dimensional lateral distribution function (2D-LDF) [@2dldf] taking into account asymmetries due to the combined geomagnetic and Askaryan effect, to interpolate the energy density. The integral over this 2D-LDF corresponds to the total radiation energy. A calibration against the SD shows that the radiation energy is 15.9MeV for a cosmic ray energy of 1EeV [@energy]. It scales quadratically with the cosmic ray energy because of the coherent character of the emission. This radiation energy can therefore be used as an energy estimator. In AERA, an energy resolution of 22% for a dataset of AERA24 events, and of 17% for a subset of events with high multiplicity ($\geq$ 5 radio stations) has been found. [*Shower maximum and mass composition:*]{} The depth in the atmosphere at which the number of secondary particles is maximum is called the shower maximum X$_{\textrm{max}}$. It is strongly correlated to the mass of the primary particle. The radio emission is mainly produced around and before this X$_{\textrm{max}}$ [@Scholten:Ludwig] and therefore carries information about it. In AERA, there are different studies ongoing to reconstruct X$_{\textrm{max}}$ from the radio emission, e.g. using shape parameters of the hyperbolic radio wavefront [@Qader], the width of the radio footprint in the shower plane [@johannesICRC], or the slope of the frequency spectrum in single radio stations [@spectralSlope]. The X$_{\textrm{max}}$ measurements by the FD are used for calibration and comparison of the results. Mass composition with multi-hybrid detection ============================================ ![Example of an event measured in all four detectors of the Pierre Auger Observatory. Upper left panel: map of AERA and SD stations with signal. Colours correspond to timing, the size of the circles and crosses to the signal strength. The black lines indicate the arrival direction reconstructed from AERA and SD. Upper middle panel: radio 2D-LDF. Upper right panel: SD LDF. Lower left panel: map of muon detector stations with signal. The size and colour corresponds to the muon density. Lower middle panel: muon LDF. Middle right panel: shower trace in FD camera. Lower right panel: Longitudinal profile of the shower measured by the FD. []{data-label="fig:hybridEvent"}](example_event_fourfold.pdf){width="\textwidth"} The ratio between the number of electrons and muons in the shower is correlated to the mass of the primary particle. For heavier nuclei, the shower development is faster, i.e. it takes less interaction generations in the cascade until the shower maximum. This leads to a smaller ratio of numbers of electrons and muons at the shower maximum. Hence, measuring the electrons and muons separately yields information about the primary particle type. Due to featuring different detector types at one site, the enhancement area of the Pierre Auger Observatory is a perfect place to prove the principle of complementary detection with AERA as a radio detector for the electromagnetic component and AMIGA as a muon counting detector. While particle detectors only see a snapshot of the shower of the moment it arrives at the detector, AERA does a calorimetric measurement of the electromagnetic part of the shower. This gives the number of electrons in the whole shower and especially around the shower maximum. The results can be cross-checked and calibrated with X$_{\textrm{max}}$ measurements of AERA as well as of FD. In addition, a combined measurement with the X$_{\textrm{max}}$ of AERA around the clock leads to a higher accuracy in the mass sensitivity through improved statistics. After around two years of combined data taking, more than 200 hybrid detected events are available for analysis. In figure \[fig:hybridEvent\], an example of a multi-hybrid event is shown, which was measured by all four detectors. Another approach aims at inclined showers. The more inclined the shower, the larger the footprint of the radio emission on ground. Therefore, the detection efficiency for a radio-antenna field like AERA increases with the zenith angle [@inclined]. The shower itself is older for larger zenith angles when it reaches ground and the electromagnetic part has mainly died out. Only the muons reach ground and are measured by particle detectors like the water-Cherenkov stations of the Pierre Auger Observatory. Hence, the electrons and muons can be measured separately by a combination of radio antennas and particle detectors, which enables measurements of the cosmic ray composition even for inclined showers. Conclusion ========== AERA detects several thousand cosmic ray events per year. Polarization measurements have shown that the emission mechanisms are well understood and that the contribution of the Askaryan effect at the AERA site constitutes 14%. AERA is sensitive to all cosmic ray properties: the arrival direction, energy and mass. The cosmic ray energy is derived from the energy stored in the radio emission. It is calibrated against the energy reconstructed with the particle detector and shows a resolution of 17%. The mass of the primary particle is correlated with the shower maximum, which can be reconstructed with different methods currently developed in AERA. Different types of detectors at the Pierre Auger Observatory allow for complementary hybrid measurements of air showers. This enables a separate measurement of the electromagnetic component with AERA and the muonic component with AMIGA, whose ratio contains information about the primary mass. In addition, for more inclined air showers, AERA has a higher detection efficiency and SD measures purely the muonic component. Thus, the complementary measurements of AERA and SD give potential for a significantly increased accuracy of the derived cosmic ray composition. References {#references .unnumbered} ========== [15]{} Pierre Auger Collaboration, Aab A et al. 2015 [*Nucl.Instrum.Meth.*]{} A [**798**]{} 172-213 Wundheiler B for the Pierre Auger Collaboration 2015 [*Proc. 34th ICRC*]{}, The Hague, 144-151 \[arXiv:1509.03732\] Kahn F D and Lerche I 1966 [*Proc. Roy. Soc.*]{} A [**289.1417**]{} 206-213 Askaryan G A 1962 [*Sov. Phys JETP*]{} [**14.2**]{} 441-443 Kambeitz O for the Pierre Auger Collaboration 2015 [*Submitted to: AIP Conf. Proc.*]{} ARENA 2014, Annapolis \[arXiv:1509.08289\] Pierre Auger Collaboration, Abreu P et al. 2012 [*JINST*]{} [**7**]{} P10011 Charrier D for the CODALEMA Collaboration 2012 [*Nucl.Instrum.Meth.*]{} A [**662**]{} S142-S145 Pierre Auger Collaboration, Aab A et al. 2014 [*Phys. 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--- abstract: 'We provide a compact analytic formula to compute the spin of the black hole produced by the coalescence of two black holes following a quasi-circular inspiral. Without additional fits than those already available for binaries with aligned or antialigned spins, but with a minimal set of assumptions, we derive an expression that can model generic initial spin configurations and mass ratios, thus covering all of the 7-dimensional space of parameters. A comparison with simulations already shows very accurate agreements with all of the numerical data available to date, but we also suggest a number of ways in which our predictions can be further improved.' author: - Luciano Rezzolla - Enrico Barausse - Ernst Nils Dorband - Denis Pollney - Christian Reisswig - Jennifer Seiler - Sascha Husa bibliography: - 'published\_version.bib' title: On the final spin from the coalescence of two black holes --- The evolution of black hole binary systems is one of the most important problems for general relativity, and more recently for astrophysics, as such systems enter the realm of observation. Recent advances in numerical relativity have made it possible to cover the entire range of the inspiral process, from large separations at which post-Newtonian (PN) calculations provide accurate orbital parameters, through the highly relativistic merger, to ringdown. For many studies of astrophysical interest, such as many-body studies of galactic mergers, or heirarchical models of black-hole formation however, it is impractical to carry out evolutions with the full Einstein, or even post-Newtonian, equations. Fortunately, recent binary black-hole evolutions in full general relativity have shown that certain physical quantities can be estimated to good accuracy if the initial encounter parameters are known. In particular, this paper develops a rather simple and robust formula for determining the spin of the black-hole remnant resulting from the merger of rather generic initial binary configurations. To appreciate the spirit of our approach it can be convenient to think of the inspiral and merger of two black holes as a mechanism which takes, as input, two black holes of initial masses $M_{1}$, $M_{2}$ and spin vectors $\boldsymbol{S}_{1}$, $\boldsymbol{S}_{2}$ and produces, as output, a third black hole of mass $M_{\rm fin}$ and spin $\boldsymbol{S}_{\rm fin}$. In conditions of particular astrophysical interest, the inspiral takes place through quasi-circular orbits since the eccentricity is removed quickly by the gravitational-radiation reaction [@Peters:1964]. Furthermore, at least for nonspinning equal-mass black holes, the final spin does not depend on the value of the eccentricity as long as it is not too large [@Hinder:2007qu]. The determination of $M_{\rm fin}$ and $\boldsymbol{S}_{\rm fin}$ from the knowledge of $M_{1,2}$ and $\boldsymbol{S}_{1,2}$, is of great importance in several fields. In astrophysics, it provides information on the properties of isolated stellar-mass black holes produced at the end of the evolution of a binary system of massive stars. In cosmology, it can be used to model the distribution of masses and spins of the supermassive black holes produced through the merger of galaxies (see ref. [@Berti2008] for an interesting example). In addition, in gravitational-wave astronomy, the a-priori knowledge of the final spin can help the detection of the ringdown. What makes this a difficult problem is clear: for binaries in quasi-circular orbits the space of initial parameters for the final spin has seven dimensions (*i.e.*, the mass-ratio $q\equiv M_2/M_1$ and the six components of the spin vectors). A number of analytical approaches have been developed over the years to determine the final spin, either exploiting the dynamics of point-particles [@Hughes:2002ei; @Buonanno:07b] or the PN approximation [@Gergely:07], or using more sophisticated approaches such as the effective-one-body approximation [@Buonanno:06cd]. Ultimately, however, computing $\boldsymbol{a}_{\rm fin} \equiv \boldsymbol{S}_{\rm fin}/M^2_{\rm fin}$ accurately requires the solution of the full Einstein equations and thus the use of numerical-relativity simulations. Several groups have investigated this problem over the last couple of years [@Campanelli:2006vp; @Pollney:2007ss; @Bruegmann:2007zj; @Rezzolla-etal-2007; @Marronetti07tbgs; @Rezzolla-etal-2007b]. While the recent possibility of measuring accurately the final spin through numerical-relativity calculations represents an enormous progress, the complete coverage of the full parameter space uniquely through simulations is not a viable option. As a consequence, work has been done to derive analytic expressions for the final spin which would model the numerical-relativity data but also exploit as much information as possible either from perturbative studies, or from the symmetries of the system [@Pollney:2007ss; @Rezzolla-etal-2007; @BoyleKesdenNissanke:07; @BoyleKesden:07; @Marronetti07tbgs; @Rezzolla-etal-2007b]. In this sense, these approaches do not amount to a blind fitting of the numerical-relativity data, but, rather, use the data to construct a physically consistent and mathematically accurate modelling of the final spin. Despite a concentrated effort in this direction, the analytic expressions for the final spin could, at most, cover 3 of the 7 dimensions of the space of parameters [@Rezzolla-etal-2007b]. Here, we show that without additional fits and with a minimal set of assumptions it is possible to obtain the extension to the complete space of parameters and reproduce all of the available numerical-relativity data. Although our treatment is intrinsically approximate, we also suggest how it can be improved. Analytic fitting expressions for $\boldsymbol{a}_{\rm fin}$ have so far been built using binaries having spins that are either *aligned* or *antialigned* with the initial orbital angular momentum. This is because in this case both the initial and final spins can be projected in the direction of the orbital angular momentum and it is possible to deal simply with the (pseudo)-scalar quantities $a_1$, $a_2$ and $a_{\rm fin}$ ranging between $-1$ and $+1$. If the black holes have *equal mass* but *unequal* spins that are either *parallel* or *antiparallel*, then the spin of the final black hole has been shown to be accurately described by the simple analytic fit [@Rezzolla-etal-2007] $$\label{eqmass_uneqspin} a_{\rm fin}(a_1,a_2)=p_0 + p_1 (a_1 + a_2) + p_2 (a_1 + a_2)^2\,,$$ where $p_0 = 0.6883 \pm 0.0003$, $p_1 = 0.1530 \pm 0.0004$, and $p_2 = -0.0088 \pm 0.0005$. When seen as a power series of the initial spins, expression  suggests an interesting physical interpretation. Its zeroth-order term, in fact, can be associated with the (dimensionless) orbital angular momentum not radiated in gravitational waves and amounting to $\sim 70\%$ of the final spin at most. The first-order term, on the other hand, can be seen as the contributions from the initial spins and from the spin-orbit coupling, amounting to $\sim 30\%$ at most. Finally, the second-order term, includes the spin-spin coupling, with a contribution to the final spin which is of $\sim 4\%$ at most. If the black holes have *unequal mass* but spins that are *equal* and *parallel*, the final spin is instead given by the analytic fit [@Rezzolla-etal-2007b] $$\begin{aligned} \label{eqspin_uneqmass} &&a_{\rm fin}(a,\nu)=a+s_{4}a^2 \nu+s_{5}a \nu^2+t_{0} a\nu+ \nonumber \\ && \hskip 1.75cm 2\sqrt{3}\nu+t_2\nu^2+t_{3}\nu^3\,,\end{aligned}$$ where $\nu$ is the symmetric mass ratio $\nu \equiv M_1M_2/(M_1+M_2)^2$, and where the coefficients take the values $s_4 = -0.129 \pm 0.012$, $s_5 = -0.384 \pm 0.261$, $t_0 = -2.686 \pm 0.065$, $t_2 = -3.454 \pm 0.132$, $t_3 = 2.353 \pm 0.548$. Although obtained independently in [@Rezzolla-etal-2007] and [@Rezzolla-etal-2007b], expressions  and  are compatible as can be seen by considering  for equal-mass binaries ($\nu =1/4$) and verifying that the following relations hold within the computed error-bars $$\label{relations} p_0= \frac{\sqrt{3}}{2} + \frac{t_2}{16} + \frac{t_3}{64}\,, \quad p_1 = \frac{1}{2} + \frac{s_5}{32} + \frac{t_0}{8}\,, \quad p_2 = \frac{s_4}{16}.$$ As long as the initial spins are aligned (or antialigned) with the orbital angular momentum, expression  can be extended to *unequal-spin, unequal-mass* binaries through the substitution $$\label{substitution} a \ \to \ \tilde{a} \equiv \frac{a_1 + a_2 q^2}{1+q^2} \,.$$ To obtain this result, it is sufficient to consider  and  as polynomial expressions of the generic quantity $$\tilde{a} \equiv a_{\rm tot} \frac{(1+q)^2}{1+q^2}\,.$$ where $a_{\rm tot} \equiv (a_1 + a_2 q^2)/(1+q)^2$ is the total dimensionless spin for generic aligned binaries. In this way, expressions  and   are naturally compatible, since $\tilde{a} = (a_1+a_2)/2$ for equal-mass unequal-spin binaries, and $\tilde{a} = a$ for unequal-mass equal-spin binaries. Furthermore, the extreme mass-ratio limit (EMRL) of expression  with the substitution  yields the expected result: $a_{\rm fin}(a_1, a_2, \nu=0) = a_1$. As already commented above, the predictions of expressions  and  cover 3 of the 7 dimensions of the space of parameters for binaries in quasi-circular orbits; we next show how to to cover the remaining 4 dimensions and derive an analytic expression for the dimensionless spin *vector* $\boldsymbol{a}_{\rm fin}$ of the black hole produced by the coalescence of two generic black holes in terms of the mass ratio $q$ and of the initial dimensionless spin vectors $\boldsymbol{a}_{1,2}$. To make the problem tractable analytically, 4 assumptions are needed. While some of these are very natural, others can be relaxed if additional accuracy in the estimate of $\boldsymbol{a}_{\rm fin}$ is necessary. It should be noted, however, that removing any of these assumptions inevitably complicates the picture, introducing additional dimensions, such as the initial separation in the binary or the radiated mass, in the space of parameters. As a result, in the simplest and yet accurate description the required assumptions are as follows: *(i) The mass radiated to gravitational waves $M_{\rm rad}$ can be neglected* *i.e.*, $M_{\rm fin} = M \equiv M_1 + M_2$. We note that $M_{\rm rad}/M = 1-M_{\rm fin}/M \approx 5-7\times 10^{-2}$ for most of the binaries evolved numerically. The same assumption was applied in the analyses of [@Rezzolla-etal-2007; @Rezzolla-etal-2007b], as well as in [@Buonanno:07b]. Relaxing this assumption would introduce a dependence on $M_{\rm fin}$ which can only be measured through a numerical simulation. *(ii) At a sufficiently large but finite initial separation the final spin vector $\boldsymbol{S}_{\rm fin}$ can be well approximated as the sum of the two initial spin vectors and of a third vector $\boldsymbol{\tilde{\ell}}$* $$\label{assumption_1} \boldsymbol{S}_{\rm fin}=\boldsymbol{S}_1+ \boldsymbol{S}_2+\boldsymbol{\tilde{\ell}}\,,$$ Differently from refs. [@Hughes:2002ei] and [@Buonanno:07b], where a definition similar to was also introduced, here we will constrain $\boldsymbol{\tilde{\ell}}$ by exploiting the results of numerical-relativity calculations rather than by relating it to the orbital angular momentum of a test particle at the innermost stable circular orbit (ISCO). When viewed as expressing the conservation of the total angular momentum, eq.  also defines the vector ${ \boldsymbol{\tilde {\ell}} }$ as the difference between the orbital angular momentum when the binary is widely separated $\boldsymbol{L}$, and the angular momentum radiated until the merger $\boldsymbol{J}_{\rm rad}$, *i.e.*, ${ \boldsymbol{\tilde {\ell}} }= \boldsymbol{L} - \boldsymbol{J}_{\rm rad}$. *(iii) The vector $\boldsymbol{\tilde{\ell}}$ is parallel to $\boldsymbol{L}$*. This assumption is correct when $\boldsymbol{S}_1=-\boldsymbol{S}_2$ and $q=1$ \[this can be seen from the PN equations at 2.5 order\], or by equatorial symmetry when the spins are aligned with $\boldsymbol{L}$ or when $\boldsymbol{S}_1=\boldsymbol{S}_2=0$ (also these cases can be seen from the PN equations). For more general configurations one expects that ${ \boldsymbol{\tilde {\ell}} }$ will also have a component orthogonal to $\boldsymbol{L}$ as a result, for instance, of spin-orbit or spin-spin couplings, which will produce in general a precession of ${ \boldsymbol{\tilde {\ell}} }$. In practice, the component of ${ \boldsymbol{\tilde {\ell}} }$ orthogonal to $\boldsymbol{L}$ will correspond to the angular momentum $\boldsymbol{J}^{\perp}_{\rm rad}$ radiated in a plane orthogonal to $\boldsymbol{L}$, with a resulting error in the estimate of $\vert { \boldsymbol{\tilde {\ell}} }\vert$ which is $\sim \vert \boldsymbol{J}^{\perp}_{\rm rad} \vert^2 / \vert \boldsymbol{\tilde{\ell}} \vert^2\sim \vert \boldsymbol{J}^{\perp}_{\rm rad} \vert^2/(2 \sqrt{3} M_1 M_2)^2$[^1]. Although these errors are small in all the configurations that we have analysed, they may be larger in general configurations. Measuring $\boldsymbol{J}^{\perp}_{\rm rad}$ via numerical-relativity simulations, or estimating it via high-order PN equations, is an obvious way to improve our approach. A similar assumption was also made in ref. [@Buonanno:07b]. *(iv) When the initial spin vectors are equal and opposite ($\boldsymbol{S}_{1}=-\boldsymbol{S}_{2}$) and the masses are equal ($q=1$), the spin of the final black hole is the same as for the nonspinning binaries*. Stated differently, equal-mass binaries with equal and opposite-spins behave as nonspinning binaries, at least when it comes down to the properties of the final black hole. While this result cannot be derived from first principles, it reflects the expectation that if the spins are the same and opposite, their contributions to the final spin cancel for equal-mass binaries. Besides being physically reasonable, this expectation is met by all of the simulations performed to date, both for spins aligned with $\boldsymbol{L}$ [@Rezzolla-etal-2007; @Rezzolla-etal-2007b] and orthogonal to $\boldsymbol{L}$ [@Bruegmann:2007zj]. In addition, this expectation is met by the leading-order contributions to the spin-orbit and spin-spin point-particle Hamiltonians and spin-induced radiation flux [@Barker1970; @Buonanno:06cd]. A similar assumption is also made, although not explicitly, in ref. [@Buonanno:07b] which, for $\boldsymbol{S}_{\rm tot} = 0$, predicts $\iota=0$ and ${\vert \boldsymbol{a}_{\rm fin} \vert}= L_{\rm orb}(\iota=0,{\vert \boldsymbol{a}_{\rm fin} \vert})/M =\,$const. \[*cf.* eqs. (12)–(13) in ref. [@Buonanno:07b]\]. Using these assumptions we can now derive the analytic expression for the final spin. We start by expressing the vector relation  as $$\label{assumption_1_bis} \boldsymbol{a}_{\rm fin}=\frac{1}{(1+q)^2} \left(\boldsymbol{a}_1+\boldsymbol{a}_2q^2 + \boldsymbol{{\ell}} q \right)\,,$$ where $\boldsymbol{a}_{\rm fin}= \boldsymbol{S}_{\rm fin}/M^2$ \[*cf.* assumption *(i)*\], $\boldsymbol{{\ell}} \equiv \boldsymbol{\tilde{\ell}}/(M_1 M_2)$, , and its norm is then given by $$\begin{aligned} \label{eq:general} &\hskip -0.5cm \quad \vert \boldsymbol{a}_{\rm fin}\vert= \frac{1}{(1+q)^2}\Big[ {\vert \boldsymbol{a}_{1} \vert}^2 + {\vert \boldsymbol{a}_{2} \vert}^2 q^4+ 2 {\vert \boldsymbol{a}_2\vert}{\vert \boldsymbol{a}_1\vert} q^2 \cos \alpha + \nonumber\\ & \hskip 1.cm 2\left( {\vert \boldsymbol{a}_1\vert}\cos \beta + {\vert \boldsymbol{a}_2\vert} q^2 \cos \gamma \right) {\vert \boldsymbol{{\ell}} \vert}{q} + \vert \boldsymbol{{\ell}}\vert^2 q^2 \Big]^{1/2}\,,\end{aligned}$$ where the three (cosine) angles $\alpha, \beta$ and $\gamma$ are defined by $$\label{cosines} \cos \alpha \equiv {\boldsymbol{\hat{a}}_1\cdot\boldsymbol{\hat{a}}_2} \,, \hskip 0.3cm \cos \beta \equiv \boldsymbol{\hat a}_1\cdot\boldsymbol{\hat{{\ell}}}\,, \hskip 0.3cm \cos \gamma \equiv \boldsymbol{\hat{a}}_2\cdot\boldsymbol{\hat{{\ell}}}\,.$$ Because $\boldsymbol{a}_{1,2} \parallel \boldsymbol{S}_{1,2}$ and $\boldsymbol{{\ell}} \parallel \boldsymbol{L}$ \[*cf.* assumption *(iii)*\], the angles $\alpha, \beta$ and $\gamma$ are also those between the initial spin vectors and the initial orbital angular momentum, so that it is possible to replace $\boldsymbol{\hat{a}}_{1,2}$ with $\boldsymbol{\hat{S}}_{1,2}$ and $\boldsymbol{\hat{\ell}}$ with $\boldsymbol{\hat{L}}$ in . Note that $\alpha, \beta$ and $\gamma$ are well-defined if the initial separation of the two black holes is sufficiently large \[*cf.* assumption *(ii)*\] and that the error introduced by assumption *(iii)* in the measure of $\cos\alpha, \cos\beta$ and $\cos\gamma$ is also of the order of $\vert \boldsymbol{J}^{\perp}_{\rm rad} \vert / \vert \boldsymbol{\tilde{\ell}} \vert$. The angle $\theta_{\rm fin}$ between the final spin vector and the initial orbital angular momentum can be easily calculated from ${\vert \boldsymbol{a}_{\rm fin} \vert}$. Because of assumption *(iii)*, the component of the final spin in the direction of $\boldsymbol{L}$ is \[*cf.* eq. \] $$\label{eq:a_z} a_{\rm fin}^{\parallel}\equiv \boldsymbol{a}_{\rm fin}\cdot \boldsymbol{\hat{{\ell}}} = \frac{{\vert \boldsymbol{a}_{1} \vert} \cos\beta + {\vert \boldsymbol{a}_{2} \vert}q^2 \cos\gamma + {\vert \boldsymbol{ {\ell}} \vert}q}{(1+q)^2}\,,$$ so that $\cos\theta_{\rm fin}={a_{\rm fin}^{\parallel}}/{ \vert \boldsymbol{a}_{\rm fin}\vert}$, and the component orthogonal to the initial orbital angular momentum is $a_{\rm fin}^{\perp} = {\vert \boldsymbol{a}_{\rm fin} \vert}\sin\theta_{\rm fin}$. In essence, therefore, our approach consists of considering the dimensionless spin vector of the final black hole as the sum of the two initial spins and of a third vector parallel to the initial orbital angular momentum when the binaries are widely separated. Implicit in the assumptions made, and in the logic of mapping an initial-state of the binary into a final one, is the expectation that the length of this vector is an intrinsic “property” of the binary, depending on the initial spin vectors and mass ratio, but not on the initial separation. This is indeed a consequence of assumption *(ii)*: because the vector $\boldsymbol{\tilde{\ell}} $ measures the orbital angular momentum that cannot be radiated, it can be thought of as the angular momentum of the binary at the “effective” ISCO and, as such, it cannot be dependent on the initial separation. A very important consequence of our assumptions is that $\boldsymbol{a}_{\rm fin}$ for a black-hole binary is already fully determined by the set of coefficients $s_4, s_5, t_0, t_2, t_3$ computed to derive expression . The latter, in fact, is simply the final spin for a special set of values for the cosine angles; since the fitting coefficients are constant, they must hold also for generic binaries. In view of this, all that is needed is to measure $\vert \boldsymbol{\ell} \vert$ in terms of the fitting coefficients computed in refs. [@Rezzolla-etal-2007; @Rezzolla-etal-2007b]. This can be done by matching expression  with  \[with the condition \] for parallel and aligned spins ($\alpha=\beta=\gamma=0$), for parallel and antialigned spins ($\alpha=0$, $\beta=\gamma=\pi$), and for antiparallel spins which are aligned or antialigned ($\alpha=\beta=\pi$, $\gamma=0$ or $\alpha=\gamma=\pi$, $\beta=0$). This matching is not unique, but the degeneracy can be broken by exploiting assumption *(iv)* and by requiring that ${\vert \boldsymbol{ {\ell}} \vert}$ depends linearly on $\cos\alpha$, $\cos\beta$ and $\cos\gamma$. We therefore obtain $$\begin{aligned} \label{eq:L} && \hskip -0.5cm {\vert \boldsymbol{ {\ell}} \vert}= \frac{s_4}{(1+q^2)^2} \left({\vert \boldsymbol{a}_{1} \vert}^2 + {\vert \boldsymbol{a}_{2} \vert}^2 q^4 + 2 {\vert \boldsymbol{a}_{1} \vert} {\vert \boldsymbol{a}_{2} \vert} q^2 \cos\alpha\right) + \nonumber \\ && \hskip 0.5cm \left(\frac{s_5 \nu + t_0 + 2}{1+q^2}\right) \left({\vert \boldsymbol{a}_{1} \vert}\cos\beta + {\vert \boldsymbol{a}_{2} \vert} q^2 \cos\gamma\right) + \nonumber \\ && \hskip 0.5cm 2 \sqrt{3}+ t_2 \nu + t_3 \nu^2 \,.\end{aligned}$$ 1.0cm -0.5cm -0.5cm We now consider some limits of expressions  and . First of all, when $q\to0$,  and  yield the correct EMRL, *i.e.*, ${\vert \boldsymbol{a}_{\rm fin} \vert}= {\vert \boldsymbol{a}_{1} \vert}$. Secondly, for equal-mass binaries having spins that are equal and antiparallel,  and  reduce to $$\label{2nd_check_bis} {\vert \boldsymbol{a}_{\rm fin} \vert}= \frac{{\vert \boldsymbol{ {\ell}} \vert}}{4} = \frac{\sqrt{3}}{2} + \frac{t_2}{16} + \frac{t_3}{64} = p_0 \simeq 0.687 \,.$$ This result allows us now to qualify more precisely a comment made before: because for equal-mass black holes which are either nonspinning or have equal and opposite spins, the vector ${\vert \boldsymbol{ {\ell}} \vert}$ does not depend on the initial spins, expression  states that $\vert \boldsymbol{{\ell}}\vert M_{\rm fin}^2/4=\vert \boldsymbol{{\ell}}\vert M^2/4=\vert \boldsymbol{{\ell}}\vert M_1 M_2$ is, for such systems, the orbital angular momentum at the effective ISCO. We can take this a step further and conjecture that $\vert \boldsymbol{{\ell}}\vert M_1 M_2 =\vert \boldsymbol{\tilde{\ell}}\vert$ is the series expansion of the dimensionless orbital angular momentum at the ISCO also for *unequal-mass* binaries which are either nonspinning or with equal and opposite spins. The zeroth-order term of this series (namely, the term $2\sqrt{3} M_1 M_2$) is exactly the one predicted from the EMRL. We note that although numerical simulations do not reveal the presence of an ISCO, the concept of an effective ISCO can nevertheless be useful for the construction of gravitational-wave templates [@Ajith:2007kx; @Hanna2008]. Finally, we consider the case of equal, parallel and aligned/antialigned spins (${\vert \boldsymbol{a}_{2} \vert}={\vert \boldsymbol{a}_{1} \vert}$, $\alpha=0$, $\beta=\gamma=0,\,\pi$), for which expressions  and  become $$\begin{aligned} \label{3rd_check} a_{\rm fin} &=& {\vert \boldsymbol{a}_{1} \vert}\cos\beta \left[ 1 + \nu (s_4{\vert \boldsymbol{a}_{1} \vert}\cos\beta + t_0 + s_5 \nu )\right] + \nonumber \\ && \hskip 2.5cm \nu( 2 \sqrt{3}+ t_2 \nu + t_3 \nu^2 )\,,\end{aligned}$$ where $\cos\beta = \pm 1$ for aligned/antialigned spins. As expected, expression  coincides with  when ${\vert \boldsymbol{a}_{1} \vert}\cos\beta=a$ and with  \[through the coefficients \] when $q=1$ and $2{\vert \boldsymbol{a}_{1} \vert}\cos\beta = a_1 + a_2$. Similarly, and reduce to  for equal, antiparallel and aligned/antialigned spins (${\vert \boldsymbol{a}_{2} \vert}={\vert \boldsymbol{a}_{1} \vert}$, $\alpha=0$, $\beta=0, \gamma=\pi$, or $\beta=\pi, \gamma=0$).   [$a^y_1$]{} [$a^z_1$]{} [$a^x_2$]{} [$a^y_2$]{} [$a^z_2$]{} [$\nu$]{} [${\vert \boldsymbol{a}_{\rm fin} \vert}$]{} [$\theta_{\rm fin}(^\circ)$]{} ----- ------- ------------- ------------- ------------- ------------- ------------- ----------- ------------------------------------------------ -------------------------------- $~$ 0.151 0.000 -0.563 0.000 0.000 0.583 0.250 0.692 2.29 $~$ 0.151 0.000 0.564 0.000 0.151 0.564 0.250 0.846 3.97 $~$ 0.413 0.000 0.413 0.000 0.413 0.413 0.250 0.815 7.86 : \[tableone\]Initial parameters of the new misaligned AEI binaries. -0.25cm -0.6cm The only way to assess the validity of expressions  and  is to compare their predictions with the numerical-relativity data. This is done in Figs. \[fig:align\_res\] and \[fig:misalign\], which collect all of the published data, together with the three additional binaries computed with the `CCATIE` code [@Pollney:2007ss] and reported in Table \[tableone\]. In these plots, the “binary order number” is just a dummy index labelling the different configurations. The left panel of Fig. \[fig:align\_res\], in particular, shows the rescaled residual, *i.e.*, $({\vert \boldsymbol{a}_{\rm fin} \vert}_{\rm fit} - {\vert \boldsymbol{a}_{\rm fin} \vert}_{\rm num.})\times 100$, for aligned binaries. The plot shows the numerical-relativity data with circles referring to equal-mass, equal-spin binaries from refs. [@Rezzolla-etal-2007; @Marronetti07tbgs; @Berti:2007snb; @Berti:2007sb; @Buonanno:2007ft; @Rezzolla-etal-2007b], triangles to equal-mass, unequal-spin binaries from refs. [@Rezzolla-etal-2007; @Berti:2007sb], and squares to unequal-mass, equal-spin binaries from refs. [@Berti:2007snb; @Buonanno:2007ft; @Rezzolla-etal-2007b; @Berti:2007sb]. Although the data is from simulations with different truncation errors, the residuals are all very small and with a scatter of $\sim 1\%$. A more stringent test is shown in the right panel of Fig. \[fig:align\_res\], which refers to misaligned binaries. In the top part, hexagons indicate the numerical values for ${\vert \boldsymbol{a}_{\rm fin} \vert}$ from ref. [@Campanelli:2006vp], squares the ones in Table \[tableone\], circles those from ref. [@Tichy:2007gso] and triangles those from ref. [@Herrmann:2007ex]; note that these latter data points refer to the aligned component $a_{\rm fin}^{\parallel}$ since this is the only component available from ref. [@Herrmann:2007ex]. The agreement is again very good, with errors of a couple of percent (see bottom part of the same panel), even if the binaries are generic and for some the initial and final spins differ by almost $180^\circ$ [@Campanelli:2006vp]. Finally, Fig. \[fig:misalign\] reports the angle between the final spin vector and the initial orbital angular momentum $\theta_{\rm fin}$ using the same data (and convention for the symbols) as in the right panel of Fig. \[fig:align\_res\]. Measuring the final angle accurately is not trivial, particularly due to the fact that the numerical evolutions start at a finite separation which does not account for earlier evolution of the orbital angular momentum vector. The values reported in [@Campanelli:2006vp] (and the relative error-bars) are shown with hexagons, while the squares refer to the binaries in Table \[tableone\], and have been computed using a new approach for the calculation of the Ricci scalar on the apparent horizon [@Jasiulek:2008dt]. Shown with asterisks and circles are instead the values predicted for the numerical data (as taken from refs. [@Campanelli:2006vp; @Tichy:2007gso; @Herrmann:2007ex] and from Table \[tableone\]) by our analytic fit (asterisks) and by the point-particle approach suggested in ref. [@Buonanno:07b] (circles). -0.5cm -0.5cm Clearly, when a comparison with numerical data is possible, the estimates of our fit are in reasonable agreement with the data and yield residuals in the final angle (*i.e.*, $(\theta_{\rm fin})_{\rm fit} - (\theta_{\rm fin})_{\rm num.}$) which are generally smaller than those obtained with the point-particle approach of ref. [@Buonanno:07b]. However, for two of the three binaries from ref. [@Campanelli:2006vp] the estimates are slightly outside the error-bars. Note that the reported angles are relative to the orbital plane at a small initial binary-separation, and thus are likely to be underestimates as they do not take into account the evolution from asymptotic distances; work is in progress to clarify this. When the comparison with the numerical data is not possible because $\theta_{\rm fin}$ is not reported (as for the data in ref. [@Herrmann:2007ex]), our approach and the one in ref. [@Buonanno:07b] yield very similar estimates. In summary: we have considered the spin vector of the black hole produced by a black-hole binary merger as the sum of the two initial spins and of a third vector, parallel to the initial orbital angular momentum, whose norm depends only on the initial spin vectors and mass ratio, and measures the orbital angular momentum not radiated. Without additional fits than those already available to model aligned/antialigned binaries, we have measured the unknown vector and derived a formula that accounts therefore for all of the 7 parameters describing a black-hole binary inspiralling in quasi-circular orbits. The equations  and , encapsulate the near-zone physics to provide a convenient, but also robust and accurate over a wide range of parameters, determination of the merger product of rather generic black-hole binaries. Testing the formula against all of the available numerical data has revealed differences between the predicted and the simulated values of a few percent at most. Our approach is intrinsically approximate and it has been validated on a small set of configurations, but it can be improved, for instance: by reducing the $\chi^2$ of the fitting coefficients as new simulations are carried out; by using fitting functions that are of higher-order than those in expressions  and ; by estimating $\boldsymbol{J}^{\perp}_{\rm rad}$ through PN expressions or by measuring it via numerical simulations. It is a pleasure to thank Peter Diener, Michael Jasiulek and Erik Schnetter for valuable discussions. EB gratefully acknowledges the hospitality of the AEI, where part of this work was carried out. The computations were performed on the clusters Belladonna and Damiana at the AEI. [^1]: Assumption *(iii)* can be equivalently interpreted as enforcing that the component of the final spin $\boldsymbol{S}_{\rm fin}$ in the orbital plane equals the one of the total initial spin $\boldsymbol{S}_1+ \boldsymbol{S}_2$ in that plane.

--- author: - | Chang-han Rhee\ Peter W. Glynn\ Stanford University\ Stanford, CA 94305, U.S.A. title: 'A NEW APPROACH TO UNBIASED ESTIMATION FOR SDE’S' --- ABSTRACT {#abstract .unnumbered} ======== In this paper, we introduce a new approach to constructing unbiased estimators when computing expectations of path functionals associated with stochastic differential equations (SDEs). Our randomization idea is closely related to multi-level Monte Carlo and provides a simple mechanism for constructing a finite variance unbiased estimator with “square root convergence rate" whenever one has available a scheme that produces strong error of order greater than 1/2 for the path functional under consideration. INTRODUCTION {#sec:intro} ============ We have recently developed a general approach to constructing unbiased estimators, given a family of biased estimators. It turns out that the conditions guaranteeing its validity are closely related to those associated with multi-level Monte Carlo methods; see Rhee and Glynn (2012) for details and a more complete discussion of the theory. In this paper, we briefly describe the idea in the setting of computing solutions of stochastic differential equations and provide an initial numerical exploration intended to shed light on the method’s potential effectiveness. As we will see below, the conditions under which our estimator produces an algorithm with “square root convergence rate" essentially coincide with the conditions required by multi-level Monte Carlo to converge at the same rate. In particular, suppose that we wish to compute an expectation of the form $\alpha = E k(X)$, where $X =(X(t): t\geq0)$ is the solution to the SDE $$\begin{aligned} dX(t) = \mu(X(t))dt + \sigma(X(t))dB(t), \label{eq:1}\end{aligned}$$ $B =(B(t):t\geq0)$ is m-dimensional standard Brownian motion, $k: C[0, \infty) \to R$, and $C[0, \infty)$ is the space of continuous functions mapping $[0,\infty)$ into $R^d$. In general, the random variable (rv) $k(X)$ can not be simulated exactly, because the underlying infinite-dimensional object $X$ can not be generated exactly. Instead, one typically approximates $X$ via a discrete-time approximation $X_h(\cdot)$. For example, the simplest such approximation is the Euler time-stepping algorithm given by $$\begin{aligned} X_h((k+1)h) = X_h(kh) + \mu(X_h(kh))h + \sigma(X_h(kh)) (B(k+1)h) - B(kh)) \label{eq:2}\end{aligned}$$ that defines $X_h$ at the time points $0, h, 2h, ...,$ with $X_h$ defined at intermediate values via (for example) linear interpolation. Because (\[eq:2\]) is only an approximation to the dynamics represented by (\[eq:1\]), the rv $k(X_h)$ is only an approximation to $k(X)$, and consequently $k(X_h)$ is a biased estimator for the purpose of computing $\alpha$. The traditional means of dealing with this is to intelligently select the step size $h$ and number of independent replications $R$ as a function of the computational budget $c$, so as to maximize the rate of convergence. However, as pointed out by Duffie and Glynn (1995), such biased numerical schemes inevitably lead to Monte Carlo estimators for $\alpha$ that exhibit slower convergence rates than the “canonical" order $c^{-1/2}$ rate associated with Monte Carlo in the presence of unbiased finite variance estimators. However, several years ago, Giles (2008) introduced an intriguing multi-level idea to deal with such biased settings that can dramatically improve the rate of convergence and can even, in some settings, achieve the canonical “square root" convergence rate associated with unbiased Monte Carlo. His approach does not construct an unbiased estimator, however. Rather, the idea is to construct a family of estimators (indexed by the desired error tolerance $\epsilon$) that has controlled bias. In this paper, we show how it is possible, in a similar computational setting, to go one step further and to produce (exactly) unbiased estimators. The remainder of this paper is organized as follows: We discuss the idea in Section 2 of this paper, while Section 3 is devoted to an initial computational exploration of this approach. THE BASIC IDEA ============== We consider here a sequence $(X_{h_n}: n>=0)$ of discrete-time time-stepping approximations to $X$ that are all constructed on a common probability space in such a way that: 1. $Ek(X_{h_n}) = Ek(X) + O(h_n)$ as $h_n \to 0$; 2. $E| k(X_{h_n}) - k(X) |^2 = O(h_n^{2r})$ as $h_n \to 0$ for some $r >0$, where $O(f(n))$ represents a function which is bounded by some constant multiple of $f(\cdot)$ as $h_n \to 0$. Assuming, as is often the case for such discretization schemes, that the scheme generates normal rv’s that are intended to mirror the Brownian increments of the process $B$ driving the SDE (as in the Euler scheme (1.2) above), the easiest way to algorithmically obtain an approximating sequence $X_{h_n}$ to $X$ in which the $X_{h_n}$’s are jointly defined on the same probability space is by successive binary refinement, so that $h_n = 2^{-n}$. In this setting, the new Brownian motion values ($B(j2^{-(n+1)})$: $j$ odd) required at discretization $2^{-(n+1)}$ can be obtained from the existing values ($B(j2^{-n}): j \geq0$) by generating $B((2k+1) 2^{-(n+1)})$ from its conditional distribution given $B(k2^{-n})$ and $B((k+1)2^{-n})$. On the other hand, one’s ability to obtain i and ii depends both on the path functional $k$ and on one’s choice of discretization scheme. In particular, suppose that one has established that the discretization $X_h$ exhibits strong order $r$. This implies that $$\begin{aligned} E \sup\{ | X_h(kh) - X(kh) |^{2r} : 0\leq k \leq \lfloor t/h\rfloor \} = O(h^{2r}).\end{aligned}$$ Thus, if $k$ is (for example) a “Lipschitz final value" expectation so that $k(x) = g(x(1))$ for some Lipschitz function $g: R^d \to R$, ii is satisfied. In addition, if $k$ is further assumed to be smooth with $| k(X) |$ integrable, then i is satisfied whenever the discretization $X_h$ is known to be of weak order 1 or higher. It should be noted that these conditions are (very) closely related to those that appear in the literature on multi-level Monte Carlo for SDEs. Note that each of the $k(X_{2^{-n}})$’s is a biased estimator for $\alpha = E k(X)$. To obtain an unbiased estimator, observe that ii) implies the existence of $p > 0$ such that $$\begin{aligned} \sum_{n = 1}^\infty E 2^{np} | k(X_{2^{-n}}) - k(X_{2^{-(n-1)}}) | ^{2r} < \infty.\end{aligned}$$ Consequently, $$\begin{aligned} \sum_{ n=1}^\infty 2^{np} | k(X_{2^{-n}}) - k(X_{2^{-(n-1)}}) |^{2r} < \infty\quad a.s.\end{aligned}$$ from which it follows that $$\begin{aligned} | k(X_{2^{-n}}) - k(X_{2^{-(n-1)}}) | = O(2^{-np})\quad a.s.\end{aligned}$$ as $n \to \infty$, and hence (in view of ii), $$\begin{aligned} k(X) = k(X_1) + \sum_{ n=1}^\infty k(X_{2^{-n}}) - k(X_{2^{-(n-1)}})\end{aligned}$$ We now introduce a rv $N$, independent of $B$, that takes values in the positive integers and has a distribution with unbounded support (so that $P(N>n)>0$ for $n\geq1$). For such a rv $N$, $$\begin{aligned} Ek(X) &= E k(X_1) + \sum_{n=1}^\infty E (k(X_{2^{-n}}) - k(X_{2^{-(n-1)}})) I(N \geq n)/ P(N \geq n)\\ &= E \left[k(X_1) + \sum _{n=1} ^N (k(X_{2^{-n}}) - k(X_{2^{-(n-1)}})) / P(N \geq n)\right]\\ &\triangleq E Z.\end{aligned}$$ Note that $Z$ is an unbiased estimator for $\alpha$. This suggests computing $\alpha$ by generating iid replicates of the rv $Z$. Of course, the “square root" convergence rate of such an estimator is not guaranteed. Given the role that finiteness of the variance plays in obtaining such convergence rates, we next study this issue. Set $\Delta_i = k(X_{2^{-i}}) - k(X_{2^{-(i-1)}})$ for $i \geq 1$ and observe that $$\begin{aligned} E Z^2 &= E k^2 (X_1) + E \sum_{i=1}^N \Delta_i^2 / P(N\geq i)^2 + 2 E k(X_1) \sum_{i=1}^N \Delta_i / P(N \geq i) + 2 E \sum_{i=1}^N \sum_{j = i+1}^N \Delta_i \Delta_j /(P(N\geq i) P(N\geq j))\\ &= E k^2 (X_1) + \sum_{i=1}^\infty E \Delta_i^2 / P(N\geq i) + 2E k(X_1) \sum_{i=1}^\infty \Delta_i + 2 \sum_{i=1}^\infty \sum_{j= i+1}^\infty E \Delta_i \Delta_j / P(N\geq i) \\ &= E k^2(X_1) + \sum_{i=1}^\infty E \Delta_i^2/P(N\geq i)+ 2 E k(X_1)(k(X)-k(X_1)) + 2 \sum_{i=1}^\infty E\Delta_i (k(X) - k(X_{2^{-i}}))/P(N\geq i)\\ &\leq E k^2(X_1) + 2 E k(X_1) (k(X) - k(X_1)) + \sum_{i=1}^\infty O(2^{-2ri})/P(N\geq i),\end{aligned}$$ so that $\text{var} Z < \infty$ if $P(N\geq i) \sim c 2^{- \gamma i}$ as $i \to \infty$, for $0 < \gamma < 2r$ (where $a_i \sim b_i$ means that $a_i/b_i \to 1$ as $i \to \infty$). Finally, Glynn and Whitt (1992) prove that “square root convergence rate" ensues if $\text{var} Z < \infty $ and if the expected computational effort required per replication of $Z$ is finite. The expected computational “work" required for each $Z$ is (roughly) given by $$E \sum_{i=0}^N t_i,$$ where $t_i$ is the incremental effort required to compute $k(X_{2^{-i}})$ (given $k(X_1), \ldots , k(X_{2^{-(i-1)}})$), and hence can be expressed as $$\begin{aligned} \sum_{i=0}^\infty t_i P(N\geq i). \label{eq:3}\end{aligned}$$ An approximation to $t_i$ is $t_i = 2^{i-1} $ (the number of additional Gaussian rv’s needed to generate $X_{2^{-i}}$). In order that (\[eq:3\]) be finite, we require that $\gamma > 1$. Consequently, a square root convergence rate is ensured when $2r> 1$ ( in which case we can, for example, choose $\gamma = (1+2r)/2$). A PRELIMINARY COMPUTATIONAL INVESTIGATION ========================================= In this section, we implement our method and compare it to the multilevel Monte Carlo algorithm suggested in Giles (2008). We consider two examples:\ [*Geometric Brownian Motion*]{} (GBM): The process under consideration here is the solution to $$dX(t) = rX(t) dt + \sigma X(t) dB(t)$$ subject to $X(0)=1$, $r=0.05$, $\sigma = 0.2$, the functional $k$ is $k(x) = x(1)$. For this set of parameters, $$E k(X) = 0.104506.$$ [*Cox-Ingersoll-Ross process*]{} (CIR): Here, $X$ solves $$dX(t) = \kappa (\theta - X(t))dt + \sigma \sqrt{X(t)} dB(t),$$ subject to $X(0)= 0.04$, $\kappa = 5$, $\theta = 0.04$, and $\sigma = 0.25$. The functional $k$ is taken here to be $k(x) = x(1)$. For this example, $$E k(X) = 0.04.$$ The numerical scheme used to solve each of the above SDE’s was the Milstein scheme; see Kloeden and Platen (1992). For the above problems, we expect $r=1$. For the purpose of this paper, we do not try to optimize the distribution of $N$, and instead choose $N$ so that $$P(N \geq i) = 2^{3i/2}$$ for $i\geq 1$. (In other words, we choose $\gamma$ as the midpoint between $1$ and $2r$, although any choice in $(1,2)$ would provide “square root convergence rate”.) To compare our method to the multi-level Monte Carlo (MLMC) mehtod, we take the view (as in Giles, 2008) that the root mean square error (RMSE) $\epsilon$ to be achieved by the algorithm is given. Giles (2008) provides a complete description of how to construct a MLMC estimator achieving approximate RMSE $\epsilon$; we have implemented that version of MLMC here. For our unbiased estimator, we generate independent and identically distributed (iid) replications of the rv $Z$ until such time as the approximate RMSE is less than or equal to $\epsilon$. In other words, our estimator for $\alpha$ is $$\begin{aligned} \frac{1}{N(\epsilon)} \sum_{i=1}^{N(\epsilon)} Z_i, \label{eq:4}\end{aligned}$$ where the $Z_i$’s are iid replicates of $Z$, and $$N(\epsilon) = \inf \left\{ n \geq 1: \frac{1}{n(n-1)} \sum_{i=1}^n \left(Z_i - \frac{1}{n}\sum_{j=1}^N Z_j\right)^2 \leq \epsilon^2\right\}$$ is the first time that the sample RMSE of the sample mean drops below $\epsilon$. We use the stopping rule $N(\epsilon)$ in order to permit easy computation for our estimator, although its use is somewhat unnatural for our estimator (since its use induces bias in our estimator). For each of our two examples, we provide two tables. The first table for each example concerns our new estimator (\[eq:4\]); IRE stands for “intended relative error", and $k\%$ then corresponds to setting $\epsilon = (k/100) |\alpha|$. The 90% confidence interval is then obtained by taking 100 replications of (4) for a given $\epsilon$, computing the sample mean and sample standard deviation of the 100 observations, and constructing a confidence interval based on the normal approximation. The column corresponding to RMSE is the square root of the average, over the 100 observations, of the square of (4) minus $E Z$. (Thus, RMSE is reporting the actual root mean square error of the estimator, rather than the intended RMSE that the estimator has been designed to attain asymptotically.) The final column, denoted “work", reports a 90% confidence interval for the expected number of normal rv’s generated to construct (\[eq:4\]), based on our 100 samples. The second table for each example provides a corresponding set of values for the MLMC estimator. IRE 90% Confidence Interval RMSE Work ------ ------------------------- --------- ------------------------ 25% 0.108318 $\pm$ 0.004503 0.02759 388.3 $\pm$ 23.3 10% 0.105360 $\pm$ 0.001874 0.01140 2850.8 $\pm$ 145.7 5% 0.105244 $\pm$ 0.000916 0.00560 11278.3 $\pm$ 221.0 2% 0.104799 $\pm$ 0.000388 0.00237 73451.9 $\pm$ 1198.8 1% 0.104500 $\pm$ 0.000174 0.00105 291488.1 $\pm$ 2157.1 0.5% 0.104492 $\pm$ 0.000088 0.00054 1170663.9 $\pm$ 6067.8 : Unbiased estimation for GBM\[tab: UBGBM\] IRE 90% Confidence Interval RMSE Work ------ ------------------------- --------- ----------------------- 25% 0.103753 $\pm$ 0.002693 0.01635 3130.3 $\pm$ 2.7 10% 0.104648 $\pm$ 0.001179 0.00716 3842.5 $\pm$ 10.5 5% 0.104908 $\pm$ 0.000551 0.00337 6378.9 $\pm$ 31.9 2% 0.104465 $\pm$ 0.000218 0.00133 24901.3 $\pm$ 196.0 1% 0.104293 $\pm$ 0.000129 0.00081 93767.1 $\pm$ 735.3 0.5% 0.104468 $\pm$ 0.000060 0.00037 377783.8 $\pm$ 4248.2 : MLMC estimation for GBM\[tab: MLMCGBM\] IRE 90% Confidence Interval RMSE Work ------ ------------------------- --------- ------------------------ 25% 0.039914 $\pm$ 0.001779 0.01080 883.1 $\pm$ 61.4 10% 0.040272 $\pm$ 0.000741 0.00450 5548.5 $\pm$ 180.7 5% 0.040206 $\pm$ 0.000342 0.00208 22693.3 $\pm$ 690.8 2% 0.039880 $\pm$ 0.000143 0.00087 142480.8 $\pm$ 1790.4 1% 0.039953 $\pm$ 0.000069 0.00042 563996.7 $\pm$ 3609.4 0.5% 0.040022 $\pm$ 0.000030 0.00018 2259485.0 $\pm$ 8235.4 : Unbiased estimation for CIR\[tab: UBCIR\] IRE 90% Confidence Interval RMSE Work ------ ------------------------- --------- ------------------------ 25% 0.040543 $\pm$ 0.000973 0.00593 3731.2 $\pm$ 13.1 10% 0.040186 $\pm$ 0.000415 0.00252 11518.9 $\pm$ 44.8 5% 0.039880 $\pm$ 0.000208 0.00127 42503.8 $\pm$ 139.4 2% 0.039862 $\pm$ 0.000100 0.00062 265948.6 $\pm$ 535.7 1% 0.040005 $\pm$ 0.000044 0.00027 1098672.3 $\pm$ 6866.5 0.5% 0.040008 $\pm$ 0.000023 0.00014 4572949.2 $\pm$ 5221.9 : MLMC estimation for CIR\[tab: MLMCCIR\] Our results are reasonably comparable to those associated with MLMC, despite the fact that we have done essentially no tuning to optimize the distribution of $N$. In addition, our estimator is (arguably) easier to implement than MLMC, since (in its current form) there are no algorithmic parameters that are estimated “on the fly" within the algorithm (in contrast to MLMC). Thus, the unbiased estimators introduced here offer a promising computational alternative to MLMC in the presence of SDE numerical schemes having a strong order greater than 1/2. REFERENCES {#references .unnumbered} ========== Duffie, D. and P. W. Glynn. 1995. Efficient Monte Carlo simulation of security prices. [*Annals of Applied Probability*]{} 5(4):897-905. Giles, M. B. 2008. Multilevel Monte Carlo path simulation. [*Operations Research*]{} 56(3):607–617. Glynn, P. W. and W. Whitt. 1992. The asymptotic efficiency of simulation estimators. [*Operations Research*]{} 40:505-520 Kloeden, P.E. and E. Platen. 1992. [*Numerical Solution of Stochastic Differential Equations*]{}. Berlin: Springer-Verlag. Rhee, C. and P. W. Glynn. 2012. From low bias to no bias: Application to SDE’s. Submitted for publication. AUTHOR BIOGRAPHIES {#author-biographies .unnumbered} ================== [**CHANG-HAN RHEE**]{} is currently a Ph.D. student in the Institute for Computational and Mathematical Engineering at Stanford University. He graduated with B.Sc. in the Department of Mathematics and Department of Computer Science at Seoul National University, South Korea. His research interests include simulation, computational probability, sensitivity analysis and stochastic control. His email address is and his web page is <http://www.stanford.edu/~chrhee/>.\ [**PETER W. GLYNN**]{} is the currently Chair of the Department of Management Science and Engineering at Stanford University. He received his Ph.D in Operations Research from Stanford University in 1982. He then joined the faculty of the University of Wisconsin at Madison, where he held a joint appointment between the Industrial Engineering Department and Mathematics Research Center, and courtesy appointments in Computer Science and Mathematics. In 1987, he returned to Stanford, where he joined the Department of Operations Research. He is now the Thomas Ford Professor of Engineering in the Department of Management Science and Engineering, and also holds a courtesy appointment in the Department of Electrical Engineering. From 1999 to 2005, he served as Deputy Chair of the Department of Management Science and Engineering, and was Director of Stanford’s Institute for Computational and Mathematical Engineering from 2006 until 2010. He is a Fellow of INFORMS and a Fellow of the Institute of Mathematical Statistics, has been co-winner of Best Publication Awards from the INFORMS Simulation Society in 1993 and 2008, was a co-winner of the Best (Biannual) Publication Award from the INFORMS Applied Probability Society in 2009, and was the co-winner of the John von Neumann Theory Prize from INFORMS in 2010. In 2012, he was elected to the National Academy of Engineering. His research interests lie in simulation, computational probability, queueing theory, statistical inference for stochastic processes, and stochastic modeling. His email address is and his web page is <http://www.stanford.edu/~glynn/>.

--- abstract: 'In this paper, we propose a state-of-the-art video denoising algorithm based on a convolutional neural network architecture. Until recently, video denoising with neural networks had been a largely under explored domain, and existing methods could not compete with the performance of the best patch-based methods. The approach we introduce in this paper, called FastDVDnet, shows similar or better performance than other state-of-the-art competitors with significantly lower computing times. In contrast to other existing neural network denoisers, our algorithm exhibits several desirable properties such as fast runtimes, and the ability to handle a wide range of noise levels with a single network model. The characteristics of its architecture make it possible to avoid using a costly motion compensation stage while achieving excellent performance. The combination between its denoising performance and lower computational load makes this algorithm attractive for practical denoising applications. We compare our method with different state-of-art algorithms, both visually and with respect to objective quality metrics.' author: - | Matias Tassano\ GoPro France\ [mtassano@gopro.com]{} - | Julie Delon\ MAP5, Université de Paris & IUF\ [julie.delon@parisdescartes.fr]{} - | Thomas Veit\ GoPro France\ [tveit@gopro.com]{} title: 'FastDVDnet: Towards Real-Time Deep Video Denoising Without Flow Estimation' --- Introduction {#sec:intro} ============ Despite the immense progress made in recent years in photographic sensors, noise reduction remains an essential step in video processing, especially when shooting conditions are challenging (low light, small sensors, etc.). Although image denoising has remained a very active research field through the years, too little work has been devoted to the restoration of digital videos. It should be noted, however, that some crucial aspects differentiate these two problems. On the one hand, a video contains much more information than a still image, which could help in the restoration process. On the other hand, video restoration requires good temporal coherency, which makes the restoration process much more demanding. Furthermore, since all recent cameras produce videos in high definition—or even larger—very fast and efficient algorithms are needed. In this paper we introduce another network for deep video denoising: FastDVDnet. This algorithm builds on DVDnet [@Tassano2019], but at the same time introduces a number of important changes with respect to its predecessor. Most notably, instead of employing an explicit motion estimation stage, the algorithm is able to implicitly handle motion thanks to the traits of its architecture. This results in a state-of-the-art algorithm which outputs high quality denoised videos while featuring very fast running times—even thousands of times faster than other relevant methods. Image denoising {#sec:image-denoising} --------------- Contrary to video denoising, image denoising has enjoyed consistent popularity in past years. A myriad of new image denoising methods based on deep learning techniques have drawn considerable attention due to their outstanding performance. Schmidt and Roth proposed in [@Schmidt2014a] the cascade of shrinkage fields method. The trainable nonlinear reaction diffusion model proposed by Chen and Pock in [@Chen2017] builds on the former. In [@Burger2012], a multi-layer perceptron was successfully applied for image denoising. Methods such as these achieve performances comparable to those of well-known patch-based algorithms such as BM3D [@Dabov2007a] or non-local Bayes (NLB [@Lebrun2013c]). However, their limitations include performance restricted to specific forms of prior, or the fact that a different set of weights must be trained for each noise level. Another widespread approach involves the use of convolutional neural networks (CNN), e.g. RBDN [@Santhanam2016], MWCNN [@Liu2018], DnCNN [@Zhang2017], and FFDNet [@Zhang2017a]. Their performance compares favorably to other state-of-the-art image denoising algorithms, both quantitatively and visually. These methods are composed of a succession of convolutional layers with nonlinear activation functions in between them. A salient feature that these CNN-based methods present is the ability to denoise several levels of noise with only one trained model. Proposed by Zhang in [@Zhang2017], DnCNN is an end-to-end trainable deep CNN for image denoising. One of its main features is that it implements residual learning [@He2016], i.e. it estimates the noise existent in the input image rather than the denoised image. In a following paper [@Zhang2017a], Zhang proposed FFDNet, which builds upon the work done for DnCNN. More recently, the approaches proposed in [@Plotz2018; @Liu2018non] combine neural architectures with non-local techniques. Video denoising {#sec:video-denoising} --------------- Video denoising is much less explored in the literature. The majority of recent video denoising methods are patch-based. We note in particular an extension of the popular BM3D to video denoising, V-BM4D [@Maggioni2012], and Video non-local Bayes (VNLB [@Arias2018]). Neural network methods for video denoising have been even rarer than patch-based approaches. The algorithm in [@chen2016deep] by Chen is one of the first to approach this problem with recurrent neural networks. However, their algorithm only works on grayscale images and it does not achieve satisfactory results, probably due to the difficulties associated with training recurring neural networks [@pascanu2013difficulty]. Vogels proposed in [@vogels2018denoising] an architecture based on kernel-predicting neural networks able to denoise Monte Carlo rendered sequences. The Video Non-Local Network (VNLnet [@Davy2019]) fuses a CNN with a self-similarity search strategy. For each patch, the network finds the most similar patches via its first non-trainable layer, and this information is later used by the CNN to predict the clean image. In [@Tassano2019], Tassano proposed DVDnet, which splits the denoising of a given frame in two separate denoising stages. Like several other methods, it relies on the estimation of motion of neighboring frames. Other very recent blind denoising approaches include the work by Ehret  [@Ehret2019] and ViDeNN [@Claus2019]. The latter shares with DVDnet the idea of performing denoising in two steps. However, contrary to DVDnet, ViDeNN does not employ motion estimation. Similarly to both DVDnet and ViDeNN, the use of spatio-temporal CNN blocks in restoration tasks has been also featured in [@vogels2018denoising; @Caballero2017]. Nowadays, the state-of-the-art is defined by DVDnet, VNLnet and VNLB. VNLB and VNLnet show the best performances for small values of noise, while DVDnet yields better results for larger values of noise. Both DVDnet and VNLnet feature significantly faster inference times than VNLB. As we will see, the performance of the method we introduce in this paper compares to the performance of the state-of-the-art, while featuring even faster runtimes. FastDVDnet {#sec:method} ========== For video denoising algorithms, temporal coherence and flickering removal are crucial aspects in the perceived quality of the results [@Seybold2018; @Seshadrinathan2010]. In order to achieve these, an algorithm must make use of the temporal information existent in neighboring frames when denoising a given frame of an image sequence. In general, most previous approaches based on deep learning have failed to employ this temporal information effectively. Successful state-of-the-art algorithms rely mainly on two factors to enforce temporal coherence in the results, namely the extension of search regions from spatial neighborhoods to volumetric neighborhoods, and the use of motion estimation. The use of volumetric (i.e. spatio-temporal) neighborhoods implies that when denoising a given pixel (or patch), the algorithm is going to look for similar pixels (patches) not only in the reference frame, but also in adjacent frames of the sequence. The benefits of this are two-fold. First, the temporal neighbors provide additional information which can be used to denoise the reference frame. Second, using temporal neighbors helps to reduce flickering as the residual error in each frame will be correlated. Videos feature a strong temporal redundancy along motion trajectories. This fact should facilitate denoising videos with respect to denoising images. Yet, this added information in the temporal dimension also creates an extra degree of complexity which could be difficult to tackle. In this context, motion estimation and/or compensation has been employed in a number of video denoising algorithms to help to improve denoising performance and temporal consistency [@Liu2015; @Tassano2019; @Arias2018; @Maggioni2012; @Buades2016a]. We thus incorporated these two elements into our architecture. However, our algorithm does not include an explicit motion estimation/compensation stage. The capacity of handling the motion of objects is inherently embedded into the proposed architecture. Indeed, our architecture is composed of a number of modified U-Net [@Ronneberger2015] blocks (see \[sec:denoising-blocks\] for more details about these blocks). Multi-scale, U-Net-like architectures have been shown to have the ability to learn misalignment [@Wu2018; @Dosovitskiy2015]. Our cascaded architecture increases this capacity of handling movement even further. In contrast to [@Tassano2019], our architecture is trained end-to-end without optical flow alignment, which avoids distortions and artifacts due to erroneous flow. As a result, we are able to eliminate a costly dedicated motion compensation stage without sacrificing performance. This leads to an important reduction of runtimes: our algorithm runs three orders of magnitude faster than VNLB, and an order of magnitude faster than DVDnet and VNLnet. displays a diagram of the architecture of our method. When denoising a given frame at time $ t $, $ \tilde{\mathbf{I}}_t $, its $ 2T=4 $ neighboring frames are also taken as inputs. That is, the inputs of the algorithm will be $ \left \{ \tilde{\mathbf{I}}_{t-2},\, \tilde{\mathbf{I}}_{t-1},\, \tilde{\mathbf{I}}_{t},\, \tilde{\mathbf{I}}_{t+1},\, \tilde{\mathbf{I}}_{t+2} \right \} $. The model is composed of different spatio-temporal denoising blocks, assembled in a cascaded two-step architecture. These denoising blocks are all similar, and consist of a modified [U-Net]{} model which takes three frames as inputs. The three blocks in the first denoising step share the same weights, which leads to a reduction of memory requirements of the model and facilitates the training of the network. Similar to [@Zhang2017a; @Gharbi2016], a noise map is also included as input, which allows the processing of spatially varying noise [@Tassano2019a]. In particular, the noise map is a separate input which provides information to the network about the distribution of the noise at the input. This information is encoded as the expected per-pixel standard deviation of this noise. For instance, when denoising Gaussian noise, the noise map will be constant; when denoising Poisson noise, the noise map will depend on the intensity of the image. Indeed, the noise map can be used as a user-input parameter to control the trade-off between noise removal vs. detail preservation (see for example the online demo in [@Tassano2019a]). In other cases, such as JPEG denoising, the noise map can be estimated by means of an additional CNN [@Guo2019]. The use of a noise map has been shown to improve denoising performance, particularly when treating spatially variant noise [@Brooks2019]. Contrary to other denoising algorithms, our denoiser takes no other parameters as inputs apart from the image sequence and the estimation of the input noise. Observe that experiments presented in this paper focus on the case of additive white Gaussian noise (AWGN). Nevertheless, this algorithm can be extended to other types of noise, e.g. spatially varying noise (e.g. Poissonian). Let $ \mathbf{I} $ be a noiseless image, while $\tilde{\mathbf{I}}$ is its noisy version corrupted by a realization of zero-mean white Gaussian noise $ \mathbf{N} $ of standard deviation $ \sigma $, then $$\label{eq:noisemod} \tilde{\mathbf{I}}=\mathbf{I}+\mathbf{N} \text{ .}$$ Denoising blocks {#sec:denoising-blocks} ---------------- Both denoising blocks displayed in \[fig:diagram-overview\], *Denoising Block 1* and *Denoising Block 2*, consist of a modified U-Net architecture. All the instances of *Denoising Block 1* share the same weights. U-Nets are essentially a multi-scale encoder-decoder architecture, with skip-connections [@He2016] that forward the output of each one of the encoder layers directly to the input of the corresponding decoder layers. A more detailed diagram of these blocks is shown in \[fig:diagram-denoiser\]. Our denoising blocks present some differences with respect to the standard U-Net: - [The encoder has been adapted to take three frames and a noise map as inputs]{} - [The upsampling in the decoder is performed with a *PixelShuffle* layer [@Shi2016], which helps reducing gridding artifacts. Please see the supplementary materials for more information about this layer.]{} - [The merging of the features of the encoder with those of the decoder is done with a pixel-wise addition operation instead of a channel-wise concatenation. This results in a reduction of memory requirements]{} - [Blocks implement residual learning—with a residual connection between the central noisy input frame and the output—, which has been observed to ease the training process [@Tassano2019a]]{} The design characteristics of the denoising blocks make a good compromise between performance and fast running times. These denoising blocks are composed of a total of $ D = 16 $ convolutional layers. In most layers, the outputs of its convolutional layers are followed by point-wise *ReLU* [@Krizhevsky2012] activation functions $ ReLU(\cdot) = \max (\cdot, 0) $, except for the last layer. Batch normalization layers (*BN* [@Ioffe2015]) are placed between the convolutional and *ReLU* layers. Discussion {#sec:discussion} ========== Explicit flow estimation is avoided in FastDVDnet. However, in order to maintain performance, we needed to introduce a number of techniques to handle motion and to effectively employ temporal information. These techniques are discussed further in this section. Please see the supplementary materials for more details about ablation studies. Two-step denoising {#ssec:two-step} ------------------ Similarly to DVDnet and ViDeNN, FastDVDnet features a two-step cascaded architecture. The motivation behind this is to effectively employ the information existent in the temporal neighbors, and to enforce the temporal correlation of the remaining noise in output frames. To prove that the two-step denoising is a necessary feature, we conducted the following experiment: we modified a *Denoising Block* of FastDVDnet (see \[fig:diagram-denoiser\]) to take five frames as inputs instead of three, which we will refer to as *Den\_Block\_5inputs*. In this way, the same amount of temporal neighboring frames are considered and the same information as in FastDVDnet is processed by this new denoiser. A diagram of the architecture of this model is shown in \[fig:architecture-5in\]. We then trained this new model and compared the results of denoising of sequences against the results of FastDVDnet (see \[sec:training-details\] for more details about the training process). It was observed that the cascaded architecture of FastDVDnet presents a clear advantage on *Den\_Block\_5inputs*, with differences in PSNR of up to $ 0.9dB $. Please refer to the supplementary materials for more details. Additionally, results by *Den\_Block\_5inputs* present a sharp increase on temporal artifacts—flickering. Despite it being a multi-scale architecture, *Den\_Block\_5inputs* cannot handle the motion of objects in the sequences as well as the two-step architecture of FastDVDnet can. Overall, the two-step architecture shows superior performance with respect to the one-step architecture. ![Architecture of the *Den\_Block\_5inputs* denoiser.[]{data-label="fig:architecture-5in"}](diagram-overview-5in){width="0.9\linewidth"} Multi-scale architecture and end-to-end training {#ssec:ms-end-to-end} ------------------------------------------------ In order to investigate the importance of using multi-scale denoising blocks in our architecture, we conducted the following experiment: we modified the FastDVDnet architecture by replacing its *Denoising Blocks* by the denoising blocks of DVDnet. This results in a two-step cascaded architecture, with single-scale denoising blocks, trained end-to-end, and with no compensation of motion in the scene. In our tests, it was observed that the usage of multi-scale denoising blocks improves denoising results considerably. Please refer to the supplementary materials for more details. We also experimented with training the multi-scale denoising blocks in each step of FastDVDnet separately—as done in DVDnet. Although the results in this case certainly improved with respect to the case of the single-scale denoising blocks described above, a noticeable flickering remained in the outputs. Switching from this separate training to an end-to-end training helped to reduce temporal artifacts considerably. Handling of motion {#ssec:mc} ------------------ Apart from the reduction of runtimes, avoiding the use of motion compensation by means of optical flow has an additional benefit. Video denoising algorithms that depend explicitly on motion estimation techniques often present artifacts due to erroneous flow in challenging cases, such as occlusions or strong noise. The different techniques discussed in this section—namely a multi-scale of the denoising blocks, the cascaded two-step denoising architecture, and end-to-end training—not only provide FastDVDnet the ability to handle motion, but also help avoid artifacts related to erroneous flow estimation. Also, and similarly to [@Zhang2017; @Tassano2019; @Tassano2019a], the denoising blocks of FastDVDnet implement residual learning, which helps to improve the quality of results a step further. shows an example on artifacts due to erroneous flow on three consecutive frames and of how the multi-scale architecture of FastDVDnet is able to avoid them. Training details {#sec:training-details} ================ The training dataset consists of input-output pairs $$P_t^j = \left \{ \left( ( \,S_t^j ,\: \mathbf{M}^j\, ), \, \mathbf{{I}}_t^j \right )\right \}_{j=0}^{m_t} \, ,$$ where $ S_t^j = (\mathbf{\tilde{I}}_{t-2}^j,\:\mathbf{\tilde{I}}_{t-1}^j,\:\mathbf{\tilde{I}}_{t}^j,\:\mathbf{\tilde{I}}_{t+1}^j,\:\mathbf{\tilde{I}}_{t+2}^j) $ is a collection of $ 2T+1=5 $ spatial patches cropped at the same location in contiguous frames, and $ \mathbf{{I}}^j $ is the clean central patch of the sequence. These are generated by adding AWGN of $ \sigma \in [5, \, 50] $ to clean patches of a given sequence, and the corresponding noise map $ \mathbf{M}^j $ is built in this case constant with all its elements equal to $ \sigma $. Spatio-temporal patches are randomly cropped from randomly sampled sequences of the training dataset. A total of $ m_t = 384000 $ training samples are extracted from the training set of the DAVIS database [@KhoRohrSch_ACCV2018]. The spatial size of the patches is $ 96 \times 96 $, while the temporal size is $ 2T+1 = 5 $. The spatial size of the patches was chosen such that the resulting patch size in the coarser scale of the *Denoising Blocks* is $ 32\times 32 $. The loss function is $$\label{eq:temp-loss} \mathcal{L} (\theta)= \frac{1}{2{m_t}} \sum_{j=1}^{m_t} \left \| \mathbf{\hat{I}}_{t}^j - \mathbf{{I}}_{t}^j \right \|^2 \, ,$$ where $ \mathbf{\hat{I}}_{t}^j = \mathcal{F} (( \,S_t^j ,\: \mathbf{M}^j\, );\,\theta) $ is the output of the network, and $ \theta $ is the set of all learnable parameters. The architecture has been implemented in PyTorch [@Paszke2017], a popular machine learning library. The ADAM algorithm [@Kingma2015] is applied to minimize the loss function, with all its hyper-parameters set to their default values. The number of epochs is set to $ 80 $, and the mini-batch size is $ 96 $. The scheduling of the learning rate is also common to both cases. It starts at $ 1\mathrm{e}{-3} $ for the first $ 50 $ epochs, then changes to $ 1\mathrm{e}{-4} $ for the following $ 10 $ epochs, and finally switches to $ 1\mathrm{e}{-6} $ for the remaining of the training. In other words, a learning rate step decay is used in conjunction with ADAM. The mix of learning rate decay and adaptive rate methods has also been applied to other deep learning projects [@Szegedy2015; @Wilson2017], usually with positive results. Data is augmented by introducing rescaling by different scale factors and random flips. During the first $ 60 $ epochs, the orthogonalization of the convolutional kernels is applied as a means of regularization. It has been observed that initializing the training with orthogonalization may be beneficial to performance [@Zhang2017a; @Tassano2019a]. Results {#sec:results} ======= Two different testsets were used for benchmarking our method: the DAVIS-test testset, and Set8, which is composed of $ 4 $ color sequences from the *Derf’s Test Media collection*[^1] and $ 4 $ color sequences captured with a GoPro camera. The DAVIS set contains $ 30 $ color sequences of resolution $ 854 \times 480 $. The sequences of Set8 have been downscaled to a resolution of $ 960 \times 540 $. In all cases, sequences were limited to a maximum of $ 85 $ frames. We used the DeepFlow algorithm [@weinzaepfel:hal-00873592] to compute flow maps for DVDnet and VNLB. VNLnet requires models trained for specific noise levels. As no model is provided for $ \sigma =30 $, no results are shown for this noise level in either of the tables. We also compare our method to a commercial blind denoising software, Neat Video (NV [@neatvideo19]). For NV, its automatic noise profiling settings were used to manually denoise the sequences of Set8. Note that values shown are the average for all sequences in the testset, the PNSR of a sequence is computed as the average of the PSNRs of each frame. In general, both DVDnet and FastDVDnet output sequences which feature remarkable temporal coherence. Flickering rendered by our methods is notably small, especially in flat areas, where patch-based algorithms often leave behind low-frequency residual noise. An example can be observed in \[fig:results-snow\] (which is best viewed in digital format). Temporally decorrelated low-frequency noise in flat areas appears as particularly bothersome for the viewer. More video examples can be found in the supplementary materials and on the website of the algorithm. The reader is encouraged to watch these examples to compare the visual quality of the results of our methods. Patch-based methods are prone to surpassing DVDnet and FastDVDnet in sequences with a large portion of repetitive structures as these methods exploit the non-local similarity prior. On the other hand, our algorithms handle non-repetitive textures very well, see e.g. the clarity of the denoised text and vegetation in \[fig:results-motorbike\]. shows a comparison of PSNR and ST-RRED on the Set8 and DAVIS dataset, respectively. The Spatio-Temporal Reduced Reference Entropic Differences (ST-RRED) is a high performing reduced-reference video quality assessment metric [@Soundararajan2013]. This metric not only takes into account image quality, but also temporal distortions in the video. We computed the ST-RRED scores with the implementation provided by the *scikit-video* library[^2]. It can be observed that for smaller values of noise, VNLB performs better on Set8. Indeed, DVDnet tends to over denoise in some of these cases. FastDVDnet and VNLnet are the best performing algorithms on DAVIS for small sigmas in terms of PSNR and ST-RRED, respectively. However, for larger values of noise DVDnet surpasses VNLB. FastDVDnet performs consistently well in all cases, which is a remarkable feat considering that it runs $ 80 $ times faster than DVDnet, $ 26 $ times faster than VNLnet, and more than $ 4000 $ times faster than VNLB (see \[sec:running-times\]). Contrary to other denoisers based on CNNs—e.g. VNLnet—, our algorithms are able to denoise different noise levels with only one trained model. On top of this, the use of methods involve no hand-tuned parameters, since they only take the image sequence and the estimation of the input noise as inputs. displays a comparison with ViDeNN. This algorithm has not actually been trained for AWGN, but for clipped AWGN. Then, a FastDVDnet model to denoise clipped AWGN was trained for this case, which we call *FastDVDnet\_clipped*. It can be observed that the performance of FastDVDnet\_clipped is superior to the performance of ViDeNN by a wide margin. **Set8** VNLB V-BM4D NV VNLnet DVDnet FastDVDnet ----------------- -------------------------------------------- ----------------------- ----------------------- ----------------------------------- ----------------------------------- ------------------------------------------- $ \sigma = 10 $ $ \color{blue}37.26 $ / $ \textbf{2.86} $ $ 36.05 $ / $ 3.87 $ $ 35.67 $ / $ 3.42 $ $ \color{red}37.10 $ / $ 3.43 $ $ 36.08 $ / $ 4.16 $ $ 36.44 $ / $ 3.00 $ $ \sigma = 20 $ $ \color{blue}33.72 $ / $ \textbf{6.28} $ $ 32.19 $ / $ 9.89 $ $ 31.69 $ / $ 12.48 $ $ \color{red}33.88 $ / $ 6.88 $ $ 33.49 $ / $ 7.54 $ $ 33.43 $ / $ 6.65 $ $ \sigma = 30 $ $ \color{red}31.74 $ / $ \textbf{11.53} $ $ 30.00 $ / $ 19.58 $ $ 28.84 $ / $ 33.19 $ - $ \color{blue}31.79 $ / $ 12.61 $ $ 31.68 $ / $ 11.85 $ $ \sigma = 40 $ $ 30.39 $ / $ 18.57 $ $ 28.48 $ / $ 32.82 $ $ 26.36 $ / $ 47.09 $ $ \color{blue}30.55 $ / $ 19.71 $ $ \color{blue}30.55 $ / $ 19.05 $ $ \color{red}30.46 $ / $ \textbf{18.45} $ $ \sigma = 50 $ $ 29.24 $ / $ 27.39 $ $ 27.33 $ / $ 49.20 $ $ 25.46 $ / $ 57.44 $ $ 29.47 $ / $ 29.78 $ $ \color{blue}29.56 $ / $ 27.97 $ $ \color{red}29.53 $ / $ \textbf{26.75} $ **DAVIS** VNLB V-BM4D VNLnet DVDnet FastDVDnet ----------------- ----------------------------------- ----------------------- ------------------------------- -------------------------------------------- ------------------------------------ $ \sigma = 10 $ $ \color{blue}38.85 $ / $ 3.22 $ $ 37.58 $ / $ 4.26 $ $ 35.83 $ / $ \textbf{2.81} $ $ 38.13 $ / $ 4.28 $ $ \color{red}38.71 $ / $ 3.49 $ $ \sigma = 20 $ $ 35.68 $ / $ 6.77 $ $ 33.88 $ / $ 11.02 $ $ 34.49 $ / $ \textbf{6.11} $ $ \color{red}35.70 $ / $ 7.54 $ $ \color{blue}35.77 $ / $ 7.46 $ $ \sigma = 30 $ $ 33.73 $ / $ 12.08 $ $ 31.65 $ / $ 21.91 $ - $ \color{blue}34.08 $ / $ 12.19 $ $ \color{red}34.04 $ / $ 13.08 $ $ \sigma = 40 $ $ 32.32 $ / $ 19.33 $ $ 30.05 $ / $ 36.60 $ $ 32.32 $ / $ 18.63 $ $ \color{blue}32.86 $ / $ \textbf{18.16} $ $ \color{red}32.82 $ / $ 20.39 $ $ \sigma = 50 $ $ 31.13 $ / $ 28.21 $ $ 28.80 $ / $ 54.82 $ $ 31.43 $ / $ 28.67 $ $ \color{red}31.85 $ / $ \textbf{25.63} $ $ \color{blue}31.86 $ / $ 28.89 $ ----------------- ----------- --------------------- **DAVIS** ViDeNN FastDVDnet\_clipped $ \sigma = 10 $ $ 37.13 $ $ \textbf{38.45} $ $ \sigma = 30 $ $ 32.24 $ $ \textbf{33.52} $ $ \sigma = 50 $ $ 29.77 $ $ \textbf{31.23} $ ----------------- ----------- --------------------- : \[tbl:results-videnn\]Comparison with ViDeNN for clipped AWGN. See the text for more details. For PSNR: larger is better; best results are shown in bold. Running times {#sec:running-times} ============= Our method achieves fast inference times, thanks to its design characteristics and simple architecture. Our algorithm takes only $ 100ms $ to denoise a $ 960 \times 540 $ color frame, which is more than $ 3 $ orders of magnitude faster than V-BM4D and VNLB, and more than an order of magnitude faster than other CNN algorithms which run on GPU, DVDnet and VNLnet. The algorithms were tested on a server with a Titan Xp NVIDIA GPU card. compares the running times of different state-of-the-art algorithms. ![\[fig:running-times\]*Comparison of running times.* Time to denoise a color frame of resolution $ 960 \times 540 $. Note: values displayed for VNLB do not include the time required to estimate motion.](jpeg/conv/runtimes_all){width="0.9\linewidth"} Conclusion {#sec:conclusions} ========== In this paper, we presented FastDVDnet, a state-of-the-art video denoising algorithm. Denoising results of FastDVDnet feature remarkable temporal coherence, very low flickering, and excellent detail preservation. This level of performance is achieved even without a flow estimation step. The algorithm runs between one and three orders of magnitude faster than other state-of-the-art competitors. In this sense, our approach proposes a major step forward towards high quality real-time deep video noise reduction. Although the results presented in this paper hold for Gaussian noise, our method could be extended to denoise other types of noise. Acknowledgments {#acknowledgments .unnumbered} =============== Julie Delon would like to thank the support of NVIDIA Corporation for providing us with the Titan Xp GPU used in this research. We thank Anna Murray and José Lezama for their valuable contribution. This work has been partially funded by the French National Research and Technology Agency (ANRT) and GoPro Technology France. **Supplemental Materials** Two-step denoising {#ssec:two-step-supl} ================== FastDVDnet features a two-step cascaded architecture. The motivation behind this is to effectively employ the information existent in the temporal neighbors, and to enforce the temporal correlation of the remaining noise in output frames. To prove that the two-step denoising is a necessary feature, we conducted the following experiment: we modified a *Denoising Block* of FastDVDnet (see the associated paper) to take five frames as inputs instead of three, which we will refer to as *Den\_Block\_5inputs*. In this way, the same amount of temporal neighboring frames are considered and the same information as in FastDVDnet is processed by this new denoiser. A diagram of the architecture of this model is shown in \[fig:architecture-5in-suppl\]. We then trained this new model and compared the results of denoising of sequences against the results of FastDVDnet. displays the PSNRs on four $ 854 \times 480 $ color sequences for both denoisers. It can be observed that the cascaded architecture of FastDVDnet presents a clear advantage on *Den\_Block\_5inputs*, with an average difference of PSNRs of $ 0.95dB $. Additionally, results by *Den\_Block\_5inputs* present a sharp increase on temporal artifacts—flickering. Despite it being a multi-scale architecture, *Den\_Block\_5inputs* cannot handle the motion of objects in the sequences as well as the two-step architecture of FastDVDnet can. Overall, the two-step architecture shows superior performance with respect to the one-step architecture. ![Architecture of the *Den\_Block\_5inputs* denoiser.[]{data-label="fig:architecture-5in-suppl"}](diagram-overview-5in){width="\linewidth"} FastDVDnet Den\_Block\_5inputs ----------------- ------------- ------------ --------------------- $ \sigma = 10 $ hypersmooth **37.34** 35.64 motorbike **34.86** 34.00 rafting **36.20** 34.61 snowboard **36.50** 34.27 $ \sigma = 30 $ hypersmooth **32.17** 31.21 motorbike **29.16** 28.77 rafting **30.73** 30.03 snowboard **30.59** 29.67 $ \sigma = 50 $ hypersmooth **29.77** 28.92 motorbike **26.51** 26.19 rafting **28.45** 27.88 snowboard **28.08** 27.37 : \[tbl:results-temp-5in-suppl\]Comparison of $ PSNR $ of two denoisers on four sequences. Best results are shown in bold. Note: for this test in particular, neither of these denoisers implement residual learning. Multi-scale architecture and end-to-end training {#ssec:ms-end-to-end-supl} ================================================ In order to investigate the importance of using multi-scale denoising blocks in our architecture, we conducted the following experiment: we modified the FastDVDnet architecture by replacing its *Denoising Blocks* by the denoising blocks of DVDnet. This results in a two-step cascaded architecture, with single-scale denoising blocks, trained end-to-end, and with no compensation of motion in the scene. We will call this new architecture FastDVDnet\_Single. shows the PSNRs on four $ 854 \times 480 $ color sequences for both FastDVDnet and FastDVDnet\_Single. It can be seen that the usage of multi-scale denoising blocks improves denoising results considerably. In particular, there is an average difference of PSNRs of $ 0.55dB $ in favor of the multi-scale architecture. FastDVDnet FastDVDnet\_Single ----------------- ------------- ------------ -------------------- $ \sigma = 10 $ hypersmooth **37.34** 36.61 motorbike **34.86** 34.30 rafting **36.20** 35.54 snowboard **36.50** 35.50 $ \sigma = 30 $ hypersmooth **32.17** 31.54 motorbike **29.16** 28.82 rafting **30.73** 30.36 snowboard **30.59** 30.04 $ \sigma = 50 $ hypersmooth **29.77** 29.14 motorbike **26.51** 26.22 rafting **28.45** 28.11 snowboard **28.08** 27.56 : \[tbl:results-fastdvdnet-ffdnet-suppl\]Comparison of $ PSNR $ of a single-scale denoiser against a multi-scale denoiser on four sequences. Best results are shown in bold. Note: for this test in particular, neither of these denoisers implement residual learning. Ablation studies {#ssec:ablation} ================ A number of modifications with respect to the baseline architecture discussed in the associated paper have been tested, namely: - the use of *Leaky ReLU* [@Maas2013] or *ELU* [@Clevert2016] instead of *ReLU*. In neither case significant changes in performance were observed, with average differences in PSNR of less than $ 0.05dB $ on all the sequences and standard deviation of noise considered. - optimizing with respect to the Huber loss [@Girshick2015] instead of the $ L_2 $ norm. No significant change of performance was observed. The mean difference in PSNR on all the sequences and standard deviation of noise considered was $ 0.04dB $ in favor of the $ L_2 $ norm case. - removing batch normalization layers. An drop in performance of $ 0.18dB $ on average was observed for this case. - taking more input frames. The baseline model was modified to take 7 and 9 input frames instead of 5. No improvement in performance was observed in neither case. It was also observed an increased difficulty of these models, which have more parameters, to converge during training with respect to the case with 5 input frames. Upscaling layers ================ In the multi-scale denoising blocks, the upsampling in the decoder is performed with a *PixelShuffle* layer [@Shi2016]. This layer repacks its input of dimension $ 4n_{ch} \times h/2 \times w/2 $ into an output of size $ n_{ch} \times h \times w $, where $ ch,\,h,\,w $ are the number of channels, the height, and the width, respectively. In other words, this layer constructs all the $ 2 \times 2 $ non-overlapping patches of its output with the pixels of different channels of the input, as shown in \[fig:down-up\] ![Upscaling layer.[]{data-label="fig:down-up"}](tikz/diag-fast.pdf){width="0.9\linewidth"} Gaussian noise model ==================== Recently, a number of algorithms have been proposed for video and burst denoising in low-light conditions, e.g. [@Chen2019; @Wang2019; @Hasinoff2016]. What is more, some of these works argue that real noise cannot be accurately modeled with a simple Gaussian model. Yet, the algorithm we propose here has been developed for Gaussian denoising because although Gaussian i.i.d. noise is not utterly realistic, it eases the comparison with other methods on comparable datasets—one of our primary goals. We believe Gaussian denoising is a middle ground where different denoising architectures can be compared fairly. Some networks which are proposed to denoise a specific low-light dataset are designed and overfitted given the image processing pipe of said dataset. In some cases, the comparison against other methods which have not been designed for the given dataset—e.g. the current version of our method—might not be accurate. Nonetheless, low-light denoising is not the main objective of our submission. Rather, it is to show that a simple, yet carefully designed architecture can outperform other more complex methods. We believe that the main challenge to denoising algorithms is the input signal-to-noise ratio. In this regard, the presented results have similar characteristics to low-light videos. Permutation invariance ====================== The algorithm proposed for burst deblurring and denoising in [@Aittala2018] features invariance to the permutation of the ordering of its input frames. One might be tempted to replicate its characteristics in an architecture such as ours to benefit from the advantages of the permutation invariance. However, the application of our algorithm is video denoising—which is not identical to burst denoising. Actually, the order in the input frames is a prior exploited by our algorithm to enforce the temporal coherence in the output sequence. In other words, permutation invariance is not necessarily desirable in our case. Recursive processing ==================== As previously discussed, in practice, the processing of our algorithm is limited to five input frames. Given this limitation, one would wonder if the theoretic performance bound might be lower to that of other solutions based on recursive processing (i.e. using the output frame in time $ t $ as input to the next frame in time $ t+1 $). Yet, our experience with recursive filtering of videos is that it is difficult for the latter methods to be on par with methods which employ multiple frames as input. Although, in theory, recursive methods are asymptotically more powerful in terms of denoising than multi-frame methods, in practice the performance of recursive methods suffers due to temporal artifacts. Any misalignment or motion compensation artifact which might appear in the output frame at a given time is very likely to appear in all subsequent outputs. An interesting example to illustrate this point is the comparison of the method in [@Ehret2018] versus the video non-local Bayes denoiser (VNLB [@Arias2018]). The former implements a recursive version of VNLB, which results in a lower complexity algorithm, but with very inferior performance with respect to the latter. [^1]: <https://media.xiph.org/video/derf> [^2]: http://www.scikit-video.org

--- abstract: 'In this paper, we explore when a locally finite triangulated category has dimension zero or finite representation type. We also study generation of derived categories by orthogonal subcategories.' address: - 'Department of Mathematics, Tokyo Gakugei University, 4-1-1 Nukuikita-machi, Koganei, Tokyo 184-8501, Japan' - 'Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, Aichi 464-8602, Japan' author: - Takuma Aihara - Ryo Takahashi title: Remarks on dimensions of triangulated categories --- [^1] [^2] Introduction ============ In this paper, we mention two remarks regarding the notion of dimensions of triangulated categories, which has been introduced by Rouquier [@R]. The first subject of this paper, which is discussed in Section 2, is to describe “smallest” triangulated categories. We focus on three kinds of smallness of triangulated categories: local finiteness, dimension zero and finite representation type. It is natural to ask if there exist implications among these conditions. We give the following answer to this question. 1. Let $\Lambda$ be an Iwanaga–Gorenstein algebra over a complete local ring with an isolated singularity. The stable category $\sCM(\Lambda)$ of Cohen–Macaulay modules is locally finite if and only if $\Lambda$ has finite CM representation type. 2. Let $\T$ be a Krull–Schmidt triangulated category. If $\T$ is finitely generated and locally finite, then it has dimension zero. If $\T$ is Ext-finite and has dimension zero, then it is locally finite. Applying the first assertion of this theorem, we deduce that for an isolated hypersurface singularity $R$, the stable category $\sCM(R)$ is locally finite if and only if it has dimension zero, if and only if $R$ has finite CM representation type (Corollary \[ihs\]). The second subject of this paper, which is discussed in Section 3, is to find a generator $\G$ of a derived category $\D$ such that the dimension of $\D$ with respect to $\G$ in the sense of [@ddcm] is as small as possible. This brings us a lower bound of the dimension of the derived category. The main result on this subject is the following. Let $\A$ be an abelian category and $\X$ a full subcategory. Let $M$ be an object of $\A$ admitting an exact sequence with $X_0,\dots,X_n\in\X$ $$\begin{aligned} &0\to X_n\to \cdots \to X_0 \to N\to M\to0\\ \text{(resp. }&0\to M\to N\to X_0\to\cdots\to X_n\to0\text{)}\end{aligned}$$ such that the corresponding element in $\Ext_\A^{n+1}(M,X_n)$ (resp. $\Ext_\A^{n+1}(X_n, M)$) is nonzero. Then $M$ is outside $\langle {}^\perp\X \rangle_{n+1}$ (resp. $\langle \X^\perp \rangle_{n+1}$) in the derived category $\da$ of $\A$. This theorem recovers the lower bounds of the derived dimension given by Rouquier [@R Proposition 7.14], Krause and Kussin [@KK Lemma 2.4] and Yoshiwaki [@Y2 Theorem 1.1]. Dimension zero versus local finiteness ====================================== Throughout this section, let $\T$ be a Krull–Schmidt triangulated category. We use $k$ and $R$ in this section as an algebraically closed field and a commutative noetherian complete local ring, respectively. Let $\Lambda$ be a *noetherian $R$-algebra*, i.e., there is a ring homomorphism $R$ to $\Lambda$ with the image contained in the center of $\Lambda$ such that $\Lambda$ is a finitely generated $R$-module. Note that $\Lambda$ is semiperfect as $R$ is complete (see [@A]). Local finiteness ---------------- We recall the definition of locally finite triangulated categories. \[def:LF\] We say that $\T$ is *locally finite* if for each object $Y$ of $\T$, (i) there are only finitely many indecomposable objects $X$ with $\Hom_\T(X,Y)\neq0$, and (ii) for every indecomposable object $X$ of $\T$, the right $\End_\T(X)$-module $\Hom_\T(X, Y)$ has finite length. Note that these are equivalent to the dual conditions; see [@A1; @K]. For an additive category $\C$ we denote by $\underline\C$ its *stable category*, i.e., the quotient of $\C$ by the projective objects. This is triangulated if $\C$ is Frobenius [@H]. We denote by $\mod\Lambda$ the category of finitely generated (right) $\Lambda$-modules; it is Frobenius (and hence the stable category $\smod\Lambda$ is triangulated) if $\Lambda$ is selfinjective. Denote by $\Db(\mod\Lambda)$ the bounded derived category of $\mod\Lambda$. Here are examples of a locally finite triangulated category. \[exLF\] Let $\Lambda$ be a finite dimensional $k$-algebra. (1) The derived category $\Db(\mod\Lambda)$ is locally finite if and only if $\Lambda$ is a piecewise hereditary algebra of finite representation type. (2) Suppose that $\Lambda$ is a selfinjective algebra. Then the stable category $\smod\Lambda$ is locally finite if and only if $\Lambda$ is of finite representation type. A triangulated category $\T$ is called *finitely generated* if it admits a thick generator $T$, that is, if $\T=\thick T$. Here, $\thick T$ is the smallest thick subcategory of $\T$ containing $T$. Important properties of locally finite triangulated categories are stated in [@K], including: [@K Proposition 4.5]\[bpLF\] Let $\T$ be a finitely generated locally finite triangulated category. Then there are only finitely many thick subcategories of $\T$. In representation theory of rings, the notion of representation types is one of the most classical subjects, and the first step is to understand finite representation type. We study the relationship of local finiteness with finiteness of representation type. For an additive category $\C$, $\ind\C$ denotes the set of isomorphism classes of indecomposable objects of $\C$. \[LFRF\] Let $\F$ be a Krull–Schmidt Frobenius category whose stable category $\sF$ satisfies Definition \[def:LF\](ii). Assume that $\proj\F=\{X\in \F\ |\ \Ext_\F^1(M,X)=0 \}$ for some $M\in\F$. Then $\sF$ is locally finite if and only if it admits an additive generator. The ‘if’ part is evident. For the ‘only if’ part, the local finiteness of $\sF$ implies that $$\X=\{X\in\ind\sF\ |\ \sHom_\F(M[-1], X)\ne0\}$$ is a finite set; we write $\X=\{X_1,\dots,X_n\}$. Let $Y$ be an indecomposable object of $\F$ which does not belong to $\X$. Then $\Ext_\F^1(M, Y)$ vanishes. By assumption $Y$ has to be projective, which implies that $Y=0$ in $\sF$. Thus, we obtain $\sF=\add(X_1\oplus\cdots\oplus X_n)$. For a prime ideal $\p$ of $R$, set $(-)_\p:=-\otimes_RR_\p$. We say that $\Lambda$ has an *isolated singularity* if $\gldim\Lambda_\p=\dim R_\p$ for any nonmaximal primes $\p$ of $R$ (cf. [@IW]). Denote by $\CM(\Lambda)$ the full subcategory of $\mod\Lambda$ consisting of *Cohen–Macaulay $\Lambda$-modules*, i.e., finitely generated $\Lambda$-modules $X$ with $\Ext_\Lambda^i(X, \Lambda)=0$ for all $i>0$. We say that $\Lambda$ is *Iwanaga–Gorenstein* if it has finite right and left selfinjective dimension. An Iwanaga–Gorenstein algebra $\Lambda$ is said to have *finite Cohen–Macaulay (abbr. CM) representation type* if $\CM(\Lambda)$ has an additive generator. Let us recall a fundamental fact. \[koremo\] Let $\Lambda$ be an Iwanaga–Gorenstein $R$-algebra with an isolated singularity. Then the stable category $\sCM(\Lambda)$ is a Krull–Schmidt triangulated category whose Hom-sets have finite length as $R$-modules. As is well-known, $\CM(\Lambda)$ is Frobenius, and $\sCM(\Lambda)$ is triangulated [@H]. Moreover, $\CM(\Lambda)$ (and $\sCM(\Lambda)$) inherits the Krull–Schmidt property from $\mod\Lambda$. Let $M$ and $N$ be in $\CM(\Lambda)$. Put $d:=\dim R$ and take a nonmaximal prime ideal $\p$ of $R$. We get isomorphisms $$\sHom_\Lambda(M, N[d])_\p\simeq \sHom_{\Lambda_\p}(M_\p, N_\p[d])\simeq \Ext_{\Lambda_\p}^d(M_\p, N_\p)=0.$$ Here, the first isomorphism comes from the fact that the functor $(-)_\p$ is exact and the last equality holds since $\gldim \Lambda_\p=\dim R_\p<d$. Therefore, we see that $\sHom_\Lambda(M, N[d])$ has finite length as an $R$-module, whence so does any Hom-set of $\sCM(\Lambda)$. We deduce the following from Propositions \[LFRF\] and \[koremo\], which extends Example \[exLF\](2). \[Glr\] Let $\Lambda$ be an Iwanaga–Gorenstein $R$-algebra with an isolated singularity. Then $\sCM(\Lambda)$ is locally finite if and only if $\Lambda$ has finite CM representation type. Put $S:=\Lambda/\rad\Lambda$, where $\rad\Lambda$ stands for the Jacobson radical of $\Lambda$. As $d:=\id\Lambda_\Lambda<\infty$, we observe that $\Omega^dS$ belongs to $\CM(\Lambda)$. We show that $\Omega^dS$ plays the role of $M$ as in Proposition \[LFRF\]. Let $X\in\CM(\Lambda)$ satisfy $\Ext_\Lambda^1(\Omega^dS, X)=0$. Then $\Ext_\Lambda^{d+1}(S, X)=0$. Thanks to Auslander’s result, $X$ has finite injective dimension; see [@GN Corollary 3.5(3)] for instance. Hence $X$ is projective. Now Propositions \[LFRF\] and \[koremo\] complete the proof. Dimension zero -------------- For $X\in\T$ set $\langle X\rangle:=\add\{X[i]\mid i\in\mathbb{Z} \}$. We say *$\T$ has dimension zero* and write $\dim\T=0$, if $\T=\langle X\rangle$ for some $X\in\T$. The following are well-known. \[exd0\] Let $\Lambda$ be a finite dimensional $k$-algebra. (1) [@CYZ Theorem] One has $\dim\Db(\mod\Lambda)=0$ if and only if $\Lambda$ is a piecewise hereditary algebra of finite representation type. (2) [@Y Corollary 3.10] Suppose that $\Lambda$ is selfinjective. Then $\dim(\smod\Lambda)=0$ if and only if $\Lambda$ is of finite representation type. Comparing this example with Example \[exLF\], we have a natural question. \[question\] Is it true that $\T$ is locally finite if and only if it has dimension zero? A $k$-linear triangulated category $\T$ is called *Ext-finite* if $\sum_{i\in\mathbb{Z}}\dim_k\Hom_\T(X, Y[i])<\infty$ for any objects $X$ and $Y$ of $\T$. Here are our answers to Question \[question\]. \[main\] 1. If $\T$ is finitely generated and locally finite, then it has dimension $0$. 2. Assume that $\T$ is Ext-finite. If $\T$ has dimension $0$, then it is locally finite. \(1) Let $T$ be a thick generator of $\T$ and put $\M:=\{M\in \ind\T\ |\ \Hom_\T(T, M)\neq0\}$. Let $N$ be an indecomposable object of $\T$. Since $T$ is a thick generator, we observe that if $\Hom_\T(T, N[i])=0$ for all integers $i$, then $N=0$, contrary to the choice of $N$. Hence $N[\ell]$ belongs to $\M$ for some integer $\ell$. This shows that every object of $\T$ is in $\langle \M\rangle$, and thus $\T=\langle \M\rangle$. As $\T$ is locally finite, $\M$ is a finite set. Therefore, putting $M$ as the direct sum of all objects in $\M$, one has $\T=\langle M\rangle$, and obtains $\dim\T=0$. \(2) As $\dim\T=0$, there is an object $M\in\T$ with $\T=\langle M \rangle$. Taking an indecomposable decomposition $M=M_1\oplus M_2\oplus\cdots\oplus M_r$, we obtain $\ind\T=\{M_i[j]\ |\ i=1,\dots,r\ \mbox{and } j\in\mathbb{Z} \}$ by the Krull–Schmidtness of $\T$. Let $Y\in\T$. Since $\T$ is Ext-finite, there exist only finitely many indecomposable objects $X$ with $\Hom_\T(X, Y)\neq0$, whence $\T$ is locally finite. The following is an extension of Proposition \[bpLF\] from the viewpoint of Theorem \[main\](1). If $\dim\T=0$, then $\T$ possesses only fnitely many thick subcategories. By assumption, we have $\T=\langle X\rangle$ for some $X\in\T$. Write $X=X_1\oplus X_2\oplus\cdots\oplus X_n$ with each $X_i$ indecomposable. Let $\U$ be a thick subcategory of $\T$. Then we may assume that $X_1,\dots,X_m$ are in $\U$, and $X_{m+1},\dots,X_n$ are not in $\U$. Now it is evident that $\U=\langle X_1\oplus\cdots\oplus X_m\rangle$, and the assertion follows. Applications ------------ In this subsection, let $\Lambda$ be an Iwanaga–Gorenstein $R$-algebra with an isolated singularity. Here is a conjecture, which seems to be folklore: \[conjecture\] The stable category $\sCM(\Lambda)$ has dimension zero if and only if the algebra $\Lambda$ has finite CM representation type. In view of Theorem \[Glr\], we get the following. \[questionconjecture\] Question \[question\] is affirmative for $\T=\sCM(\Lambda)$ if and only if Conjecture \[conjecture\] holds true for $\Lambda$. This gives a positive answer to Questoin \[question\] in the case where $R$ is a hypersurface. \[ihs\] The following are equivalent for an isolated hypersurface singularity $R$: 1. $\sCM(R)$ is locally finite; 2. $\sCM(R)$ has dimension zero; 3. $R$ is of finite CM representation type. In particular, Question \[question\] is affirmative for $\T=\sCM(R)$. It follows from [@DT Propositions 2.4(2) and 2.5] that Conjecture \[conjecture\] holds for $\Lambda=R$, whence Question \[question\] is affirmative for $\T=\sCM(R)$ by Corollary \[questionconjecture\]. Generation by orthogonal subcategories ====================================== Throughout this section, let $R$ be a ring (not necessarily noetherian or commutative). Let $\A$ be an abelian category and $\X$ a full subcategory. We denote by ${}^\perp\X$ (resp. ${}^{\perp_n}\X$) the full subcategory of $\A$ consisting of objects $M$ such that $\Ext_\A^i(M,\X)=0$ for all $i\ge1$ (resp. $1\le i\le n$). Dually, we denote by $\X^\perp$ (resp. $\X^{\perp_n}$) the full subcategory of $\A$ consisting of objects $M$ such that $\Ext_\A^i(\X,M)=0$ for all $i\ge1$ (resp. $1\le i\le n$). The following is the main result of this section. \[17121\] Let $\X$ be a full subcategory of $\A$, $M$ an object of $\A$ and $n\ge0$ an integer. 1. Suppose that there exists an exact sequence in $\A$ of the form $$0 \to X_n \to \cdots \to X_0 \to N \to M \to 0$$ with $X_0,\dots,X_n\in\X$ such that the corresponding element in $\Ext_\A^{n+1}(M,X_n)$ is nonzero. Then $M\notin{\langle{}^\perp\X\rangle}_{n+1}$ in $\da$. 2. Suppose that there exists an exact sequence in $\A$ of the form $$0 \to M \to N \to X^0 \to \cdots \to X^n \to 0$$ with $X^0,\dots,X^n\in\X$ such that the corresponding element in $\Ext_\A^{n+1}(X_n,M)$ is nonzero. Then $M\notin{\langle\X^\perp\rangle}_{n+1}$ in $\da$. We only prove the first assertion, because the second one can be shown by a dual argument. Let $L=L_0$ be the image of the morphism $X_0\to N$, and for each $1\le i\le n$ let $L_i$ be the image of the morphism $X_i\to X_{i-1}$. Then there is an exact sequence $$\label{17123} 0 \to X_n \to \cdots \to X_i \to L_i \to 0$$ for $0\le i\le n$. Set $L_{-1}=M$ and $X_{-1}=N$. For each $-1\le i\le n-1$ the exact sequence $0\to L_{i+1}\to X_i\to L_i\to0$ induces a morphism $\delta_i:L_i\to L_{i+1}[1]$ in $\da$, and we obtain the composite morphism $$\eta:M[-1]=L_{-1}[-1]\xrightarrow{\delta_{-1}[-1]}L=L_0\xrightarrow{\delta_0}L_1[1]\xrightarrow{\delta_1[1]}\cdots\xrightarrow{\delta_{n-1}[n-1]}L_n[n]=X_n[n]$$ in $\da$. The morphism $\eta$ corresponds to the original exact sequence $0 \to X_n \to \cdots \to X_0 \to N \to M \to 0$ via the isomorphism $\Ext_\A^{n+1}(M,X_n)\cong\Hom_{\da}(M[-1],X_n[n])$. By assumption $\eta$ is nonzero. Fix any $C\in{}^\perp\X$ and $j\in\Z$. Using , we easily see that $\Ext_\A^k(C,L_i)=0$ for all $0\ne k\in\Z$ and $0\le i\le n$, which implies $\Hom_{\da}(C[j],L_i[h])\cong\Ext_\A^{h-j}(C,L_i)=0$ for all $0\le i\le n$ and $j\ne h\in\Z$. Note also that $\Hom_{\da}(C[j],L_{-1}[h])\cong\Ext_\A^{h-j}(C,M)=0$ for all $j>h\in\Z$. Now it is easy to check that the induced map $$\Hom_{\da}(C[j],\delta_i[i]):\Hom_{\da}(C[j],L_i[i])\to\Hom_{\da}(C[j],L_{i+1}[i+1])$$ is zero for $-1\le i\le n-1$ since either the domain or target of the map vanishes. This shows $\Hom_{\da}({\langle{}^\perp\X\rangle}_1,\delta_i[i])=0$ for all $-1\le i\le n-1$. Applying the ghost lemma, we deduce $\Hom_{\da}({\langle{}^\perp\X\rangle}_{n+1},\eta)=0$, which implies $M[-1]\notin{\langle{}^\perp\X\rangle}_{n+1}$ as $\eta\ne0$. To give our next result, we introduce some notation. Let $\X$ be a full subcategory of $\A$. We denote by $\X_n$ the full subcategory of $\A$ consisting of objects $M$ such that there exists an exact sequence $0 \to X_n \to \cdots \to X_0 \to M \to 0$ in $\A$ with $X_i\in\X$ for all $0\le i\le n$. Dually, we denote by $\X^n$ the full subcategory of $\A$ consisting of objects $M$ such that there exists an exact sequence $0 \to M \to X^0 \to \cdots \to X^n \to 0$ in $\A$ with $X^i\in\X$ for all $0\le i\le n$. The first assertion of Theorem \[17121\] immediately recovers [@Y2 Theorem 1.1] as follows. Here, ${}^\perp T\text{-}\mathrm{tri.dim}\,\Db(\mod R)$ is the (Rouquier) dimension of the triangulated category $\Db(\mod R)$ with respect to the full subcategory ${}^\perp T$ of $\mod R$; see [@ddcm]. Let $R$ be a noetherian ring and $T$ a cotilting $R$-module of injective dimension $d>0$. Then ${}^\perp T\text{-}\mathrm{tri.dim}\,\Db(\mod R)=\mathrm{inj.dim}\,T$. Choose an $R$-module $M$ with $\Ext_R^d(M,T)\ne0$, and take a Cohen–Macaulay approximation $0\to L\to N\to M\to0$, that is, an exact sequence of $R$-modules such that $N\in{}^\perp T$ and $L\in(\add T)_d$. Applying Theorem \[17121\](1) to $\A=\mod R$, $\X=\add T$ and $n=d-1$, we have $M\notin{\langle{}^\perp T\rangle}_d$. Combine this with [@ddcm Theorem 5.3]. Our Theorem \[17121\] also gives rise to the following result. \[17126\] Let $\X$ be a full subcategory of $\A$, $M$ an object of $\A$ and $n>0$ an integer. 1. If there is an exact sequence $0\to X_n\to\cdots\to X_0\to M\to0$ which is nonzero in $\Ext_\A^n(M,X_n)$, then $M\notin{\langle{}^\perp\X\rangle}_n$ in $\da$. 2. If there is an exact sequence $0\to M\to X^0\to\cdots\to X^n\to0$ which is nonzero in $\Ext_\A^n(X^n,M)$, then $M\notin{\langle\X^\perp\rangle}_n$ in $\da$. For (1), as $n\ge1$, we can apply Theorem \[17121\](1) to the exact sequence $0\to X_n\to\cdots\to X_1\to X_0\to M\to0$ to deduce that $M\notin{\langle{}^\perp\X\rangle}_n$, and (2) is shown dually. Let $M$ be an object $M$ of $\A$, and let $n$ be a nonnegative integer. We say that *$M$ has projective (resp. injective) dimension $n$* and denote it by $\pd M=n$ (resp. $\id M=n$) if there exists an exact sequence $0\to P_n\to P_{n-1}\to\cdots\to P_0\to M\to0$ (resp. $0\to M\to I^0\to\cdots\to I^n\to0$) with the $P_i$ projectives (resp. $I^i$ injectives), and no shorter such sequence exists. The above result deduces the following corollary, whose partial assertion “$M\notin{\langle\proj\A\rangle}_{\pd M}$ if $\pd M<\infty$” is nothing but [@KK Lemma 2.4]. \[17124\] For an object $M\in\A$ one has: $$M\notin \begin{cases} {\langle\proj\A\rangle}_{\pd M} & \text{if $\pd M<\infty$},\\ {\langle\inj\A\rangle}_{\id M} & \text{if $\id M<\infty$}, \end{cases} \qquad M\notin \begin{cases} {\langle{}^\perp(\proj\A)\rangle}_{\pd M} & \text{if $\pd M<\infty$},\\ {\langle(\inj\A)^\perp\rangle}_{\id M} & \text{if $\id M<\infty$}. \end{cases}$$ The first assertion follows from the second one since $\proj\A$ and $\inj\A$ are contained in ${}^\perp(\proj\A)$ and $(\inj\A)^\perp$, respectively. In what follows, we show the second assertion. Setting $\pd M=n<\infty$, we have an exact sequence $\sigma:0\to P_n\xrightarrow{f}P_{n-1}\to\cdots\to P_0\to M\to0$ with $P_i\in\proj\A$ for $0\le i\le n$. Then the morphism $$\begin{CD} P\ @. =\ @. (0 @>>> P_n @>f>> P_{n-1} @>>> \cdots @>>> P_0 @>>> 0)\\ @V{\eta}VV @. @. @| @VVV @. @VVV\\ P_n[n]\ @. =\ @. (0 @>>> P_n @>>> 0 @>>> \cdots @>>> 0 @>>> 0) \end{CD}$$ in $\da$ corresponds to $\sigma$ via the isomorphism $\Hom_{\da}(P,P_n[n])\cong\Ext_\A^n(M,P_n)$. All the objects of $\A$ appearing in the above diagram are in $\proj\A$, and we can regard $\eta$ as a morphism in $\k(\proj\A)$. Suppose that $\eta$ is zero in $\da$. Then $\eta$ is zero in $\k(\proj\A)$, which means that $\eta$ is null-homotopic. It is easy to observe from this that $f$ is a split monomorphism in $\A$. This implies that $M$ has a projective resolution of length $n-1$, which contradicts the fact that $\pd M=n$. Therefore $\eta$ is nonzero in $\da$, and it follows from Corollary \[17126\] that $M$ in not in ${\langle{}^\perp(\proj\A)\rangle}_n$. The assertion for $\inj\A$ is shown dually. Let $R$ be a commutative ring. A finitely generated $R$-module $C$ is called *semidualizing* if the homothety map $R\to\End_R(C)$ is an isomorphism and $\Ext_R^i(C,C)=0$ for all $i>0$. We give an application of Corollaries \[17126\] and \[17124\], which involves semidualizing modules. \[17127\] Let $R$ be a commutative noetherian ring, and let $C$ be a semidualizing $R$-module. The following hold for a (not necessarily finitely generated) $R$-module $M$. 1. If $\pd_R(\Hom_R(C,M))=n<\infty$, then $M\notin{\langle{}^\perp C\rangle}_n$ in $\d(\Mod R)$. 2. If $\id_R(C\otimes_RM)=n<\infty$, then $M\notin{\langle C^\perp\rangle}_n$ in $\d(\Mod R)$. Let $\sigma:0\to P_n\to\cdots\to P_0\to\Hom_R(C,M)\to0$ be a projective resolution of $\Hom_R(C,M)$ in $\Mod R$. Tensoring $C$ gives an exact sequence $$C\otimes_R\sigma:0\to C\otimes_RP_n\to\cdots\to C\otimes_RP_0\to M\to0$$ by (the proof of) [@cdim Theorem 2.11(c)]. Suppose that $C\otimes_R\sigma$ is zero in $\Ext_R^n(M,C\otimes_RP_n)$. Then $\sigma=\Hom_R(C,C\otimes_R\sigma)$ is zero in $\Ext_R^n(\Hom_R(C,M),P_n)$, and as we have already seen in the proof of Corollary \[17124\], this yields a contradiction. Hence $C\otimes_R\sigma$ is nonzero in $\Ext_R^n(M,C\otimes_RP_n)$, and we can apply Corollary \[17126\] for $\X=\add C$ to deduce that $M$ is not in ${\langle{}^\perp C\rangle}_n$. This shows (1), and a dual argument implies (2). Let $R$ be a commutative noetherian local ring. Recall that a finitely generated $R$-module $M$ is called [*(maximal) Cohen–Macaulay*]{} if $\depth M\ge\dim R$. If $R$ is furthermore (Iwanaga–)Gorenstein, then the two definitions of a Cohen–Macaulay module coincide. We use the same notation $\CM(R)$ as in Section 2 to denote the full subcategory of $\mod R$ consisting of Cohen–Macaulay modules. From Corollary \[17124\] or \[17127\], we immediately obtain: Let $R$ be a noetherian ring, and let $M$ be a finitely generated $R$-module. Let $n$ be a nonnegative integer. Then the following statements hold in $\Db(\mod R)$. 1. If $M$ has projective dimension $n$, then $M\notin{\langle{}^\perp R\rangle}_n$. 2. If $R$ is Iwanaga–Gorenstein and $M$ has projective dimension $n$, then $M\notin{\langle\CM(R)\rangle}_n$. 3. If $R$ is a commutative Cohen–Macaulay local ring with canonical module $\omega$ and $\Hom_R(\omega,M)$ has projective dimension $n$, then $M\notin{\langle\CM(R)\rangle}_n$. To prove our next result, we establish a lemma which should be well-known to experts. \[17128\] Let $R$ be a commutative noetherian local ring. Let $M,N$ be finitely generated $R$-modules with $\Ext_R^{>0}(M,N)=0$. Then $\depth_R\Hom_R(M,N)=\depth_RN$. Let $k$ be the residue field of $R$, and put $t=\depth_RN$. There are isomorphisms $$\rhom_R(k,\Hom_R(M,N))\cong\rhom_R(k,\rhom_R(M,N))\cong\rhom_R(k\otimes_R^\mathbf{L}M,N)$$ which yield a spectral sequence $$E_2^{pq}=\Ext_R^p(\Tor^R_q(k,M),N)\Rightarrow H^{p+q}=\Ext_R^{p+q}(k,\Hom_R(M,N)).$$ Since $E_2^{pq}=0$ for all $(p,q)$ with $p<t$, we see that $H^t=E_2^{t0}$ and $H^i=0$ for all $i<t$. Hence $\Ext_R^i(k,\Hom_R(M,N))=0$ for all $i<t$, and $\Ext_R^t(k,\Hom_R(M,N))$ is isomorphic to $\Ext_R^t(k\otimes_RM,N)$. As $k\otimes_RM$ is a nonzero $k$-vector space and $\Ext_R^t(k,N)$ is nonzero, so is $\Ext_R^t(k\otimes_RM,N)$, and so is $\Ext_R^t(k,\Hom_R(M,N))$. Thus $\depth_R\Hom_R(M,N)=t$. In the case of local rings, the number $n$ in Corollary \[17127\](1) can be computed by using the well-established invariant of depth. Let $R$ be a commutative noetherian local ring, and let $C$ be a semidualizing $R$-module. Let $M$ be a finitely generated $R$-module such that $\Hom_R(C,M)$ has finite projective dimension. Then $M\notin{\langle{}^\perp C\rangle}_n$ in $\Db(\mod R)$, where $n=\depth R-\depth M$. It follows from [@cdim Theorem 2.11(c) and Corollary 2.9(a)] that $\Ext_R^{>0}(C,M)=0$, and by Lemma \[17128\] we have $\depth\Hom_R(C,M)=\depth M$. Using the Auslander–Buchsbaum formula, we get $\pd\Hom_R(C,M)=\depth R-\depth\Hom_R(C,M)=\depth R-\depth M$. The assertion now follows from Corollary \[17127\](1). We also detect relationships among orthogonal subcategories. \[17122\] Let $\X$ be a full subcategory of $\A$. One then has: $${}^{\perp_n}\X\cap{\langle{}^\perp\X\rangle}_{n+1}\subseteq{}^{\perp_1}(\X_n),\qquad \X^{\perp_n}\cap{\langle\X^\perp\rangle}_{n+1}\subseteq{(\X^n)}^{\perp_1}$$ We only prove the first equality; the second one is shown dually. Pick any object $M$ in ${}^{\perp_n}\X\cap{\langle{}^\perp\X\rangle}_{n+1}$. Assume that $M$ is not in ${}^{\perp_1}(\X_n)$. Then there exists $N\in\X_n$ such that $\Ext_\A^1(M,N)\ne0$. This implies that there are exact sequences $$\sigma:\ 0 \to N \to E \to M \to 0,\qquad \tau:\ 0 \to X_n \to \cdots \to X_0 \to N \to 0$$ in $\A$ such that $\sigma$ is nonsplit and $X_0,\dots,X_n$ belong to $\X$. We have morphisms $$\Hom_\A(M,M)\xrightarrow{f}\Ext_\A^1(M,N)\xrightarrow{g}\Ext_\A^{n+1}(M,X_n),$$ where $f,g$ are induced from $\sigma,\tau$ respectively. The map $f$ sends the identity map $\id_M$ of $M$ to the element $[\sigma]\in\Ext_\A^1(M,N)$ corresponding to the short exact sequence $\sigma$. As $\sigma$ is nonsplit, $[\sigma]$ is nonzero. Splicing $\sigma$ and $\tau$, we get an exact sequence $$\upsilon:\ 0 \to X_n \to \cdots \to X_0 \to E \to M \to 0.$$ The map $g$ sends $[\sigma]$ to $[\upsilon]\in\Ext_\A^{n+1}(M,X_n)$. Since $M$ is in ${}^{\perp_n}\X$, it is easy to see that $g$ is injective. Hence $[\upsilon]$ is also nonzero. Thus we can apply Theorem \[17121\](1) to see that $M$ is not in ${\langle{}^\perp\X\rangle}_{n+1}$, contrary to the choice of $M$. This contradiction shows the assertion. We denote by $\fl R$ the full subcategory of $\mod R$ consisting of modules of finite length. Here is an application of the above result. \[17125\] Let $R$ be a commutative Cohen–Macaulay local ring of Krull dimension $d>0$ with canonical module $\omega$. Then $\fl R\cap{\langle\CM(R)\rangle}_d=0$. Note that $\CM(R)={}^\perp\omega$ in the abelian category $\mod R$. Applying Corollary \[17122\] to $\X=\add\omega$ and $n=d-1\ge0$, we see that ${}^{\perp_{d-1}}\omega\cap{\langle\CM(R)\rangle}_d$ is contained in ${}^{\perp_1}(({\add\omega})_{d-1})$. Let $L$ be an $R$-module of finite length that belongs to ${\langle\CM(R)\rangle}_d$. Then $L$ is in ${}^{\perp_{d-1}}\omega$ since $\omega$ has depth $d$. Hence $L$ is in ${}^{\perp_1}(({\add\omega})_{d-1})$, that is, $\Ext_R^1(L,({\add\omega})_{d-1})=0$. Take a minimal Cohen–Macaulay approximation $0 \to B \to A \to L \to 0$. Then $B$ belongs to $(\add\omega)_d$. 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--- abstract: 'Dynamics, the study of change, is normally the subject of mechanics. Whether the chosen mechanics is “fundamental” and deterministic or “phenomenological” and stochastic, all changes are described relative to an external time. Here we show that once we define what we are talking about, namely, the system, its states and a criterion to distinguish among them, there is a single, unique, and natural dynamical law for irreversible processes that is compatible with the principle of maximum entropy. In this alternative dynamics changes are described relative to an internal, “intrinsic” time which is a derived, statistical concept defined and measured by change itself. Time is quantified change.' author: - | Ariel Caticha\ [Department of Physics, University at Albany-SUNY, ]{}\ [Albany, NY 12222, USA.[^1]]{} title: 'Change, Time and Information Geometry[^2]' --- Introduction ============ The notion that the concepts of time, change and motion are intimately connected goes back to antiquity. According to Aristotle, “time numbers change with respect to before and after.” One aspect of this connection is the order of a sequence of changes, their temporal order. Another aspect is the use of selected motions or changes to measure the length of time intervals, their duration. We begin by considering the notion of change. In order to establish that a system has changed one must be able to distinguish between the system being in one state and its being in another state. This requires, to begin with, a clear idea of what is meant by a state. As long as one is interested in the study of phenomena that can be deliberately reproduced by controlling a few macroscopic variables it is reasonable to expect that the values – or rather, the expected values – of these few variables are all that is needed for the purposes of prediction. This limited information defines what we mean by the state or, equivalently, the macrostate of the system. Next, to measure the extent to which states can be distinguished, we assign a probability distribution to each state. The requirement that the assignment procedure itself do not introduce any information beyond that which defines the state demands we use the method of maximum entropy (ME) [@Jaynes57][@Skilling88]. In this way the problem of distinguishing between states is transformed into another problem, that of distinguishing between the corresponding distributions. The solution to the latter problem is well known. There is a uniquely natural way to quantify the extent to which one distribution can be distinguished from another: it is given by the distance between them as measured by the Fisher-Rao information metric [@Fisher25]-[@Rodriguez89]. If we think of each state as a point in a manifold, the net outcome of these considerations (Sect. 2) is that the method of ME has transformed the manifold of states into a metric space. Distinguishability and therefore change is measured by distance. There is not yet any implication that change will happen *from* one state *to* another; to this we turn next. Temporal order, as well as the notion of time itself, are the subject of dynamics. Typically, having decided on the kinematics appropriate to a certain motion, one defines the dynamics by additional postulates about the equations of motion, perhaps in the form of a variational principle. The dynamics is postulated. The dynamical law that we adopt here (Sect. 3) is a variational principle too, but there is something very peculiar about it, there is no need to postulate it. The principle is the same we had already introduced when discussing the space of states, namely, when selecting a distribution subject to certain constraints, the preferred distribution is that of maximum entropy. It is just the same old ME principle applied in a somewhat different way. (The nature of the constraints is different. For a brief account of the ME method in a form tailored to suit the needs of this paper see Ref.[@Caticha00].) We have no freedom in choosing the dynamical law; it follows from the single piece of new information available: recognizing that changes happen. Nothing else. Suppose the system is in a certain state and a small change happens; the system moves a distance $d\ell $. We cannot with certainty predict in which direction motion occurs but, according to the principle of ME, unless there is some positive evidence to the contrary, of all the states on the surface of the sphere of radius $d\ell $ there is one to be preferred above all others: it is the state of maximum entropy. As so often in the past, it seems that once more the method of ME has allowed us to get something out of nothing; yet another free lunch. But the dynamics proposed here is different in one important respect. (We refrain from saying “defficient” rather than “different” because in the end it may turn out to be an advantage.) In the conventional Hamiltonian or Lagrangian mechanics the equations of motion describe changes relative to an external time. Here changes are described relative to an internal, “intrinsic” time which is a derived, statistical concept defined and measured by the change $d\ell $ itself. Intrinsic time is quantified change. The system provides its own clock. Perhaps this is a necessary feature of any fundamental form of mechanics that generates its own notion of time, that *explains* time. The introduction of a metric in the space of states is not new; this has been done by many authors in statistical inference, where the subject is known as Information Geometry [@Amari85][@Rodriguez90], and in physics, to study both equilibrium [@Weinhold75][@Ingarden76] and nonequilibrium thermodynamics [@Balian86][@Streater95]. What is different here is the recognition that this is all one needs to define a dynamics. An interesting consequence of these ideas is that reciprocity relations of the Onsager type [@Onsager31] valid near and far from equilibrium are obtained (Sect. 4) without any hypothesis about microscopic reversibility; in fact, no mention is made of any microscopic dynamics. By analyzing specific models other authors [@Gabrielli96] have reached similar conclusions: reciprocal relations are possible even if the underlying microscopic dynamics is not reversible. It is, of course, possible to incorporate more information, that is, additional constraints into the dynamics. In Sect. 5 we consider a simple illustrative example, the intrinsic dynamics of two coupled systems as they evolve towards equilibrium along a trajectory constrained by conservation laws. Our subject can be approached from another direction. The Greeks did not draw a sharp distinction between change in general and the more special kind of change we call motion; the falling of an apple was not viewed as being in any sense more fundamental than the ripening of an apple. The modern view does draw such distinctions; deterministic motion in space and time is considered basic while other kinds of change – notably irreversible processes in macroscopic systems – are not. They must be understood in terms of the deterministic motion of microscopic constituents. Of course, this view is not wrong, but for some purposes it may be misguided, inconvenient. All theories describing irreversible processes have, in the past, invariably turned out to be rather formidable (see e.g., [@Grabert82]-[@Luzzi90]). One reason is that the phenomena to be described are themselves quite complicated. But there is another reason, which is that these theories are attempting to achieve two conflicting goals. One goal is to reach an understanding in terms of the microscopic Hamiltonian laws of motion and requires keeping track of microscopic details. The other goal is to achieve a description in terms of the few variables that matter, those that codify the crucial information relevant to making predictions. Information about the other variables, the vast majority, is totally irrelevant. Achieving such a description requires forgetting about all microscopic details. It is remarkable that theories that accomplish these two seemingly contradictory goals are at all possible. They involve a very delicate balancing act between keeping track of details, at least for a little while (Hamiltonian evolution), and then throwing them away (projections, coarse-graining, tracing over unwanted variables, etc.). Our proposal cuts through this Gordian knot. If microscopic details are truly irrelevant then the Hamiltonian evolution itself should be largely irrelevant. The information about irrelevant details should be discarded before, not after, it is computed. This requires formulating a dynamics without the benefit (or, in this case, the hindrance) of Hamiltonians. A potentially serious problem here is the loss of predictive power that stems from the possibility of being able to choose among different dynamical laws. What would make us prefer one law over another? Remarkably the problem does not arise; once we define what we are talking about, namely, the states and the criterion to distinguish among them, there is a single, unique, and natural dynamical law that is compatible with the principle of maximum entropy. The views expressed here are clearly biased in favor of the information theory approach to statistical mechanics, but they need not contradict other points of view. The basic explanation of the second law of thermodynamics was given by Boltzmann and Gibbs long ago but later contributions by many authors have generated several different versions of it. The question of which particular version is the right one remains controversial. However, provided one adopts a certain spirit of tolerance in reading the various authors (words such as entropy or probability can be used with very different meanings), one sees that the different views are not always incompatible. The point we wish to make is that irrespective of which is one’s own personal favorite reason for preferring change in the direction of entropy increase over decrease, the *same reason* should lead one to prefer a large increase over a small one. This applies whether we favor the information theory approach [@Jaynes57][@Balian86] or one of the perhaps more traditional points of view such as ergodic theory [@Lebowitz93]. For example, directing the system toward a certain region of phase space is easier and is less sensitive to external perturbations if the region is large than if it is small; hitting a large target is easier than hitting a small target; that’s all. Thus entropy should increase to the maximum extent allowed by whatever constraints are known to hold. This last statement is widely recognized as the basis for equilibrium thermostatics. But it shouldn’t just apply to the final equilibrium state; it should apply to every one of all the intermediate states along the irreversible trajectory and not just to the end point. Clarifying in precisely what sense this statement can be extended from statics to dynamics is yet another way of stating our goals. Quantifying change ================== Let the microstates of a physical system be labelled by $x$, and let $m(x)dx$ be the number of microstates in the range $dx$. We assume that a state of the system – that is, a macrostate – is defined by the known expected values $A^{{}\alpha }$ of some $n_A$ variables $a^{{}\alpha }(x)$ ($\alpha =1,2,\ldots ,n_A$), $$\left\langle a^{{}\alpha }\right\rangle =\int dx\,p(x)a^{{}\alpha }(x)=A^{{}\alpha }\,. \label{Aalpha}$$ This limited information will certainly not be sufficient to answering all questions that one could conceivably ask about the system. Choosing the right set of variables $\{a^{{}\alpha }\}$ is perhaps the most difficult problem in statistical mechanics [@Balian]. A crucial assumption is that Eq.(\[Aalpha\]) is not just any random information, that it happens to be the *right* information for our purposes. It is convenient to think of each state as a point in an $n_A$-dimensional manifold; the numerical values $A^{{}\alpha }$ associated to each point form a convenient set of coordinates. The principle of ME allows us to associate a probability distribution to each point in the space of states. The probability distribution $p(x|A)$ that best reflects the prior information contained in $m(x)$ updated by the information $A^{{}\alpha }$ is obtained by maximizing the entropy $$S[p]=-\int \,dx\,p(x)\log \frac{p(x)}{m(x)}. \label{S[p]}$$ subject to the constraints (\[Aalpha\]). The result is $$p(x|A)=\frac 1Z\,m(x)\,e^{-\lambda _{{}\alpha }a^{{}\alpha }(x)}, \label{pzero}$$ where the partition function $Z$ and the Lagrange multipliers $\lambda _{{}\alpha }$ are given by $$Z(\lambda )=\int dx\,m(x)\,e^{-\lambda _{{}\alpha }a^{{}\alpha }(x)}\quad \text{and}\quad -\frac{\partial \log Z}{\partial \lambda _{{}\alpha }}% =A^{{}\alpha }\,. \label{Z and lambda}$$ The maximized value of the entropy is $$S(A)=-\int \,dx\,p(x|A)\log \frac{p(x|A)}{m(x)}=\log Z(\lambda )+\lambda _{{}\alpha }A^{{}\alpha }\,. \label{S(A)}$$ The second prerequisite to establishing that a system has changed from one state to another is a criterion allowing us to assert that two states $A$ and $A+dA$ are not the same. Can we distinguish between the two? If $dA$ is small enough the corresponding probability distributions $p(x|A)$ and $% p(x|A+dA)$ overlap considerably and it is easy to confuse them. We seek a real positive number to provide a quantitative measure of the extent to which these two distributions can be distinguished. The following argument is intuitively appealing. Consider the relative difference, $$\frac{p(x|A+dA)-p(x|A)}{p(x|A)}=\frac{\partial \log p(x|A)}{\partial A^{{}\alpha }}\,dA^{{}\alpha }.$$ The expected value of the relative difference might seem a good candidate, but it does not work because it vanishes identically, $$\int dx\,p(x|A)\,\frac{\partial \log p(x|A)}{\partial A^{{}\alpha }}% \,dA^{{}\alpha }=dA^{{}\alpha }\,\frac \partial {\partial A^{{}\alpha }}\int dx\,p(x|A)=0.$$ However, the variance does not vanish, $$d\ell ^2=\int dx\,p(x|A)\,\frac{\partial \log p(x|A)}{\partial A^{{}\alpha }}% \,\frac{\partial \log p(x|A)}{\partial A^{{}\beta }}\,dA^{{}\alpha }dA^{{}\beta }\equiv g_{\alpha \beta }\,dA^{{}\alpha }dA^{{}\beta }\,\,.$$ This is the measure of distinguishability we seek; a small value of $d\ell ^2 $ means the points $A$ and $A+dA$ are difficult to distinguish. The $% g_{\alpha \beta }$ are recognized as elements of the Fisher information matrix [@Fisher25]. Up to now no notion of distance has been introduced on the space of states. Normally one says that the reason it is difficult to distinguish between two points in say, the real space we seem to inhabit, is that they happen to be too close together. It is very tempting to invert the logic and assert that the two points $A$ and $A+dA$ must be very close together whenever they happen to be difficult to distinguish. Thus it is natural to interpret $% g_{\alpha \beta }$ as a metric tensor [@Rao45]. It is known as the Fisher-Rao metric, or the information metric. A disadvantage of these heuristic arguments is that they do not make explicit a crucial property of the Fisher-Rao metric, except for an overall multiplicative constant this Riemannian metric is unique [@Amari85][@Rodriguez89]. To summarize: the very act of assigning a probability distribution $p(x|A)$ to each point $A$ in the space of states, automatically provides the space of states with a metric structure. The coordinates $A$ are quite arbitrary, they need not be the expected values $\left\langle a^{{}\alpha }\right\rangle $. One can freely switch from one set to another. It is then easy to check that $g_{\alpha \beta }$ are the components of a tensor, that the distance $d\ell ^2$ is an invariant, a scalar. Incidentally, $d\ell ^2$ is also dimensionless. There is, however, one special coordinate system in which the metric takes a form that is particularly simple. These coordinates are the expected values themselves, $A^{{}\alpha }=\left\langle a^{{}\alpha }\right\rangle $. In these coordinates, $$g_{\alpha \beta }=-\frac{\partial ^2S(A)}{\partial A^{{}\alpha }\partial A^{{}\beta }}\, \label{gab}$$ with $S(A)$ given in Eq.(\[S(A)\]) and the covariance is not manifest. Intrinsic dynamics and time =========================== Our basic dynamical principle is that small changes from one state to another are possible and do, in fact, happen. We do not explain why they happen but, if we are given the valuable piece of information that some change will occur, we can then venture a guess, make a prediction as to what the most likely change will be. Before giving mathematical expression to this principle we note that large changes are assumed to be the cumulative result of many small changes. As the system moves it follows a continuous trajectory in the space of states. We almost hesitate to call this self-evident fact an assumption, but as the example of quantum theory shows, trajectories need not exist. Thus in order to go from one state to another the system will have to move through intermediate states; in order to change by a distance $2d\ell $ the system must have first changed by a distance $d\ell $. Suppose the system was in the state $A_{old}^{{}\alpha }=A^{{}\alpha }$ and that it changes by a small amount $d\ell $ to a nearby state. We have to select one new state $A_{new}^{{}\alpha }=A^{{}\alpha }+dA^{{}\alpha }$ from among those that lie on the surface of an $n_A$-dimensional sphere of radius $d\ell $ centered at $A^{{}\alpha }$. This is precisely what the ME principle was designed to do [@Caticha00], namely, to select a preferred probability distribution from within a specified given set. The only difference with more conventional applications of the ME principle is the geometrical nature of the constraint. We want to maximize $S(A^{{}\alpha }+dA^{{}\alpha })$ under variations of $% dA^{{}\alpha }$ constrained by $g_{\alpha \beta }\,dA^{{}\alpha }dA^{{}\beta }=d\ell ^2$. The notation $dA^{{}\alpha }=\dot{A}^{{}\alpha }d\ell $ is slightly more convenient; we maximize $S(A^{{}\alpha }+\dot{A}^{{}\alpha }d\ell )$ under variations of $\dot{A}^{{}\alpha }$ constrained by $$g_{\alpha \beta }\,\dot{A}^{{}\alpha }\dot{A}^{{}\beta }=1\,. \label{gAA}$$ Introducing a Lagrange multiplier $\omega $, $$\delta \left[ S(A^{{}\alpha }+\dot{A}^{{}\alpha }d\ell )-\omega \,\left( g_{\alpha \beta }\dot{A}^{{}\alpha }\dot{A}^{{}\beta }-1\right) \right] =0,$$ we get $$\left[ \frac{\partial S}{\partial A^{{}\alpha }}\,d\ell -2\omega \,g_{\alpha \beta }\dot{A}^{{}\beta }\right] \delta \dot{A}^{{}\alpha }=0\,.$$ Therefore, writing $\omega =\sigma \,d\ell /2$, we get $$\dot{A}^{{}\alpha }=\frac 1\sigma \,g^{\alpha \beta }\frac{\partial S}{% \partial A^{{}\beta }},$$ where $g^{\alpha \beta }$ is the inverse of $g_{\alpha \beta }$. This is our main result; it can be rewritten as $$\dot{A}^{{}\alpha }=\frac 1\sigma \,\lambda ^{{}\alpha } \label{main1}$$ where the vector $\lambda ^{{}\alpha }$, $$\lambda ^{{}\alpha }=g^{\alpha \beta }\,\frac{\partial S}{\partial A^{{}\beta }}\,,$$ is the entropy gradient. The interpretation is clear, the system moves along the entropy gradient. This seems such an obvious result that it can hardly be new. Notice, however, the gradient *vector* refers to the direction in which there is a maximum increase *per unit distance*; one cannot talk about the gradient vector without having first introduced a metric. The differential form defined by the derivatives $S_{{},\beta }=\partial S/\partial A^{{}\beta }=\lambda _{{}\beta }$, the gradient *one-form*, does not define a direction; it is not by itself sufficient to define the trajectory. The physical significance of the Lagrange multiplier $\sigma $ derives from the constraint Eq.(\[gAA\]) which, using Eq.(\[main1\]), can be written as $$\lambda _{{}\alpha }\lambda ^{{}\alpha }=\sigma ^2\quad \text{or}\quad \sigma =\left( \lambda _{{}\alpha }\lambda ^{{}\alpha }\right) ^{1/2},$$ $\sigma $ is the magnitude of the entropy gradient. Furthermore, from this and Eq.(\[main1\]), we get $dS=\lambda _{{}\alpha }\dot{A}^{{}\alpha }d\ell =\sigma d\ell $,or $$\sigma =\frac{dS}{d\ell }\,.$$ $\sigma $ is the rate of entropy increase along the trajectory. The main result, Eq.(\[main1\]), determines the trajectory followed by the system. It determines the tangent vector $\dot{A}^{{}\alpha }=dA^{{}\alpha }/d\ell $, but not the “velocity” $dA^{{}\alpha }/dt$. To fix this something must be said about the universe external to the system, something that relates the distance $\ell $ relative to the external time $t$. This is, in part, the role normally played by the Hamiltonian, it fixes the evolution of a system relative to external clocks. If we cannot appeal to such information (presumably because we do not have it, but perhaps because we just do not want to), then the only “time” available must be internal to the system, intrinsic to the geometry of the space of states. One convenient choice of intrinsic time $\tau $ is the distance $\ell $ itself, or $d\tau =d\ell $. Intrinsic time is change. The equation of motion is very simple: the trajectory, $A^{{}\alpha }=A^{{}\alpha }(\tau )$, is along the entropy gradient, and the system moves with unit velocity, $\dot{A}% ^{{}\alpha }\dot{A}_{{}\alpha }=1$, or $d\ell /d\tau =1$. The absolute speed $d\ell /dt$ remains unknown. Interestingly, there is no guarantee that $\tau $ will elapse relative to our own external $t$, we could have a situation with $d\tau /dt=0$. A pile of sand could, if left alone, just stay at $A^{{}\alpha }(\tau _0)$ forever; its intrinsic time $% \tau $ has stopped at $\tau _0$. The pile does not change, because it did not have (intrinsic) time to change. (One can play endless word games here.) However, should a measurement of one of the variables, for example $A^{{}1}$, indicate a change from the value $A^{{}1}(\tau _0)$ to the value that one would normally associate with another state along the trajectory, say the value $A^{{}1}(\tau _1)$ at the later time $\tau _1$, then one is immediately led to infer that the system has moved along the trajectory. Most probably all the other variables have also changed from $A^{{}\alpha }(\tau _0)$ to $A^{{}\alpha }(\tau _1)$. In this case the variable $% A^{{}1}(\tau )$ is playing the role of an internal clock. The variable $% A^{{}1}$ is a good clock provided one can invert $A^{{}1}=A^{{}1}(\tau )$, to get $\tau =\tau (A^{{}1})$. Then, the changes in all other variables $% A^{{}\alpha }=A^{{}{}\alpha }[\tau (A^{{}1})]=A^{{}{}\alpha }(A^{{}1})$ can be referred, correlated to the change in $A^{{}1}$. We see that the loss of predictive power due to the unknown absolute speed $d\ell /dt$ is quite minimal, particularly for high dimensionality (large $n_A$). At this point one could agree that the notion of $\tau $ is useful, perhaps even elegant. But are we justified in *calling* it time? Perhaps these are mere word games, but if we do call $\tau $ time, then being a distance it provides us with a model of duration. Furthermore, the very definite ordering of states along the trajectory $A^{{}\alpha }(\tau )$ provides a realization of a temporal order. Finally, the dynamics is intrinsically asymmetric; the trajectory is intrinsically oriented. There is one direction in which entropy increases providing a clear distinction between earlier and later. So this is our answer: we are justified in calling $\tau $ time, because if we do, then we have a neat model, an explanation for temporal order, for time asymmetry, and for duration. What better reasons do we need? We close this section with the observation that the system does not follow a geodesic in the space of states. From Eq.(\[main1\]) we can show that the acceleration vector, given by the absolute derivative (we assume a Riemannian geometry, with the Levi-Civita connection) $$\frac{D\dot{A}^{{}\alpha }}{d\tau }=\dot{A}_{;\beta }^{{}\alpha }\,\dot{A}% ^{{}\beta }=g^{\alpha \beta }\,f_{\beta \gamma }\,\dot{A}^{{}\gamma },$$ does not vanish. The “thermodynamic force” resembles the Lorentz force law in electrodynamics. The “field strength” tensor $f_{\alpha \beta }$, given by $f_{\alpha \beta }=\dot{A}_{{}\alpha ;\beta }-\dot{A}_{\beta ;\alpha }$, is antisymmetric as needed to preserve the unit magnitude of the velocity $% \dot{A}^{{}\alpha }$. Reciprocal relations ==================== The standard theory of irreversible thermodynamics, due to Onsager [@Onsager31], is based on the usual postulates of equilibrium thermostatics supplemented by the additional postulate that the microscopic laws of motion are symmetric under time reversal. A brief ouline is the following. As the system moves along its trajectory entropy increases at a rate $$\frac{dS}{dt}=\frac{\partial S}{\partial A^{{}\alpha }}\frac{dA^{{}\alpha }}{% dt}=\lambda _{{}\alpha }\frac{dA^{{}\alpha }}{dt}$$ relative to the external time $t$; the variables $\lambda _{{}\alpha }$ are called thermodynamic forces, and $dA^{{}\alpha }/dt$ are called fluxes. In this theory linear relations between fluxes and forces are postulated, $$\frac{dA^{{}\alpha }}{dt}=L^{\alpha \beta }\lambda _{{}\beta }\,,$$ for which there is abundant experimental evidence, at least close to thermodynamic equilibrium. The significance of these relations lies in that they postulate crossed connections between a flux of type $\alpha $ and a force of type $\beta $, and vice versa. (Thus, a temperature gradient will not just generate an heat current; it may also generate electric currents, matter flows, and so on.) The strength of these effects is measured by the phenomenological Onsager coefficients $L^{\alpha \beta }$. The central result of the theory is the reciprocal relation between these crossed effects. The reciprocity theorem, proved by Onsager on the basis of microscopic reversibility, states that the matrix of phenomenological coefficients is symmetric $$L^{\alpha \beta }=L^{\beta \alpha }\,.$$ The intrinsic dynamics discussed in the previous section also leads to reciprocal relations. The equation of motion, Eq.(\[main1\]), gives $$\frac{dA^{{}\alpha }}{dt}=\frac{d\tau }{dt}\frac{dA^{{}\alpha }}{d\tau }=% \frac{d\tau }{dt}\frac 1\sigma \,g^{\alpha \beta }\lambda _{{}\beta }\,. \label{dA/dt}$$ This allows us to identify the Onsager coefficients as $$L^{\alpha \beta }=\frac{d\tau }{dt}\frac 1\sigma \,g^{\alpha \beta }.$$ These coefficients are not constants, they vary along the trajectory, $% L^{\alpha \beta }=L^{\alpha \beta }(A)$. What is interesting here is that their symmetry follows from the symmetry of the metric tensor. No hypothesis about microscopic reversibility was needed; in fact, microscopic dynamics was not mentioned at all. In addition, the validity of Eq.(\[dA/dt\]) is not restricted to the immediate vicinity of equilibrium. To the extent that the variables $A$ are the right variables to describe phenomena far from equilibrium, the reciprocal relations should still hold. Dynamics constrained by conservation ==================================== Beyond the fact that changes happen, perhaps the most common additional information that one can have about an irreversible process is that some quantities are conserved. As an illustrative example we consider two systems that are allowed to exchange some conserved quantities and evolve towards equilibrium. To fix ideas we could think of an ideal gas filling two vessels at different temperatures and chemical potentials. Once the two vessels are connected, for example by a tube, a little hole, or a a porous plug, matter and energy will flow until equilibrium is reached. To keep this as simple as possible we assume the experimental conditions are such that throughout the process the two systems remain homogeneous and independent. The first system is described by variables $A^{{}\alpha }$, the second is described by primed variables $A^{\prime {}\alpha }$, and the entropies, given by Eq.(\[S(A)\]), are additive $$S_T(A,A^{\prime {}})=S(A)+S^{\prime }(A^{\prime {}}).$$ Since the quantities $A$ are conserved the dynamics is constrained by $% A^{\prime }=A_T-A$, with $A_T$ fixed, or $\dot{A}^{\prime }=-\dot{A}$. The conservation constraint could be incorporated using Lagrange multipliers; for this simple example it is just as easy to eliminate $A^{\prime }$. In our ideal gas example, the variables could be energy, $A^1=E$, and number of molecules, $A^2=N$. This crucial part in setting the problem, choosing the description, is the one most likely to go wrong. If the hole coupling the two vessels is too large, the ME predictions below will fail. The failure is not to be blamed in the ME method, but on the choice of variables: the pair $E$, $N$ is not enough to codify the relevant information of say, a turbulent flow. The same remark applies if the connecting porous plug is such that heat can be easily exchanged but there is resistance to matter flow. In this case additional variables are needed, perhaps describing the physical state of the plug and the gas in it. Suppose the system was in the state $A$ and that it changes by a small amount $d\tau $ to a nearby state. To select one new state $A+\dot{A}d\tau $ from among those that lie on the surface of a sphere of radius $d\tau $ centered at $A$, we maximize $$S_T(A+\dot{A}d\tau )=S(A+\dot{A}d\tau )+S^{\prime }(A_T-A-\dot{A}d\tau )$$ under variations of $\dot{A}^{{}\alpha }$ constrained by $$g_{\alpha \beta }\,\dot{A}^{{}\alpha }\dot{A}^{{}\beta }=1\,.$$ where the Fisher-Rao metric, Eq.(\[gab\]), is given by $$g_{\alpha \beta }=-\frac{\partial ^2}{\partial A^{{}\alpha }\partial A^{{}\beta }}\left( S(A)+S^{\prime }(A_T-A)\right) .$$ The result is $$\frac{dA^{{}\alpha }}{d\tau }=\frac 1\sigma \,g^{\alpha \beta }\left( \lambda _{{}\beta }-\lambda _{{}\beta }^{\prime }\right) ,$$ where $\sigma $ is the rate of entropy production $\sigma =dS_T/d\tau $. The system evolves until the conjugate variables $\lambda $ are equalized. Final remarks ============= The main conclusion is simple: unless there is positive evidence to the contrary, our best prediction is that the system evolves along the entropy gradient. What is perhaps not so trivial is that, unlike other conventional forms of dynamics, this intrinsic dynamics does not require an additional postulate. It is the unique dynamics that follows from the maximum entropy principle and nothing else. Another nontrivial aspect is that the model supplies its own notion of time. Since the irreversible macroscopic motion is not explained in terms of a reversible microscopic motion there is no need to explain irreversibility, this question never arises. Similarly, there is no need to explain the second law of thermodynamics; it is the second law (in the form of the ME axioms) that explains everything else. These ideas can be explored further in a number of directions. There is, for example, the relation with other theories of irreversible processes, such as the equations of hydrodynamics. Another possibility is to extend the theory to account for fluctuations and diffusion. The intrinsic dynamics proposed above is deterministic, but to the extent that the ME principle does not completely rule out distributions of lower entropy [@Caticha00], fluctuations about equilibrium and about the deterministic motion are possible. Perhaps the most intriguing question to pursue stems from the possibility of deriving dynamics from purely entropic arguments. This is clearly valuable in areas where the microscopic dynamics may be too far removed from the phenomena of interest, say in biology or ecology, or where it may just be unknown or perhaps even inexistent, as in economics. One could argue that these theories would be phenomenological as opposed to fundamental, that within physics the search for a fundamental mechanics would still be left open. 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[^1]: E-mail: ariel@cnsvax.albany.edu or Ariel.Caticha@albany.edu [^2]: Presented at MaxEnt 2000, the 20th International Workshop on Bayesian Inference and Maximum Entropy Methods (July 8-13, 2000, Gif-sur-Yvette, France).

--- abstract: | Using $2917 \invpb$ of data accumulated at $3.773\gev$, $44.5\invpb$ of data accumulated at $3.65\gev$ and data accumulated during a $\psi(3770)$ line-shape scan with the BESIII detector, the reaction $e^+e^-\rightarrow p\bar{p}$ is studied considering a possible interference between resonant and continuum amplitudes. The cross section of $e^+e^-\rightarrow\psi(3770)\rightarrow p\bar{p}$, $\sigma(e^+e^-\rightarrow\psi(3770)\rightarrow p\bar{p})$, is found to have two solutions, determined to be $(0.059^{+0.070}_{-0.020}\pm0.012)\pb$ with the phase angle $\phi = (255.8^{+39.0}_{-26.6}\pm4.8)^\circ$ ($<0.166 \pb$ at the 90% confidence level), or $\sigma(e^+e^-\rightarrow\psi(3770)\rightarrow p\bar{p}) = (2.57^{+0.12}_{-0.13}\pm0.12)\pb$ with $\phi = (266.9^{+6.1}_{-6.3}\pm0.9)^\circ$ both of which agree with a destructive interference. Using the obtained cross section of $\psi(3770)\rightarrow p\bar{p}$, the cross section of $p\bar{p}\rightarrow \psi(3770)$, which is useful information for the future PANDA experiment, is estimated to be either $(9.8^{+11.8}_{-3.9})\nb$ $(<27.5\nb$ at 90% C.L.) or $(425.6^{+42.9}_{-43.7})\nb$. title: 'Study of $e^+e^- \rightarrow p\bar{p}$ in the vicinity of $\psi(3770)$ ' --- BESIII ,charmonium decay ,proton form factor 13.20.Gd ,13.25.Gv ,13.40.Gp ,13.66.Bc ,14.20.Gh Introduction ============ At $e^+e^-$ colliders, charmonium states with $J^{PC}=1^{--}$, such as the $J/\psi$, $\psi(3686)$, and $\psi(3770)$, are produced through electron-positron annihilation into a virtual photon. These charmonium states can then decay into light hadrons through either the three-gluon process ($e^+e^-\rightarrow \psi \rightarrow ggg \rightarrow hadrons$) or the one-photon process ($e^+e^-\rightarrow \psi \rightarrow \gamma^* \rightarrow hadrons$). In addition to the above two processes, the non-resonant process ($e^+e^-\rightarrow \gamma^* \rightarrow hadrons$) plays an important role, especially in the $\psi(3770)$ energy region where the non-resonant production cross section is comparable to the resonant one. The $\psi(3770)$, the lowest lying $1^{--}$ charmonium state above the $D\bar{D}$ threshold, is expected to decay dominantly into the OZI-allowed $D\bar{D}$ final states [@MARK; @DELCO]. However, assuming no interference effects between resonant and non-resonant amplitudes, the BES Collaboration found a large total non-$D\bar{D}$ branching fraction of $(14.5\pm1.7\pm5.8)\%$ [@BES_nonDDbar_1; @BES_nonDDbar_2; @BES_nonDDbar_3; @BES_nonDDbar_4]. A later work by the CLEO Collaboration, which included interference between one-photon resonant and one-photon non-resonant amplitudes (assuming no interference with the three-gluon amplitude), found a contradictory non-$D\bar{D}$ branching fraction of $(-3.3\pm1.4^{+6.6}_{-4.8})\%$ [@CLEO_nonDDbar]. These different results could be caused by interference effects. Moreover, it has been noted that the interference of the non-resonant (continuum) amplitude with the three-gluon resonant amplitude should not be neglected [@interfere_wangp_1]. To clarify the situation, many exclusive non-$D\bar{D}$ decays of the $\psi(3770)$ have been investigated [@non_DDbar_exclucive_1; @non_DDbar_exclucive_2]. Low statistics, however, especially in the scan data sets have not permitted the inclusion of interference effects in these exclusive studies. BESIII has collected the world’s largest data sample of $e^+e^-$ collisions at $3.773\gev$. Analyzed together with data samples taken during a $\psi(3770)$ line-shape scan, investigations of exclusive decays, taking into account the interference of resonant and non-resonant amplitudes are now possible. Recently, the decay channel of $\psi(3770)\rightarrow p\bar{p}\pi^0$ [@ppbarpi0_matthias] has been studied considering the above mentioned interference. In this Letter, we report on a study of the two-body final state $e^+e^- \rightarrow p\bar{p}$ in the vicinity of the $\psi(3770)$ based on data sets collected with the upgraded Beijing Spectrometer (BESIII) located at the Beijing Electron-Positron Collider (BEPCII) [@BESIII_BEPCII]. The data sets include $2917\invpb$ of data at $3.773\gev$, $44.5\invpb$ of data at $3.65\gev$ [@Lumi], and data taken during a $\psi(3770)$ line-shape scan in the energy range from $3.74$ to $3.90\gev$. BESIII detector =============== The BEPCII is a modern accelerator featuring a multi-bunch double ring and high luminosity, operating with beam energies between 1.0 and $2.3\gev$ and a design luminosity of $1\times10^{33}{\ensuremath{\,\mathrm{cm^{-2}\,{s}^{-1}}}}$. The BESIII detector is a high-performance general purpose detector. It is composed of a helium-gas based drift chamber (MDC) for charged-particle tracking and particle identification by specific ionization $dE/dx$, a plastic scintillator time-of-flight (TOF) system for additional particle identification, a CsI (Tl) electromagnetic calorimeter (EMC) for electron identification and photon detection, a super-conducting solenoid magnet providing a 1.0 Tesla magnetic field, and a muon detector composed of resistive-plate chambers. The momentum resolution for charged particles at $1\gevc$ is $0.5\%$. The energy resolution of $1\gev$ photons is $2.5\%$. More details on the accelerator and detector can be found in Ref. [@BESIII_BEPCII]. A [geant4]{}-based [@geant4] Monte Carlo (MC) simulation software package, which includes a description of the geometry, material, and response of the BESIII detector, is used for detector simulations. The signal and background processes are generated with dedicated models that have been packaged and customized for BESIII [@generator]. Initial-state radiation (ISR) effects are not included at the generator level for the efficiency determination, but are corrected later using a standard ISR correction procedure [@isr_1; @isr_2]. In the ISR correction, [phokhara]{} [@phokhara] is used to produce a MC-simulated sample of $e^+e^-\rightarrow \gamma_{\rm ISR}p\bar{p}$ (without $\gamma_{\rm ISR} J/\psi$ and $\gamma_{\rm ISR} \psi(3686)$). For the estimation of backgrounds from $\gamma_{\rm ISR} \psi(3686)$ and $e^+e^-\rightarrow \psi(3770)\rightarrow D\bar{D}$, MC-simulated samples with a size equivalent to 10 times the size of data samples are analyzed. Event selection =============== The final state in this decay is characterized by one proton and one antiproton. Two charged tracks with opposite charge are required. Each track is required to have its point of closest approach to the beam axis within $10{\ensuremath{\,\mathrm{cm}}}$ of the interaction point in the beam direction and within $1{\ensuremath{\,\mathrm{cm}}}$ of the beam axis in the plane perpendicular to the beam. The polar angle of the track is required to be within the region $|\cos\theta\,|<0.8$. The TOF information is used to calculate particle identification (PID) probabilities for pion, kaon and proton hypotheses [@pid_ppbar]. For each track, the particle type yielding the largest probability is assigned. Here, the momentum of proton is high ($> 1.6\gevc$). For this high momentum protons and antiprotons, the PID efficiency is about 95%. The ratio of kaons to be mis-identified as protons is about 5%. In this analysis, one charged track is required to be identified as a proton and the other one as an antiproton. The angle between the proton and antiproton ($\theta_{p\bar{p}}$) in the rest frame of the overall $e^+e^-$ CMS system is required to be greater than 179 degrees. Finally, for both tracks, the absolute difference between the measured and the expected momentum ([*e.g.*]{} $1.637\gevc$ for the $\psi(3770)$ data sample) should be less than $40\mevc$ (about 3$\sigma$). (-110,67)[(a)]{} (-110,67)[(b)]{} (-110,67)[(c)]{} (-110,67)[(d)]{} After imposing the above event selection criteria, $684\pm26$ candidate events remain from the $\psi(3770)$ data set. Comparisons between experimental and MC data are plotted in Fig. \[fig:ppbar\_signal\]. The MC simulation agrees with the experimental data. For other data sets, signal events are selected with similar selection criteria. Signal yields are listed in Table \[tab:observed\_cross\_section\]. $\sqrt{s}$(GeV) $N_{sig}$ $\epsilon$(%) $L (\invpb)$ $(1+\delta)_{dressed}$ $\sigma_{obs}$ (pb) $\sigma_{dressed} $(pb) $\sigma_{Born}$(pb) ----------------- ---------------------- --------------- -------------- ------------------------ ---------------------------------- ---------------------------------- ---------------------------------- 3.650 $26.0\pm5.1$ $62.6\pm0.4$ 44.5 0.76 $0.90\pm0.18\pm0.06$ $1.19\pm0.24\pm0.08$ $1.12\pm0.22\pm0.08$ 3.748 $1.0^{+1.8}_{-0.6}$ $61.2\pm0.4$ 3.57 0.76 $0.46^{+0.83}_{-0.28}\pm0.03$ $0.60^{+1.08}_{-0.36}\pm0.04$ $0.54^{+0.97}_{-0.32}\pm0.04$ 3.752 $3.0^{+2.3}_{-1.9}$ $60.8\pm0.4$ 6.05 0.76 $0.82^{+0.63}_{-0.52}\pm0.06$ $1.07^{+0.82}_{-0.68}\pm0.08$ $0.96^{+0.74}_{-0.61}\pm0.07$ 3.755 $4.0^{+2.8}_{-1.7}$ $61.7\pm0.4$ 7.01 0.77 $0.93^{+0.65}_{-0.39}\pm0.06$ $1.21^{+0.85}_{-0.51}\pm0.09$ $1.09^{+0.76}_{-0.46}\pm0.08$ 3.760 $4.0^{+2.8}_{-1.7}$ $62.4\pm0.4$ 8.65 0.77 $0.74^{+0.52}_{-0.32}\pm0.05$ $0.96^{+0.67}_{-0.41}\pm0.07$ $0.87^{+0.61}_{-0.37}\pm0.06$ 3.766 $0.0^{+1.3}_{-0.0}$ $62.4\pm0.4$ 5.57 0.79 $0.00^{+0.37}_{-0.00}$ ($<0.70$) $0.00^{+0.47}_{-0.00}$ ($<0.89$) $0.00^{+0.43}_{-0.00}$ ($<0.81$) 3.772 $0.0^{+1.3}_{-0.0}$ $62.5\pm0.4$ 3.68 0.80 $0.00^{+0.56}_{-0.00}$ ($<1.06$) $0.00^{+0.70}_{-0.00}$ ($<1.33$) $0.00^{+0.64}_{-0.00}$ ($<1.20$) 3.773 $684\pm26$ $62.3\pm0.4$ 2917 0.80 $0.38\pm0.01\pm0.03$ $0.47\pm0.02\pm0.04$ $0.43\pm0.02\pm0.03$ 3.778 $0.0^{+1.3}_{-0.0}$ $62.6\pm0.4$ 3.61 0.78 $0.00^{+0.57}_{-0.00}$ ($<1.08$) $0.00^{+0.74}_{-0.00}$ ($<1.39$) $0.00^{+0.66}_{-0.00}$ ($<1.25$) 3.784 $0.0^{+1.3}_{-0.0}$ $62.4\pm0.4$ 4.57 0.75 $0.00^{+0.45}_{-0.00}$ ($<0.85$) $0.00^{+0.60}_{-0.00}$ ($<1.14$) $0.00^{+0.54}_{-0.00}$ ($<1.02$) 3.791 $1.0^{+1.8}_{-0.6}$ $62.1\pm0.4$ 6.10 0.74 $0.26^{+0.48}_{-0.16}\pm0.02$ $0.35^{+0.64}_{-0.21}\pm0.02$ $0.32^{+0.57}_{-0.19}\pm0.02$ 3.798 $3.0^{+2.3}_{-1.9}$ $61.9\pm0.4$ 7.64 0.75 $0.63^{+0.49}_{-0.40}\pm0.04$ $0.85^{+0.65}_{-0.54}\pm0.06$ $0.77^{+0.59}_{-0.48}\pm0.05$ 3.805 $1.0^{+1.8}_{-0.6}$ $61.5\pm0.4$ 4.34 0.75 $0.37^{+0.67}_{-0.22}\pm0.03$ $0.50^{+0.90}_{-0.30}\pm0.04$ $0.45^{+0.81}_{-0.27}\pm0.03$ 3.810 $20.0\pm4.5$ $62.4\pm0.4$ 52.60 0.75 $0.61\pm0.14\pm0.04$ $0.81\pm0.18\pm0.06$ $0.73\pm0.16\pm0.05$ 3.819 $1.0^{+1.8}_{-0.6}$ $61.4\pm0.4$ 1.05 0.75 $1.55^{+2.79}_{-0.93}\pm0.11$ $2.06^{+3.70}_{-1.23}\pm0.14$ $1.85^{+3.34}_{-1.11}\pm0.13$ 3.900 $12.0^{+4.3}_{-3.2}$ $61.7\pm0.4$ 52.61 0.76 $0.37^{+0.13}_{-0.10}\pm0.03$ $0.49^{+0.17}_{-0.13}\pm0.03$ $0.44^{+0.16}_{-0.12}\pm0.03$ Background estimation ===================== Background from ISR to the lower lying $\psi(3686)$ resonance, which is not taken into account in the ISR correction procedure, is estimated with a sample of MC-simulated data. The number of expected background events from this process is 0.1 and is neglected in this analysis. Background from $\psi(3770)\rightarrow D\bar{D}$ is estimated with an inclusive MC sample and can also be neglected. Exclusive channels, such as $e^+e^- \rightarrow K^+K^-$, $\mu^+\mu^-$, $\tau^+\tau^-$, $p\bar{p}\pi^0$, $p\bar{p}\gamma$ are also studied. The total background contribution is estimated to be 0.4 events, which is equivalent to a contamination ratio of 0.06%. Contributions from decay channels with unmeasured branching fractions for the $\psi(3770)$ are estimated by the branching fractions of the corresponding decay channels of $\psi(3686)$. These background contributions from unmeasured decay modes are taken into account in the systematic uncertainty (0.06%) instead of being subtracted directly. The data set at $3.65\gev$ contains a contribution from the $\psi(3686)$ tail, whose cross section is estimated to be $0.136\pm0.012\nb$ [@BES_nonDDbar_4]. The normalized contribution from this tail, 0.89 events, is also statistically subtracted from the raw signal yield. Determination of cross sections =============================== The observed cross sections at the center-of-mass energies $\sqrt{s}=3.65$, $3.773\gev$ and the fourteen different energy points in the vicinity of the $\psi(3770)$ resonance are determined according to $\sigma = \frac{N_{sig}}{\epsilon L}$, where $\epsilon$ is the detection efficiency determined from MC simulation and $L$ is the integrated luminosity for each energy point. The observed cross sections are listed in Table \[tab:observed\_cross\_section\]. For energy points with no significant signal, upper limits on the cross section at 90% C.L. are given using the Feldman-Cousins method from Ref. [@statistics_paper]. The observed cross section of $e^+e^-\rightarrow p\bar{p}$ contains the lowest order Born cross section and some higher order contributions. The BaBar Collaboration [@babar_G; @babar_G_with_sa] has taken into account bremsstrahlung, $e^+e^-$ self-energy and vertex corrections in their radiative correction. Vacuum polarization is included in their reported cross section. This corrected cross section, which is the sum of the Born cross section and the contribution of vacuum polarization, is called the dressed cross section. In order to use the BaBar measurements of $\sigma(e^+e^-\rightarrow p\bar{p})$ [@babar_G; @babar_G_with_sa] in our investigation, a radiative correction is performed to calculate the dressed cross section using the method described in Refs. [@isr_1; @isr_2]. With the observed cross sections as our initial input, a fit to the line-shape equation (Eq. (\[eq:sigma\_tot\_iso\_spin\])) is performed iteratively. At each iteration, the ISR correction factors are calculated and the dressed cross sections are updated. The calculation converges after a few iterations ($\sim$ 5). The dressed cross section at each data point is listed in Table \[tab:observed\_cross\_section\]. As a reference, the Born cross sections are also calculated and given in Table \[tab:observed\_cross\_section\]. The Born cross section around 3.773 GeV is in excellent agreement with a previous measurement obtained with CLEO data [@Kamal_cleoc]. Fit to the cross section ======================== To extract the $\psi(3770)\rightarrow p\bar{p}$ cross section, the total cross section as a function of $\sqrt{s}$ is constructed and a fit to the measured values is performed. As discussed in the introduction, the measured cross section is composed of three contributions: the three-gluon resonant process ($A_{3g}$), the one-photon resonant process ($A_{\gamma}$) and the non-resonant process ($A_{con}$). For the exclusive light hadron decay of the $\psi(3770)$, the contribution of the electromagnetic process $A_{\gamma}$ is negligible compared to that of the three-gluon strong interaction $A_{3g}$ [@P_wang_A3g_Agamma]. The resonant amplitude can then be written as $A_{\psi}\equiv A_{3g} + A_{\gamma} \sim A_{3g}$. Finally, the total cross section can be constructed with only two amplitudes, $A_{\psi}$ and $A_{con}$, $$\label{eq:sigma_tot_iso_spin} \begin{aligned} &\sigma(s) = |A_{con} + A_{\psi}e^{i\phi}|^2\\ &=\left|\sqrt{\sigma_{con}(s)}+\sqrt{\sigma_\psi} \frac{m_\psi\Gamma_\psi}{s-m_\psi^2+im_\psi\Gamma_\psi}e^{i\phi}\right|^2,\\ \end{aligned}$$ where $m_\psi$ and $\Gamma_\psi$ are the mass and width of the $\psi(3770)$ [@pdg], respectively; $\phi$ describes the phase angle between the continuum and resonant amplitudes, which is a free parameter to be determined in the fit; and $\sigma_\psi$ is the resonant cross section, which is also a free parameter. The continuum cross section, $\sigma_{con}$, has been measured by many experiments [@babar_G; @babar_G_with_sa; @bes2_G; @babar_G_old]. In Ref. [@bes2_G] from the BESII Collaboration, $\sigma_{con}$ was measured from $2$ to $3.07\gev$, and is well-described with an $s$ dependence according to $$\label{eq:sigma_con} \sigma_{con}(s)=\frac{4\pi\alpha^2v}{3s}\left(1+\frac{2m_p^2}{s}\right)|G(s)|^2,$$ $$\label{eq:form_factor} |G(s)|=\frac{C}{s^2\ln^2(s/\Lambda^2)}.$$ Here $\alpha$ is the fine-structure constant; $m_p$ is the nominal proton mass; $v$ is the proton velocity in the $e^+e^-$ rest frame; $G(s)$ is the effective proton form factor [@babar_G_old]; $\Lambda = 0.3\gev$ is the QCD scale parameter; and $C$ is a free parameter. The dressed cross sections in Table \[tab:observed\_cross\_section\], together with the BaBar measurements of the cross sections between $3$ and $4\gev$, are fitted with Eq. (\[eq:sigma\_tot\_iso\_spin\]). In this fit, 26 data points are considered: 16 points from this investigation by BESIII, 5 points from Ref. [@babar_G] and 5 points from Ref. [@babar_G_with_sa]. The free parameters are the phase angle $\phi$, the resonant cross section $\sigma_\psi$, and $C$ from the form factor describing the contribution of the continuum. Fig. \[fig:fit\] shows the data points and the fit result. The fit yields a $\chi^2/ndf$ of $13.4/23$. Two solutions are found with the same $\chi^2$ and the same parameter $C$ of $(62.0\pm2.3) {\ensuremath{\,\mathrm{GeV^4}}}$. Two solutions are found because the cross section in Eq. (\[eq:sigma\_tot\_iso\_spin\]) is constructed with the square of two amplitudes. This multi-solution problem has been explained in Ref. [@zhuk_ijmp_multi_solution]. A dip indicating destructive interference is seen clearly in the fit (the red solid line in Fig. \[fig:fit\]). The first solution for the cross section is $\sigma_{dressed}(e^+e^-\rightarrow\psi(3770)\rightarrow p\bar{p}) = (0.059 ^{+0.070}_{-0.020})\pb$ with a phase angle $\phi = (255.8^{+39.0}_{-26.6})^\circ$ ($<0.166\pb$ at the 90% C.L.). The second solution is $\sigma_{dressed}(e^+e^-\rightarrow\psi(3770)\rightarrow p\bar{p}) = (2.57 ^{+0.12}_{-0.13})\pb$ with a phase angle $\phi = (266.9^{+6.1}_{-6.3})^\circ$. For comparison, an alternative fit with only the BESIII data points is performed. Two solutions are found with the same $\chi^2/ndf$ of $6.8/13$ and the same parameter $C$ of $(62.6\pm4.1) {\ensuremath{\,\mathrm{GeV}}}^4$. The first solution for the cross section is $\sigma_{dressed}(e^+e^-\rightarrow\psi(3770)\rightarrow p\bar{p}) = (0.067 ^{+0.088}_{-0.034})\pb$ with a phase angle $\phi = (253.8^{+40.7}_{-25.4})^\circ$. The second solution is $\sigma_{dressed}(e^+e^-\rightarrow\psi(3770)\rightarrow p\bar{p}) = (2.59 \pm 0.20)\pb$ with a phase angle $\phi = (266.4\pm6.3)^\circ$. These two solutions agree with those from the previous fit, but have larger uncertainties. Table \[tab:cross\_section\_3770\_ppbar\] shows a summary of the fit results, where the first error is from the fit and the second error is from the correlated systematic uncertainties. ----------------------------------------------------------------------------- Solution $\sigma^{dressed}_{(\psi(3770)\rightarrow $\phi$ ($^\circ$) p\bar{p})}$ (pb) ---------- ------------------------------------------- ---------------------- $0.059^{+0.070}_{-0.020}\pm0.012$ ($<$ 0.166 at 90% C.L.) (2) $2.57^{+0.12}_{-0.13}\pm0.12$ $266.9^{+6.1}_{-6.3} \pm0.9$ ----------------------------------------------------------------------------- : Summary of the extracted results for different solutions of the fit. Upper limits are determined at 90% C.L.[]{data-label="tab:cross_section_3770_ppbar"} Systematic uncertainty study ============================ The sources of systematic uncertainty in the cross section measurements are divided into two categories: uncorrelated and correlated uncertainties between different energy points. The former includes only the statistical uncertainty in the MC simulated samples (0.4%), which can be directly considered in the fit. The latter refers to the uncertainties that are correlated among different energy points, such as the tracking (4% for two charged tracks), particle identification (4% for both proton and antiproton), and integrated luminosity. The integrated luminosity for the data was measured by analyzing large angle Bhabha scattering events [@Lumi] and has a total uncertainty of 1.1% at each energy point. To estimate the uncertainty from the radiative corrections, a different correction procedure using the structure-function method [@isr_structure_function] is applied, and the difference in results from these two correction procedures (2%) is taken as the uncertainty. To investigate the impact of the possible inconsistency of the MC simulation and experimental data, an alternative MC simulated sample is generated with a different proton momentum resolution (15% better than the previous MC sample), and the change in the final results (1.4%) is taken as the uncertainty. In addition, the uncertainty on the reconstruction efficiency from the unmeasured angular distribution of the proton in the rest frame of the overall $e^+e^-$ CMS system is also studied. According to hadron helicity conservation, the angular distribution of $\psi\rightarrow p\bar{p}$ can be expressed as $\frac{dN}{dcos\theta} \propto 1+\alpha \cos^2\theta$, where $\theta$ is the angle between the proton and the positron beam direction in the center-of-mass system. The theoretical value of $\alpha=0.813$ [@carimalo_alpha] is used to produce the MC simulated sample in this analysis. In the case of $\psi(3686)\rightarrow p\bar{p}$, the mean value of $\alpha$ measured by E835 (0.67$\pm$0.16) [@alpha_psi2s_e385] differs by $0.13$ from the theoretical value of $0.80$. To obtain a conservative uncertainty, an alternative MC simulated sample with $\alpha=0.683$ is used and the difference in the results (1.0%) is taken as the uncertainty. The uncertainty from the angle cut between the proton and antiproton is investigated by varying the angle cut (from 178.9 to 179.5 degrees) and the difference (2.2%) is taken as the uncertainty. All of the above sources of uncertainty are applied to the observed cross section at each energy point. The total systematic uncertainty of the individual energy points is 6.7%. The systematic uncertainties on the parameters extracted from the fit, such as $\sigma^{dressed}_{(\psi(3770)\rightarrow p\bar{p})}$ and the phase angle $\phi$, are estimated by the “offset method" [@correlated_error], in which the error propagation is determined from shifting the data by the aforementioned correlated uncertainties and adding the deviations in quadrature. In addition, a $1\mev$ uncertainty for the beam energy measurements of all the data points is considered in the fit. Summary and Discussion ====================== Using $2917\invpb$ of data collected at $3.773\gev$, $44.5\invpb$ of data collected at $3.65\gev$ and data collected during a $\psi(3770)$ line-shape scan with the BESIII detector, the reaction $e^+e^-\rightarrow p\bar{p}$ has been studied. To extract the cross section of $e^+e^-\rightarrow \psi(3770) \rightarrow p\bar{p}$, a fit, taking into account the interference of resonant and continuum amplitudes, is performed. In this investigation, the measured cross sections of $e^+e^-\rightarrow p\bar{p}$ from the BaBar experiment are included in a simultaneous fit to put more constraints on the continuum amplitude. The dressed cross section of $e^+e^-\rightarrow\psi(3770)\rightarrow p\bar{p}$ is extracted from the fit and shown in Table \[tab:cross\_section\_3770\_ppbar\]. With the obtained dressed cross section of $e^+e^-\rightarrow \psi(3770)\rightarrow p\bar{p}$, the branching fraction $B_{\psi(3770)\rightarrow p\bar{p}}$ is determined to be $(7.1^{+8.6}_{-2.9})\times10^{-6}$ or $(3.1\pm0.3)\times10^{-4}$, by dividing the dressed cross section of $e^+e^-\rightarrow \psi(3770)$ [@CLEO_nonDDbar]. Even the larger solution has a relatively small branching fraction comparing to the large total non-$D\bar{D}$ branching fraction. Thus, the $p\bar{p}$ channel alone cannot explain the large non-$D\bar{D}$ branching fraction from BESII. Using the branching fraction of $\psi(3770)\rightarrow p\bar{p}$, the cross section of its time reversed reaction $p\bar{p}\rightarrow \psi(3770)$ can be estimated using the Breit-Wigner formula [@pdg]: $$\label{eq:cross_section_ppbar_to_3770} \sigma_{p\bar{p}\rightarrow\psi(3770)}(s)= \frac{4\pi(2J+1)}{(s-4m_p^2)} \frac{B_{\psi(3770)\rightarrow p\bar{p}}}{1+[2(\sqrt{s}-M_{\psi})/\Gamma_{\psi}]^2}$$ where $M_\psi$ and $\Gamma_\psi$ are the mass and width of the $\psi(3770)$ resonance, J is the spin of the $\psi(3770)$, and $m_p$ is the proton mass. For the condition $\sqrt{s}=M_\psi$, the cross section $\sigma(p\bar{p}\rightarrow \psi(3770))$ is estimated to be either $(9.8^{+11.8}_{-3.9})\nb$ ($<27.5\nb$ at 90% C.L.) or $(425.6^{+42.9}_{-43.7})\nb$. The future $\rm \bar{P}ANDA$ (anti-Proton ANnihilations at DArmstadt) experiment is one of the key projects at the Facility for Antiproton and Ion Research (FAIR), which is currently under construction at GSI, Darmstadt. It will perform precise studies of antiproton-proton annihilations with various internal proton or nuclear targets and an anti-proton beam in the momentum range from 1.5 GeV/c to 15 GeV/c. In $\rm \bar{P}ANDA$, a detailed investigation of the charmonium spectrum and the open charm channels is foreseen. For this physics program, it is important to obtain experimental information on the so far unknown open charm cross sections, both to evaluate luminosity requirements and to design detector. Theoretical estimations vary with several orders of magnitude [@ppbar_ccbar_theory_1; @ppbar_ccbar_theory_2; @ppbar_ccbar_theory_3; @ppbar_ccbar_theory_4; @ppbar_ccbar_theory_5; @ppbar_ccbar_theory_6; @ppbar_ccbar_theory_7; @ppbar_ccbar_theory_8; @ppbar_ccbar_theory_9]. In the physics performance report for $\rm \bar{P}ANDA$ [@panda_physics_report], the $D\bar{D}$ production cross section is estimated to be $6.35\nb$, with the unknown branching ratio of $\psi(3770)\rightarrow p\bar{p}$ scaled from the known ratio of $J/\psi \rightarrow p\bar{p}$. In this paper, the cross section of $\sigma(p\bar{p}\rightarrow \psi(3770))$ has been determined. As the first charmonium state above the $D\bar{D}$ threshold, $\psi(3770)$ could be used as a source of open charm production. In this paper, two solutions on the cross section of $\sigma(p\bar{p}\rightarrow \psi(3770))$ are obtained. It is impossible to distinguish these two solutions with our data. The first solution, $(9.8^{+11.8}_{-3.9}) \nb$, is compatible with a simple scaling from $J/\psi$ used in the $\rm \bar{P}ANDA$ physics performance report. The second solution, with the cross section of $(425.6^{+42.9}_{-43.7}) \nb$, is two order of magnitudes larger. 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--- abstract: 'This note provides a correct proof of the result claimed by the second author that locally compact normal spaces are collectionwise Hausdorff in certain models obtained by forcing with a coherent Souslin tree. A novel feature of the proof is the use of saturation of the non-stationary ideal on $\omega_1$, as well as of a strong form of Chang’s Conjecture. Together with other improvements, this enables the consistent characterization of locally compact hereditarily paracompact spaces as those locally compact, hereditarily normal spaces that do not include a copy of $\omega_1$.' author: - 'Alan Dow[$^1$]{} and Franklin D. Tall[$^2$]{}' bibliography: - 'normality.bib' nocite: '[@*]' title: Normality versus paracompactness in locally compact spaces --- [^1] [^2] Introduction ============ The space of countable ordinals is locally compact, normal, but not paracompact. The question of what additional conditions make a locally compact normal space paracompact has a long history. At least 45 years ago, it was recognized that subparacompactness plus collectionwise Hausdorffness would do (see e.g. [@T1]), as would perfect normality plus metacompactness [@A]. Z. Balogh proved a variety of results under MA$_{\omega_1}$ [@B1] and **Axiom R** [@B2], and was the first to realize the importance of not including a perfect pre-image of $\omega_1$ (equivalently, the one-point compactification being countably tight [@B1]). However, he assumed collectionwise Hausdorffness in order to obtain paracompactness. A breakthrough came with S. Watson’s proof that: $V = L$ implies locally compact normal spaces are collectionwise Hausdorff, and hence locally compact normal metacompact spaces are paracompact. Watson’s proof crucially involved the idea of *character reduction*: if one wants to separate a closed discrete subspace of size $\kappa$, $\kappa$ regular, in a locally compact normal space, it suffices to separate $\kappa$ compact sets, each with an *outer base* of size $\leq \kappa$. An [**outer base**]{} for a set $K \subseteq X$ is a collection ${\mathcal}{B}$ of open sets including $K$ such that each open set including $K$ includes a member of ${\mathcal}{B}$. The use of $V = L$ was to get that normal spaces of character $\leq \aleph_1$ are collectionwise Hausdorff [@F], and variations on that theme. It was known that locally compact normal non-collectionwise Hausdorff spaces could be constructed from MA$_{\omega_1}$, indeed from the existence of a $Q$-set [@T1], so it was a big surprise when G. Gruenhage and P. Koszmider proved that: MA$_{\omega_1}$ implies locally compact, normal, metacompact spaces are $\aleph_1$-collectionwise Hausdorff and (hence) paracompact. The next result involving iteration axioms and a positive “normal implies collectionwise Hausdorff" type of result was: Let $S$ be a coherent Souslin tree (obtainable from $\diamondsuit$ or a Cohen real). Force MA$_{\omega_1}(S)$, i.e. MA$_{\omega_1}$ for countable chain condition posets preserving $S$. Then force with $S$. In the resulting model, there are no first countable $L$-spaces, no compact first countable $S$-spaces, and separable normal first countable spaces are collectionwise Hausdorff. The first two statements are consequences of MA$_{\omega_1}$ [@Sz]; the last of $V=L$, indeed of $2^{\aleph_0} < 2^{\aleph_1}$. Larson and Todorcevic used this combination to solve *Katětov’s problem*. This idea of combining consequences of a iteration axiom with “normal implies collectionwise Hausdorff" consequences of $V = L$ was exploited in [@LT1] in order to prove the consistency, modulo a supercompact cardinal, of *every locally compact perfectly normal space is paracompact*. The large cardinal was later removed, so that: If ZFC is consistent, then so is ZFC plus every locally compact perfectly normal space is paracompact. In the models of [@LT1] and [@DT2], every first countable normal space is collectionwise Hausdorff. This is achieved in two stages. The novel one is: \[lem15\] Force with a Souslin tree. Then\[LT1\] normal first countable spaces are $\aleph_1$-collectionwise Hausdorff. This is obtained by showing that if a normal first countable space is not $\aleph_1$-collectionwise Hausdorff, a generic branch of the Souslin tree induces a generic partition of the unseparated closed discrete subspace which cannot be “normalized", i.e. there do not exist disjoint open sets about the two halves of the partition. The argument is a blend of the two usual methods of proving “normal implies $\aleph_1$-collectionwise Hausdorff" results, namely those of adjoining Cohen subsets of $\omega_1$ by countably closed forcing [@T1], [@T2] and using *$\diamondsuit$ for stationary systems on $\omega_1$*, a strengthening of $\diamondsuit$ that holds in $L$ [@F]. It is noteworthy that: Either force to add $\aleph_2$ Cohen subsets of $\omega_1$, or assume $\diamondsuit$ for stationary subsets of $\omega_1$. Then normal spaces of character $\leq \aleph_1$ are $\aleph_1$-collectionwise Hausdorff. Once one has normal first countable spaces are $\aleph_1$-collectionwise Hausdorff, it is easy to obtain full collectionwise Hausdorffness by starting with $L$ as the ground model and following [@F]. However, if a supercompact cardinal is involved, instead of $L$ we need to follow the method of [@LT1], based on [@T2]. Namely, first make the supercompact indestructible under countably closed forcing [@L] and then perform an Easton extension, adding $\kappa^+$ Cohen subsets of each regular $\kappa$, before forcing with the Souslin tree. In order to extend the theorems about locally compact normal spaces being paracompact beyond the realm of first countability, one first needs to get that *locally compact normal spaces are collectionwise Hausdorff*. In [@T3], the second author claimed to have done so, in the model of [@LT1]. The key was to force to expand a closed discrete subspace in a locally compact normal space to a discrete collection of compact sets with countable outer bases and then apply the methods of [@LT1]. Unfortunately the expansion argument was flawed. A corrected argument is presented below, but at the cost of using a stronger iteration axiom (but not a larger large cardinal). With the conclusion of [@T3] restored, [@T4], [@LT2], and [@T] are re-instated. We shall then proceed to improve the results of the two latter ones. PFA$(S)[S]$ and the role of $\omega_1$ ====================================== *PFA$(S)$* is the Proper Forcing Axiom (PFA) restricted to those posets that preserve the (Souslinity of the) coherent Souslin tree $S$. *PFA$(S)[S]$ implies $\varphi$* is shorthand for *whenever one forces with a coherent Souslin tree $S$ over a model of PFA$(S)$, $\varphi$ holds.* *$\varphi$ holds in a model of form PFA$(S)[S]$* is shorthand for *there is a coherent Souslin tree $S$ and a model of PFA$(S)$ such that when one forces with $S$ over that model, $\varphi$ holds.* For discussion of PFA$(S)[S]$, see [@D2], [@To], [@LT1], [@LT2], [@T4], [@T], [@FTT], [@T6]. The following results appear in [@LT2] and [@T], respectively. \[thm:paracompactcopy\] There is a model of form ${\mathrm}{PFA}(S)[S]$ in which a locally compact, hereditarily normal space is hereditarily paracompact if and only if it does not include a perfect pre-image of ${\omega}_1$. \[thm:paracompactcountablytight\] There is a model of form ${\mathrm}{PFA}(S)[S]$ in which a locally compact normal space is paracompact and countably tight if and only if its separable closed subspaces are Lindelöf and it does not include a perfect pre-image of ${\omega}_1$. **** is the assertion that every first countable perfect pre-image of $\omega_1$ includes a copy of $\omega_1$. ${\mathrm}{PFA}(S)[S]$ implies ****. $\mathbf{PPI}$ was originally proved from PFA in [@BDFN]. Using $\mathbf{PPI}$, we are able to weaken “perfect pre-image" to “copy" in the improved version of the first theorem, but provably cannot in the second theorem. \[thm:paracompactcopyallmodels\] There is a model of form PFA$(S)[S]$ in which a locally compact, hereditarily normal space is hereditarily paracompact if and only if it does not include a copy of $\omega_1$. There is a locally compact space $X$ (indeed a perfect pre-image of $\omega_1$) which is normal, does not include a copy of $\omega_1$, in which all separable closed subspaces are compact, but $X$ is not paracompact. It is clear that to establish Theorem \[thm:paracompactcopyallmodels\], it suffices to use \[thm:paracompactcopy\] and apply $\mathbf{PPI}$ after proving: \[thm34\] ${\mathrm}{PFA}(S)[S]$ implies a hereditarily normal perfect pre-image of ${\omega}_1$ includes a first countable perfect pre-image of ${\omega}_1$. This follows from: \[lxb\] Let $X$ be a perfect pre-image of $\omega_1$, and suppose separable subspaces of $X$ are Lindelöf. Then $X$ includes a first countable perfect pre-image of $\omega_1$. and \[lxc\] ${\mathrm}{PFA}(S)[S]$ implies compact, separable, hereditarily normal spaces are hereditarily Lindelöf. Here is the proof of Lemma \[lxb\]. Let $f : X\rightarrow\omega_1$, perfect and onto. Then $X$ is locally compact, countably compact, but not compact. There is a closed $Y \subseteq X$ such that $f' = f|Y$ is perfect, irreducible, and maps $Y$ onto $\omega_1$. So $Y = \bigcup_{\alpha < \omega_1}f'^{-1}(\{\beta : \beta \leq \alpha\})$. Each $D_\alpha = f'^{-1}(\{\beta : \beta \leq \alpha\})$ is clopen and hence countably compact. It suffices to show $D_\alpha$ is hereditarily Lindelöf, for then points are $G_\delta$ and $D_\alpha$ is first countable. But then $Y$ is first countable, since $D_\alpha$ is open. To show $D_\alpha$ is hereditarily Lindelöf, we need only show it is separable. $f_\alpha = f'|D_\alpha$ is irreducible, for if there were a proper closed subset $A$ of $D_\alpha$ such that $f'(A) = f'(D_\alpha)$, then $f$ would map $A \cup (Y - D_\alpha)$ onto $\omega_1$, contradicting $f$’s irreducibility. But If $f$ is a closed irreducible map of $X$ onto $Y$ and $E$ is dense in $Y$, then $f^{-1}(E)$ is dense in $X$. Thus $D_\alpha$ is separable. Let us construct the example that constrains the hoped-for improvement of Theorem \[thm:paracompactcountablytight\]. Consider a stationary, co-stationary subset $E$ of $\omega_1$ and its Stone-Čech extension $\beta E$. The identity map $\iota$ embeds $E$ into the compact space $\omega_1 + 1$. $\iota$ extends to $\hat{\iota}$ mapping $\beta E$ onto $ \omega_1+1$; we claim that $\hat{\iota}$ maps only one element – call it $z$ – of $\beta E$ to the point $\omega_1$. The reason is that every real-valued continuous function on $E$ is eventually constant. If there were another such point, say $z'$, let $f$ be a continuous real-valued function sending $z$ to $0$ and $z'$ to $1$. Let $U, V$ be open sets about the point $\omega_1$ such that $\hat{\iota}^{-1}(U) \subseteq f^{-1}\left(\left[0, \frac{1}{2}\right]\right)$ and $\hat {\iota}^{-1}(V) \subseteq f^{-1}\left(\left(\frac{1}{2}, 1\right]\right)$. Then $\hat{\iota}^{-1}(U) \cap \hat{\iota}^{- 1}(V) = \emptyset$, but $U \cap V \cap E$ is cocountable in $E$, contradiction. Our space $X$ will be $\beta E - {\{z\}}$. $\hat{\iota}|X$ maps $X$ onto $\omega_1$; we claim this map is perfect. By 3.7.16(iii) of Engelking [@E], it suffices to show that $\hat{\iota}[\beta X - X] = \beta \omega_1 - \omega_1$. But $\beta \omega_1 = \omega_1 + 1$ and $\beta X = \beta E$, so this just says $\hat{\iota}(z) = \omega_1$, which we have. If $H, K$ are disjoint closed subsets of $X$, then their closures in $\beta E$ have at most $z$ in common. Thus their images $\hat{\iota}[H]$ and $\hat{\iota}[K]$ cannot overlap in a subspace with a point of $E$ in its closure. Since $E$ is stationary, their overlap is countable. Then at least one of them is bounded, and hence compact. it is then easy to pull back disjoint open sets to establish normality. For any perfect pre-image of $\omega_1$, it is easy to see that separable closed subspaces are compact, since they are included in a pre-image of an initial closed segment of $\omega_1$. It remains to show that $X$ does not include a copy $W$ of $\omega_1$. A standard $\beta \mathbb{N}$ argument shows that no point in $X - E$ is the limit of a convergent sequence, so the set $C$ of all limits of convergent sequences from $W$ is a subset of $E$. But $C$ is homeomorphic to $\omega_1$, so cannot be included in a co-stationary $E$. There is, however, a satisfactory improvement of Theorem \[thm:paracompactcountablytight\]: \[thm38\] There is a model of form PFA$(S)[S]$ in which a locally compact, normal, countably tight space is paracompact if and only if its separable closed subspaces are Lindelöf, and it does not include a copy of $\omega_1$. This follows from: \[thmConj2\] PFA$(S)[S]$ implies a countably tight, perfect pre-image of $\omega_1$ includes a copy of $\omega_1$. The proof of Theorem \[thm38\] is essentially the same as the proof in [@T] of our Theorem 2.2. Countably tight, hereditarily normal perfect pre-images of $\omega_1$ are rather special: Suppose $\pi : X \to \omega_1$. We say $Y \subseteq X$ is **unbounded** if $\pi(Y)$ is unbounded. \[thm39\] PFA$(S)[S]$ implies that a countably tight, hereditarily normal, perfect pre-image of $\omega_1$ is the union of a paracompact space with a finite number of disjoint unbounded copies of $\omega_1$. By \[thmConj2\], the perfect pre-image $X$ includes a copy, $W_1$, of $\omega_1$. If $W_1$ were bounded, then for some $\alpha$, $W_1 \subseteq \pi^{-1}([0, \alpha])$. But $\pi^{-1}([0, \alpha])$ is compact, and $W_1$ – being a countably compact subspace of a countably tight space – is closed in $X$ and hence in $\pi^{-1}([0, \alpha])$. But then $W_1$ is compact, contradiction. Since perfect pre-images of locally compact spaces are locally compact, $X$ is locally compact. Since $W_1$ is closed, $X - W_1$ is open and so is also locally compact. If it is paracompact, we are done; if not, apply \[thmConj2\] to get a copy $W_2$ of $\omega_1$ included in $X - W_1$. Continue. The process must end at some finite stage, since: Let $X$ be a $T_5$ space, $\pi : X \to \omega_1$ continuous, $\pi^{-1}(\{\alpha\})$ countably compact for all $\alpha \in S$, a stationary subset of $\omega_1$. Then $X$ cannot include an infinite disjoint family of closed, countably compact subspaces each with unbounded range. Note that the paracompact subspace is the topological sum of $\leq \aleph_1$ $\sigma$-compact subspaces. An early version of [@DT1] used the axioms ${\mathbf{\mathop{\pmb{\sum}}}}^-$ (defined in Section 5), ${\mathbf{PPI}}$, and the $\aleph_1$-collectionwise Hausdorffness of first countable normal spaces, as well as \[thm39\] to obtain “countably compact, hereditarily normal manifolds of dimension $> 1$ are metrizable" without the $\mathbf{P}_{22}$ axiom used in [@DT1] to get the stronger assertion in which “countably compact" is omitted. Both of the conditions for paracompactness in \[thm38\] are necessary: $\omega_1$ is locally compact, normal, first countable, its separable subspaces are countable, but it is not paracompact. Van Douwen’s “honest example” [@vD] is locally compact, normal, first countable, separable, does not include a perfect pre-image of $\omega_1$ (because it has a $G_\delta$-diagonal), but is not paracompact. Strengthenings of [PFA]{}$(S)[S]$ ================================= In addition to “front-loading” a PFA$(S)[S]$ model in order to get full collectionwise Hausdorffness, it has also been useful to employ strengthenings of PFA$(S)$ so as to obtain more reflection. E.g. in [@LT2] and [@T], **** is employed. $C\subseteq[X]^{<\kappa}$ is **tight** if whenever $\{C_\alpha:\alpha<\delta\}$ is an increasing sequence from $C$ and $\omega<\text{\normalfont cf}(\delta)<\kappa$, $\bigcup\{C_\alpha:\alpha<\beta\}\in C$. **Axiom R** If $\mathcal{S}\subseteq[X]^{<\omega_1}$ is stationary and $C\subseteq[X]^{<\omega_2}$ is tight and unbounded, then there is a $Y\in C$ such that $\mathcal{P}(Y)\cap\mathcal{S}$ is stationary in $[Y] ^{<\omega_1}$. **** (due to Fleissner [@Fle]) was obtained by using what is called *PFA$^{++}(S)$* in [@LT2], before forcing with $S$ [@LT2]. PFA$^{++}(S)$ holds if PFA$(S)$ is forced in the usual Laver-diamond way. Here we shall use a conceptually simple principle, MM$(S)$, which is forced in a more complicated way, but does not require a larger cardinal. The axiom *Martin’s Maximum* was introduced in [@FMS]. Let $\mathcal{P}$ be a partial order such that forcing with $\mathcal{P}$ preserves stationary subsets of $\omega_1$. Let $\mathcal{D}$ be a collection of $\aleph_1$ dense subsets of $\mathcal{P}$. **** asserts that for each such $\mathcal{D}$, there is a $\mathcal{D}$-generic filter included in $\mathcal{P}$. Assume there is a supercompact cardinal. Then there is a revised countable support iteration establishing [MM]{}. MM$(S)$ is defined analogously to PFA$(S)$; Miyamoto [@M] proved that there is a “nice” iteration establishing MM$(S)$ but preserving $S$. One can then define MM$(S)[S]$ analogously to PFA$(S)[S]$. In order to obtain a model of PFA$(S)[S]$ in which Theorem \[thm:paracompactcopyallmodels\] holds, we need to improve the model of [@LT2] so as to not only have **Axiom R** but also: **LCN($\aleph_1$)** Every locally compact normal space is $\aleph_1$-collectionwise Hausdorff. We shall prove that MM$(S)$ implies: **NSSAT** NS$_{\omega_1}$ (the non-stationary ideal on $\omega_1$) is $\aleph_2$-saturated. **SCC** Strong Chang Conjecture. Let $\lambda>2^{\aleph_2}$ be a regular cardinal. Let $H(\lambda)$ be the collection of hereditarily $<\lambda$ sets. Let $M^*$ be an expansion of $\langle H_\lambda,\in\rangle$. Let $N\prec M^*$ (i.e. $N$ is an elementary submodel of $M^*$) be countable. Then there is an $N'$ such that $N\prec N'\prec M^*$, $N'\cap \omega_1=N\cap\omega_1$, and $|N\cap\omega_2|=\aleph_1$. We also note: $(S)$ implies $2^{\aleph_1}=\aleph_2$. With these, we can modify the proof in [@LT1] that forcing with a Souslin tree makes *first countable normal spaces $\aleph_1$-collectionwise Hausdorff* to obtain *locally compact normal spaces are $\aleph_1$-collectionwise Hausdorff*, and then, if we wish, front-load the model as in [@LT1] to obtain full collectionwise Hausdorffness, using the character reduction method of [@W]. More precisely, the crucial new step is: \[thm41\] Suppose there is a model in which there is a Souslin tree $S$ and in which ****, ****, and $2^{\aleph_1}=\aleph_2$ hold. Then $S$ forces that locally compact normal spaces are $\aleph_1$-collectionwise Hausdorff. It will be convenient to consider the following intermediate proposition, which implies the three things that we want: **SRP** Strong Reflection Principle [@To2]. Suppose $\lambda\geq\aleph_2$ and $\mathfrak{Z}\subseteq\mathcal{P}_{\omega_1}(\lambda)$ and that for each stationary $T\subseteq\omega_1$, $$\{\sigma\in\mathfrak{Z}:\sigma\cap\omega_1\in T\}$$ is stationary in $\mathcal{P}_{\omega_1}(\lambda)$. Then for all $X\subseteq\lambda$ of cardinality $\aleph_1$, there exists $Y\subseteq\lambda$ such that: - $X\subseteq Y$ and $|Y|=\aleph_1$; - $\mathfrak{Z}\cap\mathcal{P}_{\omega_1}(Y)$ contains a set which is closed unbounded in $\mathcal{P}_{\omega_1}(Y)$. With regard to **SCC**, Shelah [@S XII.2.2, XII.2.5] proves that: If there is a semi-proper forcing P changing the cofinality of $\aleph_2$ to $\aleph_0$, then **** holds. There are various versions of *Namba forcing*, e.g. two in [@S] and one in [@Lar]. All of these change the cofinality of $\aleph_2$ to $\aleph_0$. Larson states in [@Lar p.142] that his version of Namba forcing preserves stationary subsets of $\omega_1$. In [@FMS], it is shown that a principle, **SR**, implies *any forcing that preserves stationary subsets of $\omega_1$ is semi-proper*. **SR** is a consequence of MM [@FMS]. is stronger than **SR** and so: implies . $(S)$ implies . **** implies **** and $2^{\aleph_1}\leq\aleph_2$. For the proof of \[thm:paracompactcopyallmodels\] we should also remark that: **** implies ****. We use an equivalent formulation of **SRP** due to Feng and Jech [@FJ]. **SRP** For every cardinal $\kappa$ and every $S\subseteq[\kappa]^\omega$, for every regular $\theta>\kappa$, there is a continuous elementary chain $\{N_\alpha:\alpha\in\omega_1\}$ (with $N_0$ containing some given element of $H(\theta)$, e.g. $S$) such that for all $\alpha$, $N_\alpha\cap\kappa\in S$ if and only if there is a countable $M\prec H(\theta)$ such that $N_\alpha\subseteq M$, $M\cap\omega_1=N_\alpha\cap\omega_1$, and $M\cap\kappa\in S$. Let $\mathcal{S}$ and $\mathcal{C}$ be as in **Axiom R**. Choose $\theta$ sufficiently large so that $\mathcal{S},\mathcal{C}\in H(\theta)$ and so that $\theta^{\aleph_1}=\theta$. Let $\{\mathcal{S},\mathcal{C}\}\in N_0$ and let $\{N_\alpha:\alpha\in\omega_1\}$ be as in **SRP**. By induction on $\alpha\in\omega_1$, choose $Y_\alpha\in C\cap N_{\alpha+1}$ so that $\bigcup (\mathcal{C}\cap N_\alpha)\subseteq Y_\alpha$. Then $\{Y_\alpha:\alpha\in\omega_1\}$ is an increasing chain in $\mathcal{C}$. Therefore $Y=\bigcup_{\alpha\in\omega_1}(N_\alpha\cap\kappa)$ is in $\mathcal{C}$. $\mathcal{S}^+=\{M\prec H(\theta):M\cap \kappa\in\mathcal{S}\}$ is a stationary subset of $[H(\theta)]^\omega$. This is proved in the same way as 1) of Claim 1.12 on page 196 of [@S]. Since $\{N_\alpha:\alpha\in\omega_1\}$ is an element of $H(\theta)$, there is an $M\in\mathcal{S}^+$ such that $\{N_\alpha:\alpha\in\omega_1\}\in M$. Let ${M\cap\omega_1=\delta}$. Obviously $M\cap\kappa\in\mathcal{S}$, and, by continuity, $N_\delta\subseteq M$ and $M\cap\omega_1= N_\delta\cap\omega_1$. It then follows from **SRP** that $N_\delta\in\mathcal{S}$. This actually proves that $\{\alpha\in\omega_1:N_\alpha\cap\kappa\in\mathcal{S}\}$ is a stationary subset of $\omega_1$, because we could have put any cub of $\omega_1$ as an element of $M$. Now assume that $\mathfrak{Z}\subseteq[Y]^\omega$ is a cub of $[Y]^\omega$. Choose a strictly increasing $g:\omega_1\to\omega_1$ such that for each $\alpha$, there is a $Z_\alpha\in \mathfrak{Z}$ such that $N_\alpha\cap \kappa\subseteq Z_\alpha\subseteq N_{g(\alpha)}$. If limit $\delta$ satisfies that $g(\alpha)<\delta$ for all $\alpha<\delta$, then we have that $N_\delta\cap\kappa\in\mathfrak{Z}$. This finishes the proof that $\mathcal{S}\cap[Y]^\omega$ is stationary. Suppose we have a model with a Souslin tree $S$ in which **Axiom R** holds. Then, after forcing with $S$, **** still holds. This is an improvement over [@LT2], which required a stronger axiom, Axiom R$^{++}$, holding in the model. We will use *t.u.b.* as an abbreviation for *tight unbounded*. We must consider two $S$-names: $\dot{\mathcal{C}}$ and $\dot{\mathcal{X}}$ where $\dot{ \mathcal{C}}$ is forced to be a t.u.b. subset of $[\kappa]^{\omega_1}$ and $\dot{ \mathcal{X}}$ is forced to be a stationary subset of $[\kappa]^\omega$. Let us assume that some $s_0\in S$ forces there is no $Y$ in $\dot{ \mathcal{C}}$ such that $\dot{ \mathcal{X}}\cap [Y]^\omega$ is stationary. (It would make the discussion below easier if we just assumed that $s_0$ was the root of $S$ – which one can certainly immediately do if $S$ is a coherent Souslin tree.) We first show that $\dot C$ contains a t.u.b. $C$ from the ground model. Simply put $Y\in \mathcal{C}$ if every $s\in S$ forces that $Y\in \dot{ \mathcal{C}}$. It is clear that $\mathcal{C}$ is closed under increasing $\omega_1$-chains. Thus we just have to show that it is unbounded. Let us enumerate $S$ as $\{ s_\alpha : \alpha\in \omega_1\}$. Fix any $Y_0\in [\kappa]^{\omega_1}$. By recursion choose an increasing chain $\{Y_\alpha : \alpha\in \omega_1\}$ so that for each $\alpha$, $\bigcup \{Y_\beta : \beta < \alpha\}\subseteq Y_\alpha$ and there is an extension $s_\beta$ of $s_\alpha $ forcing that $Y_{\alpha+1}\in \dot{ \mathcal{C}}$. This we may do, since $s_\alpha$ forces that $\dot{ \mathcal{C}}$ is unbounded. Now let $Y$ be the union of the chain $\{ Y_\alpha : \alpha\in \omega_1\}$. Note that for each $s\in S$ and each $\beta\in \omega_1$, there is an $\beta<\alpha$ such that $s_\alpha$ is an extension of $s$. It follows that $s$ forces that $\dot{ \mathcal{C}}\,\cap \{ Y_\alpha : \alpha \in \omega_1\}$ is uncountable, hence $s\Vdash Y\in \dot{ \mathcal{C}}$. Now we let $\mathcal{X}$ be the set of $x\in [\kappa]^\omega$ such that there is some $s\in S$ extending $s_0$ with $s\Vdash x\in \dot{ \mathcal{X}}$. It is clear that $\mathcal{X}$ is a stationary subset of $[\kappa]^\omega$ because $s_0$ forces that $\mathcal{X}$ meets every cub. Now apply **Axiom R** to choose $Y\in \mathcal{C}$ so that $\mathcal{X}\cap [Y]^\omega$ is a stationary subset of $Y$. Now we obtain a contradiction (and thus a proof) by showing that there is an extension $s\in S$ of $s_0$ that forces that $\dot{ \mathcal{X}}\cap [Y]^\omega$ is stationary. Let $\{ y_\alpha :\alpha\in \omega_1\}$ be an enumeration of $Y$. Let $\mathcal{E}$ be the set of $\delta\in \omega_1$ such that $x_\delta = \{y_\alpha :\alpha\in \delta\}\in \mathcal{X}$. Notice that $\{ \{ y_\alpha : \alpha \in \delta \} : \delta\in \omega_1\}$ is a cub in $[Y]^\omega$. Thus it follows that $\mathcal{E}$ is stationary. In fact, if $\mathcal{E}'$ is any stationary subset of $\mathcal{E}$, then $\mathcal{E}'$ is also a stationary subset of $[Y]^\omega$. For each $\delta\in \mathcal{E}$ choose $s_\delta\in S$ above $s_0$ so that $s_\delta\Vdash x_\delta\in \dot{ \mathcal{X}}$ (as per the definition of $ \mathcal{X}$). Now we have a name $\dot{ \mathcal{E}} = \{ (x_\delta, s_\delta) : \delta\in \omega_1\}$. We prove that there is some $s\in S$ above $s_0$ that forces that $\dot{ \mathcal{E}}$ is stationary. Thus such an $s$ forces that $\dot{ \mathcal{X}}\cap [Y]^\omega$ is stationary as required. Let $s_0$ be on level $\alpha_0$ of $S$. There is a $\gamma>\alpha_0$ so that each member of $S_\gamma$ decides if $\dot{ \mathcal{E}}$ is stationary. Also, for each $\bar s\in S_\gamma$ that forces $\dot{ \mathcal{E}}$ is not stationary, there is a cub $\mathcal{C}_{\bar s}$ of $\omega_1$ that $\bar s$ forces is disjoint from $\dot{ \mathcal{E}}$. Choose any $\delta $ in the intersection of those countably many cubs that is also in $\mathcal{E}$. Clearly if $\bar s\in S_\gamma$ is compatible with $s_\delta$, then $\mathcal{C}_{\bar s}$ did not exist since $\bar s \cap s_\delta$ would force that $\delta \in \mathcal{C}_{\bar s}\cap \dot{ \mathcal{E}}$. This completes the proof, since that element $\bar s$ is above $s_0$ and forces that $\dot{ \mathcal{X}}\cap [Y]^\omega$ is stationary. $(S)[S]$ implies ****. We next need: \[P. Larson\]\[larson\] Suppose - ****, and - for sufficiently large $\theta$ and stationary $E\subseteq\omega_1$, for any $X\in H(\theta)$, there is a Chang model $M$ with $M\cap\omega_1\in E, X\in M$ and $|M\cap\omega_2|=\aleph_1$. Then if $\{A_\alpha:\alpha<\omega_2\}$ are stationary subsets of $\omega_1$, $M\cap\omega_1=\delta$ is in uncountably many $A_\alpha,\alpha\in M$. It is well known that ${\mathrm}{NS}_{\omega_1}$ is $\aleph_1$-complete, since the diagonal union of $\aleph_1$ non-stationary subsets of $\omega_1$ is non-stationary. It follows that $\mathcal{P}(\omega_1)/{\mathrm}{NS}_{\omega_1}$ is a complete Boolean algebra, because (1) says it satisfies the $\aleph_2$-chain condition. Since it is complete, for each $\alpha<\omega_2$ there is a stationary $B_\alpha$ which is the sup of $\{A_\beta:\beta\in(\alpha,\omega_2)\}$. Let $E$ be the inf of the family of $B_\alpha$’s. By saturation, $E$ is really the inf of an $\aleph_1$-sized family, and so is itself stationary. Given any $\alpha\in\omega_2$, we can find an $\eta(\alpha)>\alpha$ such that the diagonal union of $\{A_\beta:\beta\in(\alpha,\eta(\alpha))\}$ includes $E$, mod ${\mathrm}{NS}_{\omega_1}$. It follows that there is a cub $C\subseteq \omega_2$ such that for each $\alpha\in C$, there is a subset of $\{A_\beta:\beta\in(\alpha,\alpha^+)\}$ of cardinality $\aleph_1$ with diagonal union including $E$, mod ${\mathrm}{NS}_{\omega_1}$, where $\alpha^+$ denotes the next element of $C$ after $\alpha$. Now let $M$ be an elementary submodel of a suitable $H(\theta)$, with $\langle A_\alpha:\alpha<\omega_2\rangle$, $E$, and $C\in M$ and $\delta=M\cap\omega_1\in E$, $|M\cap\omega_2|=\aleph_1$. We claim $\delta$ is an element of uncountably many $A_\alpha,\alpha\in M$. Since the cub $C$ divides $\omega_2$ into $\aleph_2$ disjoint intervals, $C\hspace{.03cm}\cap M$ divides $\omega_2\,\cap M$ into $\aleph_1$ disjoint intervals. Choose any one of these intervals $J$. There is a family $\mathcal{F}_J=\{F_\gamma:\gamma<\omega_1\}$ in $M$ consisting of $A_\alpha$’s indexed in the interval $J$, with diagonal union including $E$, mod ${\mathrm}{NS}_{\omega_1}$. Then there is a cub $D_J$ in $M$ disjoint from $E\setminus\nabla\mathcal{F}_J$. $D_J\cap M$ is unbounded in $M$, so $\delta=M\cap\omega_1\in D_J$, so $\delta\notin E\setminus\nabla\mathcal{F}_J$. Then $\delta\in \nabla\mathcal{F}_J$ so $\delta\in F_\gamma$ for some $\gamma\in M\cap\omega_1$ and therefore $\delta$ is in some $A_\xi$ with $\xi\in J$. We shall finish the proof that MM$(S)[S]$ implies **LCN**$(\aleph_1)$ in Section 4, but first let us note another advantage of stating MM$(S)[S]$ as a hypothesis is that we can often avoid front-loading to get collectionwise Hausdorffness, since **Axiom R** provides enough reflection. For example, \[312\] $(S)[S]$ implies a locally compact, hereditarily normal space is hereditarily paracompact if and only if it does not include a copy of $\omega_1$. As usual, we may assume the space does not include a perfect pre-image of $\omega_1$. The proof for that case in [@T] uses *P-ideal Dichotomy*, ${\mathbf{\mathop{\pmb{\sum}}}}$, $\aleph_1$-collectionwise Hausdorffness, and **Axiom R**. We can get all of these from MM$(S)[S]$. (Todorcevic [@To] proved that PFA$(S)[S]$ implies P-ideal Dichotomy; a proof was published in [@D2].) Similar considerations enable us to prove: \[thm312\] $(S)[S]$ implies a locally compact, normal, countably tight space is paracompact if and only if its separable closed subspaces are Lindelöf, and it does not include a copy of $\omega_1$. We thank Paul Larson for Lemma \[larson\] and several discussions concerning the material in this section. Next, we need to do some topology. Getting locally compact normal spaces collectionwise Hausdorff ============================================================== Let $X$ be a locally compact normal space and suppose $Y$ is a closed discrete subspace of $X$ of size $\aleph_1$. Then there is a locally compact normal space $X'$ with a closed discrete subspace $Y'$ of size $\aleph_1$, such that if $Y'$ is separated in $X'$, then $Y$ is separated in $X$, but each point in $Y'$ has character $\leq\aleph_1$. By Watson’s character reduction technique [@W], there is a discrete collection of compact subsets of $X$, $\mathcal{K}=\{K_y:y\in Y\}$, such that $y\in K_y$, and each $K_y$ has character $\leq\aleph_1$. Let $X'$ be the quotient of $X$ obtained by collapsing each $K_y$ to a point $y'$. This collapse is a perfect map, so preserves normality and local compactness, and it is clear that $\{y':y'\in Y\}$ is separated if and only if $\{K_y:y\in Y\}$ is separated, and that $Y$ is separated if $\{K_y:y\in Y\}$ is. \[lem47\] Suppose $X$ is a locally compact normal space of Lindelöf degree $\aleph_1$ with an uncountable closed discrete subspace. Then there is a continuous image of $X$ of weight $\aleph_1$ enjoying the same properties. Let $\mathcal{U}$ be an open cover of $X$ of size $\aleph_1$ with each member of $\mathcal{U}$ a cozero set with compact closure. Without loss of generality, assume that for each $x\in X$ there is a $U\in\mathcal{U}$ such that $x\in U$ and $U$ meets at most one element of a given closed discrete set $D$ of size $\aleph_1$. Also without loss of generality, assume $\mathcal{U}$ is closed under finite intersections. For each $U\in\mathcal{U}$, let $f_U:X\to[0,1]$ with $U=f^{-1}_U((0,1])$. Define an equivalence relation on $X$ by letting $x_0\!\!\sim\!\! x_1$ if $f_U(x_0)=f_U(x_1)$ for all $U\in\mathcal{U}$. Let $X/\!\!\sim$ be the quotient set, with $\pi:X\to X/\!\!\sim$ the projection. Topologize $X/\!\!\sim$ by taking as base all sets of form $\pi(U)$, $U\in\mathcal{U}$. Then $X/\!\!\sim$ is $\textup{T}_{3 \frac{1}{2}}$ and of weight $\leq\aleph_1$. To see the former, consider $X$ as embedded in $[0,1]^{C^*(X)}$ by $e(x)=(f(x))_{f\in C^*(X)}$. Let $p:[0,1]^{C^*(X)}\to[0,1]^{\{f_U:U\in\mathcal{U}\}}$ be given by $(x_f)_{f\in C^*(X)}\to(x_{f_U})_{U\in\mathcal{U}}$, i.e. $p$ projects onto only those coordinates in $C^*(X)$ which are $f_U$’s. Then $X/\!\!\sim\; =p\circ e(X)$. The projection map $\pi$ is closed, for let $F\subseteq X$ be closed and suppose $y\in\overline{\pi[F]}$. Claim $y\in\pi[F]$. $y\in\pi[U]$ for some $U\in\mathcal{U}$; note $\pi^{-1}(\pi[U])=U$ for if $\pi(x)\in\pi[U]$, $x\sim x_0$ for some $x_0\in U$. Then $f_V(x)=f_V(x_0)$ for every $V\in\mathcal{U}$. But $U=f_U^{-1}((0,1])$. Thus $f_U(x)=f_U(x_0)\in(0,1]$, which implies $x\in U$. So $\stackrel{ }{\overline{U}=\overline{\pi^{-1}\left(\pi[U]\right)}}$ is compact. Suppose $y\notin\pi[F]$. Then $y\notin\pi[F\cap\overline{U}]$, which is compact. Then $\pi[U]\setminus\pi[F\cap\overline{U}]$ is a neighborhood of $y$ disjoint from $\pi[F]$. Since $\pi$ is closed and $X$ is normal, $X/\!\!\sim$ is normal. It is clear that $\pi[D]$ is closed discrete. By continuity, $\pi[\overline{U}]\subseteq\overline{\pi[U]}$; $\pi[\overline{U}]$ is a closed set including $\pi[U]$, so including $\stackrel{ }{\overline{\pi[U]}}$, so $\pi[\overline{U}]=\overline{\pi[U]}$, so $X/\!\!\sim$ is covered by open sets with compact closures, so it is locally compact. \[lem48\] In any model obtained by forcing with a Souslin tree $S$, any locally compact normal space with a dense Lindelöf subspace has countable extent. Suppose $X_0$ is a locally compact normal space with an uncountable closed discrete subspace, which we may conveniently label as $\omega_1$, and a dense Lindelöf subspace $L$. Via normality, we can find a closed subspace $X_1$ with $\omega_1$ in its interior which is covered by $\aleph_1$-many open sets with compact closures. Without loss of generality, we may assume $X_1=\overline{{\mathrm}{int}\,X_1}$. $L$ is dense in ${\mathrm}{int}\,X_1$, so $L\cap({\mathrm}{int}\,X_1)$ is dense in $X_1$. Then $\stackrel{ }{\overline{L\cap {\mathrm}{int}\,X_1}\cap X_1}$ is a dense Lindelöf subspace of $X_1$. Thus, without loss of generality, we may as well assume our original space $X_0$ has a cover by $\aleph_1$-many open sets, each with compact closure. Without loss of generality, we may assume each is a cozero set and indeed is $\sigma$-compact. By Lemma \[lem47\], there is a continuous image of $X_0$ — call it $X$ — which is also locally compact, normal, has an uncountable closed discrete subspace, and has weight $\aleph_1$. Since both density and Lindelöfness are preserved by continuous functions, $X$ also has a dense Lindelöf subspace. Thus it suffices to find a contradiction for the special case in which the weight of our space is $\aleph_1$. For $\delta\in\omega_1$ and a cub $C\subseteq\omega_1$, let $\delta^+(C)$ denote the minimum element of $C$ greater than $\delta$. Without loss of generality, we may assume our cubs only consist of limit ordinals. For a cub $C$, we use ${\mathrm}{Fix}(C)$ to denote the set $\{\delta\in C:\text{order-type}(C\cap\delta)=\delta\}$. Let $S_\delta$ be the $\delta$th level of the Souslin tree. As usual, we work in the ground model and fix names $\dot{\mathcal{B}} = \{\dot{B}_\alpha:\alpha\in\omega_1\}$ for a base of $X$ consisting of open sets with compact closures. It is convenient to assume that $\{\dot{B}_n:n\in\omega\}$ is forced to have dense union. Again, we let $\omega_1$ label a closed discrete subspace and let $\{\dot{U}(\alpha,\xi):\xi\in\omega_1\}$ be a subset of $\dot{\mathcal{B}}$ forced to be a local base at $\alpha$. Without loss of generality, assume each $B_n$ is disjoint from the closed discrete set $\omega_1$. Fix a cub $C_0$ such that for each $\delta\in C_0$ and each $s\in S_\delta$, $s$ decides all equations of the form $\dot{B}_\alpha\cap\dot{B}_\beta =\emptyset$, for $\alpha,\beta<\delta$. Also assume that for each $s\in S_\delta\;(\delta\in C_0)$ and each $\xi,\beta\in\delta$, there is an $\alpha\in\delta$ such that $s$ forces that $\dot{U}(\xi,\beta)=\dot{B}_\alpha$. It is convenient to assume that $S$ is $\omega$-branching (specifying any infinite maximal antichain above each element would serve the same purpose). We can use $C_1={\mathrm}{Fix}(C_0)$ to define a partition $\dot{f}$ of $\omega_1$ so that for each $\xi\in\omega_1$ and each $s\in S_{\xi^+(C_1)}$, $s^\smallfrown j$ forces that $\dot{f}(\xi)=j$. Now we choose two (names of) functions $\dot{h}_1$ and $\dot{h}_2$ witnessing normality as follows: - For each $j\in\omega$ and each $i\in 2$, let $\dot{W}^i_j=\bigcup\{\dot{U}(\xi,h_i(\xi)):\xi\in \dot{f}^{-1}(j)\}$, - the $\{\dot{W}^1_j:j\in\omega\}$ form a discrete family, - the closure of $\dot{W}^2_j$ is included in $\dot{W}^1_j$. Choose any countable elementary submodel $M$ with all the above as members of $M$, such that $\delta = M\cap\omega_1$ is an element of $C_1$. We know that there is a name of an integer $\dot{J}_\delta$ satisfying that it is forced that $\dot{U}(\delta,0)\cap\dot{W}_j$ is empty for all $j\geq\dot{J}_\delta$. Choose any $s\in S$ of height at least $\delta^+(C_1)$ that decides a value $J$ for $\dot{J}_\delta$. Let $\bar{s} = s\upharpoonright\delta^+(C_1)$. Notice that $\bar{s}$ decides the truth value of the equation “$\dot{U}(\delta,0)\cap\dot{B}_\alpha=\emptyset$”, for all $\alpha\in M$. For each $n,j\in\omega$, $s$ and hence $s\upharpoonright\delta$ forces that the closure of $\dot{W}^2_j\cap \dot{B}_n$ is included in $\dot{W}^1_j$. By elementarity and compactness, this implies there is a finite $\dot{F}_{j,n}\subseteq\delta$ such that $s\upharpoonright\delta$ forces that $\dot{W}^2_j\cap \dot{B}_n\subseteq\bigcup\{\dot{B}_\eta:\eta\in \dot{F}_{j,n}\}\subseteq\dot{W}^1_j$. But now $\bar{s}$ forces $\dot{U}(\delta,0)\cap(\bigcup\{\dot{B}_\eta:\eta\in \dot{F}_{j,n}\})$ is empty for all $n$ and all $j\geq J$. On the other hand, fix any $j\geq J$ and consider what $\bar{s}^\smallfrown j$ is forcing. This forces that $\dot{f}(\delta)=j$ and that $\delta\in W^2_j$, and so $\delta$ is in the closure of the union of the sequence $\{\dot{U}(\delta,0)\cap(\bigcup\{\dot{B}_\eta:\eta\in F_{j,n}\}):n\in\omega\}$. This is a contradiction. \[cor410\] In any model obtained by forcing with a Souslin tree, if $X$ is locally compact normal, $D$ is a closed discrete subspace of $X$ of size $\aleph_1$ and $\{U_\alpha:\alpha\in\omega_1\}$ are open sets with compact closures, then for any countable $T\subseteq\omega_1$, $\stackrel{}{\overline{\bigcup\{U_\alpha:\alpha\in T\}}}\cap\; D$ is countable. ${\bigcup\{{\overline{U}_\alpha}:\alpha\in T\}}$ is dense in $\overline{\bigcup\{U_\alpha:\alpha\in T\}}$, which is locally compact normal. Getting back to the proof of \[thm41\], let us assume we are in a model of ${\mathrm}{MM}(S)$ and that we have an $S$-name $\dot{X}$ for a locally compact normal space, with a closed discrete subspace labeled as $\omega_1$, with each of its points having character $\aleph_1$. Let us note that it follows from character reduction and Lemma \[LT1\] that if there is a discrete expansion of $\omega_1$ into compact $G_\delta$’s, then $\omega_1$ will have a separation. In fact, even more, it is shown in [@T3 Theorem 12] that if $\omega_1$ is forced to have an expansion by compact $G_\delta$’s that is $\sigma$-discrete, then $\omega_1$ will be separated. Since our proof is by contradiction, we will henceforth assume that it is forced (by the root of $S$) that there is no expansion of $\omega_1$ into a $\sigma$-discrete family of compact $G_\delta$’s. For each $\xi,\alpha\in\omega_1$, let $\dot{U}(\xi,\alpha)$ be the name of the $\alpha$th neighbourhood from a local base for $\xi$ with $\dot U(\xi,0)$ forced to have compact closure. Corollary \[cor410\], and the fact that $S$ is ccc, ensure that for each $\delta\in \omega_1$, every element of $S$ forces that $\omega_1 \cap \overline{\bigcup\{\dot{U}(\xi,0):\xi<\delta\}}$ is bounded by $\gamma$ for some $\gamma\in \omega_1$. Therefore there is a cub $C_0$ such that without loss of generality, we can assume that each of the following is forced by each element of $S$: 1. for each $\delta\in C_0$, $\omega_1\cap \stackrel{}{\overline{\bigcup\{\dot{U}(\xi,0):\xi<\delta\}}}$ is included in $\delta^+(C_0)$, 2. for all $\beta\neq \xi$ in $\omega_1$, $\beta\notin \dot{U}(\xi,0)$, 3. for all $\xi,\alpha\in \omega_1$ $\dot{U}(\xi,\alpha)\subseteq \dot{U}(\xi,0)$ and has compact closure, 4. for each limit $\delta\in \omega_1$, the sequence $\{\dot{U}(\xi,\alpha):\alpha<\delta\}$ is a *regular filter*, i.e. each finite intersection of these includes the closure of another. For an $S$-name $\dot{h}$ of a function from $\omega_1$ to $\omega_1$, let $\dot{U}(\xi,\dot{h})$ stand for $\dot{U}(\xi,\dot{h}(\xi))$. For limit $\delta$, let $\dot{Z}(\xi,\delta)$ denote the $S$-name of the compact $G_\delta$ equal to $\bigcap\{\dot{U}(\xi,\alpha):\alpha<\delta\}$. For a cub $C$ and ordinal $\xi$, we also use $\dot{Z}(\xi,C)$ as an abbreviation for $\dot{Z}(\xi,\xi^+(C))$. Fix an enumeration $\{C_\gamma:\gamma\in\omega_2\}$ for a base for the cubs on $\omega_1$ (each containing only limit ordinals), chosen so that $C_0$ is as above and for $0<\lambda \in \omega_2$, $C_\lambda \subseteq {\mathrm}{Fix}(C_0)$ and $C_\lambda\setminus{\mathrm}{Fix}(C_\gamma)$ is countable for all $0\leq\gamma<\lambda$. We can do this by taking diagonal intersections, since **SRP** implies $2^{\aleph_1}=\aleph_2$. For each $\delta\in C_0$, let $\beta(\delta) = \delta^+(C_0)$. Since $\dot{Z}(\xi,C_\gamma)\subseteq \dot{U}(\xi,C_\gamma)$ for all $\xi\in \omega_1$ for all $\delta\in C_\gamma$, $\beta(\delta)<\delta^+(C_\gamma)$, and so it is forced that: $$\overline{\bigcup\{\dot{Z}(\xi,C):\xi<\delta\}} \cap\omega_1\subseteq\beta(\delta).$$ We can also assume that for all cubs $C\subseteq C_0$, there is an $S$-name $\dot A$, that is forced to be a stationary subset of ${\mathrm}{Fix}(C)$ satisfying: $$(\forall s\in S)(\forall \delta) ~~ s\Vdash \left(\delta\in \dot A \ \Rightarrow (\exists\alpha\in[\delta,\beta(\delta)])\; \alpha\in\overline{\bigcup\{\dot{Z}(\xi,C):\xi<\delta\}}~\right).$$ The reason we can make this assumption is that we are assuming there is no $\sigma$-discrete expansion of $\omega_1$ by compact $G_\delta$’s. If, in the extension, the set $A = \{ \delta : \overline{\bigcup\{\dot{Z}(\xi,C):\xi<\delta\}} \not\subseteq \delta\}$ were not stationary, then there would be a $\lambda\in \omega_2$ such that $A\cap C_\lambda$ is empty. Since the cub $C_\lambda$ divides $\omega_1$ into countable pieces, we see that we can expand the points in $\omega_1$ into a $\sigma$-discrete collection of compact $G_\delta$’s. For each $\lambda \in \omega_2$, let $\dot A_\lambda$ denote the name of the stationary set just described. For any $B\subseteq\omega_1$, we will write $$\alpha\in\langle\dot{Z}(\xi,C):\xi<\delta\rangle'$$ to mean that $\alpha$ is a limit point of that sequence of sets. Fix any function $e:S\to\omega$ with the property that for all $\delta\in\omega_1$, $e\restriction S_\delta$ is one-to-one. For an ordinal $\gamma\in\omega_2$, we use $\dot{f}_\gamma$ for the $S$-name of the function from $\omega_1$ into $\omega$ given by the property that each $s\in S_{\xi^+(C_\gamma)}$ forces that $\dot{f}_\gamma(\xi)=e(s)$. Thus $\dot{f}_\gamma$ partitions $\omega_1$ into a discrete collection of countably many closed subsets. Then let $\{\dot{W}(\gamma,n):n\in\omega\}$ be a discrete collection of open sets separating the $\dot{f}_\gamma^{-1}(n)$’s. Fix $n\in\omega$. By normality, there is an open $\dot{V}_n$ such that $S$ forces $\dot{f}^{-1}_\gamma(n)\subseteq\dot{V}_n\subseteq\dot{\overline{V}}_n\subseteq\dot{W}(\gamma,n)$. For each $\xi\in\dot{f}^{-1}_\gamma(n)$, there is an $\alpha_\xi\in\omega_1$ such that $S$ forces $\dot{U}(\xi,\alpha_\xi)\subseteq\dot{V}_n$. Let $\zeta_n(\gamma)\in\omega_2$ be such that for $\xi\in\dot{f}^{-1}_\gamma(n), \xi<\rho\in C_{\zeta_n(\gamma)}$ implies $\alpha_\xi<\rho$. Then $S$ forces $\{\dot{Z}(\xi, C_{\zeta_n(\gamma)}):\xi\in\dot{f}_\gamma^{-1}(n)\}\subseteq\dot{V}_n$. We then can find a $C_{\zeta(\gamma)}$ included in each $C_{\zeta_n(\gamma)}$ such that for every $n\in\omega$, $S$ forces $\{\dot{Z}(\xi,C_{\zeta(\gamma)}):\zeta\in\dot{f}^{-1}_\gamma(n)\}\subseteq \dot{V}_n$. Thus $$\overline{\bigcup\{\dot{Z}(\xi,C_{\zeta(\xi)}):\xi\in f_\gamma^{-1}(n)\}} \subseteq\dot{W}(\gamma,n).$$ Then we can get a $\zeta(\gamma)$ that works for all $n$. By recursion on $\gamma\in\omega_2$, we can choose $\zeta(\gamma)\geq\gamma$ as above, so that the sequence $\{\zeta(\gamma):\gamma\in\omega_2\}$ is strictly increasing. For each $\gamma$, we have the $S$-name $\dot A_{\zeta(\gamma)}$ as above. It is immediate that $A_\gamma = \{ \delta : (\exists s\in S) s\Vdash \delta\in \dot A_{\zeta(\gamma)}\}$ is a stationary set. In other words, $\delta \in A_\gamma$ implies there is some $s\in S$ and $\eta \in [\delta,\beta(\delta)]$ such that $s\Vdash \eta \in \langle\dot{Z}(\xi, C_{\zeta(\gamma)}):\xi\in\delta\rangle'$. By **SCC** and \[larson\] we may assume there is an elementary submodel $M$ of some $\langle H(\theta),\{\langle \gamma,\zeta(\gamma), A_\gamma\rangle:\gamma\in\omega_2 \}\rangle$, with $M\cap\omega_1=\delta<\omega_1$, $|M\cap\omega_2|=\aleph_1$, and an uncountable $\{\gamma_\alpha:\alpha\in\omega_1\}\subseteq M\cap\omega_2$, so that $\delta\in A_{\gamma_\alpha}$ for all $\alpha\in\omega_1$. For each $\alpha\in\omega_1$ choose $s_{\alpha}\in S$, $\eta_\alpha\in[\delta,\beta(\delta)]$ such that $s_\alpha\Vdash \eta_\alpha\in\langle\dot{Z}(\xi,C_{\zeta(\gamma_\alpha)}): \xi\in\delta\rangle'$. We may assume $s_\alpha$ is on a level at least as high as $\delta^+(C_{\gamma_\alpha})$. We may also assume that if $\alpha<\beta\in\omega_1$, then $\gamma_\alpha<\gamma_\beta$. We may also assume that the height of $s_\alpha$ is less than the height of $s_\beta$, for $\alpha<\beta$, so that $\{s_\alpha:\alpha\in\omega_1\}$ is an uncountable subset of $S$. Therefore there is an $\eta\in [\delta,\beta(\delta)]$ such that $L = \{ \alpha : \eta_\alpha = \eta\}$ is uncountable. Also, as is well-known for Souslin trees, there is an $\bar{s}\in S$, such that $\{s_\alpha :\alpha\in L\}$ includes a dense subset of $\{s\in S:\bar{s}<s\}$. By passing to an uncountable subset, we may assume that $\bar{s} <s_\alpha$ for all $\alpha\in L$ and that $\bar{s}$ is on a level above $\delta$. Similarly we may assume that for all $\xi,\rho<\delta$, $\bar{s}$ has decided the statement $$\dot{U}(\eta,0)\cap\dot{Z}(\xi,\rho)\neq\emptyset\quad \text{ for all }\xi,\rho<\delta.$$ Now choose any $\alpha\in L$ (e.g.the least one), and then choose an infinite sequence $\{\beta_l:l\in\omega\}\subseteq L\setminus(\alpha+1)$ so that $s_{\beta_l}\upharpoonright \delta^+({C_{\gamma_\alpha}})$ are all distinct. For each $l$, let $e(s_{\beta_l}\upharpoonright\delta^+({C_{\gamma_\alpha}})~)=n_l$. **Main Claim:** $\quad\bar{s}\Vdash (\forall l\in\omega)\left(\dot{W}(\gamma_\alpha,n_l)\cap\dot{U}(\eta,0)\neq 0\right).$ Once this claim is proven we are done, because we then have that $\bar{s}$ forces that $\dot{U}(\eta,0)$ cannot have compact closure, because it meets infinitely many members of the discrete family $\{\dot{W}(\gamma_\alpha,n):n\in\omega\}$. To prove the claim, first note that there is a tail of $C_\zeta(\gamma_{\beta_l})\cap\delta$ included in $C_{\zeta(\gamma_{\alpha})}$. To see this, recall $C_{\zeta(\gamma_\alpha)}\setminus{\mathrm}{Fix}(C_\zeta(\gamma_{\beta_l}))$ is countable, so some tail of ${\mathrm}{Fix}(C_\zeta(\gamma_{\beta_l}))$ is included in $C_{\zeta(\gamma_\alpha)}$. By elementarity, since $\gamma_\alpha$ and $\gamma_\beta$ are in $M$, a tail of ${\mathrm}{Fix}(C_\zeta(\gamma_{\beta_l}))\cap M$ is included in $C_{\zeta(\gamma_\alpha)}\cap M$, so a tail of ${\mathrm}{Fix}(C_\zeta(\gamma_{\beta_l}))\cap\delta$ is included in $C_{\zeta(\gamma_\alpha)}$. Since there is a tail of $C_{\zeta(\gamma_{\beta_l})}\cap \delta$ included in $C_{\zeta(\gamma_\alpha)}$, $\dot{Z}(\xi, C_{\zeta(\gamma_{\beta_l})})\subseteq\dot{Z}(\xi,C_{\zeta(\gamma_\alpha)})$ for each $\xi<\delta$ (at least on a tail — which is all that matters for limits above $\delta$). Then $s_{\beta_l}$ forces that $\eta $ is a limit of the sequence $$\langle\dot{Z}(\xi, C_{\zeta(\gamma_\alpha)}): \xi\in\delta\text{ and }\dot{f}_{\gamma_\alpha}(\xi)= n_l\rangle.$$ Of course this means that $s_{\beta_l}$ forces that $\dot{U}(\eta,0)$ meets $\dot{Z}(\xi, C_{\zeta(\gamma_\alpha)})$ for cofinally many $\xi<\delta$ such that $s_{\beta_l}\upharpoonright\gamma_\alpha\Vdash \dot{f}_{\gamma_\alpha}(\xi)=n_l$. But $\bar{s}$ has already decided the value of $\dot{f}_{\gamma_\alpha}\upharpoonright\delta$, and $\bar{s}$ already forces $\dot{U}(\eta,0)\cap\dot{Z}(\xi, C_{\zeta(\gamma_\alpha)})\neq\emptyset$ whenever $s_{\gamma_\beta}$ does. In particular then, $\bar{s}$ forces there is a $\xi$ with $\dot{f}_{\gamma_\alpha}(\xi)=n_l$ (and so $\dot{Z}(\xi, C_{\zeta(\gamma_\alpha)})\subseteq\dot{W}(\gamma_\alpha,n_l)$) and $\dot{U}(\eta,0)\cap\dot{Z}(\xi, C_\zeta(\gamma_\alpha))\neq\emptyset$. For the record, let us state what we have accomplished: $(S)[S]$ implies **LCN**$(\aleph_1)$. There is a model of $(S)[S]$ in which **LCN** holds, *i.e. every locally compact normal space is collectionwise Hausdorff.* Large Cardinals and the MOP =========================== In [@DT2] we showed that large cardinals are not required to obtain the consistency of every *locally compact perfectly normal space is paracompact*. It is interesting to see which other PFA$(S)[S]$ results can be obtained without large cardinals. The standard method used was pioneered by Todorcevic in [@To3] and given several applications in [@D], all in the context of PFA results. In the context of PFA$(S)[S]$, it is referred to in [@To] and actually carried out in [@DT1] for a version of [P-ideal Dichotomy]{} and for ****. It is routine to get additionally that such models are of form MA$_{\omega_1}(S)[S]$ by interleaving additional forcing. In [@DT2] we pointed out that such methods can give models in which in addition the following holds: **${\mathbf{\mathop{\pmb{\sum}}}}^{\bm{-}}$(sequential)** In a compact sequential space, each locally countable subspace of size $\aleph_1$ is $\sigma$-discrete. A modification of such a proof produces a model in which the following proposition (see [@FTT]) holds: **${\mathbf{\mathop{\pmb{\sum}}}}$(sequential)** Let $X$ be a compact sequential space. Let $Y\subseteq X$, $|Y|=\aleph_1$. Suppose $\{W_\alpha\}_{\alpha\in\omega_1}$, $\{V_\alpha\}_{\alpha\in\omega_1}$ are open subsets of $X$ such that: - $W_\alpha\subseteq\overline{W_\alpha}\subseteq V_\alpha,$ - $|V_\alpha\cap Y|\leq\aleph_0$, - $Y\subseteq\bigcup\{W_\alpha:\alpha\in\omega_1\}$. Then $Y$ is $\sigma$-closed discrete in $\bigcup\{W_\alpha:\alpha\in\omega_1\}$. Without the parenthetical “sequential”, ${\mathbf{\mathop{\pmb{\sum}}}}^-$ and ${\mathbf{\mathop{\pmb{\sum}}}}$ refer to the corresponding propositions obtained by replacing “sequential” by countably tight”, which follow from their sequential versions if one has **Moore-Mrówka** Every compact countably tight space is sequential. It follows easily from **Moore-Mrówka** that *locally compact countably tight spaces are sequential*. A proof of **Moore-Mrówka** from PFA$(S)[S]$ is sketched in [@To] and the author remarks that, by the usual methods, large cardinals are not necessary. Thus, one can obtain a model of MA$_{\omega_1}(S)[S]$ in which, for example, both **PPI** and **${\mathbf{\mathop{\pmb{\sum}}}}$** hold, without the need for large cardinals. Working in such a model, we can establish the following proposition, the conclusion of which was proved from PFA$(S)[S]$ in [@To] and asserted to be obtainable without large cardinals. If ZFC is consistent, it’s consistent to additionally assume that locally compact, hereditarily normal, separable spaces are hereditarily Lindelöf. Let $X$ be such a space. By \[lem48\] $X$ has countable spread. So does its one-point compactification $X^*$, which hence is countably tight [@A2]. If $X$ were not hereditarily Lindelöf, it would include a right-separated subspace $\{x_\alpha:\alpha\in\omega_1\}$. Let $\{V_\alpha:\alpha\in\omega_1\}$ be open sets witnessing right-separation. Let $x_\alpha\in W_\alpha\subseteq\overline{W_\alpha}\subseteq V_\alpha$, with $W_\alpha$ open and $\overline{W_\alpha}$ compact. Applying ${\mathbf{\mathop{\pmb{\sum}}}}$ to $X^*$, we see that $\{x_\alpha:\alpha\in\omega_1\}$ is $\sigma$-closed discrete in $W=\bigcup\{W_\alpha:\alpha\in\omega_1\}$. But $W$ is locally compact, separable, and hereditarily normal, so this contradicts \[lem48\]. Also without large cardinals we obtain: \[thm510\] If ZFC is consistent, it is consistent to additionally assume that each hereditarily normal perfect pre-image of $\omega_1$ includes a copy of $\omega_1$. Using ${\mathbf{\mathop{\pmb{\sum}}}}$ and **PPI**, we can carry out the proof of Theorem \[thm34\] above. We also have: \[thm511\] If ZFC is consistent, it is consistent to assume that every locally compact, first countable, hereditarily normal space with Lindelöf number $\leq\aleph_1$ not including a copy of $\omega_1$ is paracompact. We use the model of \[thm510\]. In [@T] the second author asserted the following, but under PFA$(S)[S]$ instead of MM$(S)[S]$, which we now see should have been used. \[lem512\] $(S)[S]$ implies that if $X$ has Lindelöf number $\leq\aleph_1$ and is locally compact, normal, and does not include a perfect pre-image of $\omega_1$, then $X$ is paracompact. In addition to the topological properties mentioned, the proof used ${\mathbf{\mathop{\pmb{\sum}}}}$ and that the space was $\aleph_1$-collectionwise Hausdorff. For the purposes of \[thm511\], however, we get $\aleph_1$-collectionwise Hausdorff just from the Souslin forcing, since the space is first countable. MM$(S)[S]$ is also relevant for questions concerning the Baireness of $C_k(X)$, for locally compact $X$ (see [@GM; @MN; @T5]). A ***moving off** collection* for a space $X$ is a collection ${\mathcal}{K}$ of non-empty compact sets such that for each compact $L$, there is a $K \in {\mathcal}{K}$ disjoint from $L$. A space satisfies the **Moving Off Property** (MOP) if each moving off collection includes an infinite subcollection with a discrete open expansion. $C_k(X)$, for a space $X$, is the collection of all continuous real-valued functions on $X$, considered as a subspace of the compact-open topology on the Cartesian power $X^{\mathbb{R}}$. A locally compact space $X$ satisfies the MOP if and only if $C_k(X)$ is Baire, i.e. satisfies the Baire Category Theorem. \[lem6\] Locally compact, paracompact spaces satisfy the MOP. \[thm35\] ${\mathrm}{MM}(S)[S]$ implies that normal spaces satisfying the MOP are paracompact if they are: - locally compact, countably tight, and hereditarily normal, or - first countable and hereditarily normal, or - locally compact, countably tight with Lindelöf number $\leq\aleph_1$, or - first countable, with Lindelöf number $\leq\aleph_1$, or - locally compact, countably tight, and countable sets have Lindelöf closures. These all follow easily from \[thmConj2\], \[thm312\], and **Moore-Mrówka**, using: In a sequential space, countably compact subspaces are closed. Countably compact spaces satisfying the MOP are compact. \[lem510\] First countable spaces satisfying the MOP are locally compact. The one-point compactification of a locally compact space $X$ is countably tight if and only if $X$ does not include a perfect pre-image of $\omega_1$. If they have the MOP, sequential spaces do not include copies of $\omega_1$, so (1) follows from \[312\]. (2) follows from (1) plus \[lem510\]. (3) follows from \[lem512\] plus \[thmConj2\]. (4) follows from (3) plus \[lem510\]. (5) follows from \[thm312\], \[thmConj2\] and Balogh’s Lemma above. In the special case of a space with the MOP, we have: \[thm513\] If ZFC is consistent, then it is consistent to additionally assume that first countable normal spaces satisfying the MOP and with Lindelöf number $\leq\aleph_1$ are paracompact. Such a space is locally compact and does not include a perfect pre-image of $\omega_1$. MA$_{\omega_1}$ gives counterexamples for the conclusions of \[thm35\] and \[thm513\]. See e.g. [@T5]. If ZFC is consistent, then it is consistent to assume that first countable hereditarily normal, locally connected spaces satisfying the MOP are paracompact. The extra ingredient is that the local connectedness will enable us to decompose the space into a sum of pieces with Lindelöf number $\leq\aleph_1$. More precisely, A space $X$ is of **Type I** if $X=\bigcup\{X_\alpha:\alpha\in\omega_1\}$, where each $X_\alpha$ is open, $\alpha<\beta$ implies $\overline{X}_\alpha\subseteq X_\beta$, and each $X_\alpha$ is Lindelöf. In [@T], it is shown on page 104 that, assuming ${\mathbf{\mathop{\pmb{\sum}}}}$ and hereditary $\aleph_1$-collectionwise Hausdorffness for a locally compact hereditarily normal space not including a perfect pre-image of $\omega_1$ that the closure of a Lindelöf subspace is Lindelöf. Then we quote: If $X$ is locally compact, locally connected, and countably tight, then $X$ is a topological sum of Type I spaces if and only if every Lindelöf subspace of $X$ has Lindelöf closure. Since a topological sum of paracompact spaces is paracompact, this will complete the proof of the Theorem. It may be of interest that **SRP** implies a weaker version of the conclusion of Theorem \[thm35\].2. implies every first countable, monotonically normal space satisfying the MOP is paracompact. Suppose $S$ is a first countable stationary subspace of some regular cardinal. Then each $s\in S$ is an $\omega$-cofinal ordinal. Each $s\in S$ is either isolated in $S$ or is a limit of some subset of $S$. By first countability, in the latter case, each such $s$ is a limit of a sequence of elements of $S$. Suppose not. Then by [@BR] the space includes a copy of a stationary subset of some regular cardinal. By [@J 37.18] **SRP** implies that that stationary set includes a copy of a closed unbounded subset of $\omega_1$. That copy is closed, countably compact but not compact, contradicting the MOP. We conjecture th answer is positive. Large cardinals would be necessary to refute the existence of such a space, since an example can be constructed from the failure of the Covering Lemma for the Core Model K, which entails the consistency of measurable cardinals. We thank Peter Nyikos for referring us to [@G], where that failure is used to construct a locally compact, locally countable, normal, non-paracompact space $X$ on $\kappa^+\times\omega_1$, where $\kappa^+$ is the successor of a singular strong limit cardinal of countable cofinality, such that the spaces $X_\alpha=\alpha\times\omega$ are metrizable for all $\alpha\in\kappa^+$. It follows that closed subspaces of $X$ of size $\leq 2^{\aleph_0}$ are locally compact and metrizable, so satisfy the MOP by \[lem6\]. On the other hand, If a Hausdorff space $Z$ is locally countable, locally compact, and closed subspaces of $\leq 2^{\aleph_0}$ have the MOP, then $Z$ has the MOP. It follows that $X$ has the MOP.\ With MM$(S)[S]$ we have: $(S)[S]$ implies that if $X$ is normal, locally compact, locally countable, and closed subspaces of size $\leq 2^{\aleph_0}$ are metrizable, then $X$ is metrizable. By the preceding proof, $X$ has the MOP. By \[thm35\], to get that $X$ is paracompact, it suffices to show that countable subspaces of $X$ have Lindelöf closures. But if $Y$ is a countable subset of $X$, $|\overline{Y}|\leq 2^{\aleph_0}$ and hence is separable metrizable and hence Lindelöf. Once we have $X$ paracompact, it follows that $X$ is a topological sum of $\sigma$-compact subspaces. But each of these has size $\leq 2^{\aleph_0}$ and so is metrizable. **Axiom R** precludes stationary non-reflecting sets of $\omega$-cofinal ordinals in $\omega_s$, and hence the locally compact, $\aleph_1$-collectionwise Hausdorff ladder system space built on such a set; we can therefore ask: Examples ======== A question left open in [@LT1] is whether, as was shown for adjoining $\aleph_2$ Cohen subsets of $\omega_1$ in [@T1], forcing with a Souslin tree would make normal spaces of character $\aleph_1$ $\aleph_1$-collectionwise Hausdorff. We shall show that the answer is negative by showing: $_{\omega_1}(S)[S]$ implies that there is a normal non-$\aleph_1$-collectionwise Hausdorff space of character $\aleph_1$. Let $S\subseteq 2^{<\omega_1}$ be a coherent Souslin tree. Fix a family ${\{a_s:s\in S\}\subseteq[\omega]^\omega}$ so that for $s<t\in S$, $a_t\subseteq^* a_s$ and for each $\gamma\in\omega_1$, $\{a_s:s\in S_\gamma\}$ is pairwise disjoint. For each limit $\delta\in\omega_1$, let $L_\delta\in\delta^\omega$ be a strictly increasing function with range cofinal in $\delta$ consisting of successor ordinals. For $a\subseteq\omega,$ let $L[a]=\{L_\delta(n):n\in a \}$. The generic $g$ for $S$ will enable us to define the required topology on the set $\omega_1$. We declare each successor ordinal to be isolated. For each limit $\delta$, the neighborhood filter for $\delta$ will be $\{L_\delta[a_s]\cup\{\delta\}:s\in g \}$. The set $C_0$ of limit ordinals is then a closed discrete set. By pressing down, we see that $C_0$ cannot be separated. It remains to show that the space is normal. It suffices to show that if $f$ is an $S$-name of a function from $C_0$ to $2$, then there is a neighborhood assignment $\{\dot{U}_\delta:\delta\in C_0\}$ and a cub $C_1$, such that for each $\alpha<\delta\in C_1$, $S$ forces that if $\dot{f}(\alpha)\neq\dot{f}(\delta)$, then $\dot{U}(\alpha)$ and $\dot{U}(\delta)$ are disjoint. There is a cub $C_1\subseteq C_0$ so that for all $\delta\in C_1$ and $\alpha<\delta_1$ each $s\in S_\delta$ decides the value of $\dot{f}(\alpha)$. For each $\delta\in C_0$, let $\delta^+$ denote the minimal element of $C_1$ above $\delta$, and choose a function $f_\delta:\omega\to 2$ so that for each $s\in S_{\delta^+}$ and each $n\in a_s$, $s$ forces $\dot{f}(\delta)=f_\delta(n)$. We will define an integer $n_\delta$ such that the value of $\dot{U}_\delta$ is forced by $s\in S_{\delta^+}$ to equal $\{\delta\}\cup L_{\delta}[a_s\setminus n_\delta]$. The sequence of functions $\{f_\delta:\delta\in C_0\}$ will be in the MA$_{\omega_1}(S)$ model. Let $\mathcal{Q}$ be the poset of partial functions $h$ from $\omega_1$ into $2$ such that ${h=^*\bigcup\{f_\delta\circ L_\delta^{-1}: \delta\in H \}}$, for some $H\in[C_0]^{<\omega}$. $\mathcal{Q}$ is ordered by extension. We claim that in ZFC, $\mathcal{Q}$ is ccc. If so, there will be a generic for $\aleph_1$ dense subsets of $\mathcal{Q}$ in a model of MA$_{\omega_1}(S)$. Let $\mathcal{H}=\{(h_\alpha,H_\alpha):\alpha\in\omega_1 \}$ be a subset of $\mathcal{Q}\times[C_0]^{<\omega}$, where $h_\alpha=\bigcup\{f_\delta\circ L_\delta^{-1}:\delta\in H_\alpha \}$. Choose any countable elementary submodel $M$ with $\mathcal{Q}$ and $\mathcal{H}$ in $M$. Let $\delta=M\cap\omega_1$ and $H_\delta\cap M=H$ and $H_\delta\setminus M=\{\delta_i:i<l\}$. We may assume that $\delta_0=\delta$ and then choose $\alpha_0\in M$ so that $H\subseteq\alpha_0$ and $L_{\delta_i}\cap\delta\subseteq\alpha_0$, for $0<i<l$. Notice that $h_\delta\!\!\upharpoonright\!\!\alpha$ is an element of $M$, for all $\alpha\in M$. In $M$, recursively choose $\alpha_0<\alpha_1<\cdots$ so that $h_{\alpha_{n+1}}\!\!\upharpoonright\!\!\alpha_n = h_\delta\!\!\upharpoonright\!\!\alpha_n$ and dom$(h_{\alpha_{n+1}})\subseteq\alpha_{n+2}$. With $\beta=\sup_n\alpha_n<\delta$, we have that there is an $n\in\omega$ such that $h_\delta\!\!\upharpoonright\!\!\beta = h_\delta\!\!\upharpoonright\!\!\alpha_n$. It follows that $h_\delta\!\!\upharpoonright\!\!\alpha_n\subseteq h_{\alpha_{n+1}}$, and so $h_\delta$ and $h_{\alpha_{n+1}}$ are compatible members of $\mathcal{Q}$. MA$_{\omega_1}(S)$ implies there is a generic for $\mathcal{Q}$ that adds a function $h$ from $\omega_1$ to $2$ that mod finite extends $f_\delta\circ L_\delta^{-1}$, for all $\delta\in C_0$. Now define $n_\delta$ to be chosen so that $h$ actually extends $f_\delta\circ L_\delta^{-1}[\omega\setminus n_\delta]$. Suppose $\alpha<\delta$, with $\delta\in C_1$, and let $s\in S_{\delta^+}$. Then $s$ forces that $f_\delta\circ L^{-1}_\delta=\dot{f}_\delta$ on $a_s$, and similarly, $s\!\!\upharpoonright\!\alpha^+$ forces that $f_\alpha\circ L^{-1}_\alpha=\dot{f}_\alpha$ on $a_{s\upharpoonright\alpha^+}$. Also, $h$ agrees with $f_\delta\circ L_\delta^{-1}$ on $a_s\setminus n_\delta$ and with $f_\alpha\circ L_\alpha^{-1}$ on $a_{s\upharpoonright\alpha^+}\setminus n_\alpha$. Thus if $\beta\in L_\delta[a_s\setminus n_\delta]\cap L_{\alpha}[a_{s\upharpoonright\alpha^+}\setminus n_\alpha]$, then $h(\beta)=\dot{f}(\alpha)=\dot{f}(\delta)$. This completes the proof that the space is normal. The strategy attempted in [@T3] was to expand a closed discrete subspace of a locally compact normal space to a discrete collection of compact $G_\delta$’s. There are limitations on such an approach, given by the following example. MA$_{\omega_1}(S)[S]$ implies there is a locally compact space of character $\aleph_1$ which includes a normalized closed discrete set which does not have a normalized discrete expansion by compact $G_\delta$’s. We modify the previous example. Let $\mathcal{A}_s$ denote the Boolean subalgebra of $\mathcal{P}(\omega)$ generated by $[\omega]^{<\omega}\cup\{a_s:s\in S\}$. In the forcing extension by $S$, let $x_g$ denote the member of the Stone space $\mathcal{S}(\mathcal{A}_s/\text{FIN})$ containing $\{a_s:s\in g\}$. In the forcing extension, our space has the base set $(\omega_1\setminus C_0)\cup(C_0\times\mathcal{S}(\mathcal{A}_s))$. The points of $\omega_1\setminus C_0$ are isolated. For each $\delta\in C_0$ and $x\in\mathcal{S}(\mathcal{A}_s/\text{FIN})$, a neighborhood of $(\delta,x)$ must include $U_\delta(a)=L_\delta[a]\cup(\{\delta\}\times a^*)$ for some $a\in x$, where $a^* = \{p\in\mathcal{S}(\mathcal{A}_\mathcal{S}):a\in p\}$. Notice that $U_\delta(a)$ is disjoint from $\{\gamma\}\times\mathcal{S}(\mathcal{A}_s/\text{FIN})$, for all $\gamma\neq\delta$. It follows immediately that the sequence $D=\{(\gamma,x_g):\delta\in C_0\}$ is a closed discrete subset. It also follows from the proof of the normality of the previous example that $D$ is normalized. Now we show that $D$ does not have a normalized discrete expansion by compact $G_\delta$’s, indeed by any $G_\delta$’s. Assume that $\{\dot{Z}_\delta:\delta\in C_0\}$ is a sequence of $S$-names so that $\dot{Z}_\delta$ is forced to be a $G_\delta$ containing $(\delta,x_g)$. There is a cub $C_1$ such that for each $\alpha\in C_0$ and each $s\in S_{\alpha^+}$ (again, $\alpha^+$ is the minimal element of $C_1$ above $\alpha$), $s$ forces that $\dot{Z}_\alpha$ contains $\{\alpha\}\times a_s^*$. Since $S$ is ccc, the cub $C_1$ can be chosen to be a member of the PFA$(S)$ model. We use $C_1$ to define a partition of $C_0$: for each $\alpha\in C_0$, we define $\dot{f}(\alpha)$ to equal the value $g(\alpha^+)$ (i.e. the element of $S_{\alpha^+}$ that $g$ picks). Thus if $\delta$ is a limit of $C_1$ and $s\in S_\delta$, then $s$ forces a value for $\dot{f}\!\!\upharpoonright\!\!\delta$. Then a potential normalizing expansion would consist of a sequence $\{\dot{n}_\alpha:\alpha\in C_0\}$ of $S$-names of integers for which $L_\alpha[a_{g\upharpoonright\alpha^+}\setminus\dot{n}_\alpha]\cup(\{\alpha\}\times a_{g\upharpoonright\alpha^+}^*)$ is an open neighborhood of $\dot{Z}_\alpha$. There is a cub $C_2\subseteq C_1$ so that for each $\delta\in C_2$ and each $s\in S_\delta$, $s$ forces a value on $\dot{n}_\alpha$ for all $\alpha<\delta$. We may choose any $s_0\in g$ so that $s_0$ forces that $L_\alpha[a_{g\upharpoonright\alpha^+}\setminus\dot{n}_\alpha]\cap L_\delta[a_{g\upharpoonright\delta^+}\setminus\dot{n}_\delta]$ is empty whenever $\dot{f}(\alpha)\neq\dot{f}(\delta)$. Working in $V[g]$, we prove there is a stationary $E$ satisfying that $L_\delta[a_{g\upharpoonright\delta^+}]\cap\bigcup\{L_\alpha[a_{g\upharpoonright\alpha^+}\setminus\dot{n}_\alpha]:\alpha\in\delta\}$ is infinite, for all $\delta\in E$. If not, then there would be an assignment $\langle m_\delta:\delta\in C\rangle$ (for some cub $C$) so that $L_\delta[a_{g\upharpoonright\delta^+}\setminus m_\delta]$ would be disjoint from $\bigcup\{L_\alpha[a_{g\upharpoonright\alpha^+}\setminus\dot{n}_\alpha]:\alpha\in\delta\}$, for all $\delta\in C$. Pressing down, we would arrive at a contradiction. Let $\dot{E}$ denote the $S$-name of the stationary set whose existence was shown in the previous paragraph. Choose any $s$ above $s_0$ and any $\delta\in C_2$ such that $s$ forces that $\delta\in\dot{E}$. Without loss of generality, the height of $s$ is $\geq\delta^+$, but note that $s\!\!\upharpoonright\!\!\delta$ forces a value on $\dot{n}_\alpha$, for all $\alpha<\delta$. This means that $s\!\!\upharpoonright\!\!\delta^+$ forces that $\delta\in \dot{E}$, since it will also decide the value of $L_\delta[a_{g\upharpoonright\delta^+}]$. We also have that $s\!\!\upharpoonright\!\!\delta$ forces a value on $\dot{f}\!\!\upharpoonright\!\!\delta$ and so we can choose a value $e\in\{0,1\}$ so that $s\!\!\upharpoonright\!\!\delta$ forces that $L_\delta[a_{s\upharpoonright\delta^+}]$ intersected with $\{L_\alpha[a_{s\upharpoonright\alpha^+}\setminus\dot{n}_\alpha]:\alpha<\delta\text{ and }\dot{f}(\alpha)=e\}$ is infinite. We now have a contradiction, since $s\!\!\upharpoonright\!\!\delta^+\cup\{(\delta^+,1-e)\}$ forces that the assigned neighborhood of $\delta$ must meet the assigned neighborhood of $\alpha$, for some $\alpha<\delta$ with $\dot{f}(\alpha)=e\neq\dot{f}(\delta)$. Point-countable type ==================== There is another normal-implies-collectionwise-Hausdorff result holding in $L$ for which we don’t know whether it holds in our MM$(S)[S]$ model: A space is of **point-countable type** if each point is a member of a compact subspace which has a countable outer neighbourhood base. Spaces of point-countable type simultaneously generalize locally compact and first countable spaces, and V$=$L implies normal spaces of point-countable type are collectionwise Hausdorff [@W]. Does MM$(S)[S]$ imply normal spaces of point-countable type are $\aleph_1$-collectionwise Hausdorff? The usual arguments would show that if so, in our front-loaded model of MM$(S)[S]$, normal spaces of point-countable type would be collectionwise Hausdorff. **Acknowledgement.** We thank Peter Nyikos for catching errors in an earlier version of this manuscript. [Alan Dow, Department of Mathematics and Statistics, University of North Carolina, Charlotte, North Carolina 28223]{} [*e-mail address:*]{} [adow@uncc.edu]{} [Franklin D. Tall, Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, CANADA]{} [*e-mail address:*]{} [f.tall@math.utoronto.ca]{} [^1]: [^2]:

--- abstract: 'The goal of this mostly expository paper is to present several candidates for hyperbolic structures on irreducible Artin-Tits groups of spherical type and to elucidate some relations between them. Most constructions are algebraic analogues of previously known hyperbolic structures on Artin braid groups coming from natural actions of these groups on curve graphs and (modified) arc graphs of punctured disks.' address: - 'Matthieu Calvez, Departamento de Matemática y Estadística , Universidad de La Frontera, Francisco Salazar 1145, Temuco, Chile' - 'Bert Wiest, Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France' author: - Matthieu Calvez - Bert Wiest title: 'Hyperbolic structures for Artin-Tits groups of spherical type' --- Introduction {#S:Introduction} ============ Given a group $G$ and a generating set $X$ of $G$, the word metric $d_X$ turns $G$ into a metric space; this space is $(1,1)$-quasi-isometric to $\Gamma(G,X)$, the Cayley graph of $G$ with respect to $X$, endowed with the usual graph metric where each edge is identified to an interval of length 1. Hyperbolic structures on groups were recently introduced and studied in [@ABO]. A *hyperbolic structure* on a group $G$ is a generating set $X$ of $G$ such that $(G,d_X)$ is Gromov-hyperbolic; note that $X$ must be infinite whenever $G$ is not itself hyperbolic. In this paper, we are interested in hyperbolic structures on Artin-Tits groups of spherical type. One motivation is trying to prove that irreducible Artin-Tits groups of spherical type are hierarchically hyperbolic [@BehrstockHagenSisto2; @BehrstockHagenSisto3], where the hierarchical structure should come from the hierarchy of parabolic subgroups of these groups. After [@CalvezWiest1; @CalvezWiest2], we know that irreducible Artin-Tits groups of spherical type admit non-trivial hyperbolic structures; i.e. each irreducible Artin-Tits group of spherical type $A$ contains an (infinite) generating set $X_{abs}^A$ such that the corresponding Cayley graph $\Gamma(A,X_{abs}^A)$ is a Gromov-hyperbolic metric space with *infinite diameter*. This hyperbolic structure was defined in a purely algebraic manner, using only the Garside structure on $A$, and it consists of the set of the so-called *absorbable elements* (to which we must add the cyclic subgroup generated by the square of the so-called Garside element if $A$ is of dihedral type). We shall briefly review this construction in Section \[S:ATHyp\]. Unfortunately, these absorbable elements are poorly understood – for instance we do not know any polynomial-time algorithm which recognizes whether any given element belongs to $X_{abs}^A$ – and this makes it quite difficult to work with the graph $\Gamma(A,X_{abs}^A)$. In this paper we generalize to any irreducible Artin-Tits group of spherical type some well-known hyperbolic structures on Artin’s braid groups with $n+1$ strands $\mathcal B_{n+1}$ (a.k.a. Artin-Tits groups of type $A_n$), $n\geqslant 3$. Because it can be identified with the mapping class group of a $n+1$ times punctured disk $\mathcal D_{n+1}$, Artin’s braid group on $n+1$ strands admits nice actions on the curve graph of $\Dnpo$ (denoted by $\mathcal C(\Dnpo)$), the arc graph of $\Dnpo$ (denoted by $\mathcal A(\Dnpo)$) and the graph of arcs in $\Dnpo$ both of whose extremities lie in $\partial\Dnpo$ (denoted by $\mathcal A_{\partial}(\Dnpo)$). All these graphs can be shown to be connected and Gromov-hyperbolic; this was first shown in [@MasurMinsky1] but the circle of ideas around [@HPW; @PS] provides simpler arguments. All these actions are cobounded (actually cocompact); according to a standard argument (Lemma \[L:Main\], close in spirit to Svarc-Milnor’s lemma [@ABO Section 3.2]), we extract from each of these actions a hyperbolic structure on $\mathcal B_{n+1}$, which consists of the union of the stabilizers of a (finite) family of representatives of the orbits of vertices. Each of these generating sets can be algebraically described in terms of the *parabolic subgroups* of the Artin-Tits group of type $A_n$, allowing to extend the definitions to any irreducible Artin-Tits group of spherical type. Given an irreducible Artin-Tits group of spherical type $A$, we define: - $X_P^A$ is the union of all proper irreducible standard parabolic subgroups of $A$ and the cyclic subgroup generated by the square of the Garside element; - $X_{NP}^A$ is the union of the normalizers of all proper irreducible standard parabolic subgroups of $A$; - $X_{abs}^A$ is the set of absorbable elements (together with the cyclic subgroup generated by the square of the Garside element, if $A$ is of dihedral type: 2 generators). Note that $X_{NP}^A$ contains the cyclic subgroup generated by the square of the Garside element (which is central). Similarly for $X_{abs}^A$: any power of the Garside element can be written as a product of at most 3 absorbable elements, provided $A$ is not of dihedral type [@CalvezWiest1 Example 3]. Therefore the center of $A$ has bounded diameter with respect to the word metric on $A$ induced by any of the above generating sets. We then study the relationships between $X_P^A$, $X_{NP}^A$ and $X_{abs}^A$. Following [@ABO], given two generating sets $X,Y$ of a group $G$, we write $X\preccurlyeq Y$ if the identity map from $(G,d_Y)$ to $(G,d_X)$ is Lipschitz (or equivalently, if $\sup_{y\in Y} d_X(1_G,y)<\infty$). The sets $X$ and $Y$ are *equivalent* if both $X\preccurlyeq Y$ and $Y\preccurlyeq X$ hold (or equivalently, if the identity map is a bilipschitz equivalence between $(G,d_X)$ and $(G,d_Y)$). Table \[Table\] summarizes the main contents of this paper, for any irreducible Artin-Tits group of spherical type $A$ with at least 3 generators. Vertical arrows indicate the identity of $A$. The *graph of irreducible parabolic subgroups* and the *additional length graph* (denoted by $\mathcal C_{parab}(A)$ and $\mathcal C_{AL}(A)$, respectively) were defined in [@CGGMW] and [@CalvezWiest1], respectively. For Artin-Tits groups *of type $A$*, all the mentioned generating sets are hyperbolic structures. In any case, all spaces under consideration have infinite diameter (Corollary \[C:InfDiam\]). -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Hyperbolicity ------------- -- --------------------------------------------------------------------- ------------------------------------------------------------------------------------------- --------------------------------------- $X_P^A$ $\Gamma(A,X_P^A)$ generalized $\mathcal A_{\partial}(\Dnpo)$ Conjectured ${ {\left\downarrow\vbox to 1cm{}\right.\kern-\nulldelimiterspace} Lipschitz. Conj.\[C:StrictInequalities\](ii): not equivalent }$ $X_{NP}^A$ $\Gamma(A,X_{NP}^A)$ generalized $\mathcal C(\Dnpo)$, Conjectured q.isom. to $\mathcal C_{parab}$ [@CGGMW] ${ {\left\downarrow\vbox to 1cm{}\right.\kern-\nulldelimiterspace} Lipschitz [@CalvezWiest1; @AntolinCumplido]. Conj.\[C:StrictInequalities\](i): equivalent }$ $X_{abs}^A$ $\Gamma(A,X_{abs}^A)$ q.isom. to $\mathcal C_{AL}(A)$ [@CalvezWiest1] Proved [@CalvezWiest1; @CalvezWiest2] -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- : Summary of the main results presented in the paper when $A$ has at least 3 generators.[]{data-label="Table"} A space similar to $\Gamma(A,X_P^A)$ was defined in [@CharneyCrisp]; that paper gives necessary and sufficient conditions on an Artin-Tits group (not necessarily of spherical type) for the union of its standard parabolic subgroups of spherical type to be a hyperbolic structure. However, when the group is itself of spherical type, this results in a Cayley graph of finite diameter. It is also interesting to note that a construction analogous to $\mathcal C_{parab}(A)$ has been recently announced –and conjectured to be hyperbolic– for each Artin-Tits group of FC type [@morris]. When $A$ is of spherical dihedral type, all three sets $X_P^A$, $X_{NP}^A$ and $X_{abs}^A$ are hyperbolic structures but their orders differ slightly, namely $X_P^A$ and $X_{NP}^A$ are equivalent to each other and smaller than $X_{abs}^A$ – see Proposition \[P:Dihedral\]; also, all corresponding Cayley graphs have infinite diameter. The plan of the paper is as follows. In Section \[S:Prelim\] we review necessary definitions and facts about Artin-Tits groups of spherical type; we also prove Lemma \[L:Main\]. The latter is used in Section \[S:BnHyp\] to describe hyperbolic structures on $\mathcal B_{n+1}$; at the same time, these structures are reinterpreted algebraically in terms of parabolic subgroups of $\mathcal B_{n+1}$. In Section \[S:ATHyp\], we justify the assertions and present the conjectures from Table \[Table\]. Preliminaries {#S:Prelim} ============= Preliminaries on Artin-Tits groups of spherical type {#SS:PrelimArtin} ---------------------------------------------------- Let $S$ be a finite set. A *Coxeter matrix* over $S$ is a symmetric matrix $M=(m_{s,t})_{s,t\in S}$ with $m_{s,s}=1$ for $s\in S$ and $m_{s,t}\in \mathbb N_{\geqslant 2}\cup \{\infty\}$ for $s\neq t\in S$. The *Coxeter graph* $\Gamma=\Gamma_S=\Gamma(M)$ associated to $M$ is a labeled graph whose vertices are the elements of $S$ and in which two distinct vertices are connected by an edge labeled by $m_{s,t}$ whenever $m_{s,t}\geqslant 3$. For two symbols $a,b$ and an integer $m$, define $$\Pi(a,b;m)=\begin{cases} (ab)^k & \text{if $m=2k$,}\\ (ab)^ka & \text{if $m=2k+1$.}\\ \end{cases}$$ The Artin-Tits system associated to $M$ (or to $\Gamma$) is the pair $(A,S)$ where $A=A_S=A_{\Gamma}$ is the group presented by $$\left\langle S\right.\left|\ \Pi(s,t;m_{s,t})=\Pi(t,s;m_{s,t}), \forall s\neq t\in S, \text{with}\ m_{s,t}<\infty\right\rangle.$$ As relations are homogeneous, there is a well-defined homomorphism $\ell:A_S\longrightarrow \mathbb Z$ which sends each $s\in S$ to 1. The *rank* of $A_S$ is the number of elements of $S$. Given a subset $T$ of $S$, denote by $\Gamma_T$ the subgraph of $\Gamma_S$ induced by the vertices in $T$ and by $A_T$ the subgroup of $A_S$ generated by $T$. It is a theorem of Van der Lek [@VanDerLek] that $(A_T,T)$ is the Artin-Tits system associated to the graph $\Gamma_T$. Such a subgroup $A_T$ ($T\subset S$) is called a *standard parabolic subgroup* of $A_S$. A *parabolic subgroup* of $A_S$ is a subgroup which is conjugate to some standard parabolic subgroup. The group $A_S$ is said to be *irreducible* if the graph $\Gamma_S$ is connected, equivalently if $A_S$ cannot be expressed as the direct product of two standard parabolic subgroups. The group $A_S$ is of *spherical type* if the Coxeter group obtained as the quotient of $A_S$ by the normal closure of the squares of elements in $S$ is finite. The group $A_S$ is said to be of *dihedral type* if it has rank 2 (in this case, the corresponding Coxeter group is exactly the usual dihedral group). Irreducible Artin-Tits groups of spherical type have been classified [@Coxeter] (see also [@Humphreys] for a modern exposition) into type $A_n$ ($n\geqslant 1$), $B_n$ ($n\geqslant 2$), $D_n$ ($n\geqslant 4$), $E_6$, $E_7$, $E_8$, $F_4$, $H_3$, $H_4$ and $I_{2m}$ ($m\geqslant 5$). Figure \[F:Coxeter\] summarizes this classification. Note that for $n\geqslant 1$, the Artin-Tits group of type $A_n$ is precisely Artin’s braid group $\mathcal B_{n+1}$. In this paper, we will be concerned only with irreducible Artin-Tits groups of spherical type. For an integer $m=3,4$, we write also $I_{2m}$ for $A_2$ and $B_2$ respectively. ![Coxeter’s classification of irreducible Artin-Tits groups of spherical type[]{data-label="F:Coxeter"}](Coxeter.pdf){width="13cm"} We recall that each irreducible Artin-Tits groups of spherical type $A=A_S$ can be equipped with a finite Garside structure – this was proved in [@BrieskornSaito; @Deligne]. The reader is referred to [@GebhardtGM Section 2] for a gentle introduction to Garside structures. We just recall the existence of a Garside element $\Delta_S$; either $\Delta_S$ or $\Delta_S^2$ generates the center of $A_S$ and we denote by $\Omega_S$ this minimal central element. (It is known that $\Omega_S=\Delta_S$ for groups of type $A_1$, $B_n$ ($n\geqslant 2$), $D_n$ ($n$ even), $E_7$, $E_8$, $F_4$, $H_3$, $H_4$ and $I_{2m}$ ($m$ even), and $\Omega_S=\Delta_S^2$ for groups of type $A_n$ ($n\geqslant 2)$, $D_n$ ($n$ odd), $E_6$, and $I_{2m}$ ($m$ odd)). Similarly, for each irreducible standard parabolic subgroup $A_T$ ($T\subset S$), there is a Garside element $\Delta_T$ and $\Omega_T=\Delta_T$ or $\Delta_T^2$ generates the center of $A_T$. Moreover, if $P$ is any irreducible parabolic subgroup of $A_S$, then $P=a^{-1}A_T a$ for some element $a\in A$ and some $T\subset S$ – note that this expression is by no means unique. However there is a well-defined element $\Omega_P=a^{-1}\Omega_Ta$ which does not depend on $a$ nor on $T$ in the above expression of $P$ and generates the center of $P$ (see [@CGGMW]). The two following results on conjugacy of parabolic subgroups will be useful later. [@Paris Theorem 6.1]\[P:Paris\] Let $A=A_S$ be an irreducible Artin-Tits group of spherical type and let $T\subset S$ such that $A_T$ is an irreducible Artin-Tits group. Then $$N_{A_S}(A_T)=Z_{A_S}(\Omega_T).$$ [@CGGMW Section 11]\[P:SimultStandard\] Let $A=A_S$ be an irreducible Artin-Tits group of spherical type. Let $P,Q$ be two irreducible parabolic subgroups of $A_S$ and suppose that $\Omega_P$ and $\Omega_Q$ commute. Then there exists $g\in A$ and $T_1,T_2\subset S$ such that $g^{-1}Pg=A_{T_1}$ and $g^{-1}Qg=A_{T_2}$. We conclude this section with a few words on the case of dihedral type Artin-Tits groups. \[P:PureFree\] Let $m\geqslant 3$ be an integer. Let $A$ be an Artin-Tits group of type $I_{2m}$. Then the quotient $A/\negthinspace\left\langle \Delta^2\right\rangle$ has a finite index subgroup isomorphic to a free group of rank $m-1$. \[C:Dihedral\] Let $m$ and $A$ be as in Proposition \[P:PureFree\]. Let $X$ be any finite generating set of $A$. Then the generating set $X\cup\langle \Delta^2\rangle$ is a hyperbolic structure on $A$. All hyperbolic structures obtained in this way are equivalent. Let $\overline X$ be the set of cosets of elements of $X$ mod $\langle \Delta^2\rangle$; this generates $A/\langle\Delta^2\rangle$. Observe that $\Gamma(A,X\cup\langle\Delta^2\rangle)$ and $\Gamma(A/\langle \Delta^2\rangle, \overline X)$ are quasi-isometric spaces. From Proposition \[P:PureFree\] it follows applying Svarc-Milnor lemma that $\Gamma(A/\langle \Delta^2\rangle, \overline X)$ is quasi-isometric to a $m-1$-regular tree, hence hyperbolic. Therefore, for any $X$, $\Gamma(A,X\cup\langle \Delta^2\rangle)$ is quasi-isometric to a regular tree, whence the claim. Cocompact actions on connected graphs ------------------------------------- The unique result in this subsection will allow us to exhibit a number of hyperbolic structures on Artin’s braid group $\mathcal B_{n+1}$. Although this is a standard fact, we include a stand-alone proof for the convenience of the reader. [@MasurMinsky1 Lemma 3.2][@KimKoberda Lemma 10]\[L:Main\] Let $X$ be a connected graph and denote by $d_X$ the graph-metric on $X$ obtained by identifying each edge to an interval of length 1. Let $G$ be a group acting by isometries on $X$ such that - there exists a *finite* set $A$ of vertices of $X$ such that every vertex orbit has a (non-necessarily unique) representative in $A$, - there exists a *finite* set $B$ of edges of $X$ such that every edge orbit has a (non-necessarily unique) representative in $B$, - the set $T=\bigcup_{a\in A} Stab_G(a)$ generates $G$. Then $(G,d_T)$ is quasi-isometric to $(X,d_X)$. It is known that $(X,d_X)$ is (1,1)-quasi-isometric to $(\mathcal V_X,d_X)$, the vertex set of $X$ endowed with the restriction of the graph metric on $X$; we focus on $(\mathcal V_X,d_X)$ rather than $X$ itself. Let $M_1= Diam (A)$. Let $A_0\subset A$ having exactly one representative of each vertex orbit under the action of $G$; for each $a\in A$, choose $g_a\in G$ such that $g_a\cdot a\in A_0$ and let $M_2=\max\{\|g_a\|_T, \ a\in A\}$ (where we denote $\|g_a\|_T=d_T(1_G,g_a)$). Let $\{\{u_i,v_i\},1\leqslant i\leqslant m\}$ be an enumeration of the finite set $B$ and fix $\alpha_i,\beta_i\in G$ so that $\alpha_i^{-1}\cdot u_i\in A$ and $\beta_i^{-1}\cdot v_i\in A$. Let $$M_3=\max_{1\leqslant i\leqslant m}\{\|\alpha_i\|_T, \|\beta_i\|_T\}.$$ Fix $a_0\in A$ and define a map $\psi\co G\longrightarrow \mathcal V_X$ by $g\mapsto g\cdot a_0$. We first prove that the map $\psi\co (G,d_T)\to (\mathcal V_X,d_X)$ is Lipschitz. Let $g,h\in G$, let $n=\|g^{-1}h\|_T$ and write $g^{-1}h=t_1\ldots t_n$, with $t_i\in T$. We have $$d_X(\psi(g),\psi(h))=d_X(a_0,g^{-1}h\cdot a_0)=d_X(a_0,t_1\ldots t_n\cdot a_0)$$ and $$\begin{aligned} d_X(a_0,t_1\ldots t_n\cdot a_0) & \leqslant & d_X(a_0,t_1\cdot a_0)+d_X(t_1\cdot a_0,t_1t_2\cdot a_0)+\ldots\\ & & \quad \quad\ldots + d_X(t_1\ldots t_{n-1}\cdot a_0,t_1\ldots t_n\cdot a_0)\\ & = & \sum_{i=1}^n d_X(a_0,t_i\cdot a_0). \end{aligned}$$ But for any $t\in T$ there is at least some $a_t\in A$ so that $t\cdot a_t=a_t$; so we have $$d_X(a_0,t\cdot a_0)\leqslant d_X(a_0,a_t)+d_X(a_t,t\cdot a_0)=d_X(a_0,a_t)+d_X(t\cdot a_t,t\cdot a_0)=2d_X(a_0,a_t)\leqslant 2M_1.$$ In summary, we have $$d_X(\psi(g),\psi(h))\leqslant 2M_1d_T(g,h).$$ On the other hand, given any vertex $x$ of $X$, there exist $a\in A$ and $g\in G$ so that $x=g\cdot a$. If $a,a'\in A$ and $g,g'\in G$ are such that $g\cdot a=g'\cdot a'=x$, we have $g_a\cdot a=g_{a'}\cdot a'\in A_0$ whence $g'^{-1}gg_a^{-1}g_a'$ fixes $a'$ and so lies in $T$. It follows that the set of elements $g\in G$ such that $g\cdot a=x$ for some $a\in A$ has diameter at most $1+2M_2$ in $(G,d_T)$. We define $\varphi:\mathcal V_X\longrightarrow G$ in the following way: to each $x\in \mathcal V_X$ we associate some $g$ in $G$ such that $g^{-1}\cdot x\in A$. It is now clear that $ {\varphi}\circ\psi$ is at distance at most $2M_2+1$ from $Id_G$ and $\psi\circ{\varphi}$ is at distance at most $M_1$ from $Id_{\mathcal V_X}$. We now show that ${\varphi}\co(\mathcal V_X,d_X)\longrightarrow (G,d_T) $ is also Lipschitz. Let $x_1,x_2$ be adjacent vertices of $X$. There is some $g$ in $G$ and $1\leqslant i\leqslant m$ such that $\{g\cdot x_1,g\cdot x_2\}=\{u_i,v_i\}$ is an edge belonging to $B$. By construction, $\alpha_i^{-1}g\cdot x_1=\alpha_i^{-1}\cdot u_i\in A$ and $\beta_i^{-1}g\cdot x_2=\beta_i^{-1}\cdot v_i\in A$; it follows that $d_T({\varphi}(x_1),g^{-1}\alpha_i)\leqslant 2M_2+1$ and $d_T({\varphi}(x_2),g^{-1}\beta_i)\leqslant 2M_2+1$. We then have $$\begin{aligned} d_T({\varphi}(x_1),{\varphi}(x_2)) & \leqslant & d_t({\varphi}(x_1),g^{-1}\alpha_i)+d_T(g^{-1}\alpha_i,g^{-1}\beta_i)+d_T(g^{-1}\beta_i,{\varphi}(x_2))\\ & \leqslant & 4M_2+2+d_T(\alpha_i,\beta_i)\\ & \leqslant & 4M_2+2+2M_3. \end{aligned}$$ In summary, if $x_1, x_2$ are arbitrary vertices of $X$ then $$d_T({\varphi}(x_1),{\varphi}(x_2))\leqslant 2(2M_2+M_3+1)\cdot d_X(x_1,x_2).\qedhere$$ Hyperbolic structures for $\mathcal B_{n+1}$ {#S:BnHyp} ============================================ In this section, we review some Gromov-hyperbolic graphs on which Artin’s braid group $\Bnpo$ acts. Lemma \[L:Main\] is put to work in order to derive hyperbolic structures on $\Bnpo$ from these actions. These hyperbolic structures are interpreted in an algebraic context. The curve graph {#Subsection:CurveGraph} --------------- Fix an integer $n\geqslant 3$. Artin’s braid group on $n+1$ strands is identified with the mapping class group of a closed, $n+1$ times punctured disk $\Dnpo$ (with boundary fixed pointwise). We consider the following model for $\Dnpo$: the disk of diameter $[0,n+2]$ in $\mathbb C$ with the standard orientation, and $n+1$ punctures at $1,\ldots, n+1$ numbered accordingly $p_1,\ldots, p_{n+1}$. For $1\leqslant i\leqslant n$, write $s_i$ for the half-Dehn twist along an horizontal arc connecting $p_i$ and $p_{i+1}$. Then $(\Bnpo,\{s_1,\ldots, s_n\})$ is exactly the Artin-Tits system associated to the Coxeter graph $A_n$. The *curve graph* $\mathcal C(\Dnpo)$ of the $n+1$ times punctured disk $\Dnpo$ is the (locally-infinite) graph defined as follows: - its vertices are isotopy classes of essential simple closed curves in $\Dnpo$, that is isotopy classes of simple closed curves in $\Dnpo$ surrounding at least 2 and at most $n$ punctures; - two distinct vertices are connected by an edge if and only if the corresponding isotopy classes can be realized disjointly. According to [@MasurMinsky1], this graph is connected and Gromov-hyperbolic; a simpler proof can be found in [@HPW]. The group $\mathcal B_{n+1}$ acts naturally by isometries (simplicial automorphisms) on $\mathcal C(\Dnpo)$. An essential simple closed curve in $\Dnpo$ (a vertex of $\mathcal C(\Dnpo)$) is said to be *round* or *standard* if it is isotopic to a geometric circle. Note that there are exactly $\frac{n(n+1)}{2}-1$ such vertices: each of them can be described by two integers $1\leqslant i<j\leqslant n+1$, where $p_i$ ($p_j$ respectively) is the first (the last respectively) puncture surrounded; we denote this curve by $c_{ij}$. For $1\leqslant i<j\leqslant n+1$, we denote by $\mathcal B_{ij}$ the set of braids whose support is enclosed by $c_{ij}$; this is the standard irreducible parabolic subgroup generated by $\{s_t, i\leqslant t\leqslant j-1\}$, which is isomorphic to a braid group on $j-i+1$ strands. The square of the Garside element of this group $\Delta_{ij}^2$ is the Dehn twist around the curve $c_{ij}$. \[L:StructureXC\] Let $X_{\mathcal C}$ be the union of stabilizers of all standard curves. Then $\Bnpo$ equipped with the word metric $d_{X_{\mathcal C}}$ is quasi-isometric to the curve graph of the $n+1$ times punctured disk. First observe that the action of each generator $s_i$ fixes some standard curve; therefore, $X_{\mathcal C}$ generates $\mathcal B_{n+1}$. According to [@FarbMargalit Section 1.3.1], the set $A$ of all standard curves contains a representative of each vertex orbit under the action of $\Bnpo$. According to [@LeeLee Proposition 4.4], the *finite* set of edges in $\mathcal C (\Dnpo)$ both of whose extremities are standard curves contains a representative of each edge orbit under the action of $\Bnpo$. Hence we see that the isometric action of $\Bnpo$ on the curve graph $\mathcal C(\Dnpo)$ satisfies the hypotheses of Lemma \[L:Main\]. \[P:Curves\] Let $X_{NP}^{\Bnpo}$ be the union of all normalizers of proper irreducible standard parabolic subgroups of $\mathcal B_{n+1}$. Then $\Bnpo$ equipped with the word metric $d_{X_{NP}^{\Bnpo}}$ is quasi-isometric to the curve graph of the $n+1$ times punctured disk. In particular, $X_{NP}^{\Bnpo}$ is a hyperbolic structure on $\Bnpo$. According to [@FarbMargalit Fact 3.8], the stabilizer of any curve $c$ is exactly the centralizer of the Dehn twist around $c$; if $c=c_{ij}$ is standard, this is the centralizer of the square of the Garside element $\Delta_{ij}$ of $\mathcal B_{ij}$ (this is the same as the centralizer of $s_i$ if $j=i+1$). In other words, in view of Proposition \[P:Paris\], $Stab_{\Bnpo}(c_{ij})$ is the same as the normalizer of the standard parabolic subgroup $\mathcal B_{ij}$. The boundary connecting arc graph --------------------------------- Again, we fix an integer $n\geqslant 3$; we keep the same model for the punctured disk $\Dnpo$ and $s_i$, $1\leqslant i\leqslant n$, are the usual Artin generators of $\Bnpo$. The *boundary connecting arc graph* $\mathcal A_{\partial}(\Dnpo)$ of the $n+1$ times punctured disk $\Dnpo$ is the (locally-infinite) graph defined as follows: - its vertices are isotopy classes of essential (i.e. not homotopic into a subset of $\partial \Dnpo$) simple arcs in $\mathcal D_{n+1}$ both of whose endpoints lie in the boundary $\partial \mathcal D_{n+1}$ and where in addition the endpoints are allowed to slide freely along the boundary during the isotopy. - two distinct vertices are connected by an edge if and only if the corresponding isotopy classes of arcs can be realized disjointly. The argument given in [@HPW] can be easily adapted to show that $\mathcal A_{\partial}(\Dnpo)$ is a Gromov-hyperbolic connected graph. The group $\Bnpo$ acts by isometries on $\mathcal A_{\partial}(\Dnpo)$. Given $1\leqslant i\leqslant n$, let $a_i$ be the arc consisting simply of the vertical line with real part $i+\frac{1}{2}$; let $A=\{a_i, 1\leqslant i \leqslant n\}$. Then $A$ exhausts all the orbits of vertices of $\mathcal A_{\partial}(\Dnpo)$ under the action of $\Bnpo$ (the orbit of a boundary connecting arc $a$ is uniquely characterized by the number of punctures in any of the two connected components of $\Dnpo\setminus a$). Two different disjoint boundary connecting arcs (forming an edge of $\mathcal A_{\partial}(\Dnpo)$) can be transformed simultaneously into two arcs $a_i,a_j$ for some $1\leqslant i\neq j\leqslant n$ by the action of a braid: this says that there is a finite system $B$ of representatives of the orbits of edges. Finally, each half-twist $s_i$ stabilizes some arc in $A$. (Indeed, $s_i$ fixes $a_{i-1}$ if $i\geqslant 2$, and $s_1$ fixes $a_2$.) Therefore, all three hypotheses of Lemma \[L:Main\] are satisfied. It follows from Lemma \[L:Main\] that the union $X_{\mathcal A_{\partial}}$ of stabilizers of arcs $a_i, \ 1\leqslant i \leqslant n$, is a hyperbolic structure on $\Bnpo$ and that $(\mathcal B_{n+1}, d_{X_{\mathcal A_{\partial}}})$ is quasi-isometric to $\mathcal A_{\partial}(\Dnpo)$. [**[Claim.]{}**]{} If $n$ is odd and $i=\frac{n+1}{2}$, the stabilizer of $a_i$ is generated by $\Delta$ and the direct product of standard parabolic subgroups $\langle s_1,\ldots,s_{i-1}\rangle\times \langle s_{i+1}\ldots s_n\rangle$. In any other case, the stabilizer of $a_i$ is generated by $\Delta^2$ and the direct product of standard parabolic subgroups $\langle s_1,\ldots,s_{i-1}\rangle\times \langle s_{i+1}\ldots s_n\rangle$. The arc $a_i$ cuts $\mathcal D_{n+1}$ into two connected components which are punctured disks with $i$ and $n+1-i$ punctures respectively. Except if $2i=n+1$, an element $g$ in the stabilizer of $a_i$ must preserve the boundary of each of these two disks. Following the conventions in [@LeeLee], this means that $g$ can be written as a product of an interior braid (which is an element of $\langle s_1,\ldots,s_{i-1}\rangle\times \langle s_{i+1}\ldots s_n\rangle$) and a tubular braid of the form $$\left((s_i\ldots s_1)(s_{i+1}\ldots s_2)\ldots (s_n\ldots s_{n-i+1})(s_{n-i+1}\ldots s_1)\ldots (s_n\ldots s_{i+1})\right)^k$$ for some $k\in \mathbb Z$ (a pure braid on 2 fat strands). We observe however that the above braid can be written as $$\Delta^{2k}\Delta^{-2k}_{\langle s_1,\ldots,s_{i-1}\rangle}\Delta^{-2k}_{\langle s_{i+1}\ldots s_{n}\rangle}.$$ If $2i=n+1$, an element of the stabilizer of $a_i$ can be written as the product of an interior braid as above and a tubular braid (on 2 fat strands) which needs not be pure, that is a power of $$(s_i\ldots s_1)(s_{i+1}\ldots s_2)\ldots (s_n\ldots s_{n-i+1}),$$ which we can rewrite as $$\Delta\Delta^{-1}_{\langle s_1,\ldots,s_{i-1}\rangle}\Delta^{-1}_{\langle s_{i+1}\ldots s_{n}\rangle}.\qedhere$$ \[P:PartialA\] Let $X_{P}^{\Bnpo}$ be the subset of $\Bnpo$ consisting of the union of all proper irreducible standard parabolic subgroups together with the center $\langle \Delta^2\rangle$. Then $\Bnpo$ equipped with the word metric $d_{X_{P}^{\Bnpo}}$ is quasi-isometric to $\mathcal A_{\partial}(\Dnpo)$. In particular, $X_{P}^{\Bnpo}$ is a hyperbolic structure on $\Bnpo$. Let $X_{\mathcal A_{\partial}}$ be the union of stabilizers of all arcs $a_i$, $1\leqslant i\leqslant n$, as above. Then $X_{\mathcal A_{\partial}}$ contains the generating set $X_{P}^{\Bnpo}$ (every element in a proper irreducible standard parabolic subgroup fixes some arc $a_i$). On the other hand, each element in $X_{\mathcal A_{\partial}}$ can be written as a product of at most 6 elements of $X_P^{\Bnpo}$, according to the above claim (observe that $\Delta$ can be decomposed as a product of three elements of proper irreducible standard parabolic subgroups). This shows that both generating sets are equivalent. The remaining part of the statement follows as $X_{\mathcal A_{\partial}}$ is a hyperbolic structure and $(\mathcal B_{n+1}, d_{X_{\mathcal A_{\partial}}})$ is quasi-isometric to $\mathcal A_{\partial}(\Dnpo)$. Generalizing to Artin-Tits groups of spherical type {#S:ATHyp} =================================================== Throughout this section, $A=A_S$ is some fixed non-cyclic irreducible Artin-Tits group of spherical type. Definitions ----------- ### Generalized boundary connecting arc graph Let $X_P^A$ be the union of all proper irreducible standard parabolic subgroups of $A$ and the cyclic subgroup generated by the square of $\Delta_S$. It is clear that $X_P^A$ generates $A$ because each generator in $S$ belongs to some proper irreducible standard parabolic subgroup. Suppose that one of the following holds. - $A$ is of dihedral type, or - $A$ is of type $A_n$ ($n\geqslant 3$). Then $X_P^A$ is a hyperbolic structure on $A$. We refer to Proposition \[P:Dihedral\] for the proof of (i). For (ii), this is Proposition \[P:PartialA\]: in this case, the Cayley graph $\Gamma(A, X_P^A)$ is quasi-isometric to the graph $\mathcal A_{\partial}(\mathcal D_{n+1})$. Let $A$ be any non-cyclic irreducible Artin-Tits group of spherical type. The set $ X_P^A$ is a hyperbolic structure on $A$. ### Generalized curve graph Let $X_{NP}^A$ be the union of the normalizers of all proper irreducible standard parabolic subgroups of $A$. Of course, $X_{NP}^A$ generates $A$, as each $s\in S$ normalizes any parabolic subgroup containing it and hence belongs to $X_{NP}^A$. Note also that any power of $\Delta_S^2$ normalizes every parabolic subgroup. Recall from [@CGGMW] the definition of the *graph of parabolic subgroups* of $A$, denoted $\mathcal C_{parab}(A)$: - its vertices are proper irreducible parabolic subgroups of $A$, - two distinct vertices $P,Q$ are connected by an edge if the minimal central elements $\Omega_P$ and $\Omega_Q$ commute. The group $A$ acts naturally (by conjugation on vertices) by isometries on $\mathcal C_{parab}(A)$. Observing that the set of *standard* parabolic subgroups is finite and using Proposition \[P:SimultStandard\], Lemma \[L:Main\] implies along the same lines as Lemma \[L:StructureXC\]: The graphs $\Gamma(A,X_{NP}^A)$ and $\mathcal C_{parab}$ are quasi-isometric. In the special case where $A_S$ is of type $A_n$ ($n\geqslant 3$), it is shown in [@CGGMW] that $\mathcal C_{parab}(A_S)$ is exactly the same graph (isometric) as the curve graph of a $n+1$ times punctured disk. We have: Suppose that one of the following holds. - $A$ is of dihedral type, or - $A$ is of type $A_n$ ($n\geqslant 3$). Then $X_{NP}^A$ is a hyperbolic structure on $A$. We refer to Proposition \[P:Dihedral\] for the proof of (i). For (ii), this is Proposition \[P:Curves\]: in this case, the Cayley graph $\Gamma(A, X_{NP}^A)$ is quasi-isometric to the graph $\mathcal C(\mathcal D_{n+1})$. Let $A$ be any non-cyclic irreducible Artin-Tits group of spherical type. The set $X_{NP}^A$ is a hyperbolic structure on $A$. ### Absorbable elements and additional length graph Let $X_{abs}^A$ be the set of absorbable elements (together with the cyclic subgroup generated by $\Delta^2$ if $A$ is of dihedral type). The definition of absorbable element of a Garside group was proposed in [@CalvezWiest1]. It is not difficult to see that each element in $S$ is absorbable, hence $X_{abs}$ generates $A$. Recall from [@CalvezWiest1] the definition of the *additional length graph* $\mathcal C_{AL}(A)$: its vertices are right cosets of $A$ modulo the cyclic subgroup generated by $\Delta$; there is an edge between the vertices $a\langle \Delta\rangle$ and $b\langle\Delta\rangle$ if there is some $u$ which is either simple (i.e. a positive prefix of $\Delta$) or absorbable such that $au$ belongs to the coset $b\langle\Delta\rangle$. \[P:CALQI\] $\mathcal C_{AL}(A)$ and $\Gamma(A,X_{abs}^A)$ are quasi-isometric. It is enough to check the quasi-isometry of the respective vertex sets. Any coset $a\langle\Delta\rangle$ has diameter at most 4 in $\Gamma(A,X_{abs}^A)$: any two elements of such a coset differ by a power of $\Delta$; and any power of $\Delta$ can be written as a product of at most 4 elements of $X_{abs}^A$ in the dihedral case (because $\Delta^q=\Delta^{2\lfloor \frac{q}{2}\rfloor} \cdot a \cdot b \cdot b^{-1}a^{-1}\Delta$), and of at most 3 absorbable elements in the general case: (the argument of [@CalvezWiest1 Example 3] for braid groups with at least 4 strands can be easily adapted to any Artin-Tits group of rank $\geqslant 3$). For any coset $a\langle\Delta\rangle$, choose an arbitrary representative; this defines a map ${\varphi}$ from the vertices of $\mathcal C_{AL}$ to $A$. Conversely, given $a\in A$, we associate to $a$ the coset $\psi(a)=a\langle\Delta\rangle$. It is straightforward to check that these maps are quasi-inverses of each other with respect to the graph metrics on $\mathcal C_{AL}(A)$ and $\Gamma(A,X_{abs}^A)$. It is also clear that $\psi$ is 1-Lipschitz. On the other hand, as the set of simple elements (positive divisors of $\Delta$) is finite, we can define $M$ as the maximum of $d_{X_{abs}^A}(1,s\Delta^j)$, for $s$ simple and $j\in \mathbb Z$. It then follows that ${\varphi}$ is $M$-Lipschitz. The graph $\mathcal C_{AL}(A)$ was shown to be hyperbolic in [@CalvezWiest1]; therefore Lemma \[P:CALQI\] shows: Let $A$ be any non-cyclic irreducible Artin-Tits group of spherical type. The set $X_{abs}^A$ is a hyperbolic structure on $A$. Comparisons and further results ------------------------------- In this subsection, we compare the different generating sets for $A$ described above. We will first look at Artin-Tits groups of dihedral type, and then at all other types. \[P:Dihedral\] Assume that $A$ is of type $I_{2m}$ ($m\geqslant 3$). Then we have $$X_P^A \sim X_{NP}^A\preccurlyeq X_{abs}^A\sim S\cup\langle\Delta^2\rangle,$$ where the middle inequality is strict. All the sets mentioned are hyperbolic structures on $A$ and the corresponding Cayley graphs all have infinite diameter. Denote by $a,b$ the generators of $A$. In order to prove that $X_P^A \sim X_{NP}^A$, we first observe that $X_P^A\subset X_{NP}^A$ whence $X_{NP}^A\preccurlyeq X_P^A$. For the converse inequality, it will be enough to show that the normalizer of $\langle a\rangle$ ($\langle b\rangle$ respectively) is generated by $\Omega$ and $a$ ($\Omega$ and $b$ respectively). In order to do so, we first observe that for any $g\in N_A(\langle a\rangle)$ we have $g^{-1} a g\in \langle a\rangle$, and using the abelianisation we see that $g^{-1} a g=a$, i.e., the normalizer of $\langle a\rangle$ coincides with $Z_A(a)$, the centraliser of $a$. This centraliser can be very easily computed using the algorithm in [@FrancoGM]. For $m$ even, we get that $Z_A(a)=\langle a, \Delta\rangle$. For $m$ odd, we obtain $\{a, \Pi(b,a;m-2)a^2\Pi(b,a;m-2)\}$ as a generating set; up to right and left multiplying the second generator by $a$, this gives the generating set $\{a,\Delta^2\}$. In any case, an element of $N_A(\langle a \rangle)$ can be written as $\Omega^p a^q$, which has length at most $(2+\|\Delta\|_{X_P^A})$ with respect to $d_{X_P^A}$ – and similarly for $N_A(\langle b\rangle)$. It follows that the identity map from $(A,d_{X_{NP}^A})$ to $(A,d_{X_{P}^A})$ is $(2+\|\Delta\|_{X_P^A})$-Lipschitz. This shows the first equivalence. In order to prove the middle inequality, we note that the set of absorbable elements is finite with $4m-8$ elements (see [@CalvezWiest1 Example 2.2] for $m=3$). Letting $M=\sup\{\|x\|_{X_P ^A},\ x\in X_{abs}^A\setminus\langle \Delta^2\rangle\}$, we see that the identity is an $M$-Lipschitz map from $(A,d_{X_{abs}^A})$ to $(A, d_{X_P^A})$, whence the middle inequality. This inequality is not an equivalence, because we have $d_{X_P^A}(Id, a^n)=1$ for all $n$ but $d_{X_{abs}^A}(Id, a^n)\to_{n\to \infty} +\infty$. For the second equivalence, we use again that there are only finitely many absorbable elements. It follows from Corollary \[C:Dihedral\] that $X_{abs}^A$ and $S\cup\langle\Delta^2\rangle$ are equivalent hyperbolic structures on $A$. Next we prove that $X_P^A$ is a hyperbolic structure. The proof uses the “guessing geodesics lemma” [@MS Theorem 3.15]. Given $g,h$ vertices of $X_P^A$, consider the subgraph $A_{gh}$ consisting of the projections to $\Gamma(A,X_P^A)$ of the set of geodesics in $\Gamma(A,S\cup \langle \Delta^2 \rangle)$ between $g$ and $h$. Because the identity from $\Gamma(A,S\cup \langle \Delta^2 \rangle)$ to $\Gamma(A,X_P^A)$ is Lipschitz and $\Gamma(A,S\cup \langle \Delta^2 \rangle)$ is hyperbolic, the different subgraphs $A_{g,h}$ form thin triangles in $\Gamma(A,X_P^A)$ so they satisfy the second condition of [@MS Theorem 3.15]. What happens if $g$, $h$ are 1 apart in $\Gamma(A,X_P^A)$? Observe first that if $g,h\in A$ satisfy $g^{-1}h=a^n$ (or $b^n$, with $n\in \mathbb Z$), then in $\Gamma(A,S\cup \langle \Delta^2 \rangle)$, there is a unique geodesic between $g$ and $h$, namely $g, ga, ga^2,\ldots, h$. Thus the subgraph $A_{gh}$ has diameter 1 as well; hence the first condition in [@MS Theorem 3.15] is also satisfied. We conclude that $\Gamma(A,X_P^A)$ is hyperbolic. Finally, we have to prove that $\Gamma(A,X_P^A)$ has infinite diameter. Let us assume, for a contradiction, that there exists some number $L$ such that every element $x$ of $A$ can be written in the form $$x=\Delta^{k_0} a^{k_1} b^{k_2} a^{k_3} b^{k_4} \ldots c^{k_L} \text{ \ \ or \ \ } x=\Delta^{k_0} b^{k_1} a^{k_2} b^{k_3} a^{k_4} \ldots c^{k_L}$$ with $k_0, k_1, \ldots, k_L\in\mathbb Z$, and $c=a$ or $c=b$, according to the parity of $L$. Now, we recall from Proposition \[P:PureFree\] that $\Gamma(A,S\cup \langle \Delta^2 \rangle)$ is quasi-isometric to an infinite regular $2(m-1)$-valent tree – in particular, as a metric space it has uncountably many ends. Also, powers of $a$ and powers of $b$ represent quasi-geodesics in this quasi-tree. However, the set of points in the tree represented by words as above has only countably many ends, leading to a contradiction. For the rest of the section, we now assume that $A$ has rank at least 3. \[P:Comparation3\] Suppose that $A$ is an irreducible Artin-Tits group of spherical type with rank at least 3. Then $X_{abs}^A\preccurlyeq X_{NP}^A\preccurlyeq X_P^A$. For the second inequality, it is enough to observe that $X_P^A\subset X_{NP}^A$ whence the identity map $(A,d_{X_P^A})\longrightarrow (A,d_{X_{NP}^A})$ is 1-Lipschitz. The first inequality was shown for braids in [@CalvezWiest1]: any braid in the stabilizer of a standard curve can be written as a product of 9 absorbable braids [@CalvezWiest1 Lemma 11]. This result has been generalized to the case of a general Artin-Tits group $A$ of spherical type in the forthcoming paper [@AntolinCumplido], where it is shown that any element which normalizes a proper irreducible standard parabolic subgroup of $A$ is a product of at most 9 absorbable elements. This is to say that the identity map from $(A,d_{X_{NP}^A})$ to $(A,d_{X_{abs}^A})$ is 9-Lipschitz. \[C:InfDiam\] The metric spaces $(A,d_{X_{NP}^A})$ and $(A,d_{X_P^A})$ have infinite diameter. This is just a combination of Proposition \[P:Comparation3\] and Lemma \[P:CALQI\] together with [@CalvezWiest2 Theorem 1.1], which asserts that $\mathcal C_{AL}(A)$ has infinite diameter. Open problems ------------- \[C:StrictInequalities\] - The inequality $X_{abs}^A\preccurlyeq X_{NP}^A$ is not strict: the converse inequality $X_{NP}^A\preccurlyeq X_{abs}^A$ also holds, and the identity map $\Gamma(A,X_{NP}^A) \to \Gamma(A,X_{abs}^A)$ is a quasi-isometry. (In the case of Artin braid group, this is claiming that $\mathcal C_{AL}(\mathcal{B}_n)$ is quasi-isometric to the curve graph of the punctured disk $\mathcal C(\Dn)$.) - The inequality $X_{NP}^A\preccurlyeq X_P^A$ is strict, i.e., the identity map $\Gamma(A,X_{P}^A)\to \Gamma(A,X_{NP}^A)$ is Lipschitz but *not* a quasi-isometry. The truth of Conjecture \[C:StrictInequalities\](i) would of course imply that $X_{NP}^A$ is a hyperbolic structure (and hence hyperbolicity of the graph of irreducible parabolic subgroups $\mathcal C_{parab}(A)$). It would also imply that Garside normal forms are unparameterized quasi-geodesics in $\mathcal C_{parab}(A)$ (since they are in $\mathcal C_{AL}(A)$); this would contrast with the recently announced example in [@RafiVerberne] of a family of geodesics in a mapping class group whose shadows in the corresponding curve graph are not unparameterized quasi-geodesics. Note that Conjecture \[C:StrictInequalities\](i) is not even known to hold in the specific case of Artin’s braid groups – see [@CalvezWiest1 Conjecture 1]. Indeed, as pointed out to us by Ursula Hamenstädt, it is very much conceivable that $\mathcal C_{AL}(\mathcal{B}_n)$ is quasi-isometric to a tree. Conjecture \[C:StrictInequalities\](ii) does hold when $A$ is a braid group with at least 4 strands: for $n\geqslant 3$, the natural map $\mathcal A_{\partial}(\Dnpo) \to \mathcal C(\Dnpo)$ is not a quasi-isometry. We refer to [@MS Section 5]. Consider the open disk $D$ of radius 1 centered at $\frac{3}{2}$ containing the first two punctures of $\mathcal D_{n+1}$ (see the beginning of Section \[Subsection:CurveGraph\]); let $X =\mathcal D_{n+1}\setminus D$: this subsurface is a hole for $\mathcal A_{\partial}(\mathcal D_{n+1})$ according to [@MS Definition 5.2]. Note also that $X$ is homeomorphic to $\mathcal D_n$. Now, let $\beta$ be a pseudo-Anosov braid on $n$ strands and consider the $n+1$-strand braid $\hat\beta$ obtained from $\beta$ by doubling the first strand. Fix $c$ greater than the constant $C_0$ relative to the complex $\mathcal A_{\partial}(\mathcal D_{n+1})$ in [@MS Theorem 5.14]. It is known [@MasurMinsky1 Proposition 4.6] that $\beta$ acts in a loxodromic way on the curve graph of $\mathcal D_n$, hence it acts loxodromically on $\mathcal C(X)$, the curve graph of $X$. In other words, we can find $\alpha>0$ so that $$d_{\mathcal C(X)}(1,\beta^k)\geqslant \alpha k,$$ for every integer $k$, and in particular $d_{\mathcal C(X)}(1,\beta^k)\geqslant c,$ for $k$ big enough. Now, [@MS Theorem 5.14] implies that there exists a constant $A=A(c)\geqslant 1$ such that for $k$ big enough, $d_{\mathcal A_{\partial}(\mathcal D_{n+1})}(1,{\hat\beta}^k)\geqslant \frac{\alpha k-A}{A}$, whence $\hat \beta$ acts loxodromically on $\mathcal A_{\partial}(\mathcal D_{n+1})$. However, as it preserves the round curve bounding $D$, $\hat\beta$ acts elliptically on $\mathcal C(\mathcal D_{n+1})$. This finishes the proof of Conjecture \[C:StrictInequalities\](ii) for braid groups. Is every large piece of 2-dimensional quasi-flat in the Cayley graph of a mapping class group sent to a subset of bounded diameter by the projection to the curve complex? This question is deliberately vague, but to the best of our knowledge, no result along these lines is known, except that *maximal-dimensional* quasi-flats are known to be squashed down to bounded diameter in the curve complex [@BehrstockMinsky; @BKMM]. We will give a more precise version of the question below, and we will show that a positive answer would imply Conjecture \[C:StrictInequalities\](i). Let us look at the Cayley graph of $A$ with respect to the Garside generators, modulo the $\Delta$-action: for any $z\in A$, all the vertices corresponding to elements of the form $x\Delta^k$ ($k\in\mathbb Z$) get identified, and if any two edges in the Cayley graph have the same endpoints after the $\Delta$-action, then they get identified as well. The quotient space is quasi-isometric to the Cayley graph of $A/Z(A)$, and we will denote it $Cay(A)/\langle\Delta\rangle$ – see [@CalvezWiest1] for a detailed account. Now suppose an element $y$ is absorbed by $x$, which by [@CalvezWiest1 Lemma 3] can be supposed to have the same length as $y$; thus $\inf(x)=\inf(y)=\inf(xy)=0$ and $\sup(x)=\sup(y)=\sup(xy)=L$. We will study the implications for the geometry of the graph $Cay(A)/\langle\Delta\rangle$. There is an equilateral triangle with corners $1$, $x$ and $xy$ (red in Figure \[F:AbsTriang\]), and with sides of length $L$ (blue in the figure) representing the Garside normal form words for $x$, $y$, and $xy$. Let us denote the Garside normal form words by $x=x_1 x_2 \ldots x_L$, and $y=y_1 y_2\ldots y_L$. Now, if $1\leqslant \ell \leqslant L$, then $x$ also absorbs $y_1\ldots y_\ell$. This means that in $Cay(A)/\langle\Delta\rangle$ we have $d(1,xy_1\ldots y_\ell)=L$: every vertex on the edge between $x$ and $xy$ is at the same distance (namely $L$) from the opposite corner of the triangle (namely $1$). More generally, if $1\leqslant \ell^y \leqslant \ell^x\leqslant L$, then $x_{L-\ell^x+1}\ldots x_L$ absorbs $y_1\ldots y_{\ell^y}$, and $d(x_1\ldots x_{L-\ell^x}, xy_1\ldots y_{\ell^y})=\ell^x$. (0,-) – (0,5+); (.5,-) – (.5,5+); (,-) – (,5+); (1.5,-) – (1.5,5+); (2,-) – (2,5+); (2.5,-) – (2.5,5+); (-,-) – (2.5+,2.5+); (-,1-) – (2.5+,3.5+); (-,2-) – (2.5+,4.5+); (-,3-) – (2+,5+); (-,4-) – (1+,5+); (-,5-) – (,5+); (1-,-) – (2.5+,1.5+); (2-,-) – (2.5+,.5+); (-,5+) – (2.5+,2.5-); (-,4+) – (2.5+,1.5-); (-,3+) – (2.5+,.5-); (-,2+) – (2+,-); (-,1+) – (1+,-); (-,+) – (,-); (1-,5+) – (2.5+,3.5-); (2-,5+) – (2.5+,4.5-); [triangle]{} (0,0) – (0,5) – (2.5,2.5) – (0,0); (0,1) circle \[radius=\]; (0,2) circle \[radius=\]; (0,3) circle \[radius=\]; (0,4) circle \[radius=\]; (.5,.5) circle \[radius=\]; (1,1) circle \[radius=\]; (1.5,1.5) circle \[radius=\]; (2,2) circle \[radius=\]; (.5,4.5) circle \[radius=\]; (1,4) circle \[radius=\]; (1.5,3.5) circle \[radius=\]; (2,3) circle \[radius=\]; [vertices]{} (0,0) circle \[radius=\]; (0,5) circle \[radius=\]; (2.5,2.5) circle \[radius=\]; (-0.4,0) node[$1$]{}; (-0.5,5) node[$xy$]{}; (2.5+0.4,2.5) node[$x$]{}; Another crucial observation is that the three sides of the triangle play completely symmetrical roles: if $x$ absorbs $y$, then $y$ absorbs $(xy)^{-1}\Delta^L$ and $\Delta^L (xy)^{-1}$ absorbs $x$. In particular, every vertex on every side of the triangle (between 1 and $x$, or between $x$ and $xy$, or between $xy$ and $1$) is at distance $L$ from the opposite corner. More generally, if two vertices of $Cay(A)/\langle\Delta\rangle$ lie on two different sides of the triangle, and if they are at distance $d_1$ and $d_2$, respectively, from the corner of the triangle shared by the two sides, then their distance in $Cay(A)/\langle\Delta\rangle$ is $\max(d_1,d_2)$. Let us call such a triangle “fat” – indeed, such a triangle is not at all thin (in the sense of $\delta$-hyperbolicity), but on the contrary all distances are at least as large as in a Euclidean comparison triangle. \[Q:FatTriangleQuestion\] We have seen that every absorbable element gives rise to a *fat* equilateral triangle in $Cay(A)/\langle\Delta\rangle$. Are the images of all such triangles in $\Gamma(A,X_{NP}^A)$ (or, equivalently, in the complex of irreducible parabolic subgroups) of uniformly bounded diameter? For $A=\Bnpo$, do the images in $\mathcal C(\Dnpo)$ of all fat equilateral triangles have uniformly bounded diameter? Notice that for every fat triangle induced from an absorbable element, the image in $\mathcal C_{AL}(A)$ is of diameter at most $2$ (because the image of every edge is of diameter $1$). A positive answer to Question \[Q:FatTriangleQuestion\] would imply that Conjecture \[C:StrictInequalities\](i) is true. [00]{} , [*Hyperbolic structures on groups*]{}, arXiv:1710.05197 , [*Parabolic subgroups acting on the additional length graph*]{}, arXiv:1906.06325 , arXiv:1509.00632 , [*Quasiflats in hierarchically hyperbolic spaces*]{}, arXiv:1704.04271 , [*Dimension and rank for mapping class groups*]{}, Annals of Mathematics 167 (2008), 1055–1077 , [*Geometry and rigidity of mapping class groups*]{}, Geometry and Topology 16 (2012), 781–888 , [*Artin-Gruppen und Coxeter-Gruppen*]{}, Invent. Math. 17 (1972), 245–271 , [*Curve graphs and Garside groups*]{}, Geometriae Dedicata 188(1) (2017), 195–213 , [*Acylindrical hyperbolicity and Artin-Tits groups of spherical type*]{}, Geometriae Dedicata 191(1) (2017), 199–215 , [*Relative hyperbolicity and Artin groups*]{}, Geometriae Dedicata 129 (2007), 1–13 , [*The complete enumeration of finite groups of the form $r_i^2=(r_ik_i)^{k_{i,j}}=1$*]{}, J. Lond. Math. Soc., 1-10(1) (1935), 21–25 , On parabolic subgroups of Artin-Tits groups of spherical type, arXiv:1712.06727 , [*Les immeubles des groupes de tresses généralisés*]{}, Invent. Math. 17 (1972), 273–302 , [*A primer on mapping class groups*]{}, Princeton Mathematical Series 49, Princeton University Press, 2011 , [*Computation of Centralizers in Braid groups and Garside groups*]{}, Revista Matemática Iberoamericana 19 (2) (2003), 367–384 , [*The cyclic sliding operation in Garside groups*]{}, Math. Zeitschrift 265 (2010), 85–114 , [*1-slim triangles and uniform hyperbolicity for arc graphs and curve graphs*]{}, J. European Math. Soc. 17(4) (2015), 755–762 , [*Reflection Groups and Coxeter Groups*]{}, Cambridge University Press, 1992 , [*The geometry of the curve graph of a right-angled Artin group*]{}, Internat. J.  Algebra and Computation 24(2) (2014), 121–169 , [*A Garside theoretic approach to the reducibility problem in braid groups*]{}, J. Algebra 320 (2) (2008), 783–820 , [*Geometry of the complex of curves I: Hyperbolicity*]{}, Invent. Math. 138 (1999), 103–149 , [*The geometry of the disk complex*]{}, J. Amer. Math. Soc. 26 (1) (2013), 1–62 , [*Combinatorial descriptions of multi-vertex 2-complexes*]{}, Illinois J. Math. 54 (1) (2010), 137–154 , [*Parabolic subgroups of Artin groups of FC type*]{}, arXiv:1906.07058 , [*Parabolic subgroups of Artin groups*]{}, Journal of Algebra 196 (1997), 369–399 , [*A note on acylindrical hyperbolicity of mapping class groups*]{}, in Hyperbolic Geometry and Geometric Group Theory, Adv. Stud. Pure Math. 73, Math. Soc. Japan, Tokyo, 2017 , [*Geodesics in the mapping class group*]{}, arXiv:1810.12489 , [*The Homotopy Type of Complex Hyperplane Complements*]{}, Ph.D. thesis (1983), Nijmegen

--- abstract: 'The relative density of visible points of the integer lattice ${\mathbb{Z}}^d$ is known to be $1/\zeta(d)$ for $d\geq 2$, where $\zeta$ is Riemann’s zeta function. In this paper we prove that the relative density of visible points in the Ammann-Beenker point set is given by $2(\sqrt{2}-1)/\zeta_K(2)$, where $\zeta_K$ is Dedekind’s zeta function over $K={\mathbb{Q}}(\sqrt{2})$.' author: - Gustav Hammarhjelm bibliography: - 'bibl.bib' title: 'The density of visible points in the Ammann-Beenker point set' --- Introduction {#secIntro} ============ A locally finite point set $\mathcal{P}\subset {\mathbb{R}}^d$ has an *asymptotic density* (or simply *density*) $\theta(\mathcal{P})$ if $$\lim_{R\to\infty}\frac{\#(\mathcal{P}\cap RD)}{{\mathrm{vol}}(RD)}=\theta(\mathcal{P})$$ holds for all Jordan measurable $D\subset {\mathbb{R}}^d$. The density of a set can be interpreted as the asymptotic number of elements per unit volume. For instance, for a lattice $\mathcal{L}\subset {\mathbb{R}}^d$ we have $\theta(\mathcal{L})=\frac{1}{{\mathrm{vol}}({\mathbb{R}}^d/\mathcal{L})}$. Let $\widehat{\mathcal{P}}=\{x\in \mathcal{P}\mid tx\notin \mathcal{P}, \forall t\in (0,1)\}$ denote the subset of the *visible* points of $\mathcal{P}$. If $\mathcal{P}$ is a regular cut-and-project set (see below) then it is known that $\theta(\mathcal{P})$ exists. In [@marklof2014visibility Theorem 1], J. Marklof and A. Strömbergsson proved that $\theta(\widehat{\mathcal{P}})$ also exists and that $0<\theta(\widehat{\mathcal{P}})\leq\theta(\mathcal{P})$ if $\theta(\mathcal{P})>0$. In particular, for such $\mathcal{P}$ the *relative density of visible points* $\kappa_\mathcal{P}:=\frac{\theta(\widehat{\mathcal{P}})}{\theta(\mathcal{P})}$ exists, but is not known explicitly in most cases. For $d\geq 2$ we have $\widehat{{\mathbb{Z}}^d}=\{(n_1,\ldots,n_d)\in {\mathbb{Z}}^d\mid \gcd(n_1,\ldots,n_d)=1\}$ and $\theta(\widehat{{\mathbb{Z}}^d})=1/\zeta(d)$ gives the probability that $d$ random integers share no common factor. This can be derived in several ways, see for instance [@nymann1972probability]; we sketch another proof in below. More generally, $\theta(\widehat{\mathcal{L}})=\frac{1}{{\mathrm{vol}}({\mathbb{R}}^d/\mathcal{L})\zeta(d)}$ for a lattice $\mathcal{L}\subset {\mathbb{R}}^d$, see e.g. [@baake2000diffraction Prop. 6]. A well-known point set, which can be realised both as the vertices of a substitution tiling and as a cut-and-project set, is the Ammann-Beenker point set. The goal of this paper is to prove that the relative density of visible points in the Ammann-Beenker point set is $2(\sqrt{2}-1)/\zeta_K(2)$. This density was computed by B. Sing in the presentation [@singppt1], but he has not published a proof of this result. The density of the visible points of ${\mathbb{Z}}^d$ {#secDensZn} ===================================================== In this section we show that $\theta(\widehat{{\mathbb{Z}}^d})=1/\zeta(d)$. We shall see that a lot of inspiration can be drawn from this example when calculating the density of the visible points in the Ammann-Beenker point set. Fix $R>0$, a Jordan measurable $D\subset {\mathbb{R}}^d$ and let ${\mathbb{P}}\subset {\mathbb{Z}}_{>0}$ denote the set of prime numbers. For each *invisible* point $n\in {\mathbb{Z}}^d\setminus \widehat{{\mathbb{Z}}^d}$, there is $p\in{\mathbb{P}}$ such that $\frac{n}{p}\in{\mathbb{Z}}^d$. Setting ${\mathbb{Z}}^d_*={\mathbb{Z}}^d{\setminus\{(0,\ldots,0)\}}$ there are only finitely many $p_1,\ldots,p_n\in{\mathbb{P}}$ such that $p_i{\mathbb{Z}}^d_*\cap RD\neq \emptyset$. By inclusion-exclusion counting we have $$\begin{aligned} \#(\widehat{{\mathbb{Z}}^d}\cap RD)&=\#\left(({\mathbb{Z}}^d_*\cap RD)\setminus \bigcup_{p\in{\mathbb{P}}}(p{\mathbb{Z}}^d_*\cap RD)\right)=\#\left(({\mathbb{Z}}^d_*\cap RD)\setminus \bigcup_{i=1}^n(p_i{\mathbb{Z}}^d_*\cap RD)\right)\\ &=\#({\mathbb{Z}}^d_*\cap RD)+\sum_{k=1}^{m}(-1)^{k}\left(\sum_{1\leq i_1<\ldots <i_k\leq m}\#(p_{i_1}{\mathbb{Z}}^d_*\cap\cdots\cap p_{i_k}{\mathbb{Z}}^d_*\cap RD)\right). \end{aligned}$$ The last sum can be rewritten to $$\sum_{n\in{\mathbb{Z}}_{>0}}\mu(n)\cdot\#(n{\mathbb{Z}}^d_*\cap RD),$$ where $\mu$ is the Möbius function. Hence $$\frac{\#(\widehat{{\mathbb{Z}}^d}\cap RD)}{{\mathrm{vol}}(RD)}=\sum_{n\in{\mathbb{Z}}_{>0}}\frac{\mu(n)\cdot\#(n{\mathbb{Z}}^d_*\cap RD)}{{\mathrm{vol}}(RD)}=\sum_{n\in{\mathbb{Z}}_{>0}}\frac{\mu(n)}{n^d}\frac{\#({\mathbb{Z}}^d_*\cap {n^{-1}}RD)}{{\mathrm{vol}}({n^{-1}}RD)}.$$ Letting $R\to\infty$, switching order of limit and summation (for instance justified by finding a constant $C$ depending on $D$ such that $\#({\mathbb{Z}}_*^d\cap RD)\leq C{\mathrm{vol}}(RD)$ for all $R$), using $\theta({\mathbb{Z}}^d_*)=1$ and $1/\zeta(s)=\sum_{n\in{\mathbb{Z}}_{>0}}\frac{\mu(n)}{n^s}$ for $s>1$, we find that $$\theta(\widehat{{\mathbb{Z}}^d})=\lim_{R\to\infty}\frac{\#(\widehat{{\mathbb{Z}}^d}\cap RD)}{{\mathrm{vol}}(RD)}=1/\zeta(d).$$ Cut-and-project sets and the Ammann-Beenker point set {#secCPS} ===================================================== The Ammann-Beenker point set can be obtained as the vertices of the Ammann-Beenker tiling, a substitution tiling of the plane using a square and a rhombus as tiles, see e.g. [@baake2013aperiodic Chapter 6.1]. In this paper however, the Ammann-Beenker set is realised as a *cut-and-project set*, a certain type of point set which we will now define. Cut-and-project sets are sometimes called (Euclidean) model sets. We will use the same notation and terminology for cut-and-project sets as in [@marklof2014free Sec. 1.2]. For an introduction to cut-and-project sets, see e.g. [@baake2013aperiodic Ch. 7.2]. If ${\mathbb{R}}^n={\mathbb{R}}^d\times {\mathbb{R}}^m$, let $$\begin{aligned} {2} \pi:& ~{\mathbb{R}}^n\longrightarrow {\mathbb{R}}^d & \pi_{\mathrm{int}}: & ~ {\mathbb{R}}^n\longrightarrow {\mathbb{R}}^m \\ & (x_1,\ldots,x_n)\longmapsto (x_1,\ldots,x_d)\hspace{1cm}& & (x_1,\ldots,x_n)\longmapsto (x_{d+1},\ldots, x_n) \end{aligned}$$ denote the natural projections. \[defnCPS\] Let $\mathcal{L}\subset {\mathbb{R}}^n$ be a lattice and $\mathcal{W}\subset \overline{\pi_{\mathrm{int}}(\mathcal{L})}$ be a set. Then the *cut-and-project* set of $\mathcal{L}$ and $\mathcal{W}$ is given by $\mathcal{P}(\mathcal{W},\mathcal{L})=\{\pi(y)\mid y\in \mathcal{L},\pi_{\mathrm{int}}(y)\in\mathcal{W}\}$. If $\partial W$ has measure zero with respect to any Haar measure on $\overline{\pi_{\mathrm{int}}(\mathcal{L})}$ we say that $\mathcal{P}(\mathcal{W},\mathcal{L})$ is *regular*. If the interior of $\mathcal{W}$ (the *window*) is non-empty, $\mathcal{P}(\mathcal{W},\mathcal{L})$ is relatively dense and if $\mathcal{W}$ is bounded, $\mathcal{P}(\mathcal{W},\mathcal{L})$ is uniformly discrete (cf. [@marklof2014free Prop. 3.1]). To realise the Ammann-Beenker point set in this way, let $K$ be the number field ${\mathbb{Q}}(\sqrt{2})$, with algebraic conjugation $x\mapsto \overline{x}$ (we will also write $\overline{x}=(\overline{x_1},\ldots,\overline{x_n})$ for $x=(x_1,\ldots,x_n)\in K^n$) and norm $N(x)=x\overline{x}$. The ring of integers $\mathcal{O}_K={\mathbb{Z}}[\sqrt{2}]$ of $K$ is a Euclidean domain with fundamental unit $\lambda:=1+\sqrt{2}$. With $\zeta:=e^{\tfrac{\pi i}{4}}$ and $\star:K\longrightarrow K$, $x\mapsto x^\star$ the automorphism generated by $\zeta \mapsto \zeta^3$, the Ammann-Beenker point set is in [@baake2013aperiodic Example 7.7] realised as $$\{x=x_1+x_2\zeta\mid x_1,x_2\in\mathcal{O}_K,x^\star\in W_8\},$$ where $W_8\subset {\mathbb{C}}$ is the regular octagon of side length $1$ centered at the origin, with sides perpendicular to the coordinate axes. Let $$\mathcal{L}=\{(x,\overline{x})\mid x=(x_1,x_2)\in \mathcal{O}_K^2\}\subset {\mathbb{R}}^4$$ be the Minkowski embedding of $\mathcal{O}_K^2$ and let $$\widetilde{\mathcal{L}}=\{(x,\overline{x})\in \mathcal{L}\mid (x_1-x_2)/\sqrt{2}\in \mathcal{O}_K\}.$$ Then, after a straight-forward translation it is seen that the Ammann-Beenker point set $\mathcal{A}$ can be realised in ${\mathbb{R}}^2$ as $\mathcal{A}=\frac{1}{\sqrt{2}}\mathcal{P}(\mathcal{W}_\mathcal{A},\widetilde{\mathcal{L}})$, where $\mathcal{W}_\mathcal{A}:=\sqrt{2}W_8$, i.e. $\mathcal{A}$ is the scaling of a cut-and-project set according to . The density of visible points of $\mathcal{A}$ {#secDensAB} ============================================== All notation used in this section is defined in and taken from . Since, for any $\mathcal{P}\subset {\mathbb{R}}^d$ whose density exists, and any $c>0$ it holds that $\theta(c\mathcal{P})=c^{-d} \theta(\mathcal{P})$ and $c\widehat{\mathcal{P}}=\widehat{c\mathcal{P}}$, finding $\theta(\widehat{\mathcal{A}'})$ with $\mathcal{A}':=\sqrt{2}\mathcal{A}=\mathcal{P}(\mathcal{W}_\mathcal{A},\widetilde{\mathcal{L}})$ will give the value of $\theta(\widehat{\mathcal{A}})$. As a first step, in , the asymptotic density of the visible points of the simpler set $\mathcal{B}=\mathcal{P}(\mathcal{W}_\mathcal{A},\mathcal{L})=\{x\in\mathcal{O}_K^2\mid \overline{x}\in\mathcal{W}_\mathcal{A}\}\subset\mathcal{O}_K^2$ will be calculated. In this result will be used to obtain $\theta(\widehat{\mathcal{A}})$. The density of visible points of $\mathcal{B}$ {#subsecDensAB'} ---------------------------------------------- The following general counting formula for bounded subsets of visible points of a point set $\mathcal{P}$ will be needed. Let $\mathcal{P}_*=\mathcal{P}\setminus\{(0,\ldots,0)\}$. \[propInclExcl\] Let $\mathcal{P}\subset {\mathbb{R}}^d$ be locally finite and fix a set $C\subset{\mathbb{R}}_{>1}$ such that for each $x\in \mathcal{P}\setminus\widehat{\mathcal{P}}$ there exists $c\in C$ with $x/c\in \mathcal{P}$. Let $R>0$ and a bounded set $D\subset {\mathbb{R}}^d$ be given. Then $$\#(\widehat{\mathcal{P}}\cap RD)=\sum_{\substack{F\subset C\\\#F<\infty}}(-1)^{\#F}\#\left(\left(\mathcal{P}_*\cap \bigcap_{c\in F}c\mathcal{P}_*\right)\cap RD\right).$$ The set $C_R:=\{c\in C\mid \mathcal{P}_*\cap c\mathcal{P}_*\cap RD\neq \emptyset\}$ is finite. Indeed, suppose this is not true and pick distinct $c_1,c_2,\ldots\in C_R$ and corresponding $x_i\in \mathcal{P}_*\cap c_i\mathcal{P}_*\cap RD$. Since $\mathcal{P}$ is locally finite, the sequence $x_1,x_2,\ldots$ contains only finitely many distinct elements. Thus, a subsequence $x_{k_1},x_{k_2},\ldots$ which is constant can be extracted, so that $x_{k_i}/c_{k_i}\in \mathcal{P}_*\cap \frac{RD}{c_{k_i}}\subset \mathcal{P}_*\cap RD$ are all distinct, contradiction to $\mathcal{P}$ being locally finite. Thus, we can write $C_R=\{c_1,\ldots,c_n\}$ for some $c_1,\ldots,c_n\in C$. Then $$\begin{aligned} &\#(\widehat{\mathcal{P}}\cap RD)=\#\left((\mathcal{P}_*\cap RD)\setminus\bigcup_{c\in C}(\mathcal{P}_*\cap c\mathcal{P}_*\cap RD)\right)\\ &=\#(\mathcal{P}_*\cap RD)-\#\left(\bigcup_{i=1}^n(\mathcal{P}_*\cap c_i\mathcal{P}_*\cap RD)\right), \end{aligned}$$ from which the result follows from the inclusion-exclusion counting formula for finite unions of finite sets. A set $C$ as in for $\mathcal{B}$ will be needed, and to this end a visibility condition for the elements of $\mathcal{B}$ is required. Given $x_1,x_2\in \mathcal{O}_K$, let $\gcd(x_1,x_2)$ be a fixed generator of the ideal generated by $x_1,x_2$ and write $\gcd(x_1,x_2)=1$ when $x_1,x_2$ are relatively prime. In the following proposition a visibility condition of the complex realisation of the Ammann-Beenker point set given in [@baake2014radial p. 477] is adapted to our situation. \[propVisCond\] The visible points of $\mathcal{B}$ are given by $$\widehat{\mathcal{B}}=\{x=(x_1,x_2)\in \mathcal{B}\mid \gcd(x_1,x_2)=1,\lambda \overline{x}\notin \mathcal{W}_\mathcal{A}\}.$$ First the necessity of the visibility conditions is established. Take $x=(x_1,x_2)$ and suppose that $\gcd(x_1,x_2)\neq 1$ so that there exists $c\in\mathcal{O}_K$ with $|N(c)|>1$ and $c\mid x_1,x_2$. Scaling $c$ by units we may assume that $1<c<\lambda$. Suppose first that $|N(c)|=|\overline{c}|c\geq 3$, which implies $|\overline{c}|>1$. By noting that $\mathcal{W}_\mathcal{A}$ is star-shaped with respect to the origin and $\mathcal{W}_\mathcal{A}=-\mathcal{W}_\mathcal{A}$ it follows that $x/c\in\mathcal{B}$, so $x$ is invisible. If $|N(c)|=2$, then each prime factor of $c$ must divide $2=\sqrt{2}\cdot \sqrt{2}$, so it can be assumed that $c=\sqrt{2}$ and hence $x$ is occluded by $x/\sqrt{2}$. If $\lambda \overline{x}\in\mathcal{W}_\mathcal{A}$ it follows immediately that $x/\lambda\in\mathcal{B}$. We now turn to the sufficiency of the visibility conditions. Take $x=(x_1,x_2)\in\mathcal{B}\setminus\widehat{\mathcal{B}}$ and $c>1$ such that $x/c\in\mathcal{B}$. As $\mathcal{B}$ is uniformly discrete, we may assume that $y:=x/c\in \widehat{\mathcal{B}}$. This implies, by necessity above, that $\gcd(y_1,y_2)=1$. Now, since $x_i=cy_i$ it follows that $c\in K$. Write $c=a/b$ with $a,b\in\mathcal{O}_K$ relatively prime. If $b$ is not a unit, $\gcd(y_1,y_2)=1$ is contradicted, hence $c\in\mathcal{O}_K$. If $|N(c)|\neq 1$ then $\gcd(x_1,x_2)\neq1$. Otherwise, $c>1$ is a unit, i.e. $c=\lambda^k$ for some integer $k>0$. Thus $\frac{\overline{x}}{\overline{c}}=\frac{\overline{x}}{\overline{\lambda}^k}\in\mathcal{W}_\mathcal{A}$. Since $\frac{1}{\overline{\lambda}}=-\lambda$ we get $(-\lambda)^k\overline{x}\in\mathcal{W}_\mathcal{A}$ and thus also $\lambda \overline{x}\in\mathcal{W}_\mathcal{A}$ as $\mathcal{W}_\mathcal{A}$ is star-shaped with respect to the origin and $-\mathcal{W}_\mathcal{A}=\mathcal{W}_\mathcal{A}$. This establishes sufficiency of the visibility conditions. Note that the proof works just as well for more general windows, that is, $\widehat{\mathcal{P}(\mathcal{W},\mathcal{L})}=\{x\in \mathcal{P}(\mathcal{W},\mathcal{L})\mid \gcd(x_1,x_2)=1,\lambda \overline{x}\notin \mathcal{W}\}$ if $\mathcal{W}\subset{\mathbb{R}}^2$ is bounded with non-empty interior, star-shaped with respect to the origin and $-\mathcal{W}=\mathcal{W}$. Let now $${\mathbb{P}}=\{\pi\in\mathcal{O}_K\mid \pi ~\mathrm{prime}, 1<\pi<\lambda\}~\text{and}~ C={\mathbb{P}}\cup \{ \lambda\}$$ so that ${\mathbb{P}}$ is a set that contains precisely one associate of every prime of $\mathcal{O}_K$. Then we have the following proposition. \[propOcclSet\] For each $x\in\mathcal{B}\setminus\widehat{\mathcal{B}}$ there is $c\in C$ such that $x/c\in \mathcal{B}$. Fix $x\in\mathcal{B}\setminus\widehat{\mathcal{B}}$. As seen in the proof of there is $c\in \mathcal{O}_K$, $c>1$, such that $x/c\in \mathcal{B}$. If $c$ is not a unit, fix $\pi\in {\mathbb{P}}$ so that $\pi\mid c$. It can be verified by hand that $\{(x,\overline{x})\mid x\in\mathcal{O}_K\}\cap ((1,\lambda)\times (-1,1))=\emptyset$, hence $|\pi|>1$ and $x/\pi\in \mathcal{B}$. If $c$ is a unit, $x/\lambda\in\mathcal{B}$ is immediate. Given a finite set $F\subset \mathcal{O}_K$ let $I_F$ be the (principal) ideal generated by the elements of $F$ if $F\neq \emptyset$ and $I_F=\mathcal{O}_K$ otherwise. Let $\ell_F$ denote a fixed *least common multiple* of $F$, that is, a generator of the ideal $\bigcap_{c\in F}c\mathcal{O}_K$. Let also $m_F=\min\{1,\min_{c\in F}|\overline{c}|\}$ and $\mathcal{L}_F=\{(\ell_Fx,\overline{\ell_Fx})\mid x\in \mathcal{O}_K^2\}$. Write $I\triangleleft \mathcal{O}_K$ when $I\subset \mathcal{O}_K$ is an ideal and define the *absolute norm* $N(I)$ of $I$ by $|N(x)|$, where $x$ is any generator of $I$. Recall *Dedekind’s zeta function* $\zeta_K(s)=\sum_{I\triangleleft \mathcal{O}_K}\frac{1}{N(I)^s}$ for $s\in {\mathbb{C}}$ with $\mathrm{Re}(s)>1$. Given a finite set $F\subset C$ it is verified that $\mathcal{B}_*\cap \bigcap_{c\in F}c\mathcal{B}_*=\mathcal{P}(m_F\mathcal{W}_\mathcal{A},\mathcal{L}_F){\setminus\{0\}}$. For any $R>0$ and bounded $D\subset {\mathbb{R}}^2$, Propositions \[propInclExcl\], \[propOcclSet\] imply that $$\label{eqA'1} \#(\widehat{\mathcal{B}}\cap RD)=\sum_{\substack{F\subset C\\\#F<\infty}}(-1)^{\#F}\#\left((\mathcal{P}(m_F\mathcal{W}_\mathcal{A},\mathcal{L}_F){\setminus\{0\}})\cap RD\right).$$ Since $\ell_F\mathcal{O}_K^2\subset\pi_{\mathrm{int}}(\mathcal{L}_F)\subset{\mathbb{R}}^2$ is dense we have $$\theta(\mathcal{P}(m_F\mathcal{W}_\mathcal{A},\mathcal{L}_F){\setminus\{0\}})=\frac{{\mathrm{vol}}(m_F\mathcal{W}_\mathcal{A})}{{\mathrm{vol}}({\mathbb{R}}^4/\mathcal{L}_F)}$$ from [@marklof2014free Prop. 3.2]. Dividing by ${\mathrm{vol}}(RD)$, letting $R\to\infty$ and switching order of limit and summation (to be justified in below) we find that $$\theta(\widehat{\mathcal{B}})=\sum_{\substack{F\subset C\\\#F<\infty}}(-1)^{\#F}\frac{{\mathrm{vol}}(m_F\mathcal{W}_\mathcal{A})}{{\mathrm{vol}}({\mathbb{R}}^4/\mathcal{L}_F)}=\sum_{\substack{F\subset C\\\#F<\infty}}(-1)^{\#F}\frac{m_F^2(1+\sqrt{2})}{2N(\ell_F)^2},$$ since ${\mathrm{vol}}(\mathcal{W}_\mathcal{A})=4(1+\sqrt{2})$ and ${\mathrm{vol}}({\mathbb{R}}^4/\mathcal{L}_F)=8N(\ell_F)^2$. The value of the right hand sum will be shown to be $1/\zeta_K(2)$ in below. The following lemma gives a bound on the number of points in the intersection of a lattice and a box in terms of the volume of the box, provided that the box is “not too thin”. \[lemLatticeBoxBound\] Let $\mathcal{L}\subset {\mathbb{R}}^d$ be a lattice and let $c>0$ be given. For any $a_i,b_i\in{\mathbb{R}}$ with $b_i-a_i>c$ set $B=\prod_{i=1}^{d}[a_i,b_i]$. Then there is a constant $L$ depending only on $\mathcal{L}$ and $c$ such that $\#(B\cap\mathcal{L})\leq L{\mathrm{vol}}(B)$. Let $n_i={\lceil\frac{b_i-a_i}{c}\rceil}\in{\mathbb{Z}}_+$. Then $\frac{b_i-a_i}{c}\leq n_i< \frac{b_i-a_i}{c}+1=\frac{b_i-a_i+c}{c}<\frac{2(b_i-a_i)}{c}.$ Hence, with $n=\prod_{i=1}^dn_i$ it follows that $n\leq \frac{2^d{\mathrm{vol}}(B)}{c^d}$. From $b_i\leq a_i+cn_i$ also $B\subset\prod_{i=1}^d[a_i,a_i+cn_i].$ Let $a=(a_1,\ldots,a_d)$ and consider $-a+\prod_{i=1}^d[a_i,a_i+cn_i]=\prod_{i=1}^d[0,cn_i]$. We have $\prod_{i=1}^d[0,cn_i]=\bigcup_{m\in{\mathbb{N}}^d,0\leq m_i<n_i}(mc+[0,c]^d).$ Hence, $B\subset \bigcup_{m\in{\mathbb{N}}^d,0\leq m_i<n_i}(a+mc+[0,c]^d)=:B'$. Find now $D>0$ depending on $\mathcal{L}$ and $c$ such that $\sup_{t\in{\mathbb{R}}^d}\#(\mathcal{L}\cap (t+[0,c]^d))=D$. Hence $\#(B\cap \mathcal{L})\leq \#(B'\cap\mathcal{L})\leq n D\leq \frac{2^dD}{c^d}{\mathrm{vol}}(B)$, so one can take $L=\frac{2^dD}{c^d}$. The following bound will be crucial in the justification of interchanging limit and summation in after division by ${\mathrm{vol}}(RD)$. \[lem-Estimate1\] Let $D\subset {\mathbb{R}}^2$ be Jordan measurable. Then there is a constant $\widetilde{L}>0$ depending only on $D$ such that for every $R>0$ and $F\subset C$ with $\#F<\infty$, $$\#((\mathcal{P}(m_F\mathcal{W}_\mathcal{A},\mathcal{L}_F)\cap RD){\setminus\{0\}})\leq \frac{\widetilde{L}R^2}{N(\ell_F)^2}.$$ By definition $$\#((\mathcal{P}(m_F\mathcal{W}_\mathcal{A},\mathcal{L}_F){\setminus\{0\}})\cap RD)=\#(\{x\in \ell_F\mathcal{O}_K^2\mid\overline{x}\in m_F\mathcal{W}_\mathcal{A}\}{\setminus\{0\}})\cap RD).$$ Note that this number is independent of the choice of $\ell_F$. There is a bijection $$(\{x\in \ell_F\mathcal{O}_K^2\mid \overline{x}\in m_F\mathcal{W}_\mathcal{A}\}{\setminus\{0\}})\cap RD\longrightarrow(\{x\in\mathcal{O}_K^2\mid \overline{x}\in \tfrac{m_F\mathcal{W}_\mathcal{A}}{|\overline{\ell_F}|}\}{\setminus\{0\}})\cap \tfrac{RD}{\ell_F}$$ given by $x\mapsto \tfrac{x}{\ell_F}$, so it suffices to estimate the number of elements in the latter set. Since $m_F\leq 1$ it follows that $\left(\mathcal{L}\cap \left(\tfrac{RD}{\ell_F}\times\tfrac{m_F\mathcal{W}_\mathcal{A}}{|\overline{\ell_F}|} \right)\right){\setminus\{0\}}\subset \left(\mathcal{L}\cap \left(\tfrac{RD}{\ell_F}\times\tfrac{\mathcal{W}_\mathcal{A}}{|\overline{\ell_F}|} \right)\right){\setminus\{0\}}$. Fix real numbers $m_1,m_2>1$ so that $D\subset [-m_1,m_1]^2=:B_1$ and $\mathcal{W}_\mathcal{A}\subset [-m_2,m_2]^2=:B_2$. Fix a number $c$ so that $c'<c$ implies $(\mathcal{L}\cap(\lambda D\times c'\mathcal{W}_\mathcal{A})){\setminus\{0\}}=\emptyset$. This can be done, for otherwise $(\mathcal{L}\cap(\lambda D\times c'\mathcal{W}_\mathcal{A})){\setminus\{0\}}$ would be non-empty for each $c'>0$, hence $\mathcal{L}\cap (\lambda D\times\mathcal{W}_\mathcal{A})$ would contain infinitely many points, contradiction, since $\mathcal{L}$ is a lattice and $\lambda D\times\mathcal{W}_\mathcal{A}$ is bounded. Suppose first that $\frac{R}{|\ell_F\overline{\ell_F}|}<c$. Scale $\ell_F$ by units so that $1\leq \tfrac{R}{\ell_F}<\lambda$ which gives $\tfrac{1}{|\overline{\ell_F}|}<c$. Hence $\left(\mathcal{L}\cap\left(\tfrac{RD}{\ell_F}\times \tfrac{\mathcal{W}_\mathcal{A}}{|\overline{\ell_F}|}\right)\right){\setminus\{0\}}\subset \left(\mathcal{L}\cap\lambda D\times\left( \tfrac{\mathcal{W}_\mathcal{A}}{|\overline{\ell_F}|}\right)\right){\setminus\{0\}}=\emptyset$ and therefore $\#\left(\left(\mathcal{L}\cap\left(\tfrac{RD}{\ell_F}\times \tfrac{\mathcal{W}_\mathcal{A}}{|\overline{\ell_F}|}\right)\right){\setminus\{0\}}\right)=0$. Suppose now that $\frac{R}{|\ell_F\overline{\ell_F}|}\geq c$. Scale $\ell_F$ so that $\sqrt{c}\leq \tfrac{R}{\ell_F}<\lambda \sqrt{c}$. This implies that $\tfrac{1}{|\overline{\ell_F}|}\geq \tfrac{1}{\lambda}\sqrt{c}>\sqrt{c}$. Thus, $[0,\sqrt{c}]^4\subset \tfrac{RB_D}{\ell_F}\times \tfrac{B_\mathcal{W}}{|\overline{\ell_F}|}=:B$. From we get a constant $L$ only depending on $\mathcal{L}$, $\sqrt{c}$ such that $ \#(B\cap \mathcal{L})\leq L{\mathrm{vol}}(B)=L\cdot 16m_1^2m_2^2\frac{R^2}{N(\ell_F)^2}.$ Now, since $\tfrac{RD}{\ell_F}\times \tfrac{\mathcal{W}_\mathcal{A}}{|\overline{\ell_F}|}\subset B$ we get that $$\#\left(\left(\mathcal{L}\cap\tfrac{RD}{\ell_F}\times \left(\tfrac{\mathcal{W}}{|\overline{\ell_F}|}\right)\right){\setminus\{0\}}\right)\leq\frac{\widetilde{L}R^2}{N(\ell_F)^2}$$ with $\widetilde{L}:=16m_1^2m_2^2L$. \[propInterchange\] The equality $$\lim_{R\to\infty}\sum_{\substack{F\subset C\\\#F<\infty}}\frac{(-1)^{\#F}\#\left((\mathcal{P}(m_F\mathcal{W}_\mathcal{A},\mathcal{L}_F){\setminus\{0\}})\cap RD\right)}{{\mathrm{vol}}(RD)}=\sum_{\substack{F\subset C\\\#F<\infty}}\frac{(-1)^{\#F}m_F^2(1+\sqrt{2})}{2N(\ell_F)^2}$$ holds for all Jordan measurable $D\subset {\mathbb{R}}^2$. For a finite $F\subset C$ let $N(R,F)=\#\left((\mathcal{P}(m_F\mathcal{W}_\mathcal{A},\mathcal{L}_F){\setminus\{0\}})\cap RD\right)$. We know that $\lim\limits_{R\to\infty}\frac{N(R,F)}{{\mathrm{vol}}(RD)}=\frac{m_F^2(1+\sqrt{2})}{2N(\ell_F)^2}$ so $$\label{eqnSwitchLimitsJustification} \lim\limits_{R\to\infty}\sum_{\substack{F\subset C\\\#F<\infty}}\frac{(-1)^{\#F}N(R,F)}{{\mathrm{vol}}(RD)}=\sum_{\substack{F\subset C\\\#F<\infty}}\lim\limits_{R\to\infty}\frac{(-1)^{\#F}N(R,F)}{{\mathrm{vol}}(RD)}$$ must be justified. In view of $$\sum_{\substack{F\subset C\\\#F<\infty}}\left|\frac{(-1)^{\#F}N(R,F)}{{\mathrm{vol}}(RD)}\right|\leq \frac{\widetilde{L}}{{\mathrm{vol}}(D)}\sum_{\substack{F\subset C\\\#F<\infty}}\frac{1}{N(\ell_F)^2}$$ and we note that $$\sum_{\substack{F\subset C\\\#F<\infty}}\frac{1}{N(\ell_F)^2}=\sum_{\substack{F\subset C\\\#F<\infty\\\lambda\in F}}\frac{1}{N(\ell_F)^2}+\sum_{\substack{F\subset C\\\#F<\infty\\\lambda\notin F}}\frac{1}{N(\ell_F)^2}\leq 2\sum_{I\triangleleft \mathcal{O}_K}\frac{1}{N(I)^2},$$ hence the sums of both sides of are absolutely convergent. Fix $\Delta>0$. We claim that there is only a finite number of non-empty $F\subset C$, $\#F<\infty$, such that $|N(\ell_F)|<\Delta$. Given such $F$ let $\ell_F=\prod_{c\in F}c>1$. Also, since $|\overline{\pi}|>1$ for all $\pi\in {\mathbb{P}}$ we have $|\overline{\ell_F}|\geq |\overline{\lambda}|$. Hence, $|N(\ell_F)|=\ell_F|\overline{\ell_F}|\leq \Delta$ implies $\ell_F\leq \frac{\Delta}{|\overline{\ell_F}|}\leq \lambda \Delta$ and $|\overline{\ell_F}|\leq \frac{\Delta}{\ell_F}\leq \Delta<\lambda\Delta$ so $(\ell_F,\overline{\ell_F})\in \{(x,\overline{x})\mid x\in\mathcal{O}_K\}\cap \lambda[-\Delta,\Delta]^2$ which is a finite set, thus elements of $F$ can only contain prime factors that occur as factors in the components of elements in this finite set, giving only finitely many possibilities for $F$. It follows that $$\begin{aligned} &\left|\lim\limits_{R\to\infty}\sum_{\substack{F\subset C\\\#F<\infty}}\frac{(-1)^{\#F}N(R,F)}{{\mathrm{vol}}(RD)}-\sum_{\substack{F\subset C\\\#F<\infty}}\frac{m_F^2(1+\sqrt{2})}{2N(\ell_F)^2}\right|\\ &\leq \left(\frac{\widetilde{L}}{{\mathrm{vol}}(D)}+\frac{1+\sqrt{2}}{2}\right)\sum_{\substack{F\subset C\\\#F<\infty\\|N(\ell_F)|\geq \Delta}}\frac{1}{N(\ell_F)^2} \end{aligned}$$ where the right hand side tends to $0$ as $\Delta\to\infty$ since $\sum\limits_{F\subset C,\#F<\infty,|N(\ell_F)|\geq \Delta}\frac{1}{N(\ell_F)^2}$ is the tail of an absolutely convergent sum, hence has been justified. From it follows that $\theta(\widehat{\mathcal{B}})=\sum_{\substack{F\subset C\\\#F<\infty}}\frac{(-1)^{\#F}m_F^2(1+\sqrt{2})}{2N(\ell_F)^2}$, and it will now be shown that the right hand side is equal to $1/\zeta_K(2)$. Define the function $\omega:\mathcal{O}_K\longrightarrow{\mathbb{C}}$, $\omega(x)=\#\{\pi\in{\mathbb{P}}\mid x/\pi\in\mathcal{O}_K\}$, so that $\omega(x)$ is the number of non-associated prime divisors of $x$. Given $I\triangleleft \mathcal{O}_K$, let $\omega(I)=\omega(x)$ for any generator $x$ of $I$ and define a Möbius function on the ideals of $\mathcal{O}_K$ by $$\mu(I)=\begin{cases}0 & \text{ if }\exists \pi\in{\mathbb{P}}\text{ such that } I\subset \pi^2\mathcal{O}_K,\\ (-1)^{\omega(I)} &\text{ otherwise.} \end{cases}$$ One verifies that $\mu(I_1I_2)=\mu(I_1)\mu(I_2)$ for relatively prime ideals $I_1,I_2$. The function $\zeta_K$ can be expressed as an Euler product for $s$ with $\mathrm{Re}(s)>1$ as $$\zeta_K(s)=\prod_{P\triangleleft \mathcal{O}_K,\,P\,\mathrm{prime}}\frac{1}{1-N(P)^{-s}}$$ and in analogy with the reciprocal formula for Riemann’s zeta function we have $$\label{eqnRecDedekind} \frac{1}{\zeta_K(s)}=\sum_{I\triangleleft \mathcal{O}_K}\frac{\mu(I)}{N(I)^s}.$$ \[thmAsDensA’\] The density of visible points of $\mathcal{B}$ is given by $$\theta(\mathcal{B})=\frac{1}{\zeta_K(2)}=\frac{48\sqrt{2}}{\pi^4}.$$ By we have $$\theta(\mathcal{B})=\sum_{\substack{F\subset C\\\#F<\infty}}\frac{(-1)^{\#F}m_F^2(1+\sqrt{2})}{2N(\ell_F)^2}.$$ Splitting the sum into two depending on whether $\lambda \in F$ or not, and using that $m_F$ is $1$ unless $\lambda\in F$, in which case $m_F=|\overline{\lambda}|=\sqrt{2}-1$, we get $$\begin{aligned} \theta(\mathcal{B})&=\sum_{\substack{F\subset C\\\#F<\infty\\\lambda\notin F}}\frac{(-1)^{\#F}(1+\sqrt{2})}{2N(\ell_F)^2}+\sum_{\substack{F\subset C\\\#F<\infty\\\lambda\in F}}\frac{(-1)^{\#F}|\overline{\lambda}|^2(1+\sqrt{2})}{2N(\ell_F)^2}\\ &=\frac{(1-|\overline{\lambda}|^2)(1+\sqrt{2})}{2}\sum_{I\triangleleft\mathcal{O}_K}\frac{\mu(I)}{N(I)^2}=\frac{1}{\zeta_K(2)}, \end{aligned}$$ last equality by . From [@washington1997introduction Theorem 4.2] one can calculate $\zeta_K(-1)=\frac{1}{12}$ and by the functional equation for Dedekind’s zeta function (cf. e.g. [@washington1997introduction p. 34]) one finds that $\zeta_K(2)=\frac{\pi^4}{48\sqrt{2}}$ which proves the claim. The density of visible points of $\mathcal{A}$ {#subsecDensAB} ---------------------------------------------- Observe that $\mathcal{A}'=\sqrt{2}\mathcal{A}\subset \mathcal{B}$. It is now shown that $C$ is also an occluding set for $\mathcal{A}'$. \[propOcclSet2\] For each $x\in\mathcal{A}'\setminus\widehat{\mathcal{A}'}$ there is $c\in C$ such that $x/c\in \mathcal{A}'$. Since $\mathcal{A}'\subset \mathcal{B}$ we have $\mathcal{A}'\setminus\widehat{\mathcal{A}'}\subset \mathcal{B}\setminus\widehat{\mathcal{B}}$ and so for each $x\in \mathcal{A}'\setminus\widehat{\mathcal{A}'}$ there exists $c\in C$ such that $x/c\in \mathcal{B}$. If $c\neq \sqrt{2}$ then $\sqrt{2}\mid \frac{x_1-x_2}{c}$ so $x/c\in\mathcal{A}'$. Take now $x\in \mathcal{A}'\setminus\widehat{\mathcal{A}'}$ such that for all $c\in C{\setminus\{\sqrt{2}\}}$ we have $x/c\notin \mathcal{B}$. Then $x/\sqrt{2}\in\mathcal{B}$, hence $\gcd(x_1,x_2)=\sqrt{2}^n$ for some $n\geq1$. Since $x\in \mathcal{A}'\setminus\widehat{\mathcal{A}'}$ there is $c\in {\mathbb{Q}}(\sqrt{2})\cap{\mathbb{R}}_{>1}$ such that $x/c\in \mathcal{A}'$. Writing $c=a/b$ with $\gcd(a,b)=1$, the only possible $\pi\in{\mathbb{P}}$ with $\pi\mid a$ is $\pi=\sqrt{2}$. If $\sqrt{2}\mid a$, then it follows that $x/\sqrt{2}\in\mathcal{A}'$. It remains to check the case where $a$ is a unit, i.e. $c=\frac{\lambda^n}{\prod_{p\in {\mathbb{P}}}\pi^{m(\pi)}}$ for some $m:{\mathbb{P}}\longrightarrow{\mathbb{Z}}_{\geq 0}$ with finite support. The facts that $c>1$ and $\pi>1$ for all $\pi\in{\mathbb{P}}$ imply $n>0$. We have $x/\lambda\notin\mathcal{B}$, hence $\overline{x}\notin |\overline{\lambda}| \mathcal{W}_\mathcal{A}$. Since $x/c\in\sqrt{2}\mathcal{A}$ it follows that $\overline{x}\in |\overline{c}|\mathcal{W}_\mathcal{A}$ and hence $|\overline{c}|>|\overline{\lambda}|$. However $$|\overline{c}|=\frac{|\overline{\lambda}|^n}{\prod_{\pi\in {\mathbb{P}}}|\overline{\pi}|^{k(\pi)}}\leq |\overline{\lambda}|^n\leq |\overline{\lambda}|,$$ contradiction. We have $\theta(\widehat{\mathcal{A}'})=\frac{1}{2\zeta_K(2)}$, hence $\theta(\widehat{\mathcal{A}})=\frac{1}{\zeta_K(2)}$. Propositions \[propInclExcl\], \[propOcclSet2\] imply $$\label{eqnB1} \frac{\#(\widehat{\mathcal{A}'}\cap RD)}{{\mathrm{vol}}(RD)}=\sum_{\substack{F\subset C\\ \#F<\infty}}\frac{(-1)^{\#F}\#\left(\left(\mathcal{A}_*'\cap \bigcap_{c\in F}c\mathcal{A}_*'\right)\cap RD\right)}{{\mathrm{vol}}(RD)}$$ and it is straight-forward to verify that $\mathcal{A}_*'\cap \bigcap_{c\in F}c\mathcal{A}_*'=\mathcal{P}(m_F\mathcal{W}_\mathcal{A},\widetilde{\mathcal{L}}_F){\setminus\{0\}}$ with $\widetilde{\mathcal{L}}_F=\{(\ell_Fx,\overline{\ell_Fx})\mid x\in \mathcal{O}_K^2,(x_1-x_2)/\sqrt{2}\in\mathcal{O}_K\}$ a sublattice of $\mathcal{\mathcal{L}}_F$ of index $2$. Hence, by [@marklof2014free Prop. 3.2], when letting $R\to\infty$ inside the sum one obtains $$\sum_{\substack{F\subset C\\ \#F<\infty}}\frac{(-1)^{\#F}{\mathrm{vol}}(m_F\mathcal{W}_\mathcal{A})}{16N(\ell_F)},$$ whence $\theta(\widehat{\sqrt{2}\mathcal{A}})=\frac{1}{2\zeta_K(2)}$ follows by and , and the other result is immediate as $\sqrt{2}\mathcal{A}=\mathcal{A}'$. The data of Table 2 of [@baake2014radial] shows that $\#(\widehat{\mathcal{A}}\cap RD)/\#(\mathcal{A}\cap RD)\approx 0.577$ for a particular $D$ and fairly large $R$. This agrees with our results, since $$\kappa_\mathcal{A}=\lim_{R\to\infty}\frac{\#(\widehat{\mathcal{A}}\cap RD)}{\#(\mathcal{A}\cap RD)}=\frac{\theta(\widehat{\mathcal{A}})}{\theta(\mathcal{A})}=\frac{\tfrac{1}{\zeta_K(2)}}{\tfrac{2{\mathrm{vol}}(\mathcal{W}_\mathcal{A})}{16}}=\frac{2(\sqrt{2}-1)}{\zeta_K(2)}=0.5773\ldots$$

--- abstract: 'In this paper, a point-to-point Orthogonal Frequency Division Multiplexing (OFDM) system with a decode-and-forward (DF) relay is considered. The transmission consists of two hops. The source transmits in the first hop, and the relay transmits in the second hop. Each hop occupies one time slot. The relay is half-duplex, and capable of decoding the message on a particular subcarrier in one time slot, and re-encoding and forwarding it on a different subcarrier in the next time slot. Thus each message is transmitted on a pair of subcarriers in two hops. It is assumed that the destination is capable of combining the signals from the source and the relay pertaining to the same message. The goal is to maximize the weighted sum rate of the system by jointly optimizing subcarrier pairing and power allocation on each subcarrier in each hop. The weighting of the rates is to take into account the fact that different subcarriers may carry signals for different services. Both total and individual power constraints for the source and the relay are investigated. For the situations where the relay does not transmit on some subcarriers because doing so does not improve the weighted sum rate, we further allow the source to transmit new messages on these idle subcarriers. To the best of our knowledge, such a joint optimization inclusive of the destination combining has not been discussed in the literature. The problem is first formulated as a mixed integer programming problem. It is then transformed to a convex optimization problem by continuous relaxation, and solved in the dual domain. Based on the optimization results, algorithms to achieve feasible solutions are also proposed. Simulation results show that the proposed algorithms almost achieve the optimal weighted sum rate, and outperform the existing methods in various channel conditions.' author: - Chih Ning Hsu - '\' bibliography: - 'IEEEabrv.bib' - 'aning\_thesis.bib' title: 'Joint Subcarrier Pairing and Power Allocation for OFDM Transmission with Decode-and-Forward Relaying' --- OFDM, decode-and-forward relay, power allocation, subcarrier pairing, optimization, continuous relaxation, Lagrange dual problem. Introduction ============ For an Orthogonal Frequency Division Multiplexing (OFDM) system with relay, identifying a proper way to allocate resources to the source and the relay is the main bottleneck for achieving good performance. In this paper, we consider a point-to-point OFDM system with a decode-and-forward (DF) half-duplex relay. Each message is transmitted in two hops each occupying one time slot. A message transmitted by the source on one subcarrier in the first time slot is, if successfully decoded by the relay, forwarded by the relay to the destination on one (not necessarily the same) subcarrier in the second time slot. With the assumption that the channel state information (CSI) is known at the source, many works have been done to make resource utilization of this system more efficient. A general downlink Orthogonal Frequency Division Multiple Access (OFDMA) relay system with individual power constraints at one source and many relays was considered in [@DF_individual_downlink]. In that work, joint optimization of the subcarrier selection and power allocation was done. However, that work assumed that a message is received by a destination either directly from the source, or from a relay which forwarded the message. Destination combining of the signals directly from the source and forwarded by the relay pertaining to the same message was not considered. In addition, as each relay collectively uses its active subcarriers to forward messages to different destinations, a more complicated re-encoding scheme has to be used by the relay to fit the received message for a particular destination into the subcarriers designated to that destination. In [@DF_OFDM_total_individual_power; @relay_DF_individual_power_constraint; @Vandendorpe_J], optimal power allocation for OFDM with DF relaying and fixed source and relay subcarrier pairing was proposed. [@DF_OFDM_total_individual_power][@Vandendorpe_J] considered two kinds of power constraints: one is that the total transmit power is shared between the source and the relay; the other has individual power constraints for the source and the relay. In [@relay_eq_channel_gain; @relay_AF_DF_eq_channel_gain; @Li_AFDF_OFDM], both power allocation and subcarrier pairing were considered for OFDM systems with relaying under the total power constraint. However, power allocation and subcarrier pairing were optimized separately. [@relay_eq_channel_gain] proposed a subcarrier pairing method by sorting the subcarriers of the source-relay (SR) link and the relay-destination (RD) link, respectively, according to their channel gains. The SR subcarrier and the RD subcarrier with the same respective ranks are then paired together. The optimality of this sorted channel pairing (SCP) scheme, in the absence of the source-destination (SD) link, for both DF and amplify-and-forward (AF) relaying schemes were proved in [@relay_AF_DF_eq_channel_gain][@Li_AFDF_OFDM]. SCP was also proposed in [@Herdin; @wittneben; @AF_pairing_without_diversity; @relay_AF_OPT_SUBCHANNEL_ASSIGNMENT] for OFDM AF relaying systems without the SD link, and in [@AF_OFDM_total_individual_power] when the SD link and destination combining are present. Power allocation with total and individual power constraints for OFDM AF relaying systems were considered in [@AF_pairing_without_diversity] and [@AF_OFDM_total_individual_power], while [@wittneben] focused on only the total power constraint. The above works dealing with power allocation for the OFDM AF relaying systems usually used approximations to relax the problem into a solvable one. Without making any approximations, [@relay_AF_total_optimal_power] investigated the optimal power allocation problem for the OFDM AF relaying systems with fixed subcarrier pairing and total power constraint in the absence of the SD link. In view of the lack of joint optimization of power allocation and subcarrier pairing for OFDM systems with DF relaying in the literature, the goal of this paper is to solve this problem with the presence of the SD link and destination combining of signals from the source and the relay. Both the total power constrained system and the individual power constrained system are considered. For the total power constrained system, we formulate the joint power allocation and subcarrier pairing problem as a mixed integer programming problem whose optimal solution is hard to obtain. We then use some special properties of the system and the continuous relaxation [@DF_individual_downlink][@concave_function] to reform the problem and solve the dual problem by the subgradient method [@subgradient_method]. With both the power and subcarrier pairing constraints, the optimization problem becomes very complicated, and the duality gap may not be zero. However, as verified by [@Yu_dual][@zero_gap_SUB_infinity] and our own simulation, the duality gap is virtually zero when the number of subcarriers is reasonably large. Thus the dual optimum value becomes a very tight upper bound for the primal optimum for most practical systems. In addition to the duality gap, some other practical issues such as algorithm design and complexity comparison are also discussed. We then extend the formulation to have individual power constraints, and find that the complications caused by individual power constraints can be alleviated in the dual domain. The dual optimum value is again a very tight upper bound for the primal optimum. Finally, we relax the constraint that only the relay can transmit in the second time slot. Therefore, additional messages may be transmitted on the idle subcarriers in the SD link in the second time slot, when it is deemed that relaying on these subcarriers does not improve the weighted sum rate. Such a model was also considered in [@Vandendorpe_J]. However, [@Vandendorpe_J] optimized power allocation (and relaying modes) only for a particular subcarrier pairing scheme without weighting of the rates. These conditions made the problem easier to solve. In this paper, we consider joint optimization of power allocation and subcarrier pairing with weighted rates. The problem is more general and difficult. However, by defining an additional indicator, we can formulate the problem similarly as in the case without the second-slot SD transmission. The problem is then solved in the dual domain. Simulation shows that, for this problem, the duality gap is also nearly zero. Based on the optimization results, algorithms to achieve feasible subcarrier pairing and power allocation are also proposed. Simulation results show that the proposed algorithms almost achieve the optimal weighted sum rate, and outperform the SCP proposed in [@relay_eq_channel_gain] in various channel conditions. The rest of this paper is organized as follows. Section \[Sec\_system\_model\] describes the system model. Section \[Sec\_maximization\_typeI\] solves the optimization problem under the total power constraint. Detailed discussions on the practical issues are also presented in this section. Section \[sec\_maximization\_individual\] solves the optimization problem under the individual power constraints. Section \[sec\_maximization\_total\_extra\_direct\] formulates and solves the optimization problem for the system with additional messages transmitted on the SD link in the second time slot, under both total and individual power constraints. Section \[Sec\_simulation\] summarizes our results and observations. Section \[Sec\_conclusion\] concludes this paper. System Model {#Sec_system_model} ============ We consider a two-hop DF relay system consisting of one source, one relay, and one destination. OFDM with the same spectral occupancy is used for all links. The total frequency band is divided into $M$ subcarriers. To avoid interference, for each subcarrier, only one node (the source or the relay) transmits in a given time slot. All time slots are of the same duration. The source transmits in the first time slot while the relay and the destination receive. The relay is half-duplex that receives in the first time slot and transmits in the second time slot. Each subcarrier used by the source in the first time slot is paired with one subcarrier used by the relay in the second time slot to convey a message. Therefore the number of subcarrier pairs in transmission is $M$. If subcarrier $k$ in the first time slot and subcarrier $m$ in the second time slot are paired, we call them subcarrier pair (SP) $(k,m)$. It is assumed that the relay re-encodes the received message with the same codebook as the one used by the source. The destination maximum ratio combines (MRC) the signals from the source in the first time slot and from the relay in the second time slot pertaining to the same message to exploit the spatial diversity. The messages transmitted on different SPs are assumed to be independent. The channel model associated with SP $(k,m)$ is shown in Fig. \[Fig\_channel\_model\]. We use $h^{SD}_{k}$, $h^{SR}_{k}$, and $h^{RD}_{m}$ to denote the channel gains of the SD link, SR link, and RD link on subcarriers $k$, $k$, and $m$, respectively. $\sigma^2_{SD,k}$, $\sigma^2_{SR,k}$, and $\sigma^2_{RD,m}$ are the variances of the additive white Gaussian noises (AWGN) in the corresponding channels. As shown in Fig. \[Fig\_channel\_model\], we use $a^{SD}_{k}=\frac{| h^{SD}_{k}|^2}{\sigma^2_{SD,k}}$, $a^{SR}_{k}=\frac{|h^{SR}_{k}|^2}{\sigma^2_{SR,k}}$, and $a^{RD}_{m}=\frac{|h^{RD}_{m}|^2}{\sigma^2_{RD,m}}$ to denote the normalized channel gains. The channels are assumed to remain constant in a two-slot period. All the normalized channel gains are assumed known at the source which will perform subcarrier pairing and power allocation. The source then informs the relay and the destination of the corresponding parameters via proper control signaling before the data transmission. These assumptions are reasonable for the situations where the channel coherence time is longer than the sum of the CSI measurement and feedback time, the control signaling time, and the data transmission duration. In practical implementation, the channel gains can be measured at the relay and the destination during the training period preceding the data transmission period. The training period has a similar structure as the data transmission period in which the source transmits training signals during the first time slot while the relay and the destination measure the SR and SD channels, respectively. The relay then transmits training signals in the second time slot to let the destination measure the RD channel. A training slot could be shorter than a data transmission slot. The measured channel gains can be fed back to the source on dedicated reverse control channels. After the source has done subcarrier pairing and power allocation, it can embed the pairing and power allocation parameters in the beginning of the first-slot data transmission. This embedded control signal is transmitted with stronger power and/or more reliable coding. So it can be guaranteed that the relay and destination can successfully decode the relevant parameters to figure out how to receive (and for the relay, how to forward as well) the upcoming data. Taking the same assumption as in [@DF_OFDM_total_individual_power; @relay_DF_individual_power_constraint; @relay_eq_channel_gain; @relay_AF_DF_eq_channel_gain; @Li_AFDF_OFDM], in Section \[Sec\_maximization\_typeI\] and Section \[sec\_maximization\_individual\] we first consider the scenario where for each SP $(k,m)$, the source only transmits in the first time slot. Even if it is decided that the relay will not transmit on subcarrier $m$, the source is not allowed to use this idle subcarrier in the second time slot. In Section \[sec\_maximization\_total\_extra\_direct\], this restriction is relaxed and the source is allowed to transmit additional messages in the second time slot on the subcarriers not used by the relay. This model has also been investigated in [@Vandendorpe_J] which assumed fixed subcarrier pairing with SPs $(k,k), k=1, 2, \ldots, M$. Together with unweighted rates, the $(k,k)$ subcarrier pairing makes determination of whether the relay will be active for SP $(k,k)$ and optimal power allocation among the SPs easier to solve. However, it is inferior and less general than the joint optimization of subcarrier pairing and power allocation considered in Section \[sec\_maximization\_total\_extra\_direct\]. For the sake of generality, we consider weighted sum rate as the performance metric. A weighting factor $w_{k} \geq 0$ is assigned to the rate transmitted by the source on subcarrier $k$ to reflect different priorities or quality-of-service (QoS) requirements. Weighted Sum Rate Maximization under Total Power Constraint {#Sec_maximization_typeI} =========================================================== In this section, we consider joint optimization of subcarrier pairing and power allocation to achieve the highest weighted sum rate under the total power constraint. We first give the problem formulation. Then a solution in the dual domain is given. The duality gap and achievability of the optimal solution, together with some practical algorithm design issues, will be discussed. Primal Problem Formulation {#primal_total} -------------------------- For a given SP $(k,m)$, let $R_{k,m}$ be its achievable weighted rate, and $p^{S}_{k,m}$ and $p^{R}_{k,m}$ be the source power in the first time slot and the relay power in the second time slot, respectively. Depending on whether the relay is active, this SP may work in either the relay mode or the direct-link mode. In the relay mode, the half-duplex relay forwards the message on subcarrier $m$ in the second time slot. In the direct-link mode, the relay does not forward, and only subcarrier $k$ of the SD link in the first time slot is used to transmit the message. Thus the weighted rate achievable with Gaussian codebooks for SP $(k,m)$ can be expressed as [@relay_capacity_theorem] $$\label{Eq_pair_rate_typeI} R_{k,m}=\left\{ \begin{aligned} &\frac{w_{k}}{2}\log(1+a^{SD}_{k}p^{S}_{k,m}), &\text{direct-link mode}\\ &\frac{w_{k}}{2}\min\left\{\log\left(1+a^{SR}_{k}p^{S}_{k,m}\right),\; \log\left(1+a^{SD}_{k}p^{S}_{k,m}+a^{RD}_{m}p^{R}_{k,m}\right)\right\}, &\text{relay mode}\\ \end{aligned} \right.$$ where the rate is scaled by $\frac{1}{2}$ because the transmission takes two time slots. Under the total power constraint of $p_{k,m}=p_{k,m}^S+p_{k,m}^R$ for the SP $(k,m)$, using relay is advantageous in terms of maximizing the achievable rate when [@DF_OFDM_total_individual_power] $$\label{relay_condition_total} a^{SR}_{k}>a^{SD}_{k}~ \text{ and } ~a^{RD}_{m}>a^{SD}_{k}.$$ In addition, based on the fact that, for the relay mode, the achievable rate is maximized when the amounts of received information at the relay and the destination are the same, the expressions in (\[Eq\_pair\_rate\_typeI\]) can be unified as [@relay_eq_channel_gain] $$\label{unified_rate} \begin{aligned} R_{k,m}=\frac{w_k}{2}\log(1+a_{k,m}p_{k,m}). \end{aligned}$$ This is obtained by letting $$\label{power_ratio} \begin{aligned} &p^{S}_{k,m}=\left\{ \begin{aligned} &\frac{a^{RD}_{m}}{a^{SR}_{k}+a^{RD}_{m}-a^{SD}_{k}}p_{k,m},&\text{relay mode}\\ &p_{k,m},&\text{direct-link mode}\\ \end{aligned} \right.\\ &p^{R}_{k,m}=\left\{ \begin{aligned} &\frac{a^{SR}_{k}-a^{SD}_{k}}{a^{SR}_{k}+a^{RD}_{m}-a^{SD}_{k}}p_{k,m},&\text{relay mode}\\ &0,&\text{direct-link mode} \end{aligned} \right. \end{aligned}$$ in (\[Eq\_pair\_rate\_typeI\]), and defining $a_{k,m}$ as the equivalent channel gain given by $$\label{Eq_ECG} a_{k,m}=\left\{ \begin{aligned} &\frac{a^{SR}_{k}a^{RD}_{m}}{a^{SR}_{k}+a^{RD}_{m}-a^{SD}_{k}},&\text{relay mode}\\ &a^{SD}_{k},&\text{direct-link mode}. \end{aligned} \right.$$ Thus, when the channel gains are known, for any possible pairing, whether a SP $(k,m)$ should be in the relay mode or the direct-link mode, and the maximum achievable weighted rate of this SP as a function of the total power $p_{k,m}$, can be derived immediately. Define an indicator $t_{k,m}$ which is 1 if SP $(k,m)$ is selected, and 0 otherwise. The weighted sum rate optimization problem can be formulated as $$\begin{aligned} \label{objective_Rate_I_MIP} \max_{{\bm p},{\bm t}}\text{ }&\sum^M_{k=1}\sum^M_{m=1}t_{k,m}\frac{w_{k}}{2}\log(1+a_{k,m}p_{k,m}) \\ \label{constraint_power_consumption} \text{s.t.}\text{ }&\sum^M_{k=1}\sum^M_{m=1}p_{k,m}\leq P, \\ \label{constraint_t_k} &\sum^M_{k=1}t_{k,m}=1,\text{ }\forall m, \\ \label{constraint_t_m} &\sum^M_{m=1}t_{k,m}=1,\text{ }\forall k, \\ \label{constraint_power_0} &p_{k,m}\geq~0,\text{ }\forall k, m, \\ \label{constraint_t_binary} &t_{k,m}\in\{0,1\},\text{ }\forall k, m,\end{aligned}$$ where $P$ is the total power constraint, ${\bm p}\in \mathds{R}_+^{M \times M}$ (with $\mathds{R}_+$ denoting the set of nonnegative real numbers) and ${\bm t}\in\{0,1\}^{M \times M}$ are matrices with entries $p_{k,m}$ and $t_{k,m}$, respectively. Since the power allocated to the unselected SPs does not contribute to the weighted sum rate, it is obvious that the optimal solution will only allocate non-zero power to the selected SPs. Although similar in the approach, there are some significant differences between the above problem formulation and the ones in [@DF_individual_downlink] and [@relay_AF_OPT_SUBCHANNEL_ASSIGNMENT]. [@DF_individual_downlink] and [@relay_AF_OPT_SUBCHANNEL_ASSIGNMENT] both did not consider the SD link and destination combining when the relay is used. In [@relay_AF_OPT_SUBCHANNEL_ASSIGNMENT], the power allocated to each subcarrier is fixed. As mentioned in Section \[introduction\], the relays in [@DF_individual_downlink] have to use complicated re-encoders with codebooks different from that of the source. These differences make our optimization problem distinct from [@DF_individual_downlink] and [@relay_AF_OPT_SUBCHANNEL_ASSIGNMENT]. The above problem is a mixed integer programming (MIP) problem which is hard to solve. Therefore, as in [@concave_function][@multilevel_waterfilling], we relax the integer constraint of (\[constraint\_t\_binary\]) as $t_{k,m}\in \mathds{R}_+, \forall k, m$. This continuous relaxation makes $t_{k,m}$ the time sharing factor of each SP. The relaxed problem then becomes $$\begin{aligned} \label{objective_Rate_I_relaxed} \max_{{\bm p},{\bm t}} ~\frac{1}{2}\sum^M_{k=1}\sum^M_{m=1}t_{k,m}~w_{k}\log\left(1+a_{k,m}\frac{p_{k,m}}{t_{k,m}}\right)~~\text{s.t.}~~ (\ref{constraint_power_consumption}), (\ref{constraint_t_k}), (\ref{constraint_t_m}), (\ref{constraint_power_0}), ~\text{and}\\ \label{constraint_t_0} t_{k,m}\geq 0,\text{ }\forall k, m.\end{aligned}$$ Note that the value of the objective function (\[objective\_Rate\_I\_relaxed\]) is the same as that of the original objective function (\[objective\_Rate\_I\_MIP\]) when $t_{k,m}\in\{0,1\}, \forall k, m$. This objective function is concave because it is a nonnegative weighted sum of concave functions in the form of $x\log(1+\frac{y}{x})$ which is concave in $(x, y)$ [@concave_function]. Since (\[objective\_Rate\_I\_relaxed\]) is a standard convex programming problem, it can be solved by numerical search algorithms such as the interior-point method [@Book_Boyd]. However, the optimal $t_{k,m}$ may not be integer-valued. Therefore, we opt to solve this problem by the dual method which can provide an upper bound for problem (\[objective\_Rate\_I\_relaxed\]) (by the weak duality [@Book_Boyd]). In Section \[sec\_dual\_type\_I\_total\], it will be shown that the solution obtained by the dual method has $t_{k,m}\in\{0,1\}, \forall k, m$. Dual Problem {#sec_dual_type_I_total} ------------ By dualizing constraints (\[constraint\_power\_consumption\]) and (\[constraint\_t\_k\]), we obtain the Lagrangian as follows: $$\label{Lagrangian_typeI} \begin{aligned} L( {\bm p}, {\bm t},\mu, {\bm \alpha})=&\frac{1}{2}\sum_{k=1}^M\sum_{m=1}^Mt_{k,m}~w_{k}\log\left(1+a_{k,m}\frac{p_{k,m}}{t_{k,m}}\right)\\ &+\mu\left(P-\sum_{k=1}^M\sum_{m=1}^M~p_{k,m}\right)+\sum_{m=1}^M\alpha_m\left(1-\sum_{k=1}^M~t_{k,m}\right), \end{aligned}$$ where $\mu\in \mathds{R}_+$ and ${\bm \alpha} ~(\text{the vector of} ~\alpha_m) \in\mathds{R}^M$ are the dual variables, with $\mathds{R}$ denoting the set of real numbers. The dual objective function is $$\label{P_Lagrange_typeI} h(\mu, {\bm \alpha})= \max_{{\bm p}, {\bm t}} ~L({\bm p},{\bm t},\mu, {\bm \alpha}) ~~\text{s.t.}~~ (\ref{constraint_t_m}),(\ref{constraint_power_0}),(\ref{constraint_t_0})$$ and the dual problem is $$\label{objective_dual} \min_{\mu, {\bm \alpha}} ~h(\mu, {\bm \alpha}) ~~\text{s.t.}~~ \mu\geq 0.$$ It is well known that a function can be maximized by first maximizing over some of the variables, and then maximizing over the remaining ones [@Book_Boyd Sec 4.1.3]. Thus we first solve $p_{k,m}$ for (\[P\_Lagrange\_typeI\]) by $$\label{Eq_L_partial_p_typeI} \frac{\partial{L}}{\partial{p_{k,m}}}=\frac{t_{k,m}w_{k}}{2}\frac{\frac{a_{k,m}}{t_{k,m}}}{1+a_{k,m}\frac{p_{k,m}}{t_{k,m}}}-\mu=\frac{w_{k}}{2}\frac{1}{\frac{1}{a_{k,m}}+\frac{p_{k,m}}{t_{k,m}}}-\mu =0$$ with constraint (\[constraint\_power\_0\]). The optimal solution is $$\begin{aligned} \label{optimal power_typeI} p_{k,m}^\ast=t_{k,m}\left[\frac{w_{k}}{2\mu}-\frac{1}{a_{k,m}}\right]^+,\end{aligned}$$ where $x^+\triangleq\max\{x,0\}$. This is similar to the result of multi-level water-filling[@multilevel_waterfilling]. We then rewrite (\[Lagrangian\_typeI\]) as $$\begin{aligned} L({\bm p}^*,{\bm t},\mu, {\bm \alpha})=\sum_{k=1}^M\sum_{m=1}^M~t_{k,m}X_{k,m}+K(\mu, {\bm \alpha}), \end{aligned}$$ where $$\label{compute X} X_{k,m}=\frac{w_{k}}{2}\log\left(1+a_{k,m}\left[\frac{w_{k}}{2\mu}-\frac{1}{a_{k,m}}\right]^+\right)-\alpha_m -\mu\left(\left[\frac{w_{k}}{2\mu}-\frac{1}{a_{k,m}}\right]^+\right),$$ $$K(\mu, {\bm \alpha})=\mu P+\sum_{m=1}^M \alpha_m.$$ We give an intuitive explanation for each term in $X_{k,m}$. The first term can be viewed as the rate obtained by selecting subcarrier $m$ in the second time slot for subcarrier $k$ in the first time slot. $\alpha_m$ is the penalty of selecting subcarrier $m$ in the second time slot. The last term is the price of power consumption. Due to the fact that $K(\mu, {\bm \alpha})$ and $X_{k,m}$ are independent of ${\bm t}$, we can easily find the optimal ${\bm t}$ for (\[P\_Lagrange\_typeI\]) with constraints (\[constraint\_t\_m\]) and (\[constraint\_t\_0\]) as $$\label{optimal_t_typeI} t_{k,m}^\ast=\left\{ \begin{aligned} &1,\text{ }m=\arg\max_{m=1,...,M}X_{k,m}\\ &0,\text{ otherwise} \end{aligned}, ~~~~\forall k \right. .$$ In operation, we first assume that $\mu$ and $\alpha_m$’s are given. Then the power allocation for every possible SP can be computed by (\[optimal power\_typeI\]) (with $t_{k,m}$ ignored). These power allocation values are used in (\[compute X\]) to compute $X_{k,m}$’s. After that, each subcarrier $k$ in the first time slot will independently select the subcarrier in the second time slot that gives the largest $X_{k,m}$ to maximize the the dual objective function (\[P\_Lagrange\_typeI\]). The last step is to find the values of $\mu$ and ${\bm \alpha}$ which minimize $h(\mu,{\bm \alpha})$. Using the subgradient method [@subgradient_method], the values of $\mu$ and ${\bm \alpha}$ can be found iteratively as $$\label{sub_gradient_typeI_total} \begin{aligned} &\mu^{(i+1)}=\mu^{(i)}-y^{(i)}\left(P-\sum_{k=1}^M\sum_{m=1}^M~p^{(i)}_{k,m}\right),\\ &\alpha_m^{(i+1)}=\alpha_m^{(i)}-z^{(i)}\left(1-\sum_{k=1}^M~t_{k,m}^{(i)}\right)\text{, }m=1,...,M, \end{aligned}$$ where the superscript $(i)$ denotes the iteration index, and $y^{(i)}$ and $z^{(i)}$ are the sequences of step sizes designed properly. With the new $\mu$ and ${\bm \alpha}$ in each iteration, the subcarrier pairing and power allocation can be updated with (\[optimal\_t\_typeI\]) and (\[optimal power\_typeI\]), respectively, for the next iteration. As the number of iterations increases, (\[sub\_gradient\_typeI\_total\]) will converge to the dual optimum variables [@subgradient_method]. The optimal ${\bm \alpha}$, together with (\[optimal\_t\_typeI\]), make $t_{k,m}^\ast$’s satisfy (\[constraint\_t\_k\]) and (\[constraint\_t\_m\]). Note that with the optimal power allocation given in (\[optimal power\_typeI\]), the achievable rate for SP $(k, m)$ is $$\label{impact_weighting} \begin{aligned} \frac{1}{2}\log\left(1+a_{k,m}\left[\frac{w_{k}}{2\mu}-\frac{1}{a_{k,m}}\right]^+\right)=\frac{1}{2}\log\left(1+a_{k,m}w_{k}\left[\frac{1}{2\mu}-\frac{1}{w_{k}a_{k,m}}\right]^+\right). \end{aligned}$$ From (\[impact\_weighting\]), the impact of the weighting factors can be viewed as weighting the channel gain of SP $(k, m)$ by $w_{k}$. A higher weighting factor results in more power allocated to the corresponding SP. Discussion on the Duality Gap {#duality_gap_total} ----------------------------- For problem (\[objective\_Rate\_I\_MIP\]), the optimal subcarrier pairing scheme may change as the total power constraint varies. Thus the maximum weighted sum rate as a function of the total power constraint may have discrete changes in the slope at the transition points where the optimal subcarrier pairing scheme changes. An example is shown in the circled region in Fig. \[Fig\_rate\_power\_concave\] for $M=2$ subcarriers. This phenomenon is similar to that observed in the optimal resource allocation for OFDMA downlink systems [@zero_gap_SUB_infinity]. However, in our case, this phenomenon is observed even when the weighting factors for all subcarriers are set to the same. As discussed in [@Yu_dual][@zero_gap_SUB_infinity], the nonconcavity shown in Fig. \[Fig\_rate\_power\_concave\] may result in nonzero duality gap. Let us denote the optimal values of the original problem (\[objective\_Rate\_I\_MIP\]), the relaxed problem (\[objective\_Rate\_I\_relaxed\]), and the relaxed dual problem (\[objective\_dual\]) by $R_B$, $R_R$, and $D_R$, respectively. The relationship between them is $R_B\leq~R_R\leq~D_R$. Since the optimal $t_{k,m}^\ast$’s found by solving (\[P\_Lagrange\_typeI\]) and (\[objective\_dual\]) satisfy (\[constraint\_t\_k\]), (\[constraint\_t\_m\]) and (\[constraint\_t\_binary\]), we conclude that $D_R$ is also the dual optimum value for problem (\[objective\_Rate\_I\_MIP\]). According to [@Yu_dual][@Chiang_zero_gap][@zero_gap_SUB_infinity], the duality gap is zero if the optimal value of the optimization problem is a concave function of the constraints. [@Yu_dual] and [@zero_gap_SUB_infinity] also showed analytically and through simulations that the concavity will be satisfied as the number of subcarriers becomes large. In our case, we found that the concavity is mostly satisfied when the number of subcarriers is reasonably large. Specifically, when $M=2$, we have observed in simulation that only about $1\%$ of the possible channel realizations will result in the nonconcavity shown in Fig. \[Fig\_rate\_power\_concave\]. When $M=4$, the probability of nonconcavity is about $0.4\%$. For $M \geq 6$, the maximum weighted sum rate is almost always concave in the total power constraint. An example is shown in Fig. \[Fig\_rate\_power\_concave\] for $M=8$ subcarriers. Thus, for practical OFDM systems, the duality gap is virtually zero, and $R_B\approx D_R$. We can then conclude that $R_B\approx R_R\approx D_R$ for most practical OFDM systems. This will be verified by the simulation results in Section \[Sec\_simulation\]. Algorithm Design {#Sec_achieave_feasible} ---------------- Combining (\[optimal\_t\_typeI\]), (\[optimal power\_typeI\]) and (\[sub\_gradient\_typeI\_total\]), the algorithm to find the optimal subcarrier pairing and power allocation can be designed as in the upper part of Table \[Table\_algo\_obtain\_feasible\_typeI\]. However, through simulation, we have observed that although (\[optimal\_t\_typeI\]) guarantees that each row of ${\bm t}$ has only one “1", some of the “1"s may be on the same column. This corresponds to the situation where more than one source subcarriers select the same relay subcarrier. As a result, the constraint (\[constraint\_t\_k\]) is violated, and the solution is not feasible. This situation usually arises in two scenarios. The first is when there are more than one source subcarriers with very strong SD gains, such that no matter which relay subcarrier they are paired with, the direct-link mode will be selected. For any of these source subcarriers, the power related terms in $X_{k,m}$ (\[compute X\]) are the same for all relay subcarriers. Thus the relay subcarrier selection (\[optimal\_t\_typeI\]) depends only on $\alpha_m$. The source subcarriers with this property will select the same relay subcarrier. The other scenario is when a source subcarrier gets a low equivalent SP channel gain $a_{k,m}$ no matter which relay subcarrier it is paired with. All the possible SPs formed by this source subcarrier will be allocated very little power, thus their $X_{k,m}$’s are dominated by the corresponding $\alpha_m$’s. Similarly, the source subcarriers with this property will most likely select the same relay subcarrier. To handle this situation, we include an amendment algorithm in the original algorithm as shown in the lower part of Table \[Table\_algo\_obtain\_feasible\_typeI\]. Based on the above discussion, the basic idea of the amendment algorithm is to each time move a “1" in a column of ${\bm t}$ with more than one “1"s to the column with no “1" that will cause the minimum change in the value of $\alpha_m$. By moving a “1" to another column with a similar $\alpha_m$ value, the weighted sum rate will not be lowered much. When doing so, the amendment algorithm will make sure to keep the “1" corresponding to the largest $X_{k,m}$ for each column with more than one “1"s. It will also move the redundant “1"s to the columns with no “1" that will result in as large $X_{k,m}$ values as possible. Thus the resultant weighted sum rate will be maximized. Eventually the pairing scheme ${\bm t}$ altered by the amendment algorithm will meet the constraints (\[constraint\_t\_k\]) and (\[constraint\_t\_m\]). The amendment algorithm is triggered when the dual variables converge to a certain degree (for the example in Table \[Table\_algo\_obtain\_feasible\_typeI\], within 1%). Once the amendment algorithm is triggered, the algorithm will continue to run for another 10% of iterations. For example, if the amendment algorithm is triggered at the 1000th iteration, the algorithm will run another 100 iterations before it outputs the solution. For each of these 10% of iterations, a feasible pairing scheme will be obtained by the amendment algorithm. Using this pairing scheme, regular water-filling over parallel channels will be applied to obtain the optimal power allocation and the corresponding weighted sum rate. The best pairing scheme and power allocation among these iterations that achieve the highest weighted sum rate will be the outputs of this algorithm. As shown in Section \[Sec\_simulation\], the weighted sum rate obtained by the algorithm in Table \[Table\_algo\_obtain\_feasible\_typeI\] is quite close to the optimal. Complexity Comparison --------------------- The total number of all possible pairing schemes is $O(M!)$. With a fixed subcarrier pairing scheme, the complexity of computing the optimal power allocation (\[optimal power\_typeI\]) for the selected pairs is $O(M)$ in terms of multiplications. The complexity of computing the resulting weighted sum rate (weighted sum of (\[unified\_rate\])) is also $O(M)$ in terms of $\log(\cdot)$ operations and multiplications. Thus the complexity of exhaustive search is $O(M\cdot M!)$ which is prohibitively high. On the other hand, in each iteration of the algorithm in Table \[Table\_algo\_obtain\_feasible\_typeI\], the complexity is dominated by the computation of $X_{k,m}, \forall k,m$, in (\[compute X\]). That complexity is $O(M^2)$ in terms of $\log(\cdot)$ operations and multiplications. For the amendment algorithm in the last 10% of iterations, alteration of the pairing scheme ${\bm t}$ takes only $O(M^2)$ additions and $\max(\cdot)$ and $\min(\cdot)$ operations. The complexity of computing the optimal power allocation and the resulting weighted sum rate is $O(M)$ multiplications and $\log(\cdot)$ operations. Therefore the overall complexity for the algorithm in Table \[Table\_algo\_obtain\_feasible\_typeI\] is $O(JM^2)$, where $J$ is the number of iterations. This complexity is much more feasible and tractable. Weighted Sum Rate Maximization under Individual Power Constraints {#sec_maximization_individual} ================================================================= When the source and the relay have individual power constraints, the weighted sum rate maximization problem becomes $$\begin{aligned} \label{objective_Rate_I_individual} \max_{{\bm p}_{S},{\bm p}_{R},{\cal S}_S,{\cal S}_R} & ~\frac{w_{k}}{2} \left(\sum_{(k,m) \in {\cal S}_S}\log\left(1+a^{SD}_{k}p^{S}_{k,m}\right) \right.\\ &\left.+\sum_{(k,m) \in {\cal S}_R} \min\left\{\log\left(1+a^{SR}_{k}p^{S}_{k,m}\right),\; \log\left(1+a^{SD}_{k}p^{S}_{k,m}+a^{RD}_{m}p^{R}_{k,m}\right)\right\} \right) \\ \label{constraint_individual_power_0} \text{s.t.}\text{ }&p^S_{k,m}, ~p^{R}_{k,m}\geq~0,\forall k, m \\ \label{constraint_source_power_consumption} &\sum_{(k,m) \in {\cal S}_S \cup {\cal S}_R} p^S_{k,m}\leq P_{S} \\ \label{constraint_relay_power_consumption} &\sum_{(k,m) \in {\cal S}_R} p^R_{k,m}\leq P_{R}\end{aligned}$$ where $P_{S}$ and $P_{R}$ are the source and the relay power constraints, respectively, and ${\bm p}_{S} \in \mathds{R}_+^{M\times M}$ and ${\bm p}_{R} \in \mathds{R}_+^{M\times M}$ are the matrices of $p^S_{k,m}$ and $p^R_{k,m}$, respectively. ${\cal S}_S$ and ${\cal S}_R$ denote the sets of SPs operating in the direct-link mode and the relay mode, respectively. If we let $t_{k,m}=1$ when $(k,m) \in {\cal S}_S \cup {\cal S}_R$ and $t_{k,m}=0$ otherwise, ${\cal S}_S$ and ${\cal S}_R$ must satisfy the additional constraints (\[constraint\_t\_k\]) and (\[constraint\_t\_m\]). This problem is very complicated. Because the condition to use relay depends not only on the channel condition, but also indirectly on the source power and relay power constraints [@DF_OFDM_total_individual_power][@relay_DF_individual_power_constraint][@Vandendorpe_J], it is not possible to classify the SPs into the direct-link mode or the relay mode in advance to use the unified weighted rate formulation (\[unified\_rate\]) and the equivalent channel gain (\[Eq\_ECG\]). In Section \[unified\_rate\_individual\], we will first investigate optimal power allocation with fixed subcarrier pairing under individual power constraints considered in [@DF_OFDM_total_individual_power][@relay_DF_individual_power_constraint][@Vandendorpe_J]. Through some insightful observations on the results of [@relay_DF_individual_power_constraint], we will find that the unified weighted rate formulation (\[unified\_rate\]) and the equivalent channel gain (\[Eq\_ECG\]) can, in fact, be applied to the dual problem of (\[objective\_Rate\_I\_individual\]). After that, (\[objective\_Rate\_I\_individual\]) can be solved similarly as in the total power constrained case. Unified Rate Formulation {#unified_rate_individual} ------------------------ For (\[objective\_Rate\_I\_individual\]), assuming fixed subcarrier pairing, a Lagrangian similar to [@relay_DF_individual_power_constraint eq. (8)] with slight changes can be obtained $$\label{Louveaux_Lagrange} \begin{aligned} L&=\sum_{(k,m) \in {\cal S}_S}\frac{w_{k}}{2}\log\left(1+a^{SD}_{k}p^{S}_{k,m}\right)\\ &+\sum_{(k,m) \in {\cal S}_R}\frac{w_{k}}{2}\log\left(1+a^{SD}_{k}p^{S}_{k,m}+a^{RD}_{m}p^{R}_{k,m}\right)\\ &+\mu_{S}\left(P_{S}-\sum_{k=1}^M p^{S}_{k,m}\right)+\mu_{R}\left(P_{R}-\sum_{(k,m) \in {\cal S}_R} p^{R}_{k,m}\right)\\ &+\sum_{(k,m) \in {\cal S}_R}\rho_{k,m}\left(a^{SR}_{k}p^{S}_{k,m}-a^{SD}_{k}p^{S}_{k,m}-a^{RD}_{m}p^{R}_{k,m}\right), \end{aligned}$$ where $\mu_S\geq 0$ and $\mu_R \geq 0$ denote the Lagrange multipliers corresponding to the source power constraint $P_S$ and the relay power constraint $P_R$, respectively. The third Lagrange multiplier $\rho_{k,m} \geq 0$ corresponds to the condition $$\label{Louveaux_relay_condition} a^{SR}_{k}p^{S}_{k,m} \geq a^{SD}_{k}p^{S}_{k,m}+a^{RD}_{m}p^{R}_{k,m}$$ for the SP $(k,m)$ to operate in the relay mode [@relay_DF_individual_power_constraint][@Vandendorpe_J]. If (\[Louveaux\_relay\_condition\]) is not valid for a relay mode SP, apparently some of that SP’s relay power can be reallocated, without reducing its rate, to other relay subcarriers to improve their rates, or simply to be conserved. Following the same procedures as in [@relay_DF_individual_power_constraint], a SP $(k,m)$ can be further classified into the following three modes given that $\mu_S$ and $\mu_R$ are fixed: $$\label{mode_conditions_individual} \left\{ \begin{aligned} &\text{Direct-link mode}:&&a^{SD}_{k}\geq~a^{SR}_{k}\text{ or }a^{RD}_{m}<~a^{SD}_{k}\frac{\mu_{R}}{\mu_{S}}\\ &\text{Relay mode}:&&a^{SR}_{k}>a^{SD}_{k}\text{ and } a^{RD}_{m}>a^{SD}_{k}\frac{\mu_{R}}{\mu_{S}}\\ &\text{Intermediate mode}:&&a^{SR}_{k}>a^{SD}_{k}\text{ and } a^{RD}_{m}=a^{SD}_{k}\frac{\mu_{R}}{\mu_{S}},\\ \end{aligned} \right.$$ where the intermediate mode is a special case of the relay mode with the condition (\[Louveaux\_relay\_condition\]) satisfied with strict inequality (and the corresponding $\rho_{k,m}=0$). That is, the relay receives more information than the destination. Thus the relay mode here is redefined to include only the SPs that satisfy (\[Louveaux\_relay\_condition\]) with equality. That is, the amounts of received information are the same at the relay and the destination. According to [@relay_DF_individual_power_constraint], usually there is at most one SP in the intermediate mode. For both the relay mode and the intermediate mode SPs, the solutions that maximize the Lagrangian (\[Louveaux\_Lagrange\]) will make its last term zero. The first condition $a^{SD}_{k}\geq a^{SR}_{k}$ for selecting the direct-link mode over the relay mode for SP $(k,m)$ is based on the fact that, in this situation, the destination will receive more information than the relay. Then there is no need to use the relay. The second condition $a^{RD}_{m}<a^{SD}_{k}\frac{\mu_{R}}{\mu_{S}}$ ensures that the direct-link mode will contribute more to the Lagrangian (\[Louveaux\_Lagrange\]) than the relay mode. For example, for SP $(k,m)$, if the power “cost" for selecting the direct-link mode, $\mu_S p^{S}_{k,m}$, in (\[Louveaux\_Lagrange\]) is kept the same as the power cost for selecting the relay mode, $\mu_S p^{S}_{k,m}+\mu_R p^{R}_{k,m}$, when $a^{RD}_{m}<a^{SD}_{k}\frac{\mu_{R}}{\mu_{S}}$, selecting the direct-link mode will result in a higher weighted rate (hence larger value of (\[Louveaux\_Lagrange\])) than selecting the relay mode. This is implied by the “single sum power constraint” approach in [@Vandendorpe_J eqs. (39) and (40)] with modified power and channel gain variables. By solving (\[Louveaux\_Lagrange\]) for weighted sum rate maximization, the power allocation can be obtained $$\label{Louveaux_power_allocation} \left\{ \begin{aligned} &\text{Direct-link mode}:&& p^{S}_{k,m}=\left[\frac{w_{k}}{2\mu_{S}}-\frac{1}{a^{SD}_{k}}\right]^+, ~p^{R}_{k,m}=0\\ &\text{Relay mode}:&& p^{S}_{k,m}=\left[\frac{w_{k}}{2\left(\mu_{S}+\mu_{R}/\beta_{k,m}\right)}-\frac{1}{a^{SR}_{k}}\right]^+, ~p^{R}_{k,m}=\left[\frac{w_{k}}{2\left(\mu_{S}\beta_{k,m}+\mu_{R}\right)}-\frac{1}{a^{SR}_{k}\beta_{k,m}}\right]^+\\ &\text{Intermediate mode}:&& \text{According to } ~1+a^{SD}_{k}p^{S}_{k,m}+a^{RD}_{m}p^{R}_{k,m}=\frac{w_{k}a^{SD}_{k}}{2\mu_{S}}, ~~~a^{RD}_{m}=a^{SD}_{k}\frac{\mu_{R}}{\mu_{S}},\\ &&&\text{and source and relay power constraints}, \\ \end{aligned} \right.$$ where $$\beta_{k,m}=\frac{a^{RD}_m}{a^{SR}_k-a^{SD}_k}.$$ Power allocation for the intermediate mode SP can be computed after the power allocations for the direct-link mode and relay mode SPs are done. Note that since the relay mode SPs must satisfy (\[Louveaux\_relay\_condition\]) with equality, it is clear that $p^S_{k,m}=\beta_{k,m}p^R_{k,m}$. From (\[mode\_conditions\_individual\]) we know that for the relay mode, $\beta_{k,m}>0$. Thus $p^{S}_{k,m}$ and $p^{R}_{k,m}$ must be zero or positive simultaneously. This can also be seen from the relay mode power allocation in (\[Louveaux\_power\_allocation\]). This observation allows us to allocate total power $p_{k,m}=p^{S}_{k,m}+ p^{R}_{k,m}$ to the relay mode SPs first according to $$\label{power_allocation_relay_individual} p_{k,m}=\left(\beta_{k,m}+1\right)\left[\frac{w_{k}}{2\left(\mu_{S}\beta_{k,m}+\mu_{R}\right)}-\frac{1}{a^{SR}_{k}\beta_{k,m}}\right]^+,$$ then obtain the corresponding $p^{S}_{k,m}$ and $p^{R}_{k,m}$ using the relay mode power distribution in (\[power\_ratio\]). To this end, the unified weighted rate expression in (\[unified\_rate\]) with (\[power\_ratio\]) and the equivalent channel gain (\[Eq\_ECG\]) can be applied here as well to the direct-link mode and relay mode SPs, when $\mu_S$ and $\mu_R$ are fixed. As to the intermediate mode SP, we examine its contributions to the rate and cost in the Lagrangian (\[Louveaux\_Lagrange\]) and find that, with $\rho_{k,m}=0$ [@relay_DF_individual_power_constraint], they are $$\begin{aligned} \text{rate}&=\frac{w_{k}}{2}\log\left(1+a^{SD}_{k}p^{S}_{k,m}+a^{RD}_{m}p^{R}_{k,m}\right)=\frac{w_{k}}{2}\log\left(\frac{w_{k}a^{SD}_{k}}{2\mu_{S}}\right)\\ \text{cost}&=\mu_{S}p^{S}_{k,m}+\mu_{R}p^{R}_{k,m}=\mu_{S}\left(p^{S}_{k,m}+\frac{\mu_{R}}{\mu_{S}}p^{R}_{k,m}\right)\\ &=\mu_{S}\left(p^{S}_{k,m}+\frac{a^{RD}_{m}}{a^{SD}_{k}}p^{R}_{k,m}\right)=\frac{\mu_{S}}{a^{SD}_{k}}\left(a^{SD}_{k}p^{S}_{k,m}+a^{RD}_{m}p^{R}_{k,m}\right)\\ &=\frac{\mu_{S}}{a^{SD}_{k}}\left(\frac{w_{k}a^{SD}_{k}}{2\mu_{S}}-1\right)=\mu_{S}\left(\frac{w_{k}}{2\mu_{S}}-\frac{1}{a^{SD}_{k}}\right) \end{aligned}.$$ If the intermediate mode SP was classified as a direct-link mode SP, it would contribute to the Lagrangian (\[Louveaux\_Lagrange\]) with $$\begin{aligned} \text{rate}&=\frac{w_{k}}{2}\log\left(1+a^{SD}_{k}p^{S}_{k,m}\right)=\frac{w_{k}}{2}\log\left(1+a^{SD}_{k}\left(\frac{w_{k}}{2\mu_{S}}-\frac{1}{a^{SD}_{k}}\right)\right)\\ &=\frac{w_{k}}{2}\log\left(\frac{w_{k}a^{SD}_{k}}{2\mu_{S}}\right)\\ \text{cost}&=\mu_{S}p^{S}_{k,m}=\mu_{S}\left(\frac{w_{k}}{2\mu_{S}}-\frac{1}{a^{SD}_{k}}\right) \end{aligned}.$$ On the other hand, if it was classified as a relay mode SP, with $\rho_{k,m}=\frac{a^{RD}_{m}\mu_S-a^{SD}_{k}\mu_R}{a^{SR}_{k}a^{RD}_{m}}=0$, its contributions to the Lagrangian (\[Louveaux\_Lagrange\]) would be $$\begin{aligned} \text{rate}&=\frac{w_{k}}{2}\log\left(1+a^{SD}_{k}p^{S}_{k,m}+a^{RD}_{m}p^{R}_{k,m}\right)\\ &=\frac{w_{k}}{2}\log\left(1+\frac{w_{k}a^{SD}_{k}}{2\left(\mu_{S}+\mu_{R}/\beta_{k,m}\right)}-\frac{a^{SD}_{k}}{a^{SR}_{k}}+\frac{w_{k}a^{RD}_{m}/\beta_{k,m}}{2\left(\mu_{S}+\mu_{R}/\beta_{k,m}\right)}-\frac{a^{RD}_{m}/\beta_{k,m}}{a^{SR}_{k}}\right)\\ &=\frac{w_{k}}{2}\log\left(\frac{w_{k}a^{SR}_{k}}{2\left(\mu_{S}+\mu_{S}\frac{a^{RD}_{m}}{a^{SD}_{k}\beta_{k,m}}\right)}\right)=\frac{w_{k}}{2}\log\left(\frac{w_{k}a^{SD}_{k}}{2\mu_{S}}\right)\\ \text{cost}&=\mu_{S}p^{S}_{k,m}+\mu_{R}p^{R}_{k,m}=\frac{\mu_{S}}{a^{SD}_{k}}\left(a^{SD}_{k}p^{S}_{k,m}+a^{RD}_{m}p^{R}_{k,m}\right)=\frac{\mu_{S}}{a^{SD}_{k}}\left(a^{SR}_{k}p^{S}_{k,m}\right)\\ &=\frac{\mu_{S}a^{SR}_{k}}{a^{SD}_{k}}\left(\frac{w_{k}}{2\left(\mu_{S}+\mu_{R}/\beta_{k,m}\right)}-\frac{1}{a^{SR}_{k}}\right)\\ &=\mu_{S}\left(\frac{w_{k}}{2\mu_{S}}-\frac{1}{a^{SD}_{k}}\right) \end{aligned}.$$ Interestingly, with given $\mu_S$ and $\mu_R$, the intermediate mode SP’s contributions to the Lagrangian (\[Louveaux\_Lagrange\]) remain the same no matter it is classified to the direct-link mode or the relay mode. Thus, in terms of maximizing the Lagrangian, we can assign the intermediate mode SP to either mode without affecting the result. However, once the optimal $\mu_S$ and $\mu_R$ are obtained, we still need to identify the intermediate mode SP and allocate its powers according to (\[Louveaux\_power\_allocation\]). In the following, we will assign the intermediate mode SP to the relay mode. Together with the conclusion that the unified weighted rate and equivalent channel gain expressions can be applied when $\mu_S$ and $\mu_R$ are fixed, the dual problem of (\[objective\_Rate\_I\_individual\]) can be formulated with unified expressions. Dual Problem {#sec_dual_type_I_individual} ------------ By dualizing (\[constraint\_t\_k\]), (\[constraint\_source\_power\_consumption\]), (\[constraint\_relay\_power\_consumption\]), (\[Louveaux\_relay\_condition\]), letting ${\bm t}$ and ${\bm \rho}$ be the matrices of $t_{k,m}$ and $\rho_{k,m}$, respectively, and applying continuous relaxation to $t_{k,m}$’s as in Section \[primal\_total\], we have the following Lagrangian $$\label{Lagrangian_individual_OK_typeI} \begin{aligned} L({\bm p},{\bm t},\mu_{S},\mu_{R}, {\bm \alpha}, {\bm \rho})=&\sum_{k=1}^M\sum_{m=1}^M~t_{k,m}\frac{w_{k}}{2}\log\left(1+a_{k,m}\frac{p_{k,m}}{t_{k,m}}\right)\\ &+\mu_{S}\left(P_{S}-\sum_{k=1}^M\sum_{m=1}^M~c^{S}_{k,m}~p_{k,m}\right)\\ &+\mu_{R}\left(P_{R}-\sum_{k=1}^M\sum_{m=1}^M~c^{R}_{k,m}~p_{k,m}\right)\\ &+\sum_{m=1}^M\alpha_m\left(1-\sum_{k=1}^M~t_{k,m}\right)\\ &+\sum_{(k,m) \in {\cal S}_R}\rho_{k,m}\left(a^{SR}_{k}p^{S}_{k,m}-a^{SD}_{k}p^{S}_{k,m}-a^{RD}_{m}p^{R}_{k,m}\right), \end{aligned}$$ where $$\label{Eq_ECG_individual} a_{k,m}=\left\{ \begin{aligned} &\frac{a^{SR}_{k}a^{RD}_{m}}{a^{SR}_{k}+a^{RD}_{m}-a^{SD}_{k}},&&\text{when }a^{SR}_{k}>a^{SD}_{k} \text{ and } a^{RD}_{m}\geq~a^{SD}_{k}\frac{\mu_{R}}{\mu_{S}}\\ &a^{SD}_{k},&&\,\,\,\text{otherwise} \end{aligned} \right.$$ $$\label{source_power_ratio_individual} c^{S}_{k,m}=\left\{ \begin{aligned} &\frac{a^{RD}_{m}}{a^{SR}_{k}+a^{RD}_{m}-a^{SD}_{k}},&&\text{when }a^{SR}_{k}>a^{SD}_{k} \text{ and } a^{RD}_{m}\geq~a^{SD}_{k}\frac{\mu_{R}}{\mu_{S}}\\ &1,&&\,\,\,\text{otherwise} \end{aligned} \right.$$ $$\label{relay_power_ratio_individual} c^{R}_{k,m}=\left\{ \begin{aligned} &\frac{a^{SR}_{k}-a^{SD}_{k}}{a^{SR}_{k}+a^{RD}_{m}-a^{SD}_{k}},&&\text{when }a^{SR}_{k}>a^{SD}_{k} \text{ and } a^{RD}_{m}\geq~a^{SD}_{k}\frac{\mu_{R}}{\mu_{S}}\\ &0,&&\,\,\,\text{otherwise} \end{aligned} \right.$$ are the equivalent channel gain, the portions of $p_{k,m}$ distributed to source power and relay power, respectively, for the two modes specified in the conditions. Similar to (\[objective\_dual\]), the dual problem associated with (\[Lagrangian\_individual\_OK\_typeI\]) can be expressed as $$\label{objective_dual_typeI_indivudual} \min_{\mu_{S},\mu_{R}, {\bm \alpha}} ~h(\mu_{S},\mu_{R}, {\bm \alpha}) ~~\text{s.t.}~~ \mu_{S}\geq 0, ~\mu_{R}\geq 0$$ with $$\label{P_Lagrange_typeI_indivudial} h(\mu_{S},\mu_{R}, {\bm \alpha})= \max_{{\bm p},{\bm t},{\bm \rho}} ~L({\bm p},{\bm t},\mu_{S},\mu_{R}, {\bm \alpha}, {\bm \rho}) ~~\text{s.t.}~~ (\ref{constraint_t_m}), (\ref{constraint_power_0}), (\ref{constraint_t_0}).$$ Note that the source and relay power distribution (\[source\_power\_ratio\_individual\]), (\[relay\_power\_ratio\_individual\]) satisfy the constraint $a^{SR}_{k}p^{S}_{k,m} = a^{SD}_{k}p^{S}_{k,m}+a^{RD}_{m}p^{R}_{k,m}$ for the relay mode. Thus the last term in (\[Lagrangian\_individual\_OK\_typeI\]) is always zero. Following the same procedure as in Section \[sec\_dual\_type\_I\_total\] and applying the results in Section \[unified\_rate\_individual\], the optimal power allocation for (\[P\_Lagrange\_typeI\_indivudial\]) can be solved as $$\label{optimal power_typeI_individual} p_{k,m}^\ast=t_{k,m}\left[\frac{w_{k}}{2(c^{S}_{k,m}\mu_{S}+c^{R}_{k,m}\mu_{R})}-\frac{1}{a_{k,m}}\right]^+.$$ The optimal $t_{k,m}$ can be solved as $$\begin{aligned} \label{optimal_t_typeI_individual} t_{k,m}^\ast=&\left\{ \begin{aligned} &1,\text{ }m=\arg\max_{m=1,...,M}Z_{k,m}\\ &0,\text{ otherwise} \end{aligned} \right., ~~~~\forall k.\end{aligned}$$ where $$\begin{aligned}\label{compute Z} Z_{k,m}=&\frac{w_{k}}{2}\log\left(1+a_{k,m}\left[\frac{w_{k}}{2(c^{S}_{k,m}\mu_{S}+c^{R}_{k,m}\mu_{R})}-\frac{1}{a_{k,m}}\right]^+\right)-\alpha_m\\ &-\mu_{S}\left(c^{S}_{k,m}\left[\frac{w_{k}}{2(c^{S}_{k,m}\mu_{S}+c^{R}_{k,m}\mu_{R})}-\frac{1}{a_{k,m}}\right]^+\right)\\ &-\mu_{R}\left(c^{R}_{k,m}\left[\frac{w_{k}}{2(c^{S}_{k,m}\mu_{S}+c^{R}_{k,m}\mu_{R})}-\frac{1}{a_{k,m}}\right]^+\right). \end{aligned}$$ Since assigning the intermediate mode SP to the relay mode does not change the dual value, we can approach the dual optimal value by the subgradient method. The Lagrange multipliers $\mu_{S}$, $\mu_{R}$, and ${\bm \alpha}$ are updated by $$\label{sub_gradient_typeI individual} \begin{aligned} &\mu_{S}^{(i+1)}=\mu_{S}^{(i)}-y_{S}^{(i)}\left(P_{S}-\sum_{k=1}^M\sum_{m=1}^M~c^{S}_{k,m}p_{k,m}^{(i)}\right),\\ &\mu_{R}^{(i+1)}=\mu_{R}^{(i)}-y_{R}^{(i)}\left(P_{R}-\sum_{k=1}^M\sum_{m=1}^M~c^{R}_{k,m}p_{k,m}^{(i)}\right),\\ &\alpha_m^{(i+1)}=\alpha_m^{(i)}-z^{(i)}\left(1-\sum_{k=1}^M~t_{k,m}^{(i)}\right)\text{, }m=1,...,M, \end{aligned}$$ where $y^{(i)}_S$, $y^{(i)}_R$ and $z^{(i)}$ are the sequences of step sizes designed properly. When the optimal subcarrier pairing and power allocation do not include an intermediate mode SP, (\[sub\_gradient\_typeI individual\]) will converge to the optimal values. However, when an intermediate mode SP is present in the optimal solution, (\[sub\_gradient\_typeI individual\]) may oscillate around the optimal values. Specifically, due to assigning the intermediate mode SP to the relay mode with power allocation (\[optimal power\_typeI\_individual\]), the relay power for that SP is increased, while the source power is decreased, to make (\[Louveaux\_relay\_condition\]) satisfied with equality instead of strict inequality. Thus, even when $\mu_S$ and $\mu_R$ are already at their optimal, the total source power consumption will be smaller than the source power constraint, and the total relay power consumption will be larger than the relay power constraint. This will result in $\mu_S$ decreased and $\mu_R$ increased in the next iteration. Then $\mu_R/\mu_S$ will be increased, and the intermediate mode SP may fall in the direct-link mode according to (\[mode\_conditions\_individual\]). Similarly, this will make $\mu_R/\mu_S$ decreased, and the intermediate mode SP may fall in the relay mode in the next iteration. As a result, (\[sub\_gradient\_typeI individual\]) oscillates. Similar oscillation was also observed in [@relay_DF_individual_power_constraint]. Thus, like in [@relay_DF_individual_power_constraint Section 3.2], the zero-crossing of the difference between the total source power consumption and the source power constraint can be used to determine the optimal $\mu_R/\mu_S$ and the corresponding mode classification and power allocation. However, due to the issues discussed in Section \[Sec\_achieave\_feasible\], we have found that the optimal zero-crossing is very difficult to trace when subcarrier pairing, mode classification and power allocation are updated at the same time. In the algorithm given in Table \[Table\_algo\_obtain\_feasible\_typeI\_individual\], similar to Table \[Table\_algo\_obtain\_feasible\_typeI\], the amendment algorithm is used to obtain a feasible pairing scheme when the subgradient method converges to a ceratin degree. With diminishing step sizes, we found that the subgradient method will eventually be stuck at assigning the intermediate mode SP (if it exists in the optimal solution) to either the direct-link mode or the relay mode. In both cases, the obtained subcarrier pairing scheme is near optimal. With fixed subcarrier pairing, and $\mu_R/\mu_S$ given by the amendment algorithm which is already very close to the optimal, the zero-crossing method in [@relay_DF_individual_power_constraint Section 3.2] can be used to quickly obtain the optimal $\mu_R/\mu_S$. Then the corresponding mode classification and power allocation can be done according to (\[mode\_conditions\_individual\]) and (\[Louveaux\_power\_allocation\]), respectively. The algorithm in Table \[Table\_algo\_obtain\_feasible\_typeI\_individual\] has the same order of complexity as that of the algorithm in Table \[Table\_algo\_obtain\_feasible\_typeI\]. Through simulation, we have also found that the duality gap for this problem approaches zero when the number of subcarriers is reasonably large. Weighted Sum Rate Maximization with Extra Direct-Link Transmission {#sec_maximization_total_extra_direct} ================================================================== In the previous sections, only the relay can transmit in the second time slot. Therefore, for the SPs operating in the direct-link mode, the second time slot is not used. It is possible to allow the source to transmit extra messages in the second time slot on these idle subcarriers. We consider this modified system with both total and individual power constraints. Total Power Constraint {#sec_extraSD_total} ---------------------- Under the total power constraint, the achievable weighted sum rate for SP $(k,m)$ for this system is $$\label{Eq_pair_rate_typeII} R_{k,m}=\left\{ \begin{aligned} &\frac{w_{k}}{2}\log\left(1+a^{SD}_{k}p^{S}_{k,m}\right)+\frac{w_{m}}{2}\log\left(1+a^{SD}_{m}q^{S}_{k,m}\right),&\text{direct-link mode}\\ &\frac{w_{k}}{2}\min\left\{\log\left(1+a^{SR}_{k}p^{S}_{k,m}\right),\; \log\left(1+a^{SD}_{k}p^{S}_{k,m}+a^{RD}_{m}p^{R}_{k,m}\right)\right\}, &\text{relay mode},\\ \end{aligned} \right.$$ where $p^{S}_{k,m}$, $p^{R}_{k,m}$, and $q^{S}_{k,m}$ represent the source power in the first time slot, relay power in the second time slot, and source power in the second time slot, respectively. By comparing the achievable weighted rate for these two modes, we find that the condition for using the relay depends not only on the channel gains but also on the power allocation. Thus we introduce an additional indicator $s_{k,m}$ related to the use of the relay as a variable to be jointly optimized. When $s_{k,m}=1$, the relay is used for SP $(k,m)$. When $s_{k,m}=0$, the relay is not used. In addition, we again make continuous relaxation for the indicators and the same adjustment to the sum rate function. The relaxed weighted sum rate maximization problem is expressed as follows $$\begin{aligned} \label{objective_Rate_II_relaxed} \max_{{\bm p},{\bm s},{\bm t}}\text{ } &\frac{1}{2}\sum^M_{k=1}\sum^M_{m=1}t_{k,m}~\left\{s_{k,m}w_{k}\log\left(1+a_{k,m}\frac{p_{k,m,1}}{t_{k,m}s_{k,m}}\right)\right.\notag\\ &+(1-s_{k,m})\left[w_{k}\log\left(1+a^{SD}_{k}\frac{p_{k,m,2}}{t_{k,m}(1-s_{k,m})}\right)\right.\notag\\ &\left.\left.+w_{m}\log\left(1+a^{SD}_{m}\frac{p_{k,m,3}}{t_{k,m}(1-s_{k,m})}\right)\right]\right\} \\ \text{s.t.}\text{ }&(\ref{constraint_t_k}), (\ref{constraint_t_m}), (\ref{constraint_t_0}) \notag\\ \label{constraint_power_k,m,r} &\sum^M_{k=1}\sum^M_{m=1}\sum^3_{r=1}p_{k,m,r}\leq P \\ \label{constraint_power_k,m,r_0} &p_{k,m,r}\geq 0,\forall k, m, r \\ \label{constraint_s_1_0} &0\leq s_{k,m}\leq 1,\forall k, m,\end{aligned}$$ where $p_{k,m,1}$ and $a_{k,m}$ are the sum power and equivalent channel gain, respectively, of the relay mode SP $(k,m)$ taking the form of the relay mode expressions in (\[power\_ratio\]) and (\[Eq\_ECG\]). $p_{k,m,2}$ and $p_{k,m,3}$ are the powers used by the direct-link mode SP $(k,m)$ in the first and second time slots, respectively. ${\bm p}\in \mathds{R}_+^{M\times M\times3}$, ${\bm t}\in \mathds{R}_+^{M\times M}$, and ${\bm s}\in \mathds{R}_+^{M\times M}$ are the matrices of $p_{k,m,r}$, $t_{k,m}$, and $s_{k,m}$, respectively. $P$ is the total power constraint. Similarly, by dualizing constraints (\[constraint\_t\_k\]) and (\[constraint\_power\_k,m,r\]), we obtain the Lagrangian as $$\label{Lagrangian_typeII} \begin{aligned} L({\bm p},{\bm t},{\bm s},\mu, {\bm \alpha})=&\frac{1}{2}\sum^M_{k=1}\sum^M_{m=1}t_{k,m}~\left\{s_{k,m}w_{k}\log\left(1+a_{k,m}\frac{p_{k,m,1}}{t_{k,m}s_{k,m}}\right)\right.\\ &+(1-s_{k,m})\left[w_{k}\log\left(1+a^{SD}_{k}\frac{p_{k,m,2}}{t_{k,m}(1-s_{k,m})}\right)\right.\\ &\left.\left.+w_{m}\log\left(1+a^{SD}_{m}\frac{p_{k,m,3}}{t_{k,m}(1-s_{k,m})}\right)\right]\right\}\\ &+\mu\left(P-\sum_{k=1}^M\sum_{m=1}^M\sum_{r=1}^3~p_{k,m,r}\right)+\sum_{m=1}^M\alpha_m\left(1-\sum_{k=1}^M~t_{k,m}\right), \end{aligned}$$ where $\mu\in \mathds{R}_+$ and ${\bm \alpha}\in\mathds{R}^M$ are the dual variables. Then the dual objective function is computed as $$\label{P_Lagrange_typeII} h(\mu,{\bm \alpha})= \max_{{\bm p},{\bm t},{\bm s}} ~L({\bm p},{\bm t},{\bm s},\mu,{\bm \alpha}) ~~\text{s.t.}~~ (\ref{constraint_t_m}), (\ref{constraint_t_0}), (\ref{constraint_power_k,m,r_0}), (\ref{constraint_s_1_0}).$$ The dual problem is given as $$\label{objective_dual_typeII} \min_{\mu,{\bm \alpha}} ~h(\mu, {\bm \alpha}) ~~\text{s.t.}~~ \mu\geq 0.$$ The solution to (\[P\_Lagrange\_typeII\]) is $$\label{optimal power_typeII} \begin{aligned} &p_{k,m,1}^\ast=t_{k,m}s_{k,m}\left[\frac{w_{k}}{2\mu}-\frac{1}{a_{k,m}}\right]^+,\\ &p_{k,m,2}^\ast=t_{k,m}(1-s_{k,m})\left[\frac{w_{k}}{2\mu}-\frac{1}{a^{SD}_{k}}\right]^+,\\ &p_{k,m,3}^\ast=t_{k,m}(1-s_{k,m})\left[\frac{w_{m}}{2\mu}-\frac{1}{a^{SD}_{m}}\right]^+, \end{aligned}$$ $$\label{Eq_s_choose} s^\ast_{k,m}=\left\{ \begin{aligned} 1&,\text{ }a^{SR}_{k}>a^{SD}_{k}\text{ and }Y^{R}_{k,m}>Y^{D}_{k,m}\\ 0&,\text{ otherwise}, \end{aligned} \right.$$ where $$\label{compute Y_R} Y^{R}_{k,m}=\frac{w_{k}}{2}\log\left(1+a_{k,m}\left[\frac{w_{k}}{2\mu}-\frac{1}{a_{k,m}}\right]^+\right)-\mu\left[\frac{w_{k}}{2\mu}-\frac{1}{a_{k,m}}\right]^+\\$$ $$\label{compute Y_D} \begin{aligned} Y^{D}_{k,m}=&\frac{w_{k}}{2}\log\left(1+a^{SD}_{k}\left[\frac{w_{k}}{2\mu}-\frac{1}{a^{SD}_{k}}\right]^+\right)\\ &+\frac{w_{m}}{2}\log\left(1+a^{SD}_{m}\left[\frac{w_{m}}{2\mu}-\frac{1}{a^{SD}_{m}}\right]^+\right)\\ &-\mu\left(\left[\frac{w_{k}}{2\mu}-\frac{1}{a^{SD}_{k}}\right]^+~+\left[\frac{w_{m}}{2\mu}-\frac{1}{a^{SD}_{m}}\right]^+\right)\\ \end{aligned}$$ are the SP $(k,m)$’s contribution to the Lagrangian when it is in the relay mode or the direct-link mode, respectively. The condition $a^{SR}_{k}>a^{SD}_{k}$ in (\[Eq\_s\_choose\]) is necessary. The reason is that the value of $Y^{R}_{k,m}$ is meaningless when $a^{SR}_{k}<a^{SD}_{k}$, since it is impossible to make the relay receive more information than the destination. The $s^\ast_{k,m}$ tells us whether it is better to use relay for the SP ($k,m$). The SP selection variable is given as follows $$\label{optimal_t_typeII} t_{k,m}^\ast=\left\{ \begin{aligned} &1,\text{ }m=\arg\max_{m=1,...,M}Y_{k,m}\\ &0,\text{ }\text{otherwise} \end{aligned} \right., ~~~~\forall k$$ where $$\label{compute Y} \begin{aligned} Y_{k,m}=s^\ast_{k,m}Y^{R}_{k,m}+(1-s^\ast_{k,m})Y^{D}_{k,m}-\alpha_m. \end{aligned}$$ Again, the dual optimal value is reached by the subgradient method. The Lagrange multipliers $\mu$ and ${\bm \alpha}$ are updated by $$\label{sub_gradient_typeII_total} \begin{aligned} &\mu^{(i+1)}=\mu^{(i)}-y^{(i)}\left(P-\sum_{k=1}^M\sum_{m=1}^M\sum_{r=1}^3~p^{(i)}_{k,m,r}\right),\\ &\alpha_m^{(i+1)}=\alpha_m^{(i)}-z^{(i)}\left(1-\sum_{k=1}^M~t_{k,m}^{(i)}\right)\text{, }m=1,...,M, \end{aligned}$$ where $y^{(i)}$ and $z^{(i)}$ are the sequences of step sizes designed properly. The algorithm to obtain feasible solutions is given in Table \[Table\_algo\_obtain\_feasible\_typeII\] where $s_{k,m}$’s found in an iteration are directly used, together with the subcarrier pairing scheme ${\bm t}$ obtained by the amendment algorithm, to compute the power allocation and weighted sum rate. Doing so is suboptimal, as $s_{k,m}$ in fact depends on the power allocation. However, this saves the complexity involved in joint optimization of $s_{k,m}$ and power allocation given fixed subcarrier pairing. The algorithm in Table \[Table\_algo\_obtain\_feasible\_typeII\] also has the same order of complexity as that of the algorithm in Table \[Table\_algo\_obtain\_feasible\_typeI\]. We have found that the duality gap for this problem is virtually zero when the number of subcarriers is reasonably large. Individual Power Constraints {#sec_extraSD_individual} ---------------------------- With individual power constraints for the source and the relay, the problem can be solved by combining the results in Section \[sec\_maximization\_individual\] and Section \[sec\_extraSD\_total\] with some crucial modifications. Due to limited space, we will discuss only these crucial points. As in Section \[sec\_maximization\_individual\], in addition to the direct-link mode and the relay mode, there may also be an intermediate mode in which the relay receives more information than the destination, and (\[Louveaux\_relay\_condition\]) is satisfied with strict inequality. On the other hand, the relay mode should satisfy (\[Louveaux\_relay\_condition\]) with equality. Note that given the same power, the rate of the direct-link mode in (\[Eq\_pair\_rate\_typeII\]) with extra second-slot SD transmission should be no less than the rate of the direct-link mode in (\[Eq\_pair\_rate\_typeI\]). This is because the latter is a special case of the former with the second time slot allocated zero power. Therefore, the necessary condition $a^{SR}_{k}>a^{SD}_{k}$ in (\[mode\_conditions\_individual\]) for the relay to be active (in both the relay and the intermediate modes) is also necessary in this case. The second necessary condition $a^{RD}_{m}=a^{SD}_{k}\frac{\mu_{R}}{\mu_{S}}$ in (\[mode\_conditions\_individual\]) for a SP to be in the intermediate mode, as derived in [@relay_DF_individual_power_constraint], is directly related to having (\[Louveaux\_relay\_condition\]) as a strict inequality. Thus it is also necessary in this case. The second condition $a^{RD}_{m}<a^{SD}_{k}\frac{\mu_{R}}{\mu_{S}}$ in (\[mode\_conditions\_individual\]) for selecting the direct-link mode over the relay mode was derived by comparing the achievable rates of the direct-link mode and the relay mode when they have the same power cost in the Lagragian (\[Louveaux\_Lagrange\]) (see the discussion after (\[mode\_conditions\_individual\]) and [@Vandendorpe_J]). With the extra second-slot SD transmission that can improve the rate for the direct-link mode, this condition may change. In fact, with the extra second-slot SD transmission, the direct-link mode may possibly be selected even when $a^{RD}_{m}>a^{SD}_{k}\frac{\mu_{R}}{\mu_{S}}$. For the special case with fixed $(k,k)$ subcarrier paring, [@Vandendorpe_J] has derived the exact condition which also depends on the allocated power. In our case, the subcarrier paring is variable and may not be the trivial $(k,k)$ pairing. Due to this reason and different weighting factors in the rate of the direct-link mode (\[Eq\_pair\_rate\_typeII\]), the exact condition based on having the same power cost is complicated and dependent also on the weighting factors. However, we may simplify the condition by comparing the contributions of the direct-link mode and the relay mode to the Lagrangian. This approach is similar to using (\[Eq\_s\_choose\]) to select modes to maximize the Lagrangian (\[P\_Lagrange\_typeII\]). The fact that the direct-link mode may also be selected when $a^{RD}_{m}>a^{SD}_{k}\frac{\mu_{R}}{\mu_{S}}$ implies that the region for the intermediate mode to occur may be encompassed by the region for selecting the direct-link mode. That is, the intermediate mode may no longer exist, except in the special situation where the optimal power allocation for a direct-link mode SP results in zero power for the second-slot SD transmission. For a SP with this property, there will be no second-slot SD transmission if the direct-link mode is selected. Then the situation becomes the same as in Section \[sec\_maximization\_individual\]. Thus (\[mode\_conditions\_individual\]) can be used to select modes, and the unified rate formulation discussed in Section \[unified\_rate\_individual\] can be applied with the intermediate mode SP assigned to the relay mode. In summary, we can assume that there are only the direct-link mode and the relay mode, and apply the results in Section \[sec\_extraSD\_total\] with the following changes. The relay mode power allocation $p_{k,m,1}^\ast$ in (\[optimal power\_typeII\]) takes the expression of the right-hand side (RHS) of (\[optimal power\_typeI\_individual\]) multiplied by $s_{k,m}$, where $c_{k,m}^S$ and $c_{k,m}^R$ are the portions of $p_{k,m,1}^\ast$ distributed to source power and relay power defined by the relay mode expressions in (\[source\_power\_ratio\_individual\]) and (\[relay\_power\_ratio\_individual\]), respectively. The RHS of (\[compute Y\_R\]) is replaced by the RHS of (\[compute Z\]) with $-\alpha_m$ removed. $\mu$ in (\[compute Y\_D\]) is replaced by $\mu_S$. In each iteration, $\mu_S$, $\mu_R$ and ${\bm \alpha}$ are updated as in (\[sub\_gradient\_typeI individual\]) using the power allocation computed in (\[optimal power\_typeII\]) ($p_{k,m,1}^\ast$ computed by the RHS of (\[optimal power\_typeI\_individual\]) multiplied by $s_{k,m}$). If there is an intermediate mode SP in the optimal solution, it must belong to the situation where the conditions (\[mode\_conditions\_individual\]) are applicable. Then, like in Section \[sec\_dual\_type\_I\_individual\], the zero-crossing method in [@relay_DF_individual_power_constraint Section 3.2] can be used to obtain the optimal $\mu_R/\mu_S$. The corresponding mode classification and power allocation can then be done accordingly. Simulation Results {#Sec_simulation} ================== This section provides the simulation results for the rates obtained by the proposed algorithms, and the dual optimum values which serve as the performance upper bounds. The performances with fixed subcarrier pairing and with the SCP proposed in [@relay_eq_channel_gain] are also presented for comparison. Note that the original SCP in [@relay_eq_channel_gain] considered only the unweighted sum rate. It first sorts the subcarriers of the SR link and the RD link, respectively, according to their normalized channel gains, then pairs the SR subcarrier with the RD subcarrier having the same rank. For weighted sum rate, according to (\[impact\_weighting\]) and the discussion right after it, we modify the SCP such that $w_{k}a^{SR}_{k}$ and $a^{RD}_{m}$ are sorted first. Then the SR subcarrier and the RD subcarrier with the same rank are paired. The RD link channel gains are not weighted for the reason that we do not know the actual subcarrier pairing scheme in advance. The channels of different links are assumed to be independent of one another. The channels of the subcarriers are independent and identically distributed (i.i.d.) Rician fading channels with $K$-factor $=1$. They are assumed constant within each two-slot period, and varying independently from one period to another. The AWGN variance is assumed to be one. The total power constraint is set as $P=5$. As for the cases with individual power constraints, the source power constraint is $P_S=4$ and the relay power constraint is $P_R=1$. These constraints are set with the practical consideration that the relay usually plays the role of assisting the transmission and/or extending the coverage, and has a smaller power than the source. In addition, when the relay is allowed more power and the achievable rate becomes limited by the source power constraint, some of the relay power will not be used. Setting $P_S=4$ and $P_R=1$ reduces the occurrence of this situation and makes the comparison with the total power constrained case fairer. For all cases, the SD link is present, and the destination performs MRC whenever the relay is used. The SCP schemes first establish subcarrier pairing using SCP. Then, in the total power constrained cases (including the case with extra direct-link transmission), (\[relay\_condition\_total\]) is used as the condition to use relay. In the individual power constrained case without extra direct-link transmission, the method in [@relay_DF_individual_power_constraint] is used for mode classification and power allocation. For the individual power constrained case with extra direct-link transmission, the method in [@relay_DF_individual_power_constraint] cannot be used because the optimal mode classification conditions are no longer (\[mode\_conditions\_individual\]). Naively using (\[mode\_conditions\_individual\]) and (\[Louveaux\_power\_allocation\]) may result in invalid power allocation as they are not the solutions in this case, and will affect the $\mu_S$, $\mu_R$ values through the iterations. On the other hand, modifying the method in [@Vandendorpe_J] to accommodate weighted rates is tedious. Thus, the algorithm discussed in Section \[sec\_extraSD\_individual\] is used with fixed subcarrier pairing from the SCP. The fixed pairing schemes use the same mode classification and power allocation procedures as that of the SCP schemes. In Figs. \[Fig\_WSR\_channel\_3\_1\_3\_noweight\_total\_individual\], \[Fig\_WSR\_channel\_3\_1\_3\_noweight\_total\_individual\_extra\], \[Fig\_WSR\_channel\_5\_1\_1\_noweight\_total\_individual\], \[Fig\_WSR\_channel\_5\_1\_1\_noweight\_total\_individual\_extra\], \[Fig\_WSR\_channel\_1\_1\_5\_noweight\_total\_individual\] and \[Fig\_WSR\_channel\_1\_1\_5\_noweight\_total\_individual\_extra\], the unweighted sum rate is considered. That is, the all-one weighting factor is used. In Figs. \[Fig\_WSR\_channel\_3\_1\_3\_weight\], \[Fig\_WSR\_channel\_5\_1\_1\_weight\] and \[Fig\_WSR\_channel\_1\_1\_5\_weight\], the weighted sum rate is considered with $w_{k}=1+\frac{k-1}{M-1},\forall k$, which is used only as an example with a concise expression. For the proposed algorithms, $\mu$, $\mu_R$, $\mu_S$ and $\alpha_m$’s were randomly initialized to be between 0 and 2 for each two-slot period. For each number of subcarriers $(\in \{4, 8, 16, 32, 64\})$, 1000 such two-slot periods were simulated, and the results averaged to avoid favoring certain initial conditions. The step sizes for the subgradient method were all set as $\frac{0.05}{\sqrt{i}}$, where $i$ is the iteration index. In the simulation, we observed that the number of iterations before the amendment algorithm was triggered depends on the number of subcarriers. The number of iterations needed ranged roughly from a few hundreds for small numbers of subcarriers ($<10$) to slightly more than 10000 for 64 subcarriers. We investigate three system configurations corresponding to different scenarios. In Figs. \[Fig\_WSR\_channel\_3\_1\_3\_noweight\_total\_individual\], \[Fig\_WSR\_channel\_3\_1\_3\_weight\] and \[Fig\_WSR\_channel\_3\_1\_3\_noweight\_total\_individual\_extra\], the mean square channel gains of the SR, SD and RD links are 3, 1, 3, respectively. This corresponds to the situation where the relay is placed between the source and the destination. In Figs. \[Fig\_WSR\_channel\_5\_1\_1\_noweight\_total\_individual\], \[Fig\_WSR\_channel\_5\_1\_1\_weight\] and \[Fig\_WSR\_channel\_5\_1\_1\_noweight\_total\_individual\_extra\], the mean square channel gains of the SR, SD and RD links are 5, 1 and 1, respectively, which means that the relay is close to the source. In Figs. \[Fig\_WSR\_channel\_1\_1\_5\_noweight\_total\_individual\], \[Fig\_WSR\_channel\_1\_1\_5\_weight\] and \[Fig\_WSR\_channel\_1\_1\_5\_noweight\_total\_individual\_extra\], the mean square channel gains of the SR, SD and RD links are 1, 1 and 5, respectively, which means that the relay is close to the destination. In these figures, we find that, in all cases, the rates obtained by the proposed algorithms are almost equal to the corresponding dual optimum values. This validates the arguments in Section \[duality\_gap\_total\], Section \[sec\_dual\_type\_I\_individual\] and Section \[sec\_extraSD\_total\] that the duality gap is virtually zero when the number of subcarriers is reasonably large. Even when the number of subcarriers is 4, the duality gap is hardly noticeable from the averaged results, because it is zero with a very high probability. These results also show that the proposed algorithms can almost achieve the optimal weighted sum rates. There are some other general trends that can be observed from these figures. One of them is that fixed subcarrier pairing incurs a significant performance loss. In addition, the weighted and unweighted sum rates increase with the number of subcarriers due to frequency diversity and more flexibility in pairing. As to the performance under different constraints, the performance under total power constraint is better than the performance under individual power constraints, due to the flexibility in power allocation. By comparing Figs. \[Fig\_WSR\_channel\_3\_1\_3\_noweight\_total\_individual\] and \[Fig\_WSR\_channel\_3\_1\_3\_noweight\_total\_individual\_extra\], \[Fig\_WSR\_channel\_5\_1\_1\_noweight\_total\_individual\] and \[Fig\_WSR\_channel\_5\_1\_1\_noweight\_total\_individual\_extra\], \[Fig\_WSR\_channel\_1\_1\_5\_noweight\_total\_individual\] and \[Fig\_WSR\_channel\_1\_1\_5\_noweight\_total\_individual\_extra\], it is clear that extra direct-link transmission always improves the performance. The SCP was proved in [@relay_AF_DF_eq_channel_gain][@Li_AFDF_OFDM] to be optimal for the unweighted system without the SD link under the total power constraint. When the SD link is present and/or when weighted sum rate is considered, the performance of the SCP depends on the link qualities. The SCP almost achieves the optimal unweighted sum rate for the cases with total power constraint and no extra direct-link transmission in Fig. \[Fig\_WSR\_channel\_3\_1\_3\_noweight\_total\_individual\] and Fig. \[Fig\_WSR\_channel\_1\_1\_5\_noweight\_total\_individual\], but becomes noticeably worse than the optimal in Fig. \[Fig\_WSR\_channel\_5\_1\_1\_noweight\_total\_individual\]. For the scenario in Fig. \[Fig\_WSR\_channel\_3\_1\_3\_noweight\_total\_individual\], this is reasonable because the SD link is relatively weak compared to the other two links. Thus the direct-link mode is rarely used, and the SCP is nearly optimal for the relay-mode SPs given that their SD subcarriers are weak. For the scenario in Fig. \[Fig\_WSR\_channel\_1\_1\_5\_noweight\_total\_individual\], the RD link is the strongest and seldom becomes the bottleneck for mode selection. For the SCP as well as the proposed algorithm, mode selection is mainly determined by the SR and SD links. For the direct-link mode SPs, the SCP and the proposed algorithm have similar performances. For the relay mode SPs, the SCP is nearly optimal because the SD link is the weakest among the three links. Overall, the SCP has a very similar performance to that of the proposed algorithm which is almost optimal. As to the case of Fig. \[Fig\_WSR\_channel\_5\_1\_1\_noweight\_total\_individual\], we can see that since the SR link is much stronger than the SD link, the condition for using relay (\[relay\_condition\_total\]) is dominated by the relation between the channel gains of the SD and the RD links. However, the SCP does not consider the SD link in establishing subcarrier pairing. As a result, the SCP is almost equivalent to random pairing in terms of optimizing the mode selection and sum rate. Thus its sum rate is smaller than that of the proposed algorithm. The SCP is still better than fixed pairing because it helps the SPs that are in the relay mode. For the individual power constrained cases, or when weighted sum rate is considered, as shown in Figs. \[Fig\_WSR\_channel\_3\_1\_3\_noweight\_total\_individual\], \[Fig\_WSR\_channel\_5\_1\_1\_noweight\_total\_individual\], and \[Fig\_WSR\_channel\_3\_1\_3\_weight\], \[Fig\_WSR\_channel\_5\_1\_1\_weight\], the gaps between the SCP and the proposed algorithms become larger. This is due to the mismatches between the SCP and these scenarios. To show that our modification to the original SCP is meaningful, we show the performance of the original (unweighted) SCP together with that of the “weighted SCP" in Fig. \[Fig\_WSR\_channel\_3\_1\_3\_weight\] and Fig. \[Fig\_WSR\_channel\_5\_1\_1\_weight\]. These two figures clearly show that the original SCP is not suitable when weighted sum rate is considered. The performance gap between the “weighted SCP" and the proposed algorithm in Fig. \[Fig\_WSR\_channel\_5\_1\_1\_weight\] is due to the aforementioned “random pairing" effect of the SCP (as in the total power constrained case in Fig. \[Fig\_WSR\_channel\_5\_1\_1\_noweight\_total\_individual\]). However, these trends do not appear in Fig. \[Fig\_WSR\_channel\_1\_1\_5\_noweight\_total\_individual\] and Fig. \[Fig\_WSR\_channel\_1\_1\_5\_weight\]. For Fig. \[Fig\_WSR\_channel\_1\_1\_5\_noweight\_total\_individual\], this is because the strong RD link makes the sum rate not limited by the low relay power constraint. Therefore, for the SCP, the situation is very similar to that with the total power constraint. As a result, the SCP is almost optimal. For Fig. \[Fig\_WSR\_channel\_1\_1\_5\_weight\], the strong RD link makes mode selection dependent almost only on the channel gains of the SR and SD links. Thus, mode selection is almost independent of the pairing scheme and weighting factors. For the source subcarriers that have relatively lower SR gains and are in the direct-link mode, all schemes yield similar performances. On the other hand, for the subcarriers in the relay mode, pairing better RD subcarriers with SR subcarriers having higher weighted channel gains can improve the weighted sum rate. Both the original SCP and the weighted SCP can do that for the SR subcarriers that are strong enough. Thus they both perform well and almost optimally. With possible extra direct-link transmission, the SCP is worse than the proposed algorithm for not considering the benefits of the extra direct-link transmission (such as more diversity from the additional independent channels, and more flexibility in water-filling) in subcarrier pairing and mode selection. Under the total power constraint, we find that the SCP is similar and even slightly worse than fixed pairing in Fig. \[Fig\_WSR\_channel\_5\_1\_1\_noweight\_total\_individual\_extra\] and Fig. \[Fig\_WSR\_channel\_1\_1\_5\_noweight\_total\_individual\_extra\]. This is because, without considering the possible extra direct-link transmission, the SCP’s pairing of strong SR subcarrier with strong RD subcarrier tends to satisfy (\[relay\_condition\_total\]) more than fixed pairing, and make more SPs use the relay. Thus it loses the opportunities to transmit more messages with the extra direct-link. This phenomenon does not appear in Fig. \[Fig\_WSR\_channel\_3\_1\_3\_noweight\_total\_individual\_extra\], for which the benefits of the extra direct-link transmission are not significant due to the weak SD link. Under individual power constraints, both the SCP and the fixed pairing schemes use the algorithm in Section \[sec\_extraSD\_individual\] for optimal joint mode selection and power allocation. The SCP always performs better than fixed pairing due to its better subcarrier pairing. In Fig. \[Fig\_WSR\_channel\_1\_1\_5\_noweight\_total\_individual\_extra\], the advantage of the SCP over fixed pairing is smaller than in Fig. \[Fig\_WSR\_channel\_1\_1\_5\_noweight\_total\_individual\] because the optimal mode selection assigns more SPs to the direct-link mode for which better SR-RD subcarrier pairing does not improve the rate. Conclusion {#Sec_conclusion} ========== In this paper we investigated OFDM point to point transmission, enhanced with a DF relay. We jointly optimized subcarrier pairing and power allocation to maximize the weighted sum rate with consideration of the source-destination link and destination combining. To the best of our knowledge, this problem has not been solved before. Both total power constraint and individual power constraints for the source and the relay were considered. The system that allows additional messages to be transmitted on the idle subcarriers not used by the relay, in the source-destination link in the second time slot, was also investigated. We solved the optimization problems by using some special properties of the systems, as well the continuous relaxation and the dual method. The subgradient method was adopted to find the Lagrange multipliers which also helped us to find the primal feasible solutions. Based on the optimization results, algorithms with tractable complexities to obtain feasible subcarrier pairing schemes and the corresponding power allocations were proposed. Simulation results showed that the proposed algorithms can achieve nearly optimal weighted sum rates, and outperform the method proposed in [@relay_eq_channel_gain] under various channel conditions. ![Channel model for subcarrier pair $(k,m)$.[]{data-label="Fig_channel_model"}](channel_model.eps){width="55.00000%"} ![Concavity of rate versus power constraint for systems with 2 and 8 subcarriers.[]{data-label="Fig_rate_power_concave"}](rate_power_2_8.eps){width="80.00000%"} ![Unweighted sum rates for the systems with $\mathbb{E}[|h^{SR}|^2]=3$, $\mathbb{E}[|h^{SD}|^2]=1$ and $\mathbb{E}[|h^{RD}|^2]=3$.[]{data-label="Fig_WSR_channel_3_1_3_noweight_total_individual"}](channel_3_1_3_noWeighting_total_individual.eps){width="77.00000%"} ![Weighted sum rates for the systems with total power constraint, and $\mathbb{E}[|h^{SR}|^2]=3$, $\mathbb{E}[|h^{SD}|^2]=1$, $\mathbb{E}[|h^{RD}|^2]=3$.[]{data-label="Fig_WSR_channel_3_1_3_weight"}](channel_3_1_3_Weighting_total.eps){width="77.00000%"} ![Unweighted sum rates for the systems with extra direct-link transmission, and $\mathbb{E}[|h^{SR}|^2]=3$, $\mathbb{E}[|h^{SD}|^2]=1$, $\mathbb{E}[|h^{RD}|^2]=3$.[]{data-label="Fig_WSR_channel_3_1_3_noweight_total_individual_extra"}](channel_3_1_3_noWeighting_total_individual_extra.eps){width="77.00000%"} ![Unweighted sum rates for the systems with $\mathbb{E}[|h^{SR}|^2]=5$, $\mathbb{E}[|h^{SD}|^2]=1$ and $\mathbb{E}[|h^{RD}|^2]=1$.[]{data-label="Fig_WSR_channel_5_1_1_noweight_total_individual"}](channel_5_1_1_noWeighting_total_individual.eps){width="77.00000%"} ![Weighted sum rates for the systems with total power constraint, and $\mathbb{E}[|h^{SR}|^2]=5$, $\mathbb{E}[|h^{SD}|^2]=1$, $\mathbb{E}[|h^{RD}|^2]=1$.[]{data-label="Fig_WSR_channel_5_1_1_weight"}](channel_5_1_1_Weighting_total.eps){width="77.00000%"} ![Unweighted sum rates for the systems with extra direct-link transmission, and $\mathbb{E}[|h^{SR}|^2]=5$, $\mathbb{E}[|h^{SD}|^2]=1$, $\mathbb{E}[|h^{RD}|^2]=1$.[]{data-label="Fig_WSR_channel_5_1_1_noweight_total_individual_extra"}](channel_5_1_1_noWeighting_total_individual_extra.eps){width="77.00000%"} ![Unweighted sum rates for the systems with $\mathbb{E}[|h^{SR}|^2]=1$, $\mathbb{E}[|h^{SD}|^2]=1$ and $\mathbb{E}[|h^{RD}|^2]=5$.[]{data-label="Fig_WSR_channel_1_1_5_noweight_total_individual"}](channel_1_1_5_noWeighting_total_individual.eps){width="77.00000%"} ![Weighted sum rates for the systems with total power constraint, and $\mathbb{E}[|h^{SR}|^2]=1$, $\mathbb{E}[|h^{SD}|^2]=1$, $\mathbb{E}[|h^{RD}|^2]=5$.[]{data-label="Fig_WSR_channel_1_1_5_weight"}](channel_1_1_5_Weighting_total.eps){width="77.00000%"} ![Unweighted sum rates for the systems with extra direct-link transmission, and $\mathbb{E}[|h^{SR}|^2]=1$, $\mathbb{E}[|h^{SD}|^2]=1$, $\mathbb{E}[|h^{RD}|^2]=5$.[]{data-label="Fig_WSR_channel_1_1_5_noweight_total_individual_extra"}](channel_1_1_5_noWeighting_total_individual_extra.eps){width="77.00000%"} [lll]{}\ \ &*Compute* $X_{k,m}^{(i)}, \forall k, m$, *using* $\mu^{(i)}$, ${\bm \alpha}^{(i)}$ *in* (\[compute X\])\ &$\textit{Compute }{\bm t}^{(i)}\textit{ using }X_{k,m}^{(i)} \textit{ in } (\ref{optimal_t_typeI})$\ &$\textit{Compute }{\bm p}^{(i)}\textit{ using }\mu^{(i)}, ~{\bm t}^{(i)} \textit{ in }(\ref{optimal power_typeI})$\ &$\textit{Compute }\mu^{(i+1)}, ~{\bm \alpha}^{(i+1)}\textit{ using }\mu^{(i)}, ~{\bm \alpha}^{(i)}, ~{\bm t}^{(i)}, ~{\bm p}^{(i)} \textit{ in }(\ref{sub_gradient_typeI_total})$\ &*If* (amendment = *false*) *and* $\left(\frac{\lvert\mu^{(i+1)}-\mu^{(i)}\rvert}{\lvert \mu^{(i+1)} \rvert}<\epsilon\right)$ *and* $\left(\frac{\lVert{\bm \alpha}^{(i+1)}-{\bm \alpha}^{(i)}\rVert}{\lVert {\bm \alpha}^{(i+1)} \rVert}<\epsilon\right)$\ &amendment = *true*\ &max\_it = $\lfloor 1.1 \times i \rfloor$\ &*End*\ &*If* (amendment = *true*)        *(\*\* amendment algorithm \*\*)*\ &$\hat{{\bm t}} = {\bm t}^{(i)}$, $c_m=\sum_{k=1}^M\hat{t}_{k,m}, \forall m$\ &*For (j = 1 to M)*&\ &*If* $(c_j>1)$,   $s^*=\arg\max_{\left\{s|\hat{t}_{s,j}=1\right\}}X_{s,j}^{(i)}$,   *End*\ &*While* ($c_j>1$)\ &$m^*=\arg\min_{\{m|c_m=0\}}\left|\alpha^{(i)}_j-\alpha^{(i)}_m\right|$\ &$r^*=\arg\max_{\left\{r|\hat{t}_{r,j}=1, ~r\neq s^*\right\}}X_{r,m^*}^{(i)}$\ &$\hat{t}_{r^*,j}=0$,  $\hat{t}_{r^*,m^*}=1$\ &$c_{j}=c_{j}-1, ~c_{m^*}=c_{m^*}+1$\ &*End*\ &*End*\ &*With fixed subcarrier pairing* $\hat{{\bm t}},$ *apply water-filling on the subcarrier pairs with equivalent*\ &  *channel gains in (\[Eq\_ECG\]) to compute power allocation* $\hat{{\bm p}}$ *and the weighted sum rate* $R$ *as in (\[objective\_Rate\_I\_MIP\])*.\ &*If* ($R>$ sum\_rate),   sum\_rate $=R$,  $\check{{\bm t}}=\hat{{\bm t}}$,  $\check{{\bm p}}=\hat{{\bm p}}$,   *End*\ &*End        (\*\* amendment algorithm \*\*)*\ &$i=i+1$\ \ [lll]{}\ \ &*Determine modes for all possible subcarrier pairs, and obtain* $a_{k,m}$, $c^S_{k,m}$ and $c^R_{k,m}$\ &  *by using* $\mu_S^{(i)}$, $\mu_R^{(i)}$ *in* (\[Eq\_ECG\_individual\]), (\[source\_power\_ratio\_individual\]) and (\[relay\_power\_ratio\_individual\])\ &*Compute* $Z_{k,m}^{(i)}, \forall k, m$, *using* $\mu_S^{(i)}$, $\mu_R^{(i)}$, ${\bm \alpha}^{(i)}$, $a_{k,m}$, $c^S_{k,m}$, $c^R_{k,m}$ *in* (\[compute Z\])\ &$\textit{Compute }{\bm t}^{(i)}\textit{ using }Z_{k,m}^{(i)} \textit{ in } (\ref{optimal_t_typeI_individual})$\ &$\textit{Compute }{\bm p}^{(i)}\textit{ using }\mu_S^{(i)}, ~\mu_R^{(i)}, ~{\bm t}^{(i)}, ~a_{k,m}, ~c^S_{k,m}, ~c^R_{k,m} \textit{ in }(\ref{optimal power_typeI_individual})$\ &$\textit{Compute }\mu_S^{(i+1)}, ~\mu_R^{(i+1)}, ~{\bm \alpha}^{(i+1)}\textit{ using }\mu_S^{(i)}, ~\mu_R^{(i)}, ~{\bm \alpha}^{(i)}, ~{\bm t}^{(i)}, ~{\bm p}^{(i)}, ~c^S_{k,m}, ~c^R_{k,m} \textit{ in }(\ref{sub_gradient_typeI individual})$\ &*If* (amendment = *false*) *and* $\left(\frac{\lvert\mu_S^{(i+1)}-\mu_S^{(i)}\rvert}{\lvert \mu_S^{(i+1)} \rvert}<\epsilon\right)$ *and* $\left(\frac{\lvert\mu_R^{(i+1)}-\mu_R^{(i)}\rvert}{\lvert \mu_R^{(i+1)} \rvert}<\epsilon\right)$ *and* $\left(\frac{\lVert {\bm \alpha}^{(i+1)}-{\bm \alpha}^{(i)}\rVert}{\lVert {\bm \alpha}^{(i+1)} \rVert}<\epsilon\right)$\ &amendment = *true*\ &max\_it = $\lfloor 1.1 \times i \rfloor$\ &*End*\ &*If* (amendment = *true*)        *(\*\* amendment algorithm \*\*)*\ &$\hat{{\bm t}} = {\bm t}^{(i)}$, $c_m=\sum_{k=1}^M\hat{t}_{k,m}, \forall m$\ &*For (j = 1 to M)*&\ &*If* $(c_j>1)$,   $s^*=\arg\max_{\left\{s|\hat{t}_{s,j}=1\right\}}Z_{s,j}^{(i)}$,   *End*\ &*While* ($c_j>1$)\ &$m^*=\arg\min_{\{m|c_m=0\}}\left|\alpha^{(i)}_j-\alpha^{(i)}_m\right|$\ &$r^*=\arg\max_{\left\{r|\hat{t}_{r,j}=1, ~r\neq s^*\right\}}Z_{r,m^*}^{(i)}$\ &$\hat{t}_{r^*,j}=0$,  $\hat{t}_{r^*,m^*}=1$\ &$c_{j}=c_{j}-1, ~c_{m^*}=c_{m^*}+1$\ &*End*\ &*End*\ &*With fixed subcarrier pairing* $\hat{{\bm t}},$ *and letting* $\hat{\mu}_S=\mu_S^{(i)}$, $\hat{\mu}_R=\mu_R^{(i)}$, *use the zero-crossing method in [@relay_DF_individual_power_constraint Section 3.2]*\ &  *to update $\hat{\mu}_S$, $\hat{\mu}_R$ to their optimal.*\ &*Use $\hat{\mu}_S$, $\hat{\mu}_R$ in (\[mode\_conditions\_individual\]) and (\[Louveaux\_power\_allocation\]) to obtain mode classification and power allocation* $\hat{{\bm p}}_{S}$, $\hat{{\bm p}}_{R}$,\ &  *and compute the weighted sum rate $R$*.\ &*If* ($R>$ sum\_rate),   sum\_rate $=R$,  $\check{{\bm t}}=\hat{{\bm t}}$,  $\check{{\bm p}}_{S}=\hat{{\bm p}}_{S}$,  $\check{{\bm p}}_{R}=\hat{{\bm p}}_{R}$,   *End*\ &*End        (\*\* amendment algorithm \*\*)*\ &$i=i+1$\ \ [lll]{}\ \ &*Compute* $\left(Y^R_{k,m}\right)^{(i)}, \left(Y^D_{k,m}\right)^{(i)}, ~\forall k, m$, *using* $\mu^{(i)}$ *in* (\[compute Y\_R\]) *and* (\[compute Y\_D\]).\ &$\textit{Obtain }{\bm s}^{(i)}\textit{ using }\left(Y^R_{k,m}\right)^{(i)}, \left(Y^D_{k,m}\right)^{(i)} \textit{ in } (\ref{Eq_s_choose})$\ &*Compute* $Y_{k,m}^{(i)}, ~\forall k, m$, *using* $\left(Y^R_{k,m}\right)^{(i)}$, $\left(Y^D_{k,m}\right)^{(i)}$, ${\bm s}^{(i)}$, ${\bm \alpha}^{(i)}$ *in* (\[compute Y\])\ &$\textit{Compute }{\bm t}^{(i)}\textit{ using }Y_{k,m}^{(i)} \textit{ in } (\ref{optimal_t_typeII})$\ &$\textit{Compute }{\bm p}^{(i)}\textit{ using }\mu^{(i)}, ~{\bm t}^{(i)}, ~{\bm s}^{(i)} \textit{ in }(\ref{optimal power_typeII})$\ &$\textit{Compute }\mu^{(i+1)}, ~{\bm \alpha}^{(i+1)}\textit{ using }\mu^{(i)}, ~{\bm \alpha}^{(i)}, ~{\bm t}^{(i)}, ~{\bm p}^{(i)} \textit{ in }(\ref{sub_gradient_typeII_total})$\ &*If* (amendment = *false*) *and* $\left(\frac{\lvert\mu^{(i+1)}-\mu^{(i)}\rvert}{\lvert \mu^{(i+1)} \rvert}<\epsilon\right)$ *and* $\left(\frac{\lVert {\bm \alpha}^{(i+1)}- {\bm \alpha}^{(i)}\rVert}{\lVert {\bm \alpha}^{(i+1)} \rVert}<\epsilon\right)$\ &amendment = *true*\ &max\_it = $\lfloor 1.1 \times i \rfloor$\ &*End*\ &*If* (amendment = *true*)        *(\*\* amendment algorithm \*\*)*\ &$\hat{{\bm t}} = {\bm t}^{(i)}$, $\hat{{\bm s}} = {\bm s}^{(i)}$, $c_m=\sum_{k=1}^M\hat{t}_{k,m}, \forall m$\ &*For (j = 1 to M)*&\ &*If* $(c_j>1)$,   $s^*=\arg\max_{\left\{s|\hat{t}_{s,j}=1\right\}}Y_{s,j}^{(i)}$,   *End*\ &*While* ($c_j>1$)\ &$m^*=\arg\min_{\{m|c_m=0\}}\left|\alpha^{(i)}_j-\alpha^{(i)}_m\right|$\ &$r^*=\arg\max_{\left\{r|\hat{t}_{r,j}=1, ~r\neq s^*\right\}}Y_{r,m^*}^{(i)}$\ &$\hat{t}_{r^*,j}=0$,  $\hat{t}_{r^*,m^*}=1$\ &$c_{j}=c_{j}-1, ~c_{m^*}=c_{m^*}+1$\ &*End*\ &*End*\ &*With fixed subcarrier pairing* $\hat{{\bm t}}$ *and* mode selection $\hat{{\bm s}},$ *apply water-filling on the direct-link, extra direct-link*\ &  *subcarriers and the relay mode subcarrier pairs with equivalent channel gains in (\[Eq\_ECG\]), to compute*\ &  *power allocation* $\hat{{\bm p}}$ *and the weighted sum rate* $R$.\ &*If* ($R>$ sum\_rate),   sum\_rate $=R$,  $\check{{\bm t}}=\hat{{\bm t}}$,  $\check{{\bm s}}=\hat{{\bm s}}$,  $\check{{\bm p}}=\hat{{\bm p}}$,   *End*\ &*End        (\*\* amendment algorithm \*\*)*\ &$i=i+1$\ \

--- abstract: | We determine several necessary and sufficient conditions for a closed almost-complex orbifold $Q$ with cyclic local groups to admit a nonvanishing vector field. These conditions are stated separately in terms of the orbifold Euler-Satake characteristics of $Q$ and its sectors, the Euler characteristics of the underlying topological spaces of $Q$ and its sectors, and in terms of the orbifold Euler class $e_{orb}(Q)$ in Chen-Ruan orbifold cohomology $H_{orb}^\ast (Q; {\mathbb{R}})$. address: 'Department of Mathematics and Computer Science, Rhodes College, 2000 N. Parkway, Memphis, TN 38112' author: - Christopher Seaton date: 'August 2nd, 2006' title: 'A Complete Obstruction to the Existence of Nonvanishing Vector Fields on Almost-Complex, Closed, Cyclic Orbifolds' --- Introduction ============ Orbifolds are singular spaces locally modeled by ${\mathbb{R}}^n / G$ where $G$ is a finite subgroup of $O(n)$ that acts with a fixed-point set of codimension at least 2. The original definition of an orbifold was introduced by Satake in [@satake1] under the name $V$-manifold, and the term orbifold was given by Thurston in [@thurston]. Thurston’s orbifolds included a larger class than those of Satake, for he allows the local groups $G$ to act with a fixed-point set of codimension 1. Today, the definition of an orbifold varies from author to author. Here, we retain the requirement that the local groups act with a fixed-point set of codimension at least 2, but do not require the groups to act effectively. Hence, Satake’s $V$-manifolds correspond to our [**reduced**]{} orbifolds. Let $Q$ be a closed, reduced orbifold of dimension $n$. One of the first things that was studied on orbifolds is the generalization of de Rham theory by Satake in [@satake1] and [@satake2]. In the latter of these two papers, Satake developed a generalization of the Poincaré-Hopf Theorem, that if $X$ is a vector field on $Q$ with only isolated zeros, then $$\label{eq-ph} \mbox{ind}_{orb}(X) = \chi_{orb}(Q).$$ Here, $\mbox{ind}_{orb}(X)$ is the orbifold index of the vector field and $\chi_{orb}(Q)$ the orbifold Euler-Satake characteristic of $Q$ (see Section \[sec-defs\] for a review of the definitions). More recently, the author has developed an additional generalization of the Poincaré-Hopf Theorem to orbifolds [@mythesis]. In this case, the left side of the equation is the orbifold index of the vector field $\tilde{X}$ induced by $X$ on $\tilde{Q}$, the space of sectors of the orbifold. The right side then becomes $\chi(\mathbb{X}_Q)$, the Euler characteristic of the underlying topological space $\mathbb{X}_Q$ of $Q$: $$\label{eq-myph} \mbox{ind}_{orb}(\tilde{X}) = \chi(\mathbb{X}_Q).$$ We will review these definitions in the sequel; here, we note that if $X$ is a nonvanishing vector field, then $\tilde{X}$ is nonvanishing as well. As in the case of manifolds [@phopf], it is a direct corollary of these formulae that an orbifold admits a nonvanishing vector field only if its orbifold Euler-Satake characteristic vanishes (in the case of Equation \[eq-ph\]), and the Euler characteristic of its underlying topological space vanishes (in the case of Equation \[eq-myph\]). Unlike the case of manifolds, however, the converse of both of these statements is false. It is easy to construct examples of $2$-orbifolds $Q$ such that $\chi_{orb}(Q) = 0$ or $\chi( \mathbb{X}_Q) = 0$, yet whose singular points force any vector field to vanish. While it is impossible for both of these invariants to vanish for a nontrivial $2$-orbifold, it is possible to construct a $4$-dimensional orbifold such that $\chi_{orb}(Q) = \chi( \mathbb{X}_Q) = 0$ that does not admit a nonvanishing vector field. For instance, one may take an orbifold whose underlying space is $\mathbb{T}^4$ and whose singular set is the disjoint union of $S^2$ and a surface of genus $2$, all with isotropy group ${\mathbb{Z}}_3$. In this paper, we determine necessary and sufficient conditions for a closed, almost-complex orbifold with cyclic local groups to admit a nonvanishing vector field. Our main result is the following theorem. \[thrm-mainresult\] Let $Q$ be a closed almost-complex cyclic orbifold, and then the following are equivalent: \(i) $Q$ admits a nonvanishing vector field. \(ii) $\tilde{Q}$ admits a nonvanishing vector field. \(iii) The Euler characteristic of the underlying space of each sector $\tilde{Q}_{(g)}$ is zero. \(iv) The orbifold Euler-Satake characteristic of each sector $\tilde{Q}_{(g)}$ is zero. \(v) $e_{orb}(Q)$, the orbifold Euler class of $Q$, is zero in $H_{orb}^\ast(Q ; {\mathbb{R}})$. In Section \[sec-defs\], we review the pertinent definitions and fix our notation. The main constructions we require are that of the space of sectors of an orbifold, Chen-Ruan orbifold cohomology, and the orbifold Euler class; the reader is referred to the original sources for a more detailed exposition. In Section \[sec-structure\], we study the relationship between the sectors of an orbifold. Section \[sec-mainresult\] contains the proof of our theorem. The author is pleased to acknowledge Carla Farsi, Alexander Gorokhovsky, Judith Packer, Arlan Ramsay, and Lynne Walling for useful discussions and support during the work leading to this result. Review of Definitions {#sec-defs} ===================== In this section, we briefly review the definitions we will need. For more information, the reader is referred to the original work of Satake in [@satake1] and [@satake2]. As well, [@ruangwt] contains as an appendix a thorough introduction to orbifolds, focusing on their differential geometry, and [@mythesis] contains an introduction to orbifolds with an emphasis on vector fields. A (${\mathcal C}^\infty$) [**orbifold**]{} $Q$ is a Hausdorff space $\mathbb{X}_Q$ such that each point is contained in an open set modeled by an [**orbifold chart**]{} or [**local uniformizing system.**]{} By this, we mean a triple $\{ V, G, \pi \}$ where - $V$ is an open subset of ${\mathbb{R}}^n$, - $G$ is a finite group with a ${\mathcal C}^\infty$ action on $V$ such that the fixed point set of any $\gamma \in G$ which does not act trivially on $V$ has codimension at least 2 in $V$, and - $\pi : V \rightarrow U$ is a surjective continuous map such that $\forall \gamma \in G$, $\pi \circ \gamma = \pi$ that induces a homeomorphism $\tilde{\pi} : V/G \rightarrow U$. The image $U=\pi(V)$ is called a [**uniformized set**]{} in $Q$. The group $G$ is known as a [**local group.**]{} If the local group of a chart $\{ V, G, \pi \}$ acts effectively, then the chart is said to be [**reduced**]{}; if all charts are reduced, then $Q$ is a [**reduced orbifold.**]{} In the spirit of [@chenhu] for the case of Abelian orbifolds, we adopt the convention that if each local group is cyclic, then $Q$ is a [**cyclic orbifold**]{}. It is required that if a point $p$ is contained in two uniformized sets $U_i$ and $U_j$, then there is a uniformized set $U_k$ such that $p \in U_k \subset U_i \cap U_j$. Moreover, if $U_i \subseteq U_j$ are two sets uniformized by $ \{ V_i , G_i , \pi_i \}$ and $ \{ V_j , G_j , \pi_j \}$, respectively, then we require that they are related by an injection $\lambda_{ij} : \{ V_i , G_i , \pi_i \} \rightarrow \{ V_j , G_j , \pi_j \}$. An [**injection**]{} $\lambda_{ij}$ is a pair $\{ f_{ij}, \phi_{ij} \}$ where - $f_{ij} : G_i \rightarrow G_j$ is an injective homomorphism such that if $K_i$ and $K_j$ denote the kernel of the action of $G_i$ and $G_j$, respectively, then $f_{ij}$ restricts to an isomorphism of $K_i$ onto $K_j$, and - $\phi_{ij} : V_i \rightarrow V_j$ is a smooth embedding such that $\pi_i = \pi_j \circ \phi_{ij}$ and such that for each $\gamma \in G_i$, $\phi_{ij} \circ \gamma = f_{ij}(\gamma) \circ \phi_{ij}$ (see [@satake2], [@ruangwt], or [@mythesis]). Orbifold vector bundles are defined over each chart $\{ V, G, \pi \}$ as $G$-vector bundles over $V$. In particular, the tangent bundle is defined locally to be the ordinary tangent bundles $TV$ with $G$-structure given by the differential of the $G$-action. Sections of orbifold vector bundles correspond locally to $G$-equivariant sections of the $G$-bundles over $V$. In particular, with the help of the exponential map, it is possible to replace each orbifold chart containing a point $p \in Q$ with an equivalent chart such that $G$ acts on $V$ as a subgroup of $O(n)$, and $p$ is the image under $\pi$ of the origin in $V$. Such a chart will be known as a [**chart at $p$**]{}, denoted $\{ V_p, G_p, \pi_p \}$, with $U_p := \pi_p(V_p)$, etc. Note that in a chart at $p$, $G_p$ is the isotropy group of $p$, and its isomorphism class is independent of the choice of chart. The [**orbifold index of a vector field $X$ on $Q$**]{} with isolated zeros is defined to be the sum of the indices at each zero of $X$, where the index at a zero point $p$ is the quotient of the (usual) index of the vector field in an orbifold chart and the order of the isotropy group at $p$. In other words, if $\{ V_p, G_p, \pi_p \}$ is an orbifold chart at $p$, then the index of $X$ at $p$ is $\frac{1}{|G_p|} \mbox{ind}_{\mathbf{0}} (\pi_p^\ast X)$. The [**orbifold Euler-Satake characteristic**]{} $\chi_{orb}(Q)$ is most easily defined by finding a simplicial decomposition ${\mathcal T}$ for $Q$ such that the isomorphism class of the isotropy group of each point on the interior of a simplex is constant (such a triangulation always exists; see [@moerdijk]). For each simplex $\sigma \in {\mathcal T}$, if we let $m_{\sigma}$ denote the order of this isotropy group, then $$\chi_{orb}(Q) := \sum\limits_{\sigma \in {\mathcal T}} (-1)^{\mbox{dim}\:\sigma} \frac{1}{m_{\sigma}}.$$ Note that if $A \cup B$ is a union of orbifolds, then it is straightforward to show from this definition that, as in the case of the usual Euler characteristic, $$\label{eq-addativeesk} \chi_{orb}(A \cup B) = \chi_{orb}(A) + \chi_{orb}(B) - \chi_{orb}(A \cap B).$$ The Chen-Ruan orbifold cohomology groups are defined in terms of the [**space of sectors**]{} of the orbifold (also known as the [**inertia orbifold**]{}). We recall the construction, referring the reader to [@chenruan] and [@kawasaki2] for more details. Select for each $p \in Q$ a chart $\{ V_p, G_p, \pi_p \}$ at $p$. Then the set $$\tilde{Q} := \{ (p, (g)_{G_p} ) : p \in Q, g \in G_p \}$$ (where $(g)_{G_p}$ is the conjugacy class of $g$ in $G_p$) is naturally an orbifold, with local charts $$\{ \pi_{p, g} : (V_p^g, C(g) ) \rightarrow V_p^g / C(g) : p \in Q, g \in G_p \} .$$ Here, $V_p^g$ is the fixed point set of $g$ in $V_p$, and $C(g)$ is the centralizer of $g$ in $G_p$. An equivalence relation can be placed on the conjugacy classes of the local groups $G_p$ as follows. The conjugacy class of $g \in G_p$ and that of $h \in G_q$ are equivalent if there is an injection of a chart at $q$ into a chart at $p$ such that $f_{qp}(h) = g$ for the corresponding homomorphism $f_{qp}$. Note that, as the choice of injection is not generally unique, this equivalence is defined on conjugacy classes. In the case that $G_p$ and $G_q$ are Abelian, of course, each conjugacy class contains one element (see [@chenhu] for more details in the case of Abelian local groups). Let $T$ denote the set of equivalence classes under this relation (which is finite for $Q$ compact) and $(g)$ the equivalence class of a conjugacy class $(g)_{G_p}$. Then $$\tilde{Q} = \bigsqcup\limits_{(g) \in T } \tilde{Q}_{(g)} ,$$ where $$\tilde{Q}_{(g)} = \{ (p, (g^\prime)_{G_p}) : g^\prime \in G_p , (g^\prime)_{G_p} \in (g) \} .$$ Each of the $\tilde{Q}_{(g)}$ for $(g) \neq (1)$ is called a [**twisted sector**]{}; $\tilde{Q}_{(1)}$ is the [**nontwisted sector**]{}, and is diffeomorphic to $Q$ as an orbifold. The [**sectors**]{} of the orbifold refer to both the twisted sectors and the nontwisted sector. We note that even in the case that $Q$ is connected, a twisted sector of $Q$ need not be. If $Q$ is an almost complex orbifold, a function $\iota : \tilde{Q} \rightarrow {\mathbb{Q}}$ is defined which is constant on each $\tilde{Q}$. The value of this function on $\tilde{Q}_{(g)}$, denoted $\iota_{(g)}$, is called the degree shifting number of $(g)$. The orbifold cohomology groups are defined by $$H_{orb}^d (Q; {\mathbb{R}}) := \bigoplus_{(g) \in T} H^{d - 2\iota_{(g)}} (\tilde{Q}_{(g)} ; {\mathbb{R}}),$$ where the groups on the right side are the usual de Rham cohomology groups of the orbifolds $\tilde{Q}_{(g)}$. For a vector bundle $\rho :E \rightarrow Q$, the space $E$ is naturally an orbifold, so it is possible to form $\tilde{E}$ as above. In the case that $E$ is a [**good**]{} vector bundle (see [@ruangwt], Section 4.3), $\tilde{E}$ is naturally an orbifold vector bundle over $\tilde{Q}$, although its dimension varies in general over the connected components of $\tilde{Q}$. Similarly, smooth sections $s : Q \rightarrow E$ of a good vector bundle $E$ naturally induce smooth sections $\tilde{s} : \tilde{Q} \rightarrow \tilde{E}$ in such a way that if $s$ is nonvanishing, then so is $\tilde{s}$ (see [@mythesis], Lemma 4.4.1; the definition of an orbifold vector bundle in this reference corresponds to Ruan’s definition of a good orbifold vector bundle). In particular, this is true of the orbifold tangent bundle $TQ$, which is always a good vector bundle. Moreover, $T\tilde{Q} = \widetilde{TQ}$, so that nonvanishing vector fields over $Q$ naturally induce nonvanishing vector fields over $\tilde{Q}$. A connection on a good vector bundle $E$ induces one on $\tilde{E}$. The [**orbifold Euler class**]{} of an orbifold vector bundle $E$ is defined in terms of such a connection. It can be taken to be the sum in $H^\ast_{orb}(Q ; {\mathbb{R}})$ of the usual Euler classes of the orbifold bundle $\tilde{E}$ restricted to each of the connected components of each $\tilde{Q}_{(g)}$ (see [@mythesis]). We will require the formula $$\label{eq-2ndgb} \sum\limits_{(g) \in T} \chi_{orb} ( \tilde{Q}_{(g)} ) = \chi (\mathbb{X}_Q).$$ See the proof of the Second Gauss-Bonnet Theorem for orbifolds (Theorem 4.4.2) in [@mythesis] for a verification of this formula. The Structure of the Sectors of a Cyclic Orbifold {#sec-structure} ================================================= The construction of $\tilde{Q}$ decomposes an orbifold $Q$ into multiple sectors, the largest of these being diffeomorphic to $Q$ and the others being simpler orbifolds of lower dimension. In this section, we study the structure of these connected components as they appear as subsets of a cyclic orbifold $Q$. Let $\pi : \tilde{Q} \rightarrow Q$ denote the projection with $\pi(p, (g)) = p$. Then $\pi$ is a ${\mathcal C}^\infty$ map (see [@chenruan]). In the case that each of the local groups is Abelian, clearly $C(g) = G_p$ for each $(g) \in T$, so that the local uniformized sets $V_p^g / C(g)$ are diffeomorphic to subsets of uniformized sets $V_p/G$ in $Q$ via $\pi$. Therefore, in this case, $\pi$ is an embedding when restricted to $\tilde{Q}_{(g)}$ for each $(g) \in T$ (see [@chenhu]). We define the following relation on the $\tilde{Q}_{(g)}$ via $\pi$: we say that $\tilde{Q}_{(h)} \leq \tilde{Q}_{(g)}$ whenever $\pi(\tilde{Q}_{(h)}) \subseteq \pi(\tilde{Q}_{(g)})$, $\tilde{Q}_{(h)} \equiv \tilde{Q}_{(g)}$ whenever $\pi(\tilde{Q}_{(h)}) = \pi(\tilde{Q}_{(g)})$, etc. Note that $\tilde{Q}_{(h)} \equiv \tilde{Q}_{(g)}$ does not imply that $\tilde{Q}_{(h)} = \tilde{Q}_{(g)}$, but that, through the appropriate restrictions of $\pi$ and inverses of these restrictions (see Equation \[eq-restrictdiffeo\]), they are diffeomorphic. Therefore, this relation can be thought of as a partial order of equivalence classes of the sectors under the equivalence relation $\equiv$. We will refer to the elements of the minimal equivalence classes under $\equiv$ as minimal with respect to the relation $\leq$. As well, it will be convenient for us to state this relation in terms of the elements of $T$; i.e. $(h) \leq (g)$ will mean that $\tilde{Q}_{(h)} \leq \tilde{Q}_{(g)}$, etc. With respect to this relation, the nontwisted sector $\tilde{Q}_{(1)}$ is clearly maximal, as are each of the $\tilde{Q}_{(g)}$ where the representatives of $(g)$ act trivially (in the case that $Q$ is not reduced). Similarly, we have the following: \[lemm-minmnflds\] Let $Q$ be a closed, cyclic orbifold, and let $\tilde{Q}_{(g)}$ be a sector of $Q$ that is minimal with respect to the relation given above. Then $\tilde{Q}_{(g)}$ is a manifold equipped with the trivial action of a finite group, and hence the associated reduced orbifold $(\tilde{Q}_{(g)})_{red}$ is a manifold. Let $\tilde{Q}_{(g)}$ be a minimal sector, and let $(p, (g))$ be a point in $\tilde{Q}_{(g)}$. Fix an orbifold chart at $(p, (g))$ of the form $\{ V_p^g, C(g), \pi_{p, g} \}$ induced by an orbifold chart $\{ V_p, G_p, \pi_p \}$ for $Q$ at $p$, and note that as $G_p$ is Abelian, $C(g) = G_p$. Without loss of generality, assume that the representative $g$ of $(g)$ is an element of $G_p$. The subgroup $\langle g \rangle$ of $G_p$ generated by $g$ clearly acts trivially on $V_p^g$, and a representative of $(g)$ is in the isotropy group of each point in $\pi (\tilde{Q}_{(g)})$. We argue that $\langle g \rangle$ is exactly the isotropy group of each point in $\pi (\tilde{Q}_{(g)})$. Let $(q, (g_q))$ be an arbitrary point in $\tilde{Q}_{(g)}$, and then $g_q$ is a representative of $(g)$ in the isotropy group $G_q$. Hence, there is a sequence of points $(p_i, (g_{p_i}))$, $i = 0, \ldots , s$ such that $p_0 = p$, $p_s = q$, $p_i$ and $p_{i+1}$ lie in the same chart $\{ V_i, G_i, \pi_i \}$, and $g_i = g_{i+1}$ in $G_i$ (see [@chenhu]; note that in general, $g_i$ and $g_{i+1}$ are required to be conjugate in $G_i$, but that this implies that they are equal in our case). Fix a chart $\{ V_q, G_q, \pi_q \}$ at $q \in Q$, and let $a$ be a generator of $G_q$. Then there is a $k \in {\mathbb{Z}}$ such that $a^k = g_q$. It is obvious that $V_q^a \subseteq V_q^{g_q}$. Moreover, for any injection of $\{ V_q, G_q, \pi_q \}$ (or a restriction to a smaller chart containing $q$) into another chart, the associated injective homomorphism $f$ satisfies $f(a)^k = f(g_q)$. This implies that in any chart for $Q$, a representative of $(a)$ has a fixed-point set contained in that of a representative of $(g_q)$. Therefore, $\pi(\tilde{Q}_{(a)})$ is contained in $\pi(\tilde{Q}_{(g_q)})$. As $(g_q)$ and $(g)$ represent the same equivalence class in $T$, $\tilde{Q}_{(g_q)} = \tilde{Q}_{(g)}$. This implies that $(a) \leq (g)$. Hence, as $\tilde{Q}_{(g)}$ is assumed to be minimal, we must have that $(a) \equiv (g)$. Therefore, $a = g_q \in G_q$. Now, for each pair $p_i, p_{i+1}$ of points above, the intersection of the chart at $p_i$ with that at $p_{i+1}$ produces a chart at $p_i$ contained in that at $p_{i+1}$. Hence, there is an injection of charts including an injective homomorphism $f_{i, i+1} : G_{p_i} \rightarrow G_{p_{i+1}}$, producing a sequence $$G_p = G_{p_0} \stackrel{f_{0, 1}}{\longrightarrow} G_{p_1} \stackrel{f_{1, 2}}{\longrightarrow} \cdots \stackrel{f_{s-1, s}}{\longrightarrow} G_{p_s} = G_q.$$ Let $F: G_p \rightarrow G_q$ be the composition of these maps, and then note that, as $a = g_q$ is a generator of $G_q$, and as $F(g) = g_q$, we must have that each $f_{i, i+1}$ is an isomorphism. In particular, $g$ generates $G_p$, and $G_q$ is isomorphic to $G_p$. As $(q, (g_q))$ was arbitrary, we conclude that the isotropy group of each point in $\tilde{Q}_{(g)}$ is isomorphic to $G_p = \langle g \rangle$. Therefore, $\tilde{Q}_{(g)}$ is a manifold equipped with the trivial action of the finite group $G_p$. In the case of a cyclic orbifold, the twisted sectors each decompose into twisted sectors themselves. In particular, as was noted above, for each $(g) \in T$, the map $$\pi_{(g)} := \pi_{|\tilde{Q}_{(g)}} : \tilde{Q}_{(g)} \rightarrow Q$$ is an embedding of $\tilde{Q}_{(g)}$ into $Q$. Hence, if $(h) \leq (g)$, then as $\tilde{Q}_{(h)}$ is embedded into $\pi(\tilde{Q}_{(g)})$, the composition $$\label{eq-restrictdiffeo} \tilde{Q}_{(h)} \stackrel{\pi_{(h)}}{\rightarrow} \pi(\tilde{Q}_{(h)}) \stackrel{\pi_{(g)}^{-1}}{\rightarrow} \tilde{Q}_{(g)}$$ defines an embedding of $\tilde{Q}_{(h)}$ into $\tilde{Q}_{(g)}$ (we note that the inverse $\pi_{(g)}^{-1}$ is defined on $\pi(\tilde{Q}_{(g)}) \supseteq \pi_{(h)}(\tilde{Q}_{(h)})$, and not on $Q$). Denote this composition $$\pi_{(h), (g)} := \pi_{(g)}^{-1} \circ \pi_{(h)} : \tilde{Q}_{(h)} \rightarrow \tilde{Q}_{(g)}.$$ If $(g) \equiv (h)$, then $\pi_{(h), (g)}$ is a diffeomorphism of orbifolds. We conclude this section with the following Lemma, which illustrates that the singular set of a sector decomposes into the image of embeddings of strictly smaller sectors. Note that we reduce the sector $\tilde{Q}_{(g)}$ only to distinguish between the singular and regular points of the reduced orbifold; for $(g) \neq (1)$, all of $\tilde{Q}_{(g)}$ is singular. \[lemm-singinsector\] Let $Q$ be a closed, cyclic orbifold. Fix a sector $\tilde{Q}_{(g)}$, and let $(p, (g))$ be a singular point of $(\tilde{Q}_{(g)})_{red}$. Then there is a $(g^\prime) \in T$ with $(g^\prime) < (g)$ such that $(g^\prime)$ has a representative in $G_p$. Hence, each singular point of $(\tilde{Q}_{(g)})_{red}$ is contained in the image of some such embedding $\pi_{(g^\prime), (g)}$ of such a $\tilde{Q}_{(g^\prime)}$ into $\tilde{Q}_{(g)}$. Moreover, the intersection of two such embeddings is the image of a sector. Fix a chart $\{ V_p, G_p, \pi_p \}$ for $Q$ at $p$, and then let $\{ V_p^g, C(g), \pi_{p, g} \}$ be the induced chart for $\tilde{Q}_{(g)}$. Assume without loss of generality that the representative $g$ of $(g)$ is an element of $G_p$. As $(p, (g))$ is assumed to be singular, there is an $h \in C(g) = G_p$ that acts nontrivially on $V_p^g$ and fixes $p$. Hence, $h$ has a fixed point subset $V_p^h$ in $V_p$ with $V_p^h \cap V_p^g \subsetneq V_p^g$. As $G_p$ is cyclic, the subgroup $\langle g, h \rangle$ of $G_p$ is cyclic as well. Let $g^\prime$ be a generator of $\langle g, h \rangle$, and then it is obvious that $V_p^{g^\prime} = V_p^g \cap V_p^h$. With this, we need only note that for any injection of a chart at $q$ to another chart, the corresponding injective homomorphism $f$ satisfies $\langle f(g^\prime) \rangle = \langle f(g), f(h) \rangle$, implying that the above relationships hold for any chart containing representatives of $(g)$, $(h)$, and $(g^\prime)$. Therefore, it follows that $(g^\prime) \leq (g)$ and $(g^\prime) \leq (h)$. In fact, we see that $\pi(\tilde{Q}_{(g)}) \cap \pi(\tilde{Q}_{(h)}) = \pi(\tilde{Q}_{(g^\prime)})$, and as $h$ acts nontrivially on $V^g$, we conclude that $(g^\prime) < (g)$. Moreover, as $(g^\prime)$ has a representative in the isotropy group $G_p$ of $p$, the singular point $(p, (g))$ lies in the image of $\tilde{Q}_{(g^\prime)}$ under $\pi_{(g^\prime),(g)}$. Now, suppose that $\tilde{Q}_{(h_1)}$ and $\tilde{Q}_{(h_2)}$ are two sectors with $(h_1) < (g)$ and $(h_2) < (g)$. If the embeddings of $\tilde{Q}_{(h_1)}$ and $\tilde{Q}_{(h_2)}$ intersect in $\tilde{Q}_{(g)}$, then there is a point $(p, (g))$ such that $G_p$ contains representatives of $(h_1)$ and $(h_2)$ (say $h_1$ and $h_2$). Again, $\langle g, h_1, h_2 \rangle$ is cyclic, so let $g^\prime$ be a generator. Repeating the above argument, it is clear that $$\pi(\tilde{Q}_{(h_1)}) \cap \pi(\tilde{Q}_{(h_2)}) \subseteq \pi(\tilde{Q}_{(g)})$$ and that $$\pi(\tilde{Q}_{(h_1)}) \cap \pi(\tilde{Q}_{(h_2)}) = \pi(\tilde{Q}_{(g^\prime)}).$$ Hence, the intersection of embeddings of two sectors is again a sector. Note that the above lemmas require that the orbifold is cyclic. It is easy to construct examples where the above fail in the case of orbifolds that do not have this property. The Result {#sec-mainresult} ========== Now, we return to the question of the existence of a nonvanishing vector field on a closed almost-complex cyclic orbifold $Q$. Using the results of Section \[sec-structure\], we will prove Theorem \[thrm-mainresult\]. Note that as $Q$ is almost-complex, $\tilde{Q}$ inherits an almost complex structure. In particular, each of the sectors of $Q$ are even-dimensional and oriented. \(i) $\Rightarrow$ (ii): Suppose $X$ is a nonvanishing vector field on $Q$. Then $X$ is a section of the orbifold tangent bundle $TQ$, and is hence required to be tangent to the singular strata of $Q$. Let $\tilde{X}$ be the induced section of $\widetilde{TQ}$. Again, as $T \tilde{Q} = \widetilde{TQ}$, $\tilde{X}$ is a vector field on $\tilde{Q}$. However, as $X$ must be tangent to each of the $\pi (\tilde{Q}_{(g)})$ in $Q$, and as $X$ does not vanish, $\tilde{X}$ is clearly a nonvanishing vector field on $\tilde{Q}$. \(ii) $\Rightarrow$ (iii): Fix a $(g) \in T$ and choose a nonvanishing vector field on $\tilde{Q}$; we let $X_{(g)}$ denote the restriction of this vector field to the sector $\tilde{Q}_{(g)}$. Form the space of sectors $\widetilde{\tilde{Q}_{(g)}}$ of $\tilde{Q}_{(g)}$ and let $\widetilde{X_{(g)}}$ denote the induced nonvanishing vector field on $\widetilde{\tilde{Q}_{(g)}}$. Clearly, the orbifold index of $\widetilde{X_{(g)}}$ is zero, and hence by Equation \[eq-myph\] (see also Corollary 4.4.4 of [@mythesis]), the Euler characteristic of the underlying space of $\tilde{Q}_{(g)}$ is zero. As $(g)$ was arbitrary, the Euler characteristic of the underlying space of each sectors is zero. \(iii) $\Rightarrow$ (iv): Suppose all of the Euler characteristics of the sectors of $Q$ vanish. Let $(h)$ be an element that is minimal with respect to the relation on $T$ so that by Lemma \[lemm-minmnflds\], $\tilde{Q}_{(h)}$ is a manifold with the trivial action of a finite group $C(h) = G_p$, where $G_p$ is the local group with $h \in G_p$. It is clear from Equation \[eq-2ndgb\] that $$\begin{array}{rcl} \chi_{orb}( \tilde{Q}_{(h)} ) &=& \frac{1}{|G_p|} \chi(\mathbb{X}_{\tilde{Q}_{(h)}} ) \\\\ &=& 0. \end{array}$$ Therefore, the orbifold Euler-Satake characteristics of all of the minimal elements $\tilde{Q}_{(h)}$ vanish. Now, fix $(g) \in T$, and suppose that for each $(h) < (g)$, the orbifold Euler characteristic of $\tilde{Q}_{(h)}$ is zero. Let ${\mathcal T}$ be a simplicial decomposition for $\tilde{Q}_{(g)}$ such that the isomorphism class of the isotropy group of each point on the interior of a simplex is constant (see Section \[sec-defs\]). Again, for each simplex $\sigma \in {\mathcal T}$, we let $m_{\sigma}$ denote the order of the isotropy group of the points in the interior of $\sigma$. Now, the Euler-Satake characteristic of $\tilde{Q}_{(g)}$ is given by $$\chi_{orb}(\tilde{Q}_{(g)}) = \sum\limits_{\sigma \in {\mathcal T}} (-1)^{\mbox{dim}\, \sigma} \frac{1}{m_\sigma}.$$ Note that the (isomorphism class of the) group generated by $g$ is contained in the isotropy group of each point in $\tilde{Q}_{(g)}$. However, any point of $\tilde{Q}_{(g)}$ whose isotropy group is strictly larger is a singular point of $(\tilde{Q}_{(g)})_{red}$, and hence is contained in the embedding of a $\tilde{Q}_{(h)}$ for $(h) < (g)$ by Lemma \[lemm-singinsector\]. For each simplex $\sigma$ not completely contained in the image of such an embedding, the isotropy group is isomorphic to $\langle g \rangle$. Separating the terms, we have $$\label{eq-2sum.esc} \begin{array}{rcl} \chi_{orb}(\tilde{Q}_{(g)}) &=& \sum\limits_{\sigma : m_\sigma = |g|} (-1)^{\mbox{dim}\, \sigma} \frac{1}{m_\sigma} + \sum\limits_{\sigma : m_\sigma > |g|} (-1)^{\mbox{dim}\, \sigma} \frac{1}{m_\sigma} \\\\ &=& \frac{1}{|g|} \sum\limits_{\sigma : m_\sigma = |g|} (-1)^{\mbox{dim}\, \sigma} + \sum\limits_{\sigma : m_\sigma > |g|} (-1)^{\mbox{dim}\, \sigma} \frac{1}{m_\sigma} \end{array}$$ The second sum is over all simplices contained in embeddings of $\tilde{Q}_{(h)}$ for $(h) < (g)$ into $\tilde{Q}_{(g)}$. While the images of these embeddings need not be disjoint, the intersection of any two sectors is again a sector by Lemma \[lemm-singinsector\]. Hence, applying Equation \[eq-addativeesk\], this sum can be expressed as a ${\mathbb{Z}}$-linear combination of Euler-Satake characteristics of sectors $\tilde{Q}_{(h)}$ with $(h) < (g)$. Using the fact that each such Euler-Satake characteristic vanishes by the inductive hypothesis, the second sum is clearly zero. Similarly, the sum $$\label{eq-2ndterm.ec} \sum\limits_{\sigma : m_\sigma > |g|} (-1)^{\mbox{dim}\, \sigma}$$ is the Euler characteristic of the underlying space of the image of embeddings of sectors. In the same way, this sum can be expressed as a ${\mathbb{Z}}$-linear combination of Euler characteristics of underlying spaces of sectors $\tilde{Q}_{(h)}$ with $(h) < (g)$. By hypothesis, each sector has underlying space with Euler characteristic 0, and therefore the sum in Equation \[eq-2ndterm.ec\] is zero. Hence, $$\label{eq-sumequal} \begin{array}{rcl} \sum\limits_{\sigma : m_\sigma > |g|} (-1)^{\mbox{dim}\, \sigma} \frac{1}{m_\sigma} &=& 0 \\\\ &=& \sum\limits_{\sigma : m_\sigma > |g|} (-1)^{\mbox{dim}\, \sigma} \\\\ &=& \frac{1}{|g|} \sum\limits_{\sigma : m_\sigma > |g|} (-1)^{\mbox{dim}\, \sigma} \end{array}$$ Now, returning to Equation \[eq-2sum.esc\], we have $$\begin{array}{rcl} \chi_{orb}(\tilde{Q}_{(g)}) &=& \frac{1}{|g|} \sum\limits_{\sigma : m_\sigma = |g|} (-1)^{\mbox{dim}\, \sigma} + \sum\limits_{\sigma : m_\sigma > |g|} (-1)^{\mbox{dim}\, \sigma} \frac{1}{m_\sigma} \\\\ &=& \frac{1}{|g|} \sum\limits_{\sigma : m_\sigma = |g|} (-1)^{\mbox{dim}\, \sigma} + \frac{1}{|g|} \sum\limits_{\sigma : m_\sigma > |g|} (-1)^{\mbox{dim}\, \sigma} \\\\ &&\mbox{(by Equation \ref{eq-sumequal})}\\\\ &=& \frac{1}{|g|} \sum\limits_{\sigma \in {\mathcal T}} (-1)^{\mbox{dim}\, \sigma} \frac{1}{m_\sigma} \\\\ &=& \frac{1}{|g|} \chi(\mathbb{X}_{\tilde{Q}_{(g)}} ) \end{array}$$ which is zero by hypothesis. Therefore, by induction, all of the orbifold Euler-Satake characteristics of the sectors vanish. \(iv) $\Rightarrow$ (i): Suppose the Euler-Satake characteristic of each sector of $Q$ is zero. Adorn $Q$ with a Riemannian metric and extend it in the natural way to a Riemannian metric on $\tilde{Q}$. We construct a nonvanishing vector field $X$ on $Q$ recursively. Start with the minimal $(h) \in T$. Each of the corresponding $\tilde{Q}_{(h)}$ are manifolds with the trivial action of a finite group by Lemma \[lemm-minmnflds\], and by hypothesis, the Euler-Satake characteristic of each of these manifolds is zero. Therefore, as the Euler characteristic of each of the underlying spaces is clearly also zero, it is well know that each $\tilde{Q}_{(h)}$ admits a nonvanishing vector field. We choose such a vector field $X_{(h)}$ on each of these minimal sectors. However, we require that these vector fields agree on $Q$ in the following sense: if $\tilde{Q}_{(h_1)} \equiv \tilde{Q}_{(h_2)}$, so that $\pi(\tilde{Q}_{(h_1)}) = \pi(\tilde{Q}_{(h_2)})$ in $Q$, then the vector fields on $\tilde{Q}_{(h_1)}$ and $\tilde{Q}_{(h_2)}$ correspond to the same vector field on $Q$. We can accomplish this by choosing $X_{(h_1)}$ and defining $X_{(h_2)}$ to be $$X_{(h_2)} := \pi^\ast [(\pi_{(h_1)})^{-1} ]^\ast X_{(h_1)}$$ (see Equation \[eq-restrictdiffeo\]). In this way, we define $X$ on one representative of each $\equiv$-equivalence class and extend the definition compatibly to the remaining members of the (finite) equivalent class. Now, fix some $(g) \in T$, and suppose that such a nonvanishing vector field has been given on each $\tilde{Q}_{(h)}$ with $(h) < (g)$ (in such a way that they agree when $(h_1) \equiv (h_2)$ as described above). Let $B:= \{ (p, (g)) : \exists \: (h) < (g)$ with $(p, (h)) \in \tilde{Q}_{(h)} \}$, i.e. the set of all points $(p, (g)) \in \tilde{Q}_{(g)}$ such that $p$ is fixed by a representative of some $(h)$ with $(h) < (g)$. For each such point, $(p, (h))$ is a point in $\tilde{Q}_{(h)}$ for such an $(h)$, so that $X_{(h)}$ is defined on this $\tilde{Q}_{(h)}$. By inverting restrictions of the map $\pi$ and pulling back the vector field on subsets of $\tilde{Q}_{(g)}$ as above, we define a vector field on $B$. Choose a finite set of orbifold charts that cover the (compact) set $B$. In each chart, we extend the vector field to a parallel vector field in an open set $W$ containing $B$. Recall that by Lemma \[lemm-singinsector\], each of the singular points of $(\tilde{Q}_{(g)})_{red}$ occur as fixed-points of some $h$ with $(h) < (g)$. Therefore, $W_{red}$ contains an open neighborhood of each of the singular points, and $(\tilde{Q}_{(g)})_{red} \backslash W_{red}$ contains only regular points of $(\tilde{Q}_{(g)})_{red}$. We extend the vector field to all of $(\tilde{Q}_{(g)})_{red}$ in such a way that the extension has only isolated zeros and note that this clearly defines a vector field on $\tilde{Q}_{(g)}$. With this, we may amalgamate the zeros using well-known techniques (see e.g. [@gp]) by finding a chart with trivial group action that contains multiple zeros in the image of a compact set. Such a chart can be given by choosing a simple path that passes through two zero points whose image does not intersect $B$ or any of the other zero points and taking a sufficiently small tubular neighborhood of that path. Hence, we need not change the vector field on $B$. Moreover, recall that each of the sectors are even-dimensional, so that the codimension of the image of each such $\tilde{Q}_{(h)}$ is at least 2; in particular, the preimage of the set $B$ does not separate a connected uniformized set. Using this technique, we extend the vector fields $X_{(g)}$ to larger and larger sectors until we have defined a nonvanishing vector field $X$ on $\tilde{Q}_{(1)} \cong Q$. \(i) $\Rightarrow$ (v): Let $X$ be a nonvanishing vector field on $Q$, and let $\tilde{X}$ be the induced section of $\widetilde{TQ}$. Restricted to each sector $\tilde{Q}_{(g)}$, the First Poincaré-Hopf Theorem for orbifolds (see [@satake2] and [@mythesis]) implies that the integral of the Euler curvature form $E(\Omega)$ defined with respect to a connection $\omega$ on $\tilde{Q}_{(g)}$ with curvature $\Omega$ is zero. Hence, as this closed top form is a representative of the term in $e_{orb}(Q)$ corresponding to $(g) \in T$, this term must be zero. As this is true for each $(g) \in T$, the orbifold Euler class vanishes. \(v) $\Rightarrow$ (iv): Suppose the orbifold Euler class $e_{orb}(Q)$ vanishes. This implies that the Euler curvature form $E(\Omega)$ of each of the sectors $\tilde{Q}_{(g)}$ of $Q$ has integral 0 over the corresponding sector. By the First Gauss-Bonnet Theorem for orbifolds (see [@satake2] and [@mythesis]), the Euler-Satake characteristics of the sectors all vanish. In the case that $Q$ is a manifold, the space of sectors is simply $Q$ itself, and the Euler characteristic and Euler-Satake characteristic coincide. Therefore, (i) and (ii) are the same statement, as are (iii) and (iv). Additionally, the orbifold Euler class $e_{orb}(Q)$ reduces to the ordinary Euler class, so that Theorem \[thrm-mainresult\] states that $Q$ admits a nonvanishing vector field if and only if its Euler characteristic vanishes, which is equivalent to its Euler class vanishing. Therefore, this theorem can be viewed as the generalization of the ‘Hairy Ball Theorem’ to the case of almost-complex orbifolds. We note that the requirement that $Q$ is almost-complex is crucial, and not simply required so that the Chen-Ruan cohomology groups are defined. In particular, if $Q$ is not almost-complex, then although all of the singular sets of a sector $\tilde{Q}_{(g)}$ must have codimension at least 2, it is not necessary that the image of each $\tilde{Q}_{(h)}$ with $(h) < (g)$ must have codimension at least 2 in $\pi(\tilde{Q}_{(g)})$ (note that the image of the $\tilde{Q}_{(h)}$ may contain regular points for $\tilde{Q}_{(h)}$ as well as singular points). Hence, the space formed by removing a copy of $\tilde{Q}_{(h)}$ from a uniformized set in $\tilde{Q}_{(g)}$ need not be connected, contributing an additional obstruction to the amalgamation of zeros of a vector field on $\tilde{Q}_{(g)}$. [10]{} B. Chen and S. 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Satake, *The [G]{}auss-[B]{}onnet theorem for ${V}$-manifolds*, Journ. Math. Soc. Japan **9** (1957) 464–492. C. Seaton, *Two [G]{}auss-[B]{}onnet and [P]{}oincaré-[H]{}opf Theorems for Orbifolds with Boundary*, PhD thesis, University of Colorado at Boulder, math.DG/0311075, 2004. W. Thurston, *The Geometry and Topology of 3-Manifolds*, Lecture Notes, Princeton University Math Dept., Princeton, New Jersey, 1978.

--- abstract: 'Bulk and decay properties, including deformation energy curves, charge mean square radii, Gamow-Teller (GT) strength distributions, and $\beta$-decay half-lives, are studied in neutron-deficient even-even and odd-$A$ Hg and Pt isotopes. The nuclear structure is described microscopically from deformed quasiparticle random-phase approximation calculations with residual interactions in both particle-hole and particle-particle channels, performed on top of a self-consistent deformed quasiparticle Skyrme Hartree-Fock basis. The observed sensitivity of the, not yet measured, GT strength distributions to deformation is proposed as an additional complementary signature of the nuclear shape. The $\beta$-decay half-lives resulting from these distributions are compared to experiment to demonstrate the ability of the method.' author: - 'J. M. Boillos' - 'P. Sarriguren' title: 'Effects of deformation on the beta-decay patterns of light even-even and odd-mass Hg and Pt isotopes' --- Introduction ============ Neutron-deficient isotopes in the lead region are nowadays well established examples of the shape coexistence phenomenon in nuclei [@heyde11; @julin01]. They have been subject of much experimental and theoretical interest in the last years. The first direct evidence of the shape coexistence in the region $Z\approx 82$ was obtained in neutron-deficient Hg isotopes from isotope shift measurements [@bonn72]. Those measurements showed a sharp transition in the nuclear size between the ground states of $^{187}$Hg and $^{185}$Hg that was interpreted [@frauendorf75] as a change from a weak oblate shape in the heavier isotopes to a more deformed prolate shape in the lighter ones from calculations based on Strutinsky’s shell correction method. Later, new isotope shift measurements [@ulm86] revealed a weakly oblate deformed character of the ground states of the even-mass Hg isotopes down to $A=182$, with an odd-even staggering persisting down to $^{181}$Hg. The radius of the oblate isomeric state in $^{185}$Hg follows the trend of the even-even ground-state radii. Shape evolution and shape coexistence in the region of $\beta$-unstable nuclei with $Z\approx 82$ were subsequently studied experimentally by $\gamma$-ray spectroscopy in the $\alpha$-decay of the products created in fusion-evaporation reactions (see Ref. [@julin01] and references therein). Maybe, the most singular case corresponds to $^{186}$Pb, where two excited $0^+$ states below 700 keV [@andreyev00] have been found. Furthermore, low-lying excited $0^+$ states have been experimentally observed at excitation energies below 1 MeV [@julin01; @andreyev00] in all even Pb isotopes between $A=184$ and $A=194$. Similarly, $0^+_2$ excited states below 1 MeV have been found in neutron-deficient Hg isotopes from $A=180$ up to $A=190$ [@julin01]. The spectroscopy of the Hg isotopes [@julin01; @hannachi; @lane95] shows a nearly constant behavior of the energy of the yrast states in the range $A=190-198$, which are interpreted as members of a rotational band on top of a weakly deformed oblate ground state. For lighter isotopes, $0^+_2$ excited states appear at low energies, decreasing in excitation energy up to $A=182$. They are interpreted as the band-heads of prolate configurations. Their excited states become yrast above $4^+$ for $A<186$, whereas the $2^+$ levels become close enough in energy to the weakly deformed states, opening the possibility of mixing strongly with them. Nevertheless, to determine the magnitude and type of deformation of the bands and their mixing, spectroscopy studies are not enough and the electromagnetic properties (E2 transition strengths) of the low-lying states have to be determined. Lifetime measurements in neutron-deficient Hg isotopes have been performed in the last years [@grahn09; @scheck10; @gaffney14]. More recently [@bree14], Coulomb-excitation experiments have been performed to study the electromagnetic properties of light Hg isotopes $^{182-188}$Hg. In these experiments, the deformation of the ground state and low-lying excited states were deduced, confirming the presence of two different coexisting structures in the light even-even Hg isotopes that are pure at higher spin values and mix at low excitation energy. The ground states of Hg isotopes in the mass range $A=182-188$ are found to be weakly deformed and of predominantly oblate nature, while the excited $0^+_2$ states in $^{182,184}$Hg exhibit a larger deformation. Similarly, low-lying states in light Pt isotopes have been studied experimentally with $\gamma$-ray spectroscopy [@cederwall90; @dracoulis91; @davidson99], showing that shape coexistence of states with different deformation is still present in neutron-deficient Pt isotopes with $Z=78$. Moderate odd-even staggering was also found in very light Pt isotopes from laser spectroscopy [@leblanc99]. From the theoretical point of view different types of models have been used to explain the coexistence of several $0^+$ states at low energies [@heyde11]. In a shell model picture, the excited $0^+$ states are interpreted as multi particle-hole excitations. Protons and neutrons outside the inert core interact through pairing and quadrupole interactions to generate deformed structures. Within a mean-field description of the nuclear structure, the presence of several minima at low energies in the energy surface, corresponding to different $0^+$ states, is understood as due to the coexistence of various collective nuclear shapes. In the mean-field approach, the energy of the different shape configurations can be evaluated with constrained calculations, minimizing the Hartree-Fock energy under the constraint of keeping fixed the nuclear deformation. The resulting total energy plots versus deformation are called in what follows deformation-energy curves (DEC). These calculations have become more and more refined with time, resulting in accurate descriptions of the nuclear shapes and the configurations involved. Calculations based on phenomenological mean fields and Strutinsky method [@bengtsson], are already able to predict the existence of several competing minima in the deformation-energy surface of neutron-deficient Pt, Hg, and Pb isotopes. Self-consistent mean-field calculations with non-relativistic Skyrme [@bender04; @yao13] and Gogny [@delaroche; @libert; @egido; @rayner10], as well as relativistic [@niksic02] energy density functionals have been carried out. Inclusion of correlations beyond mean field [@bender04; @yao13; @delaroche; @libert; @egido; @rayner10] are needed to obtain a detailed description of the spectroscopy. They involve symmetry restoration by means of angular momentum and particle number projection and configuration mixing within a generator coordinate method. It is shown that the underlying mean field picture of coexisting shapes is in general supported, except in those cases where the deformed mean-field structures appear at close energies. In this case mixing can be important, affecting B(E2) strengths and their corresponding $\beta$ deformation parameters. The basic picture is also confirmed from recent calculations within the interacting boson model with configuration mixing carried out for Hg [@nomura13; @gramos14hg] and Pt [@morales08; @gramos09; @gramos11; @gramos14pt] isotopes. Triaxiality in this mass region has also been explored systematically [@yao13; @rayner10; @nomura13; @gramos14pt; @nomura11], showing that although the axial deformations seem to be the basic ingredients, triaxiality may play a role in some cases. A systematic survey of energy surfaces in the $(\beta ,\gamma )$ plane with the Gogny D1S interaction can be found in the Bruyères-le-Châtel database [@web_Gogny]. On the other hand, it has been shown [@frisk95; @sarri98; @sarri99] that the decay properties of $\beta$-unstable nuclei may depend on the nuclear shape of the decaying nucleus. In particular, the Gamow-Teller (GT) strength distributions corresponding to $\beta^+$/EC-decay of proton-rich nuclei in the mass region $A\approx 70$ have been studied systematically [@sarri01prc; @sarri01npa; @sarri05epja; @sarri09] as a function of the deformation, using a deformed quasiparticle random-phase approximation (QRPA) approach built on a self-consistent Hartree-Fock (HF) mean field with Skyrme forces and pairing correlations. The study has also been extended to stable $pf$-shell nuclei [@sarri03; @sarri13] and to neutron-rich nuclei in the mass region $A\approx 100$ [@sarri_pere]. This sensitivity of the GT strength distributions to deformation has been exploited to determine the nuclear shape in neutron-deficient Kr and Sr isotopes by comparing theoretical results with $\beta$-decay measurements using the total absorption spectroscopy technique (TAS) [@isolde]. Similar studies for the decay properties of even-even neutron-deficient Pb, Po, and Hg isotopes were initiated in Refs. [@sarri05prc; @moreno06] to predict the extent to which GT strength distributions may be used as fingerprints of the nuclear shapes in this mass region. In those works, it was shown that the existence of shape isomers, as well as the location of their equilibrium deformations, are rather stable and independent on the Skyrme and pairing forces. It was also found that the GT strength distributions calculated at the various equilibrium deformations exhibit specific features that can be used as signatures of the shape isomers and, what is important, these features remain basically unaltered against changes in the Skyrme and pairing forces. In this paper we extend those calculations by studying the DECs and the GT strength distributions of neutron-deficient $^{174-204}$Hg and $^{170-192}$Pt isotopes, focusing on their dependence on deformation. In addition, we also include as a novelty the decay properties of the odd-$A$ isotopes and discuss the sensitivity of the decay patterns to the spin-parity of the decaying nucleus. The aim here is to identify possible signatures of the shape of the nucleus in the decay patterns. This study is timely because the possibility to carry out these measurements in odd-$A$ nuclei is being considered at present at ISOLDE/CERN [@algora]. A program aiming to measure the Gamow-Teller strength distributions in neutron-deficient isotopes in the lead region with TAS techniques started with $^{188,190,192}$Pb isotopes. These data have been already analyzed and submitted for publication [@submitted; @thesis]. Similar measurements have been carried out in $^{182,183,184,186}$Hg and are being presently analyzed [@algora]. The paper is organized as follows. In Sec. II we present briefly the main features of our theoretical framework. Section III contains our results for the energy deformation curves and GT strength distributions in the neutron-deficient Hg and Pt isotopes relevant for $\beta^+$/EC-decay. We also compare the experimental half-lives with our results and discuss the GT strength distributions and their sums in various ranges of excitation energies. Section IV contains the main conclusions. Theoretical Formalism {#sec2} ===================== In this section we present a summary of the theoretical formalism used in this paper to describe the $\beta$-decay properties in Hg and Pt neutron-deficient isotopes. More details of the microscopic calculations can be found in Refs. [@sarri98; @sarri99; @sarri01prc; @sarri01npa]. The method starts with a self-consistent calculation based on a deformed Hartree-Fock mean field obtained with effective two-body density-dependent Skyrme interactions including pairing correlations in BCS approximation. From these calculations we obtain energies, occupation probabilities and wave functions of the single-particle states. Most of the calculations in this work have been performed with the interaction SLy4 [@sly4], being among the most successful and extensively studied Skyrme force in the last years [@bender08; @bender09; @stoitsov]. Furthermore, comparison with other widely used Skyrme forces like the simpler Sk3 [@sk3] and SGII [@sg2] that has been shown to provide good spin-isospin properties, will be shown in some instances. The solution of the HF equation is found by using the formalism developed in Ref. [@vautherin], assuming time reversal and axial symmetry. The single-particle wave functions are expanded in terms of the eigenstates of an axially symmetric harmonic oscillator in cylindrical coordinates, using twelve major shells. The method also includes pairing between like nucleons in BCS approximation with fixed gap parameters for protons and neutrons, which are determined phenomenologically from the odd-even mass differences through a symmetric five-term formula involving the experimental binding energies [@audi12]. In those cases where experimental information for masses is still not available, same pairing gaps as the closer isotope measured are used. The DECs are analyzed as a function of the quadrupole deformation parameter $\beta$ from constrained HF calculations. Calculations for GT strengths are performed subsequently at the equilibrium shapes of each nucleus, that is, for the solutions (in general deformed) for which minima are obtained in the energy curves. It is worth mentioning some existing works in this mass region based on mean-field approaches other than the present Skyrme HF+BCS calculations. In particular, mean-field studies of structural changes with the Gogny interaction can be found in Ref. [@nomura13] for Hg isotopes and in Refs. [@rayner10; @gramos14pt; @nomura11] for Pt isotopes. The clear advantage of the finite-range Gogny force over the contact Skyrme force is that pairing correlations can be treated self-consistently using the same interaction through a Hartree-Fock-Bogoliubov (HFB) calculation. Triaxial landscapes were studied in those references, showing that the (axial) prolate and oblate minima, which are well separated by high-energy barriers in the $\beta$ degree of freedom, are in many cases softly linked along the $\gamma$ direction. Indeed, some axial minima become saddle points when the $\gamma$ degree of freedom is included in the analysis. The differences found with the present HF+BCS approach for the axial equilibrium values are not significant, but the topology of the surfaces are somewhat different. Similarities and differences of the various topologies are discussed in the next section. In the case of odd-$A$ nuclei, the ground state is expressed as a one-quasiparticle (1qp) state, which is determined by finding the blocked state that minimizes the total energy. In the present study we use the equal filling approximation (EFA), a prescription widely used in mean-field calculations to treat the dynamics of odd nuclei preserving time-reversal invariance [@rayner2]. In this approximation the unpaired nucleon is treated on equal footing with its time-reversed state by putting half a nucleon in a given orbital and the other half in the time-reversed partner. This approximation has been found to be equivalent to the exact blocking when the time-odd fields of the energy density functional are neglected and then, it is sufficiently precise for most practical applications [@schunck10]. Effects of time-odd terms in HFB calculations have also been studied in Ref. [@hellemans12]. An extension of beyond-mean-field calculations, where the generator coordinate method is built from self-consistently blocked 1qp HFB states for odd-mass nuclei has recently been presented in Ref. [@bally14]. The deformation in the decaying nuclei in both even-even and odd-$A$ cases, is self-consistently determined. In the odd-$A$ case, the core polarization induced by the odd particle is then taken into account. The effect found is however very small and we get very similar axial deformations in the even-even and neighbor odd-A nuclei. The small effect can be also observed in the Gogny database [@web_Gogny], comparing the DECs of the even-even and nearest odd-$A$ isotopes. Since the GT operator of the allowed transitions is a pure spin-isospin operator without any radial dependence, one expects the spatial functions of the parent and daughter wave functions to be as close as possible in order to overlap maximally. Then, transitions connecting different radial structures in the parent and daughter nuclei will be suppressed. Thus, we assume similar shapes for the decaying parent nucleus and for all populated states in the daughter nucleus, neglecting core polarization effects in the daughter nuclei. This is a common assumption to deformed QRPA calculations [@moller1]. That core polarization effects are small in both odd-odd case in relation to even-even parent and odd-even (even-odd) case in relation to the even-odd (odd-even) parent can be seen in the Gogny database [@web_Gogny], where potential energy surfaces obtained from Gogny HFB calculations are shown all along the nuclear chart. By comparing the surfaces of parent (Hg, Pt) and daughter (Au, Ir) isotopes considered in this work, one realizes that the profiles are very similar with practically no effect from core polarization due to the odd particles. The reduction in the transitions connecting different shapes have been quantified in the case of double $\beta$ decay [@alvarez04]. It has been shown that the overlaps between the wave functions in the intermediate nucleus reached from different shapes of the parent and daughter nuclei are dramatically reduced when the deformations differ from each other. Only with similar deformations the overlap is significant. Consequently, given the small polarization effects and the suppression of the overlaps with different deformations, we consider in this work only GT transitions between parent and daughter partners with like deformations. To describe GT transitions, a spin-isospin residual interaction is added to the mean field and treated in a deformed proton-neutron QRPA [@moller1; @moller2; @homma; @moller3; @moller08; @hir1; @hir2; @frisk95; @sarri01npa]. This interaction contains two parts, particle-hole (ph) and particle-particle (pp). The interaction in the ph channel is responsible for the position and structure of the GT resonance [@homma; @sarri01npa] and it can be derived consistently from the same Skyrme interaction used to generate the mean field, through the second derivatives of the energy density functional with respect to the one-body densities. The ph residual interaction is finally expressed in a separable form by averaging the Landau-Migdal resulting force over the nuclear volume, as explained in Ref. [@sarri98]. The pp component is a neutron-proton pairing force in the $J^\pi=1^+$ coupling channel, which is also introduced as a separable force [@hir1; @hir2; @sarri01npa]. Its strength is usually fitted to reproduce globally the experimental half-lives. Various attempts have been made in the past to fix this strength [@homma], arriving to expressions that depend on the model used to describe the mean field, Nilsson model in the above reference. In previous works we studied the sensitivity of the GT strength distributions to the various ingredients contributing to the deformed QRPA calculations, namely to the nucleon-nucleon effective interaction, to pairing correlations, and to residual interactions. We found different sensitivities to them. In this work, all of these ingredients have been fixed to the most reasonable choices found previously [@sarri05prc; @moreno06]. In particular we use the coupling strengths $\chi ^{ph}_{GT}=0.08$ MeV and $\kappa ^{pp}_{GT} = 0.02$ MeV for the ph and pp channels, respectively. The technical details to solve the QRPA equations have been described in Refs. [@hir1; @hir2; @sarri98]. Here we only mention that, because of the use of separable residual forces, the solutions of the QRPA equations are found by solving first a dispersion relation, which is an algebraic equation of fourth order in the excitation energy $\omega$. Then, for each value of the energy, the GT transition amplitudes in the intrinsic frame connecting the ground state $| 0^+\rangle $ of an even-even nucleus to one phonon states in the daughter nucleus $|\omega_K \rangle \, (K=0,1) $ are found to be $$\left\langle \omega _K | \sigma _K t^{\pm} | 0 \right\rangle = \mp M^{\omega _K}_\pm \, , \label{intrinsic}$$ where $t^+ |\pi \rangle =|\nu \rangle,\, t^- |\nu \rangle =|\pi \rangle$ and $$\begin{aligned} M_{-}^{\omega _{K}}&=&\sum_{\pi\nu}\left( q_{\pi\nu}X_{\pi \nu}^{\omega _{K}}+ \tilde{q}_{\pi\nu}Y_{\pi\nu}^{\omega _{K}} \right) , \\ M_{+}^{\omega _{K}}&=&\sum_{\pi\nu}\left( \tilde{q}_{\pi\nu} X_{\pi\nu}^{\omega _{K}}+ q_{\pi\nu}Y_{\pi\nu}^{\omega _{K}}\right) \, ,\end{aligned}$$ with $$\tilde{q}_{\pi\nu}=u_{\nu}v_{\pi}\Sigma _{K}^{\nu\pi },\ \ \ q_{\pi\nu}=v_{\nu}u_{\pi}\Sigma _{K}^{\nu\pi}, \label{qs}$$ in terms of the occupation amplitudes for neutrons and protons $v_{\nu,\pi}$ ($u^2_{\nu,\pi}=1-v^2_{\nu,\pi}$) and the matrix elements of the spin operator, $\Sigma _{K}^{\nu\pi}=\left\langle \nu\left| \sigma _{K}\right| \pi\right\rangle $, connecting proton and neutron single-particle states, as they come out from the HF+BCS calculation. $X_{\pi\nu}^{\omega _{K}}$ and $Y_{\pi\nu}^{\omega _{K}}$ are the forward and backward amplitudes of the QRPA phonon operator, respectively. Once the intrinsic amplitudes in Eq. (\[intrinsic\]) are calculated, the GT strength $B_{\omega}(GT^\pm)$ in the laboratory system for a transition $I_iK_i (0^+0) \rightarrow I_fK_f (1^+K)$ can be obtained as $$\begin{aligned} B_{\omega}(GT^\pm )& =& \sum_{\omega_{K}} \left[ \left\langle \omega_{K=0} \left| \sigma_0t^\pm \right| 0 \right\rangle ^2 \delta (\omega_{K=0}- \omega ) \right. \nonumber \\ && \left. + 2 \left\langle \omega_{K=1} \left| \sigma_1t^\pm \right| 0 \right\rangle ^2 \delta (\omega_{K=1}-\omega ) \right] \, , \label{bgt}\end{aligned}$$ in $[g_A^2/4\pi]$ units. To obtain this expression, the initial and final states in the laboratory frame have been expressed in terms of the intrinsic states using the Bohr-Mottelson factorization [@bm]. When the parent nucleus has an odd nucleon, the ground state can be expressed as a one-quasiparticle (1qp) state in which the odd nucleon occupies the single-particle orbit of lowest energy. Then two types of transitions are possible. One type is due to phonon excitations in which the odd nucleon acts only as a spectator. These are three-quasiparticle (3qp) states and the GT transition amplitudes in the intrinsic frame are basically the same as in the even-even case in Eq. (\[intrinsic\]), but with the blocked spectator excluded from the calculation. The other type of transitions are those involving the odd nucleon state (1qp), which are treated by taking into account phonon correlations in the quasiparticle transitions in first-order perturbation. The transition amplitudes for the correlated states can be found in Refs. [@hir2; @sarri01prc]. In this work we refer the GT strength distributions to the excitation energy in the daughter nucleus. In the case of even-even decaying nuclei, the excitation energy of the 2qp states in the odd-odd daughter nuclei is simply given by $$E_{\mbox{\scriptsize{ex}}\, [(Z,N)\rightarrow (Z-1,N+1)]}=\omega -E_{\pi_0} - E_{\nu_0} \, , \label{eexeven}$$ where $E_{\pi_0}$ and $E_{\nu_0}$ are the lowest quasiparticle energies for protons and neutrons, respectively. In the case of an odd-$A$ nucleus we have to deal with 1qp and 3qp transitions. For Hg and Pt isotopes we have odd-neutron parents decaying into odd-proton daughters. The excitation energies for 1qp transitions are $$E_{\mbox{\scriptsize{ex,1qp}}\, [(Z,N-1)\rightarrow (Z-1,N)]}=E_\pi-E_{\pi_0} \, . \label{eex1qp}$$ The excitation energy with respect to the ground state of the daughter nucleus for 3qp transitions is $$E_{\mbox{\scriptsize{ex,3qp}}\, [(Z,N-1)\rightarrow (Z-1,N)]}= \omega +E_{\nu,\mbox{\scriptsize{spect}}}-E_{\pi_0} \, . \label{eex3qp}$$ Therefore, the lowest excitation energy of 3qp type is of the order of twice the neutron pairing gap and then, the strength contained below typically 2-3 MeV in the odd-$A$ nuclei corresponds to 1qp transitions. The $\beta$-decay half-life is obtained by summing all the allowed transition strengths to states in the daughter nucleus with excitation energies lying below the corresponding $Q_{EC}$ energy, i.e., $Q_{EC}=Q_{\beta^+} +2m_e= M(A,Z)-M(A,Z+1)+2m_e $, written in terms of the nuclear masses $M(A,Z)$ and the electron mass ($m_e$), and weighted with the phase-space factors $f(Z,Q_{EC}-E_{ex})$, $$T_{1/2}^{-1}=\frac{\left( g_{A}/g_{V}\right) _{\rm eff} ^{2}}{D} \sum_{0 < E_{ex} < Q_{EC}}f\left( Z,Q_{EC}-E_{ex} \right) B(GT,E_{ex}) \, , \label{t12}$$ with $D=6200$ s and $(g_A/g_V)_{\rm eff}=0.77(g_A/g_V)_{\rm free}$, where 0.77 is a standard quenching factor. In this work we use experimental $Q_{EC}$ values [@audi12]. In $\beta^+$/EC decay, $f( Z,Q_{EC}-E_{ex})$ contains two parts, positron emission and electron capture. The former, $f^{\beta^\pm}$, is computed numerically for each value of the energy including screening and finite size effects, as explained in Ref. [@gove], $$f^{\beta^\pm} (Z, W_0) = \int^{W_0}_1 p W (W_0 - W)^2 \lambda^\pm(Z,W) {\rm d}W\, , \label{phase}$$ with $$\lambda^\pm(Z,W) = 2(1+\gamma) (2pR)^{-2(1-\gamma)} e^{\mp\pi y} \frac{|\Gamma (\gamma+iy)|^2}{[\Gamma (2\gamma+1)]^2}\, ,$$ where $\gamma=\sqrt{1-(\alpha Z)^2}$ ; $y=\alpha ZW/p$ ; $\alpha$ is the fine structure constant and $R$ the nuclear radius. $W$ is the total energy of the $\beta$ particle, $W_0$ is the total energy available in $m_e c^2$ units, and $p=\sqrt{W^2 -1}$ is the momentum in $m_e c$ units. The electron capture phase factors, $f^{EC}$, have also been included following Ref. [@gove]: $$f^{EC}=\frac{\pi}{2} \sum_{x} q_x^2 g_x^2B_x \, ,$$ where $x$ denotes the atomic sub-shell from which the electron is captured, $q$ is the neutrino energy, $g$ is the radial component of the bound state electron wave function at the nucleus, and $B$ stands for other exchange and overlap corrections [@gove]. ![(Color online) Deformation energy curves for even-even $^{174-196}$Hg isotopes obtained from constrained HF+BCS calculations with the Skyrme forces Sk3, SGII, and SLy4.[]{data-label="fig_e_beta_hg"}](fig1_hg_beta){width="80mm"} ![(Color online) Same as in Fig. \[fig\_e\_beta\_hg\], but for $^{170-192}$Pt isotopes.[]{data-label="fig_e_beta_pt"}](fig2_pt_beta){width="80mm"} ![(Color online) Isotopic evolution of the quadrupole deformation parameter $\beta$ of the various energy minima for Hg (a) and Pt (b) isotopes. The dashed lines join the deformations corresponding to the lowest HF+BCS minimum in the DECs obtained with SLy4.[]{data-label="fig_beta_A"}](fig3_beta_A){width="80mm"} ![(Color online) Calculated $\delta \langle r_c^2 \rangle $ in Hg isotopes with various deformations compared to experimental data from Refs. [@bonn72; @ulm86; @angeli04; @lee78]. []{data-label="fig_hg_dr2"}](fig4_hg_rc){width="80mm"} ![(Color online) Same as in Fig. \[fig\_hg\_dr2\], but for Pt isotopes. Experimental data are from Refs. [@leblanc99; @angeli04; @lee88; @sauvage00]. []{data-label="fig_pt_dr2"}](fig5_pt_rc){width="80mm"} ![(Color online) Folded GT strength distributions in $^{182,184,186}$Hg as a function of the excitation energy in the daughter nucleus for oblate and prolate shapes obtained with the Skyrme forces SGII and SLy4.[]{data-label="fig_hg_force"}](fig6_hg_force){width="85mm"} ![(Color online) Accumulated GT strengths in $^{184}$Hg calculated with the Skyrme interaction SLy4 for various values of the coupling strength of the ph residual interaction for a fixed value of the pp residual interaction. []{data-label="fig_hg184_ph"}](fig7_hg184_ph){width="85mm"} ![(Color online) Accumulated GT strengths in $^{184}$Hg calculated with the Skyrme interaction SLy4 for various values of the coupling strength of the pp residual interaction for a fixed value of the ph residual interaction. []{data-label="fig_hg184_pp"}](fig8_hg184_pp){width="85mm"} ![(Color online) (Left) Folded GT strength distributions in even Hg isotopes for prolate and oblate shapes using SLy4. (Right) Accumulated GT strength for the various shapes in the energy range below 7 MeV. The vertical lines correspond to the $Q_{EC}$ energies. No quenching factors are included.[]{data-label="fig_bgt_hg"}](fig9_bgt_hg){width="80mm"} ![(Color online) Same as in Fig. \[fig\_bgt\_hg\], but for even Pt isotopes.[]{data-label="fig_bgt_pt"}](fig10_bgt_pt){width="80mm"} ![(Color online). Same as in Fig. \[fig\_bgt\_hg\], but for odd-$A$ Hg isotopes.[]{data-label="fig_bgt_hg_odd"}](fig11_bgt_hg_odd){width="80mm"} ![(Color online) Same as in Fig. \[fig\_bgt\_pt\], but for odd-$A$ Pt isotopes.[]{data-label="fig_bgt_pt_odd"}](fig12_bgt_pt_odd){width="80mm"} ![(Color online) GT strength distribution in the odd isotope $^{181,183,185,187}$Hg for various $K^{\pi}$ values and deformations (see text).[]{data-label="fig_odd"}](fig13_hg_odd){width="85mm"} ![(Color online) Same as in Fig. \[fig\_odd\], but for Pt isotopes.[]{data-label="fig_odd_pt"}](fig14_pt_odd){width="85mm"} ![(Color online) Experimental $\beta^+$/EC-decay half-lives in Hg isotopes compared with the results of QRPA calculations with SLy4. The results obtained with the ground state shapes are connected with a dashed line.[]{data-label="fig_t12_hg"}](fig15_t12_hg){width="80mm"} ![(Color online) Same as in Fig. \[fig\_t12\_hg\], but for Pt isotopes.[]{data-label="fig_t12_pt"}](fig16_t12_pt){width="80mm"} Results and discussion {#results} ====================== In this section we first discuss the energy curves and shape coexistence expected, discussing the shape evolution in Hg and Pt isotopic chains. Then, we present the results obtained for the Gamow-Teller strength distributions in the neutron-deficient $^{176-192}$Hg and $^{172-186}$Pt isotopes with special attention to their dependence on the nuclear shape and discuss their relevance as signatures of deformation to be explored experimentally. Finally, we discuss the half-lives and compare them with the experimental values. Equilibrium deformations ------------------------ We show in Figs. \[fig\_e\_beta\_hg\] and \[fig\_e\_beta\_pt\] the DECs calculated with three Skyrme forces, Sk3, SGII, and SLy4, for Hg and Pt isotopes, respectively. The energies are shown as a function of the quadrupole deformation parameter calculated microscopically as $\beta=\sqrt{\pi/5}\ Q_p/(Z\langle r_c^2 \rangle)$, defined in terms of the proton quadrupole moment, $Q_p$, and charge m.s. radius, $\langle r_c^2 \rangle $. We get similar qualitative results with the three Skyrme forces considered. More specifically, we obtain the same patterns of shape coexistence with minima located at practically the same deformations although the relative energies may change from one force to another. Thus, we focus the discussion on the SLy4 interaction. In the case of Hg isotopes (Fig. \[fig\_e\_beta\_hg\]) we get prolate and oblate minima in all the isotopes from $A=174$ up to $A=196$. We can see that the ground state is predicted to be prolate for $^{174-182}$Hg and oblate for $^{184-196}$Hg isotopes. The transition occurs smoothly around $^{184}$Hg for SLy4, where we obtain two coexisting shapes at the same energy and it takes place around $^{186}$Hg ($^{188}$Hg) with SGII (Sk3). Similarly, in the case of Pt isotopes (Fig. \[fig\_e\_beta\_pt\]) we get prolate and oblate minima in all the isotopes from $A=170$ up to $A=192$, but in this case the ground state is always prolate except in the heavier isotopes, $^{190,192}$Pt, where the oblate shape becomes ground state with the three forces. The transition is very smooth and the two shapes are practically degenerate between $^{184}$Pt and $^{190}$Pt for SLy4. Except for the very light isotopes, we observe in both isotopic chains the existence of rather sharp oblate and prolate energy minima, close in energy and separated by very high energy barriers, giving raise to shape coexistence. These findings are in qualitative agreement with recent calculations [@yao13; @nomura13; @gramos14hg; @rayner10; @gramos14pt; @nomura11]. Looking in more detail the results from different calculations, one observes differences and similarities within the various approaches. There are robust features common to all calculations, such as the existence of oblate and prolate minima located at similar deformations and separated by spherical barriers, or the isotopic evolution from oblate shapes in the heavier isotopes to prolate shapes in the lighter ones. But there are also particular features that change according to the different calculations, such as the height of the barriers or the relative energies between the minima that finally determines the exact isotope where the shape transition takes place. Obviously, the exact location of the shape transition is very sensitive to small details of the calculation because the shape transition occurs precisely around the region where the energies of the competing shapes are practically degenerate. Thus, it is not surprising that the shape transition in Pt isotopes predicted in Ref. [@yao13] within a beyond mean field approach with the Skyrme SLy6 occurs at $A=186-188$ instead of $A=182-184$ in our calculation. In the same line triaxial D1M-Gogny calculations predict a smooth shape transition at $A=184-186$ [@nomura13] Similarly, the shape transition in Pt isotopes in our calculations takes place at $A=188-190$. This agrees with triaxial calculations with the Gogny force that exhibit a smooth transition at $A=186-190$, passing through a soft triaxial solution [@rayner10; @gramos14pt; @nomura11], as well as with the calculations in [@sarri08; @robledo09]. In particular, the DECs in Pt isotopes were studied in Ref. [@sarri08], comparing the effects of different interactions (SLy4, SLy6, Gogny) and pairing treatments (constant strength, constant pairing gaps, density-dependent zero-range forces). Little changes in the energy profiles were found within those treatments, but still enough to change the absolute minimum from one deformation to another in the transitional region around $^{188}$Pt, where the energies are practically degenerate. Nevertheless, for the purpose of this work, the exact location at which the shape transitions occur is not of relevance. The important aspect in this work is that a shape competition is taken place and whether the sensitivity of the B(GT) profiles to deformation can be used as a fingerprint of the nuclear shape. Then, we choose in this work a reasonable mean-field based on the Skyrme SLy4 with constant pairing gaps. to be used as a starting point for a QRPA calculation of the decay properties. To illustrate better the role of deformation in the isotopic evolution, we show in Fig. \[fig\_beta\_A\] the quadrupole deformation parameter $\beta$ of the various energy minima as a function of the mass number $A$, for Hg (a) and Pt (b) isotopic chains. The dashed lines join the deformations corresponding to the lowest HF+BCS minimum in the DECs obtained with SLy4. Starting from the heaviest isotopes in Fig. \[fig\_beta\_A\], we get spherical shapes, as they correspond to the $N=126$ neutron shell closure. Moving into the neutron-deficient region, we observe the appearance of both oblate and prolate shapes with increasing quadrupole moments. The shape of the minimum energy changes from oblate in the heavier isotopes to prolate in the lighter ones at $^{182-184}$Hg and $^{188-190}$Pt for SLy4. The shapes reach maximum quadrupole deformations of about $\beta=0.3$ in the prolate sector and about $\beta=-0.2$ in the oblate one. Charge radii and their differences have been shown [@rayner1; @rayner2] to be suitable quantities to study the evolution of the nuclear-shape changes as they can be measured with remarkable precision using laser spectroscopic techniques [@cheal]. They are calculated by folding the proton distribution of the nucleus with the finite size of the protons and the neutrons. The m.s. radius of the charge distribution in a nucleus can be expressed as [@negele] $$\langle r^2_c \rangle = \langle r^2_p \rangle _Z+ \langle r^2_c \rangle _p +(N/Z) \langle r^2_c \rangle _n + r^2_{CM} \, , \label{rch}$$ where $ \langle r^2_p \rangle _Z$ is the m.s. radius of the point proton distribution in the nucleus $$\langle r_p^2 \rangle _Z = \frac{ \int r^2\rho_p({\vec r})d{\vec r} } {\int \rho_p({\vec r})d{\vec r}} \, , \label{r2pn}$$ $ \langle r^2_c \rangle _p=0.80$ fm$^2$ [@sick03] and $ \langle r^2_c \rangle _n=-0.12$ fm$^2$ [@gentile11] are the m.s. radii of the charge distributions in a proton and a neutron, respectively. $r^2_{CM}$ is a small correction due to the center of mass motion. It is worth noticing that the most important correction to the point proton m.s. nuclear radius, coming from the proton charge distribution $ \langle r^2_c \rangle _p$, vanishes when isotopic differences are considered, since it does not involve any dependence on $N$. The variations of the charge radii trends in isotopic chains are related to deformation effects and can be used as signatures of shape transitions. For an axially symmetric static quadrupole deformation $\beta$ the increase of the charge radius with respect to the spherical value is given to first order by $$\langle r^2 \rangle = \langle r^2 \rangle _{\rm sph} \left( 1+\frac{5}{4\pi} \beta^2 \right) \, ,$$ where usually $\langle r^2 \rangle _{\rm sph}$ is taken from the droplet model. In this work we analyze the effect of the quadrupole deformation on the charge radii from a microscopic self-consistent approach. One should notice that our calculations at the mean-field level correspond to the oblate and prolate mean-field solutions and, consequently, they don’t correspond to the actual ground state to which the experimental radii are referred. In Figs. \[fig\_hg\_dr2\]-\[fig\_pt\_dr2\] we show the differences $\delta \langle r^2_c \rangle ^{A,{\rm ref}}= \langle r^2_c \rangle ^A - \langle r^2_c \rangle ^{{\rm ref}}$, where the reference isotope is $A=198$ ($A=194$) for the Hg (Pt) isotopic chain. Our calculations are compared with experimental data measured by laser spectroscopy and compiled in Ref. [@angeli04]. For Hg isotopes, the experiment [@bonn72; @ulm86; @lee78] shows an even-odd staggering in the lighter isotopes ($A=181-186$), with larger radii in the odd-$A$ isotopes. When we compare the data for light Hg isotopes with our calculations we see that the even-even isotopes are well described with an oblate shape, whereas the odd-$A$ isotopes are rather associated with a prolate shape. We also observe in our calculations a bump in the oblate radii around $A=190$ and a more pronounced one in the prolate radii around $A=188$ that are related to the shape variation of the energy minima. In the case of Pt isotopes, the experimental radii [@leblanc99; @lee88; @sauvage00] in the interval $A=178-188$ are in between the oblate and prolate radii of reference, pointing out that strong mixing between these two structures is necessary to describe the $0^+$ ground state. The agreement with experiment is reasonable in the heavier Hg and Pt isotopes for both oblate and prolate radii, indicating that these nuclei are approaching a spherical shape. Gamow-Teller strength distributions ----------------------------------- In this subsection we study the energy distribution of the Gamow-Teller strengths calculated at the equilibrium shapes that minimize the energy of the nucleus. But before showing the results of our calculations it is worth discussing briefly the expected sensitivity of these calculations to the choice of the nucleon-nucleon effective Skyrme interaction, as well as to the coupling strengths of the residual forces. Figure \[fig\_hg\_force\] illustrates the sensitivity of the GT strength distributions to the Skyrme interaction. We show in this figure continuous distributions obtained by folding the strength at each excitation energy with 1 MeV width Breit-Wigner functions. The results correspond to the Skyrme interactions SLy4 and SGII, and for three Hg isotopes, $^{182,184,186}$Hg. For a given type of deformation (oblate or prolate), we observe very similar decay patterns for both interactions, with slightly lower strength in the case of SGII. On the other hand, for a given Skyrme force the dependence on the deformation is manifest. This example demonstrates that the profiles of the GT strength distributions are characteristic of the nuclear shape and depend little on the details of the two-body force. This marked sensitivity to deformation can then be used to get information about the nuclear shape of the decaying nucleus, something that has been exploited in the past in other mass regions [@isolde]. In the next two figures we discuss the effect of the residual force on the GT strength distributions, using $^{184}$Hg as an example. In this case, for a better comparison, we plot the summed strengths that give us the total strength contained below a given energy. In Fig. \[fig\_hg184\_ph\] we can see the effect of the ph residual force. For that purpose we show the results obtained with a fixed value of the pp interaction ($\kappa ^{pp}_{GT}=0.02$ MeV) for $\chi ^{ph}_{GT}=0.08$ MeV (a), $\chi ^{ph}_{GT}=0.15$ MeV (b), and $\chi ^{ph}_{GT}=0.20$ MeV (c). As $\chi ^{ph}_{GT}$ increases, the strength is reduced, especially in the low-energy region, but the profiles of both prolate and oblate shapes remain basically the same. This reduction has immediate consequences on the half-lives that increase with increasing values of $\chi ^{ph}_{GT}$. Similarly, we show in Fig. \[fig\_hg184\_pp\] the effect of the pp residual force by taking fixed the ph residual interaction ($\chi ^{ph}_{GT}=0.08$ MeV) and varying the value of the pp interaction from $\kappa ^{pp}_{GT}=0$ (a) to $\kappa ^{pp}_{GT}=0.02$ MeV (b), and finally to $\kappa ^{pp}_{GT}=0.04$ MeV (c). As $\kappa ^{pp}_{GT}$ increases the strength is reduced and slightly shifted to lower energies, but again the prolate and oblate profiles persist. In the next figures, Figs. \[fig\_bgt\_hg\]-\[fig\_bgt\_pt\_odd\], we show the GT strength distributions for oblate and prolate shapes as a function of the excitation energy in the daughter nucleus obtained with SLy4 and with the residual forces written in Sec. \[sec2\]. Although a similar figure to Fig. \[fig\_bgt\_pt\] was already presented in Ref. [@moreno06], for the sake of completeness and to facilitate the comparison, we also show here those results for the Pt isotopes. In the first two figures we show the results for the even Hg and Pt isotopes, whereas in the last two figures we present the results for the odd-$A$ isotopes. In the left panels we can see continuous GT strength distributions resulting from a folding procedure using 1 MeV width Breit-Wigner functions on the discrete spectrum. On the other hand, in the right panels we plot the accumulated GT strength up to a reduced energy range that covers the $Q_{EC}$ energies represented by the vertical arrows. Thus, we can see in more detail both the strength distribution and the total GT strength contained in the energy window relevant to the $\beta$-decay and to the half-lives. In particular, the crossing of the curves with the $Q_{EC}$ vertical arrows shows us the total GT strength available by $\beta$-decay and eventually measurable. It should be noted that no quenching factor is included in these distributions and therefore one should consider a reduction of this strength prior to comparison with future experiments. The left panels in Figs. \[fig\_bgt\_hg\] and \[fig\_bgt\_pt\] show the GT strength distributions for the even-even Hg and Pt isotopes, respectively. The strength increases as we move away from the valley of stability to more and more neutron-deficient (lighter) isotopes (note the different scales). On a global scale the strength distribution from different shapes differ mainly in the low energy region. With minor exemptions, oblate shapes produce more strength at low energies and therefore smaller half-lives. In all cases we observe a strong peak (or double peak) at low excitation energy (below 5 MeV) and little strength above this energy, except in the heavier isotopes where a bump at high energy is developed. The differences between oblate and prolate shapes can be better appreciated in the accumulated plots displayed in the right-hand sides. In general we observe that the results from oblate shapes are more fragmented and the strength in the accumulated plots increases steadily. Conversely, prolate shapes produce in most cases a strong peak at low excitation energy an very little strength above. The location of the $Q_{EC}$ energies determines the GT strength distribution available in the decay and thus, contributing to the half-lives. Clearly, $Q_{EC}$ energies increase when moving away from stability. The next two figures, Figs. \[fig\_bgt\_hg\_odd\] and \[fig\_bgt\_pt\_odd\], contain the GT strength distributions for the odd-even Hg and Pt isotopes, respectively. In the case of odd nuclei the spin and parity of the nucleus are determined by those of the odd nucleon. In principle one would sit the odd nucleon in the single-particle orbit that minimizes the energy. However, it turns out that for deformed nuclei in this mass region several states with different spin projections and parities are very close to the Fermi surface at practically the same energy, and tiny details in the interaction can change the ground state from one to another. Given that the spin ($J$) and parity ($\pi $) of these Hg and Pt isotopes are known experimentally, we have chosen these assignments for our odd nucleons that correspond in all cases to states close to the Fermi energy. Namely, the experimental $J^\pi $ assignments of the odd-$A$ Hg isotopes are given by $J^\pi =7/2^-$ for $^{177,179}$Hg; $J^\pi =1/2^-$ for $^{181,183,185}$Hg; and $J^\pi =3/2^-$ for $^{187,189,191}$Hg. Similarly, for odd-$A$ Pt isotopes they are given by $J^\pi =5/2^-$ for $^{173}$Pt; $J^\pi =7/2^-$ for $^{175}$Pt; $J^\pi =5/2^-$ for $^{177}$Pt; $J^\pi =1/2^-$ for $^{179,181,183}$Pt; $J^\pi =9/2^+$ for $^{185}$Pt; and $J^\pi =3/2^-$ for $^{187}$Pt. Besides these values, for each nucleus, we also consider in our calculations the spin and parity corresponding to the energy minimum of the other nuclear shape. All of these values appear as labels in each isotope, where solid (dashed) lines stand for prolate (oblate) shapes. In the odd-$A$ isotopes we observe a displacement of the GT strength to higher excitation energies with respect to the even neighbor isotopes. This is due to the character of the excitation mentioned in the previous section, where we discussed that 3qp transitions, similar to those of the even isotopes but with the odd orbital blocked, appear only at energies above twice the pairing gap, typically 2-3 MeV. Similarly, the $Q_{EC}$ values are displaced in an equivalent amount since the mass differences involved in the $Q_{EC}$ definitions are sensitive to the pairing energy in a similar way. To show further the sensitivity of GT strength distributions to the spin and parity of the odd-A parent nucleus, we show in Fig. \[fig\_odd\] and Fig. \[fig\_odd\_pt\] the results for several more choices of spins and parities in $^{181,183,185,187}$Hg and $^{175,177,179,181}$Pt, respectively. These are nuclei that are currently being considered as candidates to measure their GT strength distributions at ISOLDE/CERN, using the TAS technique [@algora], and will complement the measurements already taken in Pb and Hg isotopes [@algora; @submitted; @thesis]. Obviously, the decay patterns should be affected by the spin and parity of the odd nucleon because they determine to a large extent the allowed spin and parities that can be reached in the daughter nucleus. This is especially true in the case of 1qp transitions where the odd nucleon involved determines the low-lying spectrum. Thus, one expects the low-lying GT strength to be especially sensitive to the spin-parity of the odd nucleon. This sensitivity translates immediately to the half-lives that depend on the strength contained below $Q_{EC}$. In the case $^{181}$Hg it is found experimentally that the ground state corresponds to $J^\pi=1/2^-$ with band heads at $J^\pi=7/2^-$ and $J^\pi=13/2^+$. The ground state in $^{183}$Hg is $J^\pi=1/2^-$ with another $J^\pi=7/2^-$ band head and a $J^\pi=13/2^+$ isomer at 266 keV. $^{185}$Hg has again a $J^\pi=1/2^-$ ground state with a $J^\pi=7/2^-$ band head at 34 keV, a $J^\pi=9/2^+$ at 213 keV, and a $J^\pi=13/2^+$ state at 99 keV. $^{187}$Hg is a $J^\pi=3/2^-$ nucleus with a $J^\pi=13/2^+$ band head and a $J^\pi=9/2^+$ state at 162 keV. In our calculations the experimental ground states $J^\pi=1/2^-$ correspond to prolate states with asymptotic quantum Nilsson numbers $[N n_z \Lambda ]K$ given by $[521]1/2$. We also consider prolate $7/2^-$ ($[514]7/2$) states, very close in energy an observed experimentally, as well as prolate $9/2^+$ ($[624]9/2$) states. Finally, we show the results for oblate shapes corresponding to $13/2^+$ ($[606]13/2$) states that are also seen experimentally. In $^{187}$Hg the experimental ground state $J^\pi=3/2^-$ corresponds to the oblate state $[521]3/2$. Besides the prolate $7/2^-$ ($[514]7/2$) with origin in the $f_{7/2}$ spherical orbital, we also consider a second prolate $7/2^-$ ($[503]7/2$) state in $^{183,185,187}$Hg with origin in the $h_{9/2}$ spherical orbital and labeled with an asterisk in Figs. \[fig\_odd\]-\[fig\_odd\_pt\]. These two states lead to quite different profiles of the GT strength distributions. Similarly, the ground state of $^{175}$Pt is experimentally found to be $J^\pi=7/2^-$ with a band head $J^\pi=13/2^+$ at an undetermined energy. The ground state of $^{177}$Pt is $J^\pi=5/2^-$ with a $J^\pi=7/2^+$ at 95 keV. $^{179}$Pt ($^{181}$Pt) has a $J^\pi=1/2^-$ ground state with a $J^\pi=9/2^+$ excited state at 299 keV (276 keV) and a $J^\pi=7/2^-$ excited state at 145 keV (117 keV). In addition to the states considered for Hg isotopes, calculations for Pt isotopes are also shown for prolate $5/2^-$ ($[512]5/2$) states and oblate $7/2^+$ ($[604]7/2$) and $9/2^+$ ($[604]9/2$) states. As we can see in the figures, the sensitivity of the distributions to the spin-parity is large because of the selection rules of allowed transitions. In these examples it is comparable to the effect from deformation and therefore, one can conclude that odd-$A$ isotopes may not be the best candidates to look for deformation signatures on the $\beta$-decay patterns. On the other hand, this sensitivity could be helpful to get information on the nuclear shape based on the spin and parity of the decaying nucleus, which are characteristic and very different for oblate or prolate shapes. As a matter of fact, the possibility of measuring the GT strength distributions in odd-$A$ nuclei corresponding to the ground and isomeric states separately [@algora], would represent a breakthrough in the sense that the decay patterns of prolate and oblate configurations could be disentangled by selecting properly the spin-parity of the decaying isotope. This information could be used thereafter to infer information on the shape of the ground state of the even-even isotopes. [cccccllr]{}isotope & && & $J{^\pi}$ & $Q_{EC}$ & $T_{1/2}^{\beta^+/EC}$ && $[N n_z \Lambda ]K^{\pi}$ && $T_{1/2}^{\beta^+/EC}$ $^{181}$Hg & $1/2^-$ & 7.210 & 4.9 && $[521]1/2^-$ & pro & 7.53 &&&&& $[514]7/2^-$ & pro & 3.39 &&&&& $[606]13/2^+$ & obl & 8.13 $^{183}$Hg & $1/2^-$ & 6.385 & 10.7 && $[521]1/2^-$ & pro & 21.55 &&&&& $[514]7/2^-$ & pro & 5.71 &&&&& $[503]7/2^-$ & pro & 45.21 &&&&& $[624]9/2^+$ & pro & 86.73 &&&&& $[606]13/2^+$ & obl & 36.18 $^{185}$Hg & $1/2^-$ & 5.690 & 52.2 && $[521]1/2^-$ & pro & 62.54 &&&&& $[514]7/2^-$ & pro & 9.95 &&&&& $[503]7/2^-$ & pro & 75.38 &&&&& $[624]9/2^+$ & pro & 71.74 &&&&& $[606]13/2^+$ & obl & 84.30 $^{187}$Hg & $3/2^-$ & 4.910 & 114 && $[521]3/2^-$ & obl & 83.19 &&&&& $[514]7/2^-$ & pro & 19.44 &&&&& $[503]7/2^-$ & pro & 194.4 &&&&& $[624]9/2^+$ & pro & 379.8 &&&&& $[606]13/2^+$ & obl & 464.4 [cccccllr]{}isotope & && & $J{^\pi}$ & $Q_{EC}$ & $T_{1/2}^{\beta^+/EC}$ && $[N n_z \Lambda ]K^{\pi}$ && $T_{1/2}^{\beta^+/EC}$ $^{175}$Pt & $7/2^-$ & 7.694 & 7.20 && $[514]7/2^-$ & pro & 2.33 &&&&& $[503]7/2^-$ & pro & 6.96 &&&&& $[512]5/2^-$ & pro & 10.06 &&&&& $[606]13/2^+$ & obl & 3.63 $^{177}$Pt & $5/2^-$ & 6.677 & 11.24 && $[521]1/2^-$ & pro & 15.98 &&&&& $[514]7/2^-$ & pro & 5.53 &&&&& $[503]7/2^-$ & pro & 23.61 &&&&& $[512]5/2^-$ & pro & 19.14 &&&&& $[604]7/2^+$ & obl & 30.91 $^{179}$Pt & $1/2^-$ & 5.811 & 21.25 && $[521]1/2^-$ & pro & 45.02 &&&&& $[514]7/2^-$ & pro & 10.38 &&&&& $[503]7/2^-$ & pro & 86.30 &&&&& $[512]5/2^-$ & pro & 53.49 &&&&& $[604]9/2^+$ & obl & 63.11 $^{181}$Pt & $1/2^-$ & 5.097 & 52.0 && $[521]1/2^-$ & pro & 64.23 &&&&& $[514]7/2^-$ & pro & 17.66 &&&&& $[503]7/2^-$ & pro & 143.6 &&&&& $[512]5/2^-$ & pro & 39.77 &&&&& $[604]9/2^+$ & obl & 51.73 Half-lives ---------- As we have seen above, the sensitivity of the GT strength distributions to the nuclear deformation could be used to get information about the nuclear shape in the neutron-deficient Hg and Pt isotopes. Unfortunately, these measurements are not yet available. However, we have experimental information on the $\beta^+/$EC-decay half-lives that summarize in a single quantity the detailed structure of the GT strength distribution profiles. As we can see from Eq. (\[t12\]), half-lives are no more that integral quantities obtained as sums of the GT strengths weighted with the energy-dependent phase-space factors given by Eq. (\[phase\]). Therefore, it is natural to contrast our calculations with this information. The experimental half-lives of the neutron-deficient Hg and Pt isotopes can be seen in Figs. \[fig\_t12\_hg\] and \[fig\_t12\_pt\], respectively. The total half-lives taken from [@audi12] contain also contributions from the competing $\alpha$ decay. Using the experimental percentage of the $\beta^+/$EC involved in the total decay, we have extracted the $\beta^+/$EC half-lives, which are displayed in the figures. These half-lives are compared to our calculations using the two shapes (oblate and prolate) that minimize the energy in each isotope. We have joined with dashed lines the results corresponding to the absolute energy minimum in our calculations. The spins and parities of the odd-$A$ isotopes are those considered in Figs. \[fig\_bgt\_hg\_odd\] and \[fig\_bgt\_pt\_odd\]. In both cases, Hg and Pt isotopes, we obtain fair agreement with the trend observed experimentally. In the heavier Hg isotopes oblate shapes reproduce better the experimental trend, whereas in the lighter Hg isotopes the results are more spread around the data and there is no clear advantage of one shape over the other. No firm conclusions can be extracted on preferences about the shape, except for the higher masses above $A=186$. In the case of Pt isotopes, the prolate shape looks more consistent with the data for $172<A<180$. The spread of results is somehow expected taking into account the uncertainties in the calculations coming from Skyrme forces, pairing gap parameters, residual interactions, $Q_{EC}$ values, and quenching factors included in the calculations. They were discussed in Refs. [@sarri05prc; @moreno06]. In the case of the heavier isotopes the agreement with experiment is somewhat worse, but one has to take into account that in these cases we are dealing with very large half-lives that are the natural consequence of very small $Q_{EC}$ energies as we approach the stable isotopes. Therefore, the half-lives are only sensitive to the very low-energy tail of the GT strength distribution and little changes in this tail can change the half-lives dramatically. In any case, the half-lives of almost stable nuclei can only constrain a tiny portion of the whole GT strength distribution and therefore their significance is minor. Table \[table.1\] (\[table.2\]) shows the half-lives in the odd-$A$ Hg (Pt) isotopes considered in Fig. \[fig\_odd\] (\[fig\_odd\_pt\]). We show the experimental $J^\pi$, $Q_{EC}$, and $T_{1/2}^{\beta^+/EC}$ values [@audi12] together with the calculated QRPA(SLy4) results obtained for various states and deformations. The dispersion of the results due to the spin and parity of the odd nucleus is apparent. CONCLUSIONS =========== In this work we have studied bulk and decay properties of even and odd neutron-deficient Hg and Pt isotopes using a deformed pnQRPA formalism with spin-isospin ph and pp separable residual interactions. The quasiparticle mean field is generated from a deformed HF approach with two-body Skyrme effective interactions, taking SLy4 as a reference and comparing with results from Sk3 and SGII. The formalism includes pairing correlations in the BCS approximation, using fixed gap parameters extracted from the experimental masses. The equilibrium deformations in each isotope are derived self-consistently within the HF procedure obtaining oblate and prolate coexisting shapes in most isotopes. These results are very robust and different schemes including non-relativistic self-consistent treatments with either Skyrme or Gogny interactions, as well as relativistic mean field approaches produce similar results. The isotopic evolution in Hg and Pt chains show a shape transition in agreement with experimental findings. In addition, we have calculated mean square charge radii differences and have compared them to data from laser spectroscopy with reasonable agreement. Then, we have focused on the main objective in this work, studying the decay properties of these isotopes. We have payed special attention to the deformation dependence of these properties in a search for additional fingerprints of nuclear shapes that would complement the information extracted by other means, such as rotational bands built on low-lying states and quadrupole transition rates among them. We have evaluated the energy distributions of the GT strength for the possible equilibrium shapes and have shown their energy profiles that will be compared with experiments already carried out on Hg isotopes that are being currently analyzed [@algora]. It is also highly encouraged to investigate experimentally the decay of odd-$A$ isotopes from both ground and shape-isomeric states. Measuring separately the decay patterns of states characterized by rather different spins and parities corresponding to different nuclear shapes would be a significant piece of information regarding deformation effects that can be later exploited to learn about the deformation in even systems. 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--- abstract: 'In this paper, from the viewpoint of the concentration theory of maps, we study a compact group and a Lévy group action to a large class of metric spaces, such as $\mathbb{R}$-trees, doubling spaces, metric graphs, and Hadamard manifolds.' address: 'Mathematical Institute, Tohoku University, Sendai 980-8578, JAPAN' author: - Kei Funano title: Concentration of maps and group action --- [^1] Introduction ============ Let a compact metric group $G$ acts on a compact metric space $X$. In [@mil4 Theorem 5.1], V. Milman considered a Hölder action (see Section 3.6.2 for the definition) and estimated the diameters of orbits from above by words of an isoperimetric property of the group $G$ and a covering property of $X$. As he refered in the introduction, his idea came from the fixed point theory of a Lévy group action by M. Gromov and Milman in [@milgro Theorem 7.1] (see Section 4 for the definition of a Lévy group). In this paper, we consider general continuous actions of a compact metric group and a Lévy group to some concrete noncompact metric spaces, such as $\mathbb{R}$-trees, doubling spaces, metric graphs, and Hadamard manifolds. Of isoperimetric inspiring, the Lévy-Milman concentration theory of maps played an important role in Milman’s estimation (and also Gromov and Milman’s theorem of a Lévy group action). Taking a point $x\in X$, he considered how concentrates the orbit map $G\ni g \to gx\in X$ to a constant map. Recent developments of the concentration theory of maps by the author ([@funano2], [@funad], [@funano1]), by Gromov ([@gromovcat], [@gromov]), and by M. Ledoux and K. Oleszkiewvicz ([@ledole]) enable us to estimate how the orbit map concentrate to a constant map in the case where $X$ is an $\mathbb{R}$-tree, a doubling space, a metric graph, and a Hadamard manifold. In stead of considering a Hölder action and a covering property, we provide an estimate of the diameters of orbits of a continuous action of a compact metric group to those metric spaces by words of the continuity of the action, an isoperimetric property of $G$, and a metric space property of $X$. Our results assert that we can measure how the action to those metric spaces is closed to the trivial action by the above words. In the same point of view, we obtain two results of a Lévy group action to the above spaces. A Lévy group was first introduced and analyzed by Gromov and Milman in [@milgro]. Gromov and Milman proved that every continuous action of a Lévy group to a compact metric space has a fixed point. They also pointed out that the unitary group $U(\ell^2)$ of the separable Hilbert space $\ell^2$ with the strong topology is a Lévy group. Many concrete examples of Lévy groups are known by the works of S. Glasner [@gla], H. Furstenberg and B. Weiss (unpublished), T. Giordano and V. Pestov [@giopes1], [@giopes2], and Pestov [@pestov1], [@pestov3]. For examples, groups of measurable maps from the standard Lebesgue measure space to compact groups, unitary groups of some von Neumann algebras, groups of measure and measure-class preserving automorphisms of the standard Lebesgue measure space, full groups of amenable equivalence relations, and the isometry groups of the universal Urysohn metric spaces are Lévy groups (see the recent monograph [@pestov2] for precise). One of our results states that there is no non-trivial uniformly continuous action of a Lévy group to the above spaces (Proposition \[th3\]). We also obtain a generalization of Gromov and Milman’s fixed point theorem (Proposition \[th2\]). Both two results are obtained by making Gromov and Milman’s argument precise. The article is organized as follows. In Section $2$, we recall basic facts about the concentration theory of maps and prepare for the Sections $3$ and $4$. In Section $3$, we estimates the diameter of orbits of a compact group action to $\mathbb{R}$-trees, doubling spaces, meric graphs, and Hadamard manifolds. Section $4$ is devoted to a Lévy group action to those spaces. Preliminaries ============= Concentration function and observable diameter ---------------------------------------------- In this subsection, we recall some basic facts in the concentration theory of 1-Lipschitz maps. We recall relationships between an isoperimetric property of an mm-space (metric measure space) and the concentration theory of $1$-Lipschitz functions. The concentration theory of $1$-Lipschitz functions was introduced by Milman in his investigations of asymptotic geometric analysis ([@mil1], [@mil2], [@mil3]). While the concentration theory of functions developed, the concentration theory of maps into general metric spaces was first studied by Gromov ([@gromovcat], [@gromov2], [@gromov]). He established the theory by introducing the observable diameter in [@gromov]. We first recall its definition. Let $Y$ be a metric space and $\nu$ a Borel measure on $Y$ such that $m:=\nu(Y)<+\infty$. We define for any $\kappa >0$ $$\begin{aligned} {\mathop{\mathrm{diam}} \nolimits}(\nu , m-\kappa):= \inf \{ {\mathop{\mathrm{diam}} \nolimits}Y_0 \mid Y_0 \subseteq Y \text{ is a Borel subset such that }\nu(Y_0)\geq m-\kappa\} \end{aligned}$$and call it the *partial diameter* of $\nu$. Let $(X,{\mathop{\mathit{d}} \nolimits}_X)$ be a complete sparable metric space equipped with a finite Borel measure $\mu_X$ on $X$. Henceforth, we call such a triple an *mm-space*. Let $(X,{\mathop{\mathit{d}} \nolimits}_X,\mu_X)$ be an mm-space with $m_X:=\mu_X(X)$ and $Y$ a metric space. For any $\kappa >0$ we define the *observable diameter* of $X$ by $$\begin{aligned} {\mathop{\mathrm{ObsDiam}} \nolimits}_Y (X; -\kappa):= \sup \{ {\mathop{\mathrm{diam}} \nolimits}(f_{\ast}(\mu_X),m_X-\kappa) \mid f:X\to Y \text{ is a }1 \text{{\rm -Lipschitz map}} \}, \end{aligned}$$where $f_{\ast}(\mu_X)$ stands for the push-forward measure of $\mu_X$ by $f$. The idea of the observable diameter comes from the quantum and statistical mechanics, that is, we think of $\mu_X$ as a state on a configuration space $X$ and $f$ is interpreted as an observable. Given sequences $\{X_n\}_{n=1}^{\infty}$ of mm-spaces and $\{ Y_n\}_{n=1}^{\infty}$ of metric spaces, observe that $\lim_{n\to \infty}{\mathop{\mathrm{ObsDiam}} \nolimits}_{Y_n}(X_n;-\kappa)=0$ for any $\kappa >0$ if and only if for any sequence $\{ f_n:X_n \to Y_n\}_{n=1}^{\infty}$ of $1$-Lipschitz maps there exists a sequence $\{ m_{f_n}\}_{n=1}^{\infty}$ of points such that $m_{f_n}\in Y_n$ and $$\begin{aligned} \lim_{n\to \infty}\mu_{X_n}(\{ x_n \in X_n \mid {\mathop{\mathit{d}} \nolimits}_{Y_n}(f_n(x_n),m_{f_n})\geq \varepsilon\})=0 \end{aligned}$$for any $\varepsilon>0$. A sequence $\{X_n\}_{n=1}^{\infty} $ of mm-spaces is said to be a *Lévy family* if $\lim_{n\to \infty}{\mathop{\mathrm{ObsDiam}} \nolimits}_{\mathbb{R}}(X_n;-\kappa)=0$ for any $\kappa>0$. The concept of Lévy families was first introduced in [@milgro]. For an mm-space $X$ with $\mu_X(X)=1$, we define the *concentration function* $\alpha_X:(0,+\infty)\to \mathbb{R}$ as the supremum of $\mu_X(X\setminus A_{+r})$, where $A$ runs over all Borel subsets of $X$ with $\mu_X(A)\geq 1/2$ and $A_{+r}$ is an open $r$-neighbourhood of $A$. This function describes an isoperimetric feature of the space $X$. We shall consider each closed Riemannian manifold as an mm-space equipped with the volume measure normalized to have the total volume $1$. \[exl1\]Let $M$ be a closed Riemannian manifold such that ${\mathop{\mathit{Ric}} \nolimits}_M \geq \widetilde{\kappa}_1>0$. By virtue of the Lévy-Gromov isoperimetric inequality, we obtain $\alpha_M(r)\leq e^{-\widetilde{\kappa}_1 r^2/2}$ (see [@milgro Section 1.2, Remark 2] or [@ledoux Theorem 2.4]). Since ${\mathop{\mathit{Ric}} \nolimits}_{SO(n)}\geq (n-1)/4$, we have $\alpha_{SO(n)}(r)\leq e^{-(n-1)r^2/8}$ for example. \[exl2\]Let $M$ be a closed Riemannian manifold. We denote by $\lambda_1(M)$ the non-zero first eigenvalue of the Laplacian on $M$. Then, for any $r >0$, we have $\alpha_M(r)\leq e^{-\sqrt{\lambda_1(M)}r/3}$ (see [@milgro Theorem 4.1] or [@ledoux Theorem 3.1]). Since the $n$-dimensional torus $\mathbb{T}^n:=\mathbb{S}^1 \times \mathbb{S}^1 \times \cdots \times \mathbb{S}^1$ satisfies $\lambda_1(\mathbb{T}^n)=\lambda_1(\mathbb{S}^1)=1$, we obtain $\alpha_{\mathbb{T}^n}(r)\leq e^{-r/3}$ for example. Let $X$ be an mm-space and $f:X\to \mathbb{R}$ a Borel measurable function. A number $m_f\in \mathbb{R}$ is called a *median* of $f$ if it satisfies that $f_{\ast}(\mu_X)((-\infty, m_f]) \geq m_X /2$ and $f_{\ast}(\mu_X)([m_f ,+\infty))\geq m_X/2$. We remark that $m_f$ does exist, but it is not unique in general. Relationships between the concentration function and the observable diameter are the following: \[vel1\]Let $X$ be an mm-space with $\mu_X(X)=1$. Then, for any $1$-Lipschitz function $f:X\to \mathbb{R}$ and $\varepsilon >0$, we have $$\begin{aligned} \mu_X (\{ x\in X \mid |f(x)-m_f|\geq \varepsilon \})\leq 2\alpha_X(\varepsilon). \end{aligned}$$ \[vel2\]Let $X$ be an mm-space with $\mu_X(X)=1$. Assume that a function $\alpha:(0,+\infty)\to \mathbb{R}$ satisfies that $$\begin{aligned} \mu_X (\{ x\in X \mid |f(x)-m_f|\geq \varepsilon \})\leq \alpha(\varepsilon) \end{aligned}$$for any $1$-Lipschitz function $f:X\to \mathbb{R}$. Then, we have $\alpha_X(\varepsilon) \leq \alpha(\varepsilon)$. By Lemmas \[vel1\] and \[vel2\], we obtain the following corollary: A sequence $\{X_n\}_{n=1}^{\infty}$ of mm-spaces is a Lévy family if and only if $\lim_{n\to \infty}\alpha_{X_n}(r)=0$ for any $r>0$. Combining Lemma \[vel1\] with Examples \[exl1\] and \[exl2\], we obtain the following corollaries: Let $M$ be a closed Riemannian manifold such that ${\mathop{\mathit{Ric}} \nolimits}_M \geq \widetilde{\kappa}_1>0$. Then, for any $\kappa >0$, we have $$\begin{aligned} {\mathop{\mathrm{ObsDiam}} \nolimits}_{\mathbb{R}}(M;-\kappa)\leq 2\sqrt{\frac{2\log \big(\frac{2}{\kappa}\big)}{\widetilde{\kappa}_1}}. \end{aligned}$$In particular, we have $$\begin{aligned} {\mathop{\mathrm{ObsDiam}} \nolimits}_{\mathbb{R}}(SO(n);-\kappa)\leq 4\sqrt{\frac{2\log\big(\frac{2}{\kappa}\big)}{n-1}}. \end{aligned}$$ Let $M$ be a closed Riemannian manifold. Then, for any $\kappa >0$, we have $$\begin{aligned} {\mathop{\mathrm{ObsDiam}} \nolimits}_{\mathbb{R}}(M;-\kappa)\leq \frac{6 \log \big( \frac{2}{\kappa}\big)}{\sqrt{\lambda_1(M)}}. \end{aligned}$$ In particular, we have $$\begin{aligned} {\mathop{\mathrm{ObsDiam}} \nolimits}_{\mathbb{R}}(\mathbb{T}^n;-\kappa)\leq 6 \log \Big(\frac{2}{\kappa}\Big). \end{aligned}$$ Concentration and separation ---------------------------- In this section, we recall the notion of the separation distance for an mm-space which was introduced in [@gromov]. We review relationships between the observable diameter and the separation distance. The separation distance plays an important role throughout this paper. Let $X$ be an mm-space. For $\kappa_1, \kappa_2\geq 0$, we define the *separation distance* ${\mathop{\mathrm{Sep}} \nolimits}(X;\kappa_1,\kappa_2)= {\mathop{\mathrm{Sep}} \nolimits}(\mu_X;\kappa_1,\kappa_2)$ of $X$ as the supremum of the distance ${\mathop{\mathit{d}} \nolimits}_X(A,B)$, where $A$ and $B$ are Borel subsets of $X$ satisfying that $\mu_X(A)\geq \kappa_1$ and $\mu_X(B)\geq \kappa_2$. Relationships between the observable diameter and the separation distance are followings. We refer to [@funad Subsection 2.2] for precise proofs. \[noranoraneko\]Let $X$ be an mm-space and $\kappa,\kappa' >0$ with $\kappa > \kappa'$. Then we have $$\begin{aligned} {\mathop{\mathrm{ObsDiam}} \nolimits}_{\mathbb{R}} (X ;-\kappa')\geq {\mathop{\mathrm{Sep}} \nolimits}(X;\kappa,\kappa). \end{aligned}$$ In [[@gromov Section $3\frac{1}{2}.33$]]{}, Lemma \[noranoraneko\] is stated as $\kappa =\kappa'$, but that is not true in general. For example, let $X:=\{ x_1 , x_2\}$, ${\mathop{\mathit{d}} \nolimits}_X (x_1,x_2):=1$, and $\mu_X (\{ x_1\})=\mu_X (\{ x_2 \}):= 1/2$. Putting $\kappa =\kappa'=1/2$, we have ${\mathop{\mathrm{ObsDiam}} \nolimits}_{\mathbb{R}} (X;-1/2)=0$ and ${\mathop{\mathrm{Sep}} \nolimits}(X;1/2,1/2)=1$. \[l2.1.2\]Let $\nu$ be a Borel measure on $\mathbb{R}$ with $m:=\nu(\mathbb{R})<+\infty$. Then, for any $\kappa >0$ we have $$\begin{aligned} {\mathop{\mathrm{diam}} \nolimits}(\nu, m-2\kappa)\leq {\mathop{\mathrm{Sep}} \nolimits}(\nu; \kappa, \kappa). \end{aligned}$$ In particular, for any $\kappa >0$ we have $$\begin{aligned} {\mathop{\mathrm{ObsDiam}} \nolimits}_{\mathbb{R}}(X;-2\kappa)\leq {\mathop{\mathrm{Sep}} \nolimits}(X; \kappa, \kappa). \end{aligned}$$ \[c2.1.1\]A sequence $\{ X_n\}_{n=1}^{\infty}$ of mm-spaces is a Lévy family if and only if $\lim_{n\to \infty}{\mathop{\mathrm{Sep}} \nolimits}(X_n;\kappa,\kappa) =0$ for any $\kappa >0$. Compact metric group action and diameter of a measure ----------------------------------------------------- Let a compact metric group $G$ continuously acts on a metric space $X$. For each $\eta >0$, we define a (possibly infinite) number $\rho(\eta)= \rho^{(G,X)}(\eta)$ as the supremum of ${\mathop{\mathit{d}} \nolimits}_X(gx,gy)$ for all $g\in G$ and $x,y\in X$ with ${\mathop{\mathit{d}} \nolimits}_X(x,y)\leq \eta$. Given a point $x\in X$, we indicate by $f_x:G\to X$ the orbit map of $x$, that is, $f_x(g):=gx$ for any $g\in G$. For the Haar measure $\mu_G$ on $G$ normalized as $\mu_G(G)=1$, we put $\nu_{G,x}:=(f_x)_{\ast}(\mu_G)$. \[p3.1\]Assume that $\nu_{G,x}(B_X(y,\delta))>1/2$ for some $y\in X$ and $\delta >0$. Then, we have $$\begin{aligned} \label{s3.1} {\mathop{\mathit{d}} \nolimits}_X(y,gy)\leq \delta + \rho(\delta) \end{aligned}$$for any $g\in G$. Moreover, there exists a point $x_0 \in Gx$ such that $$\begin{aligned} \label{s3.2} {\mathop{\mathit{d}} \nolimits}_X (x_0,gx_0) \leq \min \{ 2\delta + \rho(2\delta),2\delta + 2\rho(\delta) \} \end{aligned}$$for any $g\in G$. Taking any $g\in G$, we first prove (\[s3.1\]). Since $gB_X(y,\delta)\subseteq B_X(gy, \rho (\delta))$ and the measure $\nu_{G,x}$ is $G$-invariant, from the assumption, we have $$\begin{aligned} \nu_{G,x}(B_X(gy,\rho(\delta)))\geq \nu_{G,x}(gB_X(y,\delta))=\nu_{G,x}(B_X(y,\delta))>1/2. \end{aligned}$$Combining this with $\nu_{G,x}(B_X(y,\eta))>1/2$, we get $\nu_{G,x}(B_X(y,\delta)\cap B_X(gy,\rho(\delta)))>0$, which implies (\[s3.1\]). We next prove (\[s3.2\]). Since the orbit $Gx$ is compact, the support of the measure $\nu_{G,x}$ is included in $Gx$. Hence, there exists a point $x_0 \in B_X(y,\delta)\cap Gx$. Let $g\in G$. Since $\nu_{G,x}(B_X(x_0,2\delta))\geq \nu_{G,x}(B_X(x_0,2\delta))>0$, by using (\[s3.1\]), we obtain ${\mathop{\mathit{d}} \nolimits}_X(x_0,gx_0)\leq 2\delta +\rho(2\delta)$. We also have $$\begin{aligned} {\mathop{\mathit{d}} \nolimits}_X(x_0,gx_0)\leq \ &{\mathop{\mathit{d}} \nolimits}_X(x_0,y)+ {\mathop{\mathit{d}} \nolimits}_X(y,gy)+ {\mathop{\mathit{d}} \nolimits}_X(gy, g x_0)\\ \leq \ & \delta + (\delta + \rho(\delta))+ \rho(\delta)\\ = \ & 2\delta+ 2\rho(\delta), \end{aligned}$$which implies (\[s3.2\]). This completes the proof. \[p3.2\]Assume that $\nu_{G,x}(A)>1/2$ for some Borel subset $A\subseteq X$. Then, there exists a point $x_0\in Gx$ such that $$\begin{aligned} {\mathop{\mathit{d}} \nolimits}_X(x_0,gx_0)\leq {\mathop{\mathrm{diam}} \nolimits}A+\rho({\mathop{\mathrm{diam}} \nolimits}A) \end{aligned}$$for any $g\in G$. Since $A\cap Gx\neq \emptyset$, the claim follows from the same argument in the proof of Proposition \[p3.1\]. For any $\eta>0$, we put $\rho(+\eta):=\lim_{\eta'\downarrow \eta}\omega_x(\eta')$. \[c3.1\]There exists a point $z_x \in Gx$ such that $$\begin{aligned} {\mathop{\mathit{d}} \nolimits}_X (z_x , gz_x) \leq \lim_{\kappa \uparrow 1/2}{\mathop{\mathrm{diam}} \nolimits}(\nu_{G,x},1-\kappa) +\rho\big(+ \lim_{\kappa \uparrow 1/2}{\mathop{\mathrm{diam}} \nolimits}(\nu_{G,x},1-\kappa) \big) \end{aligned}$$for any $g\in G$. For any $\eta>0$, we define a (possibly infinite) number $\omega_x(\eta)=\omega_x^{(G,X)}(\eta)$ as the supremum of ${\mathop{\mathit{d}} \nolimits}_X(gx, g'x)$ for all $g,g'\in G$ with ${\mathop{\mathit{d}} \nolimits}_G(g,g')\leq \eta$. \[l3.1\]For any $\kappa_1,\kappa_2 >0$, we have $$\begin{aligned} {\mathop{\mathrm{Sep}} \nolimits}(\nu_{G,x};\kappa_1,\kappa_2)\leq \omega_x(+{\mathop{\mathrm{Sep}} \nolimits}(G;\kappa_1, \kappa_2)). \end{aligned}$$ Let $A$ and $B$ be two Borel subsets such that $\nu_{G,x}(A)\geq \kappa_1$ and $\nu_{G,x}(B)\geq \kappa_2$. Since $\mu_G((f_x)^{-1}(A))\geq \kappa_1$ and $\mu_G((f_x)^{-1}(B))\geq \kappa_2$, we have ${\mathop{\mathit{d}} \nolimits}_G((f_x)^{-1}(A), (f_x)^{-1}(B))\leq {\mathop{\mathrm{Sep}} \nolimits}(G;\kappa_1,\kappa_2)$. Thus, from the definition of $\omega_x$, we obtain ${\mathop{\mathit{d}} \nolimits}_X(A,B)\leq \omega_x(+{\mathop{\mathrm{Sep}} \nolimits}(G;\kappa_1,\kappa_2))$. This completes the proof. \[c3.2\]Assume that a sequence $\{ G_n\}_{n=1}^{\infty}$ of compact metric groups is a Lévy family and each $G_n$ acts on a metric space $X$. Assume also that there exist a sequence $\{x_n\}_{n=1}^{\infty}$ of points in $X$ and a function $\omega:(0,+\infty) \to [0,+\infty]$ such that $\lim_{\eta \to 0}\omega (\eta)=0$ and $\omega^{(G_n,X)}_{x_n}(\eta)\leq \omega(\eta)$ for any $n\in \mathbb{N}$ and $\eta >0$. Then, the sequence $\{ (X,{\mathop{\mathit{d}} \nolimits}_X, \nu_{G_n,x_n})\}_{n=1}^{\infty}$ of mm-spaces is a Lévy family. Estimates of the diameters of orbits ==================================== Throughout this section, we always assume that a compact metric group $G$ continuously acts on a metric space $X$. We shall consider the group $G$ as an mm-space $(G,{\mathop{\mathit{d}} \nolimits}_G,\mu_G )$, where $\mu_G$ is the Haar measure on $G$ normalized as $\mu_G(G)=1$. In this section, motivated by the work of Milman [@mil4], we shall estimate the diameters of orbits $Gx$ from above for concrete metric spaces $X$ by words of the continuity of the action, an isoperimetric property of $G$, and a metric space property of $X$. For this purpose, we use the notation $\rho=\rho^{(G,X)}$ and $\omega_x=\omega_x^{(G,X)}$ defined in Subsection 2.3. We first consider the case where the orbit map $f_x:G\ni g\mapsto gx\in X$ for some $x\in X$ is a $1$-Lipschitz map. In this case, applying Corollary \[c3.1\], we obtain the following: For any $\kappa\in (0,1/2)$, there exists a point $z_{\kappa}\in X$ such that $$\begin{aligned} {\mathop{\mathit{d}} \nolimits}_X(z_{\kappa},gz_{\kappa})\leq {\mathop{\mathrm{ObsDiam}} \nolimits}_X(G;-\kappa)+ \rho ({\mathop{\mathrm{ObsDiam}} \nolimits}_X(G;-\kappa)) \end{aligned}$$for any $g\in G$. Case of Euclidean spaces ------------------------ In this subsection, we consider the case where the metric space $X$ is the Euclidean space $\mathbb{R}^k$. Let ${\mathop{\mathrm{pr}} \nolimits}_i:\mathbb{R}^k \ni x= (x_i)_{i=1}^{k}\mapsto x_i \in \mathbb{R}$ be the projection. \[p4.1.1\]For any finite Borel measure $\nu$ on $\mathbb{R}^k$ with $m:=\nu(\mathbb{R}^k)$, we have $$\begin{aligned} {\mathop{\mathrm{diam}} \nolimits}(\nu,m-\kappa)\leq \sqrt{k}\max_{1\leq i\leq k} {\mathop{\mathrm{diam}} \nolimits}\Big(({\mathop{\mathrm{pr}} \nolimits}_i)_{\ast}(\nu), m-\frac{\kappa}{k}\Big). \end{aligned}$$ Applying Corollary \[c2.1.1\] to Proposition \[p4.1.1\], we obtain the following corollary: For any Lévy family $\{X_n\}_{n=1}^{\infty}$ and any $\kappa>0$, we have $$\begin{aligned} \lim_{n\to \infty}{\mathop{\mathrm{ObsDiam}} \nolimits}_{\mathbb{R}^k}(X_n;-\kappa)=0. \end{aligned}$$ \[p4.1.2\]Assume that a compact metric group $G$ continuously acts on the Euclidean space $\mathbb{R}^k$ and put $r:=\lim_{\kappa \uparrow 1/(4k)}{\mathop{\mathrm{Sep}} \nolimits}(G;\kappa,\kappa)$. Then, for any $x\in \mathbb{R}^k$, there exists a point $z_{x}\in Gx$ such that $$\begin{aligned} \label{s4.1.1} {\mathop{\mathit{d}} \nolimits}_{\mathbb{R}^k}(z_{x},g z_{x})\leq \sqrt{k} \omega_x (+ r )+ \rho(+\sqrt{k}\omega_x (+ r ) ) \end{aligned}$$for any $g\in G$. Combining Lemma \[l3.1\] with Proposition \[p4.1.1\], we get $$\begin{aligned} {\mathop{\mathrm{diam}} \nolimits}(\nu_{G,x},1-\kappa) \leq \ &\sqrt{k}\max_{1\leq i\leq k} {\mathop{\mathrm{diam}} \nolimits}\Big(({\mathop{\mathrm{pr}} \nolimits}_i)_{\ast}(\nu_{G,x}),1- \frac{\kappa}{k}\Big)\\ \leq \ & \sqrt{k}\max_{1\leq i\leq k} {\mathop{\mathrm{Sep}} \nolimits}\Big(({\mathop{\mathrm{pr}} \nolimits}_i)_{\ast}(\nu_{G,x});\frac{\kappa}{2k},\frac{\kappa}{2k}\Big)\\ \leq \ & \sqrt{k}{\mathop{\mathrm{Sep}} \nolimits}\Big(\nu_{G,x};\frac{\kappa}{2k},\frac{\kappa}{2k}\Big)\\ \leq \ & \sqrt{k} \omega_x \Big(+{\mathop{\mathrm{Sep}} \nolimits}\Big(G;\frac{\kappa}{2k},\frac{\kappa}{2k}\Big) \Big). \end{aligned}$$Applying this to Corollary \[c3.1\], we obtain (\[s4.1.1\]). This completes the proof. Case of compact metric spaces ----------------------------- In this subsection, we treat the case where the metric space $X$ is a compact metric space $K$. For any $\delta >0$, we denote by $N_K(\delta)$ the minimum number of Borel subsets of diameter at most $\delta$ which cover $K$. \[p4.2.1\]For any $\delta,\kappa>0$ and any finite Borel measure $\nu$ on $K$ with $m:=\nu(K)$, we have $$\begin{aligned} {\mathop{\mathrm{diam}} \nolimits}(\nu,m-\kappa)\leq {\mathop{\mathrm{Sep}} \nolimits}\Big(\nu;\frac{\kappa}{N_K(\delta)}, \frac{\kappa}{N_K(\delta)} \Big) +2\delta. \end{aligned}$$ Let $\{ X_n\}_{n=1}^{\infty}$ be a Lévy family and $K$ a compact metric space. Then, for any $\kappa >0$, we have $$\begin{aligned} \lim_{n\to \infty}{\mathop{\mathrm{ObsDiam}} \nolimits}_{K}(X_n;-\kappa)=0. \end{aligned}$$ By virtue of Proposotion \[p4.2.1\], the same proof of Proposition \[p4.1.2\] yields the following proposition: \[p4.2.2\]Assume that a compact metric group $G$ continuously acts on a compact metric space $K$ and put $r_{x,\delta}:= \omega_x(+\lim_{\kappa \uparrow 1/(2N_K(\delta))}{\mathop{\mathrm{Sep}} \nolimits}(G;\kappa,\kappa))+2\delta$ for $x\in K$ and $\delta>0$. Then, there exists a point $z_{x,\delta}\in Gx$ such that $$\begin{aligned} {\mathop{\mathit{d}} \nolimits}_K(z_{x,\delta},gz_{x,\delta})\leq \ & r_{x,\delta}+ \rho(+r_{x,\delta}) \end{aligned}$$for any $g\in G$. Proposition \[p4.2.2\] generalizes Milman’s result [@mil4 Theorem 5.1]. Case of $\mathbb{R}$-trees -------------------------- In this subsection, we consider the case where the metric space $X$ is an $\mathbb{R}$-tree $T$. For this purpose, we first recall some standard terminologies in metric geometry. Let $(X,{\mathop{\mathit{d}} \nolimits}_X)$ be a metric space. A rectifiable curve $\gamma:[0,1]\to X$ is called a *geodesic* if its arclength coincides with the distance ${\mathop{\mathit{d}} \nolimits}_X(\gamma(0),\gamma(1))$ and it has a constant speed, i.e., parameterized proportionally to the arc length. We say that $(X,{\mathop{\mathit{d}} \nolimits}_X)$ is a *geodesic space* if any two points in $X$ are joined by a geodesic between them. A complete metric space $T$ is called an *$\mathbb{R}$-tree* if it has the following properties: - Any two points in $T$ are connected by a unique unit speed geodesic. - The image of every simple path in $T$ is the image of a geodesic. To answer Gromov’s exercise in [@gromov Section $3\frac{1}{2}.32$], the author proved the following theorem: \[t4.3.1\]For any $\kappa>0$ and finite Borel measure $\nu$ on $T$ with $m:=\nu(T)$, we have $$\begin{aligned} \nu \Big(B_T \Big(x_{\nu},{\mathop{\mathrm{Sep}} \nolimits}\Big(\nu; \frac{\kappa}{2},\frac{m}{3} \Big) \Big) \Big)\geq 1-\kappa. \end{aligned}$$ Let $\{X_n\}_{n=1}^{\infty}$ be a Lévy family and $T$ an $\mathbb{R}$-tree. Then, for any $\kappa >0$, we have $$\begin{aligned} \lim_{n\to \infty}{\mathop{\mathrm{ObsDiam}} \nolimits}_T(X_n;-\kappa)=0. \end{aligned}$$ By Proposition \[p3.1\] and Theorem \[t4.3.1\], the following proposition follows from the same proof of Proposition \[p4.1.2\]. Assume that a compact metric group $G$ continuously acts on an $\mathbb{R}$-tree $T$. Then, for any $x\in T$ and $\kappa\in (0,1/4)$, there exists a point $ z_{x,\kappa}\in T$ such that $$\begin{aligned} {\mathop{\mathit{d}} \nolimits}_T(z_{x,\kappa},gz_{x,\kappa} )\leq \omega_x \Big( +{\mathop{\mathrm{Sep}} \nolimits}\Big(G;\kappa,\frac{1}{3}\Big)\Big) + \rho\Big( \omega_x \Big( +{\mathop{\mathrm{Sep}} \nolimits}\Big(G;\kappa,\frac{1}{3}\Big)\Big)\Big) \end{aligned}$$for any $g\in G$. Put $r:=\lim_{\kappa\uparrow 1/4}{\mathop{\mathrm{Sep}} \nolimits}(G;\kappa,\kappa)$. Then, there also exists a point $z_{x} \in Gx$ such that $$\begin{aligned} {\mathop{\mathit{d}} \nolimits}_T (z_{x} ,gz_{x})\leq \ &\min \{2\omega_x (+r) + \rho(+2\omega_x(+r)), 2\omega_x (+r)+ 2\rho(\omega_x (+r) \} \end{aligned}$$for any $g\in G$. Case of doubling spaces ----------------------- Throughout this subsection, we consider the case where the metric space $X$ is a doubling space. A complete metric space $X$ is called a *doubling space* if there exist $R_1>0$ and a function $D=D_X:(0,R_1]\to (0,+\infty)$ satisfying the following condition: Every closed ball with radius $2r_1\leq 2R_1$ is covered by at most $D(r_1)$ closed balls with radius $r_1$. This condition is equivalent to the following condition: There exists a function $C=C_X=C(r_1,r_2):(0, 2R_{1}]\times (0,2R_1]\to (0,+\infty)$ such that for every $(r_1,r_2)\in (0,2R_1]\times (0,2R_1]$, every $r_1$-separated subset in any closed ball in $X$ with radius $r_2$ contains at most $C(r_1,r_2)$ elements. For example, a complete Riemannian manifold with Ricci curvature bounded from below is a doubling space (see the proof of Corollary \[c4.4.3\]). Although the proof of the following theorem is the same analogue to [@funad Theorem 1.3], we give it for completeness. \[t4.4.1\]Let $X$ be a doubling space and $\nu$ a finite Borel measure on $X$ with $m:= \nu(X)$. Assume that a positive number $r_0$ satisfies $$\begin{aligned} r_0> \max \Big\{ {\mathop{\mathrm{Sep}} \nolimits}\Big(\nu;\kappa, \frac{m}{C(r_0,5r_0)}\Big), {\mathop{\mathrm{Sep}} \nolimits}\Big( \nu; \frac{m-\kappa}{3},\frac{m-\kappa}{3} \Big), {\mathop{\mathrm{Sep}} \nolimits}\Big(\nu; \frac{m-\kappa}{3}, \kappa \Big)\Big\} \end{aligned}$$for some $\kappa >0$. Then there exists a point $x_{0}\in X$ such that $\nu ( B_X(x_{0},3r_0))\geq m-\kappa$. Take a maximal $r_0$-separated set $\{\xi_{\alpha} \}_{\alpha\in \mathcal{A}}$ of $X$. From the doubling property of $X$, there exists $\alpha_0\in \mathcal{A}$ such that $$\begin{aligned} k:= \# \{ \beta\in \mathcal{A} \mid \xi_{\beta}\in B_X(\xi_{\alpha_0}, 5r_0) \} = \max_{\alpha \in \mathcal{A}} \#\{ \beta\in \mathcal{A} \mid \xi_{\beta}\in B_X(\xi_{\alpha}, 5r_0) \}\leq C(r_0,5r_0). \end{aligned}$$Putting $\{\beta_1, \beta_2 ,\cdots , \beta_k\}:=\{ \beta\in \mathcal{A} \mid \xi_{\beta}\in B_X(\xi_{\alpha_0}, 5r_0) \} $, we take a subset $J_1 \subseteq \{ \xi_{\alpha}\}_{\alpha\in \mathcal{A}}$ which is maximal with respect to the properties that $J_1$ is $5r_0$-separated and $\xi_{\beta_1}\in J_1$, $\xi_{\beta_2}\notin J_1$, $\cdots$, $\xi_{\beta_k} \notin J_1$. We then take $J_2 \subseteq \{ \xi_{\alpha}\}_{\alpha \in \mathcal{A}} \setminus J_1$ which is maximal with respect to the properties that $J_2$ is $5r_0$-separated and $ \xi_{\beta_2}\in J_2$, $\xi_{\beta_3}\notin J_2$, $\cdots$, $\xi_{\beta_k}\notin J_2$. In the same way, we pick $J_3 \subseteq \{ \xi_{\alpha}\}_{\alpha \in \mathcal{A}}\setminus (J_1 \cup J_2)$, $\cdots$, $J_k\subseteq \{ \xi_{\alpha}\}_{\alpha\in \mathcal{A}} \setminus (J_1 \cup J_2 \cup \cdots \cup J_{k-1})$. We then have \[cl4.4.1\]$\{\xi_{\alpha} \}_{\alpha \in \mathcal{A}} = J_1 \cup J_2 \cup \cdots \cup J_k$. Suppose that $\xi_{\alpha}\notin J_1 \cup J_2 \cup \cdots \cup J_k$ for some $\alpha \in \mathcal{A}$. Since each $J_i$ is maximal, there exists $\xi_{\alpha_i} \in J_i$ such that ${\mathop{\mathit{d}} \nolimits}_X(\xi_{\alpha}, \xi_{\alpha_i})<5r_0$ and $\xi_{\alpha}\neq \xi_{\alpha_i}$. We therefore obtain $$\begin{aligned} k+1 \leq \# \{ \xi_{\alpha}, \xi_{\alpha_1}, \xi_{\alpha_2}, \cdots , \xi_{\alpha_k} \}\leq \# \{ \beta \in \mathcal{A}\mid \xi_{\beta}\in B_X(\xi_{\alpha},5r_0)\}\leq k, \end{aligned}$$which is a contradiction. This completes the proof of the claim. By Claim \[cl4.4.1\], we have $X= \bigcup_{i=1}^k\bigcup_{\xi_{\alpha}\in J_i}B_X(\xi_{\alpha}, r_0)$. Hence there exists $i$, $1\leq i\leq k$ such that $$\begin{aligned} \nu\Big(\bigcup_{\xi_{\alpha}\in J_i}B_X(\xi_{\alpha},r_0)\Big)\geq \frac{m}{k}\geq \frac{m}{C(r_0,5r_0)}. \end{aligned}$$We then have \[cl4.4.2\] $$\begin{aligned} \nu \Big( \bigcup_{\xi_{\alpha}\in J_i}B_X(\xi_{\alpha},2r_0) \Big)\geq m-\kappa. \end{aligned}$$ Supposing that $\nu ( \bigcup_{\xi_{\alpha}\in J_i}B_X(\xi_{\alpha},2r_0) )< m-\kappa$, from the assumption of $r_0$, we have $$\begin{aligned} r_0 \leq {\mathop{\mathit{d}} \nolimits}_X\Big(X\setminus \bigcup_{\xi_{\alpha}\in J_i}B_X(\xi_{\alpha}, 2r_0), \bigcup_{\xi_{\alpha}\in J_i}B_X(\xi_{\alpha},r_0)\Big)\leq {\mathop{\mathrm{Sep}} \nolimits}\Big(\nu;\kappa,\frac{m}{C(r_0,5r_0)}\Big)<r_0. \end{aligned}$$This is a contradiction. This completes the proof of the claim. \[cl4.4.3\]There exists $\xi_{\gamma}\in J_i$ such that $ \nu(B_X(\xi_{\gamma}, 2r_0))\geq (m-\kappa)/3$. Suppose that $\nu(B_X(\xi_{\alpha}, 2r_0))< (m-\kappa)/3$ for any $\xi_{\alpha}\in J_i$. Then, by Claim \[cl4.4.2\], there exists $J_i'\subseteq J_i$ such that $$\begin{aligned} \frac{m-\kappa}{3}\leq \nu \Big(\bigcup_{\xi_{\alpha}\in J_i'}B_X(\xi_{\alpha}, 2r_0)\Big)< \frac{2(m-\kappa)}{3}. \end{aligned}$$Thus, putting $J_i'':= J_i \setminus J_i'$, from the assumption of $r_0$, we get $$\begin{aligned} r_0 \leq {\mathop{\mathit{d}} \nolimits}_X \Big( \bigcup_{\xi_{\alpha}\in J_i'}B_X(\xi_{\alpha}, 2r_0), \bigcup_{\xi_{\alpha}\in J_i''}B_X(\xi_{\alpha}, 2r_0) \Big)\leq {\mathop{\mathrm{Sep}} \nolimits}\Big(\nu;\frac{m-\kappa}{3}, \frac{m-\kappa}{3}\Big)<r_0. \end{aligned}$$This is a contradiction. This completes the proof of the claim. Combining Claim \[cl4.4.3\] with the same method of the proof of Claim \[cl4.4.2\], we finally obtain $\nu(B_X(\xi_{\gamma}, 3r_0))\geq 1-\kappa$. This completes the proof of the theorem. By Corollary \[c2.1.1\] and Theorem \[t4.4.1\], we get the following corollary: \[c4.4.1\]Let $\{X_n\}_{n=1}^{\infty}$ be a Lévy family and $X$ a doubling space. Then, for any $\kappa >0$, we have $$\begin{aligned} \lim_{n\to \infty}{\mathop{\mathrm{ObsDiam}} \nolimits}_X(X_n;-\kappa)=0. \end{aligned}$$ Applying Theorem \[t4.4.1\] to Proposition \[p3.1\], we obtain the following proposition: \[p4.4.1\]Let a compact metric group $G$ continuously acts on a doubling space $X$. Assume that a positive number $r_0$ satisfies $$\begin{aligned} r_0 >\ & \max \Big\{ \omega_{x}\Big(+{\mathop{\mathrm{Sep}} \nolimits}\Big(\nu;\kappa, \frac{1}{C(r_0,5r_0)}\Big)\Big), \omega_x\Big(+{\mathop{\mathrm{Sep}} \nolimits}\Big( \nu; \frac{1-\kappa}{3},\frac{1-\kappa}{3} \Big)\Big), \\ \ & \hspace{9cm}\omega_x \Big(+{\mathop{\mathrm{Sep}} \nolimits}\Big(\nu; \frac{1-\kappa}{3}, \kappa \Big) \Big) \Big\} \end{aligned}$$for some $\kappa\in (0,1/2)$. Then there exists a point $z_{x,\kappa}\in X$ such that $$\begin{aligned} {\mathop{\mathit{d}} \nolimits}_X(z_{x,\kappa},gz_{x,\kappa})\leq 3r_0 +\rho(3r_0) \end{aligned}$$for any $g\in G$. Moreover, there exists a point $z_{x,\kappa}'\in Gx$ such that $$\begin{aligned} {\mathop{\mathit{d}} \nolimits}_X (z_{x,\kappa}',gz_{x,\kappa}')\leq \min \{ 6r_0 + \rho(6r_0), 6r_0 + 2\rho(3r_0) \} \end{aligned}$$for any $g\in G$. We next consider the case where the function $D=D_X:(0,+\infty )\to (0,+\infty)$ is a constant function. This is equivalent to the following condition: The function $C=C_X:(0,+\infty)\times (0,+\infty)\to (0,+\infty)$ satisfies that $C(\alpha r, \alpha s)= C(r,s)$ for any $r,s, \alpha>0$. We call such a metric space a *large scale doubling space*. By Theorem \[t4.4.1\], we obtain the following corollary: \[c4.4.2\]Let $X$ be a large scale doubling space and $\nu$ be a finite Borel measure on $X$ with $m:=\nu(X)$ and put $$\begin{aligned} r_{\kappa}:=\max \Big\{ {\mathop{\mathrm{Sep}} \nolimits}\Big(\nu;\kappa, \frac{m}{C(1,5)}\Big), {\mathop{\mathrm{Sep}} \nolimits}\Big( \nu; \frac{m-\kappa}{3},\frac{m-\kappa}{3} \Big), {\mathop{\mathrm{Sep}} \nolimits}\Big(\nu; \frac{m-\kappa}{3}, \kappa \Big)\Big\} \end{aligned}$$for $\kappa>0$. Then, there exists a point $x_{\kappa}\in X$ such that $\nu(B_X(x_{\kappa},3r_{\kappa}))\geq m-\kappa$. Applying Corollary \[c4.4.2\] to Proposition \[p3.1\], we obtain the following proposition: \[p4.4.2\]Assume that a compact metric group $G$ continuously acts on a large scale doubling space $X$. Put $$\begin{aligned} r_{x,\kappa} := \ &\max\Big\{ \omega_x\Big(+{\mathop{\mathrm{Sep}} \nolimits}\Big(G;\kappa, \frac{1}{C(1,5)}\Big)\Big), \omega_x\Big( +{\mathop{\mathrm{Sep}} \nolimits}\Big(G;\frac{1-\kappa}{3}, \frac{1-\kappa}{3}\Big) \Big), \\ & \hspace{8cm}\omega_x\Big(+ {\mathop{\mathrm{Sep}} \nolimits}\Big( G;\frac{1-\kappa}{3},\kappa \Big)\Big) \Big\} \end{aligned}$$for $x\in X$ and $\kappa>0$. Then, for any $\kappa \in (0,1/2)$, there exists a point $z_{x,\kappa}\in X$ such that $$\begin{aligned} {\mathop{\mathit{d}} \nolimits}_X (z_{x,\kappa},gz_{x,\kappa}) \leq \ & 3r_{x,\kappa}+ \rho (3r_{x,\kappa} ) \end{aligned}$$for any $g\in G$. There also exists a point $z_{x,\kappa}' \in Gx$ such that $$\begin{aligned} {\mathop{\mathit{d}} \nolimits}_X (z_{x,\kappa}',gz_{x,\kappa}')\leq \min \{ 6r_{x,\kappa}+\rho(6r_{x,\kappa}), 6r_{x,\kappa}+2\rho(3r_{x,\kappa}) \} \end{aligned}$$for any $g\in G$. Assume that a complete metric space $X$ has a doubling measure $\nu_X$, that is, $\nu_X$ is a (not only finite) Borel measure on $X$ having the following properties: $X={\mathop{\mathrm{Supp}} \nolimits}\nu_X$ and there exists a constant $C=C(X)>0$ such that $$\begin{aligned} \nu_X(B_X(x,2r))\leq C \nu_X(B_X(x,r)) \end{aligned}$$for any $x\in X$ and $r>0$. For example, by virtue of the Bishop-Gromov volume comparison theorem, the volume measure of an $n$-dimensional complete Riemannian manifold $M$ with nonnegative Ricci curvature is a doubling measure with $C(M)=2^n$. \[l4.4.1\]Let $(X,\nu_X)$ be a complete metric space with a doubling measure $\nu_X$. Then, for any $0< r_1\leq r_2$ and $x,y\in X$ with $x\in B_X(y,r_2)$, we have $$\begin{aligned} \frac{\nu_X(B_X(x,r_1))}{\nu_X(B_X(y,r_2))}\geq \frac{1}{C^2}\Big(\frac{r_1}{r_2}\Big)^{\log_2 C}=C^{\log_2 \frac{r_1}{r_2}-2}. \end{aligned}$$ \[c4.4.3\]The space $(X,\nu_X)$ is a large scale doubling space with $C_X(r_1,r_2)\leq C^{2+\log_2 \{ (r_1+2r_2)/r_1\}}$. In particular, we have $C_X(1,5)\leq C^{2+\log_2 11}$. Given any $x\in X$ and $r_1,r_2>0$ with $r_2\geq r_1$, we let $\{ \xi_{\alpha}\}_{\alpha \in \mathcal{A}} \subseteq B_X(x,r_2)$ be an arbitrary $r_1$-separated set. Note that closed balls $B_X(\xi_{\alpha}, 2^{-1}r_1 -\varepsilon)$ are mutually dijoint for any $\varepsilon>0$. We hence have $$\begin{aligned} \nu_X(B_X(x, 2^{-1}r_1 +r_2))\geq \ &\nu_X\Big(\bigcup_{\alpha \in \mathcal{A}}B_X(\xi_{\alpha}, 2^{-1}r_1-\varepsilon) \Big)\\ =\ & \sum_{\alpha \in \mathcal{A}} \nu_X (B_X(\xi_{\alpha}, 2^{-1}r_1-\varepsilon))\\ \geq \ & \nu_X(B_X(\xi_{\alpha_0}, 2^{-1}r_1-\varepsilon))\# \mathcal{A}, \end{aligned}$$where $\nu_X(B_X(\xi_{\alpha_0}, 2^{-1}r_1-\varepsilon))= \min_{\alpha\in \mathcal{A}}\nu_X(B_X(\xi_{\alpha}, 2^{-1}r_1-\varepsilon))$. Applying this to Lemma \[l4.4.1\], we obtain $$\begin{aligned} \# \mathcal{A} \leq \frac{\nu_X(B_X(x, 2^{-1}r_1+r_2))}{\nu_X (B_X(\xi_{\alpha_0}, 2^{-1}r_1-\varepsilon))}\leq C^{2+ \log_2 \{ (r_1+2r_2)/ (r_1-2\varepsilon)\}}. \end{aligned}$$This completes the proof. Combining Corollary \[c4.4.2\] with Corollary \[c4.4.3\], we obtain the following corollary: \[c4.4.4\]Let $\nu$ be a finite Borel measure on $(X,\nu_X)$ with $m:=\nu(X)$. Put $$\begin{aligned} r_{\kappa}:= \max \Big\{ {\mathop{\mathrm{Sep}} \nolimits}(\nu;\kappa,C^{-2-\log_2 11}), {\mathop{\mathrm{Sep}} \nolimits}\Big( \nu;\frac{m-\kappa}{3}, \frac{m-\kappa}{3} \Big), {\mathop{\mathrm{Sep}} \nolimits}\Big( \nu; \frac{m-\kappa}{3} , \kappa\Big) \Big\} \end{aligned}$$for $\kappa>0$. Then, there exists a point $x_{\kappa}\in X$ such that $\nu(B_X (x_{\kappa}, 3r_{\kappa}))\geq 1-\kappa$. In particular, we have ${\mathop{\mathrm{diam}} \nolimits}(\nu,m-\kappa)\leq 6r_{\kappa}$. By using Corollary \[c4.4.4\], we obtain the following propostion: \[p4.4.3\]Assume that a compact metric group $G$ continuously acts on $(X,\nu_X)$. Put $$\begin{aligned} r_{x,\kappa}:= &\max \Big\{ \omega_x (+{\mathop{\mathrm{Sep}} \nolimits}(G;\kappa,C^{-2-\log_2 11})), \omega_x\Big(+{\mathop{\mathrm{Sep}} \nolimits}\Big( G;\frac{1-\kappa}{3}, \frac{1-\kappa}{3} \Big)\Big), \\ &\hspace{10cm} \omega_x\Big(+{\mathop{\mathrm{Sep}} \nolimits}\Big( G; \frac{1-\kappa}{3} , \kappa\Big)\Big) \Big\} \end{aligned}$$for $x\in X$ and $\kappa>0$. Then, for any $\kappa \in (0,1/2)$, there exists a point $z_{x,\kappa}\in X$ such that $$\begin{aligned} {\mathop{\mathit{d}} \nolimits}_X (z_{x,\kappa},gz_{x,\kappa}) \leq \ & 3r_{x,\kappa} + \rho(3r_{x,\kappa}) \end{aligned}$$for any $g\in G$. There also exists a point $z_{x,\kappa}' \in Gx$ such that $$\begin{aligned} {\mathop{\mathit{d}} \nolimits}_X (z_{x,\kappa}',gz_{x,\kappa}')\leq \min \{ 6r_{x,\kappa}+\rho(6r_{x,\kappa}), 6r_{x,\kappa}+2\rho(3r_{x,\kappa}) \} \end{aligned}$$for any $g\in G$. Assume that a compact metric group $G$ continuously acts on an $n$-dimensional complete Riemannian manifold $M$ with nonnegative Ricci curvature. Put $$\begin{aligned} r_{\kappa}:= &\max \Big\{ \omega_x (+{\mathop{\mathrm{Sep}} \nolimits}(G;\kappa,2^{-(2+\log_2 11)n})), \omega_x\Big(+{\mathop{\mathrm{Sep}} \nolimits}\Big( G;\frac{1-\kappa}{3}, \frac{1-\kappa}{3} \Big)\Big), \\ &\hspace{10cm} \omega_x\Big(+{\mathop{\mathrm{Sep}} \nolimits}\Big( G; \frac{1-\kappa}{3} , \kappa\Big)\Big) \Big\} \end{aligned}$$for $x\in M$ and $\kappa>0$. Then, for any $x\in M$ and $\kappa \in (0,1/2)$, there exists a point $z_{x,\kappa}\in M$ such that $$\begin{aligned} {\mathop{\mathit{d}} \nolimits}_M (z_{x,\kappa},gz_{x,\kappa}) \leq \ & 3r_{x,\kappa} + \rho(3r_{x,\kappa}) \end{aligned}$$for any $g\in G$. There also exists a point $z_{x,\kappa}' \in Gx$ such that $$\begin{aligned} {\mathop{\mathit{d}} \nolimits}_M (z_{x,\kappa}',gz_{x,\kappa}')\leq \min \{ 6r_{x,\kappa}+\rho(6r_{x,\kappa}), 6r_{x,\kappa}+2\rho(3r_{x,\kappa}) \} \end{aligned}$$for any $g\in G$. Case of metric graphs --------------------- In this subsection, we treat the case where $X$ is a metric graph. Let $\Gamma=(V,E)$ be a (possibly infinite) undirected connected combinatorial graph, that is, $\Gamma$ is a $1$-dimensional cell complex with the set $V$ of vertices and the set $E$ of edges. We allow the graph $\Gamma$ to have multiple edges and loops. For vertices $v,w\in V$ which are endpoints of an edge, we assign a positive number $a_{v,w}$ such that $a_{\Gamma}:=\inf_{v'\neq w'}a_{v'w'}>0$. Every edge is identified with a bounded closed interval or a circle in $\mathbb{R}^2$ with lengh $a_{vw}$, where $v$ and $w$ are endpoints of the edge. We then define the distance between two points in $\Gamma$ to be the infimum of the length of paths joining them. The graph $\Gamma$ together with such a distance function is called a *metric graph*. \[l4.5.1\]Let $(C,{\mathop{\mathit{d}} \nolimits}_C)$ be a circle in $\mathbb{R}^2$ with the Riemannian distance function ${\mathop{\mathit{d}} \nolimits}_C$ and $\nu $ a finite Borel measure on $C$ with $m:=\nu(C)$. Then, for any $\kappa>0$, we have $$\begin{aligned} {\mathop{\mathrm{diam}} \nolimits}(\nu,m-\kappa)\leq \frac{\pi}{\sqrt{2}} {\mathop{\mathrm{Sep}} \nolimits}\Big(\nu; \frac{\kappa}{4}, \frac{\kappa}{4} \Big) \end{aligned}$$ Note that $$\begin{aligned} {\mathop{\mathit{d}} \nolimits}_{\mathbb{R}^2}(x,y)\leq {\mathop{\mathit{d}} \nolimits}_C(x,y) \leq \frac{\pi}{2} {\mathop{\mathit{d}} \nolimits}_{\mathbb{R}^2}(x,y) \end{aligned}$$for any $x,y\in C$. Denoting by ${\mathop{\mathrm{pr}} \nolimits}_i:\mathbb{R}^2 \ni (x_1,x_2)\mapsto x_i\in \mathbb{R}$ the projection, by using Lemma \[l2.1.2\], we therefore obtain $$\begin{aligned} {\mathop{\mathrm{diam}} \nolimits}(\nu,m-\kappa)=\ & {\mathop{\mathrm{diam}} \nolimits}(\nu|_{(C,{\mathop{\mathit{d}} \nolimits}_C)},m-\kappa)\\ \leq \ &\frac{\pi}{2}{\mathop{\mathrm{diam}} \nolimits}(\nu|_{(C,{\mathop{\mathit{d}} \nolimits}_{\mathbb{R}^2})},m-\kappa)\\ \leq \ &\frac{\pi}{\sqrt{2}}\max_{i=1,2} {\mathop{\mathrm{diam}} \nolimits}\Big( ({\mathop{\mathrm{pr}} \nolimits}_i)_{\ast}( \nu|_{(C,{\mathop{\mathit{d}} \nolimits}_{\mathbb{R}^2})}),m-\frac{\kappa}{2} \Big)\\ \leq \ &\frac{\pi}{\sqrt{2}}\max_{i=1,2} {\mathop{\mathrm{Sep}} \nolimits}\Big( ({\mathop{\mathrm{pr}} \nolimits}_i)_{\ast}( \nu|_{(C,{\mathop{\mathit{d}} \nolimits}_{\mathbb{R}^2})});\frac{\kappa}{4},\frac{\kappa}{4} \Big)\\ \leq \ &\frac{\pi}{\sqrt{2}} {\mathop{\mathrm{Sep}} \nolimits}\Big( \nu|_{(C,{\mathop{\mathit{d}} \nolimits}_{\mathbb{R}^2})};\frac{\kappa}{4},\frac{\kappa}{4} \Big)\\ \leq \ & \frac{\pi}{\sqrt{2}} {\mathop{\mathrm{Sep}} \nolimits}\Big( \nu;\frac{\kappa}{4},\frac{\kappa}{4} \Big). \end{aligned}$$This completes the proof. For every edge $e\in E$ and $r>0$, we put $e_{-r}:=\{ x\in e \mid {\mathop{\mathit{d}} \nolimits}_{\Gamma}(e,v)>r \text{ and } {\mathop{\mathit{d}} \nolimits}_{\Gamma}(e,w)>r\}$, where $v$ and $w$ are endpoints of the edge $e$. \[t4.5.1\]Let $\nu$ be a finite Borel measure on a metric graph $\Gamma$ with $m:=\nu(\Gamma)$. Assume that postive numbers $a,\kappa,\kappa'$ satisfy that $\kappa'< \kappa$, $a<a_{\Gamma}$, and $$\begin{aligned} \max \Big\{ 2{\mathop{\mathrm{Sep}} \nolimits}\Big( \nu;\frac{\kappa}{3},\frac{\kappa}{3}\Big), 4{\mathop{\mathrm{Sep}} \nolimits}\Big( \nu;\frac{m-\kappa}{3}, \kappa'\Big) \Big\}<a \end{aligned}$$Then, we have $$\begin{aligned} \label{s4.5.1} {\mathop{\mathrm{diam}} \nolimits}(\nu,m-\kappa)\leq \max \Big\{ \frac{a}{2}+2{\mathop{\mathrm{Sep}} \nolimits}\Big(\nu; \frac{\kappa}{3},\kappa\Big), \frac{\pi}{\sqrt{2}}{\mathop{\mathrm{Sep}} \nolimits}\Big(\nu;\frac{\kappa-\kappa'}{4},\frac{\kappa-\kappa'}{4}\Big) \Big\}. \end{aligned}$$ We first consider the case of $\nu (\bigcup_{v\in V}B_X(v,a/4))\geq \kappa$. Since ${\mathop{\mathrm{Sep}} \nolimits}(\nu;\kappa/3,\kappa/3)<a/2$, as in the proof of Claim \[cl4.4.3\], there exists a vertex $v\in V$ such that $\nu (B_X(v,a/4))\geq \kappa /3$. We thus obtain $\nu (B_X(v,a/4 +{\mathop{\mathrm{Sep}} \nolimits}(\nu;\kappa/3,\kappa/3)))\geq m-\kappa$, which implies (\[s4.5.1\]). We consider the other case that $\nu (X\setminus \bigcup_{v\in V}B_X(v,a/4)) >m-\kappa$. By the same method of Claim \[cl4.4.3\], either the following (1) or (2) holds: \(1) There exists an edge $e\in E$ such that $e$ is not a loop and $\nu (e_{-a/4})\geq (m-\kappa)/3$. \(2) There exists a loop $\ell\in E$ with $\nu (\ell_{-a/4})\geq (m-\kappa)/3$. If (1) holds, combining the same proof of Claim \[cl4.4.2\] with ${\mathop{\mathrm{Sep}} \nolimits}(\nu;\kappa/3,\kappa')<a/4$, we then have $\nu (e)\geq m-\kappa'$. We therefore obtain $$\begin{aligned} {\mathop{\mathrm{diam}} \nolimits}(\nu,m-\kappa)\leq \ &{\mathop{\mathrm{diam}} \nolimits}(\nu|_{e},m-\kappa)\\ =\ &{\mathop{\mathrm{diam}} \nolimits}(\nu|_{e}, \nu(e)-(\nu(e)-m+\kappa))\\ \leq \ &{\mathop{\mathrm{Sep}} \nolimits}\Big(\nu|_{e}; \frac{\nu(e)-m+\kappa}{2},\frac{\nu(e)-m+\kappa}{2}\Big)\\ \leq \ & {\mathop{\mathrm{Sep}} \nolimits}\Big(\nu; \frac{\kappa-\kappa'}{2}, \frac{\kappa-\kappa'}{2}\Big). \end{aligned}$$If (2) holds, by Claim \[cl4.4.2\] and ${\mathop{\mathrm{Sep}} \nolimits}(\nu;\kappa/3,\kappa')< a/4$, we then get $\nu(\ell)\geq m-\kappa'$. Applying Lemma \[l4.5.1\], we therefore obtain $$\begin{aligned} {\mathop{\mathrm{diam}} \nolimits}(\nu,m-\kappa)\leq \ &{\mathop{\mathrm{diam}} \nolimits}(\nu|_{\ell},m-\kappa)\\ = \ & {\mathop{\mathrm{diam}} \nolimits}(\nu|_{\ell}, \nu(\ell)- (\nu(\ell)-m+\kappa))\\ \leq \ & \frac{\pi}{\sqrt{2}} {\mathop{\mathrm{Sep}} \nolimits}\Big( \nu|_{\ell}; \frac{\nu(\ell)-m+\kappa}{4}, \frac{\nu(\ell)-m+\kappa}{4}\Big)\\ \leq \ & \frac{\pi}{\sqrt{2}} {\mathop{\mathrm{Sep}} \nolimits}\Big( \nu|_{\ell}; \frac{\kappa -\kappa'}{4}, \frac{\kappa -\kappa'}{4}\Big)\\ \leq \ & \frac{\pi}{\sqrt{2}} {\mathop{\mathrm{Sep}} \nolimits}\Big( \nu ; \frac{\kappa -\kappa'}{4}, \frac{\kappa -\kappa'}{4}\Big). \end{aligned}$$This completes the proof of the theorem. Let $\{X_n\}_{n=1}^{\infty}$ be a Lévy family and $\Gamma$ a metric graph. Then, for any $\kappa >0$, we have $$\begin{aligned} \lim_{n\to \infty}{\mathop{\mathrm{ObsDiam}} \nolimits}_{\Gamma}(X_n;-\kappa)=0. \end{aligned}$$ By virtue of Theorem \[t4.5.1\], we obtain the following: Assume that a compact metric group $G$ continuously acts on a metric graph $\Gamma$. We also assume that postive numbers $a,\kappa,\kappa'$ and a point $x\in X$ satisfy that $\kappa'< \kappa$, $a<a_{\Gamma}$, and $$\begin{aligned} \max\Big\{2\omega_x\Big(+{\mathop{\mathrm{Sep}} \nolimits}\Big(G;\frac{\kappa}{3}, \frac{\kappa}{3}\Big)\Big), 4\omega_x \Big(+{\mathop{\mathrm{Sep}} \nolimits}\Big(G;\frac{1-\kappa}{3}, \kappa'\Big)\Big)\Big\}<a. \end{aligned}$$Put $$\begin{aligned} s_{x,\kappa,\kappa'}:= \max \Big\{ \frac{a}{2} + 2\omega_x \Big(+{\mathop{\mathrm{Sep}} \nolimits}\Big(G;\frac{\kappa}{3},\kappa \Big)\Big), \frac{\pi}{\sqrt{2}}\omega_x \Big(+{\mathop{\mathrm{Sep}} \nolimits}\Big( G;\frac{\kappa-\kappa'}{4},\frac{\kappa-\kappa'}{4}\Big)\Big)\Big\}. \end{aligned}$$Then, there exists a point $z_{x,\kappa,\kappa'}\in Gx$ such that $$\begin{aligned} {\mathop{\mathit{d}} \nolimits}_X( z_{x,\kappa,\kappa'}, gz_{x,\kappa,\kappa'})\leq s_{x,\kappa,\kappa'}+ \rho(s_{x,\kappa,\kappa'}) \end{aligned}$$for any $g\in G$. Case of Hadamard manifolds -------------------------- In this subsection, we consider the case where $X$ is a Hadamard manifold $N$, i.e., a complete simply connected Riemannian manifold with nonpositive sectional curvature. The following theorem was obtained in [@funano1 Theorem 1.3]. Let $\{X_n\}_{n=1}^{\infty}$ be a Lévy family and $N$ a Hadamard manifold. Then, for any $\kappa >0$, we have $$\begin{aligned} \lim_{n\to \infty}{\mathop{\mathrm{ObsDiam}} \nolimits}_N (X_n;-\kappa)=0. \end{aligned}$$ ### Central radius Let $N$ be a Hadamard manifold. For a finite Borel measure on $N$ with compact support, we indicate the center of mass of the measure $\nu$ by $c(\nu)$. Given any $\kappa>0$, putting $m:= \nu(N)$, we define the *central radius* ${\mathop{\mathrm{CRad}} \nolimits}(\nu,m-\kappa)$ of $\nu$ as the infimum of $\rho>0$ such that $\nu(B_N(c(\nu),\rho))\geq m-\kappa$. \[p4.6.1.1\]For a finite Borel measure $\nu$ on $\mathbb{R}^k$ with the compact support, we have $$\begin{aligned} c(\nu)=\frac{1}{\nu(\mathbb{R}^k)}\int_{\mathbb{R}^k}x d\nu(x). \end{aligned}$$ \[p4.6.1.2\]Let $N$ be a Hadamard manifold and $nu$ a finite Borel measure on $N$ with the compact support. Then, $x=c(\nu)$ if and only if $$\begin{aligned} \int_N \exp_x^{-1}(y)d\nu(y)=0. \end{aligned}$$In particular, identifying the tangent space of $N$ at the point $c(\nu)$ with the Euclidean space of the same dimension of $N$, we have $c((\exp_{c(\nu)}^{-1})_{\ast}(\nu))=0$. Proposition \[p3.1\] directly implies the following corollary: \[c4.6.1.1\]Assume that a compact metric group acts on a Hadamard manifold $N$ and put $r_x:=\lim_{\kappa \uparrow 1/2}{\mathop{\mathrm{CRad}} \nolimits}(\nu_{G,x},1-\kappa)$ for $x\in X$. Then, we have $$\begin{aligned} {\mathop{\mathit{d}} \nolimits}_X(c(\nu_{G,x}), gc(\nu_{G,x}))\leq r_x+ \rho(+r_x) \end{aligned}$$for any $g\in G$. Moreover, there exists a point $z_{x}\in Gx$ such that $$\begin{aligned} {\mathop{\mathit{d}} \nolimits}_X(z_{x},gz_{x}) \leq \ &\min \{ 2r_x+\rho(+2r_x), 2r_x+2\rho (+r_x) \} \end{aligned}$$for any $g\in G$. ### Hölder actions In this subsubsection, we consider a Hölder action of a compact Lie group to a Hadamard manifold. Let a compact Lie group $G$ acts on a Hadamard manifold $N$. We shall consider the case where $\omega_x(\eta)\leq C_1 \eta^{\alpha}$ holds for some $x\in N$ and $C_1,\alpha>0$. Combining Gromov’s observation in [@gromovcat Section 13] with one in [@gromov Section $3\frac{1}{2}.41$], we obtain the following theorem: \[t4.6.2.1\]Let $M$ be a compact Riemannian manifold and $N$ be a Hadamard manifold. Assume that a continuous map $f:M\to N$ satisfies that $$\begin{aligned} {\mathop{\mathit{d}} \nolimits}_N(f(x),f(y))\leq C_1 {\mathop{\mathit{d}} \nolimits}_M (x,y)^{\alpha} \end{aligned}$$for some $C_1>0$, $\alpha> 1$, and all $x,y\in M$. Then, the map $f:M\to N$ is a constant map. Put $\mathbb{E}(f):=c(f_{\ast}(\mu_M))$. We shall prove that ${\mathop{\mathrm{Supp}} \nolimits}f_{\ast}(\mu_X)= \{ \mathbb{E}(f)\}$, which implies the theorem. Suppose that ${\mathop{\mathrm{Supp}} \nolimits}f_{\ast}(\mu_X) \neq \{ \mathbb{E}(f)\}$. We identify the tangent space of $N$ at $\mathbb{E}(f)$ with the Euclidean space $\mathbb{R}^k$, where $k$ is the dimension of $N$. According to the hinge theorem (see [@sakai Chapter , Remark 2.6]), the map $\exp_{\mathbb{E}(f)}^{-1}:N\to \mathbb{R}^k$ is $1$-Lipschitz. Since the map $\exp^{-1}_{\mathbb{E}(f)}$ is isometric on rays issuing from $\mathbb{E}(f)$ and ${\mathop{\mathrm{Supp}} \nolimits}f_{\ast}(\mu_M)\neq \{ \mathbb{E}(f)\}$, we have $$\begin{aligned} \int_M |(\exp_{\mathbb{E} (f)}^{-1} \circ f)(x)|^2 d\mu_M(x)= \int_M {\mathop{\mathit{d}} \nolimits}_N(f(x),\mathbb{E}(f))^2 d\mu_M(x)>0. \end{aligned}$$Denoting by $((\exp_{\mathbb{E} (f)}^{-1} \circ f)(x))_i$ the $i$-th component of $(\exp_{\mathbb{E} (f)}^{-1} \circ f)(x)$, we hence see that there exists $i_0$ such that $$\begin{aligned} \int_M |((\exp_{\mathbb{E} (f)}^{-1} \circ f)(x))_{i_0}|^2d\mu_M(x)>0. \end{aligned}$$Putting $\varphi:= (\exp_{\mathbb{E} (f)}^{-1} \circ f)_{i_0}$, we observe that $$\begin{aligned} \| {\mathop{\mathrm{grad}} \nolimits}_x \varphi \|=\limsup_{y \to x}\frac{|\varphi(y)-\varphi (x)|}{{\mathop{\mathit{d}} \nolimits}_M(y,x)}\leq \limsup_{y\to x} \frac{C_1{\mathop{\mathit{d}} \nolimits}_{M}(y,x)^{\alpha}}{{\mathop{\mathit{d}} \nolimits}_M(y,x)}= 0 \end{aligned}$$and the function $\varphi$ has mean zero by Proposition \[p4.6.1.2\]. We therefore obtain $$\begin{aligned} 0<\lambda_1(M)= \inf \frac{\int_M \| {\mathop{\mathrm{grad}} \nolimits}_x g \|^2 d\mu_M (x)}{\int_M g(x)^2 d\mu_M(x)} \leq \frac{\int_M \| {\mathop{\mathrm{grad}} \nolimits}_x \varphi\|^2 d\mu_M(x)}{\int_M \varphi(x)^2 d\mu_M(x)}=0, \end{aligned}$$where the infimum is taken over all nontrivial Lipschitz maps $g:M\to \mathbb{R}$ with mean zero. This is a contradiction. This completes the proof. \[c4.6.2.1\]Assume that a compact Lie group $G$ continuously acts on a Hadamard manifold $N$. We also assume that there exists a point $x\in X$ such that the condition $\omega_x(\eta)\leq C_1 \eta^{\alpha}$ holds for some $\alpha>1$. Then, the point $x$ is a fixed point. Assume that a compact metric group $G$ contnuously acts on a Hadamard manifold $N$. In view of Corollary \[c4.6.2.1\], we shall consider the case of $0<\alpha\leq 1$. We assume that a compact metric group $G$ satisfies that $$\begin{aligned} \label{s4.6.2.1} \alpha_G(r)\leq C_2 e^{-C_3 r^{\beta}} \text{ for some }C_2,C_3, \beta>0. \end{aligned}$$See Examples \[exl1\] and \[exl2\] for examples. Let a compact metric group continuously acts on a metric space $X$. For any $r>0$ and $x\in X$, we define $\omega_x^{-1}(r)$ as the infimum of ${\mathop{\mathit{d}} \nolimits}_G(g,g')$, where $g$ and $g'$ run over all elements in $G$ such that ${\mathop{\mathit{d}} \nolimits}_X(gx,g'x)\geq r$. \[l4.6.2.1\]Assume that a compact metric group continuously acts on a metric space $X$. Then, for any $x\in X$, we have $$\begin{aligned} \alpha_{(X,\nu_{G,x})}(r)\leq \alpha_G(\omega_x^{-1}(r)). \end{aligned}$$ Let $A\subseteq X$ be any Borel subset such that $\nu_{G,x}(A)\geq 1/2$. From the difinition of $\omega_x^{-1}(r)$, we get $$\begin{aligned} \{ g\in G \mid gx\in A\}_{+\omega_x^{-1}(r)}\subseteq \{ g\in G \mid gx \in A_{+r}\}. \end{aligned}$$Since $\mu_G(\{ g\in G \mid gx\in A\})\geq 1/2$, we hence obtain $$\begin{aligned} \nu_{G,x}(X\setminus A_{+r})\leq \mu_G(G\setminus \{ g\in G \mid gx \in A\}_{+\omega_x^{-1}(r)})\leq \alpha_G(\omega_x^{-1}(r)). \end{aligned}$$This completes the proof. \[l4.6.2.2\]Let a compact metric group $G$ continuously acts on a metric space $X$. Assume that a point $x\in X$ satisfies the following Hölder condition: $$\begin{aligned} \label{s4.6.2.2} \omega_x (\eta)\leq C_1 \eta^{\alpha} \text{ holds for some }C_1>0 \text{ and }0< \alpha\leq 1. \end{aligned}$$We also assume that the group $G$ satisfies the condition (\[s4.6.2.1\]). Then, we have $$\begin{aligned} \alpha_{(N,\nu_{G,x})}(r)\leq C_2 e^{-C_1^{-\beta/\alpha} C_3 r^{\beta /\alpha}}. \end{aligned}$$ By the assumption (\[s4.6.2.2\]), ${\mathop{\mathit{d}} \nolimits}_{X}(gx,g'x)> C_1s^{\alpha}$ implies that ${\mathop{\mathit{d}} \nolimits}_G(g,g')>s$, that is, ${\mathop{\mathit{d}} \nolimits}_X(gx,g'x)\geq r$ yields that ${\mathop{\mathit{d}} \nolimits}_G(g,g')\geq (r/C_1)^{1/\alpha}$. We hence get $\omega_x^{-1}(r)\geq (r/C_1)^{1/\alpha}$. By using this and Lemma \[l4.6.2.1\], we obtain $$\begin{aligned} \alpha_{(X,\nu_{G,x})}(r)\leq \alpha_G(\omega_x^{-1}(r))\leq \alpha_G ((r/C_1 )^{1/\alpha})\leq C_2 e^{-C_1^{-\beta/\alpha}C_3 r^{\beta /\alpha}}. \end{aligned}$$This completes the proof. We denote by $\gamma_k$ the standard Gaussian measure on $\mathbb{R}^k$ with density $(2\pi)^{-k/2}e^{-|x|^2 /2}$. For any $p\geq 0$, we put $$\begin{aligned} M_p:=\int_{\mathbb{R}}|s|^p d\gamma_1(s)=2^{p/2}\pi^{-1/2}\Gamma \Big(\frac{p+1}{2}\Big). \end{aligned}$$The same proof of [@ledole Theorem 1] implies the following theorem: \[t4.6.2.2\]Assume that an mm-space $X$ satisfies that $\alpha_X(r)\leq C_1e^{-C_2 r^p}$ for some $C_1,C_2>0$ and some $p\geq 1$. Then, for any $1$-Lipschitz function $f:X\to \mathbb{R}^k$ with mean zero, we have $$\begin{aligned} \int_X|f(x)|^p d\mu_X(x)\leq \frac{C}{C_2 M_p}\int_{\mathbb{R}^k}|y|^p d\gamma_k(y) = \frac{C}{C_2 M_p}\cdot \frac{2^{p/2}\Gamma(\frac{p+k}{2})}{\Gamma (\frac{k}{2})} \approx \frac{Ck^{p/2}}{C_2}, \end{aligned}$$where $C$ is a constant depending only on $p$ and $C_1$. \[t4.6.2.3\]Let a compact metric group $G$ continuously acts on a $k$-dimensional Hadamard manifold $N$. Assume that a point $x\in N$ satisfies the Hölder condition (\[s4.6.2.2\]). We also assume that the group $G$ satisfies (\[s4.6.2.1\]) and $\alpha\leq \beta$. Then, there exists a point $z_{x}\in Gx$ such that $$\begin{aligned} \label{s4.6.2.3} {\mathop{\mathrm{diam}} \nolimits}(G z_{x})\leq \frac{C C_1 k^{1/2}}{(C_3)^{\alpha/\beta}}+ \rho \Big( \frac{C C_1 k^{1/2}}{(C_3)^{\alpha/\beta}}\Big), \end{aligned}$$where $C$ is a constant depending only on $\alpha / \beta$ and $C_1$. To apply Corollary \[c4.6.1.1\], we shall estimate ${\mathop{\mathrm{CRad}} \nolimits}(\nu_{G,x},1-\kappa)$ for $0<\kappa <1/2$ from the above. Putting $z:=c(\nu_{G,x})$, as in the proof of Theorem \[t4.6.2.1\], we identify the tangent space of $N$ at $z$ with the Euclidean space $\mathbb{R}^k$. Since the map $\exp_z^{-1}:N\to \mathbb{R}^k$ is a $1$-Lipschitz map, by virtue of Lemma \[l4.6.2.2\] and Theorem \[t4.6.2.2\], we have $$\begin{aligned} \int_N {\mathop{\mathit{d}} \nolimits}_N (y,z)^{\beta/ \alpha}d\nu_{G,x}(y) = \int_N |(\exp_z^{-1})(y)|^{\beta /\alpha }d\nu_{G,x}(y)\leq \frac{C C_1^{\beta /\alpha}k^{\beta/(2\alpha)}}{C_3}, \end{aligned}$$where $C$ is a constant depending only on $C_2$ and $\beta /\alpha$. Combining this inequality with the Chebyshev inequality, we hence get $$\begin{aligned} {\mathop{\mathrm{CRad}} \nolimits}(\nu_{G,x},1-\kappa)\leq \frac{CC_1k^{1/2}}{(C_3 \kappa)^{\alpha /\beta}} \end{aligned}$$for any $0<\kappa$. Applying Corollary \[c4.6.1.1\], we therefore obtain (\[s4.6.2.3\]). This completes the proof. ### Cases of finite groups In this subsubsection, we shall consider the case where $G$ is a finite group. Let $G$ be a finite group and $S\subseteq G\setminus \{e_G\}$ be a symmetric set of generators of $G$. We denote by $\Gamma (G,S)$ the *Cayley graph* of $G$ with respect to $S$. For such $S$, we shall consider the group $G$ as a metric group with respect to the Cayley graph distance function. Let $\Gamma =(V,E)$ be a simple finite graph, where *simple* means that there is at most one edge joining two vertices and no loops from a vertex to itself. The discrete Laplacian $\triangle_{\Gamma}$ act on functions $f$ on $V$ as follows $$\begin{aligned} \triangle_{\Gamma}f(x):= \sum_{y \sim x}(f(x)-f(y)), \end{aligned}$$where $x \sim y$ means that $x$ and $y$ are connected by an edge. We denote by $\lambda_1(\Gamma)$ the non-zero first eigenvalue of the Laplacian $\triangle_{\Gamma}$. As Theorem \[t4.6.2.1\], Gromov’s observation in [@gromovcat Section 13] together with one in [@gromov Section $3\frac{1}{2}.41$] imply the following lemma: \[l4.6.3.1\]Let $S\subseteq G\setminus \{e_G\}$ be a symmetric set of generators of a finite group $G$ and assume that the group $G$ continuously acts on a $k$-dimensional Hadamard manifold $N$. Then, for any $x\in N$ and $\kappa>0$, we have $$\begin{aligned} {\mathop{\mathrm{CRad}} \nolimits}(\nu_{G,x},1-\kappa)\leq \omega_x(1) \Big(\frac{k\#S}{2\kappa\lambda_1(\Gamma (G,S) )}\Big)^{1/2}. \end{aligned}$$ Suppose that $$\begin{aligned} \label{s4.6.3.1} r:={\mathop{\mathrm{CRad}} \nolimits}(\nu_{G,x},1-\kappa)> \omega_x(1) \Big(\frac{k\#S}{2\kappa\lambda_1(\Gamma (G,S) )}\Big)^{1/2}. \end{aligned}$$As in the proof of Theorem \[t4.6.2.1\], we identify the tangent space of $N$ at $z:=c(\nu_{G,x})$ with the Euclidean space $\mathbb{R}^k$. By the Chebyshev inequality, we get $$\begin{aligned} \int_{G} |(\exp_z^{-1} \circ f^x)(g) |^2d \mu_G(g)= \int_G {\mathop{\mathit{d}} \nolimits}_N(f^x(g),z)^2 d\mu_G(g)\geq \kappa r^2. \end{aligned}$$Hence, there exists $i_0$ such that $$\begin{aligned} \label{s4.6.3.2} \int_G ((\exp_z^{-1} \circ f^x)(g))_{i_0}^2d \mu_G(g)\geq \frac{\kappa r^2}{k}. \end{aligned}$$ Putting $\varphi:= (\exp_z^{-1} \circ f^x)_{i_0}$, by (\[s4.6.3.1\]) and (\[s4.6.3.2\]), we obtain $$\begin{aligned} \lambda_1(\Gamma (G,S))=\ & \inf\frac{\sum_{g,g'\in G;g\sim g'} (f(g)- f(g'))^2}{2\sum_{g\in G} f(g)^2}\\ \leq \ & \frac{\sum_{g,g'\in G;g\sim g'}(\varphi(g)-\varphi(g'))^2}{2\sum_{g\in G} \varphi (g)^2}\\ \leq \ & \frac{\sum_{g,g'\in G;g\sim g'}{\mathop{\mathit{d}} \nolimits}_N(f^x(g),f^x(g'))^2}{2\sum_{g\in G} \varphi (g)^2}\\ \leq \ & \frac{ \#G \#S \cdot \omega_x(1)^2 }{\#G\int_G \varphi(g)^2 d\mu_G(g)}\\ = \ & \frac{\omega_x(1)^2\#S}{\int_{G}\varphi(g)^2 d\mu_G(g)}\\ \leq \ & \frac{\omega_x(1)^2 k \#S}{\kappa r^2}\\ < \ & \lambda_1(\Gamma (G,S)), \end{aligned}$$where the infimum is taken over all nontrivial functions $f:G\to \mathbb{R}$ such that $\sum_{g\in G}f(g)=0$. This is a contradiction. This completes the proof. Applying Lemma \[l4.6.3.1\] to Corollary \[c4.6.1.1\], we obtain the following theorem: Let $S\subseteq G\setminus \{e_G\}$ be a symmetric set of generators of a finite group $G$ and assume that the group $G$ continuously acts on a $k$-dimensional Hadamard manifold $N$. Then, for any $x\in N$, we have $$\begin{aligned} {\mathop{\mathit{d}} \nolimits}_N(c(\nu_{G,x}),g c(\nu_{G,x})) \leq \omega_x(1) \Big(\frac{k \# S}{\lambda_1(\Gamma(G,S))}\Big)^{1/2}+ \rho \Big(+ \omega_x (1)\Big(\frac{k \# S}{\lambda_1(\Gamma(G,S))}\Big)^{1/2}\Big) \end{aligned}$$for any $g\in G$. There also exists a point $z_{x}\in Gx$ such that $$\begin{aligned} {\mathop{\mathit{d}} \nolimits}_N (z_{x},gz_{x})\leq \ &\min \Big\{ \omega_x(1) \Big(\frac{k \# S}{ \lambda_1(\Gamma(G,S))}\Big)^{1/2} + \rho \Big( +2\omega_x(1) \Big(\frac{k \# S}{ \lambda_1(\Gamma(G,S))}\Big)^{1/2}\Big),\\ & \hspace{1cm} \omega_x(1) \Big(\frac{k \# S}{ \lambda_1(\Gamma(G,S))}\Big)^{1/2}+ 2 \rho \Big( + \omega_x(1) \Big(\frac{k \# S}{\lambda_1(\Gamma(G,S))}\Big)^{1/2}\Big) \Big\} \tag*{} \end{aligned}$$for any $g\in G$. Lévy group action ================= In this section, we discuss about a Lévy group action to concrete metric spaces appeared in Section 3. A metrizable group $G$ is called a *Lévy group* if it contains an increasing chain of compact subgroups $G_1\subseteq G_2 \subseteq \cdots \subseteq G_n \subseteq \cdots$ having an everywhere dense union in $G$ and such that for some right-invariant compatible distance function ${\mathop{\mathit{d}} \nolimits}_G$ on $G$ the groups $G_n$, $n\in \mathbb{N}$, equipped with the Haar measures $\mu_{G_n}$ normalized as $\mu_{G_n}(G_n)=1$ and the restrictions of the distance function ${\mathop{\mathit{d}} \nolimits}_G$, form a Lévy family. See [@milgro], [@mil5], [@pestov2], [@pestov4] and references therein for informations about a Lévy group. Let a topological group $G$ acts on a metric space $X$. The action is called *bounded* if for any $\varepsilon >0$ there exists a neighbourhood $U$ of the identity element $e_{G}\in G$ such that ${\mathop{\mathit{d}} \nolimits}_X(x,gx)<\varepsilon$ for any $g\in U$ and $x\in X$. Note that every bounded action is continuous. \[l3.2\]Assume that a metric group $G$ with a right invariant distance function ${\mathop{\mathit{d}} \nolimits}_G$ boundedly acts on a metric space $X$. Then, orbit maps $f_x:G\to X$ for all $x\in X$ are uniformly equicontinuous. We shall consider an action of a Lévy group to a metric space $X$ satisfying the following condition: ($\lozenge $): We have $\lim_{n\to \infty}{\mathop{\mathrm{ObsDiam}} \nolimits}_{X}(X_n;-\kappa)=0$ for any $\kappa>0$ and any Lévy family $\{X_n\}_{n=1}^{\infty}$. Note that $\mathbb{R}$-trees, doubling spaces, metric graphs, and Hadamard manifolds satify the condition ($\lozenge$) (see Section 3). Any complete Riemannian manifolds satisfy the condition($\lozenge$). Let a topological group $G$ acts on a metric space $X$. We say that the topological group $G$ acts on $X$ *by uniform isomorphims* if for each $g\in G$, the map $X\ni x\mapsto gx\in X$ is uniform continuous. The action is said to be *uniformly equicontinuous* if for any $\varepsilon > 0$ there exists $\delta>0$ such that ${\mathop{\mathit{d}} \nolimits}_X(gx,gy)< \varepsilon$ for every $g\in G$ and $x,y\in X$ with ${\mathop{\mathit{d}} \nolimits}_X(x,y)< \delta$. Given a subset $S\subseteq G$ and $x\in X$, we put $S x:=\{gx \mid g\in S\}$. \[th2\]Assume that a Lévy group $G$ boundedly acts on a metric space $X$ having the property ($\lozenge$) by uniform isomorphisms. Then for any compact subset $K\subseteq G$ and any $\varepsilon >0$, there exists a point $x_{\varepsilon, K}\in X$ such that ${\mathop{\mathrm{diam}} \nolimits}(K x_{\varepsilon,K})\leq \varepsilon$. \[th3\]There are no non-trivial bounded uniformly equicontinuous actions of a Lévy group on a metric space having the property ($\lozenge$). From the definition of $G$, the group $G$ contains an increasing chain of compact subgroups $G_1\subseteq G_2 \subseteq \cdots \subseteq G_n \subseteq \cdots $ having an everywhere dense union in $G$ such that for some right-invariant compatible distance function ${\mathop{\mathit{d}} \nolimits}_G$ on $G$, the sequence $\{(X,{\mathop{\mathit{d}} \nolimits}_X,\mu_{G_n})\}_{n=1}^{\infty}$ forms a Lévy family. Let $x\in X$ be an arbitrary point. We first prove Proposition \[th2\]. Since $G$ boundedly acts on $X$ and ${\mathop{\mathit{d}} \nolimits}_G$ is right-invarinat, by vritue of Lemma \[l3.2\], for any $\varepsilon >0$ there exists $\delta >0$ such that ${\mathop{\mathit{d}} \nolimits}_X(gy,g'y)<\varepsilon/2$ for any $y\in X$ and $g,g'\in G$ with ${\mathop{\mathit{d}} \nolimits}_G(g,g')\leq \delta$. Take a subset $\{g_1,g_2, \cdots, g_N\}\subseteq G$ such that each $g\in K$ is within distance $\delta$ of the set $\{ g_1,g_2, \cdots, g_N\}$ and all $g_i$ are contained in $G_\ell$ for some large $\ell\in \mathbb{N}$. Since the orbit map $f_x:G\to X$ is uniformly continuous, by using Corollary \[c3.2\], the sequence $\{(X,{\mathop{\mathit{d}} \nolimits}_X, \nu_{G_n,x}) \}_{n=1}^{\infty}$ is a Lévy family. From the property ($\lozenge$) of the space $X$ the identity maps ${\mathop{\mathrm{id}} \nolimits}_n: (X,{\mathop{\mathit{d}} \nolimits}_X,\nu_{G_n,x})\to X$ concentrate, that is, $\lim_{n\to \infty}{\mathop{\mathrm{diam}} \nolimits}(\nu_{G_n,x},1-\kappa)=0$ for any $\kappa >0$. Hence there exist $\varepsilon_n >0$ and $x_n \in X_n$ such that $\lim_{n\to \infty}\varepsilon_n =0$ and $\lim_{n\to \infty}\nu_{G_n,x}(B_X(x_n,\varepsilon_n))=1$. Take $n_0\in \mathbb{N}$ such that $n_0\in \mathbb{N}$, $\nu_{G_{n_0},x}(B_X(x_{n_0},\varepsilon_{n_0}))>1/2$ and $\varepsilon_{n_0}\leq \rho^{(\{g_1, g_2, \cdots, g_N\},X)}(\varepsilon_{n_0})<\varepsilon /4$. The same method of the proof of (\[s3.1\]), we obtain $$\begin{aligned} {\mathop{\mathit{d}} \nolimits}_X(x_{n_0}, g_ix_{n_0})\leq \varepsilon_{n_0}+ \rho^{(\{g_1,g_2, \cdots, g_N\},X)}(\varepsilon_{n_0})< \varepsilon /2 \end{aligned}$$for any $g_i$. For any $g\in K$, choosing $g_i$ with ${\mathop{\mathit{d}} \nolimits}_G(g_i,g)<\delta$, we obtain $$\begin{aligned} {\mathop{\mathit{d}} \nolimits}_X(x_{n_0}, gx_{n_0})\leq {\mathop{\mathit{d}} \nolimits}_X(x_{n_0},g_ix_{n_0}) + {\mathop{\mathit{d}} \nolimits}_X(g_i x_{n_0}, g x_{n_0})\leq \frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon \end{aligned}$$by the definition of $\delta>0$. This completes the proof of Proposition \[th2\]. We next prove Proposition \[th3\]. Since $\lim_{\eta \to 0}\rho^{(G,X)}(\eta)=0$, by using Corollary \[c3.1\], we get $$\begin{aligned} {\mathop{\mathrm{diam}} \nolimits}(G_n x)\leq 2\lim_{\kappa \uparrow 1/2}{\mathop{\mathrm{diam}} \nolimits}(\nu_{G_n,x},1-\kappa)+ 2\rho^{(G,X)}\big(+\lim_{\kappa \uparrow 1/2}{\mathop{\mathrm{diam}} \nolimits}(\nu_{G_n ,x},1-\kappa)\big) \to 0 \end{aligned}$$as $n\to \infty$. Since $G_1 x\subseteq G_2 x\subseteq \cdots \subseteq G_n x \subseteq G_{n+1}x \subseteq \cdots$, we therefore obtain $G_n x= \{ x\}$ for any $n\in \mathbb{N}$. This completes the proof of Proposition \[th3\]. Note that every continuous action of a topological group on a compact metric space is bounded. Since a compact metric space has the property ($\lozenge$) and a Lévy group $G$ contains an increasing chain of compact subgroups $G_n$ having an everywhere dense union, Proposition \[th2\] includes the fixed point theorem ([@milgro Theorem 7.1]) by Gromov and Milman. 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--- abstract: 'We develop a method for predicting the yield of transiting planets from a photometric survey given the parameters of the survey (nights observed, bandpass, exposure time, telescope aperture, locations of the target fields, observational conditions, and detector characteristics), as well as the underlying planet properties (frequency, period and radius distributions). Using our updated understanding of transit surveys provided by the experiences of the survey teams, we account for those factors that have proven to have the greatest effect on the survey yields. Specifically, we include the effects of the surveys’ window functions, adopt revised estimates of the giant planet frequency, account for the number and distribution of main-sequence stars in the survey fields, and include the effects of Galactic structure and interstellar extinction. We approximate the detectability of a planetary transit using a signal-to-noise ratio (S/N) formulation. We argue that our choice of detection criterion is the most uncertain input to our predictions, and has the largest effect on the resulting planet yield. Thus drawing robust inferences about the frequency of planets from transit surveys will require that the survey teams impose and report objective, systematic, and quantifiable detection criteria. Nevertheless, with reasonable choices for the minimum S/N, we calculate yields that are generally lower, more accurate, and more realistic than previous predictions. As examples, we apply our method to the Trans-Atlantic Exoplanet Survey, the XO survey, and the [*Kepler*]{} mission. We discuss red noise and its possible effects on planetary detections. We conclude with estimates of the expected detection rates for future wide-angle synoptic surveys.' author: - 'Thomas G. Beatty and B. Scott Gaudi' title: PREDICTING THE YIELDS OF PHOTOMETRIC SURVEYS FOR TRANSITING EXTRASOLAR PLANETS --- Introduction ============ There are four ways by which extrasolar planets have been detected. The first method to unambiguously detect an extrasolar planet was pulsar timing, which relies on detecting periodic variations in the timing of the received radio signal that occur as the pulsar orbits about the system’s barycenter. The first system of three planets was found around PSR B1257+12 in 1992 [@wolszczan1992], followed by a single planet around PSR B1620-26 [@backer1993]. Although rare, the pulsar planets are some of the lowest mass extrasolar planets known: PSR B1257+12a is about twice the mass of the Moon. The second way to find extrasolar planets is through radial velocities (RV), which uses the Doppler shift of observed stellar spectra to look for periodic variations in the target star’s radial velocity. By estimating the mass of primary star, the observed radial velocity curve and velocity semi-amplitude can then be used to directly calculate the inclination-dependent mass ($M_p\sin i$) of the companion object. To date, RV surveys have detected more than 230 planets around other stars, making it the most successful method of extrasolar planet detection. While the large number of detected systems having an unseen companion with a mass on the order of $1 M_{Jup} \sin i$ statistically ensures that the majority of these are planetary bodies, the RV surveys are unable to provide more information than the minimum masses, periods, eccentricities, and the semi-major axes of the planets. RV surveys also have a limited ability to detect planets much smaller than a few Earth masses. The state of the art in RV surveys is the High Accuracy Radial Velocity Planet Searcher (HARPS) spectrometer at the La Silla Observatory in Chile, which is capable of radial velocity measurements with precisions better than $1 \ \mathrm{ms^{-1}}$ for extended periods of time [@lovis2006]. HARPS is therefore able to detect planets with masses on the order of $3$ to $4 M_{\oplus}$ in relatively short period orbits. Unfortunately, planets closer to an Earth mass will be increasingly difficult to detect since intrinsic stellar variability, in the form of acoustic oscillation modes and granulations on the photosphere, makes more precise spectroscopic radial velocity measurements harder to acquire. However, it may be possible to surmount this obstacle (as in the case of @lovis2006) through the selection of stars with “quiet” photospheres and long integration times which serve to average out the stellar variability. Gravitational microlensing is another technique for detecting extrasolar planets. Microlensing of a star occurs when a a star passes near the line of sight of the observer to another background star. The gravity of the foreground star acts as a lens on the light emitted by the background star, which causes the star in the background to become momentarily brighter as more light is directed towards the observer. Planetary companions to the lens star can further magnify the background star, and create short-term perturbations to the microlensing light curve. To date, six planets have been detected using microlensing [@bond2004; @udalski2005; @beaulieu2006; @gould2006a; @gaudi2008]. Unfortunately, the one-shot nature of microlensing observations means that information about systems discovered this way is generally sparser than that available for RV systems. Therefore, microlensing is most useful in determining the general statistical properties of extrasolar planets, such as their frequency and distribution, and not the detailed properties of the planetary systems. Planetary transits are a fourth method by which extrasolar planets have been discovered, and the one that provides the most complete set of information about the planetary system. Only planets with very specific orbital characteristics have a transit visible from Earth, because the orbital plane has to be aligned to within a few degrees of the line of sight. Therefore transiting planets are rare. Nevertheless, a transiting extrasolar planet offers the opportunity to determine the mass of the planet (when combined with RV measurements) since the inclination is now measurable, as well as the planetary radius, the density, the composition of the planetary atmosphere, the thermal emission from the planet, and many other properties (see [@charbonneau2007] for a review). Additionally, and unlike RV surveys, transiting planets should be readily detectable down to $1 R_\oplus$ and beyond, even for relatively long periods. Having accurate predictions of the number of detectable transiting planets is immediately important for the evaluation and design of current and future transit surveys. For the current surveys, predictions allow the operators to judge how efficient are their data-reduction and transit detection algorithms. Future surveys can use the general prediction method that we describe here to optimize their observing set-ups and strategies. More generally, such predictions allow us to test different statistical models of extrasolar planet distributions. Specifically, as more transiting planets are discovered and characterized, predictions relying on incorrect statistics of extrasolar planet properties will increasingly diverge from the observed set of distributions. Using straightforward estimates it appears that observing a planetary transit should not be too difficult, presuming that one observes a sufficient number of stars with the requisite precision during a given photometric survey. Specifically, if we assume that the probability of a short-period giant planet (as an example) transiting the disk of its parent star is 10%, and take the results of RV surveys which indicate the frequency of such planets is about 1% [@cumming08], together with the assumption that typical transit depths are also about 1%, the number of detections should be $\approx 10^{-3}N_{\leq 1\%}$, where $N_{\leq 1\%}$ is the number of surveyed stars with a photometric precision better than 1%. Unfortunately, this simple and appealing calculation fails. Using this estimate, we would expect that the TrES survey, which has examined approximately 35,000 stars with better than 1% precision, to have discovered 35 transiting short period planets. But, at the date of this writing, they have found four. Indeed, overall only 51 transiting planets have been found at this time by photometric surveys specifically designed to find planets around bright stars[^1]. This is almost one hundred times less than what was originally predicted by somewhat more sophisticated estimates [@horne2003]. Clearly then, there is something amiss with this method of estimating transiting planet detections. Several other authors have developed more complex models to predict the expected yields of transit surveys. [@pepper2003] examined the potential of all-sky surveys, which was expanded upon and generalized for photometric searches in clusters [@pepper2005]. [@gould2006b] and [@fressin2007] tested whether the OGLE planet detections are statistically consistent with radial velocity planet distributions. [@brown2003] was the first to make published estimates of the rate of false positives in transit surveys, and [@gillon2005] model transit detections to estimate and compare the potential of several ground- and space-based surveys. As has been recognized by these and other authors, there are four primary reasons why the simple way outlined above of estimating surveys yields fails. First, the frequency of planets in close orbits about their parent stars (the planets most likely to show transits) is likely lower than RV surveys would indicate. Recent examinations of the results from the OGLE-III field by [@gould2006b] and [@fressin2007] indicate that the frequency of short-period Jovian-worlds is on the order of $0.45\%$, not $1.2\%$ as is often assumed by extrapolating from RV surveys [@marcy2005a]. [@gould2006b] point out that most spectroscopic planet searches are usually magnitude limited, which biases the surveys toward more metal-rich stars, which are brighter at fixed color. These high metallicity stars are expected to have more planets than solar-metallicity stars [@santos2004; @fischer2005]. Second, a substantial fraction of the stars within a survey field that show better than 1% photometric precision are either giants or early main-sequence stars that are too large to enable detectable transit dips from a Jupiter-sized planet [@gould2003; @brown2003]. Third, robust transit detections usually require more than one transit in the data. This fact, coupled with the small fraction of the orbit a planet actually spends in transit, and the typical observing losses at single-site locations due to factors such as weather, create low window probabilities for the typical transit survey in the majority of orbital period ranges of interest [@vonbraun2007]. Lastly, requiring better than 1% photometric precision in the data is not a sufficient condition for the successful detection of transits: identifiable transits need to surpass some kind of a detection threshold, such as a signal-to-noise ratio (S/N) threshold. The S/N of the transit signal depends on several factors in addition to the photometric precision of the data, such as the depth of the transit and the number of data points taken during the transit event. Additionally, ground-based photometry typically exhibits substantial auto-correlation in the time series data points, on the timescales of the transits themselves. This additional red noise, which can come from a number of environmental and instrumental sources, substantially reduces the statistical power of the data [@pont2006]. In this paper we describe a method to statistically calculate the number of planets that a given transiting planet survey should be able to detect. We have done so in a way that maximizes the flexibility of our approach, so that it may be applied in as many circumstances as possible. Therefore, we have allowed for survey-specific quantities such as the observation bandpass to be defined arbitrarily, and have also allowed for our astrophysical assumptions to be easily altered. We account for factors such as the low frequency of Jovian planets, the increased difficulty of detecting transits at high impact parameters, as well as variations in stellar density due to Galactic structure and extinction due to interstellar dust. We also use the present-day mass function to realistically account for the number of main-sequence stars that will be in a given field of view. We do not treat, or simulate, the distribution and effects of giant stars. We approximate the detectability of a planetary transit using a S/N formulation. This differs from the previously published work in the field by virtue of the generalized nature of our approach. Other authors, such as [@pepper2003], [@gillon2005], [@gould2006b], and [@fressin2007] have restricted themselves to modeling specific transit surveys or types of transit surveys. Our approach, which allows for variations such as arbitrary telescope parameters, observing bandpasses, cadences, and fields of view, is not restricted to any one type of transit survey. Indeed, we offer examples of how our method may be applied to the four most frequently used modes for transit searches. Furthermore, we have conducted an extensive literature review of basic astrophysical relations, such as the Present Day Mass-Function (PDMF), Galactic structure, and the main sequence mass-luminosity relation, and included these in our predictions. We do not use more complicated (but presumably more accurate) Galactic models like the Besancon model [@robin1986] because these models are less flexible and considerably more time consuming to utilize. We first describe the mathematical formalism with which we have chosen to address this problem, and then move on to discuss our specific assumptions. As examples, we then offer our predictions for several different types of photometric surveys: point-and-stare, drift-scanning, space-based, and all-sky. For each, we choose a representative survey for that particular mode, and compare our predictions to actual survey results, or to predictions elsewhere in the literature. We also examine the possible effects of red noise on our predictions, and conclude by discussing what our predictions imply for the general state of photometric transiting planet surveys. General Formalism ================= In the most general sense, we can describe the average number of planets that a transit survey should detect as the probability of detecting a transit multiplied by the local stellar mass function, integrated over mass, distance, and the size of the observed field (described by the Galactic coordinates $(l,b)$: [ $$\begin{aligned} \label{eq:10} \frac{d^6 N_{det}}{dR_p\ dp\ dM\ dr\ dl\ db} &=& \rho_*(r,l,b)\ r^2 \cos b\ \frac{dn}{dM}\ \frac{df(R_p,p)}{dR_p\ dp} \nonumber \\ &&\times\ P_{det}(M,r,R_p,p), \end{aligned}$$ ]{} where $P_{det}(M,r,R_p,p)$ is the probability that a given star of mass $M$ and distance $r$ orbited by a planet with radius $R_p$ and period $p$ will present a detectable transit to the observing set-up. $\frac{df(R_p,p)}{dR_p\ dp}$ is the probability that a star will possess a planet of radius $R_p$ and period $p$. $dn/dM$ is the present day mass function in the local solar neighborhood, and $\rho_*$ is the local stellar density for the three-dimensional position defined by ($r,l,b$). We use $r^2 \cos b$ instead of the usual volume element for spherical coordinates, $r^2 \sin \phi$, because $b$ is defined opposite to $\phi$: $b=90^\circ$ occurs at the pole. First, we will integrate over both planetary radius and orbital period [ $$\begin{aligned} \label{eq:15} \frac{d^4 N_{det}}{dM\ dr\ dl\ db} &=& \rho_*(r,l,b)\ r^2 \cos b\ \frac{dn}{dM}\nonumber \\ &\times&\ \int_{R_{p,min}}^{R_{p,max}} \int_{p_{min}}^{p_{max}} \frac{df(R_p,p)}{dR_p\ dp}\nonumber \\ &\times&\ P_{det}(M,r,R_p,p) \,dR_p \,dp.\end{aligned}$$ ]{} To do this, we must first specify the detection probability $P_{det}$, as well as the distribution of planet radius and periods $df(R_p,p)/dR_p d_p$, which we treat in Section 2.2. The Detection Probability $P_{det}$ ----------------------------------- The probability that a planet orbiting a given star shows a detectable transit may be broken up into two separate probabilities: the chance that a planet with radius $R_p$ and period $p$ around a star of mass $M$ at a distance $r$ will transit with a sufficient signal-to-noise ratio to be detected, and the window probability that a transit will be visible for the particular observing set-up: $$\label{eq:22} P_{det}(M, r, R_p, p) = P_{S/N}(M, r, R_p, p)\ P_{win}(p)$$ ### $P_{S/N}$ To calculate the probability that a planetary system will show detectable transits, we first select the appropriate statistical test to determine whether or not our notional data shows a discernible transit. In the world of transiting planet searches, many different methods are used either individually or together to test the statistical significance of a possible transit and alert the researcher to its existence. For example, the Hungarian-made Automated Telescope Network (HATNet) uses a combination of the Dip Significance Parameter (DSP) and a Box-Least Squares (BLS) power spectrum (amongst others) to test its photometric data. In general though, whatever test one uses, it is generally the S/N of a transit shape in the observed data is what determines whether or not that transit will be identified as an event. As the square of the signal-to-noise can be approximated by the $\chi^2$ test, we therefore chose to use the $\chi^2$ value to determine if a given transit will be detectable for a set of given observation parameters. For a specific transit, the $\chi^2$ statistic, assuming uncorrelated white noise,[^2] is given by $$\label{eq:30} \chi^2 = N_{tr} \left(\frac{\delta}{\sigma}\right)^2.$$ Where $N_{tr}$ is the number of data points observed in a transit of fractional depth $\delta$ and with fractional standard deviation $\sigma$, assuming a boxcar transit shape with no ingress or egress. To be detected, the transit must have a $\chi^2$ higher than a certain minimum $\chi_{min}^2$ $$\label{eq:40} \chi^2 \geq \chi_{min}^2.$$ The fractional depth of the transit is given by the square of the planet-to-star radius ratio $$\label{eq:50} \delta = \left(\frac{R_p}{R_*}\right)^2.$$ While this formulation ignores the contributions of stellar limb-darkening to the depth of the transit, [@gould2006b] demonstrate that, to a first order approximation, transit detection rates are not affected by limb-darkening. Briefly, they point out that the effects of limb-darkening on the transit lightcurve will be offset by the effects of the planet’s ingress and egress from the stellar disk. Therefore, we have not included stellar limb-darkening terms in our calculations. To determine $\sigma$, we assume Poisson statistics, such that for a given exposure with an observed number of $N_S$ source photons from the target star and $N_B$ background photons, $$\label{eq:60} \sigma = \sqrt{\frac{N_S + N_B}{N_S^2} + \sigma_{scint}^2},$$ where $\sigma_{scint}$ accounts for scintillation as per [@young1967], and $N_S$ is defined as $$\label{eq:70} N_S = e_{\lambda} F_{S,\lambda} t_{exp} A.$$ Here $e_{\lambda}$ is the efficiency of the overall observing set-up, and can take values from 0 to 1. It accounts for photon losses in places such as the filter, the mirrors of the telescope, the CCDs, and passage through the atmosphere. $F_{S,\lambda}$ is the source photon flux in the observed bandpass in units of $\gamma\ m^{-2}\ s^{-1}$. $t_{exp}$ is the exposure time of each observation, and $A$ is simply the light collecting area of the telescope. $F_{S,\lambda}$, the observed photon flux from the source in a certain bandpass, can be defined as a function of the photon luminosity of the source, the distance to the source, and the effects of extinction due to interstellar dust: $$\label{eq:80} F_{S,\lambda} = \frac{\Phi_{\lambda}(M)}{4\pi r^2} 10^{-A_\lambda / 2.5},$$ where $10^{-A_\lambda / 2.5}$ accounts for extinction, and will be treated later. It has the effect of decreasing the number of observed photons from more distant stars, particularly in the bluer bands. $\Phi_{\lambda}(M)$, the photon luminosity of the source star, we set as a function of the stellar mass $M$ and the observational bandpass $\lambda$. We derive this quantity in Appendix A, and only note here that we approximate the source stars as blackbodies. While this is a simplification, we show in Appendix A that it is sufficient for our purposes. Using $\Phi_{\lambda}(M)$ in equation (\[eq:80\]) yields the observed photon flux at a distance $r$ from that same star. We thus now have an expression for $N_S$ as a function of the stellar mass, distance, and bandpass choice, that we can use in our calculation of $\sigma$. We now only need $N_B$, the number of background photons observed. Similar to equation (\[eq:70\]), $N_B$ is defined as $$\label{eq:140} N_B = S_{sky,\lambda} \Omega t_{exp} A,$$ where $S_{sky,\lambda}$ is the photon surface brightness of the sky at the particular wavelength $\lambda$ (or, in our case, for a specific band), and $\Omega$ is the effective area of the seeing disk. Assuming that the point-spread function (PSF) is a Gaussian with a full-width half-max (FWHM) of $\Theta_{FWHM}$ arcseconds, $\Omega$ is defined by $$\label{eq:150} \Omega = \frac{\pi}{\ln 4} \Theta_{FWHM}^2.$$ We can now place both of the expressions for $N_S$ and $N_B$ back into equation (\[eq:60\]) to arrive at a determination of $\sigma$, the photometric uncertainty in the data points. In certain cases, $N_S$ and $N_B$ will be so great that the CCD detectors of the telescope will be saturated by the number of photons received from a star during the exposure time. In this case, further measurements become useless. Mathematically, we may describe saturation as occurring when the number of photons collected by a single CCD pixel during an exposure exceeds the Full-Well Depth (FWD) of that pixel: $$\label{eq:75} N_{pixel} \geq N_{FWD}.$$ $N_{pixel}$ is related to $N_S$ and $N_B$ as [ $$\begin{aligned} \label{eq:76} N_{pixel} &=& N_S \left(1-\mathrm{exp}\left[-\ln 2 \left(\frac{\Theta_{pix}}{\Theta_{FWHM}}\right)^2\right]\right)\nonumber \\ &&+ \frac{N_B \Theta_{pix}^2}{\Omega}, \end{aligned}$$ ]{} where $\Theta_{pix}$ is the angular size of an individual pixel. The only part of equation (\[eq:30\]) that remains to be defined is the number of data points observed while the system is in transit, $N_{tr}$. Assuming a circular orbit, we can say that the number of points seen in transit is directly proportional to the fraction of time that the system spends in eclipse, and given a total number of observed data points $N_{tot}$, $N_{tr}$ is thus $$\label{eq:160} N_{tr} = N_{tot} \left(\frac{R_*}{\pi a} \right)\sqrt{1-b^2},$$ where $b$ is the impact parameter of the transit in units of the stellar radius, $b\equiv(a/R_*)\cos i$, $i$ is the orbital inclination, and we have assumed that $R_*<<a$. We note that this does not hold for correlated noise, see [@aigrain2007]. Taking equation \[eq:160\] and putting it back into our original formulation for $\chi^2$ gives $$\label{eq:170} N_{tot} \left(\frac{R_*}{\pi a} \right)\sqrt{1-b^2} \left(\frac{\delta}{\sigma}\right)^2 \geq \chi_{min}^2.$$ We define a new variable, $\chi_{eq}^2$, as the value of equation (\[eq:170\]) when $b=0$, corresponding to the $\chi^2$ of an equatorial transit about a given star. This gives us a new form that makes it easier to determine when a system will have a transit geometry sufficient to be detected by the given observing set-up: $$\label{eq:180} \chi_{eq}^2 \sqrt{1-b^2} \geq \chi_{min}^2.$$ Rearranging, we find that we are able to detect transits up to a $b_{max}$ of $$\label{eq:190} b_{max}(M, r, p) = \sqrt{1-\left(\frac{\chi_{min}^2}{\chi_{eq}^2}\right)^2}.$$ Therefore, $b_{max}$ is a function of $M$, $r$, and $p$ as a result of the relations contained within the $\chi_{eq}^2$ term. Finally, we can now deduce an explicit expression for $P_{S/N}$. Given that detectable transits can only occur between $0 \leq b \leq b_{max}$, we may simply integrate the normalized probability density function of allowable impact parameters from 0 to $b_{max}$ to find a value for $P_{S/N}$. Given that $b\equiv(a/R_*) \cos i$, the maximum value of the impact parameter for a circular orbit (different from the maximum at which we will detect transits) is $b=a/R_*$. The normalization for the impact parameter’s probability density function is therefore $$\label{eq:200} 1 = \int_0^{\frac{a}{R_*}} P_b(b) \,db.$$ Since impact parameters are uniformly distributed[^3], this gives the solution $P_b = R_*/a$. It should be noted that this form of $P_b$ is only valid in the case of circular orbits. For the period ranges probed so far by transit surveys, this is not a very restrictive assumption, since only $\sim 10\%$ of the known transiting planets have eccentricities inconsistent with zero. At longer periods, however, the effects of eccentricity must be considered [@barnes07; @burke08]. If we now integrate this over the range of impact parameters that yield detectable transits, we find [ $$\begin{aligned} \label{eq:210} P_{S/N}(M,r,R_p,p)&=&\int_0^{b_{max}} \frac{R_*}{a} \,db \nonumber \\ &=&\frac{R_*}{a} \sqrt{1-\left(\frac{\chi_{min}^2}{\chi_{eq}^2}\right)^2}.\end{aligned}$$ ]{} ### $P_{win}$ The window probability (or window function) for a transit survey describes the probability that a transiting planet with a certain period will be visible to the survey, given the observational cadence of the survey (i.e. - X hours of observing per night for Y nights). While many numerical routines exist that can exactly calculate the window probability for a given observing set-up, for our purposes calculating an exact numerical answer is generally too time intensive to be practical. -0.0in -0.0in We therefore developed an analytical framework that allows for the approximate calculation of a window probability. This has the virtue of requiring much less computing time. This is gained at the cost of a small loss of accuracy. We give a full description of our analytic window probability in Appendix B. Briefly, we assume randomly sampled observations, which leads to simple expressions for the average window probability. This implicitly ignores the effects of aliasing at integer and half-integer day periods. Thus we are not able to address issues related to non-uniform or biased *a posteriori* distributions of planet parameters resulting from the effect of aliasing (e.g. [@pont2006]). Nevertheless, the analytic solution does follow the “outline” of the exact calculations, and closely tracks the moving average value of the exact calculations sufficiently accurately for our purposes, as is shown in Figure 1. The Frequency of Planets $\frac{df(R_p,p)}{dR_p\ dp}$ ----------------------------------------------------- $df(R_p,p)/(dR_p\ dp)$ denotes the probability that a given star has a planet with radius $R_p$ and orbital period $p$. In the example surveys that we consider later, we chose to focus on the detection of short period Very Hot- and Hot-Jupiters (VHJs and HJs) with orbital periods of 1 to 5 days, as this is the regime being probed by these surveys. The exception of this focus is the Kepler survey, where we also examine the detectability of transiting planets over a range of radii and orbital periods. In both cases, we consider a fixed planet radius, so that $$\label{eq:230} \frac{df(R_p,p)}{dR_p\ dp} = k(p) f(p)\delta(R_p - R_p^\prime),$$ where $k(p)$ is the correct normalization for a given period range, $f(p)$ is the distribution within this range, and $R_p^\prime$ is the planet radius. For VHJs and HJs we use $R_p^\prime = 1.1 R_{Jup}$, the average radius of the transiting planets found to date[^4]. In general, we have left the exact distribution of $f(p)$ to be assigned as is deemed appropriate. For the VHJ and HJ period distributions described above, we adopt a distribution that is uniform in period, $f(p)=1$. The frequency of Hot Jupiters ($P\le 5~{\rm days}$, $M_p=0.1-10\ M_{Jup}$) is commonly taken to be $\sim 1\%$ from the results of the RV planet surveys (e.g., @marcy2005a). However, adopting frequencies from RV surveys can lead to biased estimates of the expected yields from transit surveys, because RV surveys constrain the frequency of planets as a function of mass, rather than radius. Furthermore, as argued by @gould2006b, most RV surveys are more metallicity biased than transit surveys, and therefore the planet frequency found by RV surveys will be correspondingly higher, since planet occurrence is an increasing function of metallicity [@santos2004; @fischer2005]. RV surveys are metallicity biased because the target samples for these surveys are magnitude-limited. Since metal rich stars are brighter at fixed color, they are overrepresented in magnitude-limited samples with respect to a volume-limited sample [@fischer2005]. Indeed, [@marcy2005b] note that because they use a magnitude-limited target list, their target stars have a definite metallicity bias. Transit surveys, on the other hand, are typically signal-to-noise ratio limited. At fixed color, metal-rich stars are both brighter and bigger. These two effects on the signal-to-noise ratio roughly cancel, and as a result transit surveys are very weakly biased with respect to metallicity (see Section 8.1 of @gould2006b for a quantitative discussion.) Indeed, there is evidence for the effects of this metallicity bias in the comparison of various surveys. From the California and Carnegie Keck magnitude-limited and metallicity-biased [@fischer2005; @marcy2005b] survey sample, @cumming08 find a frequency of $\sim 1.3\%$ for planets with $P\le 5~{\rm days}$ and $M\ge 0.2 M_{Jup}$. On the other hand, for the volume-limited [@udry2000b] CORALIE sample, @udry2007 report a frequency of $\sim 0.8\%$ for $M\ge 0.2 M_{Jup}$ and $P\la 11.5~{\rm days}$ $(a< 0.1~{\rm AU})$. Assuming that $\sim 75\%$ of these planets have $P\le 5~{\rm days}$, this corresponds to a frequency of 0.6% for $P\le 5~{\rm days}$ and $M\ge 0.2 M_{Jup}$, a factor of $\sim 2$ lower than that found for the Keck RV survey. @gould2006b examined the OGLE-III transit survey, and found a frequency of $\sim 0.5\%$ for $P\le 5~{\rm days}$, which was subsequently confirmed and strengthened by [@fressin2007]. Thus, as would be expected based on the arguments above, the frequencies inferred from transit surveys and volume-limited RV surveys are similar (and entirely consistent considering Poisson uncertainties and uncertainties in the mass-radius relation of giant planets), whereas they are substantially lower than the frequencies inferred from magnitude-limited (and metallicity-biased) RV surveys. For the VHJ and HJ predictions that we later make using TrES, XO, and Kepler, we will adopt the normalizations from @gould2006b of $k(p) = 1/690$ for VHJs with orbital periods uniformly distributed between 1-3 days, and $k(p) = 1/310$ for HJs uniformly distributed between 3-5 day periods. By using statistics from OGLE-III, instead of the RV statistics, we are avoiding the difficulties discussed above, and furthermore any unaccounted-for biases inherent in photometric surveys will be preserved, and therefore any systematic effects unknown to us will still be accurately captured in the final predictions. In other words, it is an “apples-to-apples” comparison, instead of “apples-to-oranges” using RV data. The Present Day Mass Function ----------------------------- The next step is to integrate equation (\[eq:10\]) over mass: $$\label{eq:240} \frac{d^3N_{det}}{dr\ dl\ db} = \rho_*(r,l,b) r^2 \cos b \int_{M_{min}}^{M_{max}} P_{det}(M,r) \frac{dn}{dM} \,dM$$ Having previously derived an expression for $P_{det}$, we need only determine the appropriate formulation for the PDMF, $\frac{dn}{dM}$. Using the results of [@reid2002], who used the Hipparcos data sets along with their own data from the Palomar/Michigan State University survey to create a volume-limited survey out to 25 pc, we adopt $$\begin{aligned} \label{eq:250} \frac{dn}{dM} &=& k_{norm} \left(\frac{M_*}{M_\odot}\right)^{-1.35}\ \mbox{ for } 0.1 \leq M_*/M_\odot \leq 1,\\ \frac{dn}{dM} &=& k_{norm} \left(\frac{M_*}{M_\odot}\right)^{-5.2}\ \ \mbox{ for } 1 < M_*/M_\odot, \nonumber\end{aligned}$$ where $k_{norm}$ is the appropriate normalization constant. A fuller discussion for our decision to use this particular form for the PDMF is in Appendix C. Adopting a stellar mass density in the Solar Neighborhood of $0.032 M_\odot \ \mathrm{pc}^{-3}$ [@reid2002], we find a normalization of $k_{norm} = 0.02124\ \mathrm{pc}^{-3}$. Galactic Structure ------------------ While our normalization of the mass function properly describes the density of stars in the Solar Neighborhood, beyond about 100 pc we must account for variations in stellar density due to Galactic structure. Additionally, as mentioned previously, we must also take into consideration the effects of extinction due to interstellar dust. This is the final stage of integration that needs to be done on equation (\[eq:10\]): $$\label{eq:260} N_{det} = \int_{0}^{r_{max}} \int_{l_{min}}^{l_{max}} \int_{b_{min}}^{b_{max}} P_{det}(r) \rho_*(r,l,b) r^2 \cos b \,dr \,dl \,db.$$ ### Galactic Density Model We adopt the Bahcall & Soneira model for the Galactic thin disk [@bahcall1980]. The model treats stellar density in the thin disk as a double exponential function involving both the distance from the Galactic center and height above the Galactic plane. Specifically, the mass density relative to the local density is given by $$\label{eq:280} \rho_*(r,l,b) = exp\left[-\frac{d - d_{gc}}{h_{d,*}} - \frac{|z|}{h_{z,*}}\right],$$ where $d_{gc}=8$ kpc is the distance from the Galactic Center. $h_{d,*}$ is the scale length, and $h_{z,*}$ is the scale height. We adopt a value of $h_{d,*} = 2.5\ \mathrm{kpc}$ for the scale length of the disk. For the vertical scale height, we use the following relation, which is a slightly modified form of that originally proposed by [@bahcall1980]: $$\begin{aligned} \label{eq:285} h_{z,*} &=& 90\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{ for } M_V \leq 2,\\ &=& 90 + 200 \left(\frac{M_V-2}{3}\right)\ \mbox{ for } 2 < M_V < 5, \nonumber \\ &=& 290\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{ for } 5 \leq M_V. \nonumber\end{aligned}$$ We calculate the absolute V-band magnitudes of stars of a given mass using the blackbody assumptions, and the mass-radius and mass-luminosity relations described in Appendix A. We defer the complete discussion of our reasoning behind our choices for $h_{d,*}$ and $h_{z,*}$ to Appendix D. The exact values for these parameters are not well constrained in the literature, and we show the effects of using different values in the Appendix. We note that within the range of values found in the literature, the final prediction results may differ by as much as 20%. We only treat the thin disk in our modeling because we decided that an expanded analysis that also looked at stars in the thick disk, halo, and bulge would not add a significant number of detections to our final result. Both the thick disk and the halo have much lower volume densities than the thin disk at distances typically surveyed by transit searches. In the same vein, the bulge is too far away. These populations are also typically older than the thin disk, and so have a fainter turn-off point, meaning that these stars are more difficult to detect. That being said, for deeper transit surveys, such as some of the all-sky surveys, these populations may play an interesting role. In that case, it would be fairly simple to add them into our model. We chose the Bahcall model instead of other more detailed models due to speed and portability considerations. While others, such as the Besancon models [@robin1986], take into account features such as density variations from the spiral structure of the Galaxy and the Galactic Bulge, they typically require over 30 minutes to simulate a given star field. Additionally, many are only accessible through webpage interfaces, and therefore difficult to include in a self-contained prediction algorithm. We do note that comparisons between our Galactic model and the Besancon model revealed star count differences of only 10% in the Kepler and TrES fields over a range of magnitude limits. ### Interstellar Extinction Modeling We also model the density of interstellar dust as a double exponential $$\label{eq:290} \rho_{dust}(r,l,b) = exp\left[-\frac{d - d_{gc}}{h_{d,dust}} - \frac{|z|}{h_{z,dust}}\right].$$ Which gives the dust density at a distance $r$ along a particular line of sight ($l,b$) relative to the density in the Solar Neighborhood (i.e. - $z=0$, $d=d_{gc}$). To calculate the extinction in a given bandpass for a star at a given distance, we then integrate equation (\[eq:290\]) over $r$ to arrive at the effective path length through an equivalent Solar Neighborhood dust density, $$\label{eq:310} \ell_{dust} = \int_0^r \rho_{dust}(r,l,b) \, dr.$$ The V-band extinction in magnitudes, $A_V$, is given by this path length multiplied by the appropriate normalization constant: $A_V = k_{dust,v} \ell_{dust}$. We adopt $k_{dust,v} = 1$ mag kpc$^{-1}$ [@struve1847]. We determine the extinction in other bands by using the extinction ratios ($A_\lambda/A_V$) from [@cox2000], assuming that $R_V = 3.1$. Assumptions =========== It is important to keep in mind that in the preceding derivations, we made several assumptions about the nature of extrasolar planetary systems, as well as some simplifications to keep the math and computing time more manageable. 1\. [*Detectability is Best Measured Through $\chi^2$:*]{} While S/N is a better metric than photometric precision to determine if a transit will be detectable, and the $\chi^2$ parameter is a good proxy for S/N, actual transit surveys use a wide variety of tests to detect transits. Many tests, such as the visual evaluation of the data by a human being, are difficult to simulate numerically. Thus, using just a $\chi^2$ test is a simplification, as human inspection may impose a much higher effective S/N threshold. As we show for typical surveys, the number of detections is a strong function of the S/N threshold. Thus a proper determination of the effective S/N threshold imposed by all cuts is necessary before any robust conclusions can be drawn about the rate of detections relative to expectations. This is a fundamental - and unavoidable - limitation of our study. 2\. [*No Stellar Limb-Darkening:*]{} We have neglected to include terms for stellar limb-darkening because, as stated above, to first-order its effects are canceled by the the effects of ingress and egress on the transit lightcurve [@gould2006b]. 3\. [*Stars are Blackbodies:*]{} We assume that the radiated energy of a star can be described as a blackbody. While in general this is a fair approximation (to within 0.5 mag, see Figures 11 and 12), it does not hold true for the bluer bandpasses we consider (U-band in particular) and for very low-mass stars. 4\. [*Simplified Galactic Structure:*]{} While we do treat variations in stellar density and interstellar dust densities in a general sense, our formalism is a simplified double exponential model that does not account for the spiral structure of the Galaxy or patchiness in the dust distribution. Therefore detailed, accurate predictions for an individual field may be better served by incorporating more sophisticated Galactic models (e.g. [@fressin2007]). 5\. [*No Binary Systems:*]{} While we only consider main sequence stars, we do not model the statistics of binary systems; in our formulation all stars are treated as single. Though this is clearly not the case, it was outside the scope of this project to go deeply into binary frequencies, the ability of planets to form in such systems, and whether or not such planets would be detectable. However, we can make a rough estimate of the expected magnitude of the effect of binaries. First, we assume that 1/3 of our stars have no binary companion [@duquennoy91]. We assume the remaining 2/3 of the stars are in binary systems. A typical wide-field survey has a PSF size of $\sim 20''$, which corresponds to a separation of $\sim 4000~{\rm AU}$, and a period of $\log (P/{\rm day})\sim 7.8$ for an equal-mass solar-mass binary. Adopting the period distribution for binary stars from @duquennoy91, we estimate that $\sim 90\%$ of the binary systems are unresolved. Of these unresolved binaries, we conservatively assume that those with separations $\la 5~{\rm AU}$, or $\log (P/{\rm day})\sim 3.5$, do not host hot Jupiters because the binary components are too close. Thus we estimate that $\sim 62\%$ of unresolved binaries can host hot Jupiters. To estimate the detectability of transiting planets in these, consider a binary where $F_1$, $R_1$, $M_1$ and $F_2$, $R_2$, $M_2$ are the flux, radius, and mass of the primary (“1”) and secondary (“2”), respectively. Additionally, define $\ell=F_2/F_1$, ${\cal R}=R_2/R_1$, and $q=M_2/M_1\le 1$. Then, for fixed planet radius and semimajor axis, assuming photon-limited uncorrelated uncertainties, the signal-to-noise ratios if the stars are resolved are $({\rm S/N})_1 \propto R_1^{-3/2} F_1^{1/2}$ and $({\rm S/N})_2 \propto R_2^{-3/2} F_2^{1/2}$, while if the stars are unresolved the signal-to-noise ratios for transits across each component are $({\rm S/N})_{B,1} \propto R_1^{-3/2} F_1^{1/2} (1+\ell)^{-1/2}$ and $({\rm S/N})_{B,2} \propto R_2^{-3/2} F_2^{1/2} (1+\ell)^{-1/2}$. Assuming a constant volume density of stars and no interstellar extinction, the number of detected planets scales as $({\rm S/N})_{\rm min}^{-3}$ [@gould2003b]. Thus the number of planet detections for a population of binaries in the unblended case can be written as $N_0 \propto ({\rm S/N})_1^3 + f(q) ({\rm S/N})_2^3$, where $f(q)$ is the mass ratio distribution. Similarly, for the blended case, this is $N_B \propto ({\rm S/N})_{B,1}^3 + f(q) ({\rm S/N})_{B,2}^3$. Using the scalings for the signal-to-noise ratios just derived, we evaluate the ratio of the number of detections in the blended case, relative to the number of detections one would expect if they were unblended, $$\frac{N_B}{N_0} = (1+\ell)^{-3/2} \frac{1+f(q)\ell^3{\cal R}^{-9/2}}{1+f(q)\ell^{3/2}{\cal R}^{-9/2}}. \label{eq:291}$$ We adopt $\ell = q^5$, which is roughly appropriate for the $R$-band (see Fig. 11), and ${\cal R} = q^{0.8}$ (Eq. \[eq:121\]). We consider two different distributions for $f(q)$. First, we adopt $f(q)$ from @duquennoy91, which is weighted toward low mass ratio systems. Second, we consider a distribution with a substantial population of ‘twins’ with $q=1$. Specifically, we choose $f(q)\propto 1 +3.5\exp[-0.5(q-1)^2/\sigma_q^2]$, with $\sigma_q=0.2$, which approximates the distribution found by @halbwachs03. When we then average Equation (\[eq:291\]) over $q$, weighting by $f(q)$, we find $\langle N_B/N_0\rangle \simeq 0.89$ for the @duquennoy91 mass ratio distribution, and $\langle N_B/N_0\rangle \simeq 0.68$ for the @halbwachs03 mass ratio distribution. Thus, we estimate that roughly 70-90% of planets in unresolved binaries will still be detectable. The basic point is that, because the mass-luminosity relation is so steep, only those binaries that have $q\sim 1$ result in a substantial reduction in the signal-to-noise ratio. Low mass ratio companions have little affect on the ability to detect transit signals. For the mass ratio distribution of @duquennoy91, these provide the bulk of the companion population, and so in this case the effects of binarity are suppressed even further. Finally, we can summarize these various components to provide a rough estimate of the fraction of planets that will be detected when we consider binary systems, relative to our fiducial assumption that all stars are single. We find $1/3 + (0.1+ 0.62 \langle N_B/N_0\rangle)\times (2/3) \simeq 0.68-0.77$. Thus we expect that our estimated yields are systematically high by at most $30\%$. This is generally of order or smaller than other sources of uncertainty in our predicted yields. We note that a corollary of our assertion that binaries have a small effect on the yields of transit surveys is that a substantial fraction of the planets that have been found in wide-field transit surveys should be in binary systems. Only two planets in confirmed common proper-motion binary systems have been reported [@burke2007; @bakos2007], however systematic surveys for stellar companions to the host stars of the known transiting planets have not been performed, and therefore the true incidence of binary systems amongst the sample of transiting planets is not known. Results ======= Using the preceding formalism, we simulated four different surveys modes used by transit searches: point-and-stare, drift-scanning, space-based, and all-sky surveys. For the point-and-stare and drift-scanning cases, we use the TrES and XO transit surveys as our examples, respectively. Both surveys have been successful in their hunt for planets, and so we are able to directly compare the results of our calculations with the actual number of detections. For VHJs and HJs, we simulate the detection of planets with $R_p=1.1R_{Jup}$, and with the period distribution from [@gould2006b] that we describe in Section 2.2. [c|c]{} Band & 550-700 nm\ $m_R$ Limit & $\approx 13$\ Observation Time & Varies\ Exposure Time & 90 seconds\ CCD Read Time & 25 seconds\ Telescope Diameter & 0.1 m\ Throughput & 0.6\ PSF (FWHM) & 25”\ $\theta_{pix}$ & 10”\ CCD Full-Well Depth & $1.2\times10^5\ e^-$\ Field of View & $36\ \mathrm{deg}^2$\ Galactic Longitude & Varies\ Galactic Latitude & Varies [c|cc]{} Field & Observation Time & Galactic Coordinates\ And0 & 765 hrs & \[126.11, -015.52\]\ Cyg1 & 609 hrs & \[084.49, +010.28\]\ Cas0 & 628 hrs & \[120.88, -013.47\]\ Per1 & 643 hrs & \[156.37, -014.04\]\ UMa0 & 497 hrs & \[168.87, +047.70\]\ Crb0 & 537 hrs & \[053.49, +048.92\]\ Lyr1 & 582 hrs & \[077.15, +017.86\]\ And1 & 435 hrs & \[109.03, -017.62\]\ And2 & 332 hrs & \[115.52, -016.21\]\ Tau0 & 406 hrs & \[169.83, -015.94\]\ UMa1 & 500 hrs & \[156.32, +054.01\]\ Her1 & 592 hrs & \[063.26, +026.42\]\ Lac0 & 686 hrs & \[097.79, -008.86\] We demonstrate our method for space-based surveys by simulating the upcoming Kepler mission, and comparing our predictions with other predictions for Kepler in the literature. Similarly for the all-sky mode, we examine the future planned observing programs of LSST, SDSS-II, and Pan-STARRS and make predictions for each. For each mode, we have tried to approximate the effect of red noise on detectability by increasing our S/N cut [@pont2006]. To show the effects of a more accurate red noise calculation, in Section 4.5 we evaluate its effects on the TrES survey. Point-and-stare - TrES ---------------------- Point-and-stare observing is the traditional observing mode of tracking one field of sky continuously for all, or part, of the night. There are several operational photometric transit surveys that use this particular method of observing, including HATnet [@bakos2004], SuperWasp [@pollacco2006], and KELT [@pepper2007], but we chose to model is the Trans-Atlantic Exoplanet Survey (TrES). TrES is composed of a network of three telescopes: STARE on the Canary Islands, PSST at Lowell Observatory in Arizona, and Sleuth on Mt. Palomar, California. All are small, 10 cm aperture, wide-field ($6^\circ$), CCD cameras that operate in unison to observe the same field of sky nearly continuously over one to two month periods. The survey specifics are described in more detail by [@dunham2004] and by [@brown1999]. The relevant quantities for our modeling are listed in Table 1. -0.0in We fit to the RMS plots posted on the Sleuth observing website[^5] to determine the background sky brightness in the TrES bandpass (550-700nm), using the TrES team’s stated photon throughput of 0.6, and we estimate that the TrES bandpass has a sky background of $19.6\ \mathrm{mag\ arcsec}^{-2}$. [c|ccccccccccccccc|c]{} & And0 & Cyg1 & Cas0 & Per1 & UMa0 & CrB0 & Lyr1 & And1 & And2 & Tau0 & UMa1 & Her1 & Lac0 & Total\ VHJs & 0.38 & 0.37 & 0.34 & 0.34 & 0.18 & 0.21 & 0.32 & 0.24 & 0.19 & 0.22 & 0.17 & 0.21 & 0.41 & 3.59\ HJs & 0.34 & 0.30 & 0.29 & 0.28 & 0.16 & 0.18 & 0.27 & 0.18 & 0.09 & 0.15 & 0.16 & 0.19 & 0.34 & 2.91\ Both & 0.72 & 0.67 & 0.63 & 0.62 & 0.34 & 0.39 & 0.59 & 0.42 & 0.28 & 0.37 & 0.33 & 0.40 & 0.75 & 6.50 In our simulations, we treat the three TrES telescopes as one single instrument. This is clearly an oversimplification. While these telescopes are quite similar, there are differences between the data quality and quantity obtained from each site. For example, Sleuth must contend with a higher sky background due to its proximity to urban areas, whereas the fraction of clear nights for PSST is typically lower than for the other two sites. However, accounting for these differences is difficult, and furthermore their effect on our predicted detections is small compared to the other uncertainties in our model. We therefore ignore these effects. In dealing with the window probability of the TrES network, we were greatly aided by the plots posted on the Sleuth website which showed the window probability for each field, as well as the total network observing time each field had received. We simulated thirteen different TrES fields using this methodology. The network observation times and the location of each field are shown in Table 2. The data for this table was collected off from the Sleuth observing website. The expected detections for each of the fields (and the total number of detections over all simulated fields) are shown in Table 3. The effects of our Galactic structure model can clearly be seen in distribution of expected detections. Both the UMa0 and Crb0 fields are substantially above the plane of the Galaxy ($b \approx 48^\circ$), and thus show a much lower detection rate than those fields directed towards denser regions of the sky. Figure 2 shows the distributions of detections over various parameters in the Lyr1 field. For Figure 2, we show two distributions, one assumes white noise in the photometry, the other assumes 3 mmag of red noise. For the white noise estimates, we used a signal-to-noise limit of S/N $\geq 30$ and a magnitude limit of $m_{R}\leq13$. Lyr1 is the field in which TrES-2 was discovered, and is adjacent to Lyr0, which contains TrES-1. We therefore have plotted the location of these two stars on the distributions shown in Figure 2. Both stars reside close to the peaks of our parent star mass and radius distributions. The third and fourth planets discovered by TrES, TrES-3 and TrES-4, are not plotted in Figure 2, as they reside in the Hercules fields, which are at a higher Galactic latitude than Lyra. Also of note is that with our assumptions half of the expected detections the F-dwarf range. Among the false positives encountered by the TrES team, the most frequent are eclipsing F/M-dwarf binaries. Figure 2 offers a partial explanation for this, showing that roughly half of TrES’ planet-like signals come from F-dwarfs. [|c|ccc|]{} & VHJ & HJ & Both\ S/N$\geq20$ & 6.06 & 5.72 & 11.78\ S/N$\geq25$ & 4.68 & 4.05 & 8.73\ S/N$\geq30$ & 3.59 & 2.91 & 6.50\ S/N$\geq35$ & 2.77 & 2.13 & 4.90\ S/N$\geq40$ & 2.16 & 1.59 & 3.75 [|c|ccc|]{} & VHJ & HJ & Both\ $m_R\leq12.0$ & 2.13 & 1.95 & 4.08\ $m_R\leq12.5$ & 2.95 & 2.58 & 5.53\ $m_R\leq13.0$ & 3.59 & 2.91 & 6.50\ $m_R\leq13.5$ & 3.86 & 3.01 & 6.87\ $m_R\leq14.0$ & 3.94 & 3.03 & 6.97\ $m_R\leq14.5$ & 3.96 & 3.05 & 7.01 [|c|ccc|]{} & VHJ & HJ & Both\ $R_p=0.9R_{Jup}$ & 1.66 & 1.17 & 2.83\ $R_p=1.0R_{Jup}$ & 2.52 & 1.90 & 4.42\ $R_p=1.1R_{Jup}$ & 3.59 & 2.91 & 6.50\ $R_p=1.2R_{Jup}$ & 4.53 & 3.88 & 8.41\ $R_p=1.3R_{Jup}$ & 5.52 & 5.04 & 10.56 Our predicted number of detections for the selected TrES fields of $N_{det} = 6.50$ covers only half of the fields observed by TrES, so we may roughly extrapolate this result to $13.00$ detections over all of the fields. This is roughly three times the four planets that TrES has actually detected, though we hesitate to draw strong conclusions from this comparison. While we have the luxury in our predictions of imposing draconian signal-to-noise ratio and magnitude cuts in the data, in real life the TrES detections are identified in data using more flexible selection effects. A better comparison between our predictions and the actual results of TrES (or any survey) would therefore require a better understanding of the selection effects in the observed data set. [c|c]{} Band & 400-700 nm\ $m_V$ Limit & $12$\ Observation Time & $\approx 400$ hrs\ Exposure Time & 54 seconds\ CCD Read Time & n/a\ Telescope Diameter & 0.11 m (two)\ Throughput & 0.16\ PSF (FWHM) & 45”\ $\theta_{pix}$ & 25.4”\ CCD Full-Well Depth & $1.5\times10^5\ e^-$\ Field Area & $7.2\times63\ \mathrm{deg}^2$\ Galactic Longitude & Varies\ Galactic Latitude & Varies To see how different selection criteria change our predictions for TrES, Tables 4-6 show the effects of changing the S/N limit, the magnitude limit, and the radius of the targeted planets on the detection numbers. A changing magnitude limit is particularly interesting, since all four of the TrES detections have been around stars with $m_R\leq12$. This might indicate that the effective magnitude limit of the TrES survey is brighter than what is described in the published literature. If this is the case, then using a limit of $m_R\leq12$ lowers the number of expected detections in our simulated fields to $N_{det} = 4.08$, and total detections to $8.16$. Drift-scanning - XO ------------------- -0.0in The XO survey is a wide-angle ground-based photometric survey that uses drift-scanning for its observations. The XO telescope is a pair of 11 cm cameras in place at the Haleakala Observatory in Hawaii. It monitors thousands of stars with $m_V\leq12$ in six fields spaced evenly around the sky in right ascension[^6]. The cameras scan within these fields, sweeping in declination from $0^\circ$ to $+63^\circ$ every ten minutes. Typically, this provides over 3000 observations per star in each of the XO fields. More on the specifics of the XO system can be found in [@mccullough2005]. The relevant parameters for our predictions are listed in Table 7. To determine the throughput of XO and the background sky magnitude in the XO band (400-700 nm), we fit to the RMS diagram shown in Figure 8 of [@mccullough2005]. With the addition of their stated calibration and scintillation error, we were able to exactly reproduce the relations shown in that figure using a background sky magnitude of 19.4 mag $\mathrm{arcsec}^{-1}$ and a system throughput of 0.16. [cccccc|c]{} 00 hrs & 04 hrs & 08 hrs & 12 hrs & 16 hrs & 20 hrs & Total\ 0.34 & 0.35 & 0.34 & 0.30 & 0.32 & 0.37 & 2.02 Additionally, we checked to ensure that the values we used for the observation time and the number of nights observed generated an analytic window probability similar to that shown in Figure 7 of [@mccullough2005]. Our ability to do this was greatly aided by the XO team’s description of how many nights they lost to weather and mechanical trouble. We simulated each of the six XO fields for our predictions. The results for each field using the base parameters of S/N$\geq30$, $m_V\leq12$ and $R_p=1.1R_{Jup}$ are shown in Table 8. The first planet discovered by the XO survey, XO-1b, [@mccullough2006] is located in in the field at 16 hrs, and so we have also plotted the distribution of detections within this field over a variety of parameters in Figure 3. XO-2b [@burke2007] and XO-3b [@johnskrull2007] are in noticeably different fields (08 hrs and 04 hrs, respectively), so were not included on the plot. [|c|ccc|]{} & VHJ & HJ & Both\ S/N$\geq20$ & 3.45 & 2.41 & 5.86\ S/N$\geq25$ & 2.20 & 1.27 & 3.47\ S/N$\geq30$ & 1.37 & 0.65 & 2.02\ S/N$\geq35$ & 0.83 & 0.33 & 1.16\ S/N$\geq40$ & 0.50 & 0.17 & 0.67 [|c|ccc|]{} & VHJ & HJ & Both\ $m_V\leq11.5$ & 1.09 & 0.55 & 1.64\ $m_V\leq12.0$ & 1.37 & 0.65 & 2.02\ $m_V\leq12.5$ & 1.50 & 0.69 & 2.19\ $m_V\leq13.0$ & 1.54 & 0.70 & 2.24\ $m_V\leq13.5$ & 1.55 & 0.71 & 2.46 [|c|ccc|]{} & VHJ & HJ & Both\ $R_p=0.9R_{Jup}$ & 0.29 & 0.09 & 0.38\ $R_p=1.0R_{Jup}$ & 0.69 & 0.25 & 0.94\ $R_p=1.1R_{Jup}$ & 1.37 & 0.65 & 2.02\ $R_p=1.2R_{Jup}$ & 2.07 & 1.18 & 3.25\ $R_p=1.3R_{Jup}$ & 2.93 & 1.92 & 4.85 Tables 9-11 show the effects of changing the S/N limit, the magnitude limit, and the radii of the targeted planets on XO’s predicted detection numbers. For our base case of $m_V\leq12$ and S/N$\geq30$, we expect that XO will detect 2.02 planets. If we consider the three actual detections from XO, it seems that, similar to TrES, the actual S/N and magnitude limit of the XO survey may differ from what is in the literature. For example, all three XO parent stars are at $m_V\leq11.5$ (XO-1, the dimmest, is at $m_V=11.3$). To see how this affected our calculated distributions, we also simulated the field at 16 hrs assuming S/N$\geq20$, $m_V\leq12$, and S/N$\geq20$, $m_V\leq11.5$. The results are included on Figure 3. In these cases, the total number of detections increases to 5.86 and 4.21, respectively. Space-based - Kepler -------------------- To make predictions for a space-based mission, we chose NASA’s Kepler mission, which is primarily designed to look for Earth-sized extrasolar planets. It consists of a single spacecraft that is currently scheduled to be launched into an Earth-trailing heliocentric orbit in early 2009. Onboard, Kepler will have 42 CCDs that will measure the light from 100,000 main-sequence stars within the telescope’s field of view. Ideally, Kepler will identify planets orbiting within the “habitable zone” of their parent stars which would be capable of possessing liquid water on their surfaces. [c|c]{} Band & 400-850 nm\ $A_{Kepler}$ & 0.86\ $m_V$ Limit & $\approx 14$\ Observation Time & 4 years\ Exposure Time & 3 seconds\ CCD Read Time & Negligible\ Telescope Diameter & 0.95 m\ Throughput & .3\ PSF (FWHM) & 10”\ $\theta_{pix}$ & 3.98”\ CCD Full-Well Depth & $1.2\times10^6\ e^-$\ Field of View & $106\ \mathrm{deg}^2$\ Galactic Longitude (new) & $76.32^\circ$\ Galactic Longitude (old) & $70^\circ$\ Galactic Latitude (new) & $13.5^\circ$\ Galactic Latitude (old) & $5^\circ$ Beyond looking for extrasolar Earths, the immense amount of photometry that Kepler is expected to produce over its four year operational lifetime will also allow for the detection of larger jovian worlds transiting other stars. Previous estimates of the number of VHJs and HJs expected to be found by Kepler have predicted the discovery of about 180 HJs (no VHJs had been detected at the time, and so they were not treated) [@jenkins2003]. Clearly, given that there are only 46 currently known transiting planets, this would be a significant increase in the number of known planets To simulate Kepler with our methodology, we used the values in Table 12 to simulate Kepler observations out to a distance of $8\ \mathrm{kpc}$, using a signal-to-noise ratio of SN$\geq7$. We gathered the information in Table 12 from both the Kepler mission website[^7] as well as published descriptions of the satellite [@borucki2004]. Three items are of note. First, our value for the photon detection efficiency of Kepler is based upon information from the Kepler website that a 12th magnitude G2 dwarf will show $7.8\times10^8\ \gamma\ \mathrm{hr}^{-1}$. Secondly, the value we use for the extinction ratio in the Kepler bandpass, $A_{Kepler}/A_V = 0.861$, is an average that we calculated by integrating the extinction curve over the Kepler bandpass (400-850nm) [@fitzpatrick1999]. Finally, the Kepler website and much of the literature states that Kepler will use a dim magnitude cutoff of $m_V\leq14$. In reality, there is no “hard” magnitude cutoff at which stars will be excluded from observations. Indeed, Kepler is expected to observe promising stars that are much dimmer than $m_V=14$ (D. W. Latham, private communication). Therefore, for our more explicit calculations of the expected number of HJ and VHJ detections for Kepler, we adjusted the magnitude limit such that the number of stars in the field agreed with the expected number of Kepler targets (100,000 in the case of the new field). For other, more general, simulations, we used several different magnitude limits to determine what effects these had on Kepler’s expected detections. While the location of the Kepler field is currently centered on Galactic coordinates $(l=76.32^\circ, b=13.5^\circ)$, up until 2004 it was instead situated closer to the Galactic plane at $(l=70^\circ, b=5^\circ)$. The switch in the position of the field to higher Galactic latitudes was done in order to avoid sorting through the larger number of giant stars that are found in fields directed along the plane of the Galaxy. Much of the previously published literature examining the Kepler mission have used the coordinates of the old field. For the purpose of comparison, we therefore simulated both the old and new Kepler fields. ### General Characterisation -0.15in -0.15in Looking in the new Kepler field with a magnitude limit of $m_V\leq14$, we simulated the number of detections as a function of orbital period for Jupiter-, Earth-, and Mars-sized worlds, assuming that every star possessed a planet with the given period and radius (Figure 4), and that red noise was negligible. Interestingly, the difference between the number Jupiter- and Earth-sized planets detected is minimal for $p \leq 10$ days, which indicates that Kepler is able to detect all the planets within this radius range, assuming that they transit. Therefore, for $m_V \leq 14$ and $p \leq 10$ days, the S/N of transiting Jupiter- and Earth-sized planets is large enough that Kepler’s S/N limit is irrelevant, and the $m_V\leq14$ magnitude cutoff is the limiting factor in the number of detections. For smaller planets the size of Mars, the S/N of their transits are in the neighborhood of the Kepler S/N limit for the brightest stars and shorter periods, and so Mars-sized worlds are detected at a much lower rate. -0.0in -0.1in To more accurately characterize the magnitude- and S/N-limited detection regimes, we plotted the number of detections as a function of planetary radius for orbital periods of 1 day and 365 days (Figure 5). In addition in a magnitude limit of $m_V=14$, we also plot the same relations for a limit of $m_V=20$. One can clearly see the flattening of these curves as the radius of the target planets increase, indicating the increasing effect of the magnitude limit on the detection numbers. As is expected in the S/N limited regime, detections go as $R_p^6$ [@pepper2003; @gaudi2005; @pepper2005], though at the lower end saturation of the Kepler CCDs serves to cut out some of these detections. ### Habitable Planets One of the major goals of the Kepler mission is the detection of planets in a star’s habitable zone that would be capable of supporting life. The habitable zone is defined as the distance from the parent star at which liquid water water could exist on a planet’s surface. We estimate the detection rate of habitable zone planets by assuming that all stars have a planet with a semi-major axis equal to $$\label{eq:415} \frac{a}{{\rm AU}} = \sqrt{\frac{L_{bol,*}}{L_{bol,\odot}}}.$$ This assumption is simple and likely optimistic. In reality, the habitable zone has a finite width, and habitable planets will have a range of radii. Thus, a better estimate would be to adopt a distribution of planet masses and radii, and integrate over the range of radii and periods considered habitable. Furthermore, the assumption of one habitable planet per star is probably optimistic, and so the actual number of detected habitable Earths will be smaller than we estimate, depending on the real frequency of Earth-like planets. For the sake of simplicity, however, we will adopt the assumptions above. With this assumption, we simulated the detection of habitable Earths in both the old and new Kepler fields (Figure 6). Surprisingly, despite there being nearly twice as many main sequence stars in the old field, the number of detected habitable Earths remained nearly the same in the two fields. This occurred because at the magnitude relevant for Kepler, the increased star count in the old field was mainly due to the larger number of A and F main-sequence stars found closer to the Galactic plane; habitable planets are doubly hard to detect around these stars due to the lessened depth of transit, and the longer periods required for a planet to be in the habitable zone. The exact distribution of stars by spectral type for the old and new fields is given in Table 13, and is discussed further below. For the 106,000 stars in the new Kepler field, we find that Kepler will be able to detect 79 habitable Earth radius planets, assuming that every star posses one. -0.125in -0.15in In general, Figure 6 provides an excellent example of the benefit of repositioning the Kepler field away from the Galactic plane. Namely, the number of large dwarfs and giant stars (which we have not treated) found in the Galactic plane can be avoided, reducing the false-positive rate, while the number of habitable Earths detected remains nearly the same [@gould2003b]. It is also possible that life may arise on non-Earth-sized worlds. A world the size of Mars might develop quite differently than it has in our own Solar System if it were closer to its parent star, and Jovian worlds in the habitable zone could possess large moons capable of supporting life. Figure 7 shows the number of detections of Jupiter-, Earth-, and Mars-sized habitable worlds, assuming that every star has a planet, against a variety of parameters in the new Kepler field. For completeness, the magnitude limit was set to $m_V\geq 20$. One can see in the mass and radius plots an explicit demonstration of why changing the number of bright, massive main-sequence stars in the field does not effect the number of habitable planet detections. As was mentioned earlier all detections drop to zero beyond $1.4\ M_\odot$. In the plot showing the number of detections binned against the absolute V-magnitude of the parent star we have additionally shown the effect of changing the apparent V-magnitude limit on the distribution of Earth-sized planets. ### Hot and Very Hot Jupiters Using the HJ and VHJ frequencies provided by [@gould2006b], we were able to make explicit predictions about the number of these worlds that Kepler should find. As mentioned previously, [@jenkins2003] (hereafter JD03) have made their own estimates of Kepler’s HJ detection ability (as noted previously, no VHJs were known at the time, and so were not considered), thus providing a useful point of comparison. As before, we used a limit of S/N$\geq7$, though for Jupiter-sized planets Kepler is magnitude, and not signal-to-noise, limited. -0.0in -0.0in [ccccccc|c]{} & B & A & F & G & K & M & All\ Jenkins & Doyle: & 11014& 36708& 81085& 61347& 15573& 831& 206558\ Calculated (old field): & 4278& 31370& 76630& 62891& 18243& 1264& 194678\ Calculated (new field): & 338& 5066& 41094& 45568& 13351& 942& 106342 [ccccccc|c]{} & B & A & F & G & K & M & All (HJs)\ Jenkins & Doyle: & 10& 32& 71& 54& 14& 1& 181 (181)\ Calculated (old field): & 4.2& 27.3& 55.5& 33& 8& 0.4& 127 (72.8)\ Calculated (new field): & 0.3& 4.4& 28.6& 23.5& 5.8& 0.33& 62.9 (36) At the time of the JD03 estimate the Kepler field was centered on its old coordinates. For the purposes of comparison, we therefore examined the old Kepler field in addition to the new field. Tables 13 and 14 show the distribution of our predictions by spectral type. To facilitate comparison with JD03, we have included not only the final prediction numbers for both old and new Kepler fields, but also the distribution of the calculated total number of stars in each field. As stated above, in both fields we set the magnitude limit of the simulation such that the total number of stars in the fields agreed with the number of expected Kepler targets. In the case of the new field, this is 100,000 stars, while for the old field we matched the number of stars predicted by JD03. This corresponded to a magnitude limit of $m_V=15.7$ in the new field and $m_V=15.9$ in the old field. Looking at the detections, one can see in Table 14 that our final prediction of $N_{det}=127$ for the old Kepler field is lower than the JD03 prediction. This is primarily a result of the fact that the [@gould2006b] results that we use for $k(p)$ (see Section 2.2) posit a HJ and VHJ frequency approximately $1/3$ of the frequency used in JD03. An important difference, however, between our work and JD03 is our consideration of VHJs, which were not treated by JD03. This has the effect of “doubling” the number of planets we expect Kepler to detect (as demonstrated by the final values in parentheses in Table 14). We therefore arrive at detection rates for VHJs and HJs that are 2/3 that of JD03 for the old Kepler field, and 1/3 that of JD03 in the new Kepler field. For the old field, we predict 54.2 VHJs and 72.8 HJs. For the new field, we predict 26.9 VHJs, and 36.0 HJs. In the new Kepler field, we see that looking away from the plane of the Galaxy reduces the number of HJs and VHJs detected by half to $N_{det}=62.9$, which matches the overall drop in the star count in the new field. The distribution of detections changes substantially, shifting more towards F-, G-, and K-dwarfs. This is the caused by our model of the Galaxy’s vertical structure, which uses reduced scale heights for earlier stars. Thus, most of the difference in star counts between the old and new fields is caused by fewer B-, A-, and F-dwarfs, leading to proportionally smaller detection numbers around these types of stars. While the prediction number of detections in the new Kepler field is lower than the corresponding JD03 prediction would be for the same star count, this is accounted for in the same way as in the old field. Indeed, we find that our predictions are consistent with those of JD03, except for the difference in our assumption about the frequency of HJs and VHJs. The eventual results from Kepler therefore offer an excellent opportunity to more robustly determine the actual prevalence of short period Jovian worlds. All-sky Surveys --------------- [c|cccc]{} Survey & LSST & SDSS-II & Pan-STARRS & Pan-STARRS Wide\ $m_V$ Limit (Sun-like) & 18.51 & 15.56 & 14.99 & 12.49\ $m_V$ Limit (M-dwarfs) & 23.13 & 20.18 & 19.61 & 17.11\ Observation Time & 10 years & 37.5 days & 5 months & 5 months\ Telescope Diameter & 6.5 m & 2.5 m & 1.8 m & 1.8 m\ Throughput & 0.5 & 0.5 & 0.5 & 0.5\ Field of View & $9.6\ \mathrm{deg}^2$ & $6.25\ \mathrm{deg}^2$ & $7\ \mathrm{deg}^2$ & $7\ \mathrm{deg}^2$\ Area Surveyed & $20,000\ \mathrm{deg}^2$ & $300\ \mathrm{deg}^2$ & $1200\ \mathrm{deg}^2$ & $12,000\ \mathrm{deg}^2$ In addition to wide-field photometric surveys, several current and planned projects aim to repeatedly gather photometry over gigantic swathes of the sky. While not all of them have extrasolar planet detection as one of their goals, the large areas covered by these surveys, together with their long durations (from 3 to 10 years), offer the possibility of detecting hundreds of transiting VHJs and HJs. We examined three all-sky surveys, LSST, SDSS-II, and Pan-STARRS. The Large Synoptic Survey Telescope (LSST) will be a large 8.4 m telescope, with an effective aperture of 6.4 m, situated in northern Chile. It is scheduled to finish construction sometime in 2014. Using a 9.6 deg$^2$ field of view, the LSST team intends to image 20,000 deg$^2$ of the sky repeatedly over the course of 10 years.[^8] SDSS-II is the second phase of the Sloan Digital Sky Survey (SDSS) which began in June, 2005 and is scheduled to last for three years until June of 2008. It is composed of three distinct surveys, the Sloan Legacy Survey, SEGUE, and the Sloan Supernova Survey [@sako2005]. The Supernova Survey is the only one of these three that will collect photometry capable of finding transiting planets; the other two survey modes are focused on non-variable objects within and outside the Galaxy. To execute the Supernova Survey, SDSS uses a 2.5 m telescope on Apache Point in New Mexico with a 6.25 deg$^2$ field of view. Over the three year run of SDSS-II, the team expects the Supernova Survey to image about 300 deg$^2$ of the sky near the Galactic poles every two nights for the three months a year the poles are visible. Given the observing cadence the survey expects to use, this translates into a total observing time of 37.5 days. [|c|ccc|]{} Mag. Limit & Gal. Plane & Gal. Poles & All-Sky Average\ $m_V \leq 20$ & 4.052 & 0.027 & 0.800\ LSST (18.51) & 1.609 & 0.027 & 0.387\ $m_V \leq 18$ & 1.125 & 0.026 & 0.293\ $m_V \leq 16$ & 0.219 & 0.025 & 0.087\ SDSS-II (15.56) & 0.146 & 0.020 & 0.063\ Pan-STARRS (14.99) & 0.083 & 0.016 & 0.041\ $m_V \leq 14$ & 0.029 & 0.009 & 0.017\ Pan-STARRS Wide (12.49) & 0.005 & 0.003 & 0.004\ $m_V \leq 12$ & 0.003 & 0.002 & 0.002 [|c|ccc|]{} Mag. Limit & Gal. Plane & Gal. Poles & All-Sky Average\ LSST (23.13) & 2.7558 & 0.0888 & 0.7764\ SDSS-II (20.18) & 0.2473 & 0.0398 & 0.1136\ $m_V \leq 20$ & 0.2081 & 0.0368 & 0.0989\ Pan-STARRS (19.61) & 0.1422 & 0.0304 & 0.0725\ $m_V \leq 18$ & 0.0257 & 0.0105 & 0.0169\ Pan-STARRS Wide (17.11) & 0.0092 & 0.0048 & 0.0068\ $m_V \leq 16$ & 0.0047 & 0.0015 & 0.0028\ $m_V \leq 14$ & 0.00017 & 0.00015 & 0.00016\ $m_V \leq 12$ & 0.00001 & 0.00001 & 0.00001 Finally, Pan-STARRS - the Panoramic Survey Telescope and Rapid Response System - will be a group of four 1.8 m telescopes atop the summit of Haleakala, Hawaii [@kaiser2004]. The first telescope, PS-1, is nearing completion. When fully operational, Pan-STARRS will embark on a 3 year-long “Medium Deep” survey that will observe 1200 square degrees of the sky near the Galactic poles for a total of five months.[^9] We also looked at a notional “wide” Pan-STARRS survey that looked at 10x the amount of sky as the Medium Deep survey, but used the same amount of observing time. To characterise each of these surveys and find how many transiting VHJs and HJs they should discover, we first calculated the V-magnitude limits at which each of the surveys will be able to detect transiting Jovian worlds. The complete derivation of this limit is given in Appendix E. Table 15 shows the results of these calculations, as well as the other relevant parameters for each of the surveys. We calculated magnitude limits for detecting giant planets around both Sun-like ($0.8 \leq M_*/M_\oplus \leq 1.2$) stars and M-dwarfs ($0.1 \leq M_*/M_\oplus \leq 0.5$) for S/N$\geq20$. -0.1in -0.1in To translate these magnitude limits into the number of transiting planets each of the surveys should detect, we used our methodology to calculate the density of VHJs and HJs for various limiting magnitudes, in numbers per square degree, along the Galactic plane, at the Galactic poles, and averaged around the sky. Table 16 shows the resultant values for a series of fiducial magnitude limits, as well as the planet densities at the magnitude limits of our modeled surveys for Sun-like stars; Table 17 shows transiting planet densities around M-dwarfs. Additionally, in Figure 8 we show the transiting planet density around Sun-like stars and M-dwarfs as a function of Galactic latitude. It is interesting to note the dip near the Galactic plane, which occurs as a result of extinction from interstellar dust. Knowing the angular size of each survey, we use these densities to predict the number of planets that will be seen. For SDSS-II and Pan-STARRS, which are both pointed towards the Galactic poles, we used the polar densities, while for LSST and the wide Pan-STARRS, with their much larger and wide-ranging survey area, we used the all-sky average density. The final predictions for both Sun-like stars and M-dwarfs are shown in Table 18. We note that these predictions are based on a simpler model than what we have used in other portions of this paper. For instance, we do not include the effects of CCD saturation or sky-background. Nevertheless, our calculations should provide a good expectation for the range of the detection numbers that these all-sky surveys can expect. The large difference between the number of planets expected to be detected by LSST and the other surveys is caused by two effects. First, SDSS-II and Pan-STARRS cover much smaller areas of the sky to shallower depths than LSST, and second, LSST is not primarily directed towards the Galactic poles, greatly increasing the density of stars for which planet detection is possible. Pan-STARRS Wide, meanwhile, has a substantially brighter magnitude limit than any of the other surveys, which prevents it from detecting as many planets as LSST, despite the similarities in the size of their survey areas. Pan-STARRS Wide does detect more planets than the planned Narrow survey, primarily because the latter “runs up” against the scale height of the Galaxy. [c|cc]{} & Sun-like (G2) & M-dwarfs (M5)\ LSST & 7740 & 15530\ SDSS-II & 6.0 & 11.9\ Pan-STARRS & 19.2 & 36.5\ Pan-STARRS Wide & 48.0 & 81.6 The real difficulty, however, in managing the 7,740 or 15,530 detections that we expect LSST to make will be properly identifying and separating the false-positives from the actual planetary transit events. Indeed, to date the wide-angle surveys have generated approximately 15 false positive candidates for every actual transiting planet they have seen (D. W. Latham, private communication). If we extend this ratio to LSST, SDSS-II, Pan-STARRS and Pan-STARRS Wide, we would expect them to create 116,000, 90, 290, and 720 false positives, respectively around Sun-like stars, and 233,000, 180, 550, and 1,200 around M-dwarfs, respectively. Working through this many transit candidates (especially in the case of LSST) is an intensive, time-consuming task. We would therefore expect the all-sky surveys, despite the high absolute number of predicted detections, to discover transiting planets at about the same rate as the operational wide-angle surveys, as the identification rate is predominantly limited by the ability to make follow-up observations of candidates to determine whether or not it is a false positive. Effects of Red Noise -------------------- As pointed out by [@pont2006], the general mathematical description we have used for the S/N of a transit, equation (\[eq:30\]), assumes uncorrelated errors on the collected photometric data points (white noise). In practice this may not be the case, as phenomenon such as changes in the seeing, airmass, telescope tracking, etc. introduce systematic trends into the observed data that are correlated on the timescales of transits (red noise). This has the effect of substantially decreasing the effective S/N displayed by a transit, thus making detections more difficult. Specifically, [@pont2006] show that to account for red noise, equation (\[eq:30\]) must be modified by adding a term to include a covariance coefficient $C_{ij}$ between the $i^{th}$ and $j^{th}$ measurements, $$\label{eq:600} \chi^2 = \delta^2 \left( \frac{\sigma^2}{N_{tr}} + \frac{1}{N_{tr}^2} \displaystyle\sum_{i\neq j} C_{ij}\right)^{-1}.$$ This is equation (4) in [@pont2006], using the variables we previously defined for our equation (\[eq:30\]). The authors give a more manageable version on page 8 of their work, which relates the S/N with red noise to the S/N without red noise: $$\label{eq:610} \chi_r^2 = \chi_w^2\ \left[ 1 + N_k \left(\frac{\sigma_r}{\sigma}\right)^2\right],$$ where $\chi_w^2$ is the white noise $\chi^2$ statistic we calculate in equation (\[eq:30\]), $N_k$ is the number of data points in the single $k^{th}$ transit (which on average is just $N_{tr}/$number of transits), and $\sigma_r$ is the red component of the noise in the data. [c|ccccccccccccc|c]{} & And0 & Cyg1 & Cas0 & Per1 & UMa0 & CrB0 & Lyr1 & And1 & And2 & Tau0 & UMa1 & Her1 & Lac0 & Total\ VHJs & 0.34 & 0.33 & 0.31 & 0.30 & 0.17 & 0.20 & 0.30 & 0.22 & 0.17 & 0.20 & 0.16 & 0.20 & 0.36 & 3.26\ HJs & 0.25 & 0.22 & 0.21 & 0.21 & 0.11 & 0.13 & 0.19 & 0.12 & 0.07 & 0.11 & 0.11 & 0.14 & 0.25 & 2.15\ Both & 0.59 & 0.55 & 0.52 & 0.51 & 0.28 & 0.33 & 0.49 & 0.34 & 0.24 & 0.31 & 0.27 & 0.34 & 0.61 & 5.41 -- -- To properly use this formulation, we must therefore have some idea of what value $\sigma_r$ takes. [@pont2006; @pont2007] infer values of 3 mmag of red noise for typical ground-based surveys, which can be reduced to approximately 2 mmag after applying de-trending algorithms to the data. We therefore took this tentative value of $\sigma_r = 0.002$ and used it to re-simulate the TrES fields to see what effect red noise would have on detection numbers. As noted in [@pont2007], we set a signal-to-noise ratio limit of $S/N_r \geq 12$, which is the $S/N_r$ value of the detection of TrES-1b; TrES-2b has a higher value of $S/N_r = 14$. The results are shown in Table 19. Figure 2 shows (in addition to TrES with only white noise) the distribution of detections using 2 mmag of red noise and $S/N_r \geq 12$. [c|c]{} $\sigma_r$ & Hab. Earths\ $10^{-4}$ & 14.37\ $5\cdot 10^{-5}$ & 32.53\ $10^{-5}$ & 72.88\ 0 & 78.89 Compared to a white S/N limit of $S/N_w \geq 30$, using $\sigma_r = 2$ mmag and $S/N_r \geq 12$ drops the total number of expected detections by about 20%. Looking at the split between HJs and VHJs, the HJ population is disproportionately effected, which is a result of the relatively fewer transits HJs show, lowering the overall S/N of their transit signal. Similarly, Figure 2 shows that red noise also cuts out the detections around higher mass stars, as these have a lower S/N overall. Without de-trending the data, a red noise of 3 mmag would cause an even further reduction in detections, down 60% as compared to white noise. We also examined the possible effect that red noise would have on the yield of habitable Earth-radius planets from Kepler mission. Given that Kepler is a space-based mission designed to detect transit signals with fractional depths at the level of $\sim 10^{-4}$, we would expect that the amount of red noise due to instrumental and environmental sources would be small, if it exists at all. However, this is generally difficult to predict before launch. Furthermore, there may exist significant astrophysical red noise due to intrinsic variability. We therefore estimated the expected Kepler yield for a set of three values for $\sigma_r$, assuming that every star has an Earth-radius planet with $a={\rm AU}(L/L_\odot)^{1/2}$ (Table 20). For a relatively large amount of red noise at $\sigma_r = 10^{-4}$, we find the number of habitable Earths-radius is drastically reduced from our baseline (uncorrelated noise) prediction, from 78.89 down to 14.37. This is not surprising, since Earth passing in front of a Sun-like star has a transit depth of only $8.5\cdot 10^{-5}$. For this amount of red noise, Kepler is restricted to detecting Earth-like planets only around K- and M-dwarfs. This shifts the expected distribution of host stars towards dimmer magnitudes (Figure 6). For smaller values of $\sigma_r = 5\cdot10^{-5}$ and $\sigma_r = 10^{-5}$, the number of habitable Earths detected is reduced to 32.53 and 72.88, respectively. Both of these estimates also assume that every target star has an Earth-sized planet in the habitable zone, and therefore must be viewed as upper limits. Thus we conclude that Kepler’s yield of habitable zones of habitable planets will be not be strongly affected by red noise, provided that the red noise is $\la 10^{-5}$ on the timescales of the transits, which is $\sim 13~{\rm hrs}(M/M_\odot)^{1.75}$ for solar-type stars. If the red noise is found to be larger than this, it may compromise the primary Kepler science by significantly reducing the number of expected habitable planet detections. For example, if the red noise is $\sigma_r = 10^{-4}$, and the frequency of habitable planets with $r=R_\oplus$ is $\sim 20\%$, there is a $\sim 5\%$ chance that Kepler will not detect any. The effects of red noise on the habitable planet yield can be mitigated by including a larger number of K and M dwarfs in the target sample. Of course, it will also be possible to extrapolate the frequency of shorter-period or more massive planets, where the effects of red noise are mitigated, into the habitable zone. Conclusion ========== In this paper we have developed a method to more accurately predict the number of transiting planets detectable by wide-field ground- and space-based photometric surveys. We have taken into account such factors as the frequency of gas giants around main sequence stars, the probability of transits, stellar density changes from Galactic structure, and the effects of interstellar extinction. We adopt a S/N detection threshold criterion to approximate the detectability of planetary transits. To test our model, we compared our predictions to two operating photometric surveys, TrES and XO. For both, we calculated the number of statistically expected detections, as well as the population characteristics of the host stars. For TrES, we overestimate the number of detections (thirteen versus the four actual detections), while for XO our simulations yield approximately the actual number of detections (two predicted versus three actual). The exact reason for the discrepancy between our predictions and the actual TrES results could be one of many. As we discuss in the TrES section, from the properties of the known TrES planets, it may be that the survey has a brighter magnitude cut off than the one we use in our fiducial case. If this were the case, and we changed the magnitude limit from $m_R\leq13$ to $m_R\leq12$, the number of expected detections would drop to 8.16 from 13. Red noise also has the potential to substantially affect TrES; a specific estimate of the magnitude of red-noise in the TrES survey would be worthwhile. Indeed, it is generally hard to draw specific statistical conclusions from the comparison of our results and those of the transit surveys because they typically do not adopt the strict detection criteria we have used. Promising planet candidates are often followed up even if they are beyond the stated S/N or magnitude limits of the survey. Understanding and quantifying how the survey teams select candidates is vital to appropriately deriving the statistical properties of extrasolar planets. Indeed, Tables 4-6 and 9-11 demonstrate that the actual predictions depend crucially on the specific magnitude and S/N threshold used. As more transiting planets are discovered, these statistical properties are increasingly becoming the frontier of research, shifting the focus of the field away from individual detections. Given that transiting planets are so far one of the only ways that we may infer the radius of extrasolar planets and their exact orbital properties, the statistical characteristics of this group of objects is one of the only foreseeable ways that the areas of planetary interiors, system dynamics, migration, and formation will acquire more data. In the future, besides using the preceding formalism to design more efficient transit surveys, we would hope that a more systematic approach towards transit surveys will allow this model to be used to make more specific statistical comparisons. As the number of known transiting planets grows, we would also expect that this formalism will be used to test different distributions of planet frequencies, periods, and radii against those observational results. This will allow us to better understand the statistics of extrasolar planetary systems, improve our ability to find new planets, and help to understand the implications of the ones that have already been detected. We would like to thank the referee for a helpful report. We would also like to thank Dave Charbonneau and Francis O’Donovan for providing useful discussions and information about the TrES survey, Peter McCullough for his correspondence about the XO survey, and Scott Schnee for discussions about interstellar dust. TGB would like to thank Dave Latham and Josh Winn for helpful discussions and encouragement. Photon Luminosity of a Source Star ================================== The photon luminosity in a bandpass of a given star of a given mass is calculated exactly by dividing the Planck radiation law for wavelength by the energy of a photon at that wavelength, multiplying the result by the surface area of the star, and then integrating over the wavelength range of the relevant bandpass: $$\label{eq:91} {\Phi_{\lambda}(M) = \int_{\lambda_{min}}^{\lambda_{max}} \frac{B_\lambda (T_*)}{E_\lambda}} 4\pi^2 R_*^2 \, d\lambda,$$ Where $B_\lambda (T_*)$ is the Planck Law, and $E_\lambda = \frac{hc}{\lambda}$ is the energy of a photon. Note that this assumes that stellar radiation can be approximated as a blackbody, a crude assumption that still suffices for our purposes, as we show below. Because the widths of the bandpasses in the visible and near-infrared that we are considering are generally narrow with respect to $\frac{dB_{\lambda}}{d\lambda}$, we can safely approximate this integral by the product of the central wavelength of each bandpass $\lambda_c$ and a bandpass width $\Delta \lambda$: $$\label{eq:101} {\Phi_{\lambda}(M) = \frac{B_\lambda (T_*,\lambda_c) \Delta \lambda}{E_{\lambda_c}}} 4\pi^2 R_*^2 = \frac{8\pi^2 c R_*^2 \lambda_c^{-4} \Delta \lambda}{\mathrm{exp}[hc / \lambda_c k T_*(M)]-1},$$ which, once we select a bandpass, only leaves the temperature of the source star $T_*(M)$ to be determined. Using the Stefan-Boltzmann Law, we may relate the temperature of the source star to its bolometric luminosity and radius: $$\label{eq:111} T_* = \left(\frac{L_{bol,*}}{4\pi \sigma R_*^2}\right)^{\frac{1}{4}},$$ where $R_*$ and $L_{bol,*}$ are the radius and the bolometric luminosity of the source star, respectively. We chose to treat both as functions of the mass of the source star, as this is an independent variable that allows the use of the present day mass function to describe the general stellar population. Using stellar data from [@harmanec1988] and [@popper1980], we fit broken power laws for both the $R_*(M)$ and $L_{bol,*}(M)$ relations. For the mass-radius relation, we fit for two breaks in the power law, while the fitting for the mass-luminosity relation was accomplished using only one break. The relations that we thus found are: $$\begin{aligned} \label{eq:121} R_*(M) = R_\odot \left(\frac{M}{M_\odot} \right)^\alpha\ \ \ \alpha&=&0.8\ \mbox{ for } M_* \leq 1,\\ \alpha&=&1.5\ \mbox{ for } 1 < M_* \leq 1.5, \nonumber \\ \alpha&=&.54\ \mbox{ for } 1.5 < M_*, \nonumber\end{aligned}$$ $$\begin{aligned} \label{eq:131} L_{bol,*}(M) = L_{bol,\odot} \left(\frac{M}{M_\odot}\right)^\beta\ \ \ \beta&=&2.5\ \mbox{ for } M_* \leq 0.4, \\ \beta&=&3.8\ \mbox{ for } 0.4 < M_*. \nonumber\end{aligned}$$ Equations (\[eq:121\]) and (\[eq:131\]) are plotted against the [@harmanec1988] and [@popper1980] data in Figure 9. The real test, however, is how well these relations allow for the calculation of the effective temperature of a star at a given mass using (\[eq:111\]), as this feeds directly into the determination of the photon luminosity in a given band. Figure 10 shows a plot of our mass vs. temperature relation along with data points from [@harmanec1988] and [@popper1980]. Additionally, we also show the analytical relation provided by [@harmanec1988]. One can see that our formulation of (\[eq:121\]) and (\[eq:131\]) affords a close approximation to both the data and the relation provided by [@harmanec1988]. Substituting back into (\[eq:101\]) allows us to calculate the photon luminosity of the source star, as well as the absolute magnitude of the source star in various bandpasses. If we compare these calculated absolute magnitudes in different bands for various stellar masses against empirical magnitudes for those same masses (Figures 11 and 12) we can see that the two relations are similar, to within half a magnitude, especially in longer wavelengths. Shorter wavelength bands such as U, B, and to a certain extent V, have calculated magnitudes that diverge from the empirical relation below about a solar mass, which is an unfortunate effect of using blackbody relations to determine the photon luminosity, as in these higher energy bands stellar spectra increasingly do not look like black bodies. Nevertheless, the blackbody assumption is sufficiently accurate for our purposes. Analytic Window Probability =========================== Roughly, if we ignore the effects of aliasing, we may say that the window probability of seeing a minimum of $N_{tr,min}$ transits of a planet with orbital period $p$ over the course of $N_n$ nights may be broken up into four different regimes. First, if the average number of transits during the total time during which the target star is observed exceeds $N_{tr,min}$, then it is statistically “certain” that the transit will occur during the times of observation . Second, it may be that the entire length of the survey is long enough that there are at least $N_{tr,min}$ transits, but that during the total time spent observing the average number of transits is less than that required for detection. Then one is not certain to detect $N_{tr,min}$ transits, and there is a decreasing window probability (for increasing planet period). Third, one may have $N_{tr,min}$ transits during an observing run, but only if the first transit occurs sufficiently early during that particular run. The probability of detecting the system then decreases at an even greater rate as the orbital period increases. Finally, the orbital period of system may be so great that there is no possible way in which one could see the minimum number of transits required for a detection, and $P_{window}=0$. Mathematically then, $$\begin{aligned} \label{eq:510} P_{window} &=& 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{ for } 1 \leq \frac{<N_{tr}>}{N_{tr,min}},\\ &=& \left(\frac{<N_{tr}>}{N_{tr,min}}\right)^2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{ for } \frac{<N_{tr}>}{N_{tr,min}} < 1 \leq \frac{(N_n-1)\Lambda+T_{n}}{N_{tr,min}P}, \nonumber \\ &=& \left(\frac{<N_{tr}>}{N_{tr,min}}\right)^2 \left(\frac{(N_n-1)\Lambda+T_{n}-(N_{tr,min}-1)P}{P}\right)^\frac{1}{2} \nonumber \\ &\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{ for } \frac{(N_n-1)\Lambda+T_{n}}{N_{tr,min}P} < 1 \leq \frac{(N_n-1)\Lambda+T_{n}}{(N_{tr,min}-1)P}, \nonumber \\ &=& 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{ for } \frac{(N_n-1)\Lambda+T_{n}}{(N_{tr,min}-1)P} < 1, \nonumber\end{aligned}$$ where $\Lambda = 1$ day. $<N_{tr}>$ is the average number of transits that will be seen during the time spent observing, and is given simply by: $$\label{eq:520} <N_{tr}> = \frac{N_g T_n}{P}.$$ $N_g$ is defined as the total number of “good” observing nights in a particular run, and may mathematically be described as $N_g=N_n-N_{lost}$, where $N_{lost}$ is the number of nights lost due to weather, technical problems, or other issues. $T_n$ is the length of each night’s observing time. Alternatively, one may simply view the quantity $N_g T_n$ as the total amount of good observing time in a given run. To test the accuracy of our formulation, we integrated and found the average window probability for both the analytic and exact formulations linearly averaged over several period ranges (1-3, 3-5, and 5-10 days). These are shown as a function of total observing nights under different viewing scenarios in Figure 13. One can see that as a result of the analytic solution not taking into account aliasing, there is divergence between the two solutions as the analytic form approaches unity, as well as in the cases requiring a minimum of three transits. The Present Day Mass Function ============================= A review of the literature shows that while there is agreement that the PDMF beyond $1M_\odot$ can be treated as a power law with a slope of $\alpha \approx5.3$, below a solar mass this agreement breaks down. This is because the low-mass end of the mass-function is unfortunately not as well constrained as it is for higher mass stars, reflecting the difficulty of observing dim M-dwarfs, as well as uncertainties in the mass-luminosity function [@henry1999]. For instance, [@reid2002], who used the Hipparcos data sets along with their own data from the Palomar/Michigan State University (PMSU) survey to create a volume-limted sample out to 25 pc, propose a traditional power law formulation of the PDMF: $$\label{eq:350} \Psi(M)=\frac{dn}{dM}= k_{PDMF} M^{-\alpha},$$ with $\alpha = 5.2\pm0.4$ for $M>M_\odot$, and $\alpha = 1.35\pm0.2$ for $M\leq M_\odot$, and $k_{PDMF}$ as the correct normalization. In a recent review, meanwhile, [@chabrier2003] argues for a log-normal function below a solar mass: $$\label{eq:360} \Psi(M)=\frac{dn}{dM}= \frac{1}{M(\ln10)} A\ exp\left[\frac{-(\log M-\log M_c)^2}{2\sigma^2}\right],$$ with $A = 0.158_{-0.046}^{+0.051}$, $M_c = 0.079_{-0.016}^{+0.021}$, and $\sigma = 0.69_{-0.01}^{+0.05}$. For comparison, if we normalize both to the [@scalo1986] normalization for 5 Gyr of $(dn/dM)_{1M_\odot} = k_{PDMF} = 0.019 M_\odot^{-1} \mathrm{pc}^{-3}$ and plot the two functions against each other (Figure 14), we can clearly see the divergence both numerically and functionally of these two forms below a solar mass. Interestingly however, for most surveys this divergence makes little difference in the final number of predicted detections: typically Reid’s relation gives only 3% fewer detections than Chabrier’s does, if both are normalized at a solar mass. That this difference is so small comes from the fact that most wide-angle surveys are not sensitive to detections around stars with $\log(M)<-0.15$. Hence, the region of greatest uncertainty in the PDMF is left unseen by these surveys. Since the difference between the two relations is small, and because the data set used by [@reid2002] is much more recent than that used by [@chabrier2003], we therefore chose to use the [@reid2002] form of the PDMF. For the purpose of comparing the Reid and Chabrier PDMFs in Figure 14, we normalized both to the value given by [@scalo1986]. Going forward, however, we change the normalization to the integrated mass density found observationally by [@reid2002] of $0.032 M_\odot \ \mathrm{pc}^{-3}$. Galactic Structure ================== The values for the scale length and scale height of the Galactic disk that we use in our modeling are $h_{d,*} = 2.5\ \mathrm{kpc}$ and $h_{z,*}$ as a function of absolute magnitude. Specifically, we calculate the vertical scale height of the disk as: $$\begin{aligned} \label{eq:370} h_{z,*} &=& 90\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{ for } M_V \leq 2,\\ &=& 90 + 200 \left(\frac{M_V-2}{3}\right)\ \mbox{ for } 2 < M_V < 5, \nonumber \\ &=& 290\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{ for } 5 \leq M_V, \nonumber\end{aligned}$$ which is a modified form of the relation found in [@bahcall1980], who instead used a value of 320 pc for the scale height of stars with $5 \leq M_V$. The large uncertainties in the value of the scale length of the disk reflects the uncertainty in the relevant literature. In their initial paper, [@bahcall1980] use $h_{d,*} = 2.5 - 3.0\ \mathrm{kpc}$. Since then various sources have proposed an even $h_{d,*} = 3.0\ \mathrm{kpc}$ [@kent1991], have covered all the bases by using several values between $h_{d,*} = 2.0 - 3.0\ \mathrm{kpc}$ [@olling2001], and have given the very exact result of $h_{d,*} = 2.264\pm.09\ \mathrm{kpc}$ [@drimmel2001]. To see what effect different scale lengths have on our final detection numbers, we simulated observations of a $6^\circ\times6^\circ$ field along the Galactic plane at intervals of $10^\circ$ in Galactic longitude, as well as fixing the Galactic longitude at certain points ($0^\circ$, $90^\circ$, and $180^\circ$) and varying the latitude of the fields. Figure 15 shows these results for scale lengths of 2, 2.5, and 3 kpc. At the extremes (directly towards or directly away from the Galactic core) the 2 and 3 kpc scale lengths differ by only 0.15 detections per field. We chose 2.5 kpc not only to split this difference, but also because more recent results place the scale length much lower than earlier 3 kpc estimates \[e.g. - [@drimmel2001]\]. In the case of the scale height of the Galactic disk, the literature divides between those who derive the scale height from optical star counts and those who rely on near-infrared observations. In the optical case, the scale height is presented as a function of absolute visual magnitude that varies from 90 pc to values between 250-320 pc. The near-infrared observations, on the other hand, lead to one single value of the scale height, generally around 290 pc. This happens because measurements in the infrared are dominated by older M-dwarfs that have dispersed in an even fashion away from the Galactic plane. To compare these different estimates, we have plotted the number of detections in a $6^\circ\times6^\circ$ field along $l=90^\circ$ as a function of Galactic latitude in Figure 16. We used two of the optically-derived scale height relations, as well as a constant scale height of 290 pc. One of the optical relations, the Bahcall model, is given by $$\begin{aligned} \label{eq:375} h_{z,*} &=& 90\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{ for } M_V \leq 2,\\ &=& 90 + 230 \left(\frac{M_V-2}{3}\right)\ \mbox{ for } 2 < M_V < 5, \nonumber \\ &=& 320\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{ for } 5 \leq M_V.\nonumber\end{aligned}$$ The other optical model is taken from Allen’s Astrophysical Quantities (AAQ) [@cox2000] and is: $$\begin{aligned} \label{eq:380} h_{z,*} &=& 90\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{ for } M_V, \leq 2\\ &=& 90 + 160 \left(\frac{M_V-2}{3}\right)\ \mbox{ for } 2 < M_V < 5, \nonumber \\ &=& 250\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{ for } 5 \leq M_V. \nonumber\end{aligned}$$ Also plotted in Figure 16 are the number of detections above and below $M_V=5$ for the three different scale height relations, as well as the residuals of the constant 290 pc and the AAQ relation against the Bahcall formulation. The exact choice of how the Galactic scale height is treated leads to rather different final predictions. Specifically, while the optical relations are similar, the constant 290 pc scale height from near-infrared observations leads to substantially more detections around stars with $M_V<5$. For our predictions, we have adopted the form of the [@bahcall1980] and Allen’s Astrophysical Quantities scale height relations. We find the gradated scale height in these models appealing, because it accounts for the fact that more luminous stars with shorter lifetimes should not be dispersed far away from star-forming regions in the plane of the Galaxy. The constant 290 pc relation, because it is based on near-infrared data, reflects the sensitivity of those observations to less massive stars that are brighter in longer wavelengths. Nevertheless, we have used the 290 pc result as our value for the scale height of late-type stars with $5<M_V$, in contrast to the values used in the AAQ and Bahcall models. All-Sky Surveys’ Limiting Magnitudes ==================================== As per equation (\[eq:30\]), we have $$\label{eq:440} \sqrt{\chi^2} = S/N = (N_{tr})^{1/2}\ \frac{\delta}{\sigma}.$$ For the all-sky surveys that we are considering, the number of in-transit data points observed is simply the number of data points acquired in each field multiplied by the fraction of time a planet will spend in transit during its orbit: $$\label{eq:450} S/N = \left(\frac{t_{obs} \Theta_{FOV}}{t_{exp} \Omega_{survey}} \frac{R_*}{\pi a} \right)^{1/2} \frac{\delta}{\sigma},$$ where $t_{obs}$ is the total observation time of the survey, $\Theta_{FOV}$ is solid angle of the survey telescope’s field of view, and $\Omega_{survey}$ is the total solid angle of the sky covered by the survey. Rearranging: $$\label{eq:460} \sigma = \left(\frac{t_{obs} \Theta_{FOV}}{t_{exp} \Omega_{survey}} \frac{R_*}{\pi a} \right)^{1/2} \frac{\delta}{S/N}$$ For Poisson, source-noise dominated errors, we have $$\label{eq:470} \sigma = N_{source}^{-1/2}$$ where $N_{source}$ is the total number of photons from the source collected during $t_{exp}$. We can then write, $$\label{eq:480} \sigma = \left(\frac{t_{exp}\ e_\lambda}{t_{exp,0}}\right)^{-1/2} \left(\frac{D_{scope}}{D_{scope,0}}\right)^{-1} \sigma_0\ 10^{0.2(m_V-m_{V,0})}$$ where $\sigma_0$ is the fiducial fractional photometric uncertainty achieved for a star with apparent magnitude $m_{V,0}$ using a telescope with diameter $D_{scope,0}$ and an exposure time of $t_{exp,0}$. We have defined $e_\lambda$ as the efficiency of the observing setup relative to the fiducial case. If we rearrange equation (\[eq:480\]) to $$m_V = 5\log{\left[ \left(\frac{t_{exp}\ e_\lambda}{t_{exp,0}}\right)^{1/2}\left(\frac{D_{scope}}{D_{scope,0}}\right) \frac{\sigma}{\sigma_0} \right]} + m_{V,0}$$ we may now substitute in equation (\[eq:470\]) for $\sigma$. Doing so, and simplifying, we find that $$m_V = 5\log{\left[ \left(\frac{t_{obs}\ e_\lambda}{t_{exp,0}} \frac{\Theta_{FOV}}{\Omega_{survey}} \frac{R_*}{\pi a} \right)^{1/2} \frac{D_{scope}\ \delta}{D_{scope,0}\ \sigma_0\ S/N} \right]} + m_{V,0}$$ Which is the limiting magnitude around which an all-sky survey will detect planets. 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See the Extrasolar Planets Encyclopedia at http://exoplanet.eu for an up-to-date list. [^2]: We consider red noise later, in Section 4.5. [^3]: The distribution of $\cos i$ is uniform, which follows from the assumption that planetary systems are randomly oriented. The distribution of $\cos i$ may be derived by considering the normalized angular momentum vector of the planetary orbit, which is free to point in any direction, and thus has a parameter space that forms an imaginary sphere around the parent star. Particular values of $\cos i$ are “hoops” along the surface of this sphere, and the ratio of their area to that of the total sphere is the probability that $\cos i$ will take that value. [^4]: http://www.inscience.ch/transits/ [^5]: http://www.astro.caltech.edu/ ftod/tres/sleuthObs.html [^6]: 00 hr, 04 hr, 08 hr, 12 hr, 16 hr, and 20 hr. [^7]: http://kepler.nasa.gov/ [^8]: http://www.lsst.org [^9]: The remaining balance of the 3 years will be taken up by the other Pan-STARRS science modes; see\ http://pan-starrs.ifa.hawaii.edu/project/reviews/PreCoDR/...\ documents/PSCoDD\_1\_4\_design\_reference\_mission.pdf

--- abstract: 'We introduce a novel playlist generation algorithm that focuses on the quality of transitions using a recurrent neural network (RNN). The proposed model assumes that optimal transitions between tracks can be modelled and predicted by internal transitions within music tracks. We introduce modelling sequences of high-level music descriptors using RNNs and discuss an experiment involving different similarity functions, where the sequences are provided by a musical structural analysis algorithm. Qualitative observations show that the proposed approach can effectively model transitions of music tracks in playlists.' author: - | Keunwoo Choi\ \ \ \ György Fazekas\ \ \ \ Mark Sandler\ \ \ \ bibliography: - 'icml\_2016\_workshop\_playlist.bib' title: | Towards Playlist Generation Algorithms Using\ RNNs Trained on Within-Track Transitions --- &lt;ccs2012&gt; &lt;concept&gt; &lt;concept\_id&gt;10010147.10010257.10010293.10010294&lt;/concept\_id&gt; &lt;concept\_desc&gt;Computing methodologies Neural networks&lt;/concept\_desc&gt; &lt;concept\_significance&gt;500&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10010405.10010469.10010475&lt;/concept\_id&gt; &lt;concept\_desc&gt;Applied computing Sound and music computing&lt;/concept\_desc&gt; &lt;concept\_significance&gt;500&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;/ccs2012&gt; =10000 Introduction {#intro} ============ In music recommendation, the quality of transitions become important particularly when the recommendation is provided in the form of a playlist. This is due to a unique aspect of music consumption. Unlike other products, music is consumed *i) instantaneously*, for instance, while listening using streaming services, *ii) repeatedly*, i.e., listeners are willing to listen to the same music multiple times, and *iii) quickly*, i.e., an item usually only lasts a few minutes. Hence, recommended items are typically consumed or played in a sequence. This behaviour introduces the need for *good transitions* between items, that is, the relevance and subjective judgement a recommended track depends on the previous track. Recommendation approaches using collaborative filtering are prone to overlook niche or new items, although the popularity bias of known items can be compensated for. This is called the *cold-start problem* [@ricci2011introduction]. Content-based approaches which are designed to solve the cold-start problem can suffer from lack of diversity when recommended items are selected simply by similarity. This is often called top-$N$ recommendation. It is well known that *unexpectedness*, *surprise* or *serendipity* play an important role in the music recommendation and discovery [@choi2015understanding]. Compared to other strategies, focusing on transitions can naturally provide these qualities. There have been approaches that primarily focus on the transitions of tracks [@liebman2015dj], [@chen2012playlist], [@mcfee2011natural]. They assumed the *Markov* property of hidden states or embeddings of tracks. Using the Markov property, it is assumed that future events only depend on the current one and does not depend on the past. This has been successfully used for sequence modelling for instance in speech [@rabiner1989tutorial] too. In music computing, playlist datasets [@mcfee2012hypergraph], [@maillet2009steerable], [@chen2012playlist] collaboratively created for reference by DJs and listeners were used for training and evaluation of sequence modelling approaches. Although these datasets consist of a large number of tracks, e.g. 101k playlists in [@mcfee2012hypergraph], the lack of audio data fundamentally limits research based on audio content analysis. Recently, recurrent neural networks (RNNs) have become widely used for sequence modelling in tasks such as speech recognition, substantially outperforming previous hidden Markov model-based approaches [@sak2014long]. The success of the application of RNNs largely relies on the introduction of Long Short-Term Memory (LSTM) units [@gers2000learning]. The merit of LSTM comes from the gate cells of LSTM units, that decide how much the units take input, release output, and forget the previous events. Especially, the forget gate improves the training efficiency by helping the gradients flow well. However, RNNs have not been used for playlists generation and modelling, due in part to the lack of sufficient training data. To solve this problem, we propose using an RNN trained on *within-track* transitions to model playlists. We assume that transitions between structural segments of music can be used as a model for generating the desired high-quality transitions between tracks. In general, segments in a track are different but coherent and their musical features can be expected to match well in succession. This is due to the careful and intentional design by the composer. Using this approach, the number of transitions can easily outnumber that of existing playlist datasets, and therefore it enables to train an RNN model. The rest of the paper is organised as follows. The proposed method is first described in Section \[sec:prop\]. We then present experimental results and discussion in Section \[sec:exp\] and conclude in Section \[sec:conclusinon\]. The Proposed Model {#sec:prop} ================== ![A block diagram of the proposed algorithm, (a,b) training of RNN and (c) prediction of a feature vector, $x_{pred}$.[]{data-label="figure:block"}](icml_2016_rnn_playlist_diagram.pdf){width="1.0\columnwidth"} Figure \[figure:block\] illustrates the procedure of dataset construction, as well as the training and prediction stages of the proposed algorithm. First, the training tracks are segmented and $x_i$, the features for each segments are extracted (Fig. \[figure:block\]a). Then an RNN of length $N$ ($N$=3 in the figure) is trained to learn the transitions of the sequence of feature vectors (Fig. \[figure:block\]b). When a seed track is provided, the features of the last $N$ segments are extracted and fed into the trained RNN to predict the feature vector $x_{pred}$ (Fig. \[figure:block\]c). The algorithm selects a track with a start segment that is most similar to $x_{pred}$. Structural Segmentation ----------------------- Structural segmentation is a task aiming to find the boundaries of different segments or parts in music, e.g. *intro, verse, bridge, chorus*. The most common approach is to take advantage of self-similarity between frames of the track [@foote2000automatic]. In the experiment, we used a basic and efficient method that is proposed in [@foote2000automatic]. Although the results introduce some errors, the feature vector sequences that are based on the imperfect segmentation still approximate the information about how each feature changes along time in each track. Feature Extraction {#sec:featext} ------------------ The proposed algorithm can use feature extraction methods that are relevant to listeners’ musical preferences and able to represent a musical segment. This includes estimated latent features from collaborative filtering [@van2013deep], tags such as genre and emotion [@dieleman2014end] or implicit features such as the weights of the last hidden layer of a neural network classifier [@liangcontent]. Using explicit labels such as genre can facilitate explaining the behaviour of the algorithm, which is important for research and also to the listener. In the experiment, an auto tagging algorithm in [@choi2016automatic] is used to predict a 50-dimensional vector whose elements correspond to the probability of each tag. The tagging algorithm is based on deep convolutional neural networks and trained on Million Song Dataset [@bertin2011million]. It shows state-of-the-art performance while the tags cover variety of categories such as genre, emotion, instrument, and era. Although some of the tags such as genre typically characterise the entire music track, they are not necessarily constant over the whole track. RNN Model --------- The goal of RNN training is to predict the feature of the following track ($x_{pred}$) that maintains consistency and fluctuations, i.e., a certain variation over the features. To this end, a 2-layer RNN with 512 hidden units is employed. LSTM units [@gers2000learning] are used as they show state-of-the-art performance among RNN variants for several sequence modelling tasks [@greff2015lstm]. Similarity Measure ------------------ A similarity measure is necessary to find the subsequent track using the feature vector predicted by the RNN. The similarity metric directly affects the properties of the generated playlists and therefore it should be carefully selected. Using the *cosine distance* may compensate for the popularity bias and result in recommending more niche items [@schedl2015music]. The *Discounted Cumulative Gain* (DCG) turned out to be effective in our experiment. DCG is a weighted version of *Cumulative Gain* (CG). CG is designed to measure ranking quality of a retrieved list and DCG weights on the top-$N$ relevant items by *discounting* lower relevant items. Applying this measure was motivated by the type of feature extraction algorithm we use. The extracted feature is a vector of probabilities of each tag and tags with large probabilities should be weighted more than the others to facilitate maintaining the consistency of the generated playlists. Because DCG weights high-ranking elements more, it can theoretically work better. Results and Discussion {#sec:exp} ====================== Configurations -------------- For training, we used a private dataset with 28,430 commercial tracks of modern popular music including Rock, Hip-Hop, and Jazz. We used 7,880 tracks from the *ILM10k* dataset for testing [@saari2015genre] [@allik:16]. The segmentation is performed using [@foote2000automatic], which is implemented in [@nietomsaf]. As mentioned in Section \[sec:prop\], an automatic tagging algorithm was used as a feature extractor [@choi2016automatic]. An RNN with a length of 50 is trained. We compared DCG with the cosine distance and the $l_2$ norm for computing similarity. Audio processing and RNN are implemented using *librosa* [@mcfee2015librosa], *Keras* [@chollet2015], and *Theano* [@team2016theano]. ![Transitions of feature vectors by cosine distance, $l_2$, and DCG on left, centre, and right, respectively. Y-axis is time (top to bottom) and x-axis refers to the 50 feature dimensions. The first track (topmost feature vectors) is the seed.[]{data-label="fig:transitions"}](fig_transitions_seed_2000.pdf){width="\columnwidth"} Figure \[fig:transitions\] shows the transitions of feature vectors in three playlists given the same seed track but different similarity metrics. The seed track is represented by a 7-by-50 matrix, i.e., 7 segments of the track. White horizontal lines indicate beginnings and ends of each track. The predicted feature vectors (1-by-50) are illustrated in between. The figure helps to explain several aspects of qualitative observations by the authors while listening to the generated playlists. Discussions ----------- **First**, we found both consistency and fluctuations in the extracted features within tracks. In general, several features show consistently large (blue) and small (red) values, while the other features vary. It supports the selection of feature extraction algorithm. However, there are still rooms for further improvements. For example, whitening of each feature dimension can be adopted to compensate the prior distributions of each feature. Although RNNs are able to adapt to such differences, the similarity measure may be affected from such pre-processing. **Second**, the transitions usually successfully keep the coherence within playlists as demonstrated by the figure. However, we noticed that the model is prone to missing overall similarity in long playlists. It may be related to the observation that the trained RNN occasionally predicts a vector that does not have near neighbours. It means the selected track is not similar enough to the predicted vector, which may result in an undesirable or suboptimal transition. This may first be due to the short lengths of segments in the training data. The majority of training tracks have fewer segments (90$\%$ of the training tracks has less than 17 segments), therefore the long-term dependency may not be learnt and the prediction may be dominated by short-term features. In future work, training with longer sequences such as concatenated features of tracks from sequences in an albums, setlists or curated playlists may be used to help learning better transitions. It can be also a typical behaviour of RNNs. Although RNNs generally model the long-term dependency of sequences, in many cases, RNNs have shown a behaviour puts more emphasis on recent inputs rather than older ones. However, the problem seems to be partly resolved when DCG is used as the similarity measure as discussed alongside out last set of observations. **Third**, DCG provides more coherent playlists compared to the cosine distance and $l_2$-norm. This phenomenon is found not only in the sequences corresponding to Figure \[fig:transitions\] but also in other playlists. This can be explained as follows. In each track, there are consistently strong, consistently weak, and fluctuating features. This pattern, especially the consistencies, can be easily learned by the RNN and the consistent features are maintained in the predictions. Finally, DCG prioritises the features that are large in predictions, resulting in successfully finding a track with those consistently large features. This improves the coherence of the resulting playlists. Conclusion {#sec:conclusinon} ========== In this paper, we proposed a novel algorithm for playlist generation that relies on learning desirable musical qualities from within-track transitions between musical segments. The proposed combination of an RNN, within-track structure and DCG showed an encouraging result. Within-track structures showed the consistency and dynamics that are assumed in playlists. An RNN model learnt the feature sequences and its predictions are successfully used for the selections of following tracks. Different similarity measures resulted in different playlists. Future work will investigate advanced architectures such as bidirectional RNNs [@schuster1997bidirectional] and more formal assessments. Using bidirectional RNNs can be used to create playlists that have more constraints e.g. start and end tracks [@flexer2008playlist] and steerability [@maillet2009steerable]. Formal assessments will include subjective measurement e.g. satisfaction and objective measures with regards to consistency, fluctuations and diversity.

--- author: - | Anna Jenčová [^1]\ [Mathematical Institute, Slovak Academy of Sciences,]{}\ [Štefánikova 49, 814 73 Bratislava, Slovakia]{}\ [jenca@mat.savba.sk]{} title: The structure of strongly additive states and Markov triplets on the CAR algebra --- Introduction ============ A remarkable property of von Neumann entropy is the strong subadditivity (SSA): For a state $\rho$ on the 3-fold tensor product $B(\mathcal H_A\otimes\mathcal H_B\otimes \mathcal H_C)$, we have $$S(\rho)+S(\rho_B)\le S(\rho_{AB})+S(\rho_{BC})$$ Here $\mathcal H_A$, $\mathcal H_B$ and $\mathcal H_C$ are finite dimensional Hilbert spaces and $\rho_B$, $\rho_{AB}$, $\rho_{BC}$ are the restrictions of $\rho$ to the respective subsystems. This was first proved by Lieb and Ruskai in [@liebruskai]. The structure of states that saturate the strong subadditivity of entropy, called strongly additive states, was studied in [@hjpw]. In was shown that a state $\rho$ is strongly additive if and only if it has the form $$\label{eq:ssaeq_hrpw} \rho=\bigoplus_n A_n\otimes B_n,$$ where $A_n\in B(\mathcal H_A\otimes \mathcal H_n)$ and $B_n\in B(\mathcal K_n\otimes \mathcal H_C)$ are positive operators and $\mathcal H_B$ has a decomposition $\mathcal H_B=\bigoplus_n \mathcal H_n\otimes\mathcal K_n$ (see also [@japetz], where this was proved also for the infinite dimensional case). Equivalently, $$\label{eq:ssaeq_ja} \rho= (D_{AB}\otimes I_C)(I_A\otimes D_{BC})$$ where $D_{AB}\in B(\mathcal H_A\otimes \mathcal H_B)$ and $D_{BC}\in B(\mathcal H_B\otimes \mathcal H_C)$ are positive matrices. The Markov property for states in the quantum (non-commutative) probability was introduced by Accardi [@accardi] and Accardi and Frigerio [@acfrig], in terms of completely positive unital maps, so-called quasiconditional expectations. For tensor products, it was shown that the Markov property is equivalent to strong additivity of the states [@ohyapetz]. The definition of the Markov property does not require the tensor product structure and can be applied in much more general situations. We are interested in the case of CAR algebras. The Markov states for CAR algebras were studied in [@afimu]. The strong subadditivity of entropy on CAR systems was recently shown and it was proved that strong additivity is equivalent to Markov property in the case of even states, see [@moriya]. For noneven states, a necessary and sufficient condition for equality in (SSA) was given in [@belpit]. The aim of the present paper is to find the structure of strongly additive states and Markov triplets on the CAR algebra. We find an analogue of (\[eq:ssaeq\_ja\]) for any states and of (\[eq:ssaeq\_hrpw\]) for even states. This is done by a similar method as in [@japetz], using the results of the theory of sufficient subalgebras. The paper is organized as follows. The preliminary section summarizes the most important results on the CAR algebra and on sufficient subalgebras. The main tool used in the sequel is the factorization Theorem \[thm:factorization\] in Section 2.1. Section 3 shows the relation between strong additivity and Markov property for any states on the CAR algebra. Section 4 contains the main results. Preliminaries ============= Sufficient subalgebras ---------------------- We first recall the definition and some characterizations of a sufficient subalgebra, which is a generalization of the classical notion of a sufficient statistic, see [@petz; @ohyapetz] for details. Let $\Ae$ be a finite dimensional algebra and let $\varphi,\psi$ be states on $\Ae$. Let $\Be\subset \Ae$ be a subalgebra and let $\varphi_0$, $\psi_0$ be the restrictions of the states to $\Be$. Then $\Be$ is sufficient for $\{\varphi,\psi\}$ is there is a completely positive, identity preserving map $E:\Ae\to \Be$, such that $\varphi_0\circ E=\varphi$, $\psi_0\circ E=\psi$. For simplicity, let us further assume that the states are faithful. Let $\rho_\varphi$, $\rho_\psi$ be the densities of $\varphi$, $\psi$ with respect to a trace $\Tr$: $$\varphi(a)=\Tr \rho_\varphi a, \quad \psi(a)=\Tr \rho_\psi a,\qquad a\in \Ae$$ The relative entropy $S(\varphi,\psi)$ is defined as $$S(\varphi,\psi)=S(\rho_\varphi,\rho_\psi)=\Tr \rho_\varphi(\log\rho_\varphi-\log \rho_\psi)$$ It is monotone, in the sense that we have $S(\varphi,\psi)\ge S(\varphi_0,\psi_0)$ for any subalgebra $\Be\subseteq \Ae$. We will also need the definition of the generalized conditional expectation $E_\psi: \Ae\to \Be$ with respect to the state $\psi$ [@acccec] $$E_\psi(a)=E_{\rho_\psi}(a)=\rho_{\psi_0}^{-1/2}E_\Be(\rho_\psi^{1/2}a\rho_\psi^{1/2})\rho_{\psi_0}^{-1/2}$$ where $E_\Be:\Ae \to \Be$ is the trace preserving conditional expectation. Then $E_\psi$ is a completely positive identity preserving map, such that $\psi_0\circ E_\psi=\psi$ and it is a conditional expectation if and only if $\rho^{it}_\psi\Be\rho^{-it}_\psi\subseteq \Be$ for all $t\in \mathbb R$. The following theorem gives several equivalent characterizations of sufficiency. \[thm:sufficiency\] [@ohyapetz] The following conditions are equivalent. 1. The subalgebra $\Be$ is sufficient for $\{\varphi,\psi\}$. 2. $S(\varphi,\psi)=S(\varphi_0,\psi_0)$. 3. $\rho_\varphi^{it}\rho_\psi^{-it}\in \Be$, for all $t\in \mathbb R$. 4. $E_\varphi=E_\psi$. Our results below are based on the following generalization of the classical factorization criterion for sufficient statistics. \[thm:factorization\] [@japetz] Let $\varphi$, $\psi$ be faithful states on $\Ae$ and let $\Be\subseteq \Ae$ be a subalgebra, such that $\rho_\psi^{it}\Be\rho_\psi^{-it}\subseteq \Be$ for all $t\in \mathbb R$. Then $\Be$ is sufficient for $\{\varphi,\psi\}$ if and only if $$\rho_\varphi=\rho_{\varphi_0}D,\qquad \rho_\psi=\rho_{\psi_0}D$$ where $\varphi_0=\varphi|_\Be$, $\psi_0=\psi|_\Be$ and $D$ is a positive element in the relative commutant $\Be'\cap \Ae$. The CAR algebra --------------- We recall some basic facts about the CAR algebra, for details see [@armo; @bratrob]. The CAR algebra $\mathcal A$ is the $C^*$- algebra generated by elements $\{a_i, i\in \mathbb Z\}$, satisfying the anticommutation relations $$\label{eq:car} a_ia_j+a_ja_i =0,\quad a_ia_j^*+a_j^*a_i=\delta_{ij},\qquad i,j\in \mathbb Z$$ For a subset $I\subset \mathbb Z$, the $C^*$-subalgebra generated by $\{a_i, i\in I\}$ is denoted by $\mathcal A(I)$. If $I$ is finite, $\mathcal A(I)$ is isomorphic to the full matrix algebra $M_{2^{|I|}}(\mathbb C)$ by the so-called Jordan-Wigner isomorphism. Since $$\mathcal A=\overline{\bigcup_{|I|<\infty}\mathcal A(I)}^{\, C^*},$$ there is a unique tracial state $\tau$ on $\mathcal A$, obtained as an extension of the unique tracial states on $\mathcal A(I)$, $|I|<\infty$. It has the following product property: $$\label{eq:product} \tau(ab)=\tau(a)\tau(b),\qquad a\in \mathcal A(I),\ b\in \mathcal A(J),\quad I\cap J=\emptyset$$ ### Graded commutation relations For $I\subseteq \mathbb Z$, we denote by $\Theta^I$ the (unique) automorphism of $\mathcal A$, such that $$\label{eq:thetaI} \Theta^I(a_i)=-a_i, \ i\in I,\qquad \Theta^I(a_i)=a_i,\ i\notin I$$ in particular, we denote $\Theta^{\mathbb Z}$ by $\Theta$. The even and odd parts of $\mathcal A$ are defined as $$\mathcal A_+:=\{ a\in \mathcal A,\ \Theta(a)=a\},\ \mathcal A_-:=\{ a\in \mathcal A,\ \Theta(a)=-a\}$$ and $\mathcal A(I)_+:=\mathcal A(I)\cap \mathcal A_+$, $\mathcal A(I)_-:= \mathcal A(I)\cap \mathcal A_-$. Let $I\cap J=\emptyset$ and $a\in \mathcal A(I)_\sigma$, $b\in \mathcal A(J)_{\sigma'}$, $\sigma,\sigma'\in \{+,-\}$. Then we have the graded commutation relations $$\label{eq:graded} ab=\epsilon(\sigma,\sigma')ba$$ where $$\begin{aligned} \epsilon(\sigma,\sigma')&=& -1\quad \mbox{if } \sigma=\sigma'=-\\ &=& +1\quad \mbox{otherwise} \end{aligned}$$ If $I$ is finite, then there is a self-adjoint unitary $v_I\in \mathcal A(I)$, such that $\Theta^I(a)=v_Iav_I$ for $a\in \mathcal A$ and $$\label{eq:vI} v_I=\Pi_{i\in I}v_i,\qquad v_i=a_i^*a_i-a_ia_i^*$$ Note that $v_iv_j=v_jv_i$ if $i\neq j$ and $\tau(v_i)=0$. Moreover, $v_I\in \mathcal A(I)_+$ and $\mathcal A(I)_+=\mathcal A_I\cap \{v_I\}'$. ### Matrix units Let $A\subset \mathbb Z$ be a finite set, $A=\{i_!,\dots,i_n\}$. The relations $$\begin{aligned} e_{11}^{(i_j)}:=a_{i_j}a_{i_j}^*,\qquad e_{12}^{(i_j)}:=V_{i_{j-1}}a_{i_j}\\ e^{(i_j)}_{21}:=V_{i_{j-1}}a_{i_j}^*,\qquad e_{22}^{(i_j)}:=a_{i_j}^*a_{i_j}\end{aligned}$$ with $V_{i_j}=\Pi_{k=1}^j(I-2a_{i_k}^*a_{i_k})$ define a family of mutually commuting $2\times 2$ matrix units. The Jordan-Wigner isomorphism is then given by$$e^{(A)}_{k_1l_1\dots k_nl_n}:=e^{(i_1)}_{k_1l_1}\dots e^{(i_n)}_{k_nl_n}\mapsto e_{k_1l_1}\otimes\dots\otimes e_{k_nl_n}$$ where $e_{kl}$ are standard matrix units in $M_2(\mathbb C)$. The elements $\{e^{(a)}_\alpha,\alpha\in \mathcal J(A):=(\{1,2\}\times \{1,2\})^n\}$ span $\Ae(A)$. Note that $e^{(A)}_\alpha$ are either even or odd, we denote the set of indices of the even resp. odd elements by $\mathcal J(A)_+$, resp. $\mathcal J(A)_-$. Moreover, the elements $p_\alpha^{(A)}:=e_\alpha^{(A)}(e_\alpha^{(A)})^*$ and $q_\alpha^{(A)}:=(e^{(A)}_\alpha)^*e^{(A)}_\alpha$ are even projections in $\Ae(A)$ and $$\label{eq:munits} p_\alpha^{(A)}e^{(A)}_\beta q_\alpha^{(A)}=\delta_{\alpha,\beta}e^{(A)}_\alpha, \qquad \alpha,\beta \in \mathcal J(A)$$ ### Conditional expectations Let $I\subseteq \mathbb Z$ be any subset. Then there is a unique conditional expectation $E_I: \Ae\to \Ae (I)$, satisfying $$\label{eq:condexp} \tau(ab)=\tau(E_I(a)b),\qquad a\in \Ae,\ b\in \Ae(I)$$ This implies that $\Theta E_I=E_I\Theta$. If $J\subseteq \mathbb Z$, then $E_I(a)\in \Ae(I\cap J)$ for $a\in \Ae(J)$ and $E_IE_J=E_JE_I=E_{I\cap J}$. Note also that the product property (\[eq:product\]) implies that for $a\in \Ae(J)$ with $I\cap J=\emptyset$, $E_I(a)=\tau(a)$. Strong additivity and Markov property ===================================== Let $A$, $B$, $C$ be disjoint finite subsets in $\mathbb Z$. Let us denote $\Ae=\Ae_{ABC}=\Ae(A\cup B\cup C)$, $\Ae_{AB}=\Ae(A\cup B)$ etc. Let $\varphi$ be a faithful state on $\Ae$ and let $\rho$ be its density, that is, $\varphi(x)=\Tr \rho x$ for $x\in \Ae$. Let $\varphi_{AB}$ denote the restriction of $\varphi$ to $\Ae_{AB}$, similarly $\varphi_{BC}$ and $\varphi_B$. Then the density of $\varphi_{AB}$ in $\Ae_{AB}$ is $$\rho_{AB}=E_{AB}(\rho),$$ where $E_{AB}=E_{A\cup B}$. As an element in $\Ae$, $\rho_{AB}$ is the density of the state $\varphi\circ E_{AB}$. Strong subadditivity of entropy ------------------------------- Let $\rho$ be the density of the state $\varphi$. Let $$S(\varphi)= -\Tr \rho(\log (\rho))$$ be the von Neumann entropy of $\varphi$. The strong subadditivity for CAR algebras$$\tag{SSA} S(\varphi)-S(\varphi_{AB})-S(\varphi_{BC})+S(\varphi_B)\le 0$$ was proved in [@moriya]. This inequality is equivalent with $$S(\rho,\rho_{BC})-S(\rho_{AB},\rho_B)\ge 0.$$ Since $\rho_{AB}=E_{AB}(\rho)$, $\rho_B=E_{AB}(\rho_{BC})$ are restrictions of $\rho$ and $\rho_{BC}$ to $\Ae_{AB}$, this holds by monotonicity of the relative entropy. Theorem \[thm:sufficiency\] (ii) then implies the following. \[thm:SSAsuff\] The equality in (SSA) is attained if and only if the subalgebra $\Ae_{AB}$ is sufficient for $\{\varphi,\varphi\circ E_{BC}\}$. Markov triplets and strong additivity ------------------------------------- The state $\varphi$ is a Markov triplet if there exists a completely positive, identity preserving map $E: \Ae \to \Ae_{AB}$, such that 1. $E(xy)=xE(y)$, for all $x\in \Ae_A$ and $y\in \Ae$. 2. $\varphi\circ E=\varphi$ 3. $E(\Ae_{BC})\subseteq \Ae_B$ The map $E$ is called a quasi-conditional expectation with respect to the triplet $\Ae_A \subset \Ae_{AB}\subset \Ae$. Let us now define the subalgebras $\Be\subset \Ce$ in $\Ae_{AB}$ by $$\Ce=\{ x\in \Ae_{AB}, \rho_{BC}^{it}x\rho_{BC}^{-it}\in \Ae_{AB}\}, \qquad \Be=\{ y\in \Ae_B, \rho_{BC}^{it}y\rho_{BC}^{-it}\in \Ae_B\}$$ Note that $\Ce$ is the fixed point subalgebra of the generalized conditional expectation $E_{\rho_{BC}}:\Ae \to \Ae_{AB}$ with respect to $\rho_{BC}$ [@acccec]. We also have $E_{BC}(\Ce)= \Be$. Indeed, if $x=E_{BC}(y)$ for some $y\in \Ce$, then $$\rho_{BC}^{it}x\rho_{BC}^{-it}=E_{BC}(\rho_{BC}^{it}y\rho_{BC}^{-it})\in \Ae_B,$$ so that $E_{BC}(\Ce)\subseteq \Be$, the converse inclusion is clear. \[thm:SSAmarkov\] The state $\varphi$ is a Markov triplet if and only if $\varphi$ satisfies equality in (SSA) and $\Ae_A\subseteq \Ce$. [*Proof.*]{} Let $\varphi$ be a Markov triplet and let $E$ be the quasi-conditional expectation. Then $E$ is a completely positive identity preserving map $\Ae\to\Ae_{AB}$ and $\varphi \circ E=\varphi$. Moreover, let $x\in \Ae_A$, $y\in \Ae_{BC}$, then $$\varphi\circ E_{BC}\circ E(xy)=\varphi\circ E_{BC}(xE(y))=\tau(x)\varphi(E(y))= \tau(x)\varphi(y)=\varphi\circ E_{BC}(xy)$$ Since by the commutation relations (\[eq:car\]) $\Ae$ is spanned by elements of the form $xy$, the above equality implies that $E$ preserves $\varphi\circ E_{BC}$ as well, so that $\Ae_{AB}$ is sufficient for $\{\varphi,\varphi\circ E_{BC}\}$ and equality in (SSA) holds by Theorem \[thm:SSAsuff\]. Let $$F=\lim_n \frac 1n \sum_{k=0}^{n-1}E^k$$ By the ergodic theorem, $F$ is a conditional expectation with range $\mathcal R(F)$ the fixed point subalgebra of $E$. By the property (i) of Markov triplets, $\Ae_A\subseteq \mathcal R(F)$. Since $F$ also preserves $\varphi\circ E_{BC}$, we have by Takesaki theorem that $\rho_{BC}^{it}\mathcal R(F) \rho_{BC}^{-it}\subseteq \mathcal R(F)$, hence also $\rho^{it}_{BC}\Ae_A\rho_{BC}^{-it}\subseteq \mathcal R(F)\subseteq \Ae_{AB}$. It follows that $\Ae_A\subseteq \Ce$. Conversely, suppose equality in (SSA) and $\Ae_A\subseteq \Ce$. Let $E=E_{\rho_{BC}}:\Ae\to\Ae_{AB}$ be the generalized conditional expectation. By Theorem \[thm:SSAsuff\], $\Ae_{AB}$ is sufficient for for $\{\varphi,\varphi\circ E_{BC}\}$ and by Theorem \[thm:sufficiency\] (iv), $E_{\rho_{BC}}=E_\rho$, hence $\varphi\circ E=\varphi$. By the assumptions, $\Ae_A\subseteq \Ce$ the fixed point subalgebra of $E$. The property (iii) of Markov triplets is clear from the definition of $E_{\rho_{BC}}$. The following Corollary was already proved in [@moriya]. \[coro:even\] Let $\varphi$ be an even state. Then $\varphi$ is a Markov triplet if and only if it satisfies equality in (SSA). [*Proof.*]{} Since $\rho$ is even, $\rho_{BC}$ is even as well and we always have $\Ae_A\subseteq \Ce$, by the graded commutation relations. The proof now follows from Theorem \[thm:SSAmarkov\]. Characterization of strongly additive states and Markov triplets ================================================================ \[thm:SSAfactor\] The state $\varphi$ satisfies equality in (SSA) if and only if there are positive elements $x\in \Ae_{AB}$, $y\in \Ae_{BC}$, such that $$\label{eq:factor} \rho=xy$$ [*Proof.*]{} Suppose that $\varphi$ satisfies equality in (SSA). Then $\Ae_{AB}$ is a sufficient subalgebra for $\{\varphi,\varphi\circ E_{BC}\}$. By Theorem \[thm:sufficiency\], this implies that $u_t:=\rho^{it}\rho_{BC}^{-it}\in \Ae_{AB}$ for all $t$. Since $\rho_{BC}^{it}u_s\rho_{BC}^{-it}=u_t^*u_{s+t}$ for $s,t\in \mathbb R$, this implies that $u_t\in \Ce$ for all $t$. Hence, $\Ce$ is a sufficient subalgebra as well, such that $\rho_{BC}^{it}\Ce\rho_{BC}^{-it}\subseteq \Ce$. By Theorem \[thm:factorization\], $$\begin{aligned} \rho&=&xy\\ \rho_{BC} &=& x_0y\end{aligned}$$ where $x,x_0\in \Ce\subseteq \Ae_{AB}$ are the densities of the restrictions $\varphi|_\Ce$ and $\varphi\circ E_{BC}|_\Ce$ and $y$ is a positive element in $\Ce'$. Note also that $\varphi\circ E_{BC}|_\Ce$ is the restriction of $\varphi$ to $E_{BC}(\Ce)=\Be\subseteq \Ae_B$, so that $x_0\in \Ae_B$. By the graded commutation relations, we have $(\Ae_A)_+ \subseteq \Ce$, so that $\Ce'\subseteq ((\Ae_A)_+)'= \Ae_{BC}+v_A\Ae_{BC}$, [@armo] (all commutants are taken in the algebra $\Ae$). Let $y\in \Ce'$, then $y=d_1+v_Ad_2$, where $d_1,d_2\in \Ae_{BC}$. We have $$x_0y=E_{BC}(x_0y)=E_{BC}(x_0(d_1+v_Ad_2))=x_0d_1.$$ Since $\varphi$, and therefore also its restriction to $\Be$ is faithful, $x_0$ is invertible, so that $y=d_1\in \Ae_{BC}$. Conversely, suppose $\rho=xy$ as above. Then $\rho_{AB}=xy_0$, $\rho_{BC}= x_0y$ and $\rho_B=x_0y_0$, where $y_0=E_{AB}(y)\in \Ae_B$, $x_0=E_{BC}(x)\in \Ae_B$. Clearly, both $x$ and $x_0$ must commute with both $y$ and $y_0$. Then $\rho^{it}\rho_{BC}^{-it}= x^{it}x_0^{-it}\in \Ae_{AB}$. By Theorem \[thm:sufficiency\] (iii), $\Ae_{AB}$ is sufficient for $\{\varphi,\varphi\circ E_{BC}\}$, so that $\varphi$ satisfies equality in (SSA). \[thm:Markovfactor\] The state $\varphi$ is a Markov triplet if and only if there are positive elements $x\in \Ae_{AB}$ and $y\in (\Ae_{BC})_+$, such that $$\rho=xy$$ [*Proof.*]{} Let $\varphi$ be a Markov triplet. By Theorem \[thm:SSAmarkov\], $\varphi$ satisfies equality in (SSA) and by Theorem \[thm:SSAfactor\] and its proof, there are positive elements $x\in \Ce$, $y\in \Ce'$, such that $\rho=xy$. Since $\Ae_A\subseteq \Ce$, $\Ce'\subseteq \Ae_A'=(\Ae_{BC})_++v_A(\Ae_{BC})_-$, [@armo]. This implies that $y=d_++v_Ad_-$, where $d_+\in (\Ae_{BC})_+$ and $d_-\in (\Ae_{BC})_-$. By the same reasoning as in the proof of Theorem \[thm:SSAfactor\], we get that $y=d_+\in (\Ae_{BC})_+$. Conversely, let $\rho=xy$ as above, then $\varphi$ satisfies equality in (SSA) by Theorem \[thm:SSAfactor\], and $\rho_{BC}=x_0y$, $x_0=E_{BC}(x)$. For $a\in \Ae_A$, $$\rho_{BC}^{it}a\rho_{BC}^{-it} =x_0^{it}ax_0^{-it}\in \Ae_{AB}$$ by the graded commutation relations, so that $\Ae_A\subseteq \Ce$. By Theorem \[thm:SSAmarkov\], $\varphi$ is a Markov triplet. Even Markov triplets -------------------- \[thm:evenmarkovfactor\] Let $\varphi$ be an even state. Then $\varphi$ is a Markov triplet if and only if there are positive elements $x\in \Ae_{AB}$ and $y\in \Ae_{BC}$, such that $$\rho=xy.$$ Moreover, $x$ and $y$ can be chosen even. [*Proof.*]{} Follows easily from Corollary \[coro:even\], Theorems \[thm:SSAfactor\] and \[thm:Markovfactor\] and the fact that $\rho$ is even. We will now describe the subalgebras $\Ce$ and $\Ce'$ for even states. Since $\rho_{BC}$ is even, both $\Ce$ and $\Be$ and their commutants $\Ce'$ and $\Be'$ are invariant under $\Theta$. \[lemma:c\] If $\varphi$ is even, then $$\Ce= \Ae_A\bigvee \Be$$ [*Proof.*]{} Since $\Ae_A\subseteq \Ce$ and clearly also $\Be\subseteq \Ce$, we have $\Ae_A\bigvee \Be\subseteq \Ce$. Conversely, any element $x\in \Ce\subseteq \Ae_{AB}$ has the form $x=\sum_\alpha e^{(A)}_\alpha b_\alpha$ for some $b_\alpha \in \Ae_B$, where $e^{(A)}_\alpha$ are the matrix units in $\Ae_A$. By (\[eq:munits\]), we have for any $\alpha$, $$p^{(A)}_\alpha x q^{(A)}_\alpha= e^{(A)}_\alpha b_\alpha,$$ since $q^{(A)}_\alpha$ is always even. As $\Ae_A\subseteq \Ce$, this implies that $e^{(A)}_\alpha b_\alpha\in \Ce$ for all $\alpha$. It follows that $$\rho^{it}_{BC}e^{(A)}_\alpha b_\alpha\rho^{-it}_{BC}=e^{(A)}_\alpha \rho_{BC}^{it}b_\alpha \rho_{BC}^{-it}\in \Ae_{AB},$$ hence $b_\alpha\in \Be$, so that $\Ce \subseteq \Ae_A\bigvee \Be$. \[lemma:ccom\] If $\varphi$ is even, then $$\Ce'=(\Be'\cap \Ae_{BC})_++(\Be'\cap \Ae_{BC})_- v_A$$ [*Proof.*]{} Since $\Ae_A\subseteq \Ce$, we have $\Ce'\subseteq \Ae_A'= (\Ae_{BC})_++ (\Ae_{BC})_-v_A$, by [@armo]. Let $d_++v_Ad_-\in \Ce'$ and let $x\in \Be\subset \Ce$. Then we must have $xd_+-d_+x=v_A(d_-x-xd_-)$. Applying $E_{BC}$ on both sides, we get $xd_+-d_+x=d_-x-xd_-=0$, hence $d_+\in \Be'\cap (\Ae_{BC})_+=(\Be'\cap \Ae_{BC})_+$, $d_-\in \Be'\cap (\Ae_{BC})_-=(\Be'\cap \Ae_{BC})_-$. Conversely, let $d_+ \in (\Be'\cap \Ae_{BC})_+$, $d_-\in (\Be'\cap \Ae_{BC})_-$ and let $a\in \Ae_A$, $b\in \Be$. Then by the graded commutation relations, $$ab(d_++v_Ad_-)=d_+ab+ v_Ad_-a_+b+v_Ad_-a_-b=(d_++v_Ad_-)ab$$ so that $d_++v_Ad_-\in \Ce'$. \[lemma:b\] Denote $\tilde {\Be} = \Be'\cap \Ae_B$. Then $$\Be'\cap \Ae_{BC}=\tilde {\Be}\bigvee ((\Ae_C)_++v_B(\Ae_C)_-)$$ [*Proof.*]{} It is easy to see that both $\tilde {\Be}$ and $(\Ae_C)_++v_B(\Ae_C)_-$ are subsets in $\Be'\cap \Ae_{BC}$. Conversely, any $y\in \Ae_{BC}$ has the form $y=\sum_\beta b_\beta e^{(C)}_\beta$, for some $b_\beta\in \Ae_B$. Let $x\in \Be$, then $$\begin{aligned} yx&=&\sum_\beta b_\beta e^{(C)}_\beta(x_++x_-)=\sum_\beta b_\beta x_+e^{C}_\beta+ \sum_{\beta\in \mathcal J(C)_+}b_\beta x_-e^{(C)}_\beta - \sum_{\beta\in \mathcal J(C)_-}b_\beta x_-e^{(C)}_\beta\\ &=& \sum_{\beta\in \mathcal J(C)_+} b_\beta xe^{(C)}_\beta + \sum_{\beta\in \mathcal J(C)_-}b_\beta\Theta(x) e^{(C)}_\beta\end{aligned}$$ It follows that $yx=xy$ only if $xb_\beta=b_\beta x$ for $\beta\in \mathcal J(C)_+$ and $xb_\beta=b_\beta\Theta(x)$ for $\beta\in \mathcal J(C)_-$. This is true for all $x\in \Be$ if and only if $b_\beta\in \Be'$ for $\beta\in \mathcal J(C)_+$ and $b_\beta v_B\in \Be'$ for $\beta\in \mathcal J(C)_-$, this implies the statement of the lemma. Let us now look at the algebra $\Be$. Let $P_1,\dots, P_m$ be the minimal central projections in $\Be$. Since $\Be$ is invariant under $\Theta$, we must have for each $i$, $\Theta(P_i)=P_j$ for some $j$. Suppose that $\Theta(P_i)=P_i$, $i=1,\dots,k$ and $\Theta(P_i)=P_{i+1}$ for $i=k+2l+1$, $l=0,\dots, \frac {m-k}2-1$. \[lemma:central\] Let us denote $P_A=\frac12 (1+v_A)$. The minimal central projections in $\Ce$ are $$\begin{aligned} Q_i&:=&P_i, \qquad i=1,\dots,k\\ Q_{k+1}&:=& P_AP_{k+1}+(1-P_A)P_{k+2}, \quad Q_{k+2}:=(1-P_A)P_{k+1}+P_AP_{k+2}\\ \dots\\ Q_{m-1}&:=& P_AP_{m-1}+(1-P_A)P_{m},\quad Q_m:=(1-P_A)P_{m-1}+P_AP_{m}\end{aligned}$$ [*Proof.*]{} Clearly, $\mathcal Z(\Ce)\subset \Ae_A'\cap \Ae_{AB}=(\Ae_B)_++v_A(\Ae_B)_-$ and it is easy to see that if $x_++v_Ax_-\in \mathcal Z(\Ce)$, then $x_+,x_-$ must be in $\mathcal Z(\Be)$. Therefore, $x_+=\sum c_j P_j$ and $x_-=\sum_j d_jP_j$, for some $c_j,d_j\in \mathbb C$. Since $x_+$ is even, we must have $c_j=c_{j+1}$ for $j=k+2l+1$, $l=0,\dots,\frac{m-k}2-1$. Similarly, we get $d_j=0$ for $j=1,\dots,k$ and $d_j=-d_{j+1}$ for $j=k+2l+1$, $l=0,\dots,\frac{m-k}2-1$. Suppose now that $P=x_++v_Ax_-$ is a projection, then we must have $x_+^*x_++x_-^*x_-=x_+$ and $x^*_+x_-+x_-^*x_+=x_-$. This implies that $c_j=|c_j|^2$ for $j=1,\dots,k$, $c_j=|c_j|^2+|d_j|^2\ge0$ and $c_j(d_j+\bar d_j)=d_j\in \mathbb R$, for $j>k$. Hence $2c_jd_j=d_j$, so that either $d_j=0$ and then $c_j=c_j^2$, or $c_j=\frac12$ and then $d_j=\pm \frac12$. It follows that any projection in $\mathcal Z(\Ce)$ is a sum of some of the following projections: $P_i$, $i=1,\dots,k$, $P_j+P_{j+1}$, $j=k+2l+1$, and $\frac 12 (P_j+P_{j+1}\pm v_A(P_j-P_{j-1}))$, $j=k+2l+1$. Since the last projection is equal to $Q_j$ or $Q_{j+1}$ and $Q_j+Q_{j+1}=P_j+P_{j+1}$, the Lemma follows. Let $\varphi$ be an even faithful state on $\Ae$. Then $\varphi$ is a Markov triplet if and only if there is an orthogonal family of projections $P_1,\dots,P_m\in \Ae_B$ and decompositions $P_j\Ae_BP_j=\Be_j\otimes \tilde \Be_j$, where $\Be_j$ and $\tilde \Be_j$ are full matrix algebras, such that 1. $\Theta(P_j)=P_j$ and $\Be_j$ and $\tilde \Be_j$ are invariant under $\Theta$ for $j=1,\dots,k$ 2. $\Theta(P_j)=P_{j+1}$ and $\Theta(\Be_j)=\Be_{j+1}$, $\Theta(\tilde \Be_j)= \tilde \Be_{j+1}$ for $j=k+2l+1$, $l=0,\dots,\frac{m-k}2-1$ 3. Let us denote $$\begin{aligned} V_j&=&P_jv_B\\ \Ce_j &=& \Ae_A\bigvee \Be_j\\ \tilde \Ce_j &=& \tilde\Be_j\bigvee ((\Ae_C)_++V_j(\Ae_C)_-)\end{aligned}$$ for $j=0,\dots,k$ and $$\begin{aligned} U_l&=&(P_{k+2l+1}+P_{k+2l+2})v_B\\ \De_l &=& \Ae_A\bigvee (P_A\Be_{k+2l+1}+(1-P_A) \Be_{k+2l+2})\\ \tilde \De_l &=& (P_A\tilde \Be_{k+2l+1}+(1-P_A)\tilde \Be_{k+2l+2})\bigvee ((\Ae_C)_+ +U_l(\Ae_C)_-)\end{aligned}$$ for $l=0,\dots,\frac{m-k}2-1$, then there is a decomposition $$\label{eq:evendecomp} \rho=\bigoplus_{j=1}^k x_j\otimes y_j\oplus \bigoplus_{l=0}^{\frac{m-k}2-1} (z_l\otimes w_l\oplus \Theta(z_l\otimes w_l)),$$ where $x_j\in \Ce_j$ and $y_j\in \tilde\Ce_j$ are positive and even for $j=1,\dots,k$, and $z_l\in \De_l$, $w_l\in \tilde \De_l$ are positive for $l=0,\dots,\frac{m-k}2-1$. [*Proof.*]{} Suppose that $\rho$ has the form (\[eq:evendecomp\]). Let us define $Q_1,\dots, Q_m$ from $P_1,\dots,P_m$ as in Lemma \[lemma:central\]. Then $Q_j$ are mutually orthogonal projections and it is easy to see that $Q_jx_j=x_j$, $Q_jy_j=y_j$ and $Q_{k+2l+1}z_l=z_l$, $Q_{k+2l+1}w_l=w_l$, $Q_{k+2l+2}\Theta(z_l)=\Theta(z_l)$, $Q_{k+2l+2}\Theta(w_l)=\Theta(w_l)$. Put $$x=\bigoplus_j x_j\oplus \bigoplus_l (z_l\oplus \Theta(z_l)), \qquad y=\bigoplus_j y_j\oplus \bigoplus_l (w_l\oplus \Theta(w_l))$$ then $x\in \Ae_{AB}$ and $y\in \Ae_{BC}$ are positive even elements and $\rho=xy$. By Theorem \[thm:evenmarkovfactor\], this implies that $\varphi$ is a Markov triplet. Conversely, suppose that $\varphi$ is an even Markov triplet. Then we have seen that $\rho=xy$, where $x\in \Ce$ and $y\in \Ce'$ are positive and even. By Lemmas \[lemma:c\], \[lemma:ccom\] and \[lemma:b\], $x\in \Ae_A\bigvee \Be$ and $y\in \tilde \Ce:= \tilde \Be\bigvee ((\Ae_C)_++v_B (\Ae_C)_-)$. Let $P_1,\dots,P_m$ be the minimal central projections in $\Be$ and let $\Be_j:= P_j\Be$, $\tilde \Be_j:=P_j\tilde \Be$. Then $\Be_j$ and $\tilde \Be_j$ are full matrix algebras and $P_j \Ae_BP_j=\Be_j\otimes \tilde \Be_j$. Moreover, we may suppose that there is some $k\le m$ such that 1. and 2. are fulfilled. The minimal central projections $Q_1,\dots,Q_m$ in $\Ce$ are given by Lemma \[lemma:central\]. Let us denote $\Ce_j=Q_j\Ce_j$, $\Ce'_j =Q_j\Ce'$. Then each $\Ce_j$, $\Ce'_j$ is isomorphic to a full matrix algebra and we have a decomposition $$\Ce=\bigoplus_j \Ce_j\otimes \tilde I_j,\qquad \Ce'=\bigoplus_j I_j\otimes \Ce'_j$$ Since we are interested only in even elements in $\Ce'$, we take the algebra $\tilde \Ce_j:=Q_j\tilde \Ce\subset \Ce'_j$. For $j=1,\dots,k$, $\Ce_j$ are invariant under $\Theta$. For $l=0,\dots, \tfrac{m-k}2-1$, let us denote $\De_l:=\Ce_{k+2l+1}$, $E_l:= Q_{k+2l+1}+Q_{k+2l+2}=P_{k+2l+1}+P_{k+2l+2}$. Then $E_l$ is an even projection, the algebra $E_l\Ce= \De_l\oplus \Theta(\De_l)$ is invariant under $\Theta$ and even elements in $E_l\Ce$ are of the form $x\oplus \Theta(x)$, for some $x\in \De_l$. Similar relation hold for $\Tilde \Ce_j$ and $\tilde \De_l:=\tilde \Ce_{k+2l+1}$. Let us denote $x_j:=Q_jx$, $y_j:=Q_jy$ for $j=1,\dots,k$ and $x_j:=E_lx$, $y_j:=E_ly$ for $j=k+2l+1$, $l=0,\dots, \frac{m-k}2-1$. Then all $x_j$, $y_j$ are positive and even and $$\rho=\bigoplus_{j=1}^k xj\otimes y_j \oplus \bigoplus_{l=0}^{\frac{m-k}2-1} x_{k+2l+1}y_{k+2l+1}$$ Moreover, for $j=k+2l+1$ we must have $x_j=z_l\oplus \Theta(z_l)$ for some positive $z_l\in \De_l$ and similarly $y_j=w_l\oplus \Theta(w_l)$ for positive $w_l\in \tilde \De_l$, $l=0,\dots,\frac{m-k}2-1$. The rest of the proof now follows from Lemmas \[lemma:c\] and \[lemma:b\]. [99]{} L. Accardi, On noncommutative Markov property, Funct. Anal. Appl. [**9**]{} (1975), 1-8 L. Accardi, C. Cecchini, Conditional expectations in von Neumann algebras and a theorem of Takesaki, J. Functional. Anal. [**45**]{}(1982), 245-273. L. Accardi and A. Frigerio, Markovian cocycles, Math. Proc. R. Ir. Acad. [**83**]{} (1983), 251-263 L. Accardi, F. Fidaleo, F. Mukhamedov, Markov states and chains on the CAR algebra, Inf. Dimen. Anal. Quantum Probab., Rel. Top. [**10**]{} (2007), 165-184 H. Araki, H. Moriya, Equilibrium statistical mechanics of fermion lattice systems, Rev. Math. Phys. [**15**]{} (2003), 93-198 J. Pitrik, V.P. Belavkin, Notes on the equality in SSA of entropy on CAR algebra, http://arxiv.org/abs/math-ph/0602035, (2006) O. Bratelli and D.W. Robinson, [*Operator Algebras and Quantum Statistical Mechanics II*]{}, Springer-Verlag, Heidelberg, 1981 P. Hayden, R. Jozsa, D. Petz, A. Winter, Structure of states which satisfy strong subadditivity of quantum entropy with equality, Commun. Math. Phys. [**246**]{} (2004), 359-374 A. Jenčová and D. Petz, Sufficiency in quantum statistical inference, Commun. Math. Phys. [**263**]{}(2006), 259-276. E.H. Lieb and M.B. Ruskai, Proof of the strong subadditivity of quantum-mechanical entropy, J. Math. Phys. [**14**]{} (1973), 1938-1941 H. Moriya, Markov property and strong additivity of von Neumann entropy for graded systems, J. Math. Phys. [**47**]{} (2006), 033510 M. Ohya and D. Petz, [*Quantum Entropy and Its Use*]{}, Springer-Verlag, Heidelberg, 1993 D. Petz, Sufficiency of channels over von Neumann algebras, Quart. J. Math. Oxford [**39**]{} (1988), 97-108 [^1]: Supported by the grants VEGA 2/0032/09 and meta-QUTE ITMS 26240120022

--- abstract: | Lamperti’s maximal branching process is revisited, with emphasis on the description of the shape of the invariant measures in both the recurrent and transient regimes. A truncated version of this chain is exhibited, preserving the monotonicity of the original Lamperti chain supported by the integers. The Brown theory of hitting times applies to the latter chain with finite state-space, including sharp strong time to stationarity. Additional information on these hitting time problems are drawn from the quasi-stationary point of view. **Running title:** Lamperti’s MBP. **Keywords**: discrete probability; maximal branching process; recurrence/ transience transition; shape of invariant measures; tails; failure rate monotonicity; truncation; sharp strong time to stationarity; generating functions. **MSC 2000 Mathematics Subject Classification**: 60 J 10, 60 J 80, 92 D 25. address: | $^{1}$Laboratoire de Physique Théorique et Modélisation\ Université de Cergy-Pontoise\ CNRS UMR-8089\ Site de Saint Martin\ 2 avenue Adolphe-Chauvin\ 95302 Cergy-Pontoise, France\ $^{2}$Depto. Ingenieria Matematica and Centro Modelamiento Matematico\ Universidad de Chile\ UMI 2807, Uchile-Cnrs\ Casilla 170-3 Correo 3\ Santiago, Chile\ E-mail: huillet@u-cergy.fr, smartine@dim.uchile.cl author: - 'Thierry Huillet$^{1}$, Servet Martinez$^{2}$' title: 'Revisiting John Lamperti’s maximal branching process' --- Introduction ============ The Lamperti’s maximal branching process (mbp) is a modification of the Galton-Watson (GW) branching process selecting at each step the descendants of the most prolific ancestor, [@L1]. As a Markov chain on the full set of non-negative integers, Lamperti ([@L1]-[@L2]) gave sharp conditions on the tails of the branching number under which this process is recurrent (either positive or null) or transient. Our contribution is to describe the corresponding shape of the invariant measures and we proceed as follows: while fixing a target invariant measure (supported by the integers) of the mbp, we show (in Proposition $2$) how to compute in general the law of the branching mechanism that gives rise to it. Several classes of distributions are supplied both in the recurrent and transient setups. In Propositions $3$, $4$ and $5$, the target invariant measures are probabilities with tails getting larger and larger, ranging from geometric, power-law with index $\alpha \in \left( 0,1\right) $ and power-law with index $0$ (the target has no moments of any positive order). In Propositions $6$ (and $7$), it is shown that the null recurrent (respectively transient) Lamperti chain has a non trivial invariant infinite and positive measure. An important feature of the Lamperti chain we also emphasize on is its failure rate monotonicity (Proposition $1$). The Lamperti’s mbp also makes sense when the branching mechanism takes values in the finite subset $\left\{ 1,...,N\right\} $ and the question of computing the law of the branching mechanism giving rise to any finitely supported target distribution makes sense. We address this point in Proposition $8$. If the target distribution is in particular the restriction to $\left\{ 1,...,N\right\} $ of the invariant measure of a mbp with full state-space, this construction allows to design a truncated version of the latter chain preserving its failure rate monotonicity feature (Proposition $% 9 $ and Corollary $10$). For failure rate monotone Markov chains with finite state-space, Brown, [@Brown], designed a theory of hitting times which thus applies to the truncated Lamperti chain. The main concern is the relationship existing between the first hitting times of both state $\left\{ N\right\} $ and the restricted invariant measure of the truncated Lamperti chain. By monotonicity, state $\left\{ N\right\} $ is the largest possible value that the truncated chain can explore. Under some technical condition on the initial distribution, it is recalled that the former hitting time exceeds stochastically the latter (Proposition $11$) which has the structure of a compound geometric random variable (Proposition $13$). The excess time is a sharp strong time to stationarity allowing to estimate the distance between the current state of the truncated chain to its equilibrium distribution. Its cumulated probability mass function up to $n$ can be computed from the probability that the truncated chain is in state $\left\{ N\right\} $ after $n$ steps, (Proposition $12$). The alternative classical quasi-stationary point of view to this problem is also addressed. In Proposition $14$, we exhibit the rate of decrease of the hitting times to state $\left\{ N\right\} $ in terms of the quasi-stationary distribution. In Proposition $15$, we show that under Brown’s conditions on the initial distribution $\mathbf{\pi }_{0}$, the ratio of the large tail probabilities for the first hitting times of state $\left\{ N\right\} $ starting from $% \mathbf{\pi }_{0}$ against the quasi-stationary distribution exceeds $1$. Proposition $16$ deals with a question raised by Brown concerning asymptotic exponentiality of the hitting times which applies to the truncated Lamperti chain and its time-reversal. Lamperti’s model ================ The Lamperti maximal branching process (mbp) process may be described as an extremal analogue of the GW branching process, where the next generation is formed by the offspring of a most productive individual, [@L1]. As a result of some selection (or detection) mechanism, iteratively in each generation, only the offspring of one of the most productive individuals of the underlying GW process with branching number $\nu $ is kept (or detected), the other ones being wiped out (or missed by the detector). This output mechanism amounts to pruning Galton-Watson trees by iterative selection of a largest family size ending up with the sub-tree of the fittest individuals. In [@L1], Lamperti relates this model to a percolation problem. With $X_{n}$ the size of such a population at generation $n$, $F_{n}\left( j\right) =\mathbf{P}\left( X_{n}\leq j\right) $ and $\nu _{j,n+1}\overset{d}{% =}\nu $ for all $j$, the dynamics under concern is $$X_{n+1}=\max_{j=1,...,X_{n}}\nu _{j,n+1}\Rightarrow F_{n+1}\left( j\right) =\sum_{i\geq 0}\mathbf{P}\left( X_{n}=i\right) \mathbf{P}\left( \nu \leq j\right) ^{i}=\mathbf{E}z^{X_{n}}\mid _{z=\mathbf{P}\left( \nu \leq j\right) }.$$ with initial condition: $X_{0}\overset{d}{\sim }\mathbf{\pi }_{0}$ with $% \mathbf{P}\left( X_{0}\leq j\right) :=F_{0}\left( j\right) .$ We denote $\mathbf{E}\left( X_{n+1}\mid X_{n}=i\right) =\mathbf{E}% \max_{j=1,...,i}\nu _{j}=\mathbf{E}\left( m_{i}\right) $ where $% m_{i}=\max_{j=1,...,i}\nu _{j}.$ Let $p\left( j\right) :=\mathbf{P}\left( \nu =j\right) $. We will assume that the set $\left\{ j:p\left( j\right) >0\right\} $ is either $\Bbb{N}% _{0}:=\left\{ 0,1,2,...\right\} $ or $\Bbb{N}:=\left\{ 1,2,...\right\} $ but, as we shall see, the finite case when $\left\{ j:p\left( j\right) >0\right\} =\left\{ 1,...,N\right\} $ for some integer $N\gg 1$, will also be of interest. We shall let $\phi \left( z\right) =\mathbf{E}z^{\nu }$ be the probability generating function (pgf) of $\nu .$ We shall distinguish two regimes for the branching number $\nu $: Branching number** **$\nu >0$**.** ---------------------------------- If $\nu >0$ ($p\left( 0\right) =\mathbf{P}\left( \nu =0\right) =0$ and $% \mathbf{E}\left( \nu \right) >1$), then $X_{n}>0,$ $\forall n\geq 0$ ($% X_{0}=1$), owing to $$F_{n+1}\left( 0\right) =\mathbf{P}\left( X_{n+1}=0\right) =\mathbf{E}% z^{X_{n}}\mid _{z=p\left( 0\right) =0}=\mathbf{P}\left( X_{n}=0\right) =0.$$ We can omit state $0$, being disconnected. One main concern in this context is whether $X_{n}\rightarrow \infty $ with probability (wp) $1$ (a case of transience) or to some limiting random variable (rv) $X_{\infty }$ (a case of recurrence): the tails of $\nu $ matter to decide. In the recurrent case, what is the shape of the invariant probability measure? In the null-recurrent and transient cases, what are the shapes of the invariant measure (no longer probability measures). In particular how are the tails of the invariant measure related to the tails of $\nu $. - **Transition matrix of** $\left\{ X_{n}\right\} $. With $F\left( j\right) =\mathbf{P}\left( \nu \leq j\right) $, $j\geq 1$, $\left\{ X_{n}\right\} $ is a time-homogeneous Markov chain (MC) on $\Bbb{N}$ with transition matrix ($\sum_{j\geq 1}P\left( i,j\right) =1-F\left( 0\right) ^{i}=1$) $$P\left( i,j\right) =F\left( j\right) ^{i}-F\left( j-1\right) ^{i}\text{, }% i,j\geq 1$$ equivalently $$\begin{aligned} \mathbf{P}\left( X_{n+1}>i\mid X_{n}=i\right) &=&1-F\left( i\right) ^{i} \\ \mathbf{P}_{X_{n}}\left( X_{n+1}>X_{n}\right) &=&1-F\left( X_{n}\right) ^{X_{n}}.\end{aligned}$$ Note $P\left( 1,j\right) =\mathbf{P}\left( \nu =j\right) .$ - **Some properties of** $\left\{ X_{n}\right\} $: - The Lamperti chain clearly is irreducible and aperiodic. - It holds that $\mathbf{P}\left( X_{n+1}\leq j\mid X_{n}=i\right) =:P^{c}\left( i,j\right) =F\left( j\right) ^{i}$ is a decreasing function of $i$, for all $j$: the Lamperti MC $\left\{ X_{n}\right\} $ is stochastically monotone (SM). Equivalently, with $\left\{ >j\right\} $ denoting the upper set $\left\{ j+1,...\right\} ,$ $\mathbf{P}\left( X_{n+1}>j\mid X_{n}=i\right) =:P\left( i,\left\{ >j\right\} \right) $ is an increasing function of $i$, for all $j$ and by induction $P^{n}\left( i,\left\{ >j\right\} \right) $ is an increasing function of $i$, for all $j$ and $n.$ In fact, it has a stronger monotonicity feature: The Lamperti Markov chain $\left\{ X_{n}\right\} $ is failure-rate monotone. *Proof:* The cumulated transition matrix : $P^{c}\left( i,j\right) :=\sum_{k=1}^{j}P\left( i,k\right) =:$ $P\left( i,\left\{ \leq j\right\} \right) $ satisfies: $$P^{c}\left( i_{1},j_{1}\right) P^{c}\left( i_{2},j_{2}\right) \geq P^{c}\left( i_{1},j_{2}\right) P^{c}\left( i_{2},j_{1}\right) ,$$ for all $i_{1}<i_{2}$ and $j_{1}<j_{2}$ (the matrix $P^{c}$ is totally positive of order $2$, viz TP$_{2}$): the MC $\left\{ X_{n}\right\} $ is failure rate monotone. Since if $P^{c}$ is TP$_{2}$, $P^{c}\left( i,j\right) $ is a decreasing function of $i$, for all $j$, (set $j_{2}=\infty $ in the last inequality to get $P^{c}\left( i_{1},j_{1}\right) \geq P^{c}\left( i_{2},j_{1}\right) $), TP$_{2}$ matrices $P^{c}$ form a subclass of SM matrices $P^{c}$. $\Box $ - **Generation:** As for all Markov chains, with $\left( \mathcal{U}% _{n};n\geq 1\right) $ a sequence of independent identically distributed (iid) uniform-$\left( 0,1\right) $ rvs: $$X_{n+1}=\sum_{j\geq 1}j\cdot \mathbf{1}\left( \mathcal{U}_{n+1}\in \left[ P^{c}\left( X_{n},j-1\right) ,P^{c}\left( X_{n},j\right) \right) \right) .$$ We can also check that, with $F^{-1}\left( y\right) =\inf \left( x:F\left( x\right) \geq y\right) $ the inverse function of $F$, one has $% X_{n+1}=F^{-1}\left( \mathcal{U}_{n+1}^{1/X_{n}}\right) .$ - **Transience versus recurrence:** Note that if $\mathbf{P}\left( \nu >i\right) \sim \lambda /i$, $\lambda >0$, for large $i$ ($\sim $ meaning that the ratio of the two terms appearing to the left and right of this symbol tend to $1$ as $i\rightarrow \infty $), $\mathbf{P}\left( X_{n+1}>i\mid X_{n}=i\right) \sim 1-e^{-\lambda }>0.$ In this case, $$\mathbf{P}\left( X_{n+1}\leq \left[ ix\right] \mid X_{n}=i\right) \sim F\left( ix\right) ^{i}\sim e^{-\lambda /x}.$$ and with $Z_{n}=\log X_{n}$$$\mathbf{P}\left( Z_{n+1}-Z_{n}\leq z\mid Z_{n}=\log i\right) \sim e^{-\lambda e^{-z}},$$ independent of $i$. This shows that for large $i$ and for this choice of $% \nu $, $\left\{ Z_{n}\right\} $ resembles a random walk with independent increments whose common law is a Gumbel distribution with mean $m=\log \lambda +\gamma $ ($\gamma $ the Euler constant). So $\left\{ Z_{n}\right\} $ (and $\left\{ X_{n}\right\} $) drifts to $\infty $ if $\lambda >e^{-\gamma }$ ($m>0$) and the basic results of Lamperti in [@L1], [@L2] are: $$\text{If }\lim \inf_{i}i\mathbf{P}\left( \nu >i\right) <c:=e^{-\gamma }\text{% , then }X_{n}\overset{a.s.}{\rightarrow }X_{\infty }\text{ (ergodicity),} \label{L1}$$ where $X_{\infty }$ is a non-degenerate rv and ergodicity means positive recurrence and aperiodicity. $$\text{If }\lim \sup_{i}i\mathbf{P}\left( \nu >i\right) >c:=e^{-\gamma }\text{% , then }X_{n}\rightarrow \infty \text{ wp }1\text{ (transience).} \label{L2}$$ In particular, if $\nu $ has tails heavier than $1/i$ ($i\mathbf{P}\left( \nu >i\right) \rightarrow \infty $), then $X_{n}\rightarrow \infty $ wp $1,$ (transience). $$\begin{array}{l} \text{Critical case, \cite{L2}: } \\ \text{If }\mathbf{P}\left( \nu >i\right) \sim e^{-\gamma }/i+d/\left( i\log i\right) \text{, the process }\left\{ X_{n}\right\} \text{ is:} \\ \text{- positive recurrent if }d<-e^{-\gamma }\pi ^{2}/12 \\ \text{- null recurrent if }d\in \left[ -e^{-\gamma }\pi ^{2}/12,e^{-\gamma }\pi ^{2}/12\right) \\ \text{- transient if }d>e^{-\gamma }\pi ^{2}/12. \end{array} \label{L3}$$ The case $d=e^{-\gamma }\pi ^{2}/12$ is left open and would require additional information on the tails of $\nu $ to decide whether here $% \left\{ X_{n}\right\} $ is transient or null recurrent$.$ Whenever the process $\left\{ X_{n}\right\} $ is ergodic, with $\Phi _{\infty }\left( z\right) :=\mathbf{E}z^{X_{\infty }}$, the functional equation $$F_{\infty }\left( j\right) =\mathbf{P}\left( X_{\infty }\leq j\right) =\Phi _{\infty }\left( \mathbf{P}\left( \nu \leq j\right) \right) \text{, }j\geq 1 \label{FE0}$$ admits a unique solution for the pair $\left( \mathbf{P}\left( X_{\infty }\leq j\right) ,\mathbf{P}\left( \nu \leq j\right) \right) $. Because $\Phi _{\infty }\left( z\right) $ is a pgf with $\Phi _{\infty }\left( 0\right) =0$, we have $\Phi _{\infty }\left( z\right) <z$ and so $X_{\infty }$ is stochastically larger than $\nu $: $$\text{For all }j\geq 1\text{: }\mathbf{P}\left( X_{\infty }\leq j\right) <% \mathbf{P}\left( \nu \leq j\right) . \label{SD}$$ Clearly, the maximal branching process asymptotically selects a family size $% X_{\infty }$ which is larger than the typical family size $\nu $ of the underlying Galton-Watson process. It is then of utmost interest to solve the functional equation (\[FE0\]). As we shall see, the position we will adopt is the following: suppose one has some initial guess of the limiting rv $% X_{\infty }$, we will identify the branching number $\nu $ of the Lamperti mbp realizing this task. An additional problem of interest: how long does it take for $\left\{ X_{n}\right\} $ to reach $X_{\infty }?$ To have an insight on this question, we shall ask how long it takes, for a suitably truncated version $% X_{n}^{\left( N\right) }$ of $X_{n}$, to reach height $N\gg 1$, which is intuitively more demanding than reaching the invariant measure of the truncated chain itself. We shall address these points. - **Time spent in the worst state.** Whenever the process $\left\{ X_{n}\right\} $ is ergodic, it visits infinitely often all the states, in particular the state $\left\{ 1\right\} $, and a sample path of it is made of iid successive non-negative excursions through that state$.$ State $% \left\{ 1\right\} $ is the worst case of the selection mechanism that the Lamperti chain realizes. By the ergodic theorem, the fraction of time spent by $\left\{ X_{n}\right\} $ in this state is $\pi \left( 1\right) =\mathbf{P}% \left( X_{\infty }=1\right) .$ The expected first return time ($\tau _{1,1}$) to state $\left\{ 1\right\} $ is $\mathbf{E}\left( \tau _{1,1}\right) =1/\pi \left( 1\right) $. Suppose $\left\{ X_{n}\right\} $ enters state $\left\{ 1\right\} $ from above at some time $n_{1}.$ The first return time $\tau _{1,1}:=\inf \left( n>n_{1}:X_{n}=1\mid X_{n_{1}}=1\right) $ to state $\left\{ 1\right\} $ is: - either $1$ if $X_{n_{1}}$ stays there with probability $P\left( 1,1\right) =F\left( 1\right) $ in the next step; this corresponds to a trivial excursion of length $1$ and height $0$. - or, with probability $1-F\left( 1\right) $, $\left\{ X_{n}\right\} $ starts a true excursion with positive height and length $\tau _{1,1}^{+}\geq 2.$ Thus $$\mathbf{E}\left( \tau _{1,1}\right) =\frac{1}{\pi \left( 1\right) }=F\left( 1\right) +\left( 1-F\left( 1\right) \right) \mathbf{E}\left( \tau _{1,1}^{+}\right) \text{ and}$$ $$\mathbf{E}\left( \tau _{1,1}^{+}\right) =\frac{1}{1-F\left( 1\right) }\left( \frac{1}{\pi \left( 1\right) }-F\left( 1\right) \right) >2,$$ entailing the relationship: $\frac{1}{\pi \left( 1\right) }>2-p\left( 1\right) $. Given $\left\{ X_{n}\right\} $ enters state $\left\{ 1\right\} $ from above at some time $n_{1}$, it stays there with probability $P\left( 1,1\right) =F\left( 1\right) $ in the next step, so $\left\{ X_{n}\right\} $ will quit state $\left\{ 1\right\} $ at time $n_{1}+G$ where $G$ is a shifted geometric random time with success probability $1-F\left( 1\right) .$ After $n_{1}+G$, the chain moves up before returning to state $\left\{ 1\right\} $ again and the time it takes is $\tau _{1,1}^{+}$. Considering two consecutive instants where $\left\{ X_{n}\right\} $ enters state $% \left\{ 1\right\} $ from above (defining an alternating renewal process), the fraction of time spent in state $\left\{ 1\right\} $ is: $$\rho =\frac{\mathbf{E}\left( G\right) }{\mathbf{E}\left( G\right) +\mathbf{E}% \left( \tau _{1,1}^{+}\right) }.$$ From the expression $\mathbf{E}\left( G\right) =F\left( 1\right) /\left( 1-F\left( 1\right) \right) $ and the value of $\mathbf{E}\left( \tau _{1,1}^{+}\right) $, we get: $$\rho =F\left( 1\right) \pi \left( 1\right) .\text{ }$$ - **Time reversal:** Suppose $\left\{ X_{n}\right\} $ is ergodic. Let $\pi \left( j\right) =% \mathbf{P}\left( X_{\infty }=j\right) $, $j\geq 1$. With $\mathbf{\pi }% ^{\prime }=\left( \pi \left( 1\right) ,\pi \left( 2\right) ,...\right) $ the transpose of the column-vector $\mathbf{\pi }$, $P^{\prime }$ the transpose of $P$ and $\pi \left( i\right) =\mathbf{P}\left( X_{\infty }=i\right) $ the stochastic matrix $$\overleftarrow{P}=D_{\mathbf{\pi }}^{-1}P^{\prime }D_{\mathbf{\pi }}$$ is the transition matrix of the time-reversed chain $\left\{ X_{n}^{\leftarrow }\right\} $. Since $\overleftarrow{P}\neq P$, there is no detailed balance. The process $\left\{ X_{n}^{\leftarrow }\right\} $ is such that its time-reversal $\left( X_{n}^{\leftarrow }\right) ^{\leftarrow }=X_{n}$ is stochastically monotone. The backward process $\left\{ X_{n}^{\leftarrow }\right\} $ can be generated as follows, with a time-reversal flavor: with $\left( J_{n};n\geq 1\right) $ an iid sequence with $J_{1}\overset{d}{\sim }\mathbf{\pi }$, independent of the $\nu $’s, consider the Markovian dynamics $$Y_{n+1}=J_{n+1}\cdot \mathbf{1}\left( \max_{k=1,...,J_{n+1}}\nu _{k,n+1}=Y_{n}^{{}}\right) , \label{TR}$$ giving $Y_{n+1}$ as a $\mathbf{\pi }-$mixture of the number of ancestors whose most productive individuals produce exactly $Y_{n}^{{}}$ descendants in a Galton-Watson process with branching number $\nu $. We have $$\begin{aligned} \mathbf{P}\left( Y_{n+1}=j\mid Y_{n}^{{}}=i\right) &=&\pi \left( j\right) \mathbf{P}\left( \max_{k=1,...,j}\nu _{k,n+1}=i\right) \\ &=&\pi \left( j\right) \left[ F\left( i\right) ^{j}-F\left( i-1\right) ^{j}\right] =\pi \left( j\right) \mathbf{P}\left( X_{n+1}=i\mid X_{n}=j\right) ,\end{aligned}$$ equivalently $$Q=P^{\prime }D_{\mathbf{\pi }}$$ where $Q\left( i,j\right) =\mathbf{P}\left( Y_{n+1}=j\mid Y_{n}^{{}}=i\right) .$ The process $Y_{n}^{{}}$ is substochastic (there is a positive probability that given $Y_{n}^{{}}$ no such index $Y_{n+1}$ exists) and a coffin state can be added to the state-space $\Bbb{N}$ where the system is sent to if $Y_{n+1}$ does not exist. Let $\tau _{i}$ be the first hitting time of the coffin state for $Y_{n}^{{}}$ started at $i$ with $% \mathbf{P}\left( \tau _{i}=1\right) =1-\pi \left( i\right) ,$ the mass defect in state $i$ of $Q$. Then $X_{n}^{\leftarrow }=Y_{n}^{{}}\mid \tau _{Y_{n}^{{}}}>1$ (upon conditioning $Y_{n}^{{}}$ stepwise on the event that the hitting time of the coffin state exceeds one time unit). The process $% \left\{ X_{n}^{\leftarrow }\right\} $ thus constructed has the transition matrix $\overleftarrow{P}$, as required. Branching number** **$\nu \geq 0$**.** -------------------------------------- If $p\left( 0\right) =\mathbf{P}\left( \nu =0\right) >0$ $:$ the above functional equation must be considered for $j\geq 0.$ We have $F_{n+1}\left( 0\right) =\sum_{i}\mathbf{P}\left( X_{n}=i\right) \mathbf{P}\left( \nu =0\right) ^{i}=\mathbf{E}z^{X_{n}}\mid _{z=p\left( 0\right) }>0.$ At each $n$, there is a positive probability that $X_{n}=0.$ If for some $n$, $X_{n}=0$, clearly $X_{n^{\prime }}=0$ for all $n^{\prime }>n:$ state $0$ is absorbing. $\left\{ X_{n}\right\} $ is again a Markov chain now on $\mathbf{N}_{0}$ with transition probability matrix $$P\left( i,j\right) =F\left( j\right) ^{i}-F\left( j-1\right) ^{i}\text{, }% i,j\geq 0 \label{P1}$$ in particular with $P\left( i,0\right) =F\left( 0\right) ^{i}>0$. Two cases arise: $\left( a\right) $ If $\mathbf{E}\left( \nu \right) \leq 1,$ there is almost sure (a.s.) extinction of the underlying branching process, say at $\tau _{% \mathbf{\pi }_{0},0},$ and also therefore of $\left\{ X_{n}\right\} $ at $% \tau _{\mathbf{\pi }_{0},0}^{X}\leq \tau _{\mathbf{\pi }_{0},0}$. We have $% \mathbf{P}\left( X_{n}=0\right) =\mathbf{P}\left( \tau _{0}^{X}\leq n\right) \rightarrow 1$ or $\mathbf{P}\left( X_{n}=0\right) =1$, $\forall n\geq \tau _{0}^{X}$ ($\tau _{\mathbf{\pi }_{0},0}^{X}$ is the absorption time of $% \left\{ X_{n}\right\} $ at $0$). In this case, $\Phi _{\infty }\left( z\right) =1$ for all $z\in \left[ 0,1\right] $ and one possible solution to the functional equation is $F_{\infty }\left( j\right) =1$, $j\geq 0.$ The only problem here is to fix the law of $\tau _{\mathbf{\pi }_{0},0}^{X}$ which (with $\mathbf{e}_{0}^{\prime }=\left( 1,0,0,...\right) $ with $1$ in position $0$), is: $$\mathbf{P}\left( X_{n}=0\right) =\mathbf{P}\left( \tau _{0}^{X}\leq n\right) =\mathbf{\pi }_{0}^{\prime }P^{n}\mathbf{e}_{0}\text{.}$$ $\left( b\right) $ If $\mathbf{E}\left( \nu \right) >1,$ there is extinction of the underlying branching process with probability $0<\rho _{e}<1$ ($\rho _{e}$ the smallest solution in $\left[ 0,1\right] $ to $\phi \left( z\right) =z$) entailing: - a.s. extinction: given the underlying branching process certainly goes extinct (an event with probability $\rho _{e}$), the branching process is generated by the branching number $\nu _{e}$ with $\mathbf{E}\left( z^{\nu _{e}}\right) =\phi \left( z\rho _{e}\right) /\rho _{e}$ and $\mathbf{E}% \left( \nu _{e}\right) \leq 1$, entailing: $X_{n}^{e}\rightarrow 0$ with probability (wp) $1$. The question is how fast and we are back to the case $% \left( a\right) $. - a.s. explosion: given the underlying branching process certainly explodes (an event wp $1-\rho _{e}$), the branching process is generated by $\nu _{% \overline{e}}$ characterized by $\mathbf{E}\left( z^{\nu _{\overline{e}% }}\right) =\left( \phi \left( \rho _{e}+z\left( 1-\rho _{e}\right) \right) -\rho _{e}\right) /\left( 1-\rho _{e}\right) $ with $\mathbf{P}\left( \nu _{% \overline{e}}=0\right) =0$ and $\mathbf{E}\left( \nu _{\overline{e}}\right) =% \mathbf{E}\left( \nu \right) >1$. We are back to the discussion of Subsection $2.1$ with $\left\{ X_{n}^{\overline{e}}\right\} $ either going to $\infty $ or to a limiting rv depending on the tails of $\nu _{\overline{e% }}$. The only two cases that really matter are thus the case developed in Subsection $2.1$ and case $\left( a\right) $ with state $0$ absorbing wp $1$, which was dealt with. We will therefore only consider the remaining first case when $\left\{ X_{n}\right\} $ has state-space $\Bbb{N}.$ Large $i$ estimates of $m_{i}\mathbf{=}\max_{j=1,...,i}\nu _{j}$ ================================================================ We will use ideas stemming from limit laws for maxima of a large sample of iid rvs in the continuum to give large $i$ estimates of $m_{i}\mathbf{=}% \max_{j=1,...,i}\nu _{j},$ [@EKM]. Let $X>0$ be some real-valued rv with density and no atom at $0$. Suppose $X$ has a finite mean $\mathbf{E}\left( X\right) $. Let $\overline{F}_{X}\left( x\right) =\mathbf{P}\left( X>x\right) ,$ $x>0,$ be its complementary probability distribution function (pdf). Define the law of some integral-valued rv $\nu \in \Bbb{N}$ by: $$\mathbf{P}\left( \nu >j\right) =\mathbf{P}\left( X>j\right) ,\text{ }% j=0,1,... \label{1}$$ Let $\overline{F}\left( j\right) =\mathbf{P}\left( \nu >j\right) $, $% j=0,1,...$. With $\mathbf{E}\left( X\right) =\int_{0}^{\infty }\mathbf{P}% \left( X>x\right) dx$ and $\mathbf{E}\left( \nu \right) =\sum_{j\geq 0}% \overline{F}\left( j\right) $ we have $\mathbf{E}\left( \nu \right) -1<% \mathbf{E}\left( X\right) <\mathbf{E}\left( \nu \right) $. This suggests that if $\mathbf{E}\left( X\right) $ is large, $\mathbf{E}\left( \nu \right) $ is very close to $\mathbf{E}\left( X\right) .$ Maxima of a large sample of iid rvs in the continuum ---------------------------------------------------- Let $M_{i}=\max \left( X_{1},...,X_{i}\right) $ with $\left( X_{i}\right) _{i\geq 1}$ iid with $X_{1}\overset{d}{=}X.$ Two cases arise: $\left( i\right) $ Von Mises case: With $a\left( x\right) >0,$ absolutely continuous (with respect to Lebesgue measure) with density $a^{\prime }\left( x\right) $ having $\lim a^{\prime }\left( x\right) =0$ as $% x\rightarrow \infty $, consider $$\mathbf{P}\left( X>x\right) =c\exp \left[ -\int^{x}\frac{dz}{a\left( z\right) }\right] \text{, }c>0.$$ Then $a\left( x\right) =\mathbf{E}\left( X-x\mid X>x\right) $ is the mean excess function with $a\left( x\right) /x\rightarrow 0$ as $x\rightarrow \infty .$ Define $d_{i}$ by $\overline{F}_{X}\left( c_{i}\right) =1/i$ and $d_{i}$ by $% c_{i}=a\left( c_{i}\right) .$ We have $$d_{i}^{-1}\left( M_{i}-c_{i}\right) \overset{d}{\rightarrow }G\text{ as }% i\rightarrow \infty ,$$ where $G$ has a Gumbel distribution $\mathbf{P}\left( G\leq x\right) =e^{-e^{-x}}$, $x$ real. The sequence $c_{i}$ is increasing with $i$ with $% c_{i}/i\rightarrow 0$ so at sublinear rate. With $\gamma $ the Euler constant, it then holds that $$d_{i}^{-1}\left( \mathbf{E}\left( M_{i}\right) -c_{i}\right) \rightarrow \mathbf{E}\left( G\right) =\gamma \text{ as }i\rightarrow \infty ,$$ so when $i$ gets large $\mathbf{E}\left( M_{i}\right) \sim c_{i}.$ $\left( ii\right) $ With $\alpha >0,$ suppose $$\mathbf{P}\left( X>x\right) =x^{-\alpha }L\left( x\right) ,$$ where $L\left( x\right) $ is some slowly varying function at $\infty $, with $L\left( tx\right) /L\left( x\right) \rightarrow 1$ as $x\rightarrow \infty $ for all $t>0$. Defining $c_{i}$ by $\overline{F}_{X}\left( c_{i}\right) =1/i$ we have $$c_{i}^{-1}M_{i}\overset{d}{\rightarrow }F\text{ as }i\rightarrow \infty ,$$ where $F$ has a Fréchet distribution $\mathbf{P}\left( F\leq x\right) =e^{-x^{-\alpha }}$, $x>0$ with $\mathbf{E}\left( F\right) =\Gamma \left( 1-1/\alpha \right) $ if $\alpha >1$, $=\infty $ if $\alpha \in \left( 0,1\right] $. If $\alpha >1,$ with $c_{i}=i^{1/\alpha }L_{1}\left( i\right) $ for some other slowly varying function $L_{1}$, it holds that $$c_{i}^{-1}\mathbf{E}\left( M_{i}\right) \rightarrow \mathbf{E}\left( F\right) =\Gamma \left( 1-1/\alpha \right) \text{ as }i\rightarrow \infty ,$$ so when $i$ gets large $\mathbf{E}\left( M_{i}\right) \sim \Gamma \left( 1-1/\alpha \right) c_{i}.$ And the sequence $c_{i}$ is increasing also at sublinear rate. Maxima of a large sample of discrete iid rvs: large $i$ estimation of $m_{i}$ ----------------------------------------------------------------------------- Let $m_{i}=\max \left( \nu _{1},...,\nu _{i}\right) $ with $\left( \nu _{i}\right) _{i\geq 1}$ iid with $\nu _{1}\overset{d}{=}\nu $ and $\nu $’s law given from $X$’s law as before$.$ Let $c_{i}$ be defined by $\overline{F}\left( c_{i}\right) =1/i$. In general, it is not true that, upon scaling $m_{i}$, there is a proper weak limit for this scaled rv, because in general, in the discrete setting $% \overline{F}\left( j\right) /\overline{F}\left( j-1\right) \nrightarrow 1$ ($% \mathbf{P}\left( \nu =j\right) /\mathbf{P}\left( \nu >j-1\right) \nrightarrow 0$) as $j\rightarrow \infty $. All that can be said is that $% m_{i}-c_{i}$ is tight (or bounded in probability), with $m_{i}/c_{i}% \rightarrow 1$ in probability as $i\rightarrow \infty $. Also, if we are interested in $$\mathbf{E}\left( m_{i}\right) =\sum_{j\geq 0}\left( 1-F\left( j\right) ^{i}\right) ,$$ using the latter argument, for large $i$, $\mathbf{E}\left( m_{i}\right) $ and $\mathbf{E}\left( M_{i}\right) $ are of the same order of magnitude. Ergodic case from (\[L1\])$:$ With $c_{i}$ defined by $\overline{F}% _{X}\left( c_{i}\right) =1/i$ in $\left( i\right) $ the Von Mises case or $% \left( ii\right) $ when $\mathbf{P}\left( X>x\right) =x^{-\alpha }L\left( x\right) $ and $\alpha >1$ in the domain of attraction of the Fréchet$% \left( \alpha \right) $ law $$\begin{aligned} \left( i\right) \text{ }\mathbf{E}\left( m_{i}\right) &\sim &c_{i} \\ \left( ii\right) \text{ }\mathbf{E}\left( m_{i}\right) &\sim &\Gamma \left( 1-1/\alpha \right) c_{i}=\Gamma \left( 1-1/\alpha \right) i^{1/\alpha }L_{1}\left( i\right) .\end{aligned}$$ In all these cases, $\mathbf{E}\left( m_{i}\right) $ grows at sublinear rate as $i$ gets large. Under the above assumptions on the law of $\nu $, there is thus an integer $% I $ such that $$\begin{aligned} \mathbf{E}\left( X_{n+1}\mid X_{n}=i\right) &\leq &i-1\text{ for all }i\geq I \\ \mathbf{E}\left( X_{n+1}\mid X_{n}=i\right) &<&\infty \text{ for all }i\text{ for which }\mathbf{E}\left( X_{n+1}\mid X_{n}=i\right) >i-1,\end{aligned}$$ which by Foster theorem implies that $\left\{ X_{n}\right\} $ is ergodic, [@Fost]. The limit law of the MC is the unique integrable solution to the corresponding functional equation for $X_{\infty }.$ Transient case from (\[L2\]): If $\nu $ is in the domain of attraction of the Fréchet law with $\alpha \in \left( 0,1\right) $, $\mathbf{E}\left( m_{i}\right) =\mathbf{E}\left( X_{n+1}\mid X_{n}=i\right) $ grows at a superlinear rate which leads to a transience case ($X_{n}\rightarrow \infty $ wp $1$, as $n\rightarrow \infty $). Such $\nu $s have infinite mean. If $% \alpha =1$, the process is transient (positive recurrent) if $\lim \sup_{i}i% \mathbf{P}\left( \nu >i\right) >e^{-\gamma }$ (respectively $<e^{-\gamma }$). Whenever the tails of $\nu $ satisfy any one of these conditions, $% \mathbf{E}\left( \nu \right) =\infty .$ This shows that transience of $% \left\{ X_{n}\right\} $ does not necessarily mean $\mathbf{E}\left( \nu \right) =\infty $. *Example:* If $\alpha =1$, there are positive recurrent examples for which $\mathbf{E}\left( \nu \right) =\infty $, for instance those obtained from $$\mathbf{P}\left( \nu >i\right) =\frac{1}{i\log ^{\beta }\left( 1+i\right) }% \text{ with }0<\beta <1\text{, }$$ with $L\left( x\right) =\log ^{\beta }\left( 1+x\right) $ slowly varying at $% \infty $. $\diamondsuit $ General approach to find solutions of the functional equation ============================================================= In the ergodic case from (\[L1\]), the invariant probability measure $\pi \left( j\right) :=\mathbf{P}\left( X_{\infty }=j\right) $ solves $$\mathbf{\pi }^{\prime }=\mathbf{\pi }^{\prime }P.$$ However, here, the pdfs of $\nu >0$ and $X_{\infty }>0$ are related by the functional equation: $$F_{\infty }\left( j\right) =\Phi _{\infty }\left( F\left( j\right) \right) , \label{FE}$$ and we shall give many examples of explicit pairs $\left( F_{\infty }\left( j\right) ,F\left( j\right) \right) $ solving it. As indicated above, it participates to the general program of finding the branching number $\nu $ of the Lamperti mbp realizing an initial target guess of the limiting rv $% X_{\infty }$. The rv $X_{\infty }$ is taken from the classical (shifted) set of probability mass functions (pmfs) supported by the integers. We will then compute explicitly the law of $\nu $ corresponding to classical target pmfs such as geometric, Sibuya, Poisson. The obtained distributions are far from classical and somewhat surprising. *Remark:* With the idea of spanning trees in the background, there exists a determinantal Kirchoff formula stating that, [@Pit]: $$\pi \left( j\right) =\det \left[ \left( I-P\right) ^{\left( j,j\right) }\right] ,$$ where $\left( I-P\right) ^{\left( j,j\right) }$ is the Laplacian matrix $I-P$ to which row $j$ and column $j$ have been removed. In view of the expression (\[P1\] with $i,j\geq 1$) of the Lamperti matrix $P$, the Kirchoff formula shows that the computation of $\mathbf{\pi }$ from $P$ (and so from $F$) is not, in principle, a simple matter. Our approach being to find $F$ (and so $% P $), starting from the knowledge of $\mathbf{\pi },$ this leads in return to non trivial determinantal identities. $\diamondsuit $ - **Lagrange inversion formula:** In the sequel, we shall denote by: $\left( n\right) _{k}=n\left( n-1\right) ...\left( n-k+1\right) $ and $% \left[ n\right] _{k}=n\left( n+1\right) ...\left( n+k-1\right) $ the falling and rising factorials (of order $k$) of $n$. Take $\Phi _{\infty }\left( z\right) =z\Psi _{\infty }\left( z\right) $ for some new (given) pgf $\Psi _{\infty }$ obeying $\Psi _{\infty }\left( 0\right) \neq 0.$ The pgf $\Psi _{\infty }$ is the one of $X_{\infty }-1$. Apply Lagrange inversion formula to solve $z\Psi _{\infty }\left( z\right) =u $. It gives the inverse $\Phi _{\infty }^{-1}\left( z\right) $ of $\Phi _{\infty }\left( z\right) $ as a power series in $z,$ with $$\varphi _{n}:=\left[ z^{n}\right] \Phi _{\infty }^{-1}\left( z\right) =\frac{% 1}{n}\left[ z^{n-1}\right] \Psi _{\infty }\left( z\right) ^{-n}.$$ Then $$F\left( j\right) =\Phi _{\infty }^{-1}\left( F_{\infty }\left( j\right) \right) =\sum_{n\geq 1}\varphi _{n}F_{\infty }\left( j\right) ^{n}$$ gives the $F\left( j\right) $ consistent with the original choice of $% F_{\infty }\left( j\right) $. With $B_{n,k}\left( x_{1},x_{2},...\right) $ (respectively $\widehat{B}_{n,k}\left( x_{1},x_{2},...\right) $) the exponential (respectively ordinary) Bell polynomials in the indeterminates $% \left( x_{1},x_{2},...\right) $, obeying $B_{n,k}\left( x_{1},x_{2},...\right) =0$ if $k>n$ and $B_{n,0}\left( x_{1},x_{2},...\right) =\delta _{n,0},$ we have in principle ([@Com], p. $161$) ($x_{k}=k!\pi \left( k+1\right) $) $$\begin{aligned} \varphi _{n} &=&\frac{1}{n!}\sum_{k=0}^{n-1}\left( -n\right) _{k}\pi \left( 1\right) ^{-\left( n+k\right) }B_{n-1,k}\left( \frac{2!\pi \left( 2\right) }{% 2},\frac{3!\pi \left( 3\right) }{3},...\right) \\ &=&\frac{1}{n}\sum_{k=0}^{n-1}\frac{\left( -n\right) _{k}}{k!}\pi \left( 1\right) ^{-\left( n+k\right) }\widehat{B}_{n-1,k}\left( \pi \left( 2\right) ,\pi \left( 3\right) ,...\right) \\ &=&\frac{\pi \left( 1\right) ^{-n}}{n}\sum_{k=0}^{n-1}\frac{\left( -n\right) _{k}}{k!}\widehat{B}_{n-1,k}\left( \frac{\pi \left( 2\right) }{\pi \left( 1\right) },\frac{\pi \left( 3\right) }{\pi \left( 1\right) },...\right)\end{aligned}$$ $$\begin{aligned} &=&\frac{1}{n!}\sum_{k=0}^{n-1}\left( -1\right) ^{k}\pi \left( 1\right) ^{-\left( n+k\right) }B_{n+k-1,k}\left( 0,2!\pi \left( 2\right) ,3!\pi \left( 3\right) ,...\right) \\ &=&\frac{\pi \left( 1\right) ^{-n}}{n}\sum_{k=0}^{n-1}\left( -1\right) ^{k}% \binom{n+k-1}{k}\widehat{B}_{n+k-1,k}\left( 0,\frac{\pi \left( 2\right) }{% \pi \left( 1\right) },\frac{\pi \left( 3\right) }{\pi \left( 1\right) }% ,...\right) .\end{aligned}$$ Owing to $\left( -n\right) _{k}=\left( -1\right) ^{k}\left[ n\right] _{k}$ and (see [@Com], p. $145$) $$\begin{aligned} B_{n-1,k}\left( x_{1},x_{2},...\right) &=&\left( n-1\right) !\sum^{*}\prod_{m\geq 1}\frac{1}{k_{m}!}\left( \frac{x_{m}}{m!}\right) ^{k_{m}}, \\ \widehat{B}_{n-1,k}\left( x_{1},x_{2},...\right) &=&k!\sum^{*}\prod_{m\geq 1}% \frac{x_{m}^{k_{m}}}{k_{m}!}\end{aligned}$$ where the star sum runs over $k_{m}\geq 0$, obeying $\sum_{m\geq 1}k_{m}=k$ and $\sum_{m\geq 1}mk_{m}=n-1$, we have equivalently $\varphi _{1}=1/\pi \left( 1\right) $ and if $n\geq 2$$$\varphi _{n}=\frac{\pi \left( 1\right) ^{-n}}{n}\sum_{k=1}^{n-1}\left( -1\right) ^{k}\left[ n\right] _{k}C_{n-1,k},$$ where, with $C_{n-1,0}=\delta _{n-1,0},$$$C_{n-1,k}=\sum^{*}\prod_{m\geq 1}\frac{\left( \pi \left( m+1\right) /\pi \left( 1\right) \right) ^{k_{m}}}{k_{m}!}. \label{C}$$ To summarize, we obtained the pdf $F$ of $\nu $ corresponding to $\pi \left( j\right) :=\mathbf{P}\left( X_{\infty }=j\right) ,$ solving (\[FE\]), as The mapping $X_{\infty }\rightarrow \nu $ is one-to-one. With $C_{n-1,k}$ given by (\[C\]) and $h_{n}=\frac{1}{n}\sum_{k=1}^{n-1}\left( -1\right) ^{k}\left[ n\right] _{k}C_{n-1,k}$ ($h_{1}=1$), $$F\left( j\right) =\Phi _{\infty }^{-1}\left( F_{\infty }\left( j\right) \right) =\sum_{n\geq 1}h_{n}\cdot \left( F_{\infty }\left( j\right) /\pi \left( 1\right) \right) ^{n} \label{FES}$$ is the cumulated mass function of $\nu $ corresponding to any given $\mathbf{% \pi }$. The obtained expression (\[FES\]) only depends on the ratio $F_{\infty }\left( j\right) /F_{\infty }\left( 1\right) $. Note that $\Phi _{\infty }^{-1}\left( z\right) $ is increasing from $z=0$ to $z=1$ and concave. From the fact that it is increasing, we conclude that if $F_{\infty }\left( j\right) $ is a pdf, then so is $F\left( j\right) $. From the concavity, we conclude $F\left( j\right) \geq F_{\infty }\left( j\right) $ for all $j$ (as already mentioned, $X_{\infty }$ is stochastically larger than $\nu $). While proceeding in this way, we observe that, given we first fix the law of $X_{\infty }$, the one of the corresponding $\nu $ follows. Suppose we were able to find a suitable pair of pdfs $\left( F\left( j\right) ,F_{\infty }\left( j\right) \right) $ by the Lagrange inversion formula. Then, with $F_{0}\left( j\right) =\mathbf{1}\left( j\leq 1\right) $ ($X_{0}\overset{d}{\sim }\delta _{1}$) and $\Phi _{0}\left( z\right) =z$, $% F_{1}\left( j\right) =\Phi _{0}\left( F\left( j\right) \right) =F\left( j\right) $ is a pdf, the one of $\nu $. Let $\Phi _{1}\left( z\right) =\sum_{j\geq 1}z^{j}\left( F_{1}\left( j\right) -F_{1}\left( j-1\right) \right) $ be the pgf of $X_{1}\overset{d}{=}\nu $. Next, $F_{2}\left( j\right) =\Phi _{1}\left( F\left( j\right) \right) $ is a pdf because $\Phi _{1}$ is monotone increasing obeying $\Phi _{1}\left( 0\right) =0$, $\Phi _{1}\left( 1\right) =1$. By recurrence $F_{n+1}\left( j\right) =\Phi _{n}\left( F\left( j\right) \right) $ is the pdf of some rv $X_{n+1}$ obtained from the one of $X_{n}$ and $F_{n}\left( j\right) \rightarrow F_{\infty }\left( j\right) $ solution to $F_{\infty }\left( j\right) =\Phi _{\infty }\left( F\left( j\right) \right) $. We shall deal with special cases of $X_{\infty }.$ - **Infinite divisibility:** suppose $\Psi _{\infty }\left( z\right) $ is the pgf of an infinitely divisible (ID) rv (meaning $X_{\infty }-1$ is ID). Then, as a compound Poisson rv, $$\Psi _{\infty }\left( z\right) =e^{-\lambda \left( 1-h\left( z\right) \right) },$$ for some rate $\lambda >0$ and pgf $h\left( z\right) $ obeying $h\left( 0\right) =0.$ If $\mathbf{P}\left( X_{\infty }=j+1\right) =\left[ z^{j}\right] \Psi _{\infty }\left( z\right) =\pi _{\lambda }\left( j+1\right) $ is a known simple function of $\lambda $, then $\left[ z^{j}\right] \Psi _{\infty }\left( z\right) ^{-n}$ is readily obtained as $% \pi _{-n\lambda }\left( j+1\right) $, a useful identity to get the $h_{n}$ in (\[FES\]) and so $F$ from $F_{\infty }$. - **Complete monotonicity:** Suppose $\overline{F}_{\infty }\left( j\right) $ defines a in $\left[ 0,1\right] $-valued completely monotone sequence of complementary pdfs, meaning $$\begin{aligned} \left( -1\right) ^{k}\Delta ^{\left( k\right) }\overline{F}_{\infty }\left( j\right) &\geq &0\text{ for all }j,k\geq 0,\text{ equivalently} \\ \left( -1\right) ^{k}\Delta ^{\left( k\right) }\mathbf{P}\left( X_{\infty }=j\right) &\geq &0\text{ for all }j\geq 1,k\geq 0,\end{aligned}$$ where $\Delta :$ $\Delta h\left( j\right) =h\left( j+1\right) -h\left( j\right) $ is the right-shift operator and $\Delta ^{\left( k\right) }$ its $% k-$th iterate. Note $\mathbf{P}\left( X_{\infty }=j\right) =\Delta F_{\infty }\left( j-1\right) =-\Delta \overline{F}_{\infty }\left( j-1\right) $. By Hausdorff representation theorem, $\overline{F}_{\infty }\left( j\right) $ is completely monotone (CM) if and only if $$\overline{F}_{\infty }\left( j\right) =\int_{0}^{1}u^{j}\lambda \left( du\right) ,$$ for some probability measure $\lambda \left( du\right) $ on $\left[ 0,1\right] .$ Equivalently, with $\Phi _{\infty }\left( z\right) =\sum_{j\geq 1}z^{j}% \mathbf{P}\left( X_{\infty }=j\right) $ the pgf of $X_{\infty }$, $$\sum_{j\geq 0}z^{j}\overline{F}_{\infty }\left( j\right) =\frac{1-\Phi _{\infty }\left( z\right) }{1-z}=\int_{0}^{1}\frac{1}{1-zu}\lambda \left( du\right) ,$$ as a Stieltjes transform of $\lambda \left( du\right) .$ Note that, with $U% \overset{d}{\sim }\lambda \left( du\right) $$$\Phi _{\infty }\left( z\right) =z\int_{0}^{1}\frac{1-u}{1-zu}\lambda \left( du\right) =\mathbf{E}\left( \frac{z\left( 1-U\right) }{1-zU}\right) ,$$ showing that $X_{\infty }-1$, with pgf $\Psi _{\infty }\left( z\right) =z^{-1}\Phi _{\infty }\left( z\right) =\mathbf{E}\left( \frac{1-U}{1-zU}% \right) $, is a $\lambda -$mixture of a shifted geometric rv, so that $% X_{\infty }-1$ is log-convex and infinitely divisible. As noted in [@Gupta], log-convex (log-concave) pmfs are decreasing (increasing) failure rate monotone, say DFR (IFR), meaning $\Delta r_{j}$ decreasing (increasing) where $r_{j}=\pi \left( j\right) /\overline{F}_{\infty }\left( j-1\right) =% \mathbf{P}\left( X_{\infty }=j\right) /\mathbf{P}\left( X_{\infty }\geq j\right) $ is a discrete failure ‘rate’. Explicit examples of $\left( \nu ,X_{\infty }\right) $ with support $\left\{ 1,...,\infty \right\} $. ----------------------------------------------------------------------------------------------------- In some cases, the computation of the pair $\left( F\left( j\right) ,F_{\infty }\left( j\right) \right) $ is obtained as a simple expression. - **Geometric** example: $X_{\infty }\overset{d}{\sim }$geom$\left( p\right) $ Suppose $X_{\infty }\overset{d}{\sim }$geom$\left( p\right) $, so with $% F_{\infty }\left( j\right) =1-q^{j}$. The sequence $\overline{F}_{\infty }\left( j\right) $ is of course CM as a result of $$\overline{F}_{\infty }\left( j\right) =q^{j}=\int_{0}^{1}u^{j}\lambda \left( du\right) ,\text{ with }\lambda \left( du\right) =\delta _{q}\left( du\right) ,$$ so $X_{\infty }-1$ is ID. $\left( i\right) $ The solution to (\[FE\]) is: $$\mathbf{P}\left( \nu \leq j\right) =F\left( j\right) =\frac{1-q^{j}}{% 1-q^{j+1}}\text{, }j=1,2,...$$ $\left( ii\right) $ The sequence $\overline{F}\left( j\right) $ is CM and so $\nu -1$ is ID. The distribution $F\left( j\right) $ has decreasing failure rate (DFR). $\left( iii\right) $ There are two ways to generate the corresponding branching number $\nu :$$$\left( iii-a\right) :\text{ }\nu =\inf \left( i\geq 1:\mathcal{B}_{i}\left( \alpha _{i}\right) =1\right) ,$$ where $\left( \mathcal{B}_{i}\left( \alpha _{i}\right) ;\text{ }i\geq 1\right) $ is an independent sequence of Bernoulli rvs with success parameter $\alpha _{i}=1/\left( 1+q+...+q^{i}\right) .$ Or: $$\left( iii-b\right) :\text{ }\nu =\max_{i=1,...,G}\xi _{i}$$ where $G\overset{d}{\sim }$geom$\left( p\right) $ independent of the iid sequence $\left( \xi _{i}\text{, }i\geq 1\right) $ with $\xi _{1}\overset{d}{% \sim }$geom$\left( p\right) .$ $\left( iv\right) $ The tails of both $\left( \nu ,X_{\infty }\right) $ are geometric with: $\mathbf{P}\left( \nu >j\right) /\mathbf{P}\left( X_{\infty }>j\right) \rightarrow p<1$. *Proof:* $\left( i\right) $ We have $$\Phi _{\infty }\left( z\right) =\mathbf{E}\left( z^{X_{\infty }}\right) =% \frac{pz}{1-qz}\text{ and so}$$ $$\Phi _{\infty }\left( F\left( j\right) \right) =\frac{p\frac{1-q^{j}}{% 1-q^{j+1}}}{1-q\left( \frac{1-q^{j}}{1-q^{j+1}}\right) }=1-q^{j}=\mathbf{P}% \left( X_{\infty }\leq j\right) =F_{\infty }\left( j\right) .$$ $\left( ii\right) $ With $\lambda \left( du\right) =p\sum_{j\geq 1}q^{j-1}\delta _{q^{j}}$, a probability measure, $$\overline{F}\left( j\right) =\frac{pq^{j}}{1-q^{j+1}}=\int_{0}^{1}u^{j}% \lambda \left( du\right) .$$ The rv $\nu \geq 1$ has finite mean $\mathbf{E}\left( \nu \right) =\sum_{j\geq 0}\mathbf{P}\left( \nu >j\right) =1+p\sum_{j\geq 1}q^{j}/\left( 1-q^{j+1}\right) <\infty $ and $\left\{ X_{n}\right\} $ is recurrent positive. For $j\geq 1$, we have $$\frac{\mathbf{P}\left( \nu =j\right) }{\mathbf{P}\left( \nu >j\right) }=% \frac{p}{q}\frac{1}{1-q^{j}}$$ which is decreasing with $j$. $\left( iii\right) $ The first statement $\left( iii-a\right) $ results from: $\mathbf{P}\left( \nu >i\right) =\prod_{j=1}^{i}\left( 1-\alpha _{j}\right) =\frac{pq^{i}}{1-q^{i+1}}.$ $\left( iii-b\right) $ results from $$\mathbf{P}\left( \max_{i=1,...,G}\xi _{i}>i\right) =\sum_{k\geq 1}pq^{k-1}q^{ik}=\frac{pq^{i}}{1-q^{i+1}}.$$ $\left( iv\right) $ The tails of $\nu $ are given by $\mathbf{P}\left( \nu >j\right) =1-\Phi _{\infty }^{-1}\left( \mathbf{P}\left( X_{\infty }\leq j\right) \right) =1-\frac{z}{p+qz}\mid _{1-p^{j}}\sim pq^{j}$ (for large $j$) with $\mathbf{P}\left( \nu >j\right) /\mathbf{P}\left( X_{\infty }>j\right) \rightarrow p<1$. For this model, $\nu $ and $X_{\infty }$ are geometric (power-law) and tail-equivalent but the tails of $\nu $ are thinner than the ones of $X_{\infty }.$ $\Box $ Related examples to the geometric one ($X_{\infty }$ having dominant geometric tails with an algebraic prefactor): - Suppose $X_{\infty }\overset{d}{\sim }$**negative-binomial ,**conditioned to be positive**:** With** **$\left[ \alpha \right] _{k}=\Gamma \left( \alpha +k\right) /\Gamma \left( \alpha \right) $, $\alpha >0$, suppose $$\Phi _{\infty }\left( z\right) =\mathbf{E}\left( z^{X_{\infty }}\right) =% \frac{\left( \frac{p}{1-qz}\right) ^{\alpha }-p^{\alpha }}{1-p^{\alpha }}$$ is the pgf of a negative-binomial** **rv**,** conditioned to be positive. Then, by direct inversion $$F\left( j\right) =\Phi _{\infty }^{-1}\left( F_{\infty }\left( j\right) \right) =\frac{1-\left( 1+\sum_{k=1}^{j}\frac{\left[ \alpha \right] _{k}}{k!}% q^{k}\right) ^{-1/\alpha }}{q},$$ which defines the pdf of $\nu $. In this case, $\Psi _{\infty }\left( z\right) =z^{-1}\Phi _{\infty }\left( z\right) $ is not the pgf of an ID rv. Plugging $\alpha =1$ gives back the latter geometric case. The tails of $% X_{\infty }$ goes, up to a constant prefactor, like $j^{\alpha -1}q^{j}$. - Suppose $X_{\infty }\overset{d}{\sim }$shifted **negative-bin (**$% \Psi _{\infty }\left( z\right) $ now is the pgf of the ID rv $X_{\infty }-1$): then $$\begin{aligned} \Phi _{\infty }\left( z\right) &=&\mathbf{E}\left( z^{X_{\infty }}\right) =z\left( \frac{p}{1-qz}\right) ^{\alpha }\text{ and }\Psi _{\infty }\left( z\right) =p^{\alpha }\left( 1-qz\right) ^{-\alpha } \\ \left[ z^{j}\right] \Psi _{\infty }\left( z\right) &=&p^{\alpha }\frac{% \left[ \alpha \right] _{j}}{j!}q^{j}\Rightarrow \left[ z^{j}\right] \Psi _{\infty }^{-n}\left( z\right) =\frac{\left[ -n\alpha \right] _{j}}{j!}% p^{-n\alpha }q^{j} \\ \frac{1}{n}\left[ z^{n-1}\right] \Psi _{\infty }\left( z\right) ^{-n} &=&% \frac{\left[ -n\alpha \right] _{n-1}}{n!}p^{-n\alpha }q^{n-1} \\ F_{\infty }\left( j\right) &=&\sum_{k=1}^{j}\mathbf{P}\left( X_{\infty }=k\right) =p^{\alpha }\sum_{k=1}^{j}\frac{\left[ \alpha \right] _{k-1}}{% \left( k-1\right) !}q^{k-1} \\ F\left( j\right) &=&\Phi _{\infty }^{-1}\left( F_{\infty }\left( j\right) \right) =\sum_{n\geq 1}F_{\infty }\left( j\right) ^{n}\frac{1}{n}\left[ z^{n-1}\right] \Psi _{\infty }\left( z\right) ^{-n}.\end{aligned}$$ The negative-binomial distribution with pgf $\Psi _{\infty }\left( z\right) =p^{\alpha }\left( 1-qz\right) ^{-\alpha }$ is CM (and so log-convex, ID and DFR) only when $\alpha \leq 1$. When $\alpha \geq 1$ it is log-concave, ID and IFR. - $X_{\infty }\overset{d}{\sim }$**Fisher log-series**. With $p\in \left( 0,1\right) $ and $c=-\log \left( 1-p\right) $, suppose $$\begin{aligned} \Phi _{\infty }\left( z\right) &=&\mathbf{E}\left( z^{X_{\infty }}\right) =-c^{-1}\log \left( 1-pz\right) =:z\Psi _{\infty }\left( z\right) \\ \mathbf{P}\left( X_{\infty }=k\right) &=&c^{-1}p^{k}/k\text{, }k\geq 1\text{ and }\mathbf{P}\left( X_{\infty }\leq j\right) =c^{-1}\sum_{k=1}^{j}p^{k}/k% \text{ }\end{aligned}$$ involving a truncated logarithm. We have $$\begin{aligned} \Phi _{\infty }^{-1}\left( z\right) &=&p^{-1}\left( 1-e^{-cz}\right) \text{ and } \\ F\left( j\right) &=&\Phi _{\infty }^{-1}\left( F_{\infty }\left( j\right) \right) =p^{-1}\left( 1-e^{-\sum_{k=1}^{j}p^{k}/k}\right) .\end{aligned}$$ For all $j\geq 1$, we have by construction $$F\left( j\right) >F_{\infty }\left( j\right) =c^{-1}\sum_{k=1}^{j}p^{k}/k.$$ The tails of $X_{\infty }$ goes, up to a constant prefactor, like $% j^{-1}p^{j}$. In addition, $$\begin{aligned} \mathbf{P}\left( X_{\infty }=i\right) &=&\int_{0}^{1}u^{i-1}\mu \left( du\right) \text{ where }\mu \left( du\right) =c^{-1}1_{u\in \left( 0,p\right) }du \\ \mathbf{P}\left( X_{\infty }>i\right) &=&\int_{0}^{1}u^{i}\lambda \left( du\right) \text{ where }\lambda \left( du\right) =c^{-1}\left( 1-u\right) ^{-1}1_{u\in \left( 0,p\right) }du,\end{aligned}$$ and both $X_{\infty }$ and $\nu $ are CM. Let us now look at situations when $X_{\infty }$ has heavy (algebraic) tails with index $\alpha >0$: - The power-law **Sibuya** example, [@Sibu]. With $\alpha \in \left( 0,1\right) $, suppose $X_{\infty }\overset{d}{\sim }$Sibuya$\left( \alpha \right) $, that is: $\Phi _{\infty }\left( z\right) =\mathbf{E}\left( z^{X_{\infty }}\right) =1-\left( 1-z\right) ^{\alpha },$ with $\mathbf{P}\left( X_{\infty }=j\right) =\pi \left( j\right) =\alpha \left[ 1-\alpha \right] _{j-1}/j!$. Then: $\left( i\right) $ The sequence $\pi \left( j\right) $ is CM, so log-convex, DFR and $X_{\infty }-1$ is ID. $\left( ii\right) $ $$X_{\infty }=\inf \left( i\geq 1:\mathcal{B}_{i}\left( \alpha _{i}\right) =1\right) , \label{ber}$$ where $\left( \mathcal{B}_{i}\left( \alpha _{i}\right) \right) _{i\geq 1}$ is a sequence of independent Bernoulli rvs obeying $\mathbf{P}\left( \mathcal{B}_{i}\left( \alpha _{i}\right) =1\right) =\alpha /i.$ $\left( iii\right) $ The solution to (\[FE\]) is: $$\mathbf{P}\left( \nu \leq i\right) =1-\left( 1-\alpha \sum_{j=1}^{i}\left[ 1-\alpha \right] _{j-1}/j!\right) ^{1/\alpha },$$ $\left( iv\right) $ Both $\left( X_{\infty },\nu \right) $ have algebraic (power-law) tails, but with tail index $\alpha $ and $1$ respectively. $\left( v\right) $ We have $$\mathbf{P}\left( \nu >i\right) \sim \frac{1}{\Gamma \left( 1-\alpha \right) ^{1/\alpha }}i^{-1}\text{ as }j\rightarrow \infty$$ and $1/\Gamma \left( 1-\alpha \right) ^{1/\alpha }<e^{-\gamma }.$ For all $% \alpha \in \left( 0,1\right) $, the Lamperti chain generated by $\nu $ is positive recurrent, with invariant probability measure $\mathbf{\pi }$. *Proof:* $\left( i\right) $ It can be checked that, with $\mu \left( du\right) \overset{d}{\sim }$Beta$\left( 1-\alpha ,\alpha \right) $$$\mathbf{P}\left( X_{\infty }=j\right) =\int_{0}^{1}u^{j}\mu \left( du\right) .$$ $\left( ii\right) $ is obvious and a well-known property of Sibuya$\left( \alpha \right) $ distributed rvs, [@Sibu]. $\left( iii\right) $ We have $\Phi _{\infty }^{-1}\left( z\right) =1-\left( 1-z\right) ^{1/\alpha }$ and so $$\mathbf{P}\left( \nu \leq i\right) =1-\left( 1-\alpha \sum_{j=1}^{i}\left[ 1-\alpha \right] _{j-1}/j!\right) ^{1/\alpha }.$$ $\left( iv\right) $ We have $\mathbf{P}\left( X_{\infty }=j\right) \sim \frac{\alpha }{\Gamma \left( 1-\alpha \right) }j^{-\left( \alpha +1\right) }$ and $\mathbf{P}\left( X_{\infty }>j\right) \sim \frac{1}{\Gamma \left( 1-\alpha \right) }j^{-\alpha }.$ Therefore $\mathbf{P}\left( \nu \leq j\right) \sim \Phi _{\infty }^{-1}\left( \mathbf{P}\left( X_{\infty }\leq j\right) \right) \sim 1-\mathbf{P}\left( X_{\infty }>j\right) ^{1/\alpha }\sim 1-\frac{1}{\Gamma \left( 1-\alpha \right) ^{1/\alpha }}j^{-1}.$ And $% \nu $ has lighter tails (of index $1$) than $X_{\infty }$ (of index $\alpha $)$.$ This is a concrete manifestation in the tails of the fact that $% X_{\infty }$ is stochastically larger than $\nu $. $\left( v\right) $ To decide whether or not $\nu $ belongs to the ergodic family, (\[L1\]), we need to compare $1/\Gamma \left( 1-\alpha \right) ^{1/\alpha }$ with $e^{-\gamma }$, $\gamma =-\Gamma ^{\prime }\left( 1\right) $ being the Euler constant. Indeed, based on Lamperti’s criterion, the chain is recurrent if $1/\Gamma \left( 1-\alpha \right) ^{1/\alpha }<e^{-\gamma }$ or $\log \Gamma \left( 1-\alpha \right) /\alpha >\gamma $ for all $\alpha \in \left( 0,1\right) $. But this is always true because $% \log \Gamma \left( 1-\alpha \right) /\alpha $ is an increasing function of $% \alpha $ with $\log \Gamma \left( 1-\alpha \right) /\alpha \rightarrow \gamma $ as $\alpha \rightarrow 0$ ($\log \Gamma \left( 1-\alpha \right) \sim \log \left( 1-\alpha \Gamma ^{\prime }\left( 1\right) \right) \sim -\alpha \Gamma ^{\prime }\left( 1\right) $)$.$ The critical upper bound $% e^{-\gamma }$ for the coefficient $1/\Gamma \left( 1-\alpha \right) ^{1/\alpha }$ is attained for $\alpha \rightarrow 0.$ $\Box $ Related examples to the Sibuya one with power-law tails are: - **Pareto** ($\alpha >0$): Suppose $\mathbf{P}\left( X_{\infty }>i\right) =\left( i+1\right) ^{-\alpha }$. Clearly, $$X_{\infty }=\inf \left( i\geq 1:\mathcal{B}_{i}\left( \alpha _{i}\right) =1\right) ,$$ where $\left( \mathcal{B}_{i}\left( \alpha _{i}\right) \right) _{i\geq 1}$ is a sequence of independent Bernoulli rvs obeying $\mathbf{P}\left( \mathcal{B}_{i}\left( \alpha _{i}\right) =1\right) =1-\left( 1+1/i\right) ^{-\alpha }$ where $\alpha >0$. Indeed, $$\mathbf{P}\left( X_{\infty }>i\right) =\prod_{j=1}^{i}\left( 1-\alpha _{j}\right) =\prod_{j=1}^{i}\left( 1+1/j\right) ^{-\alpha }=\left( i+1\right) ^{-\alpha }.$$ We have $\mathbf{P}\left( X_{\infty }=i\right) =i^{-\alpha }-\left( i+1\right) ^{-\alpha }=i^{-\alpha }\left( 1-\left( \left( i+1\right) /i\right) ^{-\alpha }\right) \sim \alpha i^{-\left( \alpha +1\right) }$ and so $\Phi _{\infty }\left( z\right) =\sum_{i\geq 1}z^{i}i^{-\alpha }-\sum_{i\geq 1}z^{i}\left( i+1\right) ^{-\alpha }=1-z^{-1}\left( 1-z\right) L_{\alpha }\left( z\right) =z\Psi _{\infty }\left( z\right) .$ When $\alpha \leq 1$, the polylog function $L_{\alpha }\left( z\right) =\sum_{i\geq 1}z^{i}i^{-\alpha }$ is not defined at $z=1$ but $z\Psi _{\infty }\left( z\right) =1-z^{-1}\left( 1-z\right) L_{\alpha }\left( z\right) $ is a true pgf taking the value $1$ at $z=1$. Lagrange inversion formula gives the power-series expansion of $\Phi _{\infty }^{-1}\left( z\right) $ giving $\mathbf{P}\left( \nu \leq j\right) =\Phi _{\infty }^{-1}\left( 1-\left( j+1\right) ^{-\alpha }\right) .$ The rv $X_{\infty }-1$ (with pgf $\Psi _{\infty }\left( z\right) $) is infinitely divisible. Indeed, the polylogarithm can be expressed in terms of the integral of the Bose-Einstein distribution $$L_{\alpha }\left( z\right) =\frac{1}{\Gamma \left( \alpha \right) }% \int_{0}^{\infty }\frac{x^{\alpha -1}}{z^{-1}e^{x}-1}dx=\frac{z}{\Gamma \left( \alpha \right) }\int_{0}^{1}\frac{\left( -\log u\right) ^{\alpha -1}}{% 1-uz}du$$ showing, by Hausdorff representation, that $$\mathbf{P}\left( X_{\infty }>i\right) =\left( i+1\right) ^{-\alpha }=\int_{0}^{1}u^{i}\lambda \left( du\right) \text{ where }\lambda \left( du\right) =\frac{1}{\Gamma \left( \alpha \right) }\left( -\log u\right) ^{\alpha -1}du\text{ }$$ is the probability density of $U=e^{-X}$, with $X\overset{d}{\sim }$Gamma$% \left( \alpha ,1\right) .$ The law of $X_{\infty }\geq 1$ is completely monotone (and $X_{\infty }-1$ is ID). Note $$\begin{aligned} \Phi _{\infty }\left( z\right) &=&1-z^{-1}\left( 1-z\right) L_{\alpha }\left( z\right) =z\int_{0}^{1}\frac{1-u}{1-uz}\lambda \left( du\right) =z\Psi _{\infty }\left( z\right) \\ \Psi _{\infty }\left( z\right) &=&\mathbf{E}\left( z^{X_{\infty }-1}\right) =\int_{0}^{1}\frac{1}{1-zu}\mu \left( du\right) \text{ where }\mu \left( du\right) =\left( 1-u\right) \lambda \left( du\right)\end{aligned}$$ - **Zipf** ($\alpha >1$): Suppose $\mathbf{P}\left( X_{\infty }=i\right) =i^{-\alpha }/\varsigma \left( \alpha \right) $ with associated pgf $\Phi _{\infty }\left( z\right) =L_{\alpha }\left( z\right) /L_{\alpha }\left( 1\right) ,$ $L_{\alpha }\left( 1\right) =\varsigma \left( \alpha \right) .$ Lagrange inversion formula gives the power-series expansion of $% \Phi _{\infty }^{-1}\left( z\right) $. We have $$\mathbf{P}\left( \nu \leq j\right) =\Phi _{\infty }^{-1}\left( 1-\mathbf{P}% \left( X_{\infty }>i\right) \right)$$ where, with $\lambda _{0}\left( du\right) =\frac{1}{\Gamma \left( \alpha \right) }\left( -\log u\right) ^{\alpha -1}du$$$\begin{aligned} \mathbf{P}\left( X_{\infty }>i\right) &=&\frac{1}{\varsigma \left( \alpha \right) }\sum_{j>i}j^{-\alpha }=\frac{1}{\varsigma \left( \alpha \right) }% \int_{0}^{1}\sum_{j>i}u^{j-1}\lambda _{0}\left( du\right) =\frac{1}{% \varsigma \left( \alpha \right) }\int_{0}^{1}u^{i}\left( 1-u\right) ^{-1}\lambda _{0}\left( du\right) \\ &=&\int_{0}^{1}u^{i}\lambda \left( du\right) \text{ where }\lambda \left( du\right) =\frac{\left( 1-u\right) ^{-1}}{\varsigma \left( \alpha \right) \Gamma \left( \alpha \right) }\left( -\log u\right) ^{\alpha -1}du\text{ }\end{aligned}$$ is the probability density of $U=e^{-X}$, with $X$ having density $$\frac{1}{\varsigma \left( \alpha \right) \Gamma \left( \alpha \right) }\frac{% e^{-x}x^{\alpha -1}}{1-e^{-x}}\text{, }x>0.$$ So $X_{\infty }$ (and $X_{\infty }-1$) is CM. Thus $X_{\infty }-1$ is infinitely divisible and even self-decomposable, say SD (see Example $12.18$ page $435$ of [@SH]). - **The critical case when** $X_{\infty }$ **has no moments of any positive order**: Suppose that with $\beta >0$ and $L_{1}\left( x\right) =\log \left( 1+x\right) >0$, slowly varying at $\infty $ $$\mathbf{P}\left( X_{\infty }=j\right) =\frac{C_{0}}{jL_{1}\left( j\right) ^{\beta +1}},\text{ }j\geq 1$$ where $C_{0}>0$ is the normalizing constant. Then $\mathbf{E}\left( X_{\infty }^{q}\right) =\infty $ for all $q>0.$ In this case, $\mathbf{P}% \left( X_{\infty }>j\right) \sim C_{0}\cdot L_{1}\left( j\right) ^{-\beta }$ as $j\rightarrow \infty $ with tails heavier than any power-law. Then: $\left( i\right) $ The rv $\nu $ whose distribution solves (\[FE\]) (as from Proposition $2$) is a well-defined rv obeying $j\mathbf{P}\left( \nu >j\right) \rightarrow e^{-\gamma }$ as $j\rightarrow \infty .$ $\left( ii\right) $ Furthermore $$\mathbf{P}\left( \nu >j\right) \underset{j\uparrow \infty }{\sim }e^{-\gamma }/j+d/\left( j\log j\right) +o\left( 1/\left( j\log j\right) \right)$$ with $$d=-\frac{\left( \beta +1\right) e^{-\gamma }\pi ^{2}}{12}<-\frac{e^{-\gamma }\pi ^{2}}{12}.$$ By (\[L3\]), the corresponding Lamperti chain is critical but it remains positive recurrent for all $\beta >0.$ *Proof:* $\left( i\right) $ This model for $X_{\infty }$ is indeed obtained in the limit $\alpha \rightarrow 0$ of the ansatz ($\alpha >0$) $$\mathbf{P}\left( X_{\infty }=j\right) =\frac{C_{0}}{j^{\alpha +1}L_{1}\left( j\right) ^{\beta +1}},\text{ }i\geq 1,$$ extending the previous Sibuya example with tail index $\alpha $. $\left( ii\right) $ In such an example of $X_{\infty }$ with logarithmic tails, we have more precisely $$\Phi _{\infty }\left( z\right) \underset{z\uparrow 1}{\sim }1-\frac{C_{0}}{% \left( -\log \left( 1-z\right) \right) ^{\beta }}$$ with local inverse: $\Phi _{\infty }^{-1}\left( z\right) \underset{z\uparrow 1}{\sim }1-e^{-\left( \frac{1-z}{C_{0}}\right) ^{-1/\beta }}$. We get $$\frac{1-\Phi _{\infty }\left( z\right) }{1-z}\underset{z\uparrow 1}{\sim }% \frac{1}{1-z}\frac{C_{0}}{\left( -\log \left( 1-z\right) \right) ^{\beta }}$$ so that [@Fla], with $C_{k}=\left( \frac{1}{\Gamma \left( \alpha \right) }\right) ^{\left( k\right) }\mid _{\alpha =1}$(in particular $C_{1}=\gamma $, $C_{2}=\gamma ^{2}-\pi ^{2}/6,$ with $C_{1}^{2}-C_{2}=\pi ^{2}/6$) $$\mathbf{P}\left( X_{\infty }>j\right) \underset{j\uparrow \infty }{\sim }% \frac{C_{0}}{\log ^{\beta }j}\left( 1-\frac{\beta C_{1}}{\log j}+\frac{\beta \left( \beta +1\right) C_{2}}{2\log ^{2}j}+o\left( \frac{1}{\log ^{2}j}% \right) \right) .$$ Observing $\left( 1-\frac{\beta C_{1}}{\log j}+\frac{\beta \left( \beta +1\right) C_{2}}{2\log ^{2}j}\right) ^{-1/\beta }\underset{j\uparrow \infty }{\sim }1+\frac{C_{1}}{\log j}+\frac{\left( \beta +1\right) }{2\left( \log j\right) ^{2}}\left( C_{1}^{2}-C_{2}\right) $, we are led to $$\begin{aligned} &&\mathbf{P}\left( \nu \leq j\right) \underset{j\uparrow \infty }{\sim }\Phi _{\infty }^{-1}\left( 1-\mathbf{P}\left( X_{\infty }>j\right) \right) \underset{j\uparrow \infty }{\sim }1-\left( \frac{1}{j}\right) ^{\left( 1-% \frac{\beta C_{1}}{\log j}+\frac{\beta \left( \beta +1\right) C_{2}}{2\log ^{2}j}\right) ^{-1/\beta }} \\ &&\underset{j\uparrow \infty }{\sim }1-e^{-\gamma }/j-d/\left( j\log j\right) +o\left( 1/\left( j\log j\right) \right)\end{aligned}$$ with $$d=-\frac{\left( \beta +1\right) e^{-\gamma }\pi ^{2}}{12}.$$ Because $d<-\pi ^{2}e^{-\gamma }/12$ for all $\beta >0,$ we conclude that $% \left\{ X_{n}\right\} $ generated by this $\nu $ just remains always positive-recurrent. $\Box $ **- Null-recurrent issues.*** * Irreducible aperiodic Markov chains may have or not a non-trivial invariant positive (infinite) measure, [@Harr]. In the null-recurrent case from (\[L2\]), the Lamperti model has a non trivial ($\neq \mathbf{0}$) invariant positive measure. *Proof:* To see a transition positive/null recurrence transition in the critical case, suppose $\delta \left( j\right) :=\Delta F_{\infty }\left( j\right) >0$ with $\Delta F_{\infty }\left( j\right) \rightarrow 0$ as $% j\rightarrow \infty ,$ $\Phi _{\infty }\left( z\right) =\sum_{j\geq 1}\Delta F_{\infty }\left( j\right) z^{j}$ convergent for all $z\in \left[ 0,1\right) $, $\Phi _{\infty }\left( 0\right) =0$ and $\Phi _{\infty }\left( 1\right) =\infty .$ In this case $\Delta F_{\infty }\left( j\right) $ no longer is a probability mass at $j$. One can search solutions of (\[FE\]) in this case as well and Proposition $2$ applies simply while substituting $\delta \left( j\right) $ to $\pi \left( j\right) $ in the obtained expression of $\mathbf{P% }\left( \nu \leq j\right) .$ Because $\mathbf{P}\left( \nu \leq j\right) $ only depends on the ratio $F_{\infty }\left( j\right) /F_{\infty }\left( 1\right) $, regardless of any normalization, such a sequence $\delta \left( j\right) $ defines an invariant positive and infinite measure in the null-recurrent case. $\Box $ The simplest example is the following: $\Delta F_{\infty }\left( j\right) =1/j$, with $\Phi _{\infty }\left( z\right) =\sum_{j\geq 1}\Delta F_{\infty }\left( j\right) z^{j}$ obeying $\Phi _{\infty }\left( 1\right) =\infty .$ We have $\Phi _{\infty }\left( z\right) =-\log \left( 1-z\right) $ so that, upon inverting $\Phi _{\infty }$$$\mathbf{P}\left( \nu \leq j\right) =1-e^{-\sum_{k=1}^{j}\frac{1}{k}}$$ a true pdf. Recalling $\sum_{k=1}^{j}\frac{1}{k}-\gamma -\log j\sim 1/\left( 2j\right) $, we get $\mathbf{P}\left( \nu >j\right) \sim e^{-\gamma }/j+O\left( j^{-2}\right) .$ The constant $d$ in (\[L3\]) is $d=0$ and the Lamperti chain with a branching number $\nu $ distributed as such is null-recurrent. This is also true if $\Delta F_{\infty }\left( j\right) =1/\left[ j\log \left( 1+j\right) ^{\beta +1}\right] $ with $\beta <0$ or $% \Delta F_{\infty }\left( j\right) =j^{-\alpha },$ $\alpha \in \left( 0,1\right) ,$ both expressions leading to a diverging series $\Phi _{\infty }\left( 1\right) $. - **Transient issues: non-unicity of the invariant measure.** Whenever** **$\left\{ X_{n}\right\} $ is transient, one obvious solution to the invariant measure equation $\mathbf{\pi }^{\prime }=\mathbf{\pi }% ^{\prime }P$ is $\mathbf{\pi }=\mathbf{0}$. This corresponds to the fact that $X_{\infty }\overset{d}{\sim }\delta _{\infty }$. However this solution is not unique and there are other invariant positive measures. The question of the existence of a non-trivial invariant measure for transient chains was raised by Harris, [@Har]. To exhibit such an invariant measure, suppose $\delta \left( j\right) :=\Delta F_{\infty }\left( j\right) >0$ with $\Phi _{\infty }\left( z\right) =\sum_{j\geq 1}\Delta F_{\infty }\left( j\right) z^{j}$ convergent for all $% z\in \left[ 0,1\right) $, $\Phi _{\infty }\left( 0\right) =0$ and $\Phi _{\infty }\left( 1\right) =\infty .$ In this case $\Delta F_{\infty }\left( j\right) $ no longer is a probability mass at $j$ either but it is no longer required $\Delta F_{\infty }\left( j\right) \rightarrow 0$ as $j\rightarrow \infty .$ In the transient case from (\[L3\]), the Lamperti model has a non trivial ($\neq \mathbf{0}$) invariant positive measure. *Proof:* One can search solutions of (\[FE\]) in this case as well and Proposition $2$ applies simply while substituting $\delta \left( j\right) $ to $\pi \left( j\right) $ in the obtained expression of $\mathbf{P% }\left( \nu \leq j\right) .$ Because, from (\[FES\]), $\mathbf{P}\left( \nu \leq j\right) $ only depends on the ratio $F_{\infty }\left( j\right) /F_{\infty }\left( 1\right) $ regardless of any normalization, such a sequence $\delta \left( j\right) $ defines an invariant measure in the transient case as well. $\Box $ - The simplest explicit example is the following counting measure one: $% \delta \left( j\right) =\Delta F_{\infty }\left( j\right) =1$, $F_{\infty }\left( j\right) =j$, with $\Phi _{\infty }\left( z\right) =\sum_{j\geq 1}\Delta F_{\infty }\left( j\right) z^{j}=z/\left( 1-z\right) $ obeying $% \Phi _{\infty }\left( 1\right) =\infty .$ There is a solution to (\[FE\]) which is $$\mathbf{P}\left( \nu \leq j\right) =\Phi _{\infty }^{-1}\left( j\right) =% \frac{j}{1+j}.$$ We have: $\mathbf{P}\left( \nu >j\right) =1/\left( 1+j\right) $ so that $j% \mathbf{P}\left( \nu >j\right) \underset{j\rightarrow \infty }{\rightarrow }% 1>e^{-\gamma }$, indeed corresponding to a transient case. - Suppose now $\Delta F_{\infty }\left( j\right) =j$, $F_{\infty }\left( j\right) =j\left( j+1\right) /2,$ so with $\Phi _{\infty }\left( z\right) =\sum_{j\geq 1}\Delta F_{\infty }\left( j\right) z^{j}=z/\left( 1-z\right) ^{2}$ obeying $\Phi _{\infty }\left( 1\right) =\infty .$ There is a solution to (\[FE\]) which is $$\mathbf{P}\left( \nu \leq j\right) =\Phi _{\infty }^{-1}\left( \frac{j\left( j+1\right) }{2}\right) =\frac{j\left( j+1\right) +1-\sqrt{1+2j\left( j+1\right) }}{j\left( j+1\right) }.$$ When inverting $\Phi _{\infty }\left( z\right) $ we have chosen the branch for which $\Phi _{\infty }^{-1}\left( 0\right) =0$. We have: $\mathbf{P}% \left( \nu >j\right) =\left( \sqrt{1+2j\left( j+1\right) }-1\right) /\left( j\left( j+1\right) \right) $ so that $j\mathbf{P}\left( \nu >j\right) \rightarrow \sqrt{2}>e^{-\gamma }$, also corresponding to a transient case. Defining the reversed failure rate of the sequence $\delta \left( j\right) $ as $$\overline{r}\left( j\right) =\frac{\delta \left( j\right) }{% \sum_{k=1}^{j}\delta \left( k\right) }=\frac{\Delta F_{\infty }\left( j\right) }{F_{\infty }\left( j\right) }\text{, }j\geq 1,$$ we conclude that in both examples, $\overline{r}\left( j\right) \asymp 1/j$ so with decreasing reversed failure rate. *Remark:* By the ergodic theorem: - in case (\[L1\]): $$n^{-1}\sum_{m=1}^{n}\mathbf{1}\left( X_{m}=j\mid X_{0}\overset{d}{\sim }% \mathbf{\pi }_{0}\right) \rightarrow \pi \left( j\right) \text{ as }% n\rightarrow \infty ,$$ - in cases (\[L2\]) and (\[L3\]): For all states $i,j\geq 1$$$\frac{\sum_{m=1}^{n}\mathbf{1}\left( X_{m}=i\mid X_{0}\overset{d}{\sim }% \mathbf{\pi }_{0}\right) }{\sum_{m=1}^{n}\mathbf{1}\left( X_{m}=j\mid X_{0}% \overset{d}{\sim }\mathbf{\pi }_{0}\right) }\rightarrow \frac{\delta \left( i\right) }{\delta \left( j\right) }\text{ as }n\rightarrow \infty .\text{ }% \diamondsuit$$ - **Poisson target:** We finally develop some additional examples in the recurrent case, not in the latter classes and related to the fundamental Poisson distribution class: - **Shifted** **Poisson:** Suppose** **$\Phi _{\infty }\left( z\right) =\mathbf{E}\left( z^{X_{\infty }}\right) =ze^{\lambda \left( z-1\right) }=z\Psi _{\infty }\left( z\right) ,$ **(**$\Psi _{\infty }\left( z\right) $ is the pgf of an ID Poisson rv which is log-concave). Then $$\begin{aligned} \frac{1}{n}\left[ z^{n-1}\right] \Psi _{\infty }\left( z\right) ^{-n} &=&% \frac{1}{n}\left[ z^{n-1}\right] e^{-n\lambda \left( z-1\right) }=\left( -1\right) ^{n-1}\frac{e^{\lambda n}}{n!}\left( n\lambda \right) ^{n-1} \\ F_{\infty }\left( j\right) &=&e^{-\lambda }\sum_{k=0}^{j-1}\frac{\lambda ^{k}% }{k!} \\ F\left( j\right) &=&\sum_{n\geq 1}\left( -1\right) ^{n-1}\frac{e^{\lambda n}% }{n!}\left( n\lambda \right) ^{n-1}F_{\infty }\left( j\right) ^{n} \\ &=&\sum_{n\geq 1}\left( -1\right) ^{n-1}\frac{\left( n\lambda \right) ^{n-1}% }{n!}\left( \sum_{k=0}^{j-1}\frac{\lambda ^{k}}{k!}\right) ^{n}=W_{\lambda }\left( \sum_{k=0}^{j-1}\frac{\lambda ^{k}}{k!}\right)\end{aligned}$$ The Lambert function, solving $x=W\left( x\right) e^{W\left( x\right) },$ is (by Lagrange inversion formula): $$\begin{aligned} W\left( x\right) &=&\sum_{n\geq 1}\left( -1\right) ^{n-1}\frac{n^{n-1}}{n!}% x^{n}\text{ hence} \\ W_{\lambda }\left( x\right) &:&=\lambda ^{-1}W\left( \lambda x\right) =\sum_{n\geq 1}\left( -1\right) ^{n-1}\frac{\left( n\lambda \right) ^{n-1}}{% n!}x^{n}.\end{aligned}$$ And $W_{\lambda }\left( x\right) $ solves: $x=W_{\lambda }\left( x\right) e^{\lambda W_{\lambda }\left( x\right) }$. It is positive and increasing when $x>0$, so $F\left( j\right) $ is a well-defined pdf if $F\left( \infty \right) =W_{\lambda }\left( e^{\lambda }\right) =1,$ which is the case. - **Poisson conditioned to be positive:** Suppose** **$\Phi _{\infty }\left( z\right) =\mathbf{E}\left( z^{X_{\infty }}\right) =\left( e^{\lambda z}-1\right) /\left( e^{\lambda }-1\right) $, leading directly to $\Phi _{\infty }^{-1}\left( z\right) =% \frac{1}{\lambda }\log \left( 1+z\left( e^{\lambda }-1\right) \right) .$ Then $$\begin{aligned} F_{\infty }\left( j\right) &=&\frac{1}{e^{\lambda }-1}\sum_{k=1}^{j}\frac{% \lambda ^{k}}{k!} \\ F\left( j\right) &=&\Phi _{\infty }^{-1}\left( F_{\infty }\left( j\right) \right) =\frac{1}{\lambda }\log \left( 1+F_{\infty }\left( j\right) \left( e^{\lambda }-1\right) \right) ,\end{aligned}$$ which defines a pdf with $F\left( \infty \right) =1$. In this case, although $\Psi _{\infty }\left( z\right) =z^{-1}\Phi _{\infty }\left( z\right) $ is not the pgf of an ID rv, the calculation of $F\left( j\right) $ is straightforward. Examples of $\nu \rightarrow \nu _{\left( N\right) }$ with finite support $\left\{ 1,...,N\right\} $. ----------------------------------------------------------------------------------------------------- In this Sub-section, we look at situations where both $\left( X_{\infty },\nu _{\left( N\right) }\right) $ have finite support $\left\{ 1,...,N\right\} $. Note that if $\nu $ has support $\left\{ 1,...,N\right\} $, so does $\left\{ X_{n}\right\} $ (defined recursively by $% X_{n+1}=\max_{j=1,...,X_{n}}\nu _{j,n+1}$) and then $X_{\infty }$. Conversely, if $X_{\infty }$ has support $\left\{ 1,...,N\right\} $, there exists $\nu $ with support $\left\{ 1,...,N\right\} $ such that $% X_{n+1}=\max_{j=1,...,X_{n}}\nu _{j,n+1}$ defines a sequence $\left( X_{n}\right) $ with finite support. In such cases, the Lamperti Markov chain will always be ergodic in view of its transition matrix $P_{\left( N\right) } $ being irreducible. We shall let $\pi _{\left( N\right) }\left( k\right) =% \mathbf{P}\left( X_{\infty }=k\right) .$ - **The general case:** Suppose $\Phi _{\infty }\left( z\right) =\sum_{k=1}^{N}\pi _{\left( N\right) }\left( k\right) z^{k}$, so that $\Psi _{\infty }\left( z\right) =\sum_{k=0}^{N-1}\pi _{\left( N\right) }\left( k+1\right) z^{k}$. We have $$\begin{aligned} \Psi _{\infty }\left( z\right) ^{-\alpha } &=&\pi _{\left( N\right) }\left( 1\right) ^{-\alpha }\left( 1+\sum_{k=1}^{N-1}\frac{\pi _{\left( N\right) }\left( k+1\right) }{\pi _{\left( N\right) }\left( 1\right) }z^{k}\right) ^{-\alpha } \\ &=&\pi _{\left( N\right) }\left( 1\right) ^{-\alpha }\sum_{l\geq 0}z^{l}\sum_{k=0}^{l}\left( -1\right) ^{k}\left[ \alpha \right] _{k}\sum_{{}}^{*}\prod_{m=1}^{N-1}\frac{\left( \pi _{\left( N\right) }\left( m+1\right) /\pi _{\left( N\right) }\left( 1\right) \right) ^{k_{m}}}{k_{m}!}\end{aligned}$$ where the star sum runs over $k_{m}\geq 0$, $m=1,...,N-1$ obeying $% \sum_{m=1}^{N-1}k_{m}=k$ and $\sum_{m=1}^{N-1}mk_{m}=l$. From this, we obtain the finite support version of (\[FES\]) as For any given $X_{\infty }$ with support $\left\{ 1,...,N\right\} $, the mapping $X_{\infty }\rightarrow \nu _{\left( N\right) }$ is one-to-one and onto. With $$C_{n-1,0}^{\left( N-1\right) }=\delta _{n-1,0}\text{ and }C_{n-1,k}^{\left( N-1\right) }:=\sum_{{}}^{*}\prod_{m=1}^{N-1}\frac{\left( \pi _{\left( N\right) }\left( m+1\right) /\pi _{\left( N\right) }\left( 1\right) \right) ^{k_{m}}}{k_{m}!}$$ where the star sum runs over $k_{m}\geq 0$, $m=1,...,N-1$ obeying $% \sum_{m=1}^{N-1}k_{m}=k$ and $\sum_{m=1}^{N-1}mk_{m}=n-1\geq k$, $$\varphi _{n}:=\left[ z^{n}\right] \Phi _{\infty }^{-1}\left( z\right) =\frac{% 1}{n}\left[ z^{n-1}\right] \Psi _{\infty }\left( z\right) ^{-n}=\frac{\pi _{\left( N\right) }\left( 1\right) ^{-n}}{n}\sum_{k=0}^{n-1}\left( -1\right) ^{k}\left[ n\right] _{k}C_{n-1,k}^{\left( N-1\right) }. \label{FS1}$$ So, with $h_{1}=1$ and $h_{n}=\frac{1}{n}\sum_{k=1}^{n-1}\left( -1\right) ^{k}\left[ n\right] _{k}C_{n-1,k}^{\left( N-1\right) },$ $n\geq 2$$$\mathbf{P}\left( \nu _{\left( N\right) }\leq j\right) =\sum_{n\geq 1}h_{n}\cdot \left( \mathbf{P}\left( X_{\infty }\leq j\right) /\pi _{\left( N\right) }\left( 1\right) \right) ^{n} \label{FS2}$$ is the pdf of $\nu _{\left( N\right) }$ associated to any $\mathbf{P}\left( X_{\infty }\leq j\right) =\sum_{k=1}^{j}\pi _{\left( N\right) }\left( k\right) $, $j=1,...,N$ obeying $\mathbf{P}\left( X_{\infty }\leq N\right) =1.$ *Remark:* $\left( i\right) $ $\varphi _{1}=1/\pi _{\left( N\right) }\left( 1\right) $ ($h_{1}=1$) and for $n\geq 2$, the sum over $k$ giving the expression of $% \varphi _{n}$ (or of $h_{n}$) can start at $k=1$. $\left( ii\right) $ if (a separable case in $\left( k,N\right) $): $\pi _{\left( N\right) }\left( k\right) =a_{k}/A_{N},$ $a_{k}\geq 0$, where $% A_{N}=\sum_{k=1}^{N}a_{k}$ is a normalization factor, the law of $\nu _{\left( N\right) }$ does not depend on $A_{N}$ because it only depends on the ratios $\pi _{\left( N\right) }\left( k\right) /\pi _{\left( N\right) }\left( 1\right) =a_{k}/a_{1}.$ $\diamondsuit $ **Examples:** Just like in the infinite-dimensional case, there are examples amenable to a straightforward calculation. $\left( i\right) $ Suppose $$\Phi _{\infty }\left( z\right) =\frac{\left( q+pz\right) ^{N}-q^{N}}{1-q^{N}}% ,$$ corresponding to a binomial model restricted to $\left\{ 1,...,N\right\} $ with $$\begin{aligned} \mathbf{P}\left( X_{\infty }=k\right) &=&\left[ z^{k}\right] \Phi _{\infty }\left( z\right) =\frac{1}{1-q^{N}}\binom{N}{k}p^{k}q^{N-k} \\ \mathbf{P}\left( X_{\infty }\leq j\right) &=&\sum_{k=1}^{j}\mathbf{P}\left( X_{\infty }=k\right) .\end{aligned}$$ By direct inversion of $\Phi _{\infty }\left( z\right) $, we have that $$\Phi _{\infty }^{-1}\left( \mathbf{P}\left( X_{\infty }\leq j\right) \right) =\frac{q}{p}\left[ \left( 1+\sum_{k=1}^{j}\binom{N}{k}\left( \frac{p}{q}% \right) ^{k}\right) ^{1/N}-1\right]$$ is the pdf $\mathbf{P}\left( \nu _{\left( N\right) }\leq j\right) $ of some rv $\nu _{\left( N\right) }.$ Note $\mathbf{P}\left( \nu _{\left( N\right) }\leq N\right) =\Phi _{\infty }^{-1}\left( \mathbf{P}\left( X_{\infty }\leq N\right) \right) =1.$ $\left( ii\right) $ Suppose $$\Phi _{\infty }\left( z\right) =z\left( q+pz\right) ^{N-1}=z\Psi _{\infty }\left( z\right)$$ corresponding to a shifted binomial model supported $\left\{ 1,...,N\right\} $$$\begin{aligned} \mathbf{P}\left( X_{\infty }=k\right) &=&\left[ z^{k}\right] \Phi _{\infty }\left( z\right) =\binom{N-1}{k-1}p^{k-1}q^{N-k} \\ \mathbf{P}\left( X_{\infty }\leq j\right) &=&\sum_{k=1}^{j}\mathbf{P}\left( X_{\infty }=k\right) .\end{aligned}$$ With $n\geq 1$, we have that $\Phi _{\infty }^{-1}\left( z\right) =\sum_{n\geq 1}\frac{z^{n}}{n}\left[ z^{n-1}\right] \Psi _{\infty }\left( z\right) ^{-n}$ with $$\begin{aligned} \varphi _{n} &=&\frac{q^{-n\left( N-1\right) }}{n}\left[ z^{n-1}\right] \left( 1+\frac{p}{q}z\right) ^{-n\left( N-1\right) } \\ &=&\left( -1\right) ^{n-1}q^{-n\left( N-1\right) }\left( \frac{p}{q}\right) ^{n-1}\frac{\left[ n\left( N-1\right) \right] _{n-1}}{n!}\end{aligned}$$ $$\mathbf{P}\left( \nu _{\left( N\right) }\leq j\right) =\sum_{n\geq 1}\varphi _{n}\mathbf{P}\left( X_{\infty }\leq j\right) ^{n}$$ The rv $\nu _{\left( N\right) }$ has support $\left\{ 1,...,N\right\} $. $\left( iii\right) $ Truncation of the infinite-dimensional model. This situation occurs if, for $\left( \pi _{\left( N\right) }\left( k\right) ,k=1,...,N\right) $, we consider the normalized restriction of the invariant measure $\mathbf{\pi }$ with full support $\Bbb{N}$ to its $N$ first entries. For example, assuming $\left( \pi \left( k\right) =pq^{k-1},k\geq 1\right) $ is geometric, we get $$\mathbf{P}\left( \nu _{\left( N\right) }\leq j\right) =\sum_{n\geq 1}h_{n}\cdot \left( \frac{1-q^{j}}{p}\right) ^{n}$$ where $h_{n}=\frac{q^{n-1}}{n}\sum_{k=1}^{n-1}\left( -1\right) ^{k}\frac{% \left[ n\right] _{k}}{k!}\sum_{{}}^{*}\frac{k!}{\prod_{m=1}^{N-1}k_{m}!}=% \frac{q^{n-1}}{n}\sum_{k=1}^{n-1}\left( -1\right) ^{k}\frac{\left[ n\right] _{k}}{k!}\left( N-1\right) ^{k}$, so that with $A_{n,N}=\sum_{k=1}^{n-1}% \left( -1\right) ^{k}\frac{\left[ n\right] _{k}}{k!}\left( N-1\right) ^{k}$, $$\mathbf{P}\left( \nu _{\left( N\right) }\leq j\right) =\frac{1}{q}% \sum_{n\geq 1}\frac{A_{n,N}}{n}\cdot \left( \frac{q\left( 1-q^{j}\right) }{p}% \right) ^{n}.\text{ }\diamondsuit$$ Take any probability measure $\mathbf{\pi }_{\left( N\right) }$ with support $\left\{ 1,...,N\right\} $. Compute $F_{\left( N\right) }\left( j\right) =% \mathbf{P}\left( \nu _{\left( N\right) }\leq j\right) $ from $\mathbf{\pi }% _{\left( N\right) }$ as from (\[FS2\]). Construct the $N\times N$ stochastic matrix $P_{\left( N\right) }$ with entries $P_{\left( N\right) }\left( i,j\right) =F_{\left( N\right) }\left( j\right) ^{i}-F_{\left( N\right) }\left( j-1\right) ^{i}$, $i,j\in \left\{ 1,...,N\right\} .$ The matrix $P_{\left( N\right) }$ is the transition matrix of some ergodic Lamperti chain $X_{n}^{\left( N\right) }$ with state-space $\left\{ 1,...,N\right\} ,$ having $\mathbf{\pi }_{\left( N\right) }$ as invariant probability measure and reproduction mechanism $\nu _{\left( N\right) }$. The MC $\left\{ X_{n}^{\left( N\right) }\right\} $ is failure rate monotone. Furthermore: $$\mathbf{P}_{\mathbf{\pi }_{0}}\left( X_{n}^{\left( N\right) }=j\right) \underset{n,N\rightarrow \infty }{\rightarrow }\pi \left( j\right)$$ *Proof:* The reasons are similar to the ones raised for the Lamperti chain taking values in $\Bbb{N}$. The probability $\mathbf{P}\left( X_{n+1}^{\left( N\right) }\leq j\mid X_{n}^{\left( N\right) }=i\right) =F_{\left( N\right) }\left( j\right) ^{i}$ is a decreasing function of $i$, for all $j$: the MC $\left\{ X_{n}^{\left( N\right) }\right\} $ is stochastically monotone. The cumulated transition matrix : $P_{\left( N\right) }^{c}\left( i,j\right) =\sum_{k=1}^{j}P_{\left( N\right) }\left( i,k\right) $ obeys: $$P_{\left( N\right) }^{c}\left( i_{1},j_{1}\right) P_{\left( N\right) }^{c}\left( i_{2},j_{2}\right) \geq P_{\left( N\right) }^{c}\left( i_{1},j_{2}\right) P_{\left( N\right) }^{c}\left( i_{2},j_{1}\right) ,$$ for all $i_{1}<i_{2}$ and $j_{1}<j_{2}$ ($P_{\left( N\right) }^{c}$ is totally positive of order $2$): the MC $\left\{ X_{n}^{\left( N\right) }\right\} $ is failure rate monotone. Note the induced Kirchoff determinantal identities for finite matrices: $\pi _{\left( N\right) }\left( j\right) =\det \left[ \left( I-P_{\left( N\right) }\right) ^{\left( j,j\right) }\right] .$ The last statement is obvious. $% \Box $ (Truncation of $X_{n}$) $\left( i\right) $ Take for $\mathbf{\pi }_{\left( N\right) }$ the restriction to $\left\{ 1,...,N\right\} $ of the invariant measure* *$% \mathbf{\pi }$ of the Lamperti model with countable state-space, so with: $% \mathbf{\pi }_{\left( N\right) }\left( k\right) =\pi \left( k\right) /\sum_{k=1,...,N}\pi \left( k\right) $*,* $k=1,...,N$*.* $\left( ii\right) $ Take for $\mathbf{\pi }_{\left( N\right) }$ the restriction to $\left\{ 1,...,N-1\right\} $ of the invariant measure* * $\mathbf{\pi }$ of the Lamperti model with countable state-space, so with: $% \mathbf{\pi }_{\left( N\right) }\left( k\right) =\pi \left( k\right) $*,* $k=1,...,N-1$, $\pi _{\left( N\right) }\left( N\right) =\sum_{k\geq N}\pi \left( k\right) $*.* Constructing the corresponding transition matrices $P_{\left( N\right) }$, in both cases, the truncations preserve the failure rate monotonicity of $P.$ The corresponding Lamperti chains $X_{n}^{\left( N\right) }$ with state-space $\left\{ 1,...,N\right\} ,$ having $\mathbf{\pi }_{\left( N\right) }$ as restricted invariant measure and reproduction mechanism $\nu _{\left( N\right) }$ are called the truncated Lamperti chains up to state $N$. *Remarks:* - The case $\left( i\right) $ is simpler because in this separable case, the corresponding law of $\nu _{\left( N\right) }$ does not depend on the normalization factor $\sum_{k=1,...,N}\pi \left( k\right) $. - Censored Markov chain** **([@ZL], [@GH]): with $% P_{11}=Q_{\left( N\right) }$ and $$P=\left[ \begin{array}{ll} P_{11} & P_{12} \\ P_{21} & P_{22} \end{array} \right] ,$$ define $$P_{\left( N\right) }=P_{11}+P_{12}\left( I-P_{22}\right) ^{-1}P_{21}.$$ Let $Q_{2,2}=\left( I-P_{22}\right) ^{-1}$ be the fundamental matrix of $% P_{22}$, with $Q_{2,2}\left( i,j\right) $ the mean number of visits to state $j$ in $\left\{ N+1,...,\infty \right\} $ starting from $i$ in $\left\{ N+1,...,\infty \right\} $, before visiting first $\left\{ 1,...,N\right\} $. The matrix element $\left( P_{12}Q_{2,2}P_{21}\right) \left( i,j\right) $ is the taboo probability of the paths from states $i$ to $j$ both in $\left\{ 1,...,N\right\} $ which are not allowed to visit $\left\{ 1,...,N\right\} $ in between. $P_{\left( N\right) }$ has invariant measure $\mathbf{\pi }% _{\left( N\right) }^{\prime }=\left( \pi _{1},...,\pi _{N}\right) /$norm (the restriction of $\mathbf{\pi }$ to its $N$ first entries). However, it is not clear that such a $P_{\left( N\right) }$ is SM (probably not) nor that $P_{\left( N\right) }^{c}$ is FRM$.$ Besides, $P_{\left( N\right) }$has a complicated structure in case of Lamperti. Truncating a Markov chain invariant measure while preserving the monotonicity properties of the original is not so straightforward.* *$\diamondsuit $ Brown’s analysis of the truncated Lamperti model ================================================ In this Section, we consider the truncated version $\left\{ X_{n}^{\left( N\right) }\right\} $ of the chain $\left\{ X_{n}\right\} $ corresponding to the one preserving the $N$ first entries of the full invariant measure $% \mathbf{\pi }$ of $\left\{ X_{n}\right\} $, meaning $\pi \left( i\right) \rightarrow \pi _{\left( N\right) }\left( i\right) =\pi \left( i\right) /\sum_{i=1}^{N}\pi \left( i\right) $, $i=1,...,N$ (the restriction to $% \left\{ 1,...,N\right\} $ of the full invariant measure supported by $\Bbb{N% }$). This MC has totally ordered state-space, with $\left\{ N\right\} $ as a maximal element. It is a separable case and this truncation preserves the failure-rate monotonicity of $P^{c}:$ $P_{\left( N\right) }^{c}$ remains FRM, else $P_{\left( N\right) }^{c}$ is TP$_{2}$. As in [@Brown], we shall be concerned by the relationship existing between the first hitting times of both state $\left\{ N\right\} $ and the restricted invariant measure $\mathbf{\pi }_{\left( N\right) },$ given $X_{0}^{\left( N\right) }% \overset{d}{\sim }\mathbf{\pi }_{0}$. We will assume $\pi _{0}\left( N\right) =0$, to ensure that $\left\{ X_{n}^{\left( N\right) }\right\} $ hits $\left\{ N\right\} $ for the first time with positive probability after at least one time unit. To illustrate his theory, Brown designs some ad hoc ($4\times 4$) FRM matrices; the truncated Lamperti chain is a more relevant example. The following general results for hitting times hold for the Lamperti truncated chain (see also [@Lorekth] for a survey). [@Brown]. Suppose $\mathbf{\pi }_{0}$ is such that $\pi _{0}\left( i\right) /\pi _{\left( N\right) }\left( i\right) $ decreases with $i$ and $% \pi _{0}\left( N\right) =0$. Then $\left( i\right) $ $\mathbf{P}\left( X_{n}^{\left( N\right) }=N\mid X_{0}^{\left( N\right) }\overset{d}{\sim }\mathbf{\pi }_{0}\right) $ is non-decreasing with $n.$ $\left( ii\right) $ Let $\tau _{i,j}=\inf \left( n\geq 1:X_{n}^{\left( N\right) }=j\mid X_{0}^{\left( N\right) }=i\right) $, with $\tau _{j,j}:=0$. With $\tau _{\mathbf{\pi }_{0},j}=\inf \left( n\geq 1:X_{n}^{\left( N\right) }=j\mid X_{0}^{\left( N\right) }\overset{d}{\sim }\mathbf{\pi }_{0}\right) :$$$\tau _{\mathbf{\pi }_{0},N}\overset{d}{=}T_{\left( N\right) }+\tau _{\mathbf{% \pi }_{\left( N\right) },N} \label{B0}$$ where $T_{\left( N\right) }\geq 1$ and $\tau _{\mathbf{\pi }_{\left( N\right) },N}\geq 0$ are independent. *Proof:* The condition that $\mathbf{\pi }_{0}$ is such that $\pi _{0}\left( i\right) /\pi _{\left( N\right) }\left( i\right) $ decreases with $i$ holds if $\pi _{0}\left( i\right) =\delta _{i,1}$ and also if $\pi _{0}\left( i\right) =z^{i}\pi _{\left( N\right) }\left( i\right) /$norm, $% i=1,...,N-1$ for some $z\in \left( 0,1\right) $). It says that the initial probability mass assigned to states near the bottom state $\left\{ 1\right\} $ should exceed the one assigned by $\mathbf{\pi }_{\left( N\right) }$. In particular: $\pi _{0}\left( 1\right) >\pi _{\left( N\right) }\left( 1\right) .$ The proof of this statement was derived in [@Brown] in a continuous-time setting and is easily adaptable to discrete-time. $\left( i\right) $ Let $\mathbf{e}_{N}^{\prime }=\left( 0,...,0,1\right) $ be an $N-$dimensional row vector, with $1$ in position $N$. Because $% P_{\left( N\right) }^{c}$ is FRM (in particular SM), $\mathbf{P}\left( X_{n}^{\left( N\right) }=N\mid X_{0}^{\left( N\right) }\overset{d}{\sim }% \mathbf{\pi }_{0}\right) =\mathbf{\pi }_{0}^{\prime }P_{\left( N\right) }^{n}% \mathbf{e}_{N}$ is non-decreasing with $n$ [@Brown]. See Lemma $4.2$ of [@Brown] where it is shown that this condition is fulfilled if $% \overline{P}_{\left( N\right) }^{n}\left( \mathbf{\pi }_{0},j\right) /% \overline{\pi }_{\left( N\right) }\left( j\right) $ is decreasing in $j$, which is the case for FRM Markov chains. Here $\overline{\pi }_{\left( N\right) }\left( j\right) =\sum_{k=j}^{N}\overline{\pi }_{\left( N\right) }\left( k\right) $ and $\overline{P}_{\left( N\right) }^{n}\left( \mathbf{% \pi }_{0},j\right) =\sum_{k=j}^{N}P_{\left( N\right) }^{n}\left( \mathbf{\pi }_{0},k\right) .$ It is needed in the proof that $\pi _{0}\left( i\right) /\pi _{\left( N\right) }\left( i\right) $ decreases with $i.$ $\left( ii\right) $ Owing to $P_{\left( N\right) }^{n}\left( \mathbf{\pi }% _{0},N\right) =\mathbf{\pi }_{0}^{\prime }P_{\left( N\right) }^{n}\mathbf{e}% _{N}\rightarrow \pi _{\left( N\right) }\left( N\right) $ as $n\rightarrow \infty $, $\mathbf{\pi }_{0}^{\prime }P_{\left( N\right) }^{n}\mathbf{e}% _{N}/\pi _{\left( N\right) }\left( N\right) $ is a probability distribution function of some rv $T_{\left( N\right) }$ with $$\mathbf{P}\left( T_{\left( N\right) }\leq n\right) =\mathbf{\pi }% _{0}^{\prime }P_{\left( N\right) }^{n}\mathbf{e}_{N}/\pi _{\left( N\right) }\left( N\right) \text{, }n\geq 0.$$ With $G_{\mathbf{\pi }_{0},N}\left( z\right) =\mathbf{\pi }_{0}^{\prime }\sum_{n\geq 0}z^{n}P_{\left( N\right) }^{n}\mathbf{e}_{N}=\mathbf{\pi }% _{0}^{\prime }\left( I-zP_{\left( N\right) }\right) ^{-1}\mathbf{e}_{N}$, a Green kernel of $P_{\left( N\right) }$, we thus have $$\mathbf{E}\left( z^{T_{\left( N\right) }}\right) =\frac{1-z}{\pi _{\left( N\right) }\left( N\right) }G_{\mathbf{\pi }_{0},N}\left( z\right) . \label{B1}$$ Now $$\mathbf{\pi }_{\left( N\right) }\left( N\right) =\mathbf{P}_{\mathbf{\pi }% _{\left( N\right) }}\left( X_{n}^{\left( N\right) }=N\right) =\sum_{m=0}^{n}P_{\left( N\right) }^{n-m}\left( N,N\right) \mathbf{P}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}=m\right) ,$$ of convolution type. Taking the generating function of both sides $$\mathbf{E}\left( z^{\tau _{\mathbf{\pi }_{\left( N\right) },N}}\right) =% \frac{\mathbf{\pi }_{\left( N\right) }\left( N\right) }{\left( 1-z\right) G_{N,N}\left( z\right) }, \label{B2}$$ where $G_{N,N}\left( z\right) =\sum_{m\geq 0}z^{m}P_{\left( N\right) }^{m}\left( N,N\right) =\left( I-zP_{\left( N\right) }\right) ^{-1}\left( N,N\right) $ is the Green kernel of $\left\{ X_{n}^{\left( N\right) }\right\} $ at $\left( N,N\right) .$ Similarly, $$\mathbf{P}_{\mathbf{\pi }_{0}}\left( X_{n}^{\left( N\right) }=N\right) =\sum_{m=0}^{n}P_{\left( N\right) }^{n-m}\left( N,N\right) \mathbf{P}\left( \tau _{\mathbf{\pi }_{0},N}=m\right)$$ leading to, $$G_{\mathbf{\pi }_{0},N}\left( z\right) =G_{N,N}\left( z\right) \mathbf{E}% \left( z^{\tau _{\mathbf{\pi }_{0},N}}\right) \label{B3}$$ Taking the product of (\[B2\]-\[B3\]), and recalling (\[B1\]), we get $$\phi _{\mathbf{\pi }_{0},N}\left( z\right) :=\mathbf{E}\left( z^{\tau _{% \mathbf{\pi }_{0},N}}\right) =\mathbf{E}\left( z^{T_{\left( N\right) }}\right) \mathbf{E}\left( z^{\tau _{\mathbf{\pi }_{\left( N\right) },N}}\right) .\text{ }\Box \label{B3a}$$ The latter equation indicates that $\tau _{\mathbf{\pi }_{0},N}\geq 1$ is stochastically larger than $\tau _{\mathbf{\pi }_{\left( N\right) },N}$: it takes a shorter time for $\left\{ X_{n}^{\left( N\right) }\right\} $ to first hit $\left\{ N\right\} $ starting from $\mathbf{\pi }_{\left( N\right) }$ than starting from $\mathbf{\pi }_{0}.$ The time to first hit state $% \left\{ N\right\} $ is important in the Lamperti context because at this instant, the progeny after selection is the maximum possible. But of course the process will not remain in that state unless one forces the chain to have $\left\{ N\right\} $ absorbing. As a result also, $T_{\left( N\right) }$ interprets as $\tau _{\mathbf{\pi }% _{0},\mathbf{\pi }_{\left( N\right) }},$ the first hitting time of $\mathbf{% \pi }_{\left( N\right) }$ starting from $\mathbf{\pi }_{0}$. So with $X_{T_{\left( N\right) }}\overset{d}{\sim }\mathbf{\pi }_{\left( N\right) }$, $X_{T_{\left( N\right) }}$ independent of $T_{\left( N\right) }$ and $\mathbf{P}\left( X_{n}=N\text{ for some }n<T_{\left( N\right) }\right) =0$. The latter equation also indicates that $\tau _{\mathbf{\pi }% _{0},N}\geq 1$ is stochastically larger than $\tau _{\mathbf{\pi }_{0},% \mathbf{\pi }_{\left( N\right) }}\geq 1$ (statistically, $\left\{ X_{n}^{\left( N\right) }\right\} $ started from $\mathbf{\pi }_{0}$ enters $% \mathbf{\pi }_{\left( N\right) }$ before first hitting state $N$)$.$ As a consequence, ([@Brown], Corollary $4.1$) For all $n\geq 0$$$\begin{aligned} \text{sep}\left( \mathbf{P}_{\mathbf{\pi }_{0}}\left( X_{n}^{\left( N\right) }=\cdot \right) ,\mathbf{\pi }_{\left( N\right) }\right) &=&\underset{k}{% \max }\left( 1-\mathbf{\pi }_{0}^{\prime }P_{\left( N\right) }^{n}\mathbf{e}% _{k}/\pi _{\left( N\right) }\left( k\right) \right) \\ &=&1-\mathbf{\pi }_{0}^{\prime }P_{\left( N\right) }^{n}\mathbf{e}_{N}/\pi _{\left( N\right) }\left( N\right) =\mathbf{P}\left( T_{\left( N\right) }>n\right) ,\end{aligned}$$ and $T_{\left( N\right) }$ is a minimal strong stationary time with separating state $N$. The separation distance sep$\left( \cdot ,\cdot \right) $ from $\mathbf{P}_{% \mathbf{\pi }_{0}}\left( X_{n}^{\left( N\right) }=\cdot \right) $ to $% \mathbf{\pi }_{\left( N\right) }$ gives an upper bound for the total variation norm between these two probability measures. $$\mathbf{E}\left( T_{\left( N\right) }\right) =1+\sum_{n\geq 1}\frac{\mathbf{% \pi }_{0}^{\prime }P_{\left( N\right) }^{n}\mathbf{e}_{N}}{\pi _{\left( N\right) }\left( N\right) }=1+\frac{1}{\pi _{_{\left( N\right) }}\left( N\right) }\mathbf{\pi }_{0}^{\prime }\left( I-P_{\left( N\right) }\right) ^{-1}P_{\left( N\right) }\mathbf{e}_{N}$$ There are some other facts pertaining to the fact that $\tau _{\mathbf{\pi }% _{\left( N\right) },N}$ has a geometric convolution representation. [@Brown] $\left( i\right) $ $\mathbf{P}\left( X_{n}^{\left( N\right) }=N\mid X_{0}^{\left( N\right) }=N\right) $ is non-increasing with $n$, so $$\mathbf{P}\left( W_{1}^{\left( N\right) }>n\right) =\frac{P_{\left( N\right) }^{n}\left( N,N\right) -\pi _{_{\left( N\right) }}\left( N\right) }{1-\pi _{_{\left( N\right) }}\left( N\right) }. \label{B40}$$ is a well defined complementary mass function of some rv $W_{1}^{\left( N\right) }$. $\left( ii\right) $$$\tau _{\mathbf{\pi }_{\left( N\right) },N}=\sum_{i=1}^{G_{N}}W_{i}^{\left( N\right) } \label{B4}$$ where $G_{N}\overset{d}{\sim }$geo$\left( \mathbf{\pi }_{_{\left( N\right) }}\left( N\right) \right) $ (viz $\mathbf{P}\left( G_{N}=j\right) =\pi _{_{\left( N\right) }}\left( N\right) \left( 1-\pi _{_{\left( N\right) }}\left( N\right) \right) ^{j},$ $j=0,1,...$), independent of $W_{i}^{\left( N\right) },$ $i\geq 1,$ an iid sequence with $W_{i}^{\left( N\right) }% \overset{d}{=}W_{1}^{\left( N\right) }.$ *Proof:* $\left( i\right) $ Because $P_{\left( N\right) }$ is SM as well, $P_{\left( N\right) }^{n}\left( N,N\right) \geq P_{\left( N\right) }^{n}\left( i,N\right) $ for all $i$ and $n$. Therefore, with $n_{2}>n_{1}$, $$P_{\left( N\right) }^{n_{2}}\left( N,N\right) =\sum_{i=1}^{N}P_{\left( N\right) }^{n_{2}-n_{1}}\left( N,i\right) P_{\left( N\right) }^{n_{1}}\left( i,N\right) \leq P_{\left( N\right) }^{n_{1}}\left( N,N\right) \sum_{i=1}^{N}P_{\left( N\right) }^{n_{2}-n_{1}}\left( N,i\right) =P_{\left( N\right) }^{n_{1}}\left( N,N\right) .$$ As a result, $\mathbf{P}\left( X_{n}^{\left( N\right) }=N\mid X_{0}^{\left( N\right) }=N\right) =\mathbf{e}_{N}^{\prime }P_{\left( N\right) }^{n}\mathbf{% e}_{N}=P_{\left( N\right) }^{n}\left( N,N\right) $ is non-increasing with $n$ so that the law of $W_{1}^{\left( N\right) }$ is well-defined. $\left( ii\right) $ $$\sum_{n\geq 0}z^{n}\mathbf{P}\left( W_{1}^{\left( N\right) }>n\right) =\frac{% 1-\mathbf{E}\left( z^{W_{1}^{\left( N\right) }}\right) }{1-z}=\frac{1}{1-\pi _{_{\left( N\right) }}\left( N\right) }\left( G_{N,N}\left( z\right) -\frac{% \pi _{_{\left( N\right) }}\left( N\right) }{1-z}\right)$$ Using (\[B2\]), we get $$\mathbf{E}\left( z^{\tau _{\mathbf{\pi }_{\left( N\right) },N}}\right) =% \frac{1}{1+\frac{1-\pi _{_{\left( N\right) }}\left( N\right) }{\pi _{_{\left( N\right) }}\left( N\right) }\left( 1-\mathbf{E}\left( z^{W_{1}^{\left( N\right) }}\right) \right) } \label{B5}$$ which is the pgf of the geometric convolution $\sum_{i=1}^{G_{N}}W_{i}^{% \left( N\right) }$. $\Box $ Note $G_{N}=0$ entails $\tau _{\mathbf{\pi }_{\left( N\right) },N}=0$, an event with probability $\pi _{_{\left( N\right) }}\left( N\right) $. *Remark:* Stochastically monotone Markov chain have a real and simple second largest eigenvalue, [@Keilson]. Suppose $1=\lambda _{1}>\lambda _{2}>\left| \lambda _{3}\right| \geq ...\geq \left| \lambda _{N}\right| >0$ where $\lambda _{k}=\lambda _{k,\left( N\right) }$ are the $N-$dependent eigenvalues of $P_{\left( N\right) }$. Then, $$\forall i,j\in \left\{ 1,...,N\right\} \text{, }\forall n\in \Bbb{N}\text{, }% \exists c>0:\text{ }\left| P_{\left( N\right) }^{n}\left( i,j\right) -\pi _{_{\left( N\right) }}\left( j\right) \right| \leq c\lambda _{2,\left( N\right) }^{n}.$$ In particular, $\left| P_{\left( N\right) }^{n}\left( N,N\right) -\pi _{_{\left( N\right) }}\left( N\right) \right| \leq c\lambda _{2,\left( N\right) }^{n}$ and $P_{\left( N\right) }^{n}\left( N,N\right) $ is getting close to $\pi _{_{\left( N\right) }}\left( N\right) $ as $n$ gets large, useful for (\[B40\]). $\diamondsuit $ - **Quasi-stationary distribution (qsd).** An alternative point of view on $\tau _{\mathbf{\pi }_{0},N}$ and $\tau _{\mathbf{\pi }_{\left( N\right) },N}$ can also be seen from the classical theory of qsd’s, [@cmsm]. With $i\neq N$, let $\tau _{i,N}=\inf \left( n\geq 1:X_{n}^{\left( N\right) }=N\mid X_{0}^{\left( N\right) }=i\right) $. We have $$\mathbf{P}\left( \tau _{i,N}>1\right) =\mathbf{P}\left( X_{1}^{\left( N\right) }\leq N-1\mid X_{0}^{\left( N\right) }=i\right) =P^{c}\left( i,N-1\right) =F_{\left( N\right) }\left( N-1\right) ^{i}$$ $$\begin{aligned} \mathbf{P}\left( \tau _{i,N}>n+1\right) &=&\sum_{1\leq j<N}\mathbf{P}% _{i}\left( X_{n}^{\left( N\right) }=j,\tau _{i,N}>n+1\right) \\ &=&\sum_{1\leq j<N}F_{\left( N\right) }\left( N-1\right) ^{j}\mathbf{P}% _{i}\left( X_{n}^{\left( N\right) }=j,\tau _{i,N}>n\right)\end{aligned}$$ $$\mathbf{P}\left( \tau _{i,N}>n+1\mid \tau _{i,N}>n\right) =\sum_{1\leq j<N}F_{\left( N\right) }\left( N-1\right) ^{j}\mathbf{P}_{i}\left( X_{n}^{\left( N\right) }=j\mid \tau _{i,N}>n\right)$$ $$\underset{n\rightarrow \infty }{\rightarrow }\sum_{1\leq j<N}F_{\left( N\right) }\left( N-1\right) ^{j}\mu _{\left( N-1\right) }\left( j\right) =:% \mathbf{E}\left( z^{Z_{\left( N-1\right) }}\right) \mid _{z=F_{\left( N\right) }\left( N-1\right) }=:\rho _{N}.$$ In the latter displayed formula, $\mathbf{\mu }_{\left( N-1\right) }\left( \cdot \right) $ is the quasi-stationary limiting distribution of $\left\{ X_{n}^{\left( N\right) }\right\} $ when state $N$ has been removed and $% Z_{\left( N-1\right) }\overset{d}{\sim }\mathbf{\mu }_{\left( N-1\right) }$. Stated differently, $\mathbf{\mu }_{\left( N-1\right) }^{\prime }$ is the $% \left( N-1\right) -$dimensional left eigenvector (associated to the dominant eigenvalue $\rho _{N}<1$) of the substochastic matrix $P_{\left( N-1\right) } $ obtained while removing the $N-$th row and column $N-$th column of $% P_{\left( N\right) }$. We have used $\mathbf{P}_{i}\left( X_{n}^{\left( N\right) }=j\mid \tau _{i,N}>n\right) \underset{n\rightarrow \infty }{% \rightarrow }\mu _{\left( N-1\right) }\left( j\right) $, $j\in \left\{ 1,...,N-1\right\} .$ Consequently, With $\rho _{N}$ the value of the pgf of $Z_{\left( N-1\right) }$ evaluated at $F_{\left( N\right) }\left( N-1\right) $, independently of $i\in \left\{ 1,...,N-1\right\} $$$\underset{n\rightarrow \infty }{\lim }-\frac{1}{n}\log \mathbf{P}\left( \tau _{i,N}>n\right) =-\log \mathbf{E}\left( z^{Z_{\left( N-1\right) }}\right) \mid _{z=F_{\left( N\right) }\left( N-1\right) }=-\log \rho _{N}.$$ Equivalently, $$\rho _{N}=\mathbf{E}\left( z^{Z_{\left( N-1\right) }}\right) \mid _{z=F_{\left( N\right) }\left( N-1\right) }$$ is the rate of decay of $\mathbf{P}\left( \tau _{i,N}>n\right) $. Similarly, - With $\mathbf{\pi }_{0,0}$ defined by $\mathbf{\pi }_{0}^{\prime }=:\left( \mathbf{\pi }_{0,0}^{\prime },0\right) $, for any initial distribution $% \mathbf{\pi }_{0,0},$$$\underset{n\rightarrow \infty }{\lim }-\frac{1}{n}\log \mathbf{P}\left( \tau _{\mathbf{\pi }_{0,0},N}>n\right) =-\log \mathbf{E}\left( z^{Z_{\left( N-1\right) }}\right) \mid _{z=F_{\left( N\right) }\left( N-1\right) },$$ giving the decay rate of $\mathbf{P}\left( \tau _{\mathbf{\pi }% _{0,0},N}>n\right) .$ - With $\mathbf{\pi }_{\left( N-1\right) }^{\prime }$ defined by $\mathbf{% \pi }_{\left( N\right) }^{\prime }=\left( \mathbf{\pi }_{\left( N-1\right) }^{\prime },\pi _{\left( N\right) }\left( N\right) \right) $, when starting from the invariant measure $$\underset{n\rightarrow \infty }{\lim }-\frac{1}{n}\log \mathbf{P}\left( \tau _{\mathbf{\pi }_{\left( N-1\right) }^{\prime },N}>n\right) =-\log \mathbf{E}% \left( z^{Z_{\left( N-1\right) }}\right) \mid _{z=F_{\left( N\right) }\left( N-1\right) }.$$ - Clearly also, when the initial distribution coincides with the quasi-stationary distribution: $\mathbf{\pi }_{0,0}=\mathbf{\mu }_{\left( N-1\right) },$ $$-\frac{1}{n}\log \mathbf{P}\left( \tau _{\mathbf{\mu }_{\left( N-1\right) },N}>n\right) =-\log \mathbf{E}\left( z^{Z_{\left( N-1\right) }}\right) \mid _{z=F_{\left( N\right) }\left( N-1\right) }=-\log \rho _{N}$$ for all $n.$ Letting $$\mathbf{\mu }_{\left( N-1\right) }^{\prime }P_{\left( N-1\right) }=\rho _{N}% \mathbf{\mu }_{\left( N-1\right) }^{^{\prime }}\text{ and }P_{\left( N-1\right) }\mathbf{\phi }_{\left( N-1\right) }=\rho _{N}\mathbf{\phi }% _{\left( N-1\right) },$$ be the $\left( N-1\right) $-dimensional left and right positive eigenvectors of $P_{\left( N-1\right) }$ chosen so as to satisfy: $\left| \mathbf{\mu }% _{\left( N-1\right) }\right| :=\sum_{j=1}^{N-1}\mu _{\left( N-1\right) }\left( j\right) =1$ and $\mathbf{\mu }_{\left( N-1\right) }^{\prime }% \mathbf{\phi }_{\left( N-1\right) }=1$, fixing the length $\left\| \mathbf{% \phi }_{\left( N-1\right) }\right\| _{2}^{1/2}$ of $\mathbf{\phi }_{\left( N-1\right) }$, then (by Perron-Frobenius theorem) $$\rho _{N}^{-n}P_{\left( N-1\right) }^{n}\rightarrow \mathbf{\phi }_{\left( N-1\right) }^{\prime }\mathbf{\mu }_{\left( N-1\right) }\text{ as }% n\rightarrow \infty .$$ Hence, with $\mathbf{\pi }_{0}^{\prime }=\left( \mathbf{\pi }_{0,0}^{\prime },0\right) ,$ $\left| \mathbf{\pi }_{0,0}\right| =1,$ and $\mathbf{\pi }% _{\left( N\right) }^{\prime }=\left( \mathbf{\pi }_{\left( N-1\right) }^{\prime },\pi _{\left( N\right) }\left( N\right) \right) $, $\left| \overline{\mathbf{\pi }}_{\left( N-1\right) }\right| <1$, and making use of $% \tau _{N,N}=0$$$\mathbf{P}\left( \tau _{\mathbf{\pi }_{0},N}>n\right) =\mathbf{\pi }% _{0,0}^{\prime }P_{\left( N-1\right) }^{n}\mathbf{1}\text{ and }\mathbf{P}% \left( \tau _{\mathbf{\pi }_{\left( N\right) },N}>n\right) =\mathbf{\pi }% _{\left( N-1\right) }^{\prime }P_{\left( N-1\right) }^{n}\mathbf{1<P}\left( \tau _{\mathbf{\pi }_{0},N}>n\right) \label{B6}$$ $$\mathbf{E}\left( \tau _{\mathbf{\pi }_{0},N}\right) =\mathbf{\pi }% _{0,0}^{\prime }\left( I-P_{\left( N-1\right) }\right) ^{-1}\mathbf{1}\text{ and }\mathbf{E}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}\right) =% \mathbf{\pi }_{\left( N-1\right) }^{\prime }\left( I-P_{\left( N-1\right) }\right) ^{-1}\mathbf{1<E}\left( \tau _{\mathbf{\pi }_{0},N}\right)$$ and $$\begin{aligned} \rho _{N}^{-n}\mathbf{P}\left( \tau _{\mathbf{\pi }_{0},N}>n\right) &\rightarrow &\mathbf{\pi }_{0,0}^{\prime }\mathbf{\phi }_{\left( N-1\right) }\text{ as }n\rightarrow \infty \\ \rho _{N}^{-n}\mathbf{P}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}>n\right) &\rightarrow &\mathbf{\pi }_{\left( N-1\right) }^{\prime }% \mathbf{\phi }_{\left( N-1\right) }\text{ as }n\rightarrow \infty .\end{aligned}$$ Suppose $\mathbf{\pi }_{0}$ is such that $\pi _{0}\left( i\right) /\pi _{\left( N\right) }\left( i\right) $ decreases with $i$ and $\pi _{0}\left( N\right) =0$. Then $$\frac{\mathbf{P}\left( \tau _{\mathbf{\pi }_{0},N}>n\right) }{\mathbf{P}% \left( \tau _{\mathbf{\pi }_{\left( N\right) },N}>n\right) }\underset{% n\rightarrow \infty }{\rightarrow }\frac{\mathbf{\pi }_{0,0}^{\prime }% \mathbf{\phi }_{\left( N-1\right) }}{\mathbf{\pi }_{\left( N-1\right) }^{\prime }\mathbf{\phi }_{\left( N-1\right) }}\geq 1.$$ *Proof:* Due to the stochastic domination of $\tau _{\mathbf{\pi }% _{0},N}$ over $\tau _{\mathbf{\pi }_{\left( N\right) },N}$ stated in Proposition $11$, the positive sequence $$u_{n}:=\frac{\mathbf{P}\left( \tau _{\mathbf{\pi }_{0},N}>n\right) }{\mathbf{% P}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}>n\right) }=\frac{\rho _{N}^{-n}\mathbf{P}\left( \tau _{\mathbf{\pi }_{0},N}>n\right) }{\rho _{N}^{-n}\mathbf{P}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}>n\right) }$$ is bounded below by $1$ ($u_{n}\geq 1$ for all $n$). The sequence $u_{n}$ is convergent with limit $u_{*}=\frac{\mathbf{\pi }_{0,0}^{\prime }\mathbf{\phi }_{\left( N-1\right) }}{\mathbf{\pi }_{\left( N-1\right) }^{\prime }\mathbf{% \phi }_{\left( N-1\right) }}$ and the limit obeys $u_{*}\geq 1$. We have $\rho _{N}^{-n}P_{\left( N-1\right) }^{n}\mathbf{1}\rightarrow \mathbf{\phi }_{\left( N-1\right) }$ as $n\rightarrow \infty $. The entries $% \phi _{\left( N-1\right) }\left( i\right) $ are decreasing with $i$, because it follows by induction that stochastic monotonicity of $P_{\left( N\right) } $ implies the one of $P_{\left( N-1\right) }^{n}$, so that $\mathbf{e}% _{i}^{\prime }P_{\left( N-1\right) }^{n}\mathbf{1}$ is decreasing with $i$. Because $\pi _{0,0}\left( i\right) /\pi _{\left( N-1\right) }\left( i\right) $ is decreasing with $i$, the initial probability mass assigned to states near the bottom state $\left\{ 1\right\} $ where $\mathbf{\phi }_{\left( N-1\right) }$ takes its largest values exceeds the one assigned by $\mathbf{% \pi }_{\left( N\right) }$. It is thus not that surprising that the numerator of $u_{*}$ exceeds its denominator. $\Box $ *Remark:* From (\[B6\]) $$\begin{aligned} \mathbf{E}\left( z^{\tau _{\mathbf{\pi }_{0},N}}\right) &=&1-\left( 1-z\right) \mathbf{\pi }_{0,0}^{\prime }\left( I-zP_{\left( N-1\right) }\right) ^{-1}\mathbf{1}\text{ and } \\ \mathbf{E}\left( z^{\tau _{\mathbf{\pi }_{\left( N\right) },N}}\right) &=&1-\left( 1-z\right) \mathbf{\pi }_{\left( N-1\right) }^{\prime }\left( I-zP_{\left( N-1\right) }\right) ^{-1}\mathbf{1}\end{aligned}$$ Comparing the expression* *of the pgf of* *$\tau _{\mathbf{\pi }% _{\left( N\right) },N}$ in terms of the Green kernel of $P_{\left( N-1\right) }$ with (\[B5\]), yields and identity for $\mathbf{\pi }% _{\left( N-1\right) }^{\prime }\left( I-zP_{\left( N-1\right) }\right) ^{-1}% \mathbf{1}$. Note that in (\[B5\]), only the values of $\pi _{_{\left( N\right) }}\left( N\right) $ and $P_{\left( N\right) }^{n}\left( N,N\right) $ matter. Comparing the expression* *of the pgf of* *$\tau _{% \mathbf{\pi }_{0},N}$ in terms of the Green kernel of $P_{\left( N-1\right) } $ with (\[B3a\]), also yields and identity for $\mathbf{\pi }% _{0,0}^{\prime }\left( I-zP_{\left( N-1\right) }\right) ^{-1}\mathbf{1}$. $% \diamondsuit $ By the definition of quasi-stationary distributions, we had $$\mathbf{P}\left( X_{n}^{\left( N\right) }=j\mid \tau _{\mathbf{\pi }% _{0},N}>n\right) \underset{n\rightarrow \infty }{\rightarrow }\mu _{\left( N-1\right) }\left( j\right) \text{, }j\in \left\{ 1,...,N-1\right\} .$$ Because $P_{\left( N\right) }$ is stochastically monotone, Siegmund-Pollack theorem holds, stating [@PS] $$\mathbf{P}\left( X_{n}^{\left( N\right) }=j\mid \tau _{\mathbf{\pi }% _{0},N}>n\right) \underset{n,N\rightarrow \infty }{\rightarrow }\pi \left( j\right) \text{, }j\geq 1.$$ As $N$ gets large, the qsd $\mathbf{\mu }_{\left( N-1\right) }$ gets very close to $\mathbf{\pi }_{\left( N-1\right) }$. - **Asymptotic exponentiality.** - The rv $\tau _{\mathbf{\mu }_{\left( N-1\right) },N}$ is geometric with success parameter $1-\rho _{N}$, $$\mathbf{E}\left( z^{\tau _{\mathbf{\mu }_{\left( N-1\right) },N}}\right) =% \frac{z\left( 1-\rho _{N}\right) }{1-\rho _{N}z},$$  so with mean and variance $\mathbf{E}\left( \tau _{\mathbf{\mu }_{\left( N-1\right) },N}\right) =1/\left( 1-\rho _{N}\right) $ and $\sigma ^{2}\left( \tau _{\mathbf{\mu }_{\left( N-1\right) },N}\right) =\rho _{N}/\left( 1-\rho _{N}\right) ^{2}.$ Suppose $\rho _{N}\rightarrow 1$ as $% N\rightarrow \infty $. Then $\tau _{\mathbf{\pi }_{\left( N\right) },N}/% \mathbf{E}\left( \tau _{\mathbf{\mu }_{\left( N-1\right) },N}\right) $becomes approximately exponential with mean $1$. We have $\mathbf{E}\left( \tau _{\mathbf{\mu }_{\left( N-1\right) },N}\right) \rightarrow \infty $ while $\sigma \left( \tau _{\mathbf{\mu }_{\left( N-1\right) },N}\right) /% \mathbf{E}\left( \tau _{\mathbf{\mu }_{\left( N-1\right) },N}\right) =\sqrt{% \rho _{N}}\rightarrow 1$, as* *$N\rightarrow \infty $*.* - Brown raised the question of asymptotic exponentiality of $\tau _{\mathbf{% \pi }_{\left( N\right) },N}/\mathbf{E}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}\right) $. If $\sigma ^{2}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}\right) <\infty $, as a scaled geometric convolution, $\tau _{\mathbf{\pi }_{\left( N\right) },N}/\mathbf{E}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}\right) $ is approximately exponential if $\mathbf{E}\left( \tau _{% \mathbf{\pi }_{\left( N\right) },N}\right) \rightarrow \infty $ while $% \sigma \left( \tau _{\mathbf{\pi }_{\left( N\right) },N}\right) /\mathbf{E}% \left( \tau _{\mathbf{\pi }_{\left( N\right) },N}\right) \rightarrow 1$, as* *$N\rightarrow \infty $ for the truncated Lamperti model with truncated target distribution $\mathbf{\pi }_{\left( N\right) }$. Error bounds can be obtained from the first two moments of $\tau _{\mathbf{\pi }% _{\left( N\right) },N}$ given by $$\begin{aligned} \mathbf{E}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}\right) &=&% \mathbf{E}\left( G_{N}\right) \mathbf{E}\left( W_{1}^{\left( N\right) }\right) =\frac{1-\pi _{_{\left( N\right) }}\left( N\right) }{\pi _{_{\left( N\right) }}\left( N\right) }\mathbf{E}\left( W_{1}^{\left( N\right) }\right) \\ &=&\frac{1}{\pi _{_{\left( N\right) }}\left( N\right) }\sum_{n\geq 0}\left( P_{\left( N\right) }^{n}\left( N,N\right) -\pi _{_{\left( N\right) }}\left( N\right) \right)\end{aligned}$$ $$\begin{aligned} \sigma ^{2}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}\right) &=&% \mathbf{E}\left( G_{N}\right) \sigma ^{2}\left( W_{1}^{\left( N\right) }\right) +\left( \mathbf{E}\left( W_{1}^{\left( N\right) }\right) \right) ^{2}\sigma ^{2}\left( G,_{N}\right) \\ \mathbf{E}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}^{2}\right) &=&2% \mathbf{E}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}\right) ^{2}+% \mathbf{E}\left( G_{N}\right) \mathbf{E}\left( \left( W_{1}^{\left( N\right) }\right) ^{2}\right) .\end{aligned}$$ The question of the approximation by an exponential distribution also arises for $\tau _{\mathbf{\pi }_{0},N}/\mathbf{E}\left( \tau _{\mathbf{\pi }% _{0},N}\right) $. In this direction indeed, ([@Brown], [@Brown2]) With $t\geq 0$ $$\begin{aligned} \sup_{t}\left| \mathbf{P}\left( \frac{\tau _{\mathbf{\pi }_{\left( N\right) },N}}{\mathbf{E}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}\right) }% >t\right) -e^{-t}\right| &\leq &\pi _{_{\left( N\right) }}\left( N\right) \frac{\mathbf{E}\left( \left( W_{1}^{\left( N\right) }\right) ^{2}\right) }{% \mathbf{E}\left( W_{1}^{\left( N\right) }\right) ^{2}} \\ &=&2\left( 1-\pi _{_{\left( N\right) }}\left( N\right) \right) \left[ \frac{% \mathbf{E}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}^{2}\right) }{2% \mathbf{E}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}\right) ^{2}}% -1\right] , \\ \sup_{t}\left| \mathbf{P}\left( \frac{\tau _{\mathbf{\pi }_{0},N}}{\mathbf{E}% \left( \tau _{\mathbf{\pi }_{0},N}\right) }>t\right) -e^{-t}\right| &\leq &% \frac{\mathbf{E}\left( T_{\left( N\right) }\right) }{\mathbf{E}\left( \tau _{% \mathbf{\pi }_{\left( N\right) },N}\right) }+2\left( 1-\pi _{_{\left( N\right) }}\left( N\right) \right) \left[ \frac{\mathbf{E}\left( \tau _{% \mathbf{\pi }_{\left( N\right) },N}^{2}\right) }{2\mathbf{E}\left( \tau _{% \mathbf{\pi }_{\left( N\right) },N}\right) ^{2}}-1\right]\end{aligned}$$ gives the sup-norm distance between respectively $\tau _{\mathbf{\pi }% _{\left( N\right) },N}/\mathbf{E}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}\right) $, $\tau _{\mathbf{\pi }_{0},N}/\mathbf{E}\left( \tau _{% \mathbf{\pi }_{0},N}\right) $ and an exponential rv with mean $1.$ This shows that if, as $N$ grows large, the mean and standard deviation of $% \tau _{\mathbf{\pi }_{\left( N\right) },N}/\mathbf{E}\left( \tau _{\mathbf{% \pi }_{\left( N\right) },N}\right) $ behave like the one of an exponential distribution that is if $\sigma \left( \tau _{\mathbf{\pi }_{\left( N\right) },N}\right) /\mathbf{E}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}\right) \rightarrow 1$, then $\mathbf{E}\left( \tau _{\mathbf{\pi }% _{\left( N\right) },N}^{2}\right) /\left( 2\mathbf{E}\left( \tau _{\mathbf{% \pi }_{\left( N\right) },N}\right) ^{2}\right) \rightarrow 1$ and the exponential approximation for the law of $\tau _{\mathbf{\pi }_{\left( N\right) },N}/\mathbf{E}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}\right) $ is valid. If in addition, as $N$ becomes large $$\mathbf{E}\left( T_{\left( N\right) }\right) /\mathbf{E}\left( \tau _{% \mathbf{\pi }_{\left( N\right) },N}\right) \ll 2\left( 1-\pi _{_{\left( N\right) }}\left( N\right) \right) \left[ \frac{\mathbf{E}\left( \tau _{% \mathbf{\pi }_{\left( N\right) },N}^{2}\right) }{2\mathbf{E}\left( \tau _{% \mathbf{\pi }_{\left( N\right) },N}\right) ^{2}}-1\right] ,$$ then the same holds true for the law of $\tau _{\mathbf{\pi }_{0},N}/\mathbf{% E}\left( \tau _{\mathbf{\pi }_{0},N}\right) $. - **Time reversal.** Consider the time-reversed** **version $% X_{n,\left( N\right) }^{\leftarrow }$ of the truncated Lamperti chain, so with one-step transition matrix $$\overleftarrow{P}_{\left( N\right) }=D_{\mathbf{\pi }_{\left( N\right) }}^{-1}P_{\left( N\right) }^{\prime }D_{\mathbf{\pi }_{\left( N\right) }}.$$ Its time-reversed transition matrix being $P_{\left( N\right) }$ which is in particular stochastically monotone, the Brown theory for hitting times applies to the time-reversed process as well (see [@Brown]), with $% \overleftarrow{\tau }_{\mathbf{\pi }_{0},N}$ and $\overleftarrow{\tau }_{% \mathbf{\pi }_{\left( N\right) },N}$ standing for the hitting times of the time-reversed chain. The time-reversed process $X_{n,\left( N\right) }^{\leftarrow }$ thus constructed is a truncated version of the process defined from (\[TR\]). **Acknowledgments:** The authors are indebted for support of the CMM-Basal Conicyt project AFB170001 and I.E.A. Cergy. T. H. acknowledges partial support from the labex MME-DII (Modèles Mathématiques et Économiques de la Dynamique, de l’ Incertitude et des Interactions), ANR11-LBX-0023-01. This work also benefited from the support of the Chair “Modélisation Mathématique et Biodiversité” of Veolia-Ecole Polytechnique-MNHN-Fondation X. [99]{} Brown, M. (1990). Consequences of Monotonicity for Markov Transition Functions. City College, CUNY Report No MB89-03, unpublished manuscript. Brown, M. (1990). Error bounds for exponential approximations of geometric convolutions. The Annals of Probability. Vol. 18, N${{}^{\circ }}$ 3, 1388-1402. Collet, P., Martínez, S., San Martín, J. *Quasi-stationary distributions*. 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--- author: - 'Miles H. Anderson' - Romain Bouchand - Junqiu Liu - Wenle Weng - | \ Ewelina Obrzud - Tobias Herr - 'Tobias J. Kippenberg' bibliography: - 'biblib\_2.bib' title: 'Photonic chip-based resonant supercontinuum ' --- **Supercontinuum generation in optical fibers is one of the most dramatic nonlinear effects discovered [@alfano_observation_1970; @ranka_visible_2000; @birks_supercontinuum_2000], allowing short pulses to be converted into multi-octave spanning coherent spectra. This process has enabled self-referencing of optical frequency combs, establishing the RF-to-optical link [@jones_carrier-envelope_2000; @udem_optical_2002]. ideally suited for optical frequency division[@xie_photonic_2017], Raman spectral imaging[@ideguchi_coherent_2013], telecommunications [@marin-palomo_microresonator-based_2017], or astro-spectrometer calibration [@murphy_high-precision_2007]. Soliton microcombs [@herr_temporal_2014; @kippenberg_dissipative_2018] by contrast, can generate octave-spanning spectra[@li_stably_2017; @pfeiffer_octave-spanning_2017], but with good conversion efficiency only at vastly higher repetition rates in the 100s of GHz[@bao_nonlinear_2014]. Here, we bridge this efficiency gap with resonant supercontinuum, allowing supercontinuum generation using input pulses with an ultra-low 6 picojoule energy, and duration of 1 picosecond, 10-fold longer than [@gaeta_photonic-chip-based_2019]. This creates a smooth, flattened 2,200 line frequency comb, with an electronically detectable repetition rate of 28 GHz, constituting the largest bandwidth-line-count product for any microcomb generated to date. Taken together, our work establishes resonant supercontinuum as a promising route to broadband and coherent spectra.** \[t\]   \[t\]  Supercontinuum generation (SCG, or ‘white light’ generation [@bellini_phase-locked_2000]) is a process where high intensity optical pulses are converted into coherent octave-spanning spectra by propagation through a dispersion-engineered waveguide, fiber, or material (Fig. \[fig:setup\](a)). Following the demonstration of dramatic broadening in optical fiber [@ranka_visible_2000], the process has been well studied in photonic crystal fibers [@russell_photonic-crystal_2006; @dudley_supercontinuum_2006], owing to their capacity for dispersion engineering. SCG is based on a combination of nonlinear phenomenon including soliton fission, dispersive wave formation, and the Raman self-frequency shift [@skryabin_colloquium:_2010]. Commonly, in order to generate a supercontinuum which is coherent as well as having ultra-high bandwidth, ultrashort pulses ($\sim$100 fs) with high peak powers (1 kW) are needed so that the pulse undergoes a process known as soliton fission [@herrmann_experimental_2002; @skryabin_colloquium:_2010], as opposed to incoherent modulation instability[@nakazawa_coherence_1998]. Dispersive wave emission (alternatively soliton Cherenkov radiation[@akhmediev_cherenkov_1995]) simultaneously serves to extend the spectrum towards other spectral regions far from the pump [@hilligsoe_initial_2003]. To achieve this, SCG has most often required the input of mode-locked laser systems operating at repetition rates of $<$1 GHz so as to provide large pulse energies. Although photonic chip-based waveguides with a high material nonlinearity have reduced required pulse energies by an order of magnitude, and have allowed lithographic dispersion engineering [@yeom_low-threshold_2008; @halir_ultrabroadband_2012; @leo_dispersive_2014; @guo_mid-infrared_2018], synthesis of octave spanning spectra with line spacing &gt;10 GHz has remained challenging. Accessing this regime has been achieved using SCG driven with electro-optic frequency combs [@wu_supercontinuum-based_2013; @beha_electronic_2017; @obrzud_broadband_2018-1; @carlson_ultrafast_2018], providing ultrabroad frequency comb formation at repetition rates of 10-30 GHz, although multiple stages of amplification and pulse-compression were required in order to replicate the same pulse duration and peak powers available from mode-locked lasers. An alternative technique for the generation of coherent frequency comb spectra is Kerr comb generation [@kippenberg_dissipative_2018], i.e. soliton microcombs. Kerr comb generation uses the resonant build-up of a continuous-wave laser to generate a frequency comb via parametric frequency conversion and the formation of dissipative Kerr solitons (DKS) [@herr_temporal_2014]. These DKS exhibit a rich landscape of dynamical states, such as breathing [@leo_dynamics_2013; @lucas_breathing_2017], chaos [@anderson_observations_2016], and bound-states [@wang_universal_2017]. In contrast to SCG, DKS circulate indefinitely and are a soliton of an ‘open system’, relying on a double balance of nonlinearity and dispersion, as well as parametric gain and dissipation [@akhmediev_dissipative_2008]. The cavity enhances the pump field, dramatically reducing the input power threshold for soliton formation. Yet, the process itself has an efficiency that reduces with decreasing repetition rate owing to the reduced overlap of the DKS and the background pump [@bao_nonlinear_2014] (Fig. \[fig:setup\](b)). As a consequence, octave-spanning soliton microcombs to date have been synthesized with 1 THz line spacing[@pfeiffer_octave-spanning_2017; @li_stably_2017], and it has proven challenging to synthesize spectra with 10-50 GHz repetition rate with either SCG or microcomb formation. However, a growing number of applications benefit from coherent supercontinua with line spacing in the microwave domain that can be easily detected and processed by electronics. Such widely-spaced comb spectra are resolvable in diffraction-based spectrometers for astrocombs [@ewelina_obrzud_microphotonic_2019; @suh_searching_2019], and are highly appropriate as sources for massively parallel wavelength-division multiplexing [@marin-palomo_microresonator-based_2017; @hu_single-source_2018]. They can also remove the ambiguity in the identification of individual comb lines. In this work we demonstrate *resonant* supercontinuum generation, a synthesis between conventional SCG and soliton microcombs (Fig. \[fig:setup\](c)). By supplying a microresonator with a pulsed input, we take equal advantage of the resonant enhancement offered by the cavity, as well as the higher peak input powers and conversion efficiency allowed by pulses as compared to CW[@malinowski_optical_2017]. Where recent works on pulse-driven Kerr cavities for DKS generation have focused on facilitating access to single soliton generation with high conversion efficiency [@obrzud_temporal_2017], and peak-power enhancement [@lilienfein_temporal_2019], [@okawachi_bandwidth_2014]. By promoting low dispersion with a strong third-order component, we generate a flattened, broadband spectrum close to 2/3 of an octave wide, using ten times lower pulse energy, and ten times longer pulse duration than in conventional [Si$_3$N$_4$ ]{}-based SCG [@carlson_ultrafast_2018; @okawachi_carrier_2018] and with an electronically detectable repetition rate of 28 GHz. **Resonant Supercontinuum Results.** The chip-based [Si$_3$N$_4$ ]{}microresonator used for this experiment (a section depicted in Fig. \[fig:setup\](h)), has a free spectral range (FSR) of 27.88 GHz and a loaded linewidth in the telecom band of 110 MHz (most probably value[@liu_ultralow-power_2018-1]). The waveguide dimensions have been selected to give a low dispersion of $\beta_2=-11$ fs$^2/$mm. The pulse-train incident on this chip is synthesized using cascaded electro-optic modulation, intensity modulation, and dispersion compensation [@kobayashi_optical_1988; @fujiwara_optical_2003] (see Fig. \[fig:setup\](d)), providing pulses with a minimum duration of 1 ps, at a repetition rate $f_\mathrm{eo}=$ 13.94 GHz. In this way, the microresonator is sub-harmonically pumped every two roundtrips[@ewelina_obrzud_microphotonic_2019]. This decreases the conversion efficiency by a factor of 2, but reduces the requirements on the microwave transmission system. A tunable RF signal generator supplies $f_\mathrm{eo}$, and we keep two alternative RF sources – with relatively high (RF-1) and low (RF-2) phase-noise respectively – in order to observe how their frequency noise is transferred to the resonant supercontinuum. DKS states are generated on the input pulses by sweeping their carrier frequency $\omega_p$ to the region of cavity bistability [@herr_temporal_2014], such that the detuning $\delta\omega=\omega_0-\omega_p>0$. Before a DKS can be formed stably, the difference between the repetition rate of the pulse-train $f_\mathrm{eo}$ and the cavity FSR has to be matched to within a ‘locking range’. Inside this range, the generated soliton becomes locked to the driving pulse [@obrzud_temporal_2017] (or modulated background [@weng_spectral_2019]), so that the comb $f_\mathrm{rep}=f_\mathrm{eo}$. A simulated example of this is depicted in Fig. \[fig:results1\](a). In this experiment, the locking range is $\sim$3050 kHz. \[t\] {width="\textwidth"} The measured output spectrum of the microresonator during single-state DKS operation are presented in Fig. \[fig:results1\](c), generated at two input powers: the minimum required to form a DKS, and a higher power generating the most energetic spectrum for this work. They both exist at the maximum accessible cavity detuning $\delta\omega$, where the spectral bandwidth of the DKS scales as $\Omega_{S}\propto\sqrt{P_0}$ [@coen_universal_2013; @lucas_detuning-dependent_2017], hence the dramatic broadening of the spectra. The first, least energetic soliton has a 3-dB bandwidth of 9.5 THz, and an estimated pulse duration of 55 fs based on a sech$^2$ fit. The energy of a soliton scales the same way [@bao_nonlinear_2014], hence this first DKS has the highest conversion efficiency from input comb to generated lines of 8%. The high-energy DKS measurably spans 64 THz or 600 nm, accounting for 2,300 measurable lines (1,400 in 10 dB), and has a conversion efficiency of 2.8%. This is the highest line-count for a single-state DKS, with a bandwidth exceeding the C+L bands, to our knowledge. replicating the measured spectrum predict a distorted DKS due to the strong dispersive wave emission, and having a duration of $\sim24$ fs. The spectrum is strongly enhanced on the long-wavelength side due to the third-order dispersion of the waveguide, forming a dispersive wave at 1957 nm, combined with the soliton Raman self-frequency shift which has shifted the spectral center towards 1590 nm [@karpov_raman_2016; @yi_theory_2016]. Importantly, no fast-tuning methods[@lamb_optical-frequency_2018] were required in order to form DKS. , suggesting a practical absence of cavity thermal relaxation, which has complicated stable soliton generation in the past [@brasch_bringing_2016; @li_stably_2017]. The number of DKS ranged from 1 to 3. The effective average power (see Methods) required to generate these single-soliton states ranges from 18 to 180 mW, which is highly efficient considering the $Q$ of the resonator as compared to recent experimental work in CW-driven [Si$_3$N$_4$ ]{}microresonators, of similar FSR, with even higher $Q$ [@liu_ultralow-power_2018-1; @liu_nanophotonic_2019]. The corresponding pulse energies range from 0.6 to 6 pJ, and we estimate the peak powers to be from 0.2 to 2 W based on the calculated EO-comb compression profile. **Coherence properties.** Fig. \[fig:results1\](d) shows the repetition rate beatnote of the DKS *excluding* the EO-comb spectrum. The beatnote corresponds exactly to 2 times the RF source frequency, and with a 1 Hz limited linewidth demonstrates high repetition rate stability. For measuring the optical coherence of the comb, optical heterodyne measurements are taken against increasing values of $\mu$, the comb line index from the center pump, from 1550 nm to the outer edge at 1908 nm plotted in Fig. \[fig:results1\](e-g). Fig \[fig:results1\](e) shows a narrow heterodyne beatnote at the edge of the EO-comb. As comb lines become further away from the center, their linewidth quickly broadens as shown in Fig. \[fig:results1\](f), where we plot heterodyne beatnotes up to a range of 70 nm from the comb center. This noise multiplication continues to the long-wavelength edge of the comb, where the heterodyne beatnote with a narrow-linewidth 1908 nm laser is plotted in Fig. \[fig:results1\](g). Here, the linewidth has expanded to around 7.5 MHz according to Gaussian fitting. When we switch our RF signal generator from RF-1 to the lower noise source RF-2, and measure the heterodyne comb beatnotes at the same wavelengths, three examples of which are plotted in Fig. \[fig:results1\](f,g), we find that linewidths are decisively more narrow. The linewidth at 1908 nm in particular has reduced by almost a factor of 10, down to 900 kHz. \[t\] {width="\textwidth"} **Noise Transfer Simulation.** If one were to assume that the generated soliton is *perfectly* locked to the input pulse, we would expect the optical frequency noise of each soliton comb line to be coupled to the RF noise on the input pulse repetition rate $f_\mathrm{eo}$, such that $S^{(\mu)}_{\delta\nu}(f)=\mu^2S_f^{\mathrm{(rf)}}(f)$, with relative comb index $\mu$ (assuming other sources of laser noise are small by comparison). However this is not the case for a DKS. The frequency-noise multiplication transfer function can be found in numerical simulations based on the Lugiato-Lefever Equation (LLE) [@lugiato_spatial_1987] with parameters similar to that of a typical [Si$_3$N$_4$ ]{}resonator, where instead of a normally CW-driving term we use a pulsed input $F(\phi)$ (where $\phi$ is the spacial coordinate of the cavity) similar in duration to that used in the experiment. It is also slightly positively chirped, in line with our experimental EO-comb compression stage (see Methods and the S.I.), giving a negative phase curvature on the pulse [@AGRAWAL199560]. Frequency noise equivalent to a uniform power-spectral density of $S_f^{\mathrm{(rf)}}(f)=$ 1.0 Hz$^2$/Hz is applied to the input pulse over a long period of ‘slow time’ ($t>2,000t_\mathrm{photon}$), and the corresponding jitter of the soliton is captured. For this first simulation, we have set $f_\mathrm{eo}$, the input pulse repetition rate, to be equal to the FSR for fully synchronous driving. The simulation results are shown in Fig. \[fig:simulation1\]. In the time domain (shown in Fig. \[fig:simulation1\](a,b)), the generated soliton is located at its ‘trapping point’ at $\sim0.6$ ps at the trailing edge of the pulse[@hendry_spontaneous_2018], but under these synchronous and symmetrical pulse conditions, it may equally find itself at -0.6 ps on the leading edge. As it’s trapped, or locked, to the background input pulse, it inherits jitter and gradually walks back and forth in the ‘fast time’ domain. The noise of the input pulse itself, and its corresponding walk in the time domain, is plotted in Fig. \[fig:simulation1\](c). The corresponding frequency domain results are shown in Fig. \[fig:simulation1\](d,e). Fig. \[fig:simulation1\](e) in particular is obtained by taking the Fourier transform over both dimensions of the optical field in Fig. \[fig:simulation1\](b), therefore plotting the power spectral densities on the y-axis of each individual comb line along the x-axis – a simulated heterodyne beatnote (normalized to peak). As is evident, the linewidth of each comb line widens considerably as they become further from the comb center. Beyond this cut-off point, the transfer function begins to be dominated by the response of the cavity [@guo_universal_2017], where we see a resonance located after the cavity bandwidth at $\kappa$, and a further strong cut-off at the cavity detuning at $\delta\omega=2\pi\cdot600$ MHz. The exact nature of these resonances is beyond the scope of this work, though Taking the $\beta$-line as a guide, these far-offset features will not factor into the linewidth of the outer comb lines [@domenico_simple_2010] in this system. **Optimization of Nonlinear Filtering.** Returning attention to the experimental heterodyne beatnote measurement at 1908 nm, we have observed an interesting effect when the driving repetition $f_\mathrm{eo}$ is varied. Fig \[fig:simulation2\](a) shows the 1908 nm beatnote as $f_\mathrm{eo}$ is swept from the minimum to the maximum of the soliton locking range (0 kHz defined as the minimum). The linewidth appears to narrow, reaching a minimum at the upper edge of the locking range, in this case 50 kHz. To characterize this narrowing phenomenon, we measured the frequency noise spectrum of this beatnote using in-phase/quadrature analysis [@schiemangk_accurate_2014] as $f_\mathrm{eo}$ is varied across the locking range, which is shown in Fig. \[fig:simulation2\](b). Also overlaid is $\mu\sqrt{S^\mathrm{(rf1)}_f}$, where $S^\mathrm{(rf1)}_f$ here is the independently measured frequency noise spectrum of the signal generator RF-1 at 14 GHz, with comb line $\mu=2\times1300$, (factor of 2 being from the half rep-rate driving). We further plot the corresponding experimental transfer functions $T^{(\mu)}(f)$, as before, in Fig. \[fig:simulation2\](c). As shown, the frequency noise level of the 1908 nm beatnote very closely follows the multiplied RF noise until a certain cut-off frequency, which varies from a maximum of 2 MHz reducing to $\sim$500 kHz at the edge of the locking range. In excellent qualitative agreement with the simulation results in Fig \[fig:simulation1\], the value of this corner frequency is on the order of 100 times less than the linear cavity bandwidth, of 110 MHz, experimentally confirming the presence of nonlinear filtering. We next carry out numerical simulations to analyze the $f_\mathrm{eo}$-dependent nonlinear filtering behavior. We apply a mismatch between the input pulse train repetition rate and the native repetition rate of the soliton ($d=f_\mathrm{eo}-F\!S\!R$). A small level of positive chirp on the input pulse is included in the simulation as per experimental condition (see Methods). Fig. \[fig:simulation2\](e) show the same type of result as Fig. \[fig:simulation1\](e), only now for a DKS comb for 4 different values of $d$ (manifesting as the gradient in the comb line centers) between -50 and 50 kHz. The effect on the comb linewidth broadening is dramatic. The simulated comb line for $\mu=-1300$ as $d$ is varied between 0 and 50 kHz (the maximum of the locking range) is displayed in Fig. \[fig:simulation2\](f), showing excellent agreement with the experimental observation in Fig. \[fig:simulation2\](a). Our experimental and simulation results reveal that as the repetition rate mismatch $d$ changes, the ‘trapping’ location of the DKS on the driving pulse, as well as the local trapping gradient can be significantly different. Previous studies have demonstrated that solitons can acquire a non-zero drift across the cavity space due to the presence of a gradient on the background driving field. Specifically, for a purely phase-modulated background, the soliton will become attracted to the *peak* of the phase profile [@jang_temporal_2015]. Conversely, for pure amplitude-variation, a soliton will become attracted to the *edge* of the pulse, at some critical intensity level $F_C$ [@hendry_spontaneous_2018; @hendry_impact_2019-1]. In our experiment, the driving field is essentially a mixture of both amplitude modulation (pulse driving) and phase modulation (additional chirping). As a result, the soliton is drawn towards an *intermediate* trapping location between the intensity-based trap at the edge of the pulse, and the phase-based trap at the peak. This trapping point will be modified by $d$, which acts as an effective force [@javaloyes_dynamics_2016]. In order for a DKS to continue to sustain itself, it must follow the pulse at its own shifted repetition rate, so that: $$2\pi d + \frac{\partial\phi_S}{\partial t} = 0$$ where $\phi_S$ is the angular coordinate of the soliton inside the resonator. If $d$ is non-zero, the soliton must move to a location in order to acquire a shifted repetition rate due the gradient in background phase and/or intensity[@jang_temporal_2015; @hendry_impact_2019-1]. Fig. \[fig:simulation2\](d) illustrates these situations from (i) to (iii), where $d<0$, $d=0$, and $d>0$. The soliton is initially positioned on the left, leading edge of the pulse. In Fig. \[fig:simulation2\]d(i), the mismatch $d$ and the intensity-based trapping force have combined to shift the soliton to the very left edge of the pulse. In d(ii), the soliton is located at its intermediate trapping point, which is symmetrical with the pulse. In d(iii), the mismatch $d$ adds to the phase-based trapping force, causing the soliton to move closer to the peak. The observed change in noise transfer bandwidth with varied $d$ can be understood intuitively as being due to the local trapping gradient[@hendry_impact_2019-1] that gradually decreases from the edge of the pulse to the center. Analogous to atoms/particles trapped by optical potential wells[@phillips_nobel_1998], a DKS trapped at a location closer to the input pulse center is subject to a shallower potential gradient, thus becoming more ‘free-running’ and less affected by the noise contained in the driving field. For Fig. \[fig:simulation2\]e(i) where the soliton is being ‘pulled’ on the edge of the pulse, the broadening is maximized. For e(iv), where the soliton is instead being ‘pushed’ near the peak of the pulse, it has almost reduced to zero. The reduced noise transfer effect is well reproduced by our simulation, as shown in Fig. \[fig:simulation2\](g). As $d$ increases, the DKS gets closer to the pulse center. Consequently the cut-off frequency of the noise transfer function decreases, showing remarkable agreement with the experimental measurement in Fig. \[fig:simulation2\](c). \[sec\_summary\]Summary {#sec_summarysummary .unnumbered} ----------------------- Using the nonlinear, dispersion-engineered [Si$_3$N$_4$ ]{}microresonator platform, we have generated a smooth, resonant supercontinuum based on dissipative soliton formation, comprising over 2,000 comb teeth. By exploiting the resonant enhancement of the high-$Q$ cavity, such a spectrum was generated with pulses 1-6 pJ in energy, $>$1 ps in duration, and on the order of single-Watt peak power. For future integration, the current EO-comb input could be replaced by an alternative provider of GHz rate, picosecond pulses, such as chip-based silicon or other semiconductor-based mode-locked lasers [@davenport_integrated_2018; @delfyett_exploring_2019]. Further tailoring of the dispersion landscape and replacing the straight-waveguide coupling section with an adiabatic or curved coupling section [@chen_broadband_2017; @moille_broadband_2019] will improve the generation and extraction of the short wavelength side of the spectrum. This way, the soliton comb bandwidth can be increased from 2/3rds of an octave to a full octave, enabling $f-2f$ self-referencing. This work further invites a full exploration of the parameter space for input pulse chirp and flatness parameters in order to find further optimization of the frequency noise transfer, allowing the use of higher-noise voltage-controlled oscillators for locking the input pulse repetition rate. Overall, this work demonstrates a new chip-based technique for direct access to broadband spectra at microwave repetition rates using a pulsed input, without the use of interleaving, or additional electro-optic modulation after the fact. Importantly, it can provide a way of balancing the fundamental efficiency restrictions between conventional supercontinuum generation, and dissipative soliton microcombs. Acknowledgments {#acknowledgments .unnumbered} --------------- This work was supported by Contract No. D18AC00032 (DRINQS) from the Defense Advanced Research Projects Agency (DARPA), and funding from the Swiss National Science Foundation under grant agreement No. 176563. This material is based upon work supported by the Air Force Office of Scientific Research, Air Force Material Command, USAF under Award No. FA9550-15-1-0250. W. W. acknowledges support from the EU’s H2020 research and innovation programme under Marie Sklodowska-Curie IF grant agreement No. 753749 (SOLISYNTH). Author Contributions {#author-contributions .unnumbered} -------------------- M.H.A. performed the experiment with the assistance of R.B., E.O., and T.H.. The sample microresonator was fabricated by J.L.. M.H.A. conducted the numerical analysis. The manuscript was prepared by M.H.A. with the assistance of R.B., W.W., T.J.K., and T.H.. T.J.K. supervised the project. Data Availability Statement {#data-availability-statement .unnumbered} --------------------------- All data and analysis files will be made available via `zenodo.org` upon publication. Methods {#methods .unnumbered} ------- **Electro-optic comb** The EO-comb is formed from a CW laser using an intensity modulator and three cascaded phase modulators, driven by an RF signal generator, creating approximately 50 spectral lines spaced by $f_\mathrm{eo}=$ 13.94 GHz. Two RF sources are used: RF-1: Rhode & Schwarz SMB100A; RF-2:, Keysight E8267D. The dispersion-based waveform compression stage amounts to 300 m of standard SMF-28, plus an additional length from 5 m of dispersion-compensating fiber (DCF) in order to purposefully leave a residual positive chirp on the pulse, increasing pulse duration up to $\sim$1.4 ps when not amplified (see S.I.). **Microresonator** The [Si$_3$N$_4$ ]{}microresonator used in this experiment has been fabricated with the *photonic Damascene process* [@pfeiffer_photonic_2018] with a 2350$\times$770 nm$^2$ cross-section, and possesses a loaded $Q$ probability-distribution in the telecom band of $1.8\pm0.3\times10^6$. The cavity mode spectrum $\omega_\mu$, per mode index $\mu$, is expressed as $\omega_\mu-\omega_0-\mu D_1=\sum_{k\ge2}\mu^k\frac{D_k}{k!},\quad k\in \mathbb{N}$, with the higher-order cavity dispersion on the right-hand side. Spectroscopic measurements[@liu_frequency-comb-assisted_2016] yield $D_1=2\pi\cdot27.88$ GHz (the FSR), and $D_2=2\pi\cdot7.2$ kHz ($\beta_2=-2\pi D_2/LD_1^3$). The location of the dispersive wave at $\omega_\mathrm{DW}=2\pi\cdot$154 THz allows us to infer $D_3\approx-3D_2D_1/(\omega_\mathrm{DW}-\omega_0)=2\pi\cdot15$ Hz ($\beta_3=-120$ fs$^3/$mm), when assuming $D_4=0$. **DKS Measurement** For the calculations of the *effective* power driving the resonant supercontinuum and conversion efficiency, we take into account the insertion loss of the lensed-fiber to chip interface (2.4 dB), and consider only every second EO-comb line, spaced by 28 GHz, as coupled into the microresonator. This further reduces the average power and conversion efficiency by 3 dB, and the effective pulse energy and pulse peak power seen by the resonator by 6 dB. The repetition rate beatnote, of the soliton comb lines only, was found by filtering out the EO-comb spectrum using a combination of a chirped fiber-Bragg grating and a wavelength-division multiplexer. **Simulation method** For the noise-transfer numerical investigation, we use the LLE as our mean-field model without higher order perturbations such as third-order and higher-order dispersion, stimulated Raman scattering, and spectral $\kappa(\omega)$ response: $$\label{eq:LLE1} \frac{\partial A(t)}{\partial t} = \left( i\delta\omega -\frac{\kappa}{2} -i\frac{D_2}{2}\frac{\partial^2}{\partial\phi^2} + ig|A|^2 \right) A + \sqrt{\kappa_\mathrm{ex}}F(\phi,t)$$ These omissions are so that the transfer of RF noise across the comb can be analyzed in its purest case. A full simulation of the generated DKS with all perturbations considered is provided in the S.I. The input pulse function $F(\phi,t)$ is expressed as a rectangular summation of lasing lines, similar to the experimental EO-comb: $$\label{eq:pulse} F(\phi,t) = \frac{\sqrt{P_0/\hbar\omega_0}}{M+1}\sum_{\mu=-M/2}^{M/2}e^{i\mu(\phi+\delta\phi(t))} e^{i\hat{D}_c}$$ with $P_0$ the pulse peak power and $M+1$ the total number of laser lines coupled to the cavity with spacing equal to the FSR. A residual dispersion is also applied to the pulse spectrum $\hat{D}_c$, which expresses the inexact dispersion compensation compressing the experimental EO-comb waveform (see S.I.). The long-term phase noise noise of the driving pulse is provided by $\delta\phi(t)$, related to the frequency noise of the RF source by $\delta\phi(t)=\int^{t}2\pi \,\delta\!f(t')dt'$. The simulation parameters are chosen to reflect a basic [Si$_3$N$_4$ ]{}resonator similar to that of the experiment: $\kappa=2\pi\cdot100$ MHz, coupling rate $\kappa_\mathrm{ex}=2\pi\cdot50$ MHz (critical coupling), dispersion $D_2=2\pi\cdot28$ kHz, and nonlinear coupling $g=2\pi\cdot0.054$ Hz. The driving parameters are $P_0=900$ mW (24$\times$ parametric oscillation threshold), $\delta\omega=6\kappa$, and $M=24$. Additional normal dispersion $\beta_c=+0.3$ ps$^2$, giving a spectral phase profile $\hat{D}_c=\beta_c(\mu D_1)^2/2$. Frequency-noise power spectra are found by Fourier-transform of the slow-time phase fluctuations of individual comb lines $\tilde{A}_\mu(t)$, as $\delta\!f_\mu(t) = \frac{d}{dt} \arg(\tilde{A}_\mu(t))$. As according to [@hendry_spontaneous_2018], the critical amplitude $F_C$ to which a soliton locks for this detuning ($\delta\omega>3\kappa$), based on pure intensity-based trapping, is close to the minimum amplitude required for DKS existence $|F_c|^2 \geq \frac{2\kappa^2\delta\omega}{\pi^2g\kappa_\mathrm{ex}}$.

--- abstract: 'We propose a sample efficient stochastic variance-reduced cubic regularization (Lite-SVRC) algorithm for finding the local minimum efficiently in nonconvex optimization. The proposed algorithm achieves a lower sample complexity of Hessian matrix computation than existing cubic regularization based methods. At the heart of our analysis is the choice of a constant batch size of Hessian matrix computation at each iteration and the stochastic variance reduction techniques. In detail, for a nonconvex function with $n$ component functions, Lite-SVRC converges to the local minimum within $\tilde{O}(n+n^{2/3}/\epsilon^{3/2})$[^1] Hessian sample complexity, which is faster than all existing cubic regularization based methods. Numerical experiments with different nonconvex optimization problems conducted on real datasets validate our theoretical results.' author: - 'Dongruo Zhou[^2]    and    Pan Xu[^3]    and    Quanquan Gu[^4]' bibliography: - 'reference.bib' date: 'May 18, 2018[^5]' title: 'Sample Efficient Stochastic Variance-Reduced Cubic Regularization Method' --- [^1]: Here $\tilde{O}$ hides poly-logarithmic factors [^2]: Department of Computer Science, University of California, Los Angeles, CA 90095, USA; e-mail: [drzhou@cs.ucla.edu]{} [^3]: Department of Computer Science, University of California, Los Angeles, CA 90095, USA; e-mail: [panxu@cs.ucla.edu]{} [^4]: Department of Computer Science, University of California, Los Angeles, CA 90095, USA; e-mail: [qgu@cs.ucla.edu]{} [^5]: The first version of this paper was submitted to UAI 2018 on March 9, 2018. This is the second version with improved presentation and additional baselines in the experiments, and was submitted to NeurIPS 2018 on May 18, 2018.

--- abstract: 'A lattice is called well-rounded if its minimal vectors span the corresponding Euclidean space. In this paper we completely describe well-rounded full-rank sublattices of ${\mathbb Z}^2$, as well as their determinant and minima sets. We show that the determinant set has positive density, deriving an explicit lower bound for it, while the minima set has density 0. We also produce formulas for the number of such lattices with a fixed determinant and with a fixed minimum. These formulas are related to the number of divisors of an integer in short intervals and to the number of its representations as a sum of two squares. We investigate the growth of the number of such lattices with a fixed determinant as the determinant grows, exhibiting some determinant sequences on which it is particularly large. To this end, we also study the behavior of the associated zeta function, comparing it to the Dedekind zeta function of Gaussian integers and to the Solomon zeta function of ${\mathbb Z}^2$. Our results extend automatically to well-rounded sublattices of any lattice $A {\mathbb Z}^2$, where $A$ is an element of the real orthogonal group $O_2({\mathbb R})$.' address: 'Department of Mathematics, Claremont McKenna College, 850 Columbia Avenue, Claremont, CA 91711-6420' author: - Lenny Fukshansky bibliography: - 'esm.bib' title: 'On distribution of well-rounded sublattices of $\mathbb Z^2$' --- Ł[[L]{}]{} [H]{} Introduction and statement of results ===================================== Let $N \geq 2$ be an integer, and let $\Lambda \subseteq \real^N$ be a lattice of full rank. Define the [*minimum*]{} of $\Lambda$ to be $$|\Lambda| = \min_{\bx \in \Lambda \setminus \{\bo\}} \|\bx\|,$$ where $\|\ \|$ stands for the usual Euclidean norm on $\real^N$. Let $$S(\Lambda) = \{ \bx \in \Lambda : \|\bx\| = |\Lambda| \}$$ be the set of [*minimal vectors*]{} of $\Lambda$. We say that $\Lambda$ is a [*well-rounded*]{} lattice (abbreviated WR) if $S(\Lambda)$ spans $\real^N$. WR lattices come up in a wide variety of different contexts, including sphere packing, covering, and kissing number problems, coding theory, and the linear Diophantine problem of Frobenius, just to name a few. Still, the WR condition is special enough so that one would expect WR lattices to be relatively sparce. However, in 2005 C. McMullen [@mcmullen] showed that in a certain sense [*unimodular*]{} WR lattices are “well distributed” among all [*unimodular*]{} lattices in $\real^N$, where a unimodular lattice is a lattice with determinant equal to 1. More specifically, he proved the following theorem, from which he derived the 6-dimensional case of the famous Minkowski’s conjecture for unimodular lattices. \[[@mcmullen]\] \[mcmullen\] Let $A \subseteq SL_N(\real)$ be the subgroup of diagonal matrices with positive diagonal entries, and let $\Lambda$ be a full-rank unimodular lattice in $\real^N$. If the closure of the orbit $A \Lambda$ is compact in the space of all full-rank unimodular lattices in $\real^N$, then it contains a WR lattice. Notice that in a certain sense this is a statement about distribution of WR lattices in the space of all unimodular lattices in a fixed dimension. Motivated by this beautiful theorem, we want to investigate the distribution of WR sublattices of $\zed^N$, which is a natural arithmetic problem. For instance, for a fixed positive integer $t$, does there necessarily exist a WR subllatice $\Lambda \subseteq \zed^N$ so that $\det(\Lambda) = t$? If so, how many different such sublattices are there? The first trivial observation is that if $t = d^N$ for some $d \in \zed_{>0}$ and $I_N$ is the $N \times N$ identity matrix, then the lattice $\Lambda = (d I_N) \zed^N$ is WR with $\det(\Lambda) = t$ and $|\Lambda| = d$. It seems however quite difficult to describe [*all*]{} WR sublattices of $\zed^N$ in an arbitrary dimension $N$. This paper is concerned with providing such a description in dimension two. From now on we will write $\WR(\Omega)$ for the set of all full-rank WR sublattices of a lattice $\Omega$; in this paper we will concentrate on $\WR(\zed^2)$. In section 3 we develop a certain parametrization of lattices in $\WR(\zed^2)$, which we then use to investigate the determinant set $\D$ of such lattices and to count the number of them for a fixed value of determinant. Specifically, let $\D$ be the set of all possible determinant values of lattices in $\WR(\zed^2)$, and let $\Mm$ be the set of all possible values of squared minima of these lattices, i.e. $\Mm = \{ |\Lambda|^2 : \Lambda \in \WR(\zed^2) \}$. It is easy to see that $\Mm$ is precisely the set of all positive integers, which are representable as a sum of two squares. Then it is interesting to understand how dense are these sets in $\zed_{>0}$. For any subset $\PP$ of $\zed$ and $M \in \zed_{>0}$, we write $$\PP(M) = \{ n \in \PP : n \leq M\}.$$ Define [*lower density*]{} of $\PP$ in $\zed$ to be $$\DL_{\PP} = \liminf_{M \rightarrow \infty} \frac{|\PP(M)|}{M},$$ and its [*upper density*]{} in $\zed$ to be $$\DU_{\PP} = \limsup_{M \rightarrow \infty} \frac{|\PP(M)|}{M}.$$ Clearly, $0 \leq \DL_{\PP} \leq \DU_{\PP} \leq 1$. If $0 < \DL_{\PP}$, we say that $\PP$ [*has density*]{}, and if $\DL_{\PP} = \DU_{\PP}$, i.e. if $\lim_{M \rightarrow \infty} \frac{|\PP(M)|}{M}$ exists, we say that $\PP$ [*has asymptotic density*]{} equal to the value of this limit, which could be 0. With this notation, we will show that $\D$ has density. More specifically, we prove the following. \[dense\] The determinant set $\D$ of lattices in $\WR(\zed^2)$ has representation $$\D = \left\{ (a^2+b^2)cd\ :\ a,b \in \zed_{\geq 0},\ \max\{a,b\} >0,\ c,d \in \zed_{>0},\ 1 \leq \frac{c}{d} \leq \sqrt{3} \right\},$$ and lower density $$\label{D_dens} \DL_{\D} \geq \frac{3^{\frac{1}{4}}-1}{2 \cdot 3^{\frac{1}{4}}} \approx 0.12008216 \dots$$ The minima set $\Mm$ has asymptotic density 0. We prove Theorem \[dense\] in section 4. Now, if $\Lambda \in \WR(\zed^2)$, let $\bx,\bwy$ be a minimal basis for $\Lambda$, and let $\theta$ be the angle between the vectors $\bx$ and $\bwy$; it is a well known fact that in dimensions $\leq 4$ a lattice is always generated by vectors corresponding to its successive minima, so such a basis certainly exists (see, for instance, [@pohst]). Then there is a simple connection between the minimum and the determinant of $\Lambda$: $$\det(\Lambda) = \|\bx\| \|\bwy\| \sin\theta = |\Lambda|^2 \sqrt{ 1 - \frac{\left( \bx^t \bwy \right)^2}{|\Lambda|^4}} = \sqrt{ |\Lambda|^4 - \left( \bx^t \bwy \right)^2 }.$$ Lemma \[gauss\] below implies that $0 \leq |\bx^t \bwy| \leq \frac{|\Lambda|^2}{2}$. Therefore we have $$\frac{\sqrt{3}\ |\Lambda|^2}{2} \leq \det(\Lambda) \leq |\Lambda|^2.$$ In view of this relation, it is especially interesting that the determinant set has positive density while the minima set has density 0. Next, for each $u \in \D$ we want to count the number of $\Lambda \in \WR(\zed^2)$ such that $\det(\Lambda) = u$. We need some additional notation. Suppose $t \in \zed_{>0}$ has prime factorization of the form $$\label{prim_fact} t = 2^w p_1^{2k_1} \dots p_s^{2k_s} q_1^{m_1} \dots q_r^{m_r},$$ where $p_i \equiv 3\ (\md 4)$, $q_j \equiv 1\ (\md 4)$, $w \in \zed_{\geq 0}$, $k_i \in \frac{1}{2} \zed_{>0}$, and $m_j \in \zed_{>0}$ for all $1 \leq i \leq s$, $1 \leq j \leq r$. Let $\alpha(t)$ be the number of representations of $t$ as a sum of two squares ignoring order and signs, that is $$\label{alpha} \alpha(t) = \left| \left\{ (a,b) \in \zed^2_{\geq 0} : a^2+b^2 = t,\ a \leq b \right\} \right|.$$ Also define $$\label{alpha*} \alpha_*(t) = \left| \left\{ (a,b) \in \zed^2_{\geq 0} : a^2+b^2 = t,\ a \leq b,\ \gcd(a,b)=1 \right\} \right|,$$ for all $t > 2$, and define $\alpha_*(1) = \alpha_*(2) = \frac{1}{2}$. It is a well-known fact that $\alpha(t)$ is given by $$\alpha(t) = \left\{ \begin{array}{ll} 0 & \mbox{if any $k_i$ is a half-integer} \\ \frac{1}{2} B & \mbox{if each $k_i$ is an integer and $B$ is even } \\ \frac{1}{2} \left( B - (-1)^w \right) & \mbox{if each $k_i$ is an integer and $B$ is odd,} \end{array} \right.$$ where $B=(m_1+1) \dots (m_r+1)$ (see, for instance [@silverman], [@weisstein]). Clearly, when $t$ is squarefree, $\alpha_*(t)=\alpha(t)$. It is also a well-known fact that for $t$ as in (\[prim\_fact\]) $$\alpha_*(t) = \left\{ \begin{array}{ll} 0 & \mbox{if $s \neq 0$ or $w > 1$} \\ 2^{r-1} & \mbox{if $s=0$ and $w = 0$ or $1$.} \end{array} \right.$$ We also define the function $$\label{beta} \beta_{\nu}(t) = \left| \left\{ d \in \zed_{>0} : d\ |\ t\ \text{and } \frac{\sqrt{t}}{\nu} \leq d \leq \sqrt{t} \right\} \right|,$$ for every $1 < \nu \leq 3^{1/4}$. The value of $\nu$ which will be particularly important to us is $\nu = 3^{1/4}$, therefore we define $$\beta(t) = \beta_{3^{1/4}}(t).$$ We discuss the function $\beta_{\nu}(t)$ in more detail in section 2; at least it is clear that for each given $t$, $\beta_{\nu}(t)$ is effectively computable for every $\nu$. Finally, for any $t \in \zed_{>0}$ define $$\delta_1(t) = \left\{ \begin{array}{ll} 1 & \mbox{if $t$ is a square} \\ 2 & \mbox{if $t$ is not a square,} \end{array} \right.$$ and $$\delta_2(t) = \left\{ \begin{array}{ll} 0 & \mbox{if $t$ is odd} \\ 1 & \mbox{if $t$ is even, $\frac{t}{2}$ is a square} \\ 2 & \mbox{if $t$ is even, $\frac{t}{2}$ is not a square.} \end{array} \right.$$ With this notation, we can state our second main theorem. \[count\] Let $u \in \zed_{>0}$, and let $\N(u)$ be the number of lattices in $\WR(\zed^2)$ with determinant equal to $u$. If $u=1$ or $2$, then $\N(u)=1$, the corresponding lattice being either $\zed^2$ or $\left( \begin{matrix} 1&-1 \\ 1&1 \end{matrix} \right) \zed^2$, respectively. Let $u>2$, and define $$t = t(u) = \left\{ \begin{array}{ll} u & \mbox{if $u$ is odd} \\ \frac{u}{2} & \mbox{if $u$ is even.} \end{array} \right.$$ Then: $$\begin{aligned} \label{N_formula} \N(u) & = & \delta_1(t) \beta(t) + \delta_2(t) \beta \left(\frac{t}{2} \right) + 4 \mathop{\sum_{n|t, 1<n<t/2}}_{n\ \text{not a square}} \alpha_* \left(\frac{t}{n} \right) \beta(n) \nonumber \\ & + & 2 \mathop{\sum_{n|t, 1 \leq n<t/2}}_{n\ \text{a square}} \alpha_* \left(\frac{t}{n} \right) (2\beta(n)-1).\end{aligned}$$ In particular, if $u \notin \D$, then the right hand side of (\[N\_formula\]) is equal to zero. Theorem \[count\] can also be easily extended to a more general class of lattices. Namely, write $O_2(\real)$ for the real orthogonal group, then for every $A \in O_2(\real)$ and every $\bx,\bwy \in \real^2$ we have $(A\bx)^t (A\bwy) = \bx^t \bwy$, i.e. $O_2(\real)$ is the isometry group of $\real^2$ with respect to the Euclidean norm. Therefore, if $A \in O_2(\real)$ then $\Lambda \in \WR(A\zed^2)$ if and only if $A^t \Lambda \in \WR(\zed^2)$. This immediately implies the following result. \[gen\] Let $A \in O_2(\real)$. Then the determinant set and the minima set of lattices in $\WR(A\zed^2)$ are $\D$ and $\Mm$ respectively, as defined above. Moreover, for each $u \in \D$ the number of lattices in $\WR(A\zed^2)$ with determinant equal to $u$ is given by $\N(u)$ as in Theorem \[count\]. We prove Theorem \[count\] in section 5. In section 6 we use Theorem \[count\] to work out simple examples of our formula in the case of prime power and product of two primes determinants. We also describe the “orthogonal” elements of $\WR(\zed^2)$, which come from ideals in Gaussian integers; these are quite sparse among all lattices in $\WR(\zed^2)$. We then derive easy to use bounds on $\N(u)$ and on the normal order of $\N(u)$. We also demonstrate examples of “extremal” sequences of determinant values, for which $\N(u)$ is especially large; see Corollary \[size\_N\]. In section 7 we derive a formula for the number of lattices in $\WR(\zed^2)$ of fixed minimum. In section 8 we study some basic properties of a zeta function, corresponding to the well-rounded sublattices of $\zed^2$. Namely, for $s \in \cee$ define $$\label{WR_zeta} \zeta_{\WR(\zed^2)}(s) = \sum_{\Lambda \in \WR(\zed^2)} (\det(\Lambda))^{-s} = \sum_{u=1}^{\infty} \N(u) u^{-s},$$ where $\N(u)$ is as above. In particular, $\N(u) \neq 0$ if and only if $u \in \D$. For a Dirichlet series $\sum_{n=1}^{\infty} c_n n^{-s}$, we say that it has a [*pole of order*]{} $\mu$ at $s=s_0$, where $\mu$ and $s_0$ are positive real numbers, if $$\label{pole_def} 0 < \lim_{s \rightarrow s_0^+} |s-s_0|^{\mu} \sum_{n=1}^{\infty} |c_n n^{-s}| < \infty.$$ We will also say that such a Dirichlet series is [*bounded from above (or below)*]{} by a Dirichlet series $\sum_{n=1}^{\infty} b_n n^{-s}$, if $\sum_{n=1}^{\infty} |c_n n^{-s}| \leq \sum_{n=1}^{\infty} |b_n n^{-s}|$ (respectively, $\geq \sum_{n=1}^{\infty} |b_n n^{-s}|$). In section 8 we prove the following result. \[zeta\] Let the notation be as above, then $\zeta_{\WR(\zed^2)}(s)$ is analytic for all $s \in \cee$ with $\Re(s) > 1$, and is bounded from below by a Dirichlet series that has a pole of order 2 at $s=1$. Moreover, for every real $\eps >0$ there exists a Dirichlet series with a pole of order $2+\eps$ at $s=1$, which bounds $\zeta_{\WR(\zed^2)}(s)$ from above. Notice that Theorem \[zeta\] provides additional information about the growth of $\N(u)$. In section 8 we prove Theorem \[zeta\] by means of considering the behavior of some related Dirichlet series, namely the generating functions of $\alpha_*$ and $\beta_{\nu}$. We should remark that we are not using the notion of a pole here in a sense that would imply the existence of an analytic continuation, but only to reflect on the growth of the coefficients; in fact, the arithmetic function $\N(u)$ behaves sufficiently erraticaly that one would doubt $\zeta_{\WR(\zed^2)}(s)$ having an analytic continuation to the left of $s=1$. We are now ready to proceed. A special divisor function ========================== As above, let $1 < \nu \leq 3^{1/4}$. In this section we briefly discuss bounds on the divisor function $\beta_{\nu}(t)$. Let $$\M_{\nu}(t) = \left\{ d \in \zed_{>0} : d\ |\ t\ \text{and } \frac{\sqrt{t}}{\nu} \leq d \leq \sqrt{t} \right\}.$$ \[beta\_gcd\] If $d_1,d_2 \in \M_{\nu}(t)$, then $\gcd(d_1,d_2)>1$. Suppose $d_1,d_2 \in \M_{\nu}(t)$, and $\gcd(d_1,d_2)=1$. Then $d_1d_2|t$, but $$\frac{t}{\nu^2} < d_1d_2 \leq t.$$ Notice that $d_1d_2 \neq t$, since this would imply $d_1=d_2=\sqrt{t}$. Then $$1 < \frac{t}{d_1d_2} < \nu^2 \leq \sqrt{3},$$ but $\frac{t}{d_1d_2} \in \zed$, which is a contradiction. Lemma \[beta\_gcd\] implies in particular that $\M_{\nu}(t)$ can contain at most one prime $q$, and in this case every $d \in \M_{\nu}(t)$ must be divisible by $q$. Write $p(t)$ for the smallest prime divisor of $t$. Another immediate consequence of Lemma \[beta\_gcd\] is that gaps between two consecuitive elements of $\M_{\nu}(t)$ must be greater or equal than $p(t)$. Therefore, since $\beta_{\nu}(t)=|\M_{\nu}(t)|$, we obtain $$\label{beta_bound1} \beta_{\nu}(t) \leq \left[ \left( \frac{\nu - 1}{\nu p(t)} \right) \sqrt{t} \right] + 1,$$ where $p(t) \geq 2$ for each $t \in \zed$, however for most $t$ better bounds are known. Let us write $\tau(t)$ for the number of distinct divisors of $t$ and $\omega(t)$ for the number of distinct prime divisors of $t$ (see [@hall] for detailed information on $\tau(t)$ and $\omega(t)$). Hooley’s $\Delta$-function of $t$ is defined by $$\Delta(t) = \max_x \left| \left\{ d \in \zed_{>0} : d|t,\ e^x < d \leq e^{x+1} \right\} \right|.$$ If we take $x = \log \frac{\sqrt{t}}{\nu}$, then it is easy to see that $$\M_{\nu}(t) \subseteq \left\{ d \in \zed_{>0} : d|t,\ e^x < d \leq e^{x+1} \right\},$$ and hence $\beta_{\nu}(t) \leq \Delta(t)$. Now, as stated in [@ten1] (see also [@hall]) a consequence of Sperner’s theorem is that $$\label{beta_bound2} \beta_{\nu}(t) \leq \Delta(t) \leq O \left( \frac{\tau(t)}{\sqrt{\omega(t)}} \right),$$ and if $t$ is squarefree the constant in $O$-notation is equal to 2. By Theorem 317 of [@hardy], for any $\eps > 0$ there exists an integer $t_0(\eps)$ such that for all $t > t_0(\eps)$, $$\label{tau_bound} \tau(t) < 2^{(1+\eps) \frac{\log t}{\log \log t}}.$$ Then combining (\[beta\_bound2\]) with (\[tau\_bound\]), we see that for any $\eps > 0$ there exists an integer $t_0(\eps)$ such that for all $t > t_0(\eps)$,$$\label{beta_bound3} \beta_{\nu}(t) \leq O \left( t^{\frac{(1+\eps) \log 2}{\log \log t}} \right),$$ which is much better than $O(\sqrt{t})$, the bound of (\[beta\_bound1\]), when $t$ is sufficiently large. In fact, for most $t$ we expect $\beta_{\nu}(t)$ to be even much smaller. A result of [@ten1] states that if $\psi(t)$ is any function such that $\psi(t) \rightarrow \infty$, as slowly as we wish, as $t \rightarrow \infty$, then $$\label{beta_av} \beta_{\nu}(t) \leq \Delta(t) < \psi(t) \log \log t$$ for all positive integers $t$ in a sequence of asymptotic density 1. The function $\beta_{\nu}(t)$ also has a geometric interpretation. Write the prime decomposition for $t$ as $$t = p_1^{e_1} \dots p_{\omega(t)}^{e_{\omega(t)}}$$ for corresponding distinct primes $p_1, \dots, p_{\omega(t)}$ and positive integers $e_1, \dots, e_{\omega(t)}$. If $d$ is a divisor of $t$, then $$d = p_1^{x_1} \dots p_{\omega(t)}^{x_{\omega(t)}}$$ for some integers $0 \leq x_i \leq e_i$ for each $1 \leq i \leq \omega(t)$. Then $\frac{\sqrt{t}}{\nu} \leq d \leq \sqrt{t}$ if and only if $$\log \frac{\sqrt{t}}{\nu} \leq \sum_{i=1}^{\omega(t)} x_i \log p_i \leq \log \sqrt{t}.$$ In other words, $\beta_{\nu}(t)$ is precisely the number of integer lattice points in the polytope $P_{\nu}(t)$ in $\real^{\omega(t)}$ bounded by the hyperplanes $$\label{poly_pt} x_i = 0,\ x_i = e_i\ \forall\ 1 \leq i \leq \omega(t),\ \sum_{i=1}^{\omega(t)} x_i = \log \frac{\sqrt{t}}{\nu},\ \sum_{i=1}^{\omega(t)} x_i = \log \sqrt{t}.$$ In other words, $\beta_{\nu}(t) = \left|P_{\nu}(t) \cap \zed^{\omega(t)}\right|$, and $P_{\nu}(t)$ may be an irrational polytope. Counting integer lattice points in irrational polytopes is a very hard problem; a generating function for this problem is defined in [@barvinok], but almost nothing seems to be known about it. Parametrization of well-rounded lattices ======================================== In this section we present an explicit description and a convenient parametrization of lattices in $\WR(\zed^2)$, which we later use to prove the main results of this paper. First we introduce some additional notation, following [@near:ort]. An ordered collection of linearly independent vectors $\{ \bx_1, \dots, \bx_k \} \subset \real^N$, $2 \leq k \leq N$, is called [*nearly orthogonal*]{} if for each $1 < i \leq k$ the angle between $\bx_i$ and the subspace of $\real^N$ spanned by $\bx_1, \dots, \bx_{i-1}$ is in the interval $\left[ \frac{\pi}{3}, \frac{2\pi}{3} \right]$. In other words, this condition means that for each $1 < i \leq k$ $$\label{near} \frac{| \bx_i^t \bwy |}{\|\bx_i\| \|\bwy\|} \leq \frac{1}{2},$$ for all non-zero vectors $\bwy \in \spn_{\real} \{ \bx_1, \dots, \bx_{i-1} \}$. The following result is Theorem 1 of [@near:ort]; in case $N=2$ this was proved by Gauss. \[[@near:ort]\] \[no\] Suppose that an ordered basis $\{ \bx_1, \dots, \bx_k \}$ for a lattice $\Lambda$ in $\real^N$ of rank $1 < k \leq N$ is nearly orthogonal. Then it contains a minimal vector of $\Lambda$. In particular, if all vectors $\bx_1, \dots, \bx_k$ of Theorem \[no\] have the same norm, then $\Lambda$ is a WR lattice; we will call such a basis $\bx_1, \dots, \bx_k$ minimal. For the rest of this paper we will restrict to the case $N=2$. Here is a first characterization of WR sublattices of $\zed^2$. \[gauss\] A sublattice $\Lambda \subseteq \zed^2$ of rank 2 is in $\WR(\zed^2)$ if and only if it has a basis $\bx,\bwy$ with $$\label{gauss_cond} \|\bx\| = \|\bwy\|,\ |\cos \theta| = \frac{| \bx^t \bwy |}{\|\bx\| \|\bwy\|} \leq \frac{1}{2},$$ where $\theta$ is the angle between $\bx$ and $\bwy$. Moreover, if this is the case, then the set of minimal vectors $S(\Lambda) = \{ \pm \bx, \pm \bwy \}$. In particular, a minimal basis for $\Lambda$ is unique up to $\pm$ signs and reordering. Suppose first that $\Lambda$ contains a basis $\bx,\bwy$ satisfying (\[gauss\_cond\]). By Theorem \[no\] this must be a minimal basis, meaning that $\Lambda$ is WR. Next assume that $\Lambda$ is WR, and let $\bx,\bwy \in S(\Lambda)$ be linearly independent vectors. It is a well known fact that for lattices of rank $\leq 4$ linearly independent minimal vectors form a basis, hence $\bx,\bwy$ is a basis for $\Lambda$, and $|\Lambda| = \|\bx\| = \|\bwy\|$. Let $\theta$ be the angle between $\bx$ and $\bwy$. We can assume without loss of generality that $\cos \theta > 0$: if not, replace $\bx$ with $-\bx$ or $\bwy$ with $-\bwy$. Notice that $\bo \neq \bx-\bwy \in \Lambda$, and $$\|\bx - \bwy\| = \sqrt{ \|\bx\|^2 + \|\bwy\|^2 - 2 \bx^t \bwy } = |\Lambda| \sqrt{ 2 (1 - \cos \theta) }.$$ If $\cos \theta > \frac{1}{2}$, then $\|\bx - \bwy\| < |\Lambda|$, which is a contradiction. This proves (\[gauss\_cond\]), and also implies that the angle between two minimal linearly independent vectors in $\Lambda$ must lie in the interval $\left[ \frac{\pi}{3}, \frac{2\pi}{3} \right]$. Now assume that $\Lambda \in \WR(\zed^2)$, and let $\bx,\bwy$ be a minimal basis for $\Lambda$, so $|\Lambda| = \|\bx\| = \|\bwy\|$. Let $$\theta_1,\theta_2,\theta_3,\theta_4$$ be angles between pairs of vectors $\{\bx,\bwy\}$, $\{\bwy,-\bx\}$, $\{-\bx,-\bwy\}$, and $\{-\bwy,\bx\}$ respectively. Since all of these vectors are in $\Lambda$ and have length $|\Lambda|$, it must be true that $$\theta_1,\theta_2,\theta_3,\theta_4 \in \left[ \frac{\pi}{3}, \frac{2\pi}{3} \right].$$ On the other hand, $$\theta_2 = \theta_4 = \pi - \theta_1,\ \theta_3 = \theta_1.$$ Assume there exists a vector $\bz \in \Lambda$ of length $|\Lambda|$ which is not equal to $\pm \bx, \pm \bwy$. Then all the angles it makes with the vectors $\pm \bx, \pm \bwy$ must lie in the interval $\left[ \frac{\pi}{3}, \frac{2\pi}{3} \right]$. This means that at least one of these angles must be equal to $\frac{\pi}{3}$, assume without loss of generality that this is the angle $\bz$ makes with $\bx$. Then $$\bz = \left( \begin{matrix} z_1 \\ z_2 \end{matrix} \right) = \left( \begin{matrix} \cos \left( \frac{\pi}{3} \right)&-\sin \left( \frac{\pi}{3} \right) \\ \sin \left( \frac{\pi}{3} \right)&\cos \left( \frac{\pi}{3} \right) \end{matrix} \right) \left( \begin{matrix} x_1 \\ x_2 \end{matrix} \right) = \left( \begin{matrix} \frac{x_1}{2} - \frac{x_2\sqrt{3}}{2} \\ \frac{x_1\sqrt{3}}{2} + \frac{x_2}{2} \end{matrix} \right),$$ where $x_1,x_2 \in \zed$ are coordinates of $\bx$. Since $\bz \in \Lambda$, it must be true that $z_1,z_2 \in \zed$, but this is not possible. Hence a vector $\bz$ like this cannot exist, and this completes the proof. Next we develop a certain convenient explicit parametrization of lattices in $\WR(\zed^2)$. We start with lemmas describing two different families of such lattices. \[abcd1\] Let $a,b,c,d \in \zed$ be such that $$\label{cond_esm1} 0 < |d| \leq |c| \leq \sqrt{3} |d|,\ \max \{|a|,|b|\} > 0.$$ Then $$\label{esm1} \Lambda = \left( \begin{matrix} ac+bd&ac-bd \\ bc-ad&bc+ad \end{matrix} \right) \zed^2$$ is in $\WR(\zed^2)$ with $$\label{det_esm1} \det(\Lambda) = 2(a^2+b^2)|cd|.$$ Suppose $a,b,c,d \in \zed$ satisfy (\[cond\_esm1\]). Let $\bx = \left( \begin{matrix} ac+bd \\ bc-ad \end{matrix} \right)$ and $\bwy = \left( \begin{matrix} ac-bd \\ bc+ad \end{matrix} \right)$, then $$\begin{aligned} \label{square} \|\bx\|^2 & = & (ac+bd)^2 + (bc-ad)^2 \nonumber \\ & = & (a^2+b^2)(c^2+d^2) \nonumber \\ & = & (ac-bd)^2 + (bc+ad)^2 = \|\bwy\|^2.\end{aligned}$$ Let $\Lambda = \spn_{\zed} \{\bx, \bwy\}$, then $\rk \Lambda = 2$. Let $\theta$ be the angle between $\bx$ and $\bwy$, and let $c = \gamma d$, where by (\[cond\_esm1\]), $1 \leq |\gamma| \leq \sqrt{3}$. Then, by (\[square\]) and (\[cond\_esm1\]) $$\begin{aligned} |\cos(\theta)| & = & \frac{| \bx^t \bwy |}{\|\bx\| \|\bwy\|} = \frac{|(a^2+b^2)(c^2-d^2)|}{(a^2+b^2)(c^2+d^2)} \\ & = & \frac{c^2-d^2}{c^2+d^2} = \frac{\gamma^2-1}{\gamma^2+1} \leq \frac{1}{2}.\end{aligned}$$ Therefore $\theta \in \left[ \frac{\pi}{3}, \frac{2\pi}{3} \right]$, and so, by Lemma \[gauss\], $\Lambda$ is WR; (\[det\_esm1\]) follows. This completes the proof. \[abcd2\] Let $a,b,c,d \in \zed$ be such that $$\label{cond_esm2} c^2+d^2 \geq 4|cd|,\ \max \{|a|,|b|\} > 0.$$ Then $$\label{esm2} \Lambda = \left( \begin{matrix} ac-bd&ad-bc \\ ad+bc&ac+bd \end{matrix} \right) \zed^2$$ is in $\WR(\zed^2)$ with $$\label{det_esm2} \det(\Lambda) = (a^2+b^2) |c^2-d^2|.$$ Suppose $a,b,c,d \in \zed$ satisfy (\[cond\_esm2\]). Let $\bx = \left( \begin{matrix} ac-bd \\ ad+bc \end{matrix} \right)$ and $\bwy = \left( \begin{matrix} ad-bc \\ ac+bd \end{matrix} \right)$, and define $\Lambda = \spn_{\zed} \{\bx, \bwy\}$. Then $\rk \Lambda = 2$, and $\|\bx\|, \|\bwy\|$ are the same as in (\[square\]). Let $\theta$ be the angle between $\bx$ and $\bwy$. Then, by (\[square\]) and (\[cond\_esm2\]) $$\begin{aligned} |\cos(\theta)| & = & \frac{| \bx^t \bwy |}{\|\bx\| \|\bwy\|} = \frac{2|cd|(a^2+b^2)}{(a^2+b^2)(c^2+d^2)} \\ & = & \frac{2|cd|}{c^2+d^2} \leq \frac{1}{2}.\end{aligned}$$ Therefore $\theta \in \left[ \frac{\pi}{3}, \frac{2\pi}{3} \right]$, and so, by Lemma \[gauss\], $\Lambda$ is WR; (\[det\_esm2\]) follows. This completes the proof. \[2-class\] Suppose $\Lambda \in \WR(\zed^2)$. Then $\Lambda$ is either of the form as described in Lemma \[abcd1\] or as in Lemma \[abcd2\]. Suppose $\Lambda \in \WR(\zed^2)$, and let $\bx,\bwy \in \Lambda$ be a minimal basis $$\label{c1} \|\bx\|^2 = x_1^2 + x_2^2 = |\Lambda|^2 = y_1^2+y_2^2 = \|\bwy\|^2.$$ Notice that due to (\[c1\]) it must be true that either the pairs $x_1,y_1$ and $x_2,y_2$, or the pairs $x_1,y_2$ and $x_2,y_1$ are of the same parity. Indeed, suppose this is not true, then we can assume without loss of generality that $x_1,x_2$ are even and $y_1,y_2$ are odd. But then $$x_1^2 + x_2^2 \equiv 0\ (\md\ 4),\ y_1^2+y_2^2 \equiv 2\ (\md\ 4),$$ which contradicts (\[c1\]). Therefore, either $$\label{c2} \frac{x_1-y_1}{2},\ \frac{x_1+y_1}{2},\ \frac{y_2-x_2}{2},\ \frac{x_2+y_2}{2} \in \zed,$$ or $$\label{c3} \frac{x_1-y_2}{2},\ \frac{x_1+y_2}{2},\ \frac{y_1-x_2}{2},\ \frac{x_2+y_1}{2} \in \zed.$$ First assume (\[c2\]) is true. Then let $$\label{c4} c = \gcd \left( \frac{x_1+y_1}{2}, \frac{x_2+y_2}{2} \right),\ a = \frac{x_1+y_1}{2c},\ b = \frac{x_2+y_2}{2c},\ d = \frac{(y_2-x_2)c}{x_1+y_1}.$$ Clearly $a,b,c \in \zed$. We will now show that $d \in \zed$. Indeed, $$d = \frac{(y_2-x_2)c}{x_1+y_1} = \frac{y^2_2-x^2_2}{(x_2+y_2)\left(\frac{x_1+y_1}{c}\right)},$$ and of course $(x_2+y_2)\ |\ (y^2_2-x^2_2)$. Also, by (\[c1\]) $$\left(\frac{x_1+y_1}{c}\right)\ |\ (x_1+y_1)\ |\ (x_1^2-y_1^2) = (y^2_2-x^2_2),$$ and by definition of $c$ in (\[c4\]), $$\gcd\left( x_2+y_2,\ \frac{x_1+y_1}{c} \right) = 1,$$ which implies that $$(x_2+y_2)\left(\frac{x_1+y_1}{c}\right)\ |\ (y^2_2-x^2_2),$$ and hence $d \in \zed$. With these definitions of $a,b,c,d$, it is easy to see that $$x_1 = ac+bd,\ x_2 = bc-ad,\ y_1 = ac-bd,\ y_2 = bc+ad,$$ and hence $\Lambda$ is precisely of the form (\[esm1\]). Moreover, since it is WR, Lemma \[gauss\] implies that it must satisfy condition (\[gauss\_cond\]), which implies (\[cond\_esm1\]). This finishes the proof in case (\[c2\]) is true. The proof in case (\[c3\]) is true is completely analogous, in which case $\Lambda$ is of type (\[esm2\]), and then (\[cond\_esm2\]) is satisfied. Suppose now that a lattice $$\Lambda = \left( \begin{matrix} ac-bd&ad-bc \\ ad+bc&ac+bd \end{matrix} \right) \zed^2$$ with $$c^2+d^2 \geq 4|cd|,\ \max \{|a|,|b|\} > 0$$ as in Lemma \[abcd2\], and $$\det(\Lambda) = (a^2+b^2) |c^2-d^2|$$ is even. We will show that in this case $\Lambda$ can be represented in the form as in Lemma \[abcd1\]. First assume that $a^2+b^2$ is even, then $a^2,b^2$, and hence $a,b$, must be of the same parity, meaning that $a+b$ and $a-b$ are even. Define $$a_1 = \frac{a-b}{2},\ b_1 = \frac{a+b}{2},$$ then $a^2+b^2 = 2(a_1^2+b_1^2)$. Let $c_1,d_1$ be such that $c_1 d_1 = c^2-d^2$, and $$|c_1| = \max\{ |c-d|, |c+d| \},\ |d_1| = \min\{ |c-d|, |c+d| \}.$$ Suppose for instance that $c_1 = c+d$ and $d_1 = c-d$ (the argument is completely analogous in case $c_1 = c-d$ and $d_1 = c+d$). Then $$c = \frac{c_1+d_1}{2},\ d = \frac{c_1-d_1}{2},$$ and so $4cd = c_1^2-d_1^2$. On the other hand $c^2+d^2 = \frac{c_1^2+d_1^2}{2}$. The fact that $c^2+d^2 \geq 4|cd|$ implies that $$\frac{c_1^2+d_1^2}{2} \geq |c_1^2 - d_1^2| = c_1^2 - d_1^2,$$ since $|c_1| \geq |d_1|$, and so $$|c_1| \leq \sqrt{3}\ |d_1|.$$ This choice of $a_1,b_1,c_1,d_1$ satisfies the conditions of (\[cond\_esm1\]), and it is easy to see that $$\label{lat_eq} \Lambda = \left( \begin{matrix} ac-bd&ad-bc \\ ad+bc&ac+bd \end{matrix} \right) \zed^2 = \left( \begin{matrix} a_1c_1+b_1d_1&a_1c_1-b_1d_1 \\ b_1c_1-a_1d_1&b_1c_1+a_1d_1 \end{matrix} \right) \zed^2,$$ and $\det(\Lambda) = 2(a_1^2+b_1^2)|c_1d_1|$. Next assume that $c^2-d^2$ is even. Then $(c+d)(c-d)$ is even and $(c+d)+(c-d) = 2c$ is even, which implies that $(c+d)$ and $(c-d)$ must both be even, in particular $c^2-d^2$ is divisible by $4$. Let $c_1,d_1$ be such that $4 c_1 d_1 = c^2-d^2$, and $$|c_1| = \frac{1}{2} \max\{ |c-d|, |c+d| \},\ |d_1| = \frac{1}{2} \min\{ |c-d|, |c+d| \}.$$ By an argument as above, we can easily deduce again that $$|d_1| \leq |c_1| \leq \sqrt{3}\ |d_1|.$$ Let $$a_1 = a-b,\ b_1 = a+b,$$ then $2(a^2+b^2) = a_1^2+b_1^2$, and so $$\det(\Lambda) = (a^2+b^2)|c^2-d^2| = 4(a^2+b^2)|c_1d_1| = 2(a_1^2+b_1^2)|c_1d_1|.$$ Once again, it is easy to check that with $a_1,b_1,c_1,d_1$ defined this way (\[lat\_eq\]) holds. Let $$\label{E1} \E' = \left\{ (a,b,c,d) \in \zed^4\ :\ 0 < |d| \leq |c| \leq \sqrt{3} |d|,\ \max \{|a|,|b|\} > 0 \right\},$$ and $$\begin{aligned} \label{O1} \OO' = \{ (a,b,c,d) \in \zed^4 & : & 0 < |d| \leq |c| \leq \sqrt{3} |d|,\ \max \{|a|,|b|\} > 0, \nonumber \\ & & \text{so that } 2 \nmid (a^2+b^2) |cd| \}.\end{aligned}$$ Define two classes of integral lattices $$\label{E} \E = \left\{ \Lambda(a,b,c,d) = \left( \begin{matrix} ac+bd&ac-bd \\ bc-ad&bc+ad \end{matrix} \right) \zed^2 : (a,b,c,d) \in \E' \right\},$$ and $$\label{O} \OO = \left\{ \Lambda(a,b,c,d) = \left( \begin{matrix} \frac{ac+ad+bd-bc}{2}&\frac{ac-ad-bc-bd}{2} \\ \frac{ac+bc+bd-ad}{2}&\frac{ac+ad+bc-bd}{2} \end{matrix} \right) \zed^2 : (a,b,c,d) \in \OO' \right\}.$$ Then for every $\Lambda = \Lambda(a,b,c,d) \in \E$, $$\det(\Lambda) = 2(a^2+b^2) |cd|,$$ is even, and for every $\Lambda = \Lambda(a,b,c,d) \in \OO$, $$\det(\Lambda) = (a^2+b^2) |cd|$$ is odd. We proved the following theorem. \[disjoint\] The set $\WR(\zed^2)$ can be represented as the disjoint union of $\E$ and $\OO$. Moreover, the set of all possible determinants of lattices in $\WR(\zed^2) = \E \cup \OO$ is $$\label{dets} \D = \{ (a^2+b^2) |cd| : a,b,c,d \in \zed,\ 0 < \max \{|a|,|b|\},\ 0 < |d| \leq |c| \leq \sqrt{3} |d| \}.$$ \[min\_rem\] Notice also that $$|\Lambda|^2 = \left\{ \begin{array}{ll} (a^2+b^2)(c^2+d^2) & \mbox{if $\Lambda \in \E$} \\ \frac{1}{2}(a^2+b^2)(c^2+d^2) & \mbox{if $\Lambda \in \OO$.} \end{array} \right.$$ Therefore the set of squared minima $\Mm(\E)$ of the lattices from $\E$ can be represented as $$\begin{aligned} \label{min_rep1} \Mm(\E) = \{ (a^2+b^2)(c^2+d^2) & : & a,b,c,d \in \zed,\ 0 < \max \{|a|,|b|\},\nonumber \\ & & 0 < |d| \leq |c| \leq \sqrt{3} |d|,\ 2 | (a^2+b^2)cd \},\end{aligned}$$ and the set of squared minima $\Mm(\OO)$ of the lattices from $\OO$ can be represented as $$\begin{aligned} \label{min_rep2} \Mm(\OO) = \Big\{ \frac{1}{2} (a^2+b^2)(c^2+d^2) & : & a,b,c,d \in \zed,\ 0 < \max \{|a|,|b|\},\nonumber \\ & & 0 < |d| \leq |c| \leq \sqrt{3} |d|,\ 2 \nmid (a^2+b^2)cd \Big\}.\end{aligned}$$ Then the set of squared minima $\Mm$ can be represented as $\Mm = \Mm(\E) \cup \Mm(\OO)$. \[monoid\] The determinant set $\D$ in (\[dets\]) and the squared minima set $\Mm$ are commutative monoids under multiplication. If $t_1 = (a_1^2+b_1^2)c_1d_1$ and $t_2 = (a_2^2+b_2^2)c_2d_2$ are in $\D$, then $t_1t_2 = (a_3^2+b_3^2)c_3d_3 \in \D$, where $a_3=a_1a_2+b_1b_2$, $b_3=b_1a_2-a_1b_2$, $c_3 = \pm \max\{|c_1d_2|,|d_1c_2|\}$, and $d_3 = \pm \min\{|c_1d_2|,|d_1c_2|\}$. It is also obvious that a product of two integers which are representable as sums of two squares is also representable as a sum of two squares. Next we will use Theorem \[disjoint\] to investigate the structure of the set $\D$ and to count the number of lattices in $\WR(\zed^2)$ of a fixed determinant. Proof of Theorem \[dense\] ========================== The description of the set $\D$ in the statement of Theorem \[dense\] follows immediately from (\[dets\]). In this section we will mostly be concerned with deriving the estimate (\[D\_dens\]) for the lower density of $\D$. For each real number $1 < \nu \leq 3^{1/4}$, define the set $$\label{prod_set} \B_{\nu} = \left\{ n \in \zed_{>0} : \exists\ d \in \zed_{>0}\ \text{such that } d\ |\ n\ \text{and } \frac{\sqrt{n}}{\nu} \leq d \leq \sqrt{n} \right\}.$$ Then notice that another description of the set $\D$ in (\[dets\]) is $$\D = \Mm \B_{3^{1/4}} = \{ m n : m \in \Mm,\ n \in \B_{3^{1/4}} \},$$ where $\Mm$ is the set of squared minima of lattices in $\WR(\zed^2)$, as before, so $$\label{sum_set} \Mm = \{ m \in \zed_{>0} : m = k^2+l^2\ \text{for some } k,l \in \zed \}.$$ By a well known theorem of Fermat, the set $\Mm$ consists precisely of those positive integers $m$ in whose prime factorization every prime of the form $(4k+3)$ occurs an even number of times. Since $1 \in \Mm \cap \B_{3^{1/4}}$, we have $\Mm, \B_{3^{1/4}} \subset \D$; on the other hand, $6 \in \B_{3^{1/4}} \setminus \Mm$ and $2 \in \Mm \setminus \B_{3^{1/4}}$, hence $\Mm \subsetneq \D$ and $\B_{3^{1/4}} \subsetneq \D$. Moreover, $\D \subsetneq \zed_{>0}$, since for instance $3 \notin \D$. It is a well-known result of Landau (see, for instance [@motohashi]) that $\Mm$ has asymptotic density equal to 0, specifically $$\lim_{M \rightarrow \infty} \frac{1}{M} \left| \{ n \in \Mm : n \leq M \} \right| = \lim_{M \rightarrow \infty} \frac{1}{\sqrt{\log M}} = 0.$$ Let us investigate the density of the sets $\B_{\nu}$ for a fixed $\nu \in (1,3^{1/4}]$. As before, for each $M \in \zed_{>0}$ we write $$\B_{\nu}(M) = \{ n \in \B_{\nu} : n \leq M \}.$$ For each $n \in \zed_{>0}$ define $$I_{\nu}(n) = \left\{n^2, n(n-1), \dots, n \left(n-\left[\left(\frac{\nu-1}{\nu}\right)n\right]\right)\right\}.$$ Notice that every $k \in \B_{\nu}(M)$ is of the form $k = n(n-i)$ for some $n$ and $i \leq \left[\left(\frac{\nu-1}{\nu}\right)n\right]$, and so if $n \leq [\sqrt{M}]$, then $k \in \bigcup_{n=1}^{[\sqrt{M}]} I_{\nu}(n)$. For each $n \in \zed_{>0}$, $$\label{I_card} |I_{\nu}(n)| = \left[\left(\frac{\nu-1}{\nu}\right)n\right] + 1.$$ There may also be some $k = n(n-i) \in \B_{\nu}(M)$ with $n > [\sqrt{M}]$ for some $i \leq \left(\frac{\nu-1}{\nu}\right)n$. Then $k = n^2 - ni \leq M$, and so $i \geq n - \frac{M}{n}$. It is easy to see that this is only possible if $n \leq \left[ \sqrt{\nu M} \right]$, and so for each $[\sqrt{M}] < n \leq \left[ \sqrt{\nu M} \right]$ define $$J_{\nu,M}(n) = \left\{ n(n-i) : \left[ n - \frac{M}{n} \right] + 1 \leq i \leq \left[ \left(\frac{\nu-1}{\nu}\right)n \right] \right\}.$$ Clearly, $J_{\nu,M}(n) \subseteq I_{\nu}(n)$ for each such $n$, and $$\label{B_contain_J} \B_{\nu}(M) = \left( \bigcup_{n=1}^{[\sqrt{M}]} I_{\nu}(n) \right) \cup \left( \bigcup_{n=[\sqrt{M}]+1}^{\left[ \sqrt{\nu M} \right]} J_{\nu,M}(n)\right).$$ For simplicity of approximation notice that $$\label{B_contain} \bigcup_{n=1}^{[\sqrt{M}]} I_{\nu}(n) \subseteq \B_{\nu}(M) \subseteq \bigcup_{n=1}^{\left[ \sqrt{\nu M} \right]} I_{\nu}(n).$$ We immediately obtain an upper bound on $|\B_{\nu}(M)|$. \[I\_upp\] For all $M \in \zed_{>0}$, $$\label{I_upp1} |\B_{\nu}(M)| \leq \frac{\nu-1}{2}\ M + \frac{\nu-1}{2 \sqrt{\nu}}\ \sqrt{M}.$$ Combining (\[B\_contain\]) and (\[I\_card\]), we obtain: $$|\B_{\nu}(M)| \leq \sum_{n=1}^{\left[ \sqrt{\nu M} \right]} |I_{\nu}(n)| \leq \frac{\nu-1}{\nu} \sum_{n=1}^{\left[ \sqrt{\nu M} \right]} n \leq \frac{\nu-1}{2 \nu} \sqrt{\nu M} \left( \sqrt{\nu M} + 1 \right).$$ The bound of (\[I\_upp1\]) follows. Next we want to produce a lower bound on $|\B_{\nu}(M)|$. For this we first consider the pairwise intersections of the sets $I_{\nu}(n)$. \[I\_int\] Let $m < n \leq [\sqrt{M}]$. (1) If $n > m \sqrt{\nu}$, or if $n \geq m \sqrt{\nu}$ and $\sqrt{\nu}$ is irrational, then $I_{\nu}(n) \cap I_{\nu}(m) = \emptyset$. (2) If $\gcd(m,n)=1$, then $I_{\nu}(n) \cap I_{\nu}(m) = \emptyset$. (3) If $m < n < m \sqrt{\nu}$, then $$|I_{\nu}(n) \cap I_{\nu}(m)| \leq \left[ \frac{\nu-1}{\nu}\ \gcd(m,n) \right] + 1.$$ Define $Q=\frac{\nu-1}{\nu}$. Let $m < n \leq [\sqrt{M}]$, and suppose that $k \in I_{\nu}(n) \cap I_{\nu}(m)$. Then $$k = n(n-x) = m(m-y),$$ for some integers $0 \leq x \leq Qn$ and $0 \leq y \leq Qm$. Define a line $$L(n,m) = \{ (x,y) \in \real^2 : nx-my=n^2-m^2 \},$$ and a rectangular box $$R(n,m) = \{ (x,y) \in \real^2 : 0 \leq x \leq Qn,\ 0 \leq y \leq Qm \}.$$ It follows immediately that $$|I_{\nu}(n) \cap I_{\nu}(m)| = |L(n,m) \cap R(n,m) \cap \zed^2|.$$ The line $L(m,n)$ passes through the points $\left( \frac{n^2-m^2}{n}, 0 \right)$ and $\left( 0, -\frac{n^2-m^2}{n} \right)$, so in particular $L(n,m) \cap R(n,m) = \emptyset$ if $\frac{n^2-m^2}{n} > Qn$, i.e. if $n > m \sqrt{\nu}$. Also if $\sqrt{\nu}$ is irrational, then $m \sqrt{\nu}$ is never an integer, and so $I_{\nu}(n) \cap I_{\nu}(m) = \emptyset$ if $n \geq m \sqrt{\nu}$, proving (1). Now suppose $m < n < m \sqrt{\nu}$, and let $(x,y) \in L(n,m) \cap R(n,m) \cap \zed^2$. Then $$y = -\frac{n(n-x)}{m} + m \in \zed_{>0},$$ hence $m\ |\ n(n-x)$. Clearly, $m \nmid n$, and so we must have $$\lcm(m,n) = \frac{mn}{\gcd(m,n)}\ |\ n(n-x),$$ meaning that $$\label{gcd1} \frac{m}{\gcd(m,n)}\ |\ n-x.$$ In particular, if $\gcd(m,n)=1$, we must have $m\ |\ n-x$, but $n-x \leq n < 2m$, meaning that in order for $n-x$ to be divisible by $m$, it must be equal to $m$. This would imply that $x=n-m < \frac{n^2-m^2}{n}$, hence $y < 0$, meaning that $(x,y) \notin R(n,m)$, which is a contradiction. This proves (2). Now assume that $(x_1,y_1),(x_1+t,y_2) \in L(n,m) \cap R(n,m) \cap \zed^2$, where $t$ is as small as possible. By (\[gcd1\]), $\frac{m}{\gcd(m,n)}\ |\ n-x_1$ and $\frac{m}{\gcd(m,n)}\ |\ n-x_1-t$, so $\frac{m}{\gcd(m,n)}\ |\ t$, and by minimality of $t$ we must have $t = \frac{m}{\gcd(m,n)}$. Therefore $x$-coordinates of points in $L(n,m) \cap R(n,m) \cap \zed^2$ must satisfy $$\frac{n^2-m^2}{n} \leq x < Qn,$$ and the distance between $x$-coordinates of any two such points must be at least $\frac{m}{\gcd(m,n)}$. Hence, $$\begin{aligned} |I_{\nu}(n) \cap I_{\nu}(m)| & = & |L(n,m) \cap R(n,m) \cap \zed^2| \leq \left[ \frac{Qn - \frac{n^2-m^2}{n}}{\frac{m}{\gcd(m,n)}} \right] + 1 \\ & = & \left[ \frac{(\nu m^2 - n^2) \gcd(m,n)}{\nu mn} \right] + 1 \leq \left[ \frac{\nu-1}{\nu}\ \gcd(m,n) \right] + 1.\end{aligned}$$ This proves (3). \[I\_low\] For all $M \in \zed_{>0}$, $$\label{I_low1} |\B_{\nu}(M)| > \frac{\nu-1}{2 \nu}\ M \left( 1 - \frac{ \log \log \sqrt{M}}{\log \sqrt{M}} \right)^2.$$ Let $N=\pi(\sqrt{M})$, i.e. the number of primes up to $\sqrt{M}$. It is a well-known fact that for all $\sqrt{M} \geq 11$, $$\label{N_bound} N \geq \frac{\sqrt{M}}{\log \sqrt{M}}.$$ Hence suppose that $M \geq 121$, and let $p_1, \dots, p_N$ be all the primes up to $\sqrt{M}$ in ascending order. By part (2) of Lemma \[I\_int\], $$I_{\nu}(p_i) \cap I_{\nu}(p_j) = \emptyset,$$ for all $1 \leq i \neq j \leq N$. As above, we let $Q=\frac{\nu-1}{\nu}$. Therefore, using (\[I\_card\]) we obtain: $$\label{low2} |\B_{\nu}(M)| \geq \sum_{i=1}^N |I_{\nu}(p_i)| \geq Q \sum_{i=1}^N p_i.$$ A result of R. Jakimczuk [@jakimczuk] implies that $$\label{prime_bound} \sum_{i=1}^N p_i > \frac{N^2}{2}\ \log^2 N.$$ The bound (\[I\_low1\]) follows upon combining (\[low2\]) with (\[prime\_bound\]) and (\[N\_bound\]). Now assume that $M < 121$, so $\sqrt{M} < 11$. A direct verification shows that in this case $I_{\nu}(n) \cap I_{\nu}(m) = \emptyset$ for all $1 \leq n \neq m \leq [\sqrt{M}]$, and so $$|\B_{\nu}(M)| \geq \sum_{n=1}^{[\sqrt{M}]} |I_{\nu}(n)| \geq Q \sum_{n=1}^{[\sqrt{M}]} n \geq \frac{\nu-1}{2 \nu}\ M.$$ This completes the proof. Combining Lemmas \[I\_upp\] and \[I\_low\], we obtain the following result. \[B\_density\] $$\frac{\nu-1}{2 \nu} \leq \DL_{\B_{\nu}} \leq \DU_{\B_{\nu}} \leq \frac{\nu-1}{2}.$$ Using (\[I\_low1\]), we see that $$\DL_{\B_{\nu}} = \liminf_{M \rightarrow \infty} \frac{|\B_{\nu}(M)|}{M} \geq \frac{\nu-1}{2 \nu} \lim_{M \rightarrow \infty} \left( 1 - \frac{ \log \log \sqrt{M}}{\log \sqrt{M}} \right)^2 = \frac{\nu-1}{2 \nu},$$ and using (\[I\_upp1\]), we see that $$\DU_{\B_{\nu}} = \limsup_{M \rightarrow \infty} \frac{|\B_{\nu}(M)|}{M} \leq \frac{\nu-1}{2} + \frac{\nu-1}{2 \sqrt{\nu}} \lim_{M \rightarrow \infty} \frac{1}{\sqrt{M}} = \frac{\nu-1}{2}.$$ This completes the proof. It is also possible to produce bounds on $|\B_{\nu}(M)|$ using decomposition (\[B\_contain\_J\]) instead of (\[B\_contain\]), and employing the fact that $$|J_{\nu,M}(n)| = \left[ \left(\frac{\nu-1}{\nu}\right)n \right] - \left[ n - \frac{M}{n} \right].$$ It is also possible to employ the full power of Lemma \[I\_int\], in particular part (3), to further refine the lower bound on $|\B_{\nu}(M)|$. These estimates however produce only marginally better constants, but much messier bounds in general. We could also lift the restriction that $\nu \leq 3^{1/4}$ with essentially no changes to the arguments, but in any case the important situation is that with $\nu$ being close to 1, and we want to emphasize that the case of utmost importance to us is that with $\nu = 3^{1/4}$. Now (\[D\_dens\]) of Theorem \[dense\] follows immediately by recalling that $\B_{3^{1/4}} \subseteq \D$, and applying Theorem \[B\_density\] with $\nu = 3^{1/4}$. The bounds on density of $\B_{\nu}$ are of independent interest, and will also be used in section 8 below to determine the order of the pole of the zeta function of well-rounded lattices. Proof of Theorem \[count\] ========================== We start with a lemma which identifies all the 4-tuples from $\E' \cup \OO'$ which parametrize the same lattices. \[transform1\] Let $(a_1,b_1,c_1,d_1), (a_2,b_2,c_2,d_2) \in \E' \cup \OO'$. Then $$\label{gamma1} \Lambda(a_1,b_1,c_1,d_1) = \Lambda(a_2,b_2,c_2,d_2)$$ if and only if there exists $0 \neq \gamma \in \que$ such that $(a_1,b_1,c_1,d_1)$ is equal to one of the following: $$\begin{aligned} \label{gamma2} & & \left( \frac{a_2}{\gamma}, \frac{b_2}{\gamma}, \gamma c_2, \gamma d_2 \right),\ \left( \frac{b_2}{\gamma}, -\frac{a_2}{\gamma}, -\gamma d_2, \gamma c_2 \right),\ \left( -\frac{b_2}{\gamma}, \frac{a_2}{\gamma}, -\gamma d_2, \gamma c_2 \right), \nonumber \\ & & \left( -\frac{a_2}{\gamma}, -\frac{b_2}{\gamma}, \gamma c_2, \gamma d_2 \right),\ \left( -\frac{a_2}{\gamma}, -\frac{b_2}{\gamma}, -\gamma c_2, \gamma d_2 \right),\ \left( \frac{b_2}{\gamma}, -\frac{a_2}{\gamma}, \gamma d_2, \gamma c_2 \right), \nonumber \\ & & \left( -\frac{b_2}{\gamma}, \frac{a_2}{\gamma}, \gamma d_2, \gamma c_2 \right), \left( \frac{a_2}{\gamma}, \frac{b_2}{\gamma}, -\gamma c_2, \gamma d_2 \right).\end{aligned}$$ If $(a_1,b_1,c_1,d_1)$ is equal to one of the 4-tuples as in (\[gamma2\]), then a direct verification shows that (\[gamma1\]) is true. Suppose, on the other hand, that (\[gamma1\]) is true. Let $$\bx_1 = \left( \begin{matrix} a_1c_1+b_1d_1 \\ b_1c_1-a_1d_1 \end{matrix} \right),\ \bwy_1 = \left( \begin{matrix} a_1c_1-b_1d_1 \\ b_1c_1+a_1d_1 \end{matrix} \right),$$ and $$\bx_2 = \left( \begin{matrix} a_2c_2+b_2d_2 \\ b_2c_2-a_2d_2 \end{matrix} \right),\ \bwy_2 = \left( \begin{matrix} a_2c_2-b_2d_2 \\ b_2c_2+a_2d_2 \end{matrix} \right),$$ if $(a_1,b_1,c_1,d_1), (a_2,b_2,c_2,d_2) \in \E'$, or $$\bx_1 = \left( \begin{matrix} \frac{a_1c_1+a_1d_1+b_1d_1-b_1c_1}{2} \\ \frac{a_1c_1+b_1c_1+b_1d_1-a_1d_1}{2} \end{matrix} \right),\ \bwy_1 = \left( \begin{matrix} \frac{a_1c_1-a_1d_1-b_1c_1-b_1d_1}{2} \\ \frac{a_1c_1+a_1d_1+b_1c_1-b_1d_1}{2} \end{matrix} \right),$$ and $$\bx_2 = \left( \begin{matrix} \frac{a_2c_2+a_2d_2+b_2d_2-b_2c_2}{2} \\ \frac{a_2c_2+b_2c_2+b_2d_2-a_2d_2}{2} \end{matrix} \right),\ \bwy_2 = \left( \begin{matrix} \frac{a_2c_2-a_2d_2-b_2c_2-b_2d_2}{2} \\ \frac{a_2c_2+a_2d_2+b_2c_2-b_2d_2}{2} \end{matrix} \right),$$ if $(a_1,b_1,c_1,d_1), (a_2,b_2,c_2,d_2) \in \OO'$. By Lemma \[gauss\], this means that the basis matrix $(\bx_1\ \bwy_1)$ for $\Lambda(a_1,b_1,c_1,d_1)$ must be equal to one of the following basis matrices for $\Lambda(a_2,b_2,c_2,d_2)$: $$\begin{aligned} & & (\bx_2\ \bwy_2),\ (-\bx_2\ \bwy_2),\ (\bx_2\ -\bwy_2),\ (-\bx_2\ -\bwy_2), \\ & & (\bwy_2\ \bx_2),\ (-\bwy_2\ \bx_2),\ (\bwy_2\ -\bx_2),\ (-\bwy_2\ -\bx_2).\end{aligned}$$ A direct verification shows that in each of these cases $(a_1,b_1,c_1,d_1)$ is equal to one of the 4-tuples as in (\[gamma2\]), in the same order. This completes the proof. \[sqr\_free\] Notice that in Lemma \[transform1\] $(a_1,b_1,c_1,d_1)$ can be equal to the second, third, sixth, or seventh 4-tuple in (\[gamma2\]) only if $$|c_1|=|\gamma||d_2|,\ |d_1|=|\gamma||c_2|,$$ but on the other hand we know that $(a_1,b_1,c_1,d_1), (a_2,b_2,c_2,d_2) \in \E' \cup \OO'$. Combining these facts we obtain $$\label{cd_eq} |c_1| = |\gamma||d_2| \leq |\gamma||c_2| = |d_1| \leq |c_1|,$$ which implies that there must be equality everywhere in (\[cd\_eq\]). In this case the determinant of the corresponding lattice $\Lambda$ is equal to $(a_1^2+b_1^2) c_1^2$ if $\Lambda \in \OO$ or to $2(a_1^2+b_1^2) c_1^2$ if $\Lambda \in \E$. [*Proof of Theorem \[count\].*]{} Let $u \in \D$. If $u=1,2$, the proof is by direct verification. Assume from here on that $u>2$, and let $$t = t(u) = \left\{ \begin{array}{ll} u & \mbox{if $u$ is odd} \\ \frac{u}{2} & \mbox{if $u$ is even.} \end{array} \right.$$ Define $$D(t) = \{ n \in \zed_{>0} : n | t \},$$ i.e. $D(t)$ is the set of positive divisors of $t$. Define $$D_1(t) = \{ (c,d) \in D(t) \times D(t) : d \leq c \leq \sqrt{3} d,\ cd | t \}.$$ For each $(c,d) \in D_1(t)$, define $$S_t(c,d) = \left\{ (a,b) \in \zed^2_{\geq 0} : a^2+b^2 = \frac{t}{cd},\ a \leq b \right\}.$$ Also let $$T(t) = \{ (a,b,c,d) \in \zed^4_{\geq 0} : (c,d) \in D_1(t),\ (a,b) \in S_t(c,d) \}.$$ Define an equivalence relation on $T(t)$ by writing $$(a_1,b_1,c_1,d_1) \sim (a_2,b_2,c_2,d_2)$$ if $(a_1,b_1,c_1,d_1) = \left( \frac{a_2}{\gamma},\frac{b_2}{\gamma},\gamma c_2,\gamma d_2 \right)$ for some $\gamma \in \que_{>0}$. Then let $T_1(t)$ be the set of all equivalence classes of elements of $T(t)$ under $\sim$, i.e. $T_1(t) = T(t)/\sim$. By abuse of notation, we will write $(a,b,c,d)$ for an element of $T_1(t)$. We first have the following lemma. \[gcd\_ab\] For each equivalence class in $T_1(t)$ it is possible to select a unique representative $(a,b,c,d)$ with $\gcd(a,b)=1$. Let $(a,b,c,d) \in T_1(t)$, and let $q=\gcd(a,b)$, then it is easy to see that $$(a,b,c,d) \sim \left( \frac{a}{q},\frac{b}{q},qc,qd \right).$$ Moreover, suppose that $(a_1,b_1,c_1,d_1),(a_2,b_2,c_2,d_2) \in T_1(t)$ are such that $$\gcd(a_1,b_1)=\gcd(a_2,b_2)=1,$$ and $$(a_1,b_1,c_1,d_1) \sim (a_2,b_2,c_2,d_2).$$ Then there exists $\gamma = \frac{s}{q} \in \que_{>0}$ with $\gcd(s,q)=1$ such that $$a_1=\frac{q}{s} a_2,\ b_1=\frac{q}{s} b_2,\ c_1=\frac{s}{q} c_2,\ d_1=\frac{s}{q} d_2.$$ Then $s|a_2$, $s|b_2$, and so $s|\gcd(a_2,b_2)=1$, hence $s=1$. Also $q|qa_2=a_1$, $q|qb_2=b_1$, and so $q|\gcd(a_1,b_1)=1$, hence $q=1$. Therefore $$(a_1,b_1,c_1,d_1) = (a_2,b_2,c_2,d_2).$$ This completes the proof. Therefore, for each $t$ we only need to count the lattices produced by the 4-tuples $(a,b,c,d) \in T_1(t)$ with $\gcd(a,b)=1$. Let $(a,b,c,d)$ be such a 4-tuple, then either $(a,b) = (0,1)$ or $a,b,c,d \neq 0$, since $\gcd(0,b)=b$. Moreover, $a \neq b$ unless $a=b=1$. First assume $c \neq d$. If $(a,b) \neq (0,1),(1,1)$. By Lemma \[transform1\] and Remark \[sqr\_free\] only the following 4-tuples produce the same lattice $\Lambda(a,b,c,d)$: $$\begin{aligned} & & (a,b,c,d),\ (-a,-b,-c,-d),\ (-a,-b,c,d),\ (a,b,-c,-d),\\ & & (a,b,-c,d),\ (a,b,c,-d),\ (-a,-b,-c,d),\ (-a,-b,c,-d).\end{aligned}$$ Then each $(a,b,c,d) \in T_1(t)$ gives rise to the four distinct lattices: $$\label{sqrfr_lat1} \Lambda(a,b,c,d),\ \Lambda(-a,b,c,d),\ \Lambda(b,a,c,d),\ \Lambda(-b,a,c,d),$$ since $\Lambda(a,b,d,c)=\Lambda(-b,a,c,d)$. Also, each of $(0,1,c,d),(1,1,c,d),(a,b,1,1) \in T_1(t)$ gives rise to the following pairs of distinct lattices, respectively: $$\begin{aligned} \label{sqrfr_lat2} & & \Lambda(0,1,c,d),\ \Lambda(1,0,c,d); \nonumber \\ & & \Lambda(1,1,c,d),\ \Lambda(-1,1,c,d); \nonumber \\ & & \Lambda(a,b,1,1),\ \Lambda(-a,b,1,1).\end{aligned}$$ Now suppose that $c=d$. Then $$\Lambda(a,b,c,c) = \Lambda(-b,a,c,c),\ \Lambda(-a,b,c,c) = \Lambda(b,a,c,c).$$ Hence, if $(a,b) \neq (0,1),(1,1)$, then each $(a,b,c,c) \in T_1(t)$ gives rise to two distinct lattices, $\Lambda(a,b,c,c)$ and $\Lambda(b,a,c,c)$. Notice also that $$\Lambda(0,1,c,c)=\Lambda(1,0,c,c),\ \Lambda(1,1,c,c)=\Lambda(-1,1,c,c).$$ Hence 4-tuples $(0,1,c,c), (1,1,c,c) \in T_1(t)$ give rise to only one lattice each. The formula for $\N(u)$, $u \in \D$, of Theorem \[count\] follows. Also notice that if $u \in \zed_{>0} \setminus \D$, then for every divisor $n$ of $u$, either $\alpha_*(t/n)$ or $\beta(n)$ is equal to zero, and so the right hand side of (\[N\_formula\]) is equal to zero by construction. This completes the proof of the theorem. \[points\] Our problem can be interpreted in terms of counting integral points on certain varieties. Let us say that two points $$\bx = (x_1,x_2,x_3,x_4)^t,\ \bwy = (y_1,y_2,y_3,y_4)^t \in \real^4$$ are equivalent if there exists $U \in GL_2(\zed)$ such that $$U \left( \begin{matrix} x_1&x_3 \\ x_2&x_4 \end{matrix} \right) = \left( \begin{matrix} y_1&y_3 \\ y_2&y_4 \end{matrix} \right).$$ Notice that the number of [*all*]{} full-rank sublattices of $\zed^2$ with determinant equal to $u$ is precisely the number of integral points on the hypersurface $$x_1x_4 - x_2x_3 = u,$$ modulo this equivalence. This number is well known: one formula, for instance, is given by (\[all\_sublattices\]) below. On the other hand, by Lemma \[gauss\], the number of [*well-rounded*]{} full-rank sublattices of $\zed^2$ with determinant equal to $u$ is the number of integral points on the subset of the variety $$x_1x_4 - x_2x_3 = u,\ x_1^2+x_2^2-x_3^2-x_4^2 = 0,$$ defined by the inequality $$2|x_1x_3+x_2x_4| \leq x_1^2+x_2^2,$$ modulo the same equivalence. This makes direct counting much harder, and so our parametrization is quite useful. Corollaries =========== The first immediate consequence of Theorem \[count\] is the following. \[zero\_even\] If $u \in \zed_{>0}$ is odd, then $\N(u) = \N(2u)$. To demonstrate some examples of our formulas at work, we derive the following simpler looking expressions for the case of prime-power determinants. \[primep\] Let $p$ be a prime, $k \in \zed_{>0}$. Let $u = p^k$ or $2p^k$. Then $$\N(u) = \left\{ \begin{array}{ll} 0 & \mbox{if $p \equiv 3\ (\md 4)$ and $k$ is odd} \\ 1 & \mbox{if $p \equiv 3\ (\md 4)$ and $k$ is even} \\ 1 & \mbox{if $p=2$} \\ k+1 & \mbox{if $p \equiv 1\ (\md 4)$} \end{array} \right.$$ First assume that $p \neq 2$. Define $t$ as in the statement of Theorem \[count\], then $t=p^k$. If $k$ is even, then by Theorem \[count\] $$\begin{aligned} \N(u) & = & \beta(p^k) + 4 \sum_{j=1}^{\frac{k}{2}} \alpha_*(p^{k+1-2j}) \beta(p^{2j-1}) + 2 \sum_{j=0}^{\frac{k}{2}-1} \alpha_*(p^{k-2j}) (2\beta(p^{2j})-1) \\ & = & 1 + 2 \sum_{j=0}^{\frac{k}{2}-1} \alpha_*(p^{k-2j}),\end{aligned}$$ since $\beta(p^{2j-1})=0$, and $\beta(p^{2j})=1$ for all $j$. If $p \equiv 3\ (\md 4)$, then $\alpha_*(p^{k-2j}) = 0$ for all $j$. If $p \equiv 1\ (\md 4)$, then $\alpha_*(p^{k-2j}) = 1$ for all $j$, in which case $$\N(u) = 1 + 2 \sum_{j=0}^{\frac{k}{2}-1} 1 = k+1.$$ Next assume $k>1$ is odd. Then, in the same manner as above, $$\begin{aligned} \N(u) & = & 2 \beta(p^k) + 4 \sum_{j=1}^{\frac{k-1}{2}} \alpha_*(p^{k+1-2j}) \beta(p^{2j-1}) + 2 \sum_{j=0}^{\frac{k-1}{2}} \alpha_*(p^{k-2j}) (2\beta(p^{2j})-1) \\ & = & 2 \sum_{j=0}^{\frac{k-1}{2}} \alpha_*(p^{k-2j}),\end{aligned}$$ which is equal to 0 if $p \equiv 3\ (\md 4)$. If $p \equiv 1\ (\md 4)$, then $$\N(u) = 2 \sum_{j=0}^{\frac{k-1}{2}} 1 = k+1.$$ If $k=1$, then by Theorem \[count\] $$\N(u) = 2 \alpha_*(p) = \left\{ \begin{array}{ll} 0 & \mbox{if $p \equiv 3\ (\md 4)$} \\ 2 & \mbox{if $p \equiv 1\ (\md 4)$}. \end{array} \right.$$ Now assume that $p=2$, $u=p^k$, and $k>1$: the case $k=1$, i.e. $u=2$ is considered separately in the statement of Theorem \[count\]. If $k$ is even, then $$\begin{aligned} \N(u) & = & 2 \beta(2^{k-1}) + \beta(2^{k-2}) + 4 \sum_{j=1}^{\frac{k-2}{2}} \alpha_*(2^{k-2j}) \beta(2^{2j-1}) \\ & + & 2 \sum_{j=0}^{\frac{k-4}{2}} \alpha_*(2^{k-1-2j}) (2\beta(2^{2j})-1) = 1,\end{aligned}$$ since $\alpha_*(2^i)=0$ for all $i>1$, and $\beta(2^i)=0$ for all odd $i$. Now let $k$ be odd. Then $$\begin{aligned} \N(u) & = & \beta(2^{k-1}) + 2 \beta(2^{k-2}) + 4 \sum_{j=1}^{\frac{k-3}{2}} \alpha_*(2^{k-2j}) \beta(2^{2j-1}) \\ & + & 2 \sum_{j=0}^{\frac{k-3}{2}} \alpha_*(2^{k-1-2j}) (2\beta(2^{2j})-1) = 1.\end{aligned}$$ This completes the proof. In precisely the same manner, we obtain the following formulas for the case when determinant is a product of two odd primes. \[two\_primes\] If $u=p_1p_2$, where $p_1 < p_2$ are odd primes, then $$\N(u) = \left\{ \begin{array}{ll} 0 & \mbox{if $p_1$ or $p_2 \equiv 3\ (\md 4)$ and $p_2>\sqrt{3}p_1$} \\ 2 & \mbox{if $p_1$ or $p_2 \equiv 3\ (\md 4)$ and $p_2 \leq \sqrt{3}p_1$} \\ 4 & \mbox{if $p_1$ and $p_2 \equiv 1\ (\md 4)$ and $p_2>\sqrt{3}p_1$} \\ 6 & \mbox{if $p_1$ and $p_2 \equiv 1\ (\md 4)$ and $p_2 \leq \sqrt{3}p_1$.} \end{array} \right.$$ Direct verification. The same way one can apply the formulas of Theorem \[count\] to obtain explicit expressions for $\N(u)$ for many other instances of $u$ as well. Notice that some of the lattices in $\WR(\zed^2)$ come from ideals in $\zed[i]$. Namely, let $u = a^2+b^2 \in \D$ and consider the lattices $\Lambda_1(a,b) = \left( \begin{matrix} a&-b \\ b&a \end{matrix} \right) \zed^2$ and $\Lambda_2(a,b) = \left( \begin{matrix} a&b \\ -b&a \end{matrix} \right) \zed^2$ with $\det(\Lambda_1) = \det(\Lambda_2) = u$. Let $I_1(a,b)$ and $I_2(a,b)$ be the ideals in $\zed[i]$ generated by $a+bi$ and $a-bi$ respectively, then $-b+ai = i(a+bi) \in I_1(a,b)$ and $b+ai = i(a-bi) \in I_2(a,b)$. Hence $I_1(a,b)$ and $I_2(a,b)$ map bijectively onto $\Lambda_1(a,b)$ and $\Lambda_2(a,b)$ respectively under the canonical mapping $x+iy \rightarrow \left( \begin{matrix} x \\ y \end{matrix} \right)$, and $\Lambda_1(a,b) = \Lambda_2(a,b)$ if and only if $b=0$, which can only happen when $u$ is a square. Notice that such representation is only possible for the determinant values $u$ which are also in the minima set $\Mm$; in other words, a full-rank WR sublattice of $\zed^2$ comes from an ideal in $\zed[i]$ if and only if it has an orthogonal basis. It is easy to see that the number of such lattices of determinant $u \in \Mm$, which is precisely the number of ideals of norm $u$ in $\zed[i]$, is equal to $2\alpha(u)$ if $u$ is not a square, and $2\alpha(u) + 1$ if $u$ is a square. With this in mind, we can now state the following immediate consequence of Corollaries \[primep\] and \[two\_primes\]. \[ideals\] If $u \in \D$ is of the form $u = p^k, 2p^k$, where $p$ is a prime, or $u = p_1p_2$ where $\sqrt{3} p_1 < p_2$ are odd primes, then all lattices in $\WR(\zed^2)$ of determinant $u$ come from ideals of norm $u$ in $\zed[i]$. This of course is not true in general, in fact the class of such lattices coming from ideals in $\zed[i]$ is quite thin. Notice in particular that in order for a lattice $\Lambda \in \WR(\zed^2)$ to come from an ideal of $\zed[i]$ it must first of all be true that $\det \Lambda \in \Mm$, which has density 0 versus the entire determinant set $\D$, which has positive density. Let $\Pp = \{p_1,p_2,\dots\}$ be the collection of all primes in the arithmetic progression $4n+1$. By Dirichlet’s theorem on primes in arithmetic progressions, $\Pp$ is infinite. For each $p_i \in \Pp$ define $\PP_i = \{p_i^k,2p_i^k\}_{k=1}^{\infty}$. Then $\bigcup_{i=1}^{\infty} \PP_i \subset \D$, and Corollary \[primep\] implies that for each $i$, $$\N(u) = \frac{\log u}{\log p_i} + 1,$$ for each $u \in \PP_i$. In other words, there are infinite sequences in $\D$ on which $\N(u)$ grows at least logarithmically in $u$. For comparison, it is a well known fact (see for instance [@bgruber]) that for any positive integer $u$ with prime factorization $u=q_1^{c_1} \dots q_m^{c_m}$ the number of [*all*]{} full-rank sublattices of $\zed^2$ with determinant $u$ is $$\label{all_sublattices} F(2,u) = \prod_{j=1}^m \frac{q_j^{c_j+1} - 1}{q_j - 1},$$ which grows linearly in $u$. It is therefore interesting to exhibit sequences of determinant values $u$ for which $\N(u)$ is especially large. Recall that for an integer $u$, $\tau(u)$ and $\omega(u)$ are numbers of divisors and of prime divisors of $u$, respectively. We can report the following consequence of Theorem \[count\]. \[size\_N\] For each $u \in \zed_{>0}$, $$\label{size_O1} \N(u) \leq O \left( \tau(u)^2 2^{\omega(u)} \right) \leq O \left( \left( \frac{\sqrt{2} \log u}{\omega(u)} \right)^{2 \omega(u)} \right).$$ Moreover, $$\label{size_O2} \N(u) < O \left( (\log u)^{\log 8} \right),$$ for all $u \in \D$ outside of a subset of asymptotic density 0. However, there exist infinite sequences $\{ u_k \}_{k=1}^{\infty} \subset \D$ such that for every $k \geq 1$ $$\label{size_O3} \N(u_k) \geq (\log u_k)^k.$$ For instance, there exists such a sequence with $u_k \leq \exp \left( O (k (\log k)^2) \right)$ and $\omega(u_k) = O(k \log k)$. Notice that the right hand side of (\[N\_formula\]) is the sum of at most $\tau(u)$ nonzero terms. Combining (\[beta\_bound2\]) with the formula for $\alpha_*$ in section 1, it follows that each of these terms is at most $O \left( \tau(u) 2^{\omega(u)} \right)$. Let $u$ have a prime decomposition of the form $u = p_1^{e_1} \dots p_n^{e_n}$, so $\omega(u)=n$, then: $$\frac{\log u}{n} = \frac{1}{n} \sum_{i=1}^n e_i \log p_i \geq \left( \prod_{i=1}^n e_i \log p_i \right)^{\frac{1}{n}} \geq \left( O \left( \prod_{i=1}^n (e_i+1) \right) \right)^{\frac{1}{n}} = \left( O(\tau(u)) \right)^{\frac{1}{n}}.$$ This proves (\[size\_O1\]), and (\[size\_O2\]) follows from (\[size\_O1\]) combined with Theorems 431 and 432 of [@hardy], which state that the normal orders of $\omega(u)$ and $\tau(u)$ are $\log \log u$ and $2^{\log \log u}$, respectively. Next, write $p_n$ for the $n$-th prime congruent to 1 mod 4. It is a well known fact that $$\label{n_prime} p_n = O(n \log n).$$ For each $n \geq 1$, define $v_n = \prod_{i=1}^n p_i^2$. Write $\N_I(v_n)$ for the number of lattices in $\WR(\zed^2)$ with determinant $v_n$ that come from ideals in $\zed[i]$, then $$\label{NI} \N(v_n) \geq \N_I(v_n) = 2\alpha(v_n)+1 = 3^n.$$ Let $k$ be a positive integer. We want to choose $n$ such that $$\label{nO_1} \N(v_n) \geq 3^n \geq \left( \log \left( \prod_{i=1}^n p_i^2 \right) \right)^k = 2^k \left( \sum_{i=1}^n \log p_i \right)^k.$$ By (\[n\_prime\]), $$\label{nO_2} \sum_{i=1}^n \log p_i = \sum_{i=1}^n \log \left( O(i \log i) \right) = \sum_{i=1}^n O \left( \log (i \log i) \right) = \sum_{i=1}^n O \left( \log i \right) \leq O(n \log n).$$ Combining (\[nO\_1\]) with (\[nO\_2\]) and taking logarithms, we see that it is sufficient to choose $n$ such that $$\frac{n}{\log n} \geq O(k),$$ hence we can take $n = O(k \log k)$. Then, by (\[nO\_2\]), for this choice of $n$ we have $$v_n = \exp \left( 2 \sum_{i=1}^n \log p_i \right) \leq \exp \left( O(n \log n) \right) = \exp \left( O (k (\log k)^2) \right).$$ Let $u_k = v_n$ for this choice of $n$, and so $n = \omega(u_k)$. This completes the proof. Let $v_n$ be as in the proof of Corollary \[size\_N\] above, i.e. $v_n = \prod_{i=1}^n p_i^2$, where $p_1, p_2, \dots$ are primes congruent to 1 mod 4; for instance, the first 9 such primes are 5, 13, 17, 29, 37, 43, 47, 53, 61. For each $k$ choose the smallest $n$ so that $v_n > (\log v_n)^k$, and let $u_k = v_n$ for this choice of $n$. Here is the actual data table for the first few values of the sequence $\{u_k\}$ computed with Maple. [*$k$*]{} [*$n$*]{} [*$u_k = v_n$*]{} [*$\N(u_k)$*]{} [*$(\log u_k)^k$*]{} ----------- ----------- ------------------------------ ----------------- ---------------------- 1 2 $4225$ 9 8.34877454 2 4 $1026882025$ 518 430.5539044 3 7 $5741913252704971225$ 215002 80589.79464 4 9 $60016136730202390980384025$ 14324372 12413026.85 Notice that the choice of $n = O(k \log k)$ as in Corollary \[size\_N\] insures that not just $\N(u_k)$, but even the much smaller $\N_I(u_k)$ (compare for instance the values of $\N(u_k)$ in the table above to $\N_I(u_k) = 3^n$) is greater than $(\log u_k)^k$, and even with this stronger restriction $u_k$ and $\omega(u_k)$ grow relatively slow as functions of $k$. Counting well-rounded lattices with fixed minimum ================================================= Let $m \in \zed_{>0}$, then, as stated in [@martinet:venkov], there exist $\left[ \frac{m+1}{2} \right]$ WR lattices $\Lambda$, not necessarily integral, of rank 2 in $\real^2$ with $|\Lambda| = \sqrt{m}$, generated by a minimal basis $\bx,\bwy$ with $0 < \bx^t \bwy \leq \left[ \frac{m-1}{2} \right]$. This information, however, does not lead to an explicit formula for the number of WR sublattices of $\zed^2$ of prescribed minimum. We derive such a formula here. Let $m \in \zed_{>0}$. Suppose that $\Lambda \in \WR(\zed^2)$ and $|\Lambda|^2=m$, then by Lemma \[gauss\] there exists a representation $\Lambda = \left( \begin{matrix} x_1&y_1 \\ x_2&y_2 \end{matrix} \right) \zed^2$ with $$\label{abcd_min} \bx = \left(\begin{matrix} x_1 \\ x_2 \end{matrix} \right),\ \bwy = \left(\begin{matrix} y_1 \\ y_2 \end{matrix} \right) \in \zed^2,\ \|\bx\|=\|\bwy\|=m,\ \theta(\bx,\bwy) \in \left[ \frac{\pi}{3}, \frac{\pi}{2} \right],$$ where $\theta(\bx,\bwy)$ is the angle between vectors $\bx$ and $\bwy$, since if $\theta(\bx,\bwy) \in \left( \frac{\pi}{2}, \frac{2\pi}{3} \right]$ we can always replace $\bx$ with $-\bx$ or $\bwy$ with $-\bwy$ to ensure that $\theta(\bx,\bwy) \in \left[ \frac{\pi}{3}, \frac{\pi}{2} \right]$. Then define $$\C_m = \{ \bx \in \zed^2 : x_2 > 0,\ \|\bx\| = m \}.$$ For each $\bx \in \C_m$ let $$E_m(\bx) = \left\{ \bwy \in \zed^2 : y_2 \geq 0,\ \|\bwy\| = m,\ \theta(\bx,\bwy) \in \left[ \frac{\pi}{3}, \frac{\pi}{2} \right] \right\},$$ and define $\eta_m(\bx) = |E_m(\bx)|$. The following result follows immediately. \[count\_min\] Let $m \in \Mm$. Let $\N'(m)$ be the number of lattices in $\WR(\zed^2)$ with minimum equal to $m$. Then $$\N'(m) = \sum_{\bx \in \C_m} \eta_m(\bx).$$ Notice that for each $\bx \in \C_m$, $\eta_m(\bx)$ is precisely the number of integer lattice points on the arc of the circle of radius $\sqrt{m}$, bounded by the points $\left( \begin{matrix} \frac{x_1}{2} - \frac{x_2\sqrt{3}}{2} \\ \frac{x_1\sqrt{3}}{2} + \frac{x_2}{2} \end{matrix} \right)$ and $\left( \begin{matrix} -x_2 \\ x_1 \end{matrix} \right)$. The angle corresponding to this arc is $\frac{\pi}{6}$. On the other hand, $\alpha(m)$ as defined by (\[alpha\]) is the number of integer lattice points on the quarter-circle of radius $\sqrt{m}$ centered at the origin and bounded by the points $(\sqrt{m},0)$, $(0, \sqrt{m})$. It is not difficult to see that for every $\bx \in \C_m$, $$\eta_m(\bx) \leq \left\{ \begin{array}{ll} \alpha(m) & \mbox{if $m$ is not a square} \\ \alpha(m)+1 & \mbox{if $m$ is a square.} \end{array} \right.$$ Indeed, for each $\left( \begin{matrix} 0 \\ \sqrt{m} \end{matrix} \right) \neq \left( \begin{matrix} y_1 \\ y_2 \end{matrix} \right) \in E_m(\bx)$, either $\left( \begin{matrix} y_1 \\ y_2 \end{matrix} \right)$ or $\left( \begin{matrix} -y_2 \\ y_1 \end{matrix} \right)$ is contained in the first quadrant and so is counted by $\alpha(m)$; if $m$ is a square, we add one to account for the point $\left( \begin{matrix} 0 \\ \sqrt{m} \end{matrix} \right)$, which is not counted by $\alpha(m)$. In general, these bounds are sharp, for instance $\alpha(13)=\eta_{13}\left( \begin{matrix} 2 \\ 3 \end{matrix} \right)=1$. However, if $\alpha(m)$ is large and the integral points $\bx$ are well distributed on the quarter-circle, it is possible to do better. For this $m$ needs to satisfy certain special conditions. More precisely, write prime decomposition of $m$ as $$m = 2^w p_1^{2l_1} \dots p_s^{2l_s} q_1^{k_1} \dots q_r^{k_r},$$ where $p_i \equiv 3\ (\md 4)$, $q_j \equiv 1\ (\md 4)$, $w \in \zed_{\geq 0}$, $l_i \in \frac{1}{2} \zed_{>0}$, and $k_j \in \zed_{>0}$ for all $1 \leq i \leq s$, $1 \leq j \leq r$. If $l_i \notin \zed$ for any $1 \leq i \leq s$, then $\alpha(m) = 0$, so let us assume that $l_i \in \zed$ for all $1 \leq i \leq s$. Then $$\alpha(m) = \alpha \left( \frac{m}{2^w p_1^{2l_1} \dots p_s^{2l_s}} \right),$$ hence we can assume that $$\label{m_cong1} m = q_1^{k_1} \dots q_r^{k_r},$$ where $q_j \equiv 1\ (\md 4)$ and $k_j \in \zed_{>0}$ for all $1 \leq j \leq r$. Define $$L(m) = \sqrt{ \frac{\log (q_1 \dots q_r)}{\log \alpha(m)} },$$ and let $$\Mm_1 = \left\{ m \in \Mm : m\ \text{as in (\ref{m_cong1}) and } L(m) \rightarrow 0 \text{ as } m \rightarrow \infty \right\}.$$ A result of Babaev [@babaev] implies that for $m \in \Mm_1$ $$\label{alphap_bound} \eta_m(\bx) = \frac{\alpha(m)}{3} + O(L(m)\alpha(m)),$$ for each $\bx \in \C_m$. For $m \in \Mm \setminus \Mm_1$ I am not aware of upper bounds on $\eta_m(\bx)$ better than $\alpha(m)$; a classical result of Jarnik on the number of integral lattice points on convex curves [@jarnik] as well as more modern results, for instance of Bombieri and Pila [@pila], imply a general bound on $\eta_m(\bx)$ which is at best $O \left(m^{\frac{1}{4}+\eps} \right)$ for each $\bx \in \C_m$. In precisely the same manner as Corollary \[gen\] follows from Theorem \[count\], the following is an immediate consequence of Theorem \[count\_min\]. \[gen\_min\] Let $A \in O_2(\real)$. Then for each $m \in \Mm$ the number of full-rank WR sublattices of $A\zed^2$ with squared minimum equal to $m$ is given by $\N'(m)$ as in Theorem \[count\_min\]. \[min\_det\] Notice that even fixing both, the minimum and the determinant, does not identify an element of $\WR(\zed^2)$ uniquely. For instance, if $u$ is representable as a sum of two squares, then the number of lattices $\Lambda \in \WR(\zed^2)$ with $|\Lambda|^2 = \det(\Lambda) = u$ is $$\N_I(u) = \left\{ \begin{array}{ll} 2\alpha(u) & \mbox{if $u$ is not a square} \\ 2\alpha(u)+1 & \mbox{if $u$ is a square,} \end{array} \right.$$ i.e. precisely the number of lattices in $\WR(\zed^2)$ of determinant $u$ coming from ideals in $\zed[i]$, as defined in section 6. Hence even this number can tend to infinity with $u$. Zeta function of well-rounded lattices ====================================== Given any finitely generated group $G$, it is possible to associate a zeta function $\zeta_G(s) = \sum_{n=1}^{\infty} a_n n^{-s}$ to it, where the coefficients $a_n$ count the number of its subgroups of index $n$ and $s \in \cee$ (see [@lubot], Chapter 15 for details). Such zeta functions are extensively studied objects, since they encode important arithmetic information about the group in question and often have interesting properties. For example, by Theorem 15.1 of [@lubot] (see also (5) of [@reiner]) $$\label{all_zeta} \zeta_{\zed^2}(s) = \sum_{\Lambda \subseteq \zed^2} (\det(\Lambda))^{-s} = \sum_{u=1}^{\infty} F(2,u) u^{-s} = \zeta(s) \zeta(s-1),$$ where the sum is taken over all sublattices $\Lambda$ of $\zed^2$ of finite index, $F(2,u)$ is given by (\[all\_sublattices\]), and $\zeta(s)$ is the Riemann zeta function. This $\zeta_{\zed^2}(s)$ is an example of Solomon’s zeta function (see [@reiner], [@solomon]). The identity (\[all\_zeta\]) holds in the half-plane $\Re(s) > 2$, where this series is absolutely convergent, and so the function $\zeta_{\zed^2}(s)$ is analytic (Proposition 1 of [@reiner]). Moreover, in this half-plane $\zeta_{\zed^2}(s) = \sum_{u=1}^{\infty} \sigma(u) u^{-s}$, where $\sigma(u) = \sum_{d|u} d$ (see Theorem 290 of [@hardy]); $\zeta_{\zed^2}(s)$ has a pole at $s=2$. In this section we study the properties of the partial zeta function corresponding not to all, but only to the [*well-rounded*]{} sublattices of $\zed^2$ as defined by (\[WR\_zeta\]). We also define the Dedekind zeta function of Gaussian integers $\zed[i]$ $$\label{dedekind} \zeta_{\zed[i]}(s) = \sum_{\Aa \subseteq \zed[i]} \Nn(\Aa)^{-s} = \mathop{\sum_{\Lambda \in \WR(\zed^2)}}_{|\Lambda|^2=\det(\Lambda)} (\det(\Lambda))^{-s} = \sum_{m=1}^{\infty} \N_I(m) m^{-s},$$ where $\N_I(m)$ is as above, and the first sum is taken over all the ideals $\Aa = (a+bi)\zed[i]$ for some $a,b \in \zed$, and $\Nn(\Aa) = a^2+b^2$ is the norm of such ideal. In other words, coefficients of $\zeta_{\zed[i]}(s)$ count the elements of $\WR(\zed^2)$ that come from ideals in $\zed[i]$ while coefficients of $\zeta_{\WR(\zed^2)}(s)$ count all elements of $\WR(\zed^2)$. We also note that $\zeta_{\zed[i]}(s)$ is analytic on $\Re(s) > 1/2$ except for a simple pole at $s=1$ (see Theorem 5 on p. 161 of [@lang]). It is clear that for all $u \in \zed_{>0}$ $$\label{zeta_c} \N_I(u) \leq \N(u) \leq F(2,u),$$ in other words $\zeta_{\WR(\zed^2)}(s)$ is “squeezed” between $\zeta_{\zed[i]}(s)$ and $\zeta_{\zed^2}(s)$. Moreover, our estimates on coefficients in the previous sections suggest that $\zeta_{\WR(\zed^2)}(s)$ should be “closer” to $\zeta_{\zed[i]}(s)$ than to $\zeta_{\zed^2}(s)$. Theorem \[zeta\], which we will now prove, makes this statement more precise. We start by studying some related Dirichlet series. \[Dir1\] Let $t = t(u)$ be as in Theorem \[count\]. The Dirichlet series $\sum_{u=1}^{\infty} \frac{2\alpha_*(t)}{u^{s}}$ is absolutely convergent at least in the half-plane $\Re(s) > 1$ with a simple pole at $s=1$. Moreover, when $\Re(s) > 1$ it has an Euler product expansion: $$\label{Euler_prod} \sum_{u=1}^{\infty} \frac{2\alpha_*(t)}{u^{s}} = \left( 1 + \frac{1}{2^s} + \frac{1}{4^s} \right) \prod_{p \equiv 1 (\md 4)} \frac{p^s+1}{p^s-1}.$$ First of all, notice that for every $u \in \zed_{>0}$, $$2\alpha_*(t) \leq 2\alpha(t) \leq 2\alpha(u) + 1 \leq \N_I(u) + 1,$$ therefore $\sum_{u=1}^{\infty} 2\alpha_*(t) u^{-s}$ is absolutely convergent at least on the half-plane $\Re(s) > 1$ with at most a simple pole at $s=1$, since $\sum_{u=1}^{\infty} (\N_I(u)+1) u^{-s} = \zeta_{\zed[i]}(s) + \zeta(s)$ is. Next, let $$\alpha_*'(n) = \left\{ \begin{array}{ll} \alpha_*(n) & \mbox{if $n$ is odd} \\ 0 & \mbox{if $n$ is even.} \end{array} \right.$$ Notice that $2\alpha'_*$ is a multiplicative arithmetic function, specifically $2\alpha'_*(1)=1$ and $2\alpha'_*(mn) = 2\alpha'_*(m) 2\alpha'_*(n)$ for all $m,n \in \zed_{>0}$ with $\gcd(m,n)=1$. Therefore, by Theorem 286 of [@hardy] the series $\sum_{u=1}^{\infty} 2\alpha'_*(u) u^{-s}$ has the following Euler-type product representation, where $p$ is always a prime: $$\begin{aligned} \sum_{u=1}^{\infty} 2\alpha'_*(u) u^{-s} & = & \prod_p \left( \sum_{k=0}^{\infty} 2\alpha'_*(p^k) p^{-ks} \right) = \prod_{p \equiv 1 (\md 4)} \left( 1 + 2 \sum_{k=1}^{\infty} p^{-ks} \right) \\ & = & \prod_{p \equiv 1 (\md 4)} \left( \frac{2}{1-p^{-s}} - 1 \right) = \prod_{p \equiv 1 (\md 4)} \frac{p^s+1}{p^s-1},\end{aligned}$$ whenever this product is convergent. Also notice that since $\alpha_*(2u) = 0$ if $2|u$ and $\alpha_*(2u) = \alpha_*(u)$ if $2 \nmid u$, we have $$\begin{aligned} \sum_{u=1}^{\infty} \frac{2\alpha_*(t)}{u^{s}} & = & \sum_{u=1}^{\infty} \frac{2\alpha'_*(u)}{u^{s}} + \sum_{u=1}^{\infty} \frac{2\alpha_*(u)}{(2u)^{s}} \\ & = & \sum_{u=1}^{\infty} \frac{2\alpha'_*(u)}{u^{s}} + \frac{1}{2^s} \left( \sum_{u=1}^{\infty} \frac{2\alpha'_*(u)}{u^{s}} + \frac{1}{2^s} \sum_{u=1}^{\infty} \frac{2\alpha'_*(u)}{u^{s}} \right) \\ & = & \left( 1 + \frac{1}{2^s} + \frac{1}{4^s} \right) \sum_{u=1}^{\infty} \frac{2\alpha'_*(u)}{u^{s}},\end{aligned}$$ which proves (\[Euler\_prod\]) when $\prod_{p \equiv 1 (\md 4)} \frac{p^s+1}{p^s-1}$ is convergent. It is easy to notice that this happens when $\Re(s) > 1$, but $\prod_{p \equiv 1 (\md 4)} \frac{p+1}{p-1}$ diverges, meaning that $\sum_{u=1}^{\infty} \frac{2\alpha_*(t)}{u^{s}}$ must have a pole at $s=1$, and by our argument above we know that it must be a simple pole. This completes the proof. For the next lemma, let $\B_{\nu}$ be as in (\[prod\_set\]) in section 4. \[Dir2\] For each $1 < \nu \leq 3^{1/4}$, the Dirichlet series $\sum_{u \in \B_{\nu}} \frac{1}{u^{s}}$ is absolutely convergent in the half-plane $\Re(s) > 1$ with a simple pole at $s=1$ in the sense of (\[pole\_def\]). First notice that $$\sum_{u \in \B_{\nu}} \left| \frac{1}{u^s} \right| \leq \sum_{n=1}^{\infty} \left| \frac{1}{n^s} \right|,$$ and so must be analytic when $\Re(s) > 1$ with at most a simple pole at $s=1$. On the other hand, let the Dirichlet lower density of the set $\B_{\nu}$ be defined as $$\liminf_{s \rightarrow 1^+} \frac{\sum_{u \in \B_{\nu}} u^{-s}}{\sum_{u \in \zed_{>0}} u^{-s}} = \liminf_{s \rightarrow 1^+} \frac{1}{\zeta(s)} \sum_{u \in \B_{\nu}} u^{-s}.$$ It is a well known fact (see for instance equation (1.6) of [@ahlswede]) that the Dirichlet lower density of a set is greater or equal than its lower density. Hence, by Theorem \[B\_density\] $$0 < \frac{\nu-1}{2 \nu} \leq \DL_{\B_{\nu}} \leq \liminf_{s \rightarrow 1^+} \frac{1}{\zeta(s)} \sum_{u \in \B_{\nu}} u^{-s},$$ which implies that $\sum_{u \in \B_{\nu}} u^{-s}$ must have a pole of the same order as $\zeta(s)$ at $s=1$. This completes the proof. \[Dir3\] For each $1 < \nu \leq 3^{1/4}$, Dirichlet series $\sum_{u=1}^{\infty} \frac{\beta_{\nu}(u)}{u^{s}}$ is absolutely convergent in the half-plane $\Re(s) > 1$, and is bounded below by a Dirichlet series with a pole of order 1 at $s=1$. Moreover, for every real $\eps >0$ there exists a Dirichlet series with a pole of order $1+\eps$ at $s=1$, which bounds $\sum_{u=1}^{\infty} \frac{\beta_{\nu}(u)}{u^{s}}$ from above. For each $1 < \nu \leq 3^{1/4}$, define $\chi_{\nu}$ to be the characteristic function of the set $\B_{\nu}$, i.e. for each $u \in \zed_{>0}$, $$\chi_{\nu}(u) = \left\{ \begin{array}{ll} 1 & \mbox{if $u \in \B_{\nu}$} \\ 0 & \mbox{if $u \notin \B_{\nu}$.} \end{array} \right.$$ Clearly, $\beta_{\nu}(u) \geq \chi_{\nu}(u)$, therefore $$\sum_{u=1}^{\infty} \left| \frac{\beta_{\nu}(u)}{u^{s}} \right| \geq \sum_{u=1}^{\infty} \left| \frac{\chi_{\nu}(u)}{u^{s}} \right| = \sum_{u \in \B_{\nu}} \left| \frac{1}{u^{s}} \right|,$$ which, combined with Lemma \[Dir2\], proves the lower bound of the lemma. On the other hand, recall that $\beta_{\nu}(u) \leq \Delta(u)$ for all $u \in \zed_{>0}$, where $\Delta(u)$ is Hooley’s $\Delta$-function, as defined in section 2, hence $\sum_{u=1}^{\infty} \left| \beta_{\nu}(u) u^{-s} \right| \leq \sum_{u=1}^{\infty} \left| \Delta(u) u^{-s} \right|$. Hooley’s $\Delta$-function is known to satisfy $$\label{hooley_log} \sum_{u=1}^{\infty} \left| \frac{\Delta(u)}{u^{s}} \right| \ll_{\eps} \sum_{u=1}^{\infty} \left| \frac{(\log u)^{\eps}}{u^{s}} \right|,$$ for every $\eps > 0$, which is a consequence of Tenenbaum’s bound on the average order of $\Delta(u)$ (see [@tenenbaum], also [@hall]), and so the upper bound of the lemma follows by observing that $\sum_{u=1}^{\infty} (\log u)^{\eps} u^{-s}$ has a pole of order $1+\eps$ at $s=1$. Since $\sum_{u=1}^{\infty} (\log u)^{\eps} u^{-s}$ is absolutely convergent in the half-plane $\Re(s) > 1$, (\[hooley\_log\]) also proves that so is $\sum_{u=1}^{\infty} \frac{\beta_{\nu}(u)}{u^{s}}$. We are now ready to prove Theorem \[zeta\]. [*Proof of Theorem \[zeta\].*]{} First of all notice that (\[zeta\_c\]) combined with comparison test for series imply that $\sum_{u=1}^{\infty} \N(u) u^{-s}$ has a pole at $s=1$, since $\zeta_{\zed[i]}(s)$ has a pole at $s=1$, and is absolutely convergent, i.e. $\zeta_{\WR(\zed^2)}(s)$ is analytic, for $\Re(s) > 2$, since $\zeta_{\zed^2}(s)$ is analytic when $\Re(s) > 2$. In fact, we can do better. Let $$\beta'(n) = \left\{ \begin{array}{ll} 2\beta(n)-1 & \mbox{if $n$ is a square} \\ 2\beta(n) & \mbox{if $n$ is not a square,} \end{array} \right.$$ for every $n \in \zed_{>0}$. Notice that for every $u \in \zed_{>0}$, $\N(u)$ can be expressed in terms of the Dirichlet convolution of arithmetic functions $2\alpha_*$ and $\beta'$: $$\N(u) = (2\alpha_* * \beta')(t) + \left( \delta_1(t)\beta(t) + \delta_2(t)\beta\left( \frac{t}{2} \right) - 2\beta'(t) - \frac{1+(-1)^t}{2} \beta' \left( \frac{t}{2} \right) \right),$$ where $t=t(u)$, $\delta_1(t)$, and $\delta_2(t)$ are as in Theorem \[count\]. Therefore, by Theorem 284 of [@hardy] $$\begin{aligned} \label{zeta_prod1} \zeta_{\WR(\zed^2)}(s) & = & \sum_{u=1}^{\infty} (2\alpha_* * \beta')(t) u^{-s} \nonumber \\ & + & \sum_{u=1}^{\infty} \left( \delta_1(t)\beta(t) + \delta_2(t)\beta\left( \frac{t}{2} \right) - 2\beta'(t) - \frac{1+(-1)^t}{2} \beta' \left( \frac{t}{2} \right) \right) u^{-s} \nonumber \\ & = & \left( \sum_{u=1}^{\infty} 2\alpha_*(t) u^{-s} - 2\right) \left( \sum_{u=1}^{\infty} \beta'(t) u^{-s} \right) \nonumber \\ & + & \sum_{u=1}^{\infty} \left( \delta_1(t)\beta(t) + \delta_2(t)\beta\left( \frac{t}{2} \right) \right) u^{-s} - \sum_{u=1}^{\infty} \frac{1+(-1)^t}{2} \beta' \left( \frac{t}{2} \right) u^{-s},\end{aligned}$$ whenever these three series are absolutely convergent. Now notice that $$\frac{1}{|2^s|} \sum_{u=1}^{\infty} \frac{\beta(u)}{|u^s|} \leq \sum_{u=1}^{\infty} \frac{\delta_1(t)\beta(t)}{|u^s|} \leq 2 \sum_{u=1}^{\infty} \frac{\beta(u)}{|u^s|},$$ and $$\frac{1}{|4^s|} \sum_{u=1}^{\infty} \frac{\beta(u)}{|u^s|} \leq \sum_{u=1}^{\infty} \frac{\delta_2(t)\beta\left(\frac{t}{2}\right)}{|u^s|} = \frac{1}{|4^s|} \sum_{u=1}^{\infty} \frac{\delta_1(u)\beta(u)}{|u^s|} \leq \frac{2}{|4^s|} \sum_{u=1}^{\infty} \frac{\beta(u)}{|u^s|},$$ as well as $$\frac{1}{|2^s|} O \left( \sum_{u=1}^{\infty} \frac{\beta(u)}{|u^s|} \right) = \frac{1}{|2^s|} \sum_{u=1}^{\infty} \frac{\beta'(u)}{|u^s|} \leq \sum_{u=1}^{\infty} \frac{\beta'(t)}{|u^s|} \leq \sum_{u=1}^{\infty} \frac{\beta'(u)}{|u^s|} = O \left( \sum_{u=1}^{\infty} \frac{\beta(u)}{|u^s|} \right),$$ whenever $\Re(s) > 0$. Also $$\sum_{u=1}^{\infty} \frac{1+(-1)^t}{2} \beta' \left( \frac{t}{2} \right) u^{-s} = 4^{-s} \sum_{u=1}^{\infty} \beta'(u) u^{-s}.$$ Now the conclusion of Theorem \[zeta\] follows by applying these observations along with Lemmas \[Dir1\], \[Dir3\] to (\[zeta\_prod1\]). \[ded\_zeta\_sq\] Notice that one implication of Theorem \[zeta\] is that $\N(u)$, the coefficient of $\zeta_{\WR(\zed^2)}(s)$, grows, roughly speaking, like the coefficient of $\zeta_{\zed[i]}(s)^2$, which is $$\sum_{mn = u} \N_I(m) \N_I(n).$$ Finally, we mention that in the same manner one can define zeta functions of well-rounded sublattices of any lattice $\Omega$ in $\real^N$ for any $N$. Studying the properties of these functions may yield interesting arithmetic information about the distribution of such sublattices. [**Acknowledgment.**]{} I would like to thank Kevin Ford, Preda Mihailescu, Baruch Moroz, Gabriele Nebe, Bogdan Petrenko, Sinai Robins, Eugenia Soboleva, Paula Tretkoff, Jeff Vaaler and Victor Vuletescu for their helpful comments on the subject of this paper. I would also like to acknowledge the wonderful hospitality of Max-Planck-Institut für Mathematik in Bonn, Germany, where a large part of this work has been done.

--- 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`.

--- abstract: 'We examine the dependence of the spatial two-point correlation function of quasars $\xi_{qq}(r,z)$ at different redshifts on the initial power spectrum in flat cosmological models. Quasars and other elements of the large-scale structure of the universe are supposed to form in the peaks of the scalar Gaussian field of density fluctuations of appropriate scales. Quasars are considered as a manifestation of short-term active processes at the centers of these fluctuations; such processes set in when dark matter counterflows and a shock wave appear in the gas. We propose a method for calculating the correlation function $\xi_{qq}(r,z)$ and show its amplitude and slope to depend on the shape of the initial power spectrum and the scale $R$ of the fluctuations in which quasars are formed. We demonstrate that in the CDM models with the initial power spectrum slope it is possible to explain, by choosing appropriate values of $R$, how the amplitudes and correlation radii of $\xi_{qq}(r,z)$ may either increase or decrease with increasing redshift $z$. In particular, the correlation radii of $\xi_{qq}(r,z)$ grow from when $R$ grows from to . The H+CDM model at realistic values of $R$ fails to account for the observational data according to which the $\xi_{qq}(r,z)$ amplitude decreases with increasing $z$.' author: - 'B. Novosyadlyj, Yu. Chornij' title: '**SPATIAL CORRELATION FUNCTION OF QUASARS AND POWER SPECTRUM OF COSMOLOGICAL MATTER DENSITY PERTURBATIONS**' --- epsf.tex KiFNT, v. 14, No 2(1998) p.156-165 Introduction ============ The most elaborated scenario of the origin of the large-scale structure in the universe is pictured by the model in which such a structure results from the evolution of the uniform isotropic Gaussian scalar field of cosmological matter density fluctuations under the effect of gravitational instability. The major unresolved problems in this case are the choice of an inflationary model for an early universe, determination of the nature of the “dark” non-baryon matter (DM) and its fraction in the total matter mass. Assumptions as to the inflationary model and DM nature define the shape of the initial (post-recombination) power spectrum of cosmological density fluctuations, $P(k)$, and thus the principal parameters of large-scale structure can be theoretically calculated. Therefore it is of importance to test cosmological models with given power spectra $P(k)$. This testing can be done by calculating spatial two-point correlation functions for large-scale structure elements on various scales and comparing them with observational data. The testing is based on the relationship between the characteristics of structure elements and their correlation functions, on the one hand, and the amplitude and slope of the initial spectrum $P(k)$ at different $k$, on the other hand. Information on the power spectrum on small and intermediate scales (, , $H_0$ being the Hubble constant at the present epoch) resides in the correlation functions of bright massive galaxies, rich clusters of galaxies, and quasars. The “observed” correlation functions of all these three types of objects are described by the approximate expression , where $r_0$ is the correlation radius equal to , \[[@bs]-[@iras]\], \[[@adr]-[@mo]\]. for the three types of objects, respectively. At the moment investigations of the “observed” correlation function of quasars in different redshift ranges give ambiguous results. For example, the correlation function amplitude found in \[[@iov2; @komb2; @mo]\] decreases with increasing $z$, it remains unchanged in \[[@adr]\], and grows at and diminishes at in \[[@komb3]\]. The common result of these studies is that the amplitude for quasars is larger than for galaxies but is smaller than for rich clusters of galaxies. This result was obtained by different authors from two quasar samples: a combined sample of all quasars observed in the $z$ range from $0.1$ to $4.5$ and the “nearby” quasars at . Here we look into the possibility of using the above results as a test for cosmological models with given power spectra $P(k)$; to this end, the theoretical correlation function of quasars has to be calculated. Theoretical methods for calculating correlation functions of galaxies and their clusters are based on the theory of Gaussian random fields \[[@dor2]-[@kais]\], they have been devised in detail. The results obtained within the scope of cosmological models with given initial power spectra were analyzed in \[[@hn1]-[@watan]\], for example. Calculation of the correlation function of quasars $\xi_{qq}$ is complicated by a number of problems which have yet to be resolved. What scale is typical of the regions where quasars formed at different $z$? What are typical physical parameters of quasars: mass, duration of formation, lifetime, etc.? What is the relation between $\xi_{qq}$ and these parameters? Answers to these questions essentially depend on the physical model chosen for the quasar phenomenon. In particular, the disk accretion of gas on a massive black hole at the center of a galaxy may be a model mechanism. Therefore, for the mass of the fluctuations in which quasars are formed we can take the mass of “parent” galaxy $M_{g/q}$ which is able to ensure a high luminosity of the nucleus (observed as a quasar) during the quasar lifetime $\tau_q$. Based on the results of \[[@efst]-[@turn]\], we may take for $M_{g/q}$ (the black hole mass ), yr for $\tau_q$, and yr for the duration of quasar formation. Whether these parameters are the same for quasars at different $z$ is still an open question. Principal assumptions and formulation of the problem ==================================================== In the cosmological scenario used by us, galaxies, rich clusters of galaxies, and quasars appear in the peaks of the scalar Gaussian field of matter density fluctuations on corresponding scales, the relative amplitude of fluctuations being ($\sigma$ is the rms amplitude, $\nu$ is the peak height). It is assumed that galaxies and their clusters come into being when counterflows arise in the DM and a shock wave arises in the gas. The amplitude $\delta$ at this moment $t$ is determined from Tolmen’s model in terms of redshift: . The amplitude corresponding to the objects that appeared earlier is , it has a normal distribution: . The probability that a galaxy or a rich cluster of galaxies occurs at a fixed $z$ is $$\label{P1} P_1(z)=\int\limits_{\delta(z)}^{\infty} p(\delta)\,d\delta.$$ The probability that two galaxies exist simultaneously at a fixed $z$ at two different points $\vec x_1$ and $\vec x_2$ ($r=|\vec x_1-\vec x_2|$) is $$\label{P2} P_2(z)=\int\limits_{\delta(z)}^{\infty}\int\limits_{\delta(z)}^{\infty} p(\delta_1,\delta_2)\,d\delta_1d\delta_2,$$ where $p(\delta_1,\delta_2)$ is the two-dimensional normal distribution of random amplitudes $\delta_1$ and $\delta_2$ \[[@ven]\]: $$p(\delta_1,\delta_2)=(2\cdot\pi)^{-1}\cdot\left(\sqrt{\xi^2(0,z)-\xi^2(r,z)}\right)^{-1}$$ $$\label{p2} \times exp\left(-\frac{\xi(0,z)\cdot\delta_1^2+\xi(0,z)\cdot\delta_2^2-2\cdot\xi(r,z)\cdot\delta_1\cdot\delta_2}{2\cdot\left(\xi^2(0,z)-\xi^2(r,z)\right)}\right),$$ here $r>0$ and $\xi(r,z)$ is the correlation function of the density fluctuations in which the objects are formed. The function is calculated from the given initial power spectrum $P\left(k,R_f\right)$ smoothed on the scale $R_f$ which corresponds to the scale of the objects: $$\label{xi_p} \xi(r,z)=\frac{1}{2\pi^2}\cdot\int\limits_0^{\infty}\\ k^2\cdot\frac{P\left(k,R_f\right)}{(1+z)^2}\cdot\frac{sin(kr)}{kr}\,dk,$$ where $$P\left(k,R_f\right)=P(k)\cdot W^2(kR_f)$$ and $$W(kR_f)=exp\left(-\frac{1}{2}\cdot k^2\cdot R_f^2\right)$$ is the smoothing function. The statistical correlation function of the fluctuation peaks in which cosmological objects are formed is, by definition, $$\label{xi_oo_o} \xi_{oo}^{st}(r,z)\equiv\frac{P_2(z)}{P_1^2(z)}-1.$$ This function for rich clusters of galaxies or for galaxies at $z=0$ is \[[@kais]\] $$\xi_{oo}^{st}(r)\equiv\xi_{oo}(r,z=0)=\sqrt{\frac{2}{\pi}}\cdot\left(erfc\left(\frac{\nu}{\sqrt{2}}\right)\right)^{-2}\times$$ $$\label{xi_oo} \times\int\limits_{\nu}^{\infty}\\ e^{-1/2\cdot y^2}\cdot erfc\left(\frac{\nu- y\cdot \xi(r)/\xi(0)}{\sqrt{2\cdot(1-\xi^2(r)/\xi^2(0))}}\right)\,dy-1.$$ Expression (6) is simplified at and $$\label{xi1_oo} \xi_{oo}^{st}(r)\approx \left(\frac{\nu}{\sigma}\right)^2\cdot \xi(r).$$ In this case $\xi_{oo}^{st}(r)$ for objects is related to $\xi(r)$ for the fluctuations on the corresponding scale in which the objects are formed. The factor is called the statistical biasing of these objects, and the procedures for its determination for galaxies and rich clusters of galaxies are described in \[[@bbks]-[@hn1; @co_ka]\]. In deriving the correlation function of quasars we take advantage of the approach proposed in \[[@kais]\] with allowance made for the specific nature of quasars. Correlation function of quasars =============================== To calculate correlation functions of galaxies and rich clusters of galaxies, we have to specify the scale of the corresponding fluctuations in which they are formed. For quasars, there are two more important quantities along with this scale: the duration of quasar formation $\Delta t_{q}$ and quasar lifetime $\tau_{q}$. The smallest mass of the galaxies which may experience the quasar phase in the course of their evolution is \[[@efst]-[@turn]\]. The duration yr is much shorter than the cosmological evolution time of fluctuations in which such galaxies are formed at , and so we may ignore this quantity in first approximation. The same studies \[[@efst]-[@turn]\] reveal that the quasar phase is short-lived yr. The quantities $P_1(z)$ and $P_2(z)$ in (1), (2) are the probabilities that the random amplitudes of corresponding fluctuations are , i. e., they determine the probability of coming into being or the probability of the existence of objects on corresponding scales with redshifts larger than a given $z$. For quasars, the interval $\Delta(z,\tau_q)$ between the limits of integration in (1) and (2) is determined as the lifetime $\tau_q$ of the quasars which came into being at the cosmological moment $t(z)$ corresponding to a given $z$ \[[@nov3]-[@nov_qso]\]: $$\label{P2_2} \Delta(z,\tau_q)=1.13\cdot (z+1)\cdot\frac{\tau_q}{t(z)}.$$ When $\tau_q\sim t(z)$, the quantities $P_1(z)$ and $P_2(z)$ are in form (1), (2) with the upper limit of integration $\delta(z)+\Delta(z,\tau_q)$. When $\tau_q\ll t(z)$, expressions (1), (2) take the form $$\label{P1_1} P_1^q(z)=p(\delta(z))\cdot \Delta(z,\tau_q),\; P_2^q(z)=p(\delta_1,\delta_2)\cdot\Delta^2(z,\tau_q).$$ in view of smallness of $\Delta(z,\tau_q)$. Then, on the basis of (5) and in view of (9), the correlation function of quasars is $$\xi_{qq}^{st}(r,z)=\left(\sqrt{1-\left(\frac{\xi(r)}{\xi(0)}\right)^2}\right)^{-1}$$ $$\label{xi_ªª} \times exp\left(\frac{2.86\cdot(z+1)^2}{\xi(0)}\cdot\left(1+\frac{\xi(0)}{\xi(r)}\right)^{-1}\right)-1,$$ As it follows from (4), the ratio is independent of $z$, and . The above expression for $\xi_{qq}^{st}(r,z)$ represents the statistical component in the correlation function of the fluctuation peaks where quasars are formed, but it takes no account of the dynamics of background large-scale inhomogeneities (the dynamical component). According to \[[@bbks]\], the complete correlation function of objects is $$\label{xi_k_p} \xi_{oo}(r,z)=\left(\sqrt{\xi_{oo}^{st}(r,z)}+\sqrt{\xi_{oo}^{d}(r,z)}\right)^2,$$ where the statistical component for quasars $\xi_{oo}^{st}(r,z)\equiv \xi_{qq}^{st}(r,z)$ is represented by (10) and the dynamical component is correlation function (4) of density fluctuations. Let us examine the approximation (. In this approximation $\xi_{qq}^{st}(r,z)$ (10) takes the form $$\label{xi_ªª_t} \xi_{qq}^{st}(r,z)\approx\left(\frac{1.69\cdot(1+z)}{\sigma^2}\right)^2\cdot\xi(r)\\ =\left(\frac{\nu}{\sigma}\right)^2\cdot\xi(r),$$ after its expansion into the Taylor series. It coincides with expression (7) for the correlation function of the galaxies and clusters which formed in high peaks (that is, at high redshifts), but in the case of quasars there is no such a restriction to high peaks only. According to (4), the dynamical component $\xi(r,z)$ is expressed in terms of $\xi(r)$ $$\label{xi_d} \xi(r,z)=\xi(r)\cdot(1+z)^{-2}.$$ Then, in view of (11)-(13), we can write $$\label{xi_kk_tp} \xi_{qq}(r,z)=\left(\frac{1.69\cdot(1+z)}{\sigma^2}+\frac{1}{1+z}\right)^2\cdot\xi(r).$$ As seen from (12) and (13), the statistical component in the correlation function of quasars increases with $z$, while the dynamical component diminishes. Therefore, when $z$ is variable and $r$ is fixed, the correlation function amplitude may have a minimum if a decrease in the dynamical component is not compensated by an increase in the statistical component at small $z$. The minimum point can be found from the condition with expression (14): $$\label{extr} z_*=\frac{\sigma}{1.3}-1.$$ Thus, at $\sigma<1.3$ we have $\partial \xi_{qq}(r,z)/\partial z>0$ within the interval $z>0$ the correlation function amplitude does nothing but grows. When , the amplitude is minimum at . Over the entire range , in which quasars are observed, the amplitude of the correlation function of quasars diminishes at , over the range it diminishes at , and over the range at . These results were derived in the approximation , which is valid at . On these scales, the large-scale inhomogeneities have the amplitude even at , so that the correction of the spectrum $P(k)$ for the nonlinear evolution does not affect the results. Calculation results =================== We calculated the correlation function of quasars $\xi_{qq}(r,z)$ within the framework of models with various initial power spectra of fluctuations $P(k)$. Among models with various $\Omega_{b}$ (fraction of baryon density in the density of the whole matter in terms of the critical density , $\Omega_{CDM}$, and $\Omega_{HDM}$ (CDM stands for the cold dark matter of axion type and HDM for the hot dark matter of massive neutrino type), the CDM models with and $\Omega_{CDM}=1-\Omega_{b}$ are believed to be the most promising at the moment; they include the “standard” CDM model with the spectrum slope ($P(k)=A\cdot k^n\cdot T^2(k)$, where $A$ is a spectrum normalization constant, $k$ is the wave number, and $T(k)$ is a transfer function depending on DM nature), “inclined” CDM models with , and the “hybrid” H+CDM model with , , , . Model parameters and methods of spectrum normalization on the COBE results \[[@benn; @smoot]\] were described in \[[@nov2; @nov_qso; @nov_tr]\]. To calculate $\xi_{qq}(r,z)$, one has to specify the characteristic scale or mass of fluctuations in which quasars are formed and smooth the power spectrum $P(k)$ by a Gaussian filter with the radius $R_f$ corresponding to this scale. It follows from the expression \[[@bbks]\] that the radius corresponding to the mass \[[@efst]-[@turn]\] is . So, we start from the assumption that quasars are an early short phase in the evolution of galaxies with this mass and calculate the correlation function of quasars at this value of $R_f$. Figure 1 shows approximations of the observed correlation functions of galaxies \[[@dav; @iras]\] and rich clusters of galaxies \[[@bs; @iras]\] and the correlation function of quasars calculated by expression (11) for $z=0.5, 1, 2, 3, 4$ in various cosmological models. =7truecm It is apparent that at large redshifts, where the correlation function is mainly specified by its statistical component and the dynamical component is of minor importance, the amplitudes of correlation functions of quasars diminish with increasing $n$ in various CDM models. This is a manifestation of the well-known tendency of Gaussian field peaks to decrease the degree of their clustering with decreasing height $\nu$. Calculations of these heights for a fixed $z$ by the expression where (4), reveal that they are the largest in the CDM model with $n = 0.7$ ($\sigma=1.67$), somewhat smaller in the CDM model with $n=0.8$ ($\sigma=2.37$), and the smallest in the CDM model with $n=1$ ($\sigma=4.78$). The above rms amplitudes $\sigma$ allow one to follow the variations of correlation function amplitudes for quasars in these models at different $z$ (see expression (15)). Thus, the amplitude $\xi_{qq}(r,z)$ in the CDM model with $n=0.7$ diminishes at small redshifts, reaches its minimum at $z$ $z\approx0.3$, and then it grows. In the CDM model with $n=0.8$, a similar minimum occurs at $z\approx0.8$, in the CDM model $n = 1$ at $z\approx2.7$, and in the H+CDM ($\sigma=1.39$) - at $z\approx0.07$. These results are in good agreement with numerical calculations (Fig. 1). When the plots in Fig. 1 are matched to observational data \[[@adr]-[@mo]\], one can see that only the CDM model with $n = 1$ and results of \[[@iov2]\] are in accord as to the correlation radius and the trend of the $\xi_{qq}(r,z)$ amplitude variations. As regards other observational data, the cosmological models discussed here under the assumption that quasars are formed in the fluctuations on the scale fail to explain the amplitudes of the observed correlation functions of quasars at various $z$ and their correlation radii. According to \[[@adr]-[@mo]\], the radii at different redshifts. These correlation function parameters in the CDM models are smaller than in the observed function, while in the H+CDM model they are too large. This may mean that the fluctuation scale chosen by us, , is too small in the CDM models and too large in the H+CDM model. The correlation functions $\xi_{qq}(r,z)$ plotted in Fig. 2 were calculated with the CDM models with such fluctuation scales $R_f$ for each model that the functions do not contradict the data of \[[@adr; @komb2; @mo]\] on the amplitude (or correlation radius) on the scales . At the amplitudes of these correlation functions of quasars are no smaller than the amplitude of the observed correlation function of galaxies, with due regard for a large scatter of that function on these scales. =7truecm Besides, to make more correct the comparison between the theoretical and observational data, we took into consideration that the observed correlation functions of quasars were determined for wide redshift intervals \[[@adr]-[@komb2; @mo]\], while the theoretical functions were calculated for fixed $z$. Therefore, the observed correlation functions at different $\Delta z$ were fitted by the theoretical functions calculated for $\bar z$, the weighted mean in these intervals found from the observed distribution of quasars $n_q(z)$ \[[@schmidt]\] in the comoving reference frame: $$\bar z= \frac{\int\limits^{z_{max}}_{z_{min}}z\cdot n(z)\cdot r^2(z)\,dr(z)} {\int\limits^{z_{max}}_{z_{min}} n(z)\cdot r^2(z)\,dr(z)},$$ where $r(z)=2c/H_0\cdot\left(1-(z+1)^{-0.5}\right)$ is the distance in the comoving reference frame, $dr(z)=c/H_0\cdot(z+1)^{-1.5}\,dz$, and $c$ is the velocity of light. The weighting made to fit $\xi_{qq}(r,z)$ to the results of \[[@iov2; @mo]\] gave $\bar z=1.3$ for the interval $\Delta z\equiv[0.1,1.5]$ and $\bar z=2.6$ for $[1.5,4.5]$. For the intervals $\Delta z$ used in studies \[[@adr; @komb2]\] we found $\bar z=1.05$ for $[0.1,1.1]$, $\bar z=1.49$ for $[1.1,1.7]$, and $\bar z=2.38$ for $[1.7,3.1]$. Based on the correlation functions of quasars which correspond to the data of \[[@adr; @komb2; @mo]\] as to correlation radius , we determined the scales $R_f$ of fluctuations in which quasars are formed and their rms amplitudes $\sigma(R_f)$ in the CDM models: $R_f\approx1.3h^{-1}$ Mpc, $\sigma(R_f)=2.24$ (in the model with $n=1$); $R_f\approx0.7h^{-1}$ Mpc, $\sigma(R_f)=1.73$ ($n = 0.8$); $R_f\approx0.45h^{-1}$ Mpc, $\sigma(R_f)=1.52$ ($n = 0.7$). Judging from the above values of $\sigma(R_f)$ and expression (15), the amplitude and the correlation radius of $\xi_{qq}(r,z)$ in these models grow with $\bar z$. Conclusion ========== We have elaborated a method for the theoretical calculation of the correlation function of quasars, it is based on the Gaussian statistics of the initial field of cosmological matter density fluctuations. The correlation functions of quasars were calculated for different redshifts in cosmological models with given initial power spectra $P(k)$ on the assumption that quasars are formed in the peaks of the fluctuations and exist over a time much shorter than the cosmological time. The amplitudes and slopes of the correlation functions are shown to depend on fluctuation scale and power $P(k)$ on corresponding scales. The amplitudes and correlation radii of the observed and theoretical correlation functions of quasars are consistent in the CDM models with the spectrum slope $n$ ranging from 0.7 to 1 when the scale $R_f$ of the corresponding fluctuations in which quasars are formed ranges from to . In this case, however, the function amplitudes increase with $z$. Although there is no evidence of any monotonic growth of the correlation function amplitude in studies \[[@adr; @iov2; @komb2; @komb3; @mo]\], the estimates are so contradictory that we cannot use them to test the cosmological models. The H+CDM model fails to explain the correlation radius derived from observational data if the scale of fluctuations in which quasars are formed exceeds . Our results are in accord with the physical models which regard the quasar phenomenon as an early short phase in the evolution of massive galaxies or as a merger of galaxies situated in groups. [99]{} 1. N. A. Bahcall, R. M. Soneira, and A. S. Shalay, Astrophys. J., 1983, V.70, N1, P.20. 2. M. Davis and P. J. E. Peebles, Astrophys. J., 1983, V.267, N2, P.465. 3. Y. P. Jing and R. Valdarnini, Astrophys. J., 1983, V.406, N.1, P.6. 4. P. Andreani and S. Cristiani, Astrophys. J., 1992, V.398, N1, L13. 5. A. Iovino, P. Shaver, and S. 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--- author: - 'Á. Sánchez-Monge' - 'R. Cesaroni' - 'M. T. Beltrán' - 'M. S. N. Kumar' - 'T. Stanke' - 'H. Zinnecker' - 'S. Etoka' - 'D. Galli' - 'C. A. Hummel' - 'L. Moscadelli' - 'T. Preibisch' - 'T. Ratzka' - 'F. F. S. van der Tak' - 'S. Vig' - 'C. M. Walmsley' - 'K.-S. Wang' date: 'Received date; accepted date' title: | A candidate circumbinary Keplerian disk in 35:\ A study with ALMA --- Introduction {#sint} ============ While different scenarios have been proposed to explain the formation of high-mass (i.e. OB-type) stars (monolithic collapse in a turbulence-dominated core – Krumholz et al. [@krumholz2009]; competitive accretion driven by a stellar cluster – Bonnell & Bate [@bonnellbate2006]; Bondi-Hoyle accretion – Keto [@keto2007]; see also the review by Zinnecker & Yorke [@ziyo]), all of them predict the formation of circumstellar disks. It is thus surprising that only a handful of disk candidates have been observed in association with massive (proto)stars. As a matter of fact, despite many observational efforts, convincing evidence of disks has been found only around early B-type (proto)stars, while circumstellar disks around O-type stars remain elusive (Wang et al. [@wang2012]; Cesaroni et al. [@cesaroni2007] and references therein). Moreover, a detailed investigation of the disk properties, comparable to that performed in disks around low-mass stars (e.g. Dutrey et al. [@dutrey2007]) is still missing due to the large distances of OB-type (proto)stars and the limited angular resolution at (sub)millimeter wavelengths. With the advent of the Atacama Large Millimeter Array (ALMA) the situation is bound to improve dramatically, as resolutions $\ll$1 will be easily obtained. With this in mind, we performed ALMA Cycle 0 observations of two IR sources containing B-type (proto)stars. These were chosen on the basis of their luminosities (on the order of $10^4~L_\odot$), presence of bipolar nebulosities/outflows, detection of broad line wings in typical jet/outflow tracers (SiO), and strong emission in hot molecular core (HMC) tracers (such as methyl cyanide, ). Here we present the most important results obtained for one of the two sources, 35. 35 is a well known star forming region located at a distance of $2.19_{-0.20}^{+0.24}$ kpc (Zhang et al. [@zhang2009]), with a luminosity of $\sim$$3\times10^4~L_\odot$[^1]. The region is characterized by the presence of a butterfly shaped reflection nebula oriented NE–SW (see Fig. \[flarge\]), as well as a bipolar molecular outflow in the same direction, observed in  by Dent et al. ([@dent1985a]), Gibb et al. ([@gibb2003]; hereafter GHLW), Birks et al. ([@birks2006]; hereafter BFG), and López-Sepulcre et al. ([@lopezsepulcre2009]). The (1–0) line emission appears to trace also a N–S collimated flow (see BFG), coinciding with a thermal radio jet (Heaton & Little [@heatonlittle1988]; GHLW) seen also at IR wavelengths (Dent et al. [@dent1985b]; Walther et al. [@walther1990]; Fuller et al. [@fuller2001]; De Buizer [@debuizer2006]; Zhang et al. [@zhang2013]). It has been proposed that the poorly collimated NE–SW outflow and the N–S jet could be manifestations of the same bipolar flow undergoing precession (Little et al. [@little1998]). However, evidence for multiple outflows in this region is provided by SiO, , , and  line observations (GHLW; Lee et al. [@lee2012]). A molecular clump elongated perpendicular to the NE–SW outflow has been mapped in dense gas tracers (, CS), whose emission exhibits a velocity gradient from NW to SE (Little et al. [@little1985]; Brebner et al. [@brebner1987]). This was first interpreted as a large ($\sim$1 or 0.6 pc) disk/toroid rotating about the NE–SW outflow axis, but GHLW, on the basis of their  and  observations, propose that this is actually a fragmented rotating envelope containing multiple young stellar objects (YSOs). Indeed, GHLW identify a core at the center of the outflow and another core, named G35MM2, offset to the SE. Observations and results {#sobs} ======================== 35 was observed with ALMA Cycle 0 at 350 GHz in May and June 2012, with baselines in the range 36–400 m, providing sensitivity to structures $\le$2 . The digital correlator was configured in 4 spectral windows (with dual polarization) of 1875 MHz and 3840 channels each (covering the ranges 334.85–338.85 GHz and 346.85–350.85 GHz), providing a resolution of $\sim$0.4 km s$^{-1}$. Flux, gain, and bandpass calibrations were obtained through observations of Neptune and J1751$+$096. The data were calibrated and imaged using CASA. A continuum map was obtained from line-free channels and subtracted from the data. The synthesized beam is $0\farcs51\times0\farcs46$, P.A.=48. The rms noise is $\sim$6 mJy beam$^{-1}$ for individual line channels, while in the continuum image it is $\sim$1.8 mJy beam$^{-1}$, implying a S/N of only $\sim$100. The latter indicates a reduced dynamic range. In Fig. \[flarge\], we present the map of the 350 GHz continuum emission overlaid on an enhanced resolution Spitzer/IRAC image at 4.5  extracted from the GLIMPSE survey (Benjamin et al. [@benj]). The sub-millimeter continuum emission is clearly tracing an elongated structure across the waist of the butterfly shaped nebula. In all likelihood, we are detecting the densest part of the flattened molecular structure observed on a larger scale by Little et al. ([@little1985]), Brebner et al. ([@brebner1987]), and GHLW. Along the elongated structure a chain of at least 5 cores is seen (Fig. \[fcont\]), lending support to GHLW’s idea that one is dealing with a fragmented structure instead of the smooth disk/toroid hypothesized by Little et al. ([@little1985]). We stress that the angular resolution of our maps ($\sim$7 times better than previous (sub)millimeter observations) reveals that the YSOs powering the outflow(s) lie inside cores A and/or B (see Fig. \[fcont\]), because these two are the only cores located close to the geometrical center of the bipolar nebula. In particular, core B lies along the N–S jet traced by the IR and radio emission, and coincides with one of the free-free sources detected by GHLW. The hypothesis by BFG that core G35MM2 could be driving the NE–SW outflow is not convincing, as this core is off-center, lying right at the border of the waist outlined by the IR emission. Methyl cyanide emission (as well as other typical HMC tracers) is clearly detected only towards cores A and B, and marginally towards G35MM2. Emission by vibrationally excited lines of  also indicates that cores A and B could be hosting massive stars. Core B coincides also with a compact free-free continuum source detected by GHLW at 6 and 3.6 cm, and by Codella et al. ([@codella2010]) at 1.3 cm. This emission could be part of the N–S thermal radio jet or might be coming from an  region ionized by an embedded early-type star. We will discuss this possibility in Sect. \[sdis\]. Faint radio emission at a $5.6\sigma$ level is detected also towards core A, consistent with the presence of embedded star(s). We now investigate the gas velocity field in the two cores by computing the first moment of a  line, a dense gas tracer. Figure \[fvelo\] plots the result for the (19–18) $K$=2 line, with overlaid the line emission averaged over the same velocity interval used to calculate the first moment. Note that the mean velocity of core B ($\sim$30 ) differs by $\sim$2  from that of core A ($\sim$32 ), which also differs by a similar amount from that ($\sim$34 ) obtained with lower angular resolution observations of other tracers (see e.g. GHLW). Discrepancies of this type are often found in high-mass star-forming regions (see, e.g., the case of AFGL5142 – Estalella et al. [@estalella1993]; Zhang et al. [@zhang2007]) and are likely related to the fragmentation process in molecular clumps. The interesting result is that both cores present velocity gradients, which may be due to expansion or rotation of the gas. Note that the sense of the gradient in core B is the opposite of that measured on larger scales (see Fig 5 of Little et al. [@little1985] and Figs. 8 and 9 of GHLW). However, the velocity range traced by the extended emission (30–41 ) differs significantly from that sampled by the  lines ($\sim$26–38 ), suggesting that we are observing cores whose velocity field is not tightly related to that of the largest, fragmenting structure where they are embedded. In the following, we will address this question for core B, which is better resolved and associated with the N–S jet/outflow. The nature of core B: A Keplerian rotating disk? {#skep} ================================================ An elongated structure with a velocity gradient along its major axis, such as that observed in core B, can be interpreted as either a collimated bipolar outflow, or a rotating ring/disk seen close to edge on. We thus first investigate the possibility that we are observing the root of a large-scale bipolar outflow. At first glance, both the orientation (see Fig. \[fcont\]) and sign of the velocity gradient seen in Fig. \[fvelo\] appear consistent with those of the N–S jet imaged by BFG (see the red-shifted emission at 40  in their Fig. 3). However, the CO emission in the northern jet lobe is detected by BFG also at 28  (see their Fig. 3), namely at blue-shifted velocities. Moreover, the direction of the  velocity gradient has a negative position angle, whereas the jet is slightly inclined to the east. Such a difference could be explained if the jet is precessing (as suggested by BFG) and lies close to the plane of the sky, because the jet direction would change from the small to the large scale. We believe that this explanation is not satisfactory, because the orientation of the  velocity gradient differs from that of the radio jet [*on the same scale*]{}. This can be seen in Fig. \[fvelo\], where the two free-free emission sources are aligned N–S, whereas the velocity gradient (and the core major axis) are inclined to the west (PA$\simeq$–20). Based on these findings, we consider the alternative possibility that the velocity gradient is tracing a disk. To analyse the kinematics of the gas in better detail, we fitted the  emission in each 0.4  velocity channel with a 2-D Gaussian. This allows us to derive the peaks of the line emission at different velocities. Then all peaks can be plotted together to obtain a picture of the gas velocity field. This is done in Fig. \[fkepfit\]a, where for each peak we also plot the corresponding 50% contour level to give an idea of the size of the gas emitting at that velocity. Two considerations are in order. The first is that the peaks outline a sort of elliptical pattern, roughly centered on the position of the continuum peak and oriented SSE–NNW. The second is that the most blue- and red-shifted peaks tend to converge towards the position of the sub-mm continuum peak. On this basis, one is tempted to hypothesize that the  emission is tracing Keplerian-like rotation, as this would explain why both the emission at systemic velocities and that at high (blue- and red-shifted) velocities are observed towards the rotation center. {width="14cm"} With this in mind, we have fitted the observed velocity pattern assuming Keplerian rotation about a point-like source. We stress that ours is a purely kinematical fit, where only the rotation velocity in a 2-D disk is calculated, and no estimate of the observed line intensity is computed. Also, no attempt is made to demonstrate the uniqueness of the model proposed and we envisage the possibility that other interpretations might be possible. Our simple model is sufficient to constrain a number of important parameters. The inputs of the model are: the LSR velocity of the system (), the central point mass () and its position ($x_0$,$y_0$), the inclination angle ($\theta$) of the rotation axis with respect to the plane of the sky, and the position angle ($\psi$) of the projected major axis of the disk. The best fit was obtained by varying all input parameters inside reasonable ranges and minimizing the expression $\sum_i{\left({V_i^{\rm observed}}-V_i^{\rm model}\right)^2}$ where $i$ indicates the peak in a generic velocity channel. For our calculations, we have used the $K$=2, 3, and 4 components of the (19–18) transition as well as other prominent lines of other species, such as ($7_{1,6}$–$6_{1,5}$) $v_{\rm t}$=1, ($14_{1,13}$–$14_{0,14}$), and HC$_3$N(37–36). All of these lines reveal a kinematical pattern very similar to that in Fig. \[fkepfit\]a, demonstrating that such a pattern does not depend on the tracer and mirrors a real physical structure. In Fig. \[fkepfit\]b we show an overlay of the velocity peaks of all the lines and a velocity map of the best-fit model. Clearly, the agreement between the computed and observed LSR velocities is remarkable. The best-fit parameters are the following: =30.0$\pm$0.3 , $x_0$=18$^{\rm h}$58$^{\rm m}$13027$\pm$0002, $y_0$=01403594$\pm$007, $\psi$=157$\pm$4, $\theta$=19$\pm$1, =18$\pm$3 . A lower limit for the disk radius is $R_{\rm disk}\simeq2500$ AU, obtained from the maximum deprojected distance of the peaks from the center. One may wonder if such a big structure can undergo Keplerian rotation. The mass of the disk can be estimated from the continuum emission. The integrated flux density from core B at 350 GHz is 0.32 Jy. Assuming a dust absorption coefficient 0.5 cm$^2$g$^{-1}~(\nu/230.6~{\rm GHz})$ (Kramer et al. [@kramer2003]) and a gas-to-dust mass ratio of 100, we obtain $\sim$3  for a dust temperature of 100 K. Despite the large uncertainties on the dust opacity and temperature, we believe that the mass of core B is significantly less than the central mass ($\sim$18 ), which satisfies the condition for Keplerian rotation. It is worth stressing that our findings are in good agreement with the recent study by Ilee et al. ([@ilee]). Through measurements of scattered light from 35, these authors find that the CO first overtone bandhead emission at 2.3  can be fitted with a Keplerian disk rotating about a 17.7  star. The distribution of the molecular peaks in Fig. \[fkepfit\]b clearly shows that our observations detect only the NE side of the disk. We speculate that this could be an opacity effect. If the disk is optically thick in the relevant lines, flared, and inclined by 19, only part of the surface heated by the star is visible to the observer. This creates an asymmetry along the direction of the projected disk axis, with line emission being more prominent on the side (in our case the NE side) where the disk surface is visible. Clearly, radiative transfer calculations are needed to confirm this scenario, but we note that in our source the NE part of the disk axis should be pointing towards the observer, consistent with the orientation of the CO outflow (blue shifted to the NE and red shifted to the SW - see GHLW) and the obscuration seen to the SW in the IR images (see Fig. 2). The stellar content of core B: A binary system? {#sdis} =============================================== An issue that is worth discussing is whether an 18  YSO is compatible with the bolometric luminosity of the region. ($3\times10^4~L_\odot$; see Sect. \[sint\]). Depending on the adopted mass–luminosity relation, the luminosity expected for an 18  star ranges from $2.5\times10^4~L_\odot$ (Diaz-Miller et al. [@diazmiller1998]), to $6.6\times10^4~L_\odot$ (Martins et al. [@martins2005]). This means that the 18  star should be the main contributor to the luminosity of the whole star forming region. Such a possibility seems quite unlikely due to the presence of multiple cores (see Fig. \[fcont\]), one of which is an HMC possibly hosting at least another high-mass star (core A). A possibility is that one is underestimating the true luminosity due to the “flashlight effect”, where part of the stellar photons are lost through the outflow cavities. According to the recent model by Zhang et al. ([@zhang2013]), when this effect is taken into account, the luminosity obtained assuming isotropic emission ($3.3\times10^4~L_\odot$) becomes as large as $7\times10^4$–$2.2\times10^5~L_\odot$ consistent with a single star of $\sim$$20$–$34~M_\odot$. While the previous explanation is possible, the isotropic estimate appears more robust than a model-dependent value, and we thus consider another hypothesis, namely that one is dealing with a binary system. In this case, the luminosity is significantly reduced with respect to that of a single 18  object and may be as low as $\sim7\times10^3~L_\odot$ for equal members. The latter is much less than the bolometric luminosity, thus allowing for a significant contribution by other YSOs. The existence of a binary system could also explain why the N–S jet associated with core B is not aligned with the disk rotation axis. The presence of a companion would in fact be sufficient to induce precession of the jet/outflow, as hypothesized by Shepherd et al. ([@shepherd2000]) to explain the observed precession of the bipolar outflow from the high-mass protostar . In this scenario, the outflow from core B would precess about an axis oriented NE–SW, i.e. along the bisector of the butterfly shaped IR nebula seen in Fig. \[flarge\]. The IR emission would arise from the cavity opened by the outflow itself during its precession, while the thermal radio jet would trace the current direction of the precessing axis. The last question we address is the origin of the free-free emission from core B (see e.g. Fig. \[fvelo\]). Could this be tracing a (hypercompact)  region? According to GHLW, this source (n. 7 in their Table 1) has a spectral index $>$1.3 between 6 and 3.6 cm, compatible with free-free emission from an optically thick  region. Extrapolation of the 3.6 cm flux density (0.5 mJy) to 1.3 cm gives $>$1.9 mJy, in agreement with the 3 mJy flux measured, with lower angular resolution, by Codella et al. ([@codella2010]). It is hence possible that the emission is partially thick at 1.3 cm and the Lyman continuum estimate of $10^{45}~{\rm s}^{-1}$ obtained under the optically thin assumption is a lower limit. A binary system with a total mass of 18  has a Lyman continuum flux (see Diaz-Miller et al. [@diazmiller1998]) ranging from $5\times10^{44}$ s$^{-1}$ (for equal masses) to $1.8\times10^{47}$ s$^{-1}$. We conclude that we could be observing an  region ionized by a binary system at the center of a Keplerian disk. Should this be confirmed, 35 would represent a unique example of radio jet coexisting with an  region powered by the same YSO(s). Investigating such a short-lived transition phase in the evolution of an OB-type star could provide us with important clues on the formation process of these objects. It is a pleasure to thank Göran Sandell for stimulating discussions on the 35 region and the anonymous referee for his/her constructive criticisms. We also acknowledge the support of the European ALMA Regional Center and the Italian ARC node. This paper makes use of the following ALMA data: ADS/JAO.ALMA\#2011.0.00275.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. This work also used observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. MSNK is supported by a Ciência 2007 contract, funded by FCT (Portugal) and POPH/FSE (EC). 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--- author: - '${}^{1,2}$Yoshiaki Maeda, ${}^3$ Akifumi Sako, ${}^4$ Toshiya Suzuki and  ${}^4$ Hiroshi Umetsu' title: Gauge Theories in Noncommutative Homogeneous Kähler Manifolds --- [ ${}^{1}$ Department of Mathematics, Faculty of Science and Technology, Keio University\ 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan\ ${}^2$ Mathematical Research Centre, University of Warwick\ Coventry CV4 7AL, United Kingdom,\ ${}^3$ Department of Mathematics, Faculty of Science Division II,\ Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan\ ${}^4$ Kushiro National College of Technology\ 2-32-1 Otanoshike-nishi, Kushiro, Hokkaido 084-0916, Japan ]{} [**MSC 2010:**]{} 53D55 , 81R60 Introduction ============ Field theories on noncommutative spaces appear in various phenomena in physics. For example, effective theories on D-branes with NS-NS B field backgrounds give rise to gauge theories on noncommutative spaces [@Seiberg:1999vs]. As another example, in matrix models [@Banks:1996vh; @Ishibashi:1996xs], noncommutative field theories corresponding to fuzzy spaces appear when one expands the models around some classical solutions. A typical noncommutative space is the noncommutative ${\mathbb R}^d$. Field theories on the noncommutative ${\mathbb R}^d$ have many intriguing properties. For example, there is work on the existence of noncommutative instantons [@Nekrasov:1998ss], noncommutative scalar solitons [@Gopakumar:2000zd], etc. as classical solutions and the appearance of UV-IR mixing [@Minwalla:1999px] at the quantum level (see also the review papers [@Douglas:2001ba; @sako_review; @Szabo:2001kg], for examples). It is important to investigate whether field theories on more generic noncommutative manifolds have similar properties. However, field theories on noncommutative manifolds are not well understood at present, except for a few examples such as, noncommutative tori, $S^2$, etc. Several methods to construct noncommutative manifolds have been proposed, including the important approach by the deformation quantization. Deformation quantization was first introduced in [@Bayen:1977ha]. After [@Bayen:1977ha], several alternative methods of deformation quantization were proposed [@DeW-Lec; @Omori; @Fedosov; @Kontsevich]. In particular, deformation quantization of Kähler manifolds was studied in [@Moreno86a; @Moreno86b; @Cahen93; @Cahen95]. We study gauge field theories on noncommutative Kähler manifolds based on the deformation quantization with separation of variables introduced by Karabegov to quantize Kähler manifolds [@Karabegov; @Karabegov1996; @Karabegov2011]. The purpose of this paper is to construct gauge theories on noncommutative homogeneous Kähler manifolds. Field theories need to define differentials on base spaces. Note that the usual differentiations by coordinates in a noncommutative space may not be derivations; in other words, they do not satisfy the Leibniz rule for star products in general. We use inner derivations as differentials, which are defined by commutators with a function $P$ under a star product, [*i.e.*]{} $[ P , \ \cdot \ ]_*$. These operators automatically satisfy the Leibniz rule. For a generic $P$, the inner derivation $[ P , \ \cdot \ ]_*$ includes higher derivative terms. The necessary and sufficient condition on $P$ such that the inner derivation includes no higher derivative terms is known [@Muller:2004]. The necessary and sufficient condition is that $P$ is the Killing potential. For homogeneous Kähler manifolds ${\cal G} / {\cal H}$, there are Killing vectors ${\cal L}_a$ which constitute the Lie algebra of the isometry group ${\cal G}$. The Killing potential $P_a$ corresponding to ${\cal L}_a$ exists, and ${\cal L}_a$ is represented by the inner derivation ${\cal L}_a = \{ P_a , \ \cdot \ \} = - \frac{i}{\hbar} [ P_a, \ \cdot \ ]_*$. Using these Killing potentials, we construct a gauge theory on noncommutative homogeneous Kähler manifolds. In our previous papers [@Sako:2012ws; @Sako:2013noa], we studied deformation quantizations with separation of variables for ${\mathbb C}P^N$ and ${\mathbb C}H^N$, and gave explicit expressions for the star products. Using these results, we describe $U(n)$ gauge theories on noncommutative ${\mathbb C}P^N$ and on noncommutative ${\mathbb C}H^N$, as examples.  (On other types of noncommutative ${\mathbb C}P^N$, different gauge theories have been constructed. For example, a gauge theory on fuzzy ${\mathbb C}P^N$ is studied in [@CarowWatamura:1998jn; @Grosse:2004wm]. ) The organization of this article is as follows. In section 2, after we review deformation quantization with separation of variables for Kähler manifolds proposed by Karabegov, we study differentials on noncommutative Kähler manifolds. The conditions under which inner derivations become vector fields (Killing vector fields) are provided. We then construct gauge theories on noncommutative homogeneous Kähler manifolds. In section 3, we discuss gauge theories on noncommutative ${\mathbb C}P^N$ and ${\mathbb C}H^N$, as concrete examples. In section 4, we summarize our results and give some further discussion. Deformation quantization of gauge theories with separation of variables ======================================================================= Deformation quantization with separation of variables {#reviewKarabegov} ----------------------------------------------------- We briefly review the deformation quantization with separation of variables for Kähler manifolds, which proposed by Karabegov [@Karabegov1996]. Let $\Phi$ be a K[" a]{}hler potential and $\omega$ a K[" a]{}hler 2-form for $N$-dimensional Kähler manifolds $M$: $$\omega := i g_{k \bar{l}} dz^{k} \wedge d \bar{z}^{l}, ~~~~ g_{k \bar{l}} := \frac{\partial^2 \Phi}{\partial z^{k} \partial \bar{z}^{l}} .$$ We denote the inverse of the metric $(g_{k \bar{l}})$ as $(g^{\bar{k} l})$, and set $g_{\bar{k}l} = g_{l \bar{k}}$, $g^{l \bar{k}} = g^{\bar{k} l} $. We use the following abbreviations $$\begin{aligned} \partial_k = \frac{\partial}{\partial z^{k}} , ~~~~ \partial_{\bar{k}} = \frac{\partial}{\partial \bar{z}^{k}}.\end{aligned}$$ Deformation quantization is defined as follows. Let $\cal F$ be a set of formal power series in $\hbar$ with coefficients of $C^{\infty}$ functions on $M$ $$\begin{aligned} {\cal F} := \left\{ f \ \Big| \ f = \sum_k \hbar^k f_k, ~f_k \in C^\infty (M) \right\} ,\end{aligned}$$ where $\hbar$ is a noncommutative parameter. A star product is defined on ${\cal F}$ by $$\begin{aligned} f * g = \sum_k \hbar^k C_k (f,g), \end{aligned}$$ such that the product satisfies the following conditions. 1. $*$ is associative product. 2. $C_k$ is a bidifferential operator. 3. $C_0$ and $C_1$ are defined as $$\begin{aligned} && C_0 (f,g) = f g, \\ &&C_1(f,g)-C_1(g,f) = i \{ f, g \}, \label{weakdeformation}\end{aligned}$$ where $\{ f, g \}$ is the Poisson bracket. 4. $ f * 1 = 1* f = f$. Moreover, $*$ is called a star product with separation of variables when it satisfies $$a * f = a f, ~~~~ f * b = f b,$$ for any holomorphic function $a$ and any anti-holomorphic function $b$. Karabegov constructed a star product with separation of variables for Kähler manifolds in terms of differential operators [@Karabegov; @Karabegov1996], as briefly explained below. For the left star multiplication by $f \in {\cal F}$, there exists a differential operator $L_f$ such that $$L_f g = f * g .$$ $L_f$ is given as a formal power series in $\hbar$ $$L_f = \sum_{n=0}^{\infty} \hbar^n A^{(n)}, \label{Lf-An}$$ where $A^{(n)}$ is a differential operator which contains only partial derivatives by $z^i$ and has the following form $$A^{(n)} = \sum_{k\geq 0} a^{(n;k)}_{\bar{i}_1 \cdots \bar{i}_k} D^{\bar{i}_1} \cdots D^{\bar{i}_k}, \label{An-a}$$ where $$D^{\bar i} = g^{{\bar i} j} \partial_j.$$ In particular, $a^{(n;0)}$ which is a $C^{\infty}$ function on $M$ acts as a multiplication operator. Note that the differential operators $D^{\bar i}$ satisfy the following relations, $$\begin{aligned} [ D^{\bar i} , D^{\bar j} ] &= 0, \\ [D^{\bar{i}}, \partial_{\bar{j}}\Phi] &= \delta_{ij}. \label{D-dphi}\end{aligned}$$ Karabegov showed the following theorem. $L_f$ is uniquely determined by requiring the following conditions, $$\begin{aligned} L_f 1 = f * 1 =f, \label{Lf-dphi1}\\ [L_f , \partial_{\bar i} \Phi + \hbar \partial_{\bar i}] = 0. \label{Lf-dphi2}\end{aligned}$$ Substituting the expression of $L_f$ in (\[Lf-An\]) to the conditions (\[Lf-dphi1\]) and (\[Lf-dphi2\]), one obtains the following recursion relations, $$\begin{aligned} & A^{(0)} = f, ~~~ A^{(r)} 1 = 0, \label{Ar-rec1} \\ & [A^{(r)}, \partial_{\bar{i}}\Phi] = [\partial_{\bar{i}}, A^{(r-1)}], \label{Ar-rec2}\end{aligned}$$ for $r \geq 1$, where $f$ is assumed to be independent of $\hbar$ (in general, we set for $f \in {\cal F}$, $A^{(0)} = f_0$ and $A^{(r)} 1 = f_r$ in the above equations). In the case of $r=1$, one can easily find $$A^{(1)} = \partial_{\bar{i}} f D^{\bar{i}}, \label{A1}$$ where (\[D-dphi\]) is used. Let us observe that $a^{(r;0)} = a^{(r;1)}_{\bar{i}}= 0$ for $r\geq 2$ in the expressions (\[An-a\]), namely, $$A^{(r)} = \sum_{k\geq 2} a^{(r;k)}_{\bar{i}_1 \cdots \bar{i}_k} D^{\bar{i}_1} \cdots D^{\bar{i}_k}, ~~~~ r\geq 2. \label{Ar}$$ From the condition (\[Ar-rec1\]), $a^{(r;0)}=0 ~(r\geq 1)$ trivially obeys. We then define the twisted symbol of $A^{(r)}$ as $a^{(r)} (\xi) = \sum a^{(r;k)}_{\bar{i}_1 \cdots \bar{i}_n} \xi^{\bar{i}_1} \cdots \xi^{\bar{i}_n}$. The twisted symbol of the left hand side in (\[Ar-rec2\]) is $\partial a^{(r)}(\xi)/\partial \xi^{\bar{i}}$ from (\[D-dphi\]). That of the right hand side in (\[Ar-rec2\]) does not contain the zeroth order term of $\xi$, because of $a^{(r;0)}=0$ for $r\geq 1$. Therefore, $a^{(r)} ~(r\geq 2)$ does not contain the first order term of $\xi$. This prove the assertion. Here is a useful theorem given by Karabegov. The differential operator $L_f$ for an arbitrary function $f$ is obtained from the operator $L_{{\bar z}^i}$, which corresponds to the left $*$ multiplication of ${\bar z}^i$, $$\label{Lf_Lz} L_f = \sum_{\alpha} \frac{1}{\alpha !} \left(\frac{\partial}{\partial {\bar z}}\right)^\alpha f (L_{\bar z} - {\bar z})^\alpha ,$$ where $\alpha$ is a multi-index. Similarly, the differential operator $\displaystyle{R_f = \sum_{n=0}^\infty \hbar^n B^{(n)}}$ corresponding to the right $*$ multiplication by a function $f$ contains only partial derivatives by $\bar{z}^i$ and is determined by the conditions $$\begin{aligned} & R_f 1 = 1*f = f, \\ & [R_f, \partial_i \Phi + \hbar{\partial_i}] = 0.\end{aligned}$$ $B^{(n)}$ has the following form, $$\begin{aligned} B^{(0)} &= f, ~~~ B^{(1)} = \partial_i f D^i, ~~~ B^{(r)} = \sum_{k\geq 2} b^{(r;k)}_{i_1 \cdots i_k} D^{i_1} \cdots D^{i_k}, \label{Br}\end{aligned}$$ where $D^i = g^{i\bar{j}}\partial_{\bar{j}}$. The differential operator $R_f$ for an arbitrary function $f$ is obtained from the operator $R_{z^i}$, which corresponds to the right $*$ multiplication by $z^i$, $$\label{Rf_Rz} R_f = \sum_{\alpha} \frac{1}{\alpha !} \left(\frac{\partial}{\partial z}\right)^\alpha f (R_z - z)^\alpha.$$ Derivations in deformation quantization {#Deri_Defo} ---------------------------------------- A differential calculus on noncommutative spaces can be constructed based on the derivations of the algebra $C^\infty(M)[[\hbar]]$ with its star product, whose derivation ${\mathbf d}$ are linear operators satisfying the Leibniz rule, i.e. $ {\mathbf d} (f*g)= {\mathbf d} f * g + f * {\mathbf d}g$ . In commutative space, vector fields are obviously derivations. However first order differential operators in noncommutative space do not satisfy the Leibniz rule in general. In this subsection, we study inner derivations ${\cal L}$, in particular, let ${\cal L}$ be a linear differential operator such that ${\cal L} (f) =[P, f]_* := P*f - f*P,$ ($P, f \in C^\infty(M)[[\hbar]]$). Note that inner derivations are not first order differential operator, since the explicit expression of the star commutator $[P, f]_*$ includes higher derivative terms of $f$ for a generic $P$. In particular, inner derivations corresponding to vector fields play an important role, when we construct field theories on noncommutative spaces. It is known that such vector fields are given as the Killing vector fields [@Muller:2004]. In this section, we review the fact to obtain the differentials on noncommutative ${\mathbb C}P^N$ and ${\mathbb C}H^N$. Let $M$ be a Kähler manifold with the $*$ product with separation of variables given in section \[reviewKarabegov\]. Let $P \in C^{\infty}(M)[[ \hbar ]]$, $f$ be an arbitrary $C^{\infty}$ function on $M$ and $[ P , f ]=P*f - f*P $ i.e. the inner derivation of the $*$-product mentioned above. Then $[P, f]_* = i \hbar\{P, f\}$ if and only if $D^i D^j P =0$ and $D^{\bar{i}} D^{\bar{j}} P=0$ for all $i, j = 1, 2, \cdots, N$. Namely, higher derivative terms of $f$ in $[P, f]_*$ vanish and this inner derivation is given by some vector field when these conditions are satisfied. From the formulas (\[Lf\_Lz\]) and (\[Rf\_Rz\]), we find $$\begin{aligned} [P, f]_* &= R_f P - L_f P \nonumber \\ &= \sum_\alpha \left[ \frac{1}{\alpha !} \left(R_z - z\right)^\alpha P \cdot \left(\frac{\partial}{\partial z} \right)^\alpha -(L_{\bar{z}} - \bar{z})^\alpha P \cdot \left(\frac{\partial}{\partial \bar{z}} \right)^\alpha \right] f. \label{[P,f]}\end{aligned}$$ The differential operators $L_{\bar{z}^i}$ and $R_{z^i}$ have the following forms, $$\begin{aligned} L_{\bar{z}^i} &= \bar{z}^i + \sum_{n=1}^\infty \hbar^n A^{(n)}_{\bar{i}}, \\ R_{z^i} &= z^i + \sum_{n=1}^\infty \hbar^n B^{(n)}_i.\end{aligned}$$ From (\[A1\]), (\[Ar\]) and (\[Br\]), $A^{(n)}_{\bar{i}}$ and $B^{(n)}_i$ are given as $$\begin{aligned} A^{(1)}_{\bar{i}} &= D^{\bar{i}}, ~~~~ A^{(r)}_{\bar{i}} = \sum_{k\geq 2} a^{(r;k)}_{\bar{i}; \bar{j}_1 \cdots \bar{j}_k} D^{\bar{j}_1} \cdots D^{\bar{j}_k}, ~~~~ r\geq 2, \label{Air} \\ B^{(1)}_i &= D^i, ~~~~ B^{(r)}_i = \sum_{k\geq 2} b^{(r;k)}_{i; j_1 \cdots j_k} D^{j_1} \cdots D^{j_k}, ~~~~ r\geq 2. \label{Bir}\end{aligned}$$ The first order terms in $\hbar$ in the right hand side of (\[\[P,f\]\]) give the Poisson bracket $i\hbar\{P, f\}$. Looking at $\left(L_{\bar{z}^{i_1}} - \bar{z}^{i_1}\right) \cdots \left(L_{\bar{z}^{i_k}} - \bar{z}^{i_k}\right) P$ for $k \geq 2$ and $P=\sum_{n=0}^\infty \hbar^n P^{(n)}$, we have $$\begin{aligned} \left(L_{\bar{z}^{i_1}} - \bar{z}^{i_1}\right) \cdots \left(L_{\bar{z}^{i_k}} - \bar{z}^{i_k}\right) P &= \sum_{n=0}^\infty \sum_{m_1=1}^\infty \cdots \sum_{m_k=1}^\infty \hbar^{n+m_1+\cdots m_k} A^{(m_1)}_{\bar{i}_1} \cdots A^{(m_k)}_{\bar{i}_k} P^{(n)}. \label{LLP} \end{aligned}$$ Assuming $[P, f]=i\hbar\{P, f\}$, namely, assuming that the all terms in (\[LLP\]) vanish, we show $D^{\bar{i}} D^{\bar{j}} P=0$. The terms of the order $\hbar^2$ in (\[LLP\]) exists only for $k=2$ and has the following form, $$\begin{aligned} A_{\bar{i}_1}^{(1)}A_{\bar{i}_2}^{(1)} P^{(0)} = D^{\bar{i}_1} D^{\bar{i}_2}P^{(0)}.\end{aligned}$$ Hence, $D^{\bar{i}_1}D^{\bar{i}_2}P^{(0)}=0$, and we find $$\begin{aligned} \sum_{m_1=1}^\infty \cdots \sum_{m_k=1}^\infty \hbar^{n+m_1+\cdots m_k} A^{(m_1)}_{\bar{i}_1} \cdots A^{(m_k)}_{\bar{i}_k} P^{(0)} = 0,\end{aligned}$$ from the explicit forms of $A_{\bar{i}}^{(r)}$, (\[Air\]). As the induction assumption, we set $D^{\bar{i}}D^{\bar{j}}P^{(n)}=0$ for $n= 0, 1, \dots, r-1$. Similar to the case of $P^{(0)}$, the following equation holds for $n=0, 1, \dots, r-1$, $$\begin{aligned} \sum_{m_1=1}^\infty \cdots \sum_{m_k=1}^\infty \hbar^{n+m_1+\cdots m_k} A^{(m_1)}_{\bar{i}_1} \cdots A^{(m_k)}_{\bar{i}_k} P^{(n)} = 0,\end{aligned}$$ and the right hand side of (\[LLP\]) becomes $$\begin{aligned} \sum_{n=r}^\infty \sum_{m_1=1}^\infty \cdots \sum_{m_k=1}^\infty \hbar^{n+m_1+\cdots m_k} A^{(m_1)}_{\bar{i}_1} \cdots A^{(m_k)}_{\bar{i}_k} P^{(n)}.\end{aligned}$$ The term of the order ${\cal O}(\hbar^{r+2})$ in this sum exists only for $k=2$ and has the following form, $$\begin{aligned} A_{\bar{i}_1}^{(1)}A_{\bar{i}_2}^{(1)} P^{(r)} = D^{\bar{i}_1} D^{\bar{i}_2}P^{(r)}.\end{aligned}$$ Thus, $D^{\bar{i}_1}D^{\bar{i}_2}P^{(r)}=0$. Therefore, it is shown that $D^{\bar{i}}D^{\bar{j}}P=0$ holds for all $i, j$. Similarly, $D^i D^j P =0$ can be derived by considering $\left(R_z - z\right)^\alpha P$. The converse is easily shown from the above equations. Real valued functions which satisfy $D^i D^j P =0$ and $D^{\bar{i}}D^{\bar{j}} P=0$ on Kähler manifolds are known as Killing potentials [@Freedman:2012zz]. The Killing potential gives a holomorphic Killing vector $\zeta^i \partial_i + \zeta^{\bar{i}}\partial_{\bar{i}} = \{P, \cdot~\}$, $$\begin{aligned} \zeta^i &= -ig^{i\bar{j}}\partial_{\bar{j}}P = -iD^i P, \\ \zeta^{\bar{i}} &= ig^{\bar{i}j} \partial_j P = iD^{\bar{i}}P .\end{aligned}$$ $\zeta^i$ is holomorphic, and $\zeta^{\bar{i}}$ is anti-holomorphic. The metric and the complex structure of the Kähler manifold are invariant under the transformations generated by the holomorphic Killing vectors, $\delta_\zeta z^i = \zeta^i, ~\delta_\zeta \bar{z}^i = \zeta^{\bar{i}}$. Summarizing these facts, we have the following corollary [([@Muller:2004])]{} In deformation quantization defined in Section \[reviewKarabegov\], the inner derivations given as vector fields are the Killing vector fields ${\cal L}_a = \zeta_a^i\partial_i + \zeta_a^{\bar{i}} \partial_{\bar{i}}$ . Deformed gauge theory --------------------- In the previous section, we studied inner derivations given as vector fields on noncommutative Kähler manifolds. Using this, we investigate gauge theories with a gauge group $G$ on noncommutative homogeneous Kähler manifolds $M$ given by the deformation quantization in Section \[reviewKarabegov\].   ( From several view points, matrix models and its related topics studied in [@Kawai:2009vb; @Kawai:2010sf; @Kitazawa] are useful for understanding the gauge theory constructed in this subsection.)  In the following, we consider $U(n)$ gauge theories for simplicity. All results in this section can be applied for any matrix groups. At first, we introduce a noncommutative $U(n)$ transformations as a deformation of the unitary transformations. If $g \in U(n)$, then $g^\dagger g = I$, where $g^{\dagger}$ is the hermitian conjugate of $g$ and $I$ is the identity matrix. As a natural extension, we define $G:= C^{\infty} (M)[[ \hbar ]] \otimes GL(n ; {\mathbb C})$ such that for $\displaystyle U= \sum_{k=0}^{\infty} \hbar^k U^{(k)}$ and $\displaystyle U^\dagger = \sum_{k=0}^{\infty} \hbar^k U^{(k)\dagger} \in G$, $$\begin{aligned} \label{3_1} U^{\dagger} * U = \sum_{n=0}^\infty \hbar^n \sum_{m=0}^n U^{(m)\dagger} * U^{(n-m)} = I. \end{aligned}$$ This condition is imposed for each order of $\hbar$. For arbitrary $U^{(0)} \in C^\infty (M)\otimes U(n) $, (\[3\_1\]) has solutions which are determined recursively at each order of $\hbar$ [@maeda_sako]. In noncommutative Kähler manifolds, the partial derivative $\partial$ does not play an essential role, since the Leibniz rule is failed; $\partial (f * g) \neq \partial f * g + f * \partial g$. To construct a covariant derivative of a gauge theory, we should adopt some derivations (operators which satisfy the Leibniz rule) instead of $\partial$. In particular, inner derivations are given by commutators of the star product. The space of inner derivations is infinite dimensional. Hence, if the whole space of inner derivations is used to construct gauge theories, the infinite number of gauge fields would be introduced. (See for example [@DuboisViolette:1988ps; @DuboisViolette:1999cj] .) In this article, we consider deformation quantization of a homogeneous Kähler manifold ${\cal G}/{\cal H}$ and choose a subalgebra of the Lie algebra of inner derivations. Here, we assume that ${\cal G}$ is a connected semisimple Lie group so that ${\cal G}/{\cal H}$ has at least nondegenerate metric. Then, we construct a deformation quantization of gauge theories on ${\cal G}/{\cal H}$ whose covariant derivatives are derived from inner derivations corresponding to the Killing vector fields. In a homogeneous Kähler manifold ${\cal G}/{\cal H}$, there are the holomorphic Killing vector fields ${\cal L}_a = \zeta_a^i(z)\partial_i + \zeta_a^{\bar{i}}(\bar{z}) \partial_{\bar{i}}$ corresponding to the Lie algebra of the isometry group ${\cal G}$, $$[{\cal L}_a, {\cal L}_b] = if_{abc} {\cal L}_c,$$ where $a$ is an index of the Lie algebra of ${\cal G}$ and $f_{abc}$ is its structure constant. There exists the Killing potential $P_a$ corresponding to ${\cal L}_a$, $ {\cal L}_a = \{P_a, \cdot~\}$. As stated in the previous section, the Killing vector ${\cal L}_a$ can be described by $*$-commutator and satisfy the Leibniz rule, $$\begin{aligned} {\cal L}_a &= -\frac{i}{\hbar} [P_a, \cdot~]_*, \\ {\cal L}_a (f*g) &= ({\cal L}_a f) * g + f * ({\cal L}_a g).\end{aligned}$$ The Killing vectors are normalized here as $$\eta^{ab} \zeta_a^i \zeta_b^{\bar{j}} = g^{i\bar{j}}, ~~~~ \eta^{ab} \zeta_a^i \zeta_b^j = 0, ~~~~ \eta^{ab} \zeta_a^{\bar{i}} \zeta_b^{\bar{j}} = 0, \label{zeta-g}$$ where $\eta^{ab}$ is the inverse of the Killing form of the Lie algebra of ${\cal G}$. We introduce gauge fields corresponding to ${\cal L}_a$ in the following. Let us consider a commutative homogeneous Kähler manifold $M={\cal G}/{\cal H}$. We denote the indices of $TM$ as $\mu = 1, 2, \cdots, 2N$ for combining the holomorphic and anti-holomorphic indices. We define ${\cal A}_a^{(0)}$ as $$\begin{aligned} {\cal A}_a^{(0)} = \zeta^\mu_a A_\mu = \zeta^i_a A_i + \zeta^{\bar{i}}_a A_{\bar{i}},\end{aligned}$$ where $A_i$ and $A_{\bar{i}}$ are gauge fields on $M$. Its curvature is defined as $$\begin{aligned} {\cal F}_{ab}^{(0)}:= {\cal L}_a {\cal A}_b^{(0)}- {\cal L}_b {\cal A}_a^{(0)} -i [ {\cal A}_a^{(0)} , {\cal A}_b^{(0)} ] -i f_{abc}{\cal A}_c^{(0)} ,\end{aligned}$$ where $[ A , B ] = AB-BA$. ${\cal F}_{ab}^{(0)}$ is related to the curvature of $A_\mu$, $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\mu A_\nu -i [A_\mu, A_\nu]$, as $$\begin{aligned} {\cal F}_{ab}^{(0)} = \zeta_a^\mu \zeta_b^\nu F_{\mu\nu}.\end{aligned}$$ By using (\[zeta-g\]), it is shown that $$\begin{aligned} \label{taioukankei} \eta^{ac} \eta^{bd} {\cal F}_{ab}^{(0)} {\cal F}_{cd}^{(0)} = g^{\mu\rho} g^{\nu\sigma} F_{\mu\nu} F_{\rho\sigma}.\end{aligned}$$ Now, we consider a noncommutative deformation of gauge theories. We define $$\begin{aligned} {\cal A}_a := \sum_{k=0}^\infty \hbar^k {\cal A}_a^{(k)}\end{aligned}$$ as a gauge field, and define its gauge transformation by $$\begin{aligned} {\cal A}_a \rightarrow {\cal A}_a' = i U^{-1}* {\cal L}_a U +U^{-1}* {\cal A}_a * U . \label{Atrans}\end{aligned}$$ Let us define a curvature of ${\cal A}_a$ by $$\begin{aligned} {\cal F}_{ab} := {\cal L}_a {\cal A}_b - {\cal L}_b {\cal A}_a -i [ {\cal A}_a , {\cal A}_b ]_* -i f_{abc}{\cal A}_c . \end{aligned}$$ ${\cal F}_{ab}$ transforms covariantly: $$\begin{aligned} {\cal F}_{ab} \rightarrow {\cal F}_{ab}' = U^{-1} * {\cal F}_{ab} *U. \label{Ftrans}\end{aligned}$$ $$\begin{aligned} {\cal F}_{ab}' &= {\cal L}_a {\cal A}_b' - {\cal L}_b {\cal A}_a ' -i [ {\cal A}_a' , {\cal A}_b' ]_* -i f_{abc}{\cal A}_c'. \label{F'} \end{aligned}$$ Using (\[Atrans\]) and $$\begin{aligned} 0= {\cal L}_a (U^{-1} * U )= {\cal L}_a U^{-1} * U+ U^{-1} * {\cal L}_a U \end{aligned}$$ which is obtained from the Leibniz rule for ${\cal L}_a$, the right hand side of (\[F’\]) is written as $$\begin{aligned} U^{-1}*{\cal F}_{ab}*U + i U^{-1} * [{\cal L}_a, {\cal L}_b] U + f_{abc} U^{-1} * {\cal L}_c U. \end{aligned}$$ Noting that $[ {\cal L}_a , {\cal L}_b ]= i f_{abc} {\cal L}_c$, we have ${\cal F}_{ab}' = U^{-1} * {\cal F}_{ab} *U$. Using this lemma, we obtain the gauge invariant action. A gauge invariant action for the gauge field is given by $$\begin{aligned} S_g := \int_{{\cal G}/{\cal H}} \mu_g ~ {\rm tr} \left( \eta^{ac}\eta^{bd} {\cal F}_{ab} * {\cal F}_{cd} \right),\end{aligned}$$ where $\mu_g$ is a trace density. The gauge invariance of the action is obtained by (\[Ftrans\]) and the cyclic symmetry of the trace density. The existence of trace density, $\int_M f*g \mu_g = \int_M g*f \mu_g$, is guaranteed in [@Karabegov:1998hm]. Scalar fields are also introduced as similar to commutative case. As an example, let us consider a complex scalar field $\displaystyle \phi= \sum_k \phi^{(k)} \hbar^k$ and its hermitian conjugate $\displaystyle \phi^{\dagger}$ which transform as the fundamental representation of the gauge group, $$\begin{aligned} \label{gauge_trans_scalar} \phi \rightarrow \phi'= U^{-1} * \phi,~~~~ \phi^\dagger \rightarrow {\phi^{\dagger}}'= \phi^{\dagger} * U .\end{aligned}$$ A covariant derivative for this scalar field is defined by $$\begin{aligned} \nabla_a \phi := {\cal L}_a \phi - i {\cal A}_a * \phi, \label{covariavt scalar}\end{aligned}$$ and then this transforms covariantly; $$\begin{aligned} {\nabla_a}' \phi' = U^{-1}* {\nabla_a} \phi .\end{aligned}$$ Therefore we obtain the gauge invariant action. Let $\phi$ be a fundamental representation complex scalar field and $\phi^{\dagger}$ be a hermitian conjugate of $\phi$ whose gauge transformations are given by (\[gauge\_trans\_scalar\]). Then, the following action is gauge invariant. $$\begin{aligned} \label{scalar_action} S_{\phi} = \int_{{\cal G}/{\cal H}} \mu_g \left\{ \eta^{ab} \nabla_a \phi^{\dagger}* \nabla_b \phi + V(\phi^{\dagger} * \phi ) \right\} ,\end{aligned}$$ where $V$ is a potential as a function of one variable. Gauge theories in noncommutative ${\mathbb C}P^N$ and ${\mathbb C}H^N$ ======================================================================== In this section, as examples of the deformed gauge theories defined in the previous section, we will construct noncommutative gauge theories on ${\mathbb C}P^N$ and ${\mathbb C}H^N$ by using deformation quantization with separation variables. Deformation quantization with separation variables of ${\mathbb C}P^N$ and ${\mathbb C}H^N$ {#Defo_Qun_CPN_CHN} ------------------------------------------------------------------------------------------- We recall the results for the deformation quantization with separation of variables for ${\mathbb C}P^N$ and ${\mathbb C}H^N$ [@Sako:2012ws]. In the inhomogeneous coordinates $z^i ~(i=1, 2, \cdots, N)$, the Kähler potential of $\mathbb{C}P^N$ is given by $$\begin{aligned} \Phi = \ln \left(1+|z|^2\right), \label{phi}\end{aligned}$$ where $|z|^2 = \sum_{k=1}^N z^k \bar{z}^k$. The metric $(g_{i\bar{j}})$ is $$\begin{aligned} ds^2 &= 2g_{i\bar{j}}dz^id\bar{z}^j, \label{ds} \\ g_{i\bar{j}} &= \partial_i \partial_{\bar{j}} \Phi = \frac{(1+|z|^2)\delta_{ij}-z^j \bar{z}^i}{(1+|z|^2)^2}, \label{metric}\end{aligned}$$ and the inverse of the metric $(g^{\bar{i}j})$ is $$\begin{aligned} g^{\bar{i}j} = (1+|z|^2)\left(\delta_{ij}+z^j\bar{z}^i\right). \label{inverse}\end{aligned}$$ Recall that the left star multiplication for a function $f$, $L_f$, is written by using $L_{{\bar z}^l}$, (\[Lf\_Lz\]). The explicit expression for $L_{{\bar z}^l}$ on $\mathbb{C}P^N$ is given by $$\begin{aligned} L_{\bar{z}^l} &= \bar{z}^l + \hbar D^{\bar{l}} + \sum_{n=2}^\infty \hbar^n \sum_{m=2}^n a^{(n)}_m \partial_{\bar{j}_1}\Phi \cdots \partial_{\bar{j}_{m-1}}\Phi D^{\bar{j}_1} \cdots D^{\bar{j}_{m-1}} D^{\bar{l}} \nonumber \\ &= \bar{z}^l + \sum_{m=1}^\infty \alpha_m(\hbar) \partial_{\bar{j}_1}\Phi \cdots \partial_{\bar{j}_{m-1}}\Phi D^{\bar{j}_1} \cdots D^{\bar{j}_{m-1}} D^{\bar{l}}, \label{L_z}\end{aligned}$$ where $$\begin{aligned} & \alpha_m (t) = \sum_{n=m}^\infty t^n a^{(n)}_m, \label{f-def} \\ & \alpha_1 (t) = t, \\ & \alpha_m(t) = t^m \prod_{n=1}^{m-1}\frac{1}{1-nt} = \frac{\Gamma(1-m+1/t)}{\Gamma(1+1/t)}, \qquad (m\geq 2). \label{fm}\end{aligned}$$ The function $\alpha_m (t)$ actually coincides with the generating function for the Stirling numbers of the second kind $S(n,k)$, and $a^{(n)}_m$ is related to $S(n,k)$ as $$\begin{aligned} a^{(n)}_{m} = S(n-1, m-1). \label{2ndS}\end{aligned}$$ One of non-trivial star products is $\bar{z}^i*z^j$, $$\begin{aligned} \bar{z}^i*z^j &= \bar{z}^iz^j + \hbar \delta_{ij} (1+|z|^2) {}_2F_1\left(1, 1; 1-1/\hbar; -|z|^2\right) \nonumber \\ & ~~~~+\frac{\hbar}{1-\hbar} \bar{z}^i z^j (1+|z|^2) {}_2F_1 \left(1, 2; 2-1/\hbar; -|z|^2\right), \label{barz-z}\end{aligned}$$ where ${}_2F_1$ is the Gauss hypergeometric function. For $\mathbb{C}H^N$, similar results are obtained. The Kähler potential and the metric are given by $$\begin{aligned} \Phi =& - \ln \left(1-|z|^2\right), \\ g_{i\bar{j}} &= \partial_i \partial_{\bar{j}} \Phi = \frac{(1-|z|^2)\delta_{ij}+\bar{z}^i z^j}{(1-|z|^2)^2}, \\ g^{\bar{i}j} &= (1-|z|^2)\left(\delta_{ij}-\bar{z}^i z^j\right).\end{aligned}$$ The operator $L_{\bar{z}^l}$ is expanded as a power series of the noncommutative parameter $\hbar$, and has the following explicit representation, $$\begin{aligned} L_{\bar{z}^l} &= \bar{z}^l + \hbar D^{\bar{l}} + \sum_{n=2}^\infty \hbar^n \sum_{m=2}^n (-1)^{n-1} a^{(n)}_m \partial_{\bar{j}_1}\Phi \cdots \partial_{\bar{j}_{m-1}}\Phi D^{\bar{j}_1} \cdots D^{\bar{j}_{m-1}} D^{\bar{l}} \nonumber \\ &= \bar{z}^l + \sum_{m=1}^\infty (-1)^{m-1} \beta_m(\hbar) \partial_{\bar{j}_1}\Phi \cdots \partial_{\bar{j}_{m-1}}\Phi D^{\bar{j}_1} \cdots D^{\bar{j}_{m-1}} D^{\bar{l}}, \label{Lz-ch}\end{aligned}$$ where $$\begin{aligned} \beta_n(t)=(-1)^n \alpha_n(-t)= \frac{\Gamma(1/t)}{\Gamma(n+1/t)}. \label{beta-n}\end{aligned}$$ Then, one of non-trivial star products is $\bar{z}^i*z^j$, $$\begin{aligned} \bar{z}^i*z^j =& \bar{z}^iz^j + \hbar \delta_{ij} (1-|z|^2) {}_2F_1\left(1, 1; 1 + 1/\hbar; |z|^2 \right)\nonumber \\ & - \frac{\hbar}{1+\hbar} \bar{z}^i z^j (1-|z|^2) {}_2F_1 \left(1, 2; 2+1/\hbar; |z|^2\right).\end{aligned}$$ We should comment on the relation between our previous results and those of preceding related works [@Balachandran; @Bordemann; @Hayasaka:2002db]. Balachandran [*et al.*]{} gave an explicit expression of $*$ product on fuzzy ${\mathbb C}P^n$, using matrix regularization [@Balachandran]. Their $*$ product is expressed as a finite series. Though our $*$ product is, in general, an infinite series in $\hbar$, it coincides with Barachandran’s $*$ product if we take $\hbar = 1/L (L \in {\mathbb N})$. On the other hand, Bordemann [*et al.*]{} obtained a star product which has a similar form of an infinite series in the noncommutative parameter $\hbar$ to our star product [@Bordemann]. In fact, their star product is shown to be equivalent to ours (see [@Sako:2012ws] section 3). Also in [@Hayasaka:2002db], an explicit expression of a star product on fuzzy $S^2$ is given as an infinite series in a noncommutative parameter, which coincides with our expression in the case of ${\mathbb C}P^1$. Differentials on noncommutative $\mathbb{C}P^N$ {#KillingCP} ------------------------------------------------ In this section, we study differentials in a noncommutative $\mathbb{C}P^N$ with the star product with separation of variables. In $\mathbb{C}P^N$, the conditions $D^i D^j P =0$ and $D^{\bar{i}}D^{\bar{j}}P=0$ can be solved as $$\begin{aligned} P = \frac{\alpha_i z^i + \bar{\alpha}_i \bar{z}^i + \beta_{ij}\bar{z}^i z^j}{1+|z|^2}, \label{P-CP}\end{aligned}$$ where $\alpha_i$ and $\beta_{ij} = \bar{\beta}_{ji}$ are complex parameters and $|z|^2 = \sum_{i=1}^N z^i \bar{z}^i$. The number of the real parameters is $N^2+2N$ and these correspond to the $SU(N+1)$ isometry transformations of $\mathbb{C}P^N$. In the following, we give concrete expressions of the Killing potentials corresponding to the generators of $su(N+1)$, the Lie algebra of $SU(N+1)$. Homogeneous coordinates of $\mathbb{C}P^N$ $$\begin{aligned} \left\{\xi^A | A = 0, 1, \cdots, N \right\} & = \left\{ \xi^0, \xi^i | i= 1, 2, \dots, N \right\}\end{aligned}$$ are related with inhomogeneous coordinates on the chart of $\xi^0 \neq 0$: $$\begin{aligned} z^i &= \frac{\xi^i}{\xi^0}, \qquad \bar{z}^i = \frac{\bar{\xi}^i}{\bar{\xi}^0}, \qquad (i= 1, 2, \dots, N).\end{aligned}$$ Since Kähler potential is given by $ \Phi = \ln (1+|z|^2) $, the isometry of $SU(N+1)$ with the homogeneous coordinates is given by $$\begin{aligned} \delta \xi^A &= i \theta^a (T_a)_{AB} \xi^B, \qquad \delta \bar{\xi}^A = -i \theta^a \bar{\xi}^B (T_a)_{BA}, \end{aligned}$$ where $ \theta^a$ are real parameters, and its Lie derivative is given by $$\begin{aligned} & {\cal L}_a = -\left(T_a\right)_{AB} \left( \xi^B \frac{\partial}{\partial \xi^A} -\bar{\xi}^A \frac{\partial}{\partial \bar{\xi}^B} \right), \\ & \left[ {\cal L}_a, {\cal L}_b \right] = if_{abc} {\cal L}_c.\end{aligned}$$ Here we introduce the generators $(T_a)_{AB}$ of $su(N+1)$ in the fundamental representation which satisfy the following relations, $$\begin{aligned} [T_a, T_b] &= if_{abc} T_c, \qquad {\rm Tr}~ T_a = 0, \\ {\rm Tr}~ T_a T_b &= \delta_{ab}, \\ (T_a)_{AB} (T_a)_{CD} &= \delta_{AD}\delta_{BC} -\frac{1}{N+1} \delta_{AB}\delta_{CD},\end{aligned}$$ where $f_{abc}$ is the structure constant of $SU(N+1)$, $a=1, 2, \dots, N^2+2N$, and $A, B=0, 1, \dots, N$. Generators of the isometry $SU(N+1)$ in the inhomogeneous coordinates are given as $$\begin{aligned} {\cal L}_a = \zeta_a^i \partial_i + \zeta_a^{\bar{i}} \partial_{\bar{i}} &= (T_a)_{00} \left( z^i \partial_i - \bar{z}^i \partial_{\bar{i}} \right) + (T_a)_{0i} \left( z^iz^j\partial_j + \partial_{\bar{i}} \right) \nonumber \\ & ~~~~+ (T_a)_{i0} \left( - \partial_i - \bar{z}^i \bar{z}^j \partial_{\bar{j}} \right) +(T_a)_{ij} \left( - z^j \partial_i + \bar{z}^i \partial_{\bar{j}} \right),\end{aligned}$$ and $$\begin{aligned} \zeta^i_a &:= (T_a)_{00} z^i + (T_a)_{0j}z^jz^i - (T_a)_{i0} - (T_a)_{ij}z^j, \\ \zeta^{\bar{i}}_a &:= -(T_a)_{00} \bar{z}^i + (T_a)_{0i} - (T_a)_{j0}\bar{z}^j\bar{z}^i + (T_a)_{ji}\bar{z}^j.\end{aligned}$$ The quadratic forms of $\zeta^i_a$ and $\zeta^{\bar{i}}_a$ become the metric, $$\begin{aligned} \zeta^i_a \zeta^{\bar{j}}_a &= -(1+|z|^2)(\delta_{ij}+z^i \bar{z}^j) = -g^{i\bar{j}}, \\ \zeta^i_a \zeta^j_a &= 0, \qquad \zeta^{\bar{i}}_a \zeta^{\bar{j}}_a = 0.\end{aligned}$$ As we saw in section 2, the Killing vector fields can be represented by star commutators with the Killing potentials. In the case of ${\mathbb C}P^N$, using the concrete expressions of the star product in section 3.1, ${\cal L}_a$ can be written as $$\begin{aligned} {\cal L}_a f = -\frac{i}{\hbar} [P_a, f]_* .\end{aligned}$$ $P_a$ are obtained as $$\begin{aligned} P_a &= -i(T_a)_{AB} \left( \frac{\bar{\xi}^A \xi^B}{|\xi|^2} - \delta_{AB} \right) \nonumber \\ &= i(T_a)_{00}\left( z^i \partial_i \Phi - 1 \right) - i (T_a)_{0i} \partial_{\bar{i}}\Phi -i (T_a)_{i0} \partial_i \Phi - i (T_a)_{ij} z^j \partial_i \Phi.\end{aligned}$$ Note that $P_a$ is determined up to an additional constant. The Killing potentials $P_a$ give a representation of the $su(N+1)$ under the star commutator, $$[P_a, P_b]_* = -\hbar f_{abc} P_c,$$ and the bilinear of $P_a$ becomes a constant, $$P_a * P_a = -N \left(\frac{1}{N+1} + \hbar \right).$$ The Killing potential $P$ in (\[P-CP\]) can be written in a linear combination of $P_a$. The star commutators between $P_a$ and a function $f$ become the Lie derivative ${\cal L}_a f$ of $f$ corresponding to the generator $T_a$, $$\begin{aligned} -\frac{i}{\hbar} [P_a, f]_* &= {\cal L}_a f \nonumber \\ &= \left[ (T_a)_{00} \left( z^i \partial_i - \bar{z}^i \partial_{\bar{i}} \right) + (T_a)_{0i} \left( z^iz^j\partial_j + \partial_{\bar{i}} \right) \right. \nonumber \\ & \left. ~~~~ + (T_a)_{i0} \left( - \partial_i - \bar{z}^i \bar{z}^j \partial_{\bar{j}} \right) +(T_a)_{ij} \left( - z^j \partial_i + \bar{z}^i \partial_{\bar{j}} \right) \right] f.\end{aligned}$$ As emphasized before, since the expression of the star product has the coordinate dependence, general vector fields do not satisfy the Leibniz rule. However, the Leibniz rule trivially holds for the Killing vector fields, because they are described as the star commutators, $$\begin{aligned} {\cal L}_a (f*g) = -\frac{i}{\hbar} [P_a, f*g]_* = -\frac{i}{\hbar} [P_a, f]_* *g - \frac{i}{\hbar} f*[P_a, g] = ({\cal L}_a f)*g + f*({\cal L}_a g).\end{aligned}$$ Differentials on noncommutative $\mathbb{C}H^N$ ----------------------------------------------- As similar to the $\mathbb{C}P^N$, we give explicit expressions of inner derivations given by the Killing potential for $\mathbb{C}H^N$. The Killing potential satisfying $D^i D^j P =0$ and $D^{\bar{i}}D^{\bar{j}}P=0$ can be solved as $$\begin{aligned} P = \frac{\alpha_i z^i + \bar{\alpha}_i \bar{z}^i + \beta_{ij}\bar{z}^i z^j}{1-|z|^2}, \label{P-CH}\end{aligned}$$ where $\alpha_i$ and $\beta_{ij} = \bar{\beta}_{ji}$ are complex parameters. In the following, we construct inner derivations corresponding to the isometry transformations. We first summarize useful facts in the isometry of $\mathbb{C}H^N$. As homogeneous coordinates of $\mathbb{C}H^N$ we denote $$\begin{aligned} \left\{\zeta^A | A = 0, 1, \cdots, N \right\} & = \left\{ \zeta^0, \zeta^i | i= 1, 2, \cdots, N \right\},\end{aligned}$$ and their relation between with inhomogeneous coordinates on the chart $\zeta^0 \neq 0$ are given by $$\begin{aligned} z^i &= \frac{\zeta^i}{\zeta^0}, \qquad \bar{z}^i = \frac{\bar{\zeta}^i}{\bar{\zeta}^0}, \qquad (i= 1, 2, \cdots, N).\end{aligned}$$ Since the Kähler potential is given by $\Phi = -\ln (1-|z|^2)$, there is an $SU(1, N)$ isometry. Let us summarize the notations of $SU(1, N)$. $SU(1, N)$ transformations preserve $$|\xi|^2 = \eta_{AB} \bar{\xi}^A \xi^B,$$ where the metric is defined by $(\eta_{AB}) = diag.(1, \overbrace{-1, \cdots, -1}^N)$. In other words, $SU(1, N)$ is defined as $$U \in SU(1, N) ~\Longleftrightarrow~ U^\dagger \eta U = \eta, ~~ \det U =1.$$ The Lie algebra $su(1, N)$ is defined by $$A \in su(1, N) ~\Longleftrightarrow~ U = e^A \in SU(1, N) ~\Longleftrightarrow~ \eta A^\dagger \eta = -A, ~~ {\rm Tr} A =0.$$ As a basis, we choose $(N+1)\times(N+1)$ matrices $T_a ~(a=1, 2, \dots, N^2+2N)$ which satisfy the following relations, $$\begin{aligned} & {\rm Tr} T_a = 0, \\ & \left(T_a^\dagger\right)_{00} = -\left(T_a\right)_{00},~~ \left(T_a^\dagger\right)_{ij} = -\left(T_a\right)_{ij}, \nonumber \\ & \left(T_a^\dagger\right)_{0i} = \left(T_a\right)_{0i},~~ \left(T_a^\dagger\right)_{i0} = \left(T_a\right)_{i0}, \\ & {\rm Tr} T_a T_b = h_{ab}, \qquad (h_{ab}) = diag.(\overbrace{-1, \cdots, -1}^{N^2}, \overbrace{1, \cdots, 1}^{2N}), \\ & T_a^\dagger = h_{ab} T_b, \\ & [T_a, T_b] = f_{abc} T_c, \qquad (f_{abc} \in \mathbb{R}), \\ & h_{ab} (T_a)_{AB}(T_b)_{CD} = \delta_{AD}\delta_{BC} - \frac{1}{N+1} \delta_{AB}\delta_{CD}. \end{aligned}$$ More explicit form of a basis is given in the appendix \[su(1,N)\]. Using these notations, transformations and generators of the isometry $SU(1, N)$ in the homogeneous coordinates are obtained as $$\begin{aligned} \delta \xi^A &= \theta^a (T_a)_{AB} \xi^B, \qquad \delta \bar{\xi}^A = \theta^a \bar{\xi}^B (T_a^\dagger)_{BA}, \\ {\cal L}_a &= -\left(T_a\right)_{AB} \xi^B \frac{\partial}{\partial \xi^A} -\left(T_a^\dagger\right)_{AB} \bar{\xi}^A \frac{\partial}{\partial \bar{\xi}^B}, \\ \left[ {\cal L}_a, {\cal L}_b \right] &= f_{abc} {\cal L}_c.\end{aligned}$$ The generators of the isometry $SU(1, N)$ in the inhomogeneous coordinates are $$\begin{aligned} {\cal L}_a = \zeta_a^i \partial_i + \zeta_a^{\bar{i}} \partial_{\bar{i}} &= (T_a)_{00} \left( z^i \partial_i - \bar{z}^i \partial_{\bar{i}} \right) + (T_a)_{0i} \left( z^iz^j\partial_j -\partial_{\bar{i}} \right) \nonumber \\ & ~~~~+ (T_a)_{i0} \left( - \partial_i + \bar{z}^i \bar{z}^j \partial_{\bar{j}} \right) +(T_a)_{ij} \left( - z^j \partial_i + \bar{z}^i \partial_{\bar{j}} \right),\end{aligned}$$ and $$\begin{aligned} \zeta^i_a &:= (T_a)_{00} z^i + (T_a)_{0j}z^jz^i - (T_a)_{i0} - (T_a)_{ij}z^j, \\ \zeta^{\bar{i}}_a &:= -(T_a)_{00} \bar{z}^i - (T_a)_{0i} + (T_a)_{j0}\bar{z}^j\bar{z}^i + (T_a)_{ji}\bar{z}^j.\end{aligned}$$ The quadratic forms of $\zeta^i_a$ and $\zeta^{\bar{i}}_a$ become the metric, $$\begin{aligned} \zeta^i_a \zeta^{\bar{j}}_b h_{ab} &= (1-|z|^2)(\delta_{ij} - z^i \bar{z}^j) = g^{i\bar{j}}, \\ \zeta^i_a \zeta^j_b h_{ab} &= 0, \qquad \zeta^{\bar{i}}_a \zeta^{\bar{j}}_b h_{ab} = 0.\end{aligned}$$ As we found in general case (or similar to the case of ${\mathbb C}P^N$), the Killing vector fields are written by commutators of the Killing potentials, $$\begin{aligned} {\cal L}_a f = -\frac{i}{\hbar} [P_a, f]_*,\end{aligned}$$ and the $P_a$ are given by $$\begin{aligned} P_a =& i(T_a)_{AB} \left( \frac{\eta_{AC}\bar{\xi}^C \xi^B}{|\xi|^2} - \delta_{AB} \right) \nonumber \\ =& i(T_a)_{00}\left( z^i \partial_i \Phi +1 \right) + i (T_a)_{0i} \partial_{\bar{i}}\Phi -i (T_a)_{i0} \partial_i \Phi - i (T_a)_{ij} z^j \partial_i \Phi.\end{aligned}$$ Note that $P_a$ is determined up to an additional constant. The following formula is also obtained as similar to ${\mathbb C}P^N$: $$\begin{aligned} P_a * P_b~ h_{ab} =- N \left( \frac{N}{N+1}- \hbar \right).\end{aligned}$$ Cyclic property of integration and actions of gauge theories ------------------------------------------------------------ In this section, we first show the cyclic property of integration, explicitly. \[cycle\] Let $M$ be $\mathbb{C}P^N$ or $\mathbb{C}H^N$, and let $F$ and $G$ be arbitrary compact supported bounded smooth functions on $M$. Then, the Riemannian volume form is a trace density with respect to the star products with separation of variables, namely we have $$\begin{aligned} \label{CP_CH_cycle} \int_M F*G \sqrt{g} dz^1 \cdots dz^{N} d\bar{z}^1 \cdots d\bar{z}^N =\int_M G*F \sqrt{g} dz^1 \cdots dz^{N} d\bar{z}^1 \cdots d\bar{z}^N .\end{aligned}$$ Note that the star products can be written by using the Levi-Civita connection $\nabla_i$ and $\nabla_{\bar{i}}$ as $$\begin{aligned} F*G &= FG + \sum_{n=1}^\infty c_n(\hbar) g^{\bar{i}_1 j_1} \cdots g^{\bar{i}_n j_n} \left(\nabla_{\bar{i}_1} \cdots \nabla_{\bar{i}_n} F \right) \left(\nabla_{j_1} \cdots \nabla_{j_n} G \right) ,\end{aligned}$$ where $ c_n (\hbar) = {\alpha_n (\hbar)} / {n!} $ for ${\mathbb C}P^N$ and $c_n(\hbar) = \beta_n (\hbar)/n!$ for $\mathbb{C}H^N$ (see [@Sako:2012ws]).  ( Do not confuse the Levi-Civita connection $\nabla_i$ with the gauge covariant derivative (\[covariavt scalar\]).)  We use the following relations which hold for the Levi-Civita connections and the Riemannian curvature tensor on $\mathbb{C}P^N$ and $\mathbb{C}H^N$ ([@Kobayashi_Nomizu] p169): $$\begin{aligned} & [\nabla_i, \nabla_j] = 0, ~~~~ [\nabla_{\bar{i}}, \nabla_{\bar{j}}] =0, \\ & [\nabla_i, \nabla_{\bar{j}}] v_k = {R_{i\bar{j}k}}^l v_l, ~~~~ [\nabla_i, \nabla_{\bar{j}}] v_{\bar{k}} = {R_{i\bar{j}\bar{k}}}^{\bar{l}} v_{\bar{l}}, \\ & {R_{i\bar{j}k}}^l = -c(\delta_{kl}g_{i\bar{j}} +\delta_{il}g_{k\bar{j}}), ~~~~ {R_{\bar{i} j \bar{k}}}^{\bar{l}} = -c(\delta_{kl}g_{j\bar{i}} +\delta_{il}g_{j\bar{k}}), \\ & \nabla_{m} {R_{i\bar{j}k}}^l = \nabla_{\bar{m}}{R_{i\bar{j}k}}^l = \nabla_m {R_{\bar{i} j \bar{k}}}^{\bar{l}} = \nabla_{\bar{m}} {R_{\bar{i} j \bar{k}}}^{\bar{l}} = 0.\end{aligned}$$ Here $c=1$ and $c=-1$ are for $\mathbb{C}P^N$ and $\mathbb{C}H^N$, respectively. To prove the theorem \[cycle\], we use the following lemma. \[lemma\] For the arbitrary $C^{\infty} $ function $G$ on $M$, $$\begin{aligned} \nabla_{\bar{i}_1} \cdots \nabla_{\bar{i}_n} \nabla_{j_1} \cdots \nabla_{j_n} G = \nabla_{j_1} \cdots \nabla_{j_n} \nabla_{\bar{i}_1} \cdots \nabla_{\bar{i}_n} G. \label{nablaG_lem}\end{aligned}$$ The proof of this lemma is given in the appendix [\[nabla\]]{}. Theorem \[cycle\] can be shown easily by using this lemma. $$\begin{aligned} \int d\mu F*G &= \int d\mu \left[ FG + \sum_{n=1}^\infty c_n(\hbar) g^{\bar{i}_1 j_1} \cdots g^{\bar{i}_n j_n} \left(\nabla_{\bar{i}_1} \cdots \nabla_{\bar{i}_n} F \right) \left(\nabla_{j_1} \cdots \nabla_{j_n} G \right) \right] \nonumber \\ &= \int d\mu \left[ GF + \sum_{n=1}^\infty (-1)^n c_n(\hbar) g^{\bar{i}_1 j_1} \cdots g^{\bar{i}_n j_n} F \left(\nabla_{\bar{i}_1} \cdots \nabla_{\bar{i}_n} \nabla_{j_1} \cdots \nabla_{j_n} G \right) \right] \nonumber \\ &= \int d\mu \left[ GF + \sum_{n=1}^\infty (-1)^n c_n(\hbar) g^{\bar{i}_1 j_1} \cdots g^{\bar{i}_n j_n} F \left( \nabla_{j_1} \cdots \nabla_{j_n} \nabla_{\bar{i}_1} \cdots \nabla_{\bar{i}_n} G \right) \right] \nonumber \\ &= \int d\mu \left[ GF + \sum_{n=1}^\infty c_n(\hbar) g^{\bar{i}_1 j_1} \cdots g^{\bar{i}_n j_n} \left( \nabla_{\bar{i}_1} \cdots \nabla_{\bar{i}_n} G \right) \left(\nabla_{j_1} \cdots \nabla_{j_n} F\right) \right] \nonumber \\ &= \int d\mu G*F\end{aligned}$$ where $d\mu$ is the volume form on $\mathbb{C}P^N$ or $\mathbb{C}H^N$ written in (\[CP\_CH\_cycle\]). This result is possible to be extended to functions of formal power series of bounded smooth functions with compact supports. In section 2, we constructed a gauge theory on general noncommutative homogeneous Kähler manifolds. In particular, we consider gauge theory on the noncommutative ${\mathbb C}P^N \approx {\rm SU(N+1)}/{\rm S(U(1)\times U(N))} $ and ${\mathbb C}H^N \approx {\rm SU(1,N)}/{\rm S(U(1)\times U(N))} $ with separation of variables. In the previous section, the derivations for functions on noncommutative Kähler manifolds with isometry, and concrete expressions of the derivations for ${\mathbb C}P^N$ and ${\mathbb C}H^N$ are constructed. Using them, a gauge theory with gauge group $G$ on the coset space is constructed. In addition, trace density is given by usual volume density as we see in this section. Then the action for the gauge fields is given by $$\begin{aligned} S_g := \int_{{\mathbb C}P^N} \sqrt{g}dz^1 \cdots dz^N d\bar{z}^1 \cdots d \bar{z}^N ~ {\rm tr} \left( {\cal F}_{ab} * {\cal F}_{cd} \eta^{ac}\eta^{bd} \right) ,\end{aligned}$$ where ${\rm tr}$ is trace for gauge group $G$. The gauge invariance of the action is guaranteed by (\[Ftrans\]) and the cyclic symmetry. The action for the scalar field are same as (\[scalar\_action\]); $$\begin{aligned} \label{scalar_actionCP} S_{\phi} = \int_{M} \sqrt{g}dz^1 \cdots dz^N d\bar{z}^1 \cdots d \bar{z}^N \{ \nabla_a \phi^{\dagger}* \nabla_b \phi \eta^{ab} + V(\phi^{\dagger} * \phi )\} .\end{aligned}$$ Conclusions =========== We focused on a gauge theory which has derivations given by order one differential operators, by only considering inner derivations which possess vector fields expressions on general homogeneous Kähler manifolds. As examples, we constructed explicit expressions for these inner derivations on ${\mathbb C}P^N$ and ${\mathbb C}H^N$. For our deformation quantization, we directly proved that integrations of $*$-products of functions with the volume form of the Kähler metric of ${\mathbb C}P^N$ and ${\mathbb C}H^N$ have a cyclic property. We then constructed an action functional having gauge symmetry on these manifolds. We note that the action functionals given in this article are gauge invariants not only for noncommutative homogeneous Kähler manifolds but also for the isometry groups of general noncommutative Kähler manifolds. In this sense, gauge theories on general noncommutative Kähler manifolds are constructed in this article. However, the relation between the usual action of gauge fields (\[taioukankei\]) and the normalization (\[zeta-g\]) is obtained only for noncommutative homogeneous Kähler manifolds. In other words, the correspondence between the gauge theories on a commutative space and the noncommutative space is clear, and it is possible to interpret the noncommutative gauge theory as a deformation of the commutative gauge theory for noncommutative homogeneous Kähler manifolds.\ [**Acknowledgments**]{}\ Y.M. was supported in part by JSPS KAKENHI No.23340018 and No.22654011, and A.S. was supported in part by JSPS KAKENHI No.23540117. We thank the referee for pointing out the relation between section 2.2 and [@Muller:2004]. The proof of the lemma \[lemma\] ================================ \[nabla\] We give the proof of the lemma \[lemma\], $$\begin{aligned} \nabla_{\bar{i}_1} \cdots \nabla_{\bar{i}_n} \nabla_{j_1} \cdots \nabla_{j_n} G = \nabla_{j_1} \cdots \nabla_{j_n} \nabla_{\bar{i}_1} \cdots \nabla_{\bar{i}_n} G. \label{nablaG}\end{aligned}$$ When $n=1$, trivially $$\begin{aligned} \nabla_{\bar{i}} \nabla_j G = \nabla_j \nabla_{\bar{i}}G.\end{aligned}$$ Assume $ \nabla_{\bar{i}_1} \cdots \nabla_{\bar{i}_{n-1}} \nabla_{j_1} \cdots \nabla_{j_{n-1}} G = \nabla_{j_1} \cdots \nabla_{j_{n-1}} \nabla_{\bar{i}_1} \cdots \nabla_{\bar{i}_{n-1}} G.$ We here use the following notation for simplicity, $$\begin{aligned} [k, l] \equiv g_{\bar{i}_k j_l} \nabla_{\bar{i}_1} \cdots \hat{\nabla}_{\bar{i}_k} \cdots \nabla_{\bar{i}_n} \nabla_{j_1} \cdots \hat{\nabla}_{j_l} \cdots \nabla_{j_n} G,\end{aligned}$$ where “$\hat{A}$” means $A$ is removed. Then, $$\begin{aligned} \nabla_{\bar{i}_1} \cdots \nabla_{\bar{i}_n} \nabla_{j_1} \cdots \nabla_{j_n} G =& \nabla_{\bar{i}_1} \nabla_{j_1} \left( \nabla_{\bar{i}_2} \cdots \nabla_{\bar{i}_n} \nabla_{j_2} \cdots \nabla_{j_n} G \right) \nonumber \\ & + \sum_{k=2}^n \nabla_{\bar{i}_1} \cdots \nabla_{\bar{i}_{k-1}} [\nabla_{\bar{i}_k}, \nabla_{j_1}] \nabla_{\bar{i}_{k+1}} \cdots \nabla_{\bar{i}_n} \nabla_{j_2} \cdots \nabla_{j_n} G \nonumber \\ =& \nabla_{\bar{i}_1} \nabla_{j_1} \left( \nabla_{j_2} \cdots \nabla_{j_n} \nabla_{\bar{i}_2} \cdots \nabla_{\bar{i}_n} \right)G \nonumber \\ & + \sum_{k=2}^n \nabla_{\bar{i}_1} \cdots \nabla_{\bar{i}_{k-1}} \left[ \sum_{l=k+1}^n {R_{\bar{i}_k j_1 \bar{i}_l}}^{\bar{p}} \nabla_{\bar{i}_{k+1}} \cdots \mathop{\nabla_{\bar{p}}}^{(l)} \cdots \nabla_{\bar{i}_n} \nabla_{j_2} \cdots \nabla_{j_n} G \right. \nonumber \\ & \left. ~~~~ + \sum_{l=2}^n {R_{\bar{i}_k j_1 j_l}}^p \nabla_{\bar{i}_{k+1}} \cdots \nabla_{\bar{i}_n} \nabla_{j_2} \cdots \mathop{\nabla_p}^{(l)} \cdots \nabla_{j_n} G \right] \nonumber \\ =& \nabla_{\bar{i}_1} \nabla_{j_1} \left( \nabla_{j_2} \cdots \nabla_{j_n} \nabla_{\bar{i}_2} \cdots \nabla_{\bar{i}_n} \right)G \nonumber \\ & + c \sum_{k=2}^n \left[ - \sum_{l=k+1}^n \left([k, 1] + [l, 1]\right) + \sum_{l=2}^n \left([k, 1] + [k, l]\right) \right] \nonumber \\ =& \nabla_{\bar{i}_1} \nabla_{j_1} \left( \nabla_{j_2} \cdots \nabla_{j_n} \nabla_{\bar{i}_2} \cdots \nabla_{\bar{i}_n} \right) G \nonumber \\ & + c \sum_{k=2}^n (k-1) [k, 1] - c \sum_{k=2}^{n-1} \sum_{l=k+1}^n [l, 1] + c \sum_{k=2}^n \sum_{l=2}^n [k, l] \nonumber \\ =& \nabla_{\bar{i}_1} \nabla_{j_1} \left( \nabla_{j_2} \cdots \nabla_{j_n} \nabla_{\bar{i}_2} \cdots \nabla_{\bar{i}_n} \right)G \nonumber \\ & + c \sum_{k=2}^n (k-1) [k, 1] - c \sum_{l=3}^n \sum_{k=2}^{l-1} [l, 1] + c \sum_{k=2}^n \sum_{l=2}^n [k, l] \nonumber \\ =& \nabla_{\bar{i}_1} \nabla_{j_1} \left( \nabla_{j_2} \cdots \nabla_{j_n} \nabla_{\bar{i}_2} \cdots \nabla_{\bar{i}_n} \right)G \nonumber \\ & + c \sum_{k=2}^n (k-1) [k, 1] - c \sum_{l=3}^n (l-2) [l, 1] + c \sum_{k=2}^n \sum_{l=2}^n [k, l] \nonumber \\ =& \nabla_{\bar{i}_1} \nabla_{j_1} \left( \nabla_{j_2} \cdots \nabla_{j_n} \nabla_{\bar{i}_2} \cdots \nabla_{\bar{i}_n} \right)G + c \sum_{k=2}^n [k, 1] + c \sum_{k=2}^n \sum_{l=2}^n [k, l].\end{aligned}$$ Next, the first term in the last expression, $\nabla_{\bar{i}_1} \nabla_{j_1} \left( \nabla_{j_2} \cdots \nabla_{j_n} \nabla_{\bar{i}_2} \cdots \nabla_{\bar{i}_n} \right)G$ becomes $$\begin{aligned} \nabla_{\bar{i}_1} \nabla_{j_1} \nabla_{j_2} \cdots \nabla_{j_n} \nabla_{\bar{i}_2} \cdots \nabla_{\bar{i}_n} G =& \nabla_{j_1} \cdots \nabla_{j_n} \nabla_{\bar{i}_1} \cdots \nabla_{\bar{i}_n}G \nonumber \\ & + \sum_{k=1}^n \nabla_{j_1} \cdots \nabla_{j_{k-1}} \left[\nabla_{\bar{i}_1}, \nabla_{j_k}\right] \nabla_{j_{k+1}} \cdots \nabla_{j_n} \nabla_{\bar{i}_2} \cdots \nabla_{\bar{i}_n} G \nonumber \\ =& \nabla_{j_1} \cdots \nabla_{j_n} \nabla_{\bar{i}_1} \cdots \nabla_{\bar{i}_n}G \nonumber \\ & + \sum_{k=1}^n \nabla_{j_1} \cdots \nabla_{j_{k-1}} \left[ \sum_{l=k+1}^n {R_{\bar{i}_1 j_k j_l}}^p \nabla_{j_{k+1}} \cdots \mathop{\nabla_p}^{(l)} \cdots \nabla_{j_n} \nabla_{\bar{i}_2} \cdots \nabla_{\bar{i}_n} G \right. \nonumber \\ & \left. ~~~~~ + \sum_{l=2}^n {R_{\bar{i}_1 j_k \bar{i}_l}}^{\bar{p}} \nabla_{j_{k+1}} \cdots \nabla_{j_n} \nabla_{\bar{i}_2} \cdots \mathop{\nabla_{\bar{p}}}^{(l)} \cdots \nabla_{\bar{i}_n} G \right] \nonumber \\ =& \nabla_{j_1} \cdots \nabla_{j_n} \nabla_{\bar{i}_1} \cdots \nabla_{\bar{i}_n}G \nonumber \\ & + c \sum_{k=1}^n \left[ \sum_{l=k+1}^n \left([1, k] + [1, l]\right) - \sum_{l=2}^n \left([1, k] + [l, k]\right) \right] \nonumber \\ =& \nabla_{j_1} \cdots \nabla_{j_n} \nabla_{\bar{i}_1} \cdots \nabla_{\bar{i}_n}G \nonumber \\ & - c \sum_{k=2}^n (k-1) [1, k] + c \sum_{k=1}^n \sum_{l=k+1}^n [1, l] - c \sum_{k=1}^n \sum_{l=2}^n [l, k] \nonumber \\ =& \nabla_{j_1} \cdots \nabla_{j_n} \nabla_{\bar{i}_1} \cdots \nabla_{\bar{i}_n}G - c \sum_{l=2}^n [l, 1] - c \sum_{k=2}^n \sum_{l=2}^n [l, k].\end{aligned}$$ This completes the proof for the lemma. A basis of $su(1, N)$ {#su(1,N)} ===================== A concrete basis of $su(1, N)$, $T_a, (a = 1, 2, \cdots, (N+1)^2 -1)$ is given as follows; $$\begin{aligned} \{T_a\} = \{I_{ij},~ J_{ij},~ H_k, I_{0i},~ J_{0i}\},\end{aligned}$$ where $i, j, k = 1, 2, \cdots, N$ and $i<j$ in $I_{ij}, J_{ij}$. $$\begin{aligned} I_{ij} &= \frac{1}{\sqrt{2}} (E_{ij} - E_{ji}), \\ J_{ij} &= \frac{i}{\sqrt{2}} (E_{ij} + E_{ji}), \\ H_k &= \frac{i}{\sqrt{k(k+1)}} \left( \sum_{i=1}^k E_{ii} - k E_{k+1, k+1} \right), ~~~~ (E_{N+1, N+1} = E_{00}), \\ I_{0i} &= \frac{1}{\sqrt{2}}(E_{i0} + E_{0i}), \\ J_{0i} &= \frac{i}{\sqrt{2}}(E_{i0} - E_{0i}),\end{aligned}$$ where $(E_{AB})_{CD} = \delta_{AC} \delta_{BD}$ and $A, B, C, D = 0, 1, \dots N$. $I_{ij}, J_{ij}, H_k$ are anti-hermitian and $I_{0i}, J_{0i}$ are hermitian. 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--- abstract: 'Using Monte Carlo simulation, we study the influence of geometric confinement on demixing for a series of symmetric non-additive hard spheres mixtures confined in slit pores. We consider both a wide range of positive non-additivities and a series of pore widths, ranging from the pure two dimensional limit to a large pore width where results are close to the bulk three dimensional case. Critical parameters are extracted by means of finite size analysis. We find that for this particular case in which demixing is induced by volume effects, phase separation is in most cases somewhat impeded by spatial confinement. However, a non-monotonous dependence of the critical pressure and density with pore size is found for small non-additivities. In this latter case, it turns out that an otherwise stable bulk mixture can be forced to demix by simple geometric confinement when the pore width decreases down to approximately one and a half molecular diameters.' author: - 'N.G. Almarza, C. Martín, E. Lomba and C. Bores' title: Demixing and confinement in slit pores --- Introduction ============ Phase separation under confinement has been for decades a topic of primary interest both from the technological and fundamental science standpoints[@RPP_1999_62_1573]. It is obvious that the reduction in the number of neighbors of those molecules adjacent to the pore walls will induce important phase diagram shifts, whose character will be mostly dependent on the nature of the wall-fluid (or wall-adsorbate) interaction. In the limit of plain two dimensional confinement the system will exhibit bidimensional criticality, which is essentially different -e.g. as critical indices are concerned- from its bulk three dimensional counterpart[@TCritPhen93]. We assume that this bidimensional criticality also holds for the different levels of confinement studied in this work.[@Binder1992a] Many new and interesting effects can be induced by confining and the interplay between adsorbate-adsorbate and adsorbate-pore wall forces. Very recently, Severin and coworkers[@Severin2014] found evidence of a microphase separation in an otherwise fully miscible mixture of ethanol and water when adsorbed in a slit pore formed by a graphene layer deposited on a mica wall. Of utmost interest are also the effects that confinement have on enhancing or preempting crystallization of undercooled fluids[@APL_2005_86_103110; @JPCM_2006_18_R15]. This has been a key approach in the attempts to throw some light in the search for the elusive liquid-liquid critical point in undercooled water[@Biddle2014], resorting to the preemption of crystallization induced by tight confinement of water in nanopores[@Chen2006; @Bertrand2012] and extensive use of diffraction experiments in combination with computer simulations. Not long ago, Fortini and Dijsktra[@Fortini2006] explored the possibility of manipulating colloidal crystal structures by confinement in slit pores. In contrast, thorough studies on the influence of tunable confinement on demixing transitions are scarce[@Duda2003]. One of the simplest systems that illustrate demixing in binary mixtures is the non-additive hard sphere system (NAHS) with positive non-additivity, of which the limiting case of the Widom-Rowlinson model[@Widom1970] has deserved particular theoretical attention and prompted the development of specially adapted algorithms to cope with the hard-core singularities and critical slowing down of the demixing transition[@Johnson1997]. More general instances of the non-additive hard sphere mixture problem (mostly in the symmetric case) have been studied in the two-dimensional limit[@Saija2002], and in a number of detailed studies in three dimensions[@JCP_1996_104_4180; @Gozdz2003; @Jagannathan2003; @Buhot2005]. In this work, we intend to explore thoroughly the demixing transition of the symmetric non-additive hard sphere mixture under confinement in a slit pore by means of computer simulation. The model defined as mixture of A and B components, is characterized by an interaction of the type $$u_{\alpha\beta}(r) = \left\{\begin{array}{ll} \infty & \mbox{if}\; r \le \sigma (1 + (1-\delta_{\alpha\beta})\Delta) \\ 0 & \mbox{if}\; r > \sigma (1 + (1-\delta_{\alpha\beta})\Delta) \end{array}\right.$$ where $\alpha,\beta$ denote the A and B species, $\delta_{\alpha\beta}$ is Kronecker’s delta, the non-additivity parameter is $\Delta > 0$, and $r$ is the interparticle separation. We will study a series of confined non-additive hard sphere mixtures (for various $\Delta>0$ values) using extensive semi-grand ensemble Monte Carlo simulations[@Kofke1988; @FrenkelSmitbook; @Gozdz2003]. The effects of geometric confinement are modeled by the presence of hard-core walls, separated by a distance, $H$, that constrain the particle movement in one space direction (along the $z$-axis as defined here). The fluid particle with thus be subject to an external potential of the form $$V^{ext}(z) = \left\{\begin{array}{ll} 0 & {\rm if } \; \sigma/2 \le z \le H - \sigma/2 \\ \infty & {\rm otherwise}. \end{array} \right.$$ This aims at reproducing the behavior of a fluid confined in a slit pore. Since all interactions at play are purely hard-core, the demixing transition will result from the interplay of entropic and enthalpic (i.e. excluded volume) effects. Our calculations range from the pure two dimensional limit to a relatively large pore width ($10\sigma$, approaching the bulk three dimensional mixture). We have taken advantage of the particular nature of the interaction to implement cluster algorithms[@PRL_1987_58_86; @PRL_1989_62_261; @Buhot2005] in order to cope with the critical slowing down when approaching the consolute point. Finite size scaling techniques have been applied in order to provide accurate estimates of the critical points[@Gozdz2003]. These systems were previously studied by Duda et al.[@Duda2003] by means of mean-field theory and Monte Carlo simulations, considering two values of the slit width, $H$, and different values of $\Delta$. In most of the cases they simulate just one system size, corresponding to a number of particles $N=1000$. Here we will perform a comprehensive analysis of the phase diagram for different values of $H$, and $\Delta$. In addition, for each case several values of $N$ will be considered, which will allow us to get more reliable estimates of the phase diagram of these systems, and in particular of the critical points. The rest of the paper is sketched as follows: in the next section we briefly summarize the computer simulation techniques we have used, and our main results are presented together with conclusions and future prospects in Section III. Methodology =========== Given the particular symmetry of our model, the most appropriate simulation approach to study the phase equilibria is the use of semi-grand canonical Monte Carlo (MC) simulations [@Kofke1988; @FrenkelSmitbook; @Gozdz2003]. We impose the difference between the chemical potentials of the two components $\Delta \mu \equiv \mu_B - \mu_A$, the volume $V$, the temperature $T$ and keep the total number of particles, $N (= N_{\rm A} + N_{\rm B})$ fixed; $x = N_{\rm A}/N$ is the concentration of particle species A. The total number density $\rho = N/V$ is thus fixed. In addition to the conventional MC moves, particles can also modify their identity (i.e. the species to which they belong)[@JCP_1996_104_4180]. The identity sampling can be performed through an efficient cluster algorithm that involves all the particles in the systems and that will be presented later in the paper. After $5 \times 10^5$ MC sweeps for equilibration, our simulations were typically extended over $2\times 10^6$ MC sweeps to perform averages. A sweep involves $N$ single-particle translation attempts, and one cluster move. Note that due to symmetry the critical mole fraction of component $A$ (and $B$) will be $x_c=1/2$, and the demixing transition will occur at $\Delta \mu=0$. When demixing occurs, the mole fraction, $X$ of the components in the two phases, are computed through the ensemble averages of the order parameter $$\theta=2 x-1, \label{theta}$$ as $X = 1/2 \pm \sqrt{<\theta^2>}/2$. Given the symmetry of the model and the efficiency of the cluster algorithm, the average of $x$ from the simulations at $\Delta \mu=0$ will be $<x> \simeq 1/2$, independently of the presence or absence of demixing at the simulation conditions. By analysis of the mole fraction histograms for a series of binary mixtures at different total densities, $\rho = \rho_A+\rho_B$, one can obtain a series of phase diagrams for each sample size, as illustrated in Figure \[dphas2\], where the extreme size dependence of the results on the sample size in the neighborhood of the critical point can be readily appreciated. It is well known that as the critical point is approached, larger samples are needed, correlations become long ranged and critical slowing down must be dealt with somehow. To that aim we have complemented single particle moves with cluster moves[@PRL_1987_58_86; @PRL_1989_62_261; @Buhot2005] following the Swendseng-Wang strategy[@PRL_1987_58_86]. Two particles of the same species are considered linked within the same cluster when their separation is less than $\sigma (1+\Delta)$. Note that due to the linking criteria and the hard-core interactions all the particles belonging to a given cluster are of the same species, and two particles lying at a distance below $\sigma(1+\Delta)$ are necessarily included in the same cluster. As a consequence, cluster identity swaps do not lead to particle overlaps, and for the symmetric case, $\Delta \mu=0$, the procedure leads to a rejection-free algorithm of composition sampling for a fixed set of particle positions. This algorithm rests on two key elements: (1) Clusters are built following the rules defined above, and (2) For each cluster one of the two possible identities ($A$ or $B$) is independently chosen with equal probabilities. Along the simulations the fraction of configurations containing percolating clusters is monitored as an additional signal of the presence of a phase transition[@Stauffer2003]. Another issue that has to be addressed is the calculation of the pressure in the confined system with discontinuous interactions. The scheme proposed by de Miguel and Jackson[@Miguel2006] and further exploited for the Widom-Rowlinson mixture in Reference turns out to be the simplest approach in the present case. In order to estimate the pressure, we perform virtual compressions of the system (both in the $z$ direction –orthogonal to the pore walls– and in the $x,y$ directions). The virial pressure is then computed as $$\beta P^{z,xy} = \lim_{\Delta V_{z,xy}\to 0} \langle\rho + \frac{1}{\Delta V_{z,xy}}n_{o}(\Delta V) \rangle \label{pres}$$ where $\beta= 1/k_BT$ as usual, $\Delta V$ is the change of the volume in the compression, $\Delta V = V - V^{\rm test}$, with $\Delta V >0$, and $n_{o}(\Delta V)$ is the number of particle pairs that overlap during the virtual (test) compression of the system. In practice, the pressure is calculated by computing $n_{o}$ for a set of values of $\Delta V$ and extrapolating to the the limit $\Delta V\rightarrow 0$. Now, the demixing transition is monitored following the evolution and size dependence of a series of appropriate order parameters. Here we have considered on one hand, $\theta$, as defined in Eq. (\[theta\]), and on the other the fraction of percolating configurations, $\chi$. A configuration is defined as percolating if (and only if) at least one of its clusters becomes of infinite size when considering the periodic boundary conditions; those clusters are often denoted as [*wrapping*]{} clusters. With the $\theta$ order parameter we proceed to perform a Binder cumulant like analysis[@ZPB_1981_43_119; @landau-binder_book_2005]. This is done by considering ratios between momenta of the order parameter probability distribution given as: $$U_{2n} = \frac{\langle \theta^{2n} \rangle }{\langle \theta^2 \rangle^{n}}, \label{un}$$ where the angular brackets indicate ensemble averages, and looking at how these quantities vary with the density for different system sizes. Calculations are carried out for different samples sizes, $N$, and curves of $\chi$, $U_4$, and $U_6$ are plotted vs. total density $\rho$. According to the finite size scaling analysis[@landau-binder_book_2005], the crossing of the curves $U_{2n}(\rho)$ for different system sizes, should define the critical point and be size independent for sufficiently large samples. In practice, we fit the critical density estimates, $\rho_c(N)$, obtained from different crossings. This is done by taking pairs of system sizes, $N_i < N_j$, and looking for the density $\rho_c(N_j|N_i)$ where the curves of the analyzed property for the two system sizes cross. The results $\rho_c(N_j|N_i)$ for a given $N_i$ are taken as estimates for the pseudo-critical densities for the system size $N_j$, and from then one can extrapolate the critical density in the thermodynamic limit $(1/N_j \rightarrow 0$). These extrapolations were done by fitting the results to straight lines of the form[@Gozdz2003] $\rho_c(N) = \rho_c + a N^{-1/(2\nu)}$, where we took $\nu=1$, according to the assumed bidimensional criticality. Notice that a more rigorous finite-size scaling analysis should be based on results from simulations carried out in either $(N,p,T,\Delta \mu$) or $(\mu_A,V,T,\Delta\mu)$ ensembles instead of resorting to $(N,V,T,\Delta\mu)$ semi-grand ensemble simulations.[@Fisher1968a; @Almarza2012c; @Lopez2012b] The estimates of $\rho_c$ obtained from the fraction of percolating configurations and the cumulants are fully consistent within statistical error bars. The corresponding critical pressures are obtained by means of a series of semi-grand canonical simulations carried out at the critical density and $\Delta \mu=0$ with varying sample sizes and extrapolating $\beta P^{xy}$ and $\beta P^z$ for $1/N\longrightarrow 0$. An example of the evolution of the order parameters for the two dimensional limit, (i.e. pore width $H=\sigma$) and non-additivity $\Delta=0.2$ is presented in Figures \[U48\] and \[chip\]. For densities about $\rho_c$, demixing occurs at $\Delta \mu=0$. The mole fractions of the coexisting phases for each given system size are computed through the order parameter $\theta$, as: $X_{\pm} = \frac{1}{2} \left[ 1 \pm \sqrt{ \langle \theta^2 \rangle}\right]$. Using the results for different system sizes, we estimate the composition in the thermodynamic limit by fitting the results to a second-order polynomial in $(1/N)$. Then, the $X-\rho$ phase diagram can be fully estimated discarding the equilibrium data close to the critical $\rho_{c}$ (much affected by sample size dependence), and using the extrapolated data $X(\rho)$, and a fit to the approximate[@Fisher1968a] scaling law $$\left|\frac{X(\rho)-x_c}{x_c}\right| \propto \left|\frac{\rho}{\rho_{c}}-1\right|^\beta, \label{scal}$$ where we assume the system to belong to the two dimensional Ising universality class[@Gozdz2003], and hence $\beta =1/8$. Results ======= We have considered systems with varying degrees of non-additivity, ranging from $\Delta = 0.1$ to $\Delta=1$, and pore widths from $\sigma$ to $10\sigma$ (see Table \[critres\] for the specific values). Semi-grand ensemble simulations were run for samples of 400, 900, 1600, 2500, 3600 and 4900 particles when $H < 5.5\sigma$. Sample sizes of 6400 particles were included for pore widths larger than $5.5\sigma$ up to $H=10\sigma$ where an additional sample size of 8100 particles was included. As mentioned in the previous section, for a given system defined by a pair ($H,\Delta$), simulations are run for a series of total densities, $\rho$, and we monitored the behavior of the order parameters (as illustrated in Figures \[U48\] and \[chip\]). Following the procedures indicated above we obtain a series of phase diagrams as illustrated in graphs of Figure \[DfaseH\] for three selected pore widths, $H=\sigma, 2.5\sigma$, and $10\sigma$. The complete set of critical properties for most of the systems studied is collected in Table \[critres\]. From Figure \[DfaseH\] one immediately appreciates that increasing the non-additivity lowers the critical density, i.e. favors demixing as expected. In contrast, we observe that confinement tends to stabilize the mixed phase. This effect is particularly visible when going from the $H=2.5\sigma$ system to the two dimensional case, where one sees that the critical density practically doubles for the two largest non-additivities. Obviously, as the non-additivity decreases demixing occurs at higher packing fractions and packing constraints necessarily limit the effects of confinement on the critical density. Interestingly, we observe that as $H>2.5\sigma$ the change on the critical density is much smaller, and practically negligible for the smallest non-additivity. In practice, as we will see later, for $H=10\sigma$ the critical values of the bulk three dimensional hard sphere mixture have almost been reproduced. This effect of stabilization of the mixture due to confinement can be easily understood when one realizes that the average number of neighbors is reduced as one goes from the bulk three dimensional system to the two dimensional one. This implies that particles of a given type A (or B) will have fewer neighbors of type B (or A) when they are close to the walls, the limiting case being the two dimensional system. As a consequence, these particles will have a lower tendency to demix as the density (or pressure) is increased. Obviously, the fraction of particles adjacent to the walls is maximum when $H=\sigma$, and this fraction decreases rapidly as $H$ increases, and as a consequence the critical density decreases. Once the pore allows for two fluid layers inside, the fall in the critical density as the pore widens is not so pronounced. Now, in Figures \[rhocH\] and \[PcH\] we observe the explicit evolution of the critical density vs $1/H$ and the critical pressure vs $H$. In Figure \[rhocH\] some values from the literature for the two and three dimensional limit are included. As mentioned before, the critical densities for $H=10\sigma$ practically have already converged to those of the unconfined system. The dependence of the critical density on pore size has two linear regimes, which for $\Delta = 1$ and $1/2$ merge continuously at $H \approx 1.5\sigma$. For smaller pore sizes the critical densities grow rapidly as the pore size shrinks due to the marked decrease in the number of neighbors induced by the presence of walls. For larger pore sizes, $H > 4\sigma$, another linear regime with a less pronounced slope sets in. An interesting feature emerges in the region $1.5\sigma < H< 4\sigma$ for $\Delta=0.1$ and 0.2: both the critical density and $\beta P^z_c$ show a clear non-monotonous dependence on H, with maxima located at $H\approx \sigma, 3\sigma$, and $4\sigma$, –the latter only visible in the pressure curve– recalling the neighbor shell structure of a pair distribution function. We find then that in the ranges $(n-1/2)\sigma \lesssim H \lesssim n\sigma$ ($n=2,3$) the critical density and pressure increase (i.e. the mixture is stabilized) when the pore widens. Note however that on $\beta P^{xy}_c$ the maxima are shifted towards larger $H$-values, and actually the minima of $\beta P^{xy}_c$ lie close to the maxima of $\beta P^{z}_c$. Somehow, the increase in the pressure against the pore walls tends to be compensated by a decrease of the pressure along the unbound directions.This mismatch is the obvious result of the lack of isotropy induced by the walls. An extreme situation occurs at $H = 1.5\sigma$ and $\Delta=0.1$, for which the critical density ($\rho_c=0.591$) is appreciably lower than that of the bulk[@Gozdz2003] ($\rho_c=0.6325(8)$). This actually implies that for certain systems (i.e. degrees of non-additivity), a stable mixture can be forced to demix by simple geometric confinement. In fact one observes that the maxima in $\rho_c$ – i.e. local stability maxima for the mixtures – occur when the pore can fit approximately an integer number of layers ($1,2,$ and 3). From these state points, increasing or decreasing the pore size induces demixing. The effect of the increase in pore size is easily explained as the result of an increasing number of neighbors of different species that will prefer to be in a single component phase. On the other hand, if we focus on the behavior of the system when going from $H\approx 2\sigma$ to $H\approx 1.5\sigma$, we realize that the number of neighbors does not dramatically change when $H$ varies within these limits, as long as $\Delta$ is small. In fact, for $\Delta\rightarrow 0$ $H = (1+\sqrt{2/3})\sigma\approx 1.8\sigma$ the pore still allows for a closed packed structure of two particle layers with 9 neighbors per particle, with A and B particles mixed. If the number of neighbors remains approximately constant, the reduction of available volume with the shrinkage of the pore width will induce demixing. The effect will still be present but less patent when going from $H\approx 3\sigma$ to $H\approx 2\sigma$. Large values of $\Delta$ will destroy this stabilizing effect, e.g. when $\Delta \approx 1/2$ volume exclusion will prevent the presence of unlike neighbors in adjacent layers. Small $\Delta$ values allow for this possibility and therefore higher packing fractions of the stable mixture can be found, by which the non-monotonous dependence of the critical properties on the pore width is explained. In summary, we have presented a detailed study of the effects of geometric confinement on symmetric mixtures of non-additive hard spheres. We have found that, as an overall trend, confinement tends to impede demixing, rising both critical densities and pressures, but interestingly for small degrees of non-additivity a non-monotonous dependence is found. In fact, for certain values of the cross interaction, it is found that confinement can induce demixing by simple packing effects. In future works, we will address the effects of competition between energetic and steric contributions to the intermolecular potential and tunable wall interactions. The authors acknowledge the support from the Dirección General de Investigación Científica y Técnica under Grants No. FIS2010-15502 and FIS2013-47350-C5-4-R. The CSIC is also acknowledged for providing support in the form of the project PIE 201080E120. [30]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , , , , ****, (). , , , , ** (, ). , ****, (). , , , ****, (). , , , , , , , ****, (). , , , , , , ****, (). , , , ****, (). , , , , , , , , ****, (). , , , ****, (). , ****, (). , , , ****, (). , ****, (). , , , , ****, (). , ****, (). , , , , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ** (, ). , ****, (). , ****, (). , ** (, , ), ed. , ****, (). , , , ****, (). , ****, (). , ** (, , ). , ****, (). , , , ****, (). , , , ****, (). ------------ ------------------ ---------------------------- ------------------------- ------------------ ---------------------------- ------------------------- ------------------ ---------------------------- ------------------------- ------------------ ---------------------------- ------------------------- $H/\sigma$ $\rho_c\sigma^3$ $\beta P_{c}^{xy}\sigma^3$ $\beta P_{c}^z\sigma^3$ $\rho_c\sigma^3$ $\beta P_{c}^{xy}\sigma^3$ $\beta P_{c}^z\sigma^3$ $\rho_c\sigma^3$ $\beta P_{c}^{xy}\sigma^3$ $\beta P_{c}^z\sigma^3$ $\rho_c\sigma^3$ $\beta P_{c}^{xy}\sigma^3$ $\beta P_{c}^z\sigma^3$ 1.00 0.841 8.30 — 0.690 3.84 – 0.460 1.314 – 0.286 0.547 – 1.05 0.802 7.91 16.95 0.657 3.66 13.85 – – – – – – 1.10 0.765 7.55 8.67 0.628 3.49 7.01 0.419 1.193 4.64 – – – 1.25 0.679 6.64 4.08 0.555 3.07 3.05 0.370 1.051 1.919 – – – 1.50 0.591 5.52 3.86 0.477 2.56 2.10 0.314 0.876 1.089 0.194 0.365 0.621 1.75 0.607 4.70 7.91 0.445 2.18 2.54 0.280 0.751 0.912 – – – 2.00 0.699 4.71 8.31 0.477 1.97 3.62 0.266 0.660 0.959 0.155 0.275 0.405 2.25 0.675 4.97 4.83 0.488 1.97 2.75 0.262 0.604 0.889 – – – 2.50 0.638 4.69 4.41 0.476 1.97 2.25 0.257 0.575 0.795 0.138 0.227 0.319 2.75 0.635 4.24 5.44 0.461 1.88 2.16 0.252 0.558 0.735 – – – 3.00 0.656 4.16 5.54 0.456 1.78 2.21 0.247 0.542 0.693 0.128 0.202 0.268 3.50 0.641 4.20 4.40 0.456 1.73 2.04 0.239 0.512 0.632 0.122 0.189 0.241 4.00 0.642 3.97 4.66 0.449 1.67 1.93 0.233 0.491 0.592 0.117 0.180 0.221 4.50 0.638 3.96 4.27 0.446 1.63 1.86 0.229 0.478 0.562 0.113 0.173 0.207 5.00 0.636 3.85 4.29 0.443 1.61 1.80 0.225 0.468 0.541 0.109 0.167 0.196 6.00 0.634 3.78 4.11 0.439 1.57 1.73 0.220 0.452 0.508 0.105 0.160 0.182 7.50 0.631 3.72 3.94 0.436 1.54 1.66 0.215 0.440 0.482 0.100 0.153 0.169 10.00 0.630 3.65 3.82 0.432 1.51 1.59 0.210 0.428 0.458 0.096 0.147 0.158 ------------ ------------------ ---------------------------- ------------------------- ------------------ ---------------------------- ------------------------- ------------------ ---------------------------- ------------------------- ------------------ ---------------------------- ------------------------- : Critical parameters for non-additive hard sphere mixtures confined in slit pores. Error estimates of critical densities and pressures are below the last significant digits in both instances. \[critres\] ![Size dependence of phase diagram of symmetric non-additive mixtures (with $\Delta=0.1$ and $\Delta=0.2$) confined in a slit pore of width $H=1.5\sigma$. The dotted line marks the estimate for the critical density as obtained from the finite-size scaling analysis. The symbol correspond to the critical point in the density-mole fraction plane. []{data-label="dphas2"}](DfasesD02H15){width="12cm"} ![Size dependence of the $U_4$, $U_6$ and $U_8$ cumulants of the order parameter $\theta$ for the symmetric non-additive hard sphere mixture\[U48\]](OrdPD02){width="15cm"} ![Size dependence of the fraction of percolating configurations, $\chi$, for the symmetric non-additive hard sphere mixture\[chip\]](PercD02H1){width="15cm"} ![Phase diagram of the non-additive hard-sphere mixtures for various pore widths and non-additivity parameters \[DfaseH\]](DfaseH){width="15cm"} ![Critical density dependence on the slit pore size as computed in this work and compared with limiting values in and 2D and 3D by Buhot[@Buhot2005] and by Góźdź[@Gozdz2003] \[rhocH\]](RhoCritica_VDeltas){width="15cm"} ![Critical pressure dependence on the slit pore size for various non-additivity parameters. Upper graph corresponds to the pressure on the pore walls and the lower graph to the corresponding transverse components\[PcH\]](PresH_TDelta){width="15cm"}

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Introduction ============ In a recent experiment [@CSR99], a strongly isolated quantum dot was charged with excess electrons, and their sequential escapes were recorded over a one hour time period. This was repeated 150 times to obtain a statistical distribution of decay times. The dot is formed in an electron gas located at a depth of $70 \ nm$ in a $GaAs-AlGaAs$ heterostructure. Its shape is defined by electrostatic confinement using a set of gates, as sketched in the insert to Fig. \[fig1\] . The gate voltages were ramped up quickly, so that the dot retained a sizeable number of excess electrons when it was well isolated from the surrounding electron gas. The observations correspond to sequential tunnelling of (seven) electrons from the dot to the surroundings. The lifetimes extracted from the escape times distribution [@CSR99] are shown in Fig. \[fig1\]. A striking quasi-linear dependence of the logarithm of the lifetime on electron number is apparent. Sequential decays have been known and studied for over a century in the context of nuclear physics. The combined instances of alpha and beta decays from the heaviest elements are responsible for most natural radioactivity. The description of alpha decay in terms of tunnelling of alpha particles through a confining potential dates back to the 1920’s (Gamow [@G28], Condon and Gurney [@CG28]). Although the basic nature of the decay as a barrier penetration is well understood, accurate predictions for radioactive lifetimes are difficult because the process by which the escaping alpha-particle is preformed within the nucleus requires an understanding of four-body correlations. As a result, it is impossible to deduce accurate information on the barrier shape. Nevertheless, alpha decays have provided useful information on nuclear radii and the range and gross features of the nuclear interaction [@PB75]. It has become commonplace to say that a quantum dot is an artificial atom, but in fact the self-consistent potential confining electrons in a large dot has more in common with the mean field potential in a heavy nucleus: flat in the interior, with abrupt walls. An artificial nucleus is a more apt description, as will become clear in this paper. Indeed, the detection of sequential decays from an isolated quantum dot is a more favourable situation for study of the decay process, as the question of preforming the electron does not arise. Hence, we can more confidently test our knowledge of the confining barriers for electrons, as well as the profile, and dependence on occupation number, of the dot potential. We will analyze these aspects in this work, and show that these measurements of the lifetimes of “radioactive quantum dots” introduce new constraints on our ability to model their structure. The present experiment has another significant advantage over nuclear decays: instead of counting incoherent decays from a large sample of identical nuclei, here a single dot is involved, and the correlation between consecutive events can be analyzed. In addition, it should be possible to design the shape, density and excitation energy of the dot within rather broad margins, so that future experiments on mesoscopic systems will be much more flexible than those in nuclear systems, where only those nuclei existing in nature, or created in sufficient numbers, can be studied. Thus, the study of electron decays from a quantum dot has the potential to reveal new features of the tunnelling process. This is a topic of currently renewed interest: see for example van Dijk and Nogami [@vDN99]. The type of simple model developed in this paper can be of great utility in such future studies. In this work we will describe the decay process using analytic models which incorporate characteristics of the confinement potential extracted from realistic numerical simulations. As the dot contains about 300 electrons, Poisson-Thomas-Fermi calculations should be adequate to describe the electron density and the confining potential of the dot. With these in hand we have developed accurate analytic approximations for the confining potential that allow us to construct an envelope approximation wavefunction for the electrons in the dot, and to compute the electron lifetimes from a fully quantal expression for the transmission amplitude across the barrier. Previous works which model a quantum dot have been concerned with the wave functions of confined states in the dot, the electron density distribution and the shape of the confining potential. For such purposes, only the inside of the barrier matters. It is when one looks at the escape of electrons from the dot that the barrier height, its width and shape become important; these are the new features explored in this paper. In section II we describe the development of our model, while in Section III we discuss the results for the sequence of lifetimes and compare them with experiment. Some details are relegated to two appendices. Modelling of Isolated Dot Decays ================================= Framework --------- The Poisson-Thomas-Fermi modelling is described in more detail elsewhere [@MS98], so here we list only the main steps:\ [*i)*]{} first, Poisson-Schrödinger (PS) and Poisson-Thomas-Fermi (PTF) simulations as described in [@MS94] are performed for the ungated heterostructure. Our inputs for the PS simulation are the thickness and composition of each layer in the heterostructure, and the dopant concentration in the donor layer. From these we predict the density of the 2DEG. The only adjustable parameter is the donor ionization energy which is set to be $e\Phi_i = 0.12 \ eV$, in order to reproduce the measured 2DEG density, $ n_e = 2.74 \, 10^{11} \ cm^{-2} $. For the simpler Poisson-Thomas-Fermi scheme we employ a common relative permitivity $\varepsilon_r = 12.2$ for all layers of the heterostructure, which, combined with the parameters already used for the PS simulation, also reproduces the experimental $n_e$. After this “fitting” the model has no other free parameters.\ [*ii)*]{} For the gated structure we use the gate layout and voltages of the experiment. To solve the Poisson equation for the gated heterostructure one has to impose as a boundary condition the value of the electrostatic potential on the exposed surface of the heterostructure, and on the gates. We assume Fermi level pinning and choose the energy of the surface states as the zero of the energy scale. In this convention, the conduction band edge is set at $e V_s = 0.67 \ eV $ on the exposed surface. Under each gate the conduction band is set at $ eV_{ms}+ eV_g$, where $V_g$ is the gate voltage and the metal semiconductor contact potential, $eV_{ms}$, is taken as $0.81 \ eV$ [@Mo95]. The electrostatic potential due to the gates is then computed using semi-analytic expressions based on the work of Davies [*et al.*]{} [@Davies88] and [@DLS95]. Added to this are: [*a)*]{} the Coulomb potential (direct term) between the electrons, and a mirror term which imposes the boundary conditions at the surface, and [*b)*]{} the contribution from the fully ionized donor layer and its mirror term (see Sect. IIA of [@MWS94] for details of a similar example.) We neglect exchange and correlation effects, which are small.\ [*iii)*]{} The connection between the confining potential defined by the conduction band edge and the electron density is completed by using the Thomas-Fermi approximation at zero temperature: $$\rho_e({\vec r}) = {1\over {3\pi^2}} \left( {{2m^*}\over \hbar^2} (E_F -eV({\vec r}))\right)^{3/2} \label{eq:1}$$ The PTF iteration is performed starting from the ungated heterostructure densities as trial values. Equilibrium dot --------------- As a first step, we examine the dot in its final state after all the excess electrons have escaped. This corresponds to a PTF simulation with the same Fermi level, $E_{F,dot} = 0$, for the electrons in the dot and in the 2DEG outside the barriers. The gate voltages are taken from ref. [@CSR99] as $V_{PL} = -0.40 \ V$, $V_{C1}= V_{C2} = -0.44 \ V$ and $V_H = -0.7 \ V$. The predicted PTF 3D electron distribution $\rho_e(x,y,z)$ is more conveniently visualized in terms of a projected 2D density: $$n_e(x,y) = \int_{z_j}^{\infty} \rho_e(x,y,z) \ dz \ , \label{eq:2}$$ where $z_j$ is the junction plane. The $n_e(x,y)$ distribution, shown in Fig. \[fig2\], has an approximately rectangular boundary, and its maximum value is close to the 2DEG density of the ungated heterostructure. In this calculation the dot contains 286 electrons. Dot with excess electrons ------------------------- To study these configurations we set the Fermi level inside the dot, $E_{F,dot}$, higher than its value outside the barriers, $E_{F,2DEG}=0$. We can do so because the dot is well pinched off from the surrounding electron gas. We ran PTF simulations with equally spaced values for $E_{F,dot}$ running from $0$ to $17.5 \ meV$ in steps of $2.5 \ meV$. The occupation $Q$ of the dot increases linearly with $E_{F,dot}$ at the rate $2.75$ electrons per meV, giving occupations $286 \le Q \le 334$. The simulations also produce the confining potential for the electrons in the dot, $e V(x,y,z)$. To reduce this to a two-dimensional function, $U(x,y)$, we take a weighted average over the density profile in the $z$ direction: $$U_{PTF}(x,y) = {{\int_{z_j}^{\infty} e V(x,y,z) \ P(z) \ dz}\over {\int_{z_j}^{\infty} P(z) \ dz}} \label{eq:3}$$ where $$P(z) = \int_\Omega \rho_e(x,y,z) \ dx \ dy \ . \label{eq:4}$$ Here the domain of integration $\Omega$ is a rectangle in the $x y$ plane which extends a short distance into the surrounding electron gas, (from $(x_l,y_l) = (-510 \ nm, -255 \ nm)$ to $(x_r,y_r) = (510 \ nm, 255 \ nm)$.) This includes an area outside the dot where the 2DEG is still depleted by the gates. Although the computed $V(x,y,z)$ is not separable, previous experience with Poisson-Schrödinger simulations of wires [@MWS94],[@MS96] and circular dots, has shown us that the factorization ansatz leads to very good approximations when the $z$ degree of freedom is integrated out as in eq. \[eq:3\]. This prescription to construct the 2D potential avoids the type of [*ad hoc*]{} assumptions often made. In Fig. \[fig3\] we show the $U_{PTF}(x,y) $ corresponding to the equilibrium dot of Fig. \[fig2\]. As expected from the gate layout shown in the inset to Fig. \[fig1\], it has two very high barriers running parallel to the $x$ axis, one centered at $ y=0 $ and the other that begins with a steep rise at $y \simeq 400 \ nm$ (and shows clearly the mark of the three-fingered gate layout labelled $C1,\, C2$, and $PL$unger in Fig. \[fig1\]). Tunnelling across these barriers is negligible. In addition there is a symmetric pair of barriers running parallel to the $y$ axis, with maxima at $x \simeq \pm 238 \ nm $ through which the electrons [ *do*]{} tunnel. In the interior, the potential is practically constant. Although these $x$-barriers have somewhat increasing height with increasing $y$, the rectangular shape of the potential suggests using a separable approximation in cartesian coordinates: $$U_{PTF} (x,y) \simeq U_s(x,y) = U(x) + W(y) \label{eq:5}$$ We will interpret the experimental decay data using this separability ansatz. For the $W(y)$ barriers, which are basically impenetrable, we use two simple models described below. As a guide to a realistic choice for the $x$-dependent term we examine in Fig. \[fig4\] the profiles of $U_{PTF}(x,y)$ at a fixed value of $y = 200 \ nm$ in the middle of the dot. The profiles shown cover a range of occupations of up to forty excess electrons. In this range, the potential at the dot center increases linearly with $Q$, according to $$U_0 = 0.347 Q - 118.4 \ meV \ . \label{eq:6}$$ At large distances outside the dot, $U_\infty = -18.8$ meV is constant. Similarly, the location of the barrier maximum and its height can be parametrized as: $$\begin{aligned} x_b &=& 238 - {{ Q - 286 }\over 16} \ nm \nonumber \\ U_b &=& 0.117 Q - 13.4 \ meV \ . \label{eq:7}\end{aligned}$$ Note that $ d U_b / d Q \approx 1/3 d U_0 / dQ$ reflects the decrease of the screened Coulomb repulsion away from the center of the dot. Furthermore, we have found that the $x$-dependence can be very well reproduced (see Fig. \[fig4\]) using the following analytic model: $$\begin{aligned} U(x) &=& U_b + U_{MF}(x) \, , \quad x > 0, \nonumber \\ &=& U(-x) \, , \qquad \qquad x < 0, \quad {\rm where} \nonumber \\ U_{MF} &\equiv& U_c {{\sinh^2 \left({{x-x_b}\over w_b}\right)}\over {\cosh^2\left({{x-x_b}\over w_b} - \mu\right)}} \label{eq:8}\end{aligned}$$ This potential form has the great advantage that the transmission coefficient for $U_{MF}$ is known analytically [@MF53]. $U_{MF}$ is an asymmetric barrier which takes one value for $x << x_b$ and another value for $x >> x_b$: $$\begin{aligned} U_{MF}(x_b) &=& 0 \nonumber \\ U_{MF}(\infty) &\equiv& \lim_{x \to \infty} U_{MF}(x) = U_c e^{2\mu} \nonumber \\ U_{MF}(-\infty) &\equiv& \lim_{x \to -\infty} U_{MF}(x) = U_c e^{-2\mu} \ . \label{eq:9}\end{aligned}$$ The parameters $U_b, U_c, \mu, x_b, w_b$ allow one to fit the barrier height, the potential floors inside and outside the dot, the barrier spacing and the barrier width. Since the barriers are spread quite far apart, in practice $x_b >> w_b$, so $U_{MF}(x=0) \approx U_{MF}(-\infty)$. In this case, $$\begin{aligned} U_{0} &\equiv& U(0) \approx U_b + U_c e^{-2\mu} \nonumber \\ U_{\infty} &\equiv& \lim_{x \to \infty} U(x) = U_b + U_c e^{2\mu} . \label{eq:10}\end{aligned}$$ Then we can solve for $$\begin{aligned} \mu &=& {1\over 4} \ln \left({{U_b - U_{\infty}}\over {U_b - U_0}}\right) \qquad {\rm and } \nonumber \\ U_c &=& -(U_b -U_0) e^{2\mu} \, . \label{eq:11}\end{aligned}$$ To determine the parameters appearing in eq. \[eq:8\], we take the values of the PTF potential at the origin, $U_0$, well beyond the barrier, $U_{\infty}$, and the value $U_{x_b}$ at the barrier maximum $x=x_b$, and then plot $U(x)$ to find the best $w_b$, which turned out to be $48 \ nm$. This gives a convenient analytic form for the confining potential, motivated by PTF, whose transmission coefficient is: $$T = {{2 \sinh (\pi k_+) \sinh(\pi k_-)}\over {\cosh(\pi(k_+ + k_-)) + \cosh(\pi \beta)}} \quad , \label{eq:12}$$ where $$\begin{aligned} k_{-/+} &=& \sqrt{ {{2m^*}\over \hbar^2}(E- U_{0/\infty}) w_b^2} \qquad {\rm and} \nonumber \\ \beta &=& \sqrt{{{2m^*}\over \hbar^2}(2U_b-2U_c-U_0-U_{\infty})w_b^2-1} \quad . \label{eq:13}\end{aligned}$$ [*Barrier shape $W(y)$*]{}: In Fig. \[fig5\] we examine a section of $U_{PTF}(x=0,y)$ through the center of the dot. We use two approximate models, the simplest one being an infinite square well, of width $w_y \approx 350 \ nm$. The slightly fancier one is a truncated harmonic oscillator: $$\begin{aligned} W_{tho}(y) &=& \qquad 0 \hskip 2.8cm {\rm (flat} \,\,\, {\rm bottom)} \nonumber \\ &=& -0.13 + {1\over 2} k_y (y - y_0)^2 \quad {\rm (walls)} \ . \label{eq:14}\end{aligned}$$ with $ y_0 = 238 \ nm$ and $k_y = 7.35 \cdot 10^{-6} \ nm^{-2} $. As can be seen in Fig. \[fig5\] this parametrization (plus the constant term $U_0$) reproduces the main features of the $x=0$ sections of the PTF potentials. By combining eqs. \[eq:6\] to \[eq:14\] we determine a separable analytic potential model for the dot containing a desired number $Q$ of electrons. This removes the necessity of repeatedly solving the PTF equations for the self-consistent field, while studying the decay process. Quasibound states of the dot ---------------------------- We construct the electron wave functions inside the dot in the envelope function approximation, using our parametrized potential, $U_s(x,y)$. The single electron energies are $$E_{n_x,n_y} = E_{n_x} + E_{n_y} \, \label{eq:15}$$ and the electron wavefunctions factorize $$\Psi_{n_x,n_y}(x,y) = \phi_{n_x}(x) \psi_{n_y}(y) \, . \label{eq:16}$$ The factors satisfy 1D Schrödinger equations: $$\begin{aligned} &-& {\hbar^2\over {2m^*}} \phi_{n_x}''(x) + U(x) \phi_{n_x}(x) = E_{n_x} \phi_{n_x}(x) \nonumber \\ &-& {\hbar^2\over {2m^*}} \psi_{n_y}''(y) + W(y) \psi_{n_y}(y) = E_{n_y} \psi_{n_y}(y) \label{eq:17}\end{aligned}$$ The second equation is for a confined wavefunction, easily solved by standard numerical methods. We label the solutions by the number of loops, $n_y$, of the eigenfunction. For example, taking $W(y)$ to have hard walls, the energy is $$E_{n_y,sw } = {\hbar^2\over {2m^*}} \left({{n_y \pi}\over w_y}\right)^2 \ . \label{eq:18}$$ For the truncated harmonic oscillator shape there is no similar analytic expression, but the dependence on $n_y$ is similar. The $x$-dependent equation describes 1D electrons confined in the dot by the “leaky barriers”. Weakly quasibound state solutions were computed using methods described in [@K77]. However, for levels corresponding to the long tunnelling lifetimes observed in the experiment, the energies and eigenfunctions can be computed well enough by the simpler prescription of setting the electron wavefunction to zero at the points $\pm x_b$ inside the barriers. Furthermore, if only the eigenvalues and lifetimes are needed, we have checked that the WKB quantization condition is adequate: $$\int_{x_l}^{x_r} \sqrt{{{2m^*}\over \hbar^2} ( E(n_x) - U(x) )} \ dx = \left(n_x - {1\over 2} \right) \pi \label{eq:19}$$ In the Appendix we describe the determination of the lifetimes, $\tau_{n_x}$. From here on the energies presented are obtained in the WKB approximation. The differences from the more accurate predictions using the true quasibound state energies can scarcely be seen on the scale of the graphs. For barrier penetrability we use eq. \[eq:12\]. We “construct" the desired dot configuration with excess electrons by generating a $U_s(x,y)$ for the chosen value of $Q$, and filling the levels as follows: a) First we list the $(E_{n_x},\tau_{n_x})$, in order of increasing $n_x$ (and therefore of increasing energy and decreasing lifetime.) This list is truncated at an $n_x = n_{x,max}$ whose lifetime is less than $0.01 \ sec.$ b) Next we form a list of 2D levels $(n_x,n_y)$ by choosing those for which $$E_{n_x} + E_{n_y} \le E_{n_{x,max}} + E_{n_y = 1} \ . \label{eq:20}$$ The levels in this list are occupied in order of increasing energy and according to Fermi statistics, see eqs. \[eq:a5\] \[eq:a6\]. We choose the dot Fermi level so that the number of electrons is the desired Q. It is supposed that, for the long lifetimes observed in the experiment, the electrons have time to lose energy by phonon collisions and occupy the quasibound states of lowest energy. Then, as described in Appendix A, we determine the lifetime for one electron to escape from the dot. This involves a weighted average of the level lifetimes, according to the occupancy of each level at the experimental temperature, $T' = 100 \ mK$. To produce a sequence of decays for comparison to experiment we proceed as follows: [*i)*]{} we start with a dot containing a number of electrons, $Q_0$, chosen large enough so that the lifetime for one electron to escape is smaller than those observed in experiment. [*ii)*]{} We redetermine the barrier and dot configuration for $Q = Q_0 -1$ electrons, as described in the above paragraph and determine again the corresponding lifetime for escape of one electron. This process is repeated to generate a sequence of decays that covers and extends beyond the range of lifetimes measured in experiment. From that list we choose as the first observed electron decay that corresponding to the $Q$ whose lifetime is the first to be larger than $t_0 = 25 $ seconds. Results and discussion ====================== In Fig. \[fig6\] we show results from our model, using parameters chosen as described above, for a range of lifetimes extending over three orders of magnitude. The stars correspond to the truncated harmonic oscillator choice for $W(y)$, whereas the $+$’s are for the square well choice (with a value $w_y = 380 \ nm$ chosen to optimize the agreement with the other prescription in the range of experimental lifetimes, from 10 to 1000 seconds.) One sees that the trends are very similar. For $Q$ in the neighbourhood of $304$, the predicted decay lifetimes fall in the experimental range. As already mentioned in Section II, our PTF simulations predict $Q=286 $ for the dot in equilibrium with the surrounding electron gas. This is also what we find with this separable model, as the curve of lifetimes shown in figure \[fig6\] extrapolates smoothly up to $Q= 287$, for which we predict a lifetime of $\log_{10}\tau = 5.2$, or 44 hours. After that, the Fermi level of the electrons inside the dot falls below that of the surrounding 2DEG and further decays are blocked. It should take almost two days for the dot to reach equilibrium with its surroundings. Before attempting a more detailed comparison with the experimental data it is useful to examine the main features in our predicted sequences. First we focus on the linear behaviour for values $ Q < 300$. (We have found similar behaviour in other ranges of $Q$ when we use slightly different sets of parameters.) Such linear dependence occurs when our model produces a sequence of decays dominated by those from a single 1D electron level; [*i.e.*]{} corresponding to a fixed value of $n_x$. To understand why, suppose that at zero temperature and for $Q$ electrons, the occupied level with shortest lifetime is $(n_{x,s},n_y)$, and that $\{n_x',n_y'\}$ are occupied levels with higher energy and longer lifetime (this requires that at least $n_x' < n_{x,s}$ for longer lifetime and $ n_y' > n_y$ for higher total energy.) When one forms the $Q-1$ electron configuration according to the rules explained above, one of the $\{n_x',n_y'\}$ levels will be empty, whereas the level $(n_{x,s},n_y)$ will again be filled. In more physical terms: all the electrons with energy above that of the level with shortest lifetime will lose energy by phonon collisions and fall into the leaky level, from which they finally escape. Since the lifetime does not depend on $n_y$, all the electrons with energy above that of the state $(n_{x,s}, n_y = 1)$ will escape through the same leaky 1D level, $n_{x,s}$, which remains the favoured decay channel as long as it is occupied. Therefore the total probability for [**one**]{} electron to escape from the occupied states with quantum number $n_{x,s}$ is the probability for a single 1D electron with energy $E_{n_{x,s}}$, multiplied by the number of electrons in occupied states with the same quantum number $n_{x,s}$: $q_{n_{x,s}}$: $$\tau(Q) = {{\displaystyle{\tau_{n_{x,s}}(Q)}} \over {\displaystyle{q_{n_{x,s}}(Q)}}} \quad , \label{eq:tm1}$$ and when the occupation $q_{n_{x,s}}$ of the leaky level is constant, the linear variation of $\log_{10}(\tau)$ reflects that of the lifetime of the leaky level. This is where the 2D nature of the quantum dot asserts its presence, even though the decay appears to proceed only in one dimension. In Fig. \[fig7\] we show the occupations of the two levels with the shortest lifetimes. One sees that when $Q < 300$ the occupation of the $n_x = 13 $ level stays practically constant and $n_x = 14 $ level remains empty. For higher values of $Q$ both levels contribute significantly to the escape lifetime. In this situation: $$\tau(Q) = {1\over{\displaystyle{ {{q_a(Q)}\over{ \tau_a(Q)}} + {{q_b(Q)} \over {\tau_b(Q)}}}}}\ . \label{eq:tm2}$$ This is shown as the dotted curve in Fig. \[fig6\], and it accounts very well for the trend of the lifetimes predicted by the separable model. Our separable model favours the appearance of the linear decay sequences, because of the degeneracy in lifetime of states with the same $n_{x,s}$. A non-separable model would lift that degeneracy and then the lifetime sequences should show a behaviour intermediate between the two situations discussed above. In particular, the sudden change of slope at $Q=302$ in Fig. \[fig6\] would presumably spread over a wider range of values of $Q$. Not surprisingly, the predicted lifetimes for the observed decays depend sensitively on details of the barrier shape. Those shown in Fig. \[fig8\] correspond to the square well choice for $W(y)$, and our standard set of parameters. In addition we show how the lifetimes vary when the barrier width is changed by amounts ranging from $+ 4\%$ to $-3\%$ (from left to right). As can be seen, the exact value of each decay lifetime depends quite strongly on the barrier width, as expected for a tunnelling process. But the number of slow decays is much more stable: four or five in most of the cases shown, and in several cases their lifetimes are quite compatible with the experimental points. In particular it is remarkable that a $2\%$ increase in the standard barrier width produces a sequence (third line from left) in excellent agreement with experiment (disconnected points shifted to extreme left). There is a clear distinction between the lifetime trends of the thicker and thinner barrier widths. In the latter one sees very clearly the transition between escape from the $n_x = 13 $ and the $n_x = 14 $ levels at $Q = 302$. For the thicker barrier widths, escape is dominated by the $n_x = 14$ levels that become progressively more occupied above $Q = 302$. We have explored the dependence of the model predictions on changes by similar percentages of the barrier heights, potential floor $U_0$, and the width, $ w_y$, of $W(y)$. The results are qualitatively similar to those shown above for the changes in the barrier width, with a number of slow decays ranging from 4 to 6, and in some cases they are very similar to the data in Fig. \[fig1\]. We therefore conclude that our model predictions are quite consistent with the experimental trends, although a quantitative comparison with the measured lifetimes is hampered by the strong sensitivity of tunnelling to any small change in the barrier shape. Finally we show in Fig. \[fig9\] predictions for the fast decays: their number and location in a graph such as that of figure \[fig1\] depends very sensitively on the time ($t_0$) beyond which the experiment measures lifetimes. That is, as the dot is isolated there must be a burst of very short lived escapes, but after some seconds one reaches the stage where separate events can be recorded and the lifetimes deduced. The two dashed horizontal lines in Fig. \[fig9\] correspond to values of $\log_{10}( 25 \ s)$ and $\log_{10}(35 \ s)$. As can be seen, when we explore the same range of barrier widths as in Fig. \[fig8\], the number of fast decays above that $t_0$ varies from 3 to 4. The overall trend seems to be consistent with experiment, in particular if the value of $t_0$ is increased towards the more pessimistic estimate of $35 \ s$. Summary and Conclusions ======================= Electron escape from a strongly isolated dot with excess electrons has been studied in the framework of the self-consistent Poisson-Schrödinger and Poisson-Thomas-Fermi approximations. Based on these calculations a rectangular separable potential model has been devised which incorporates the main features of the self-consistent field. Rearrangement effects are taken into account by recalculating the confining potential $U_s(x,y)$ after each electron escape. The use of a separable potential introduces certain correlations in the energy spectrum of the single electron orbitals. A more realistic confining potential would have a more rounded shape, which would remove the separability, and modify those correlations. In the same vein, the tunnelling in our simplified model is 1D, whereas the actual process is 2D. We find it quite remarkable that despite all these simplifications the predictions turn out to be so satisfactory. The model therefore may be reliable for extrapolating to longer times. For instance we find that the isolated dot would hold one excess electron for as long as 44 hours. On such a time scale, one could use well isolated dots containing a few long lived electrons, to study their entangled states. This would open an interesting new approach to the implementation of quantum computation in semiconductor devices. We are grateful to DGES-Spain for continued support through grants PB97-0915 and UE97-0014(JM), to EPSRC-UK (CS) and to NSERC-Canada for research grant SAPIN-3198 (DWLS). This work was carried out as part of QUADRANT, Esprit project EP-23362 funded by the EU. Lifetimes ========== We summarize here the expressions relating the lifetimes to the probability of transmission across the barrier. We follow the standard treatment and definitions for alpha decay in nuclear physics, as can be found for example in ref.[@PB75]. Our potential $U_s(x,y)$ is separable, and the electron can escape only across the barriers in the $x$-direction. Therefore, we have adapted the expressions derived in [@PB75] to the 1D situation. The lifetime $\tau = 1/\lambda$ is the inverse of the “decay constant”, defined as the number of “decays” per second per parent “dot”. For one dot the electron wavefunction is normalized to unity over the volume inside the barriers, and $\lambda$ for a given level is just the outgoing flux at large distance. When the decay probability is small, one can treat the electron as confined in the dot. Classically, its trajectory will oscillate between the right, $x_r$, and left, $x_l$ turning points, with a period $$P = 2 \int_{x_r}^{x_l} {{ dx}\over v(x) } \ , \label{eq:a1}$$ where $v(x)$ is the classical electron velocity at energy $E_x$: $$v(x) = \sqrt{ {2\over m^*}\left( E_x - U(x) \right)} \, . \label{eq:a2}$$ The flux $\lambda$ is then given by the frequency of hits against the barriers, $2/P$, times the transmission probability $T$ across a barrier, and therefore: $$\tau = {1\over \lambda} = \frac{1}{T} \int_{x_l}^{x_r} {{dx} \over v} \ . \label{eq:a3}$$ This expression is very convenient because the transmission coefficient eq. \[eq:12\] for our parametrized potential, $U(x)$, is known analytically [@MF53]. For more general barrier profiles and the long lifetimes of interest, one can use the WKB approach and its corresponding connection formulae across the barrier (see e.g. Appendix D of [@PB75]): $$\begin{aligned} T_{WKB} &\approx& e^{ 2 \omega} \nonumber \\ \omega &=& \int_{x_r}^{x_t} \kappa \ dx = \int_{x_r}^{x_t} \sqrt{ {{2m }\over \hbar^2} \left( U(x) - E_x \right) }\, \, dx \label{eq:a4}\end{aligned}$$ If the WKB wave function is used inside the well to determine the period $P$, the same decay half-life is obtained as in eq. \[eq:a3\] above. Since the dot is located inside a crystal at temperature $T'$, via phonon coupling the electrons in the dot should also be at the same temperature. The level occupations $f(E)$ are determined by Fermi statistics: $$f(E) = \bigl[ {1 + e^{{ E-E_F} \over {k_BT'}}} \bigr]^{-1} \ , \label{eq:a5}$$ where these are now 2D energies. The Fermi level is obtained from $$Q = \sum_{i=(n_x,n_y)} \ 2 f(E_i) , \label{eq:a6}$$ where the $2$ accounts for spin degeneracy. For the ensemble of electrons in the dot, the flux $\lambda$ will now be the sum of fluxes for each occupied single particle level, weighted by the level occupancy: $$\lambda = \sum_{i=(n_x,n_y)} 2 f(E_i) \lambda_i \label{eq:a7}$$ and the corresponding half-life is still $\tau = 1/\lambda$. In particular this argument applies in the $T'=0$ limit, as we implicitly assumed in Section II to explain the sequence of lifetimes. Lifetime dependence on $Q$ =========================== For a level of given $n_x$, the lifetime depends on $Q$ because the barrier characteristics change, and so does the level energy, $E_{n_x}$. The latter varies mainly because $U_0$ depends on Q, and this affects the transmission probability $T$. To good approximation $${{ d E_{n_x}}\over {d Q} } \simeq {{d U_0}\over {d Q}} \label{eq:b1}$$ Neglecting the dependence of the level lifetime on the period $P$, we can write $$\begin{aligned} \frac{ d \,\ln \tau }{dQ} &\simeq& - {{ d \ln T}\over { d E}} {{dE} \over {dQ}} \ . \label{eq:b2}\end{aligned}$$ Taking the transmission probability $T$ from the WKB expression leads to $$\begin{aligned} {{ d \ln \tau }\over {dQ}} &=& 2 {d \over {dQ}} \int_{barrier} \sqrt{{{2m^*}\over \hbar^2} (U(x) -E) }\ dx \ \nonumber \\ &=& \int_{barrier} \sqrt{{{2m^*}\over \hbar^2} {1\over {U(x) -E}}} \left( {{dU(x)}\over{dQ}}\, -\, {{dE}\over{dQ}} \right) dx \nonumber \\ \label{eq:b3}\end{aligned}$$ Noting eq. \[eq:b1\], the second contribution to the integral depends linearly on the placement of the potential floor. However, the variation of the barrier shape ($dU(x) / dQ$) cannot be neglected. Indeed, eq. \[eq:8\] gives approximately $$\begin{aligned} {{dU(x)}\over {dQ}} &=& {dU_0\over dQ} \left( {1\over 3} + {2\over 3} {{\sinh^2 \left({{x-x_b}\over w_b}\right)}\over {e^{- 2\mu}\cosh^2\left({{x-x_b}\over w_b} - \mu\right)}}\right) \nonumber \\ && \hskip 2cm {\rm when} \quad x < x_b \nonumber \\ {{dU(x)}\over {dQ}} &=& {dU_0\over dQ} \left( {1\over 3} - {1\over 3} {{\sinh^2 \left({{x-x_b}\over w_b}\right)}\over \phantom{-} {e^{2\mu}\cosh^2\left({{x-x_b}\over w_b} - \mu\right)}} \right) \nonumber \\ && \hskip 2cm {\rm when} \quad x > x_b \ . \label{eq:b4}\end{aligned}$$ Using \[eq:b4\], the contribution from $dU/dQ$ to the integral of eq. \[eq:b3\] is obtained with an accuracy better than $ 2\%$. For the standard choice of parameters, and $Q$ in the range $300$ to $310$ the computed values of $d \log_{10} \tau / dQ$ turn out to be $\simeq -0.14$ for the levels of interest. In Fig. \[fig10\] we plot the evolution of the level lifetimes with $Q$, compared to the expression ($Q_0 = 303 $) $$\log_{10} \tau_{n_x}(Q) = \log_{10} \tau_{n_x}(Q_0) -0.14 ( Q - Q_0 ) \ . \label{eq:b5}$$ J. Cooper, C.G. Smith, D.A. Ritchie, E.H. Linfield, Y. Jin and H. Launois, “ Direct Observation of Single Electron Decay from an Artificial Nucleus”, Proceedings of EP2DS-13, Physica E [ **6**]{} (2000) 457-460. G. Gamow, Z. Phys. [**51**]{} (1928) 204. E.U. Condon and R. W. Gurney, Nature [**122**]{} (1928) 439. M.A. Preston and R.K. Bhaduri, “Structure of the Nucleus” Addison Wesley (1975), see Part III and Appendix D. W. van Dijk and Y. Nogami, Phys. Rev. Lett. [**83**]{} (1999) 2867-71. J. Martorell and D.W.L. Sprung, to be published, and unpublished QUADRANT progress reports. J. Martorell and D.W.L. Sprung, Phys. Rev. B [**49**]{} (1994) 13750-59. J. Martorell, Hua Wu and D.W.L. Sprung, Phys. Rev. B [**50**]{} (1994) 17298-308. W. Mönch, “Semiconductor Surfaces and Interfaces”, Springer (1995). J. H. Davies, Semicond. Sci. Technol. [**3**]{} (1988) 995. J.H. Davies, A. Larkin and E.V. Sukhorukov, J. Appl. Phys. [**77**]{} (1995) 4504-12. J. Martorell and D.W.L. Sprung, Phys. Rev. B [**54**]{} (1996) 11386-96. “Methods of Theoretical Physics", P.M. Morse and H. Feshbach, McGraw Hill, (1953); see pp. 1651-60. J Killingbeck, J. Phys. A [**10**]{} (1977) L99-L103; Phys. Lett. A [**78**]{} (1980) 235-6.

--- abstract: 'Microscopic structure of the low-lying isovector dipole excitation mode in neutron-rich $^{26,28,30}$Ne is investigated by performing deformed quasiparticle-random-phase-approximation (QRPA) calculations. The particle-hole residual interaction is derived from a Skyrme force through a Landau-Migdal approximation. We have obtained the low-lying resonance in $^{26}$Ne at around 8.5 MeV. It is found that the isovector dipole strength at $E_{x}<10$ MeV exhausts about 6.0% of the classical Thomas-Reiche-Kuhn dipole sum rule. This excitation mode is composed of several QRPA eigenmodes, one is generated by a $\nu(2s^{-1}_{1/2} 2p_{3/2})$ transition dominantly, and the other mostly by a $\nu(2s^{-1}_{1/2} 2p_{1/2})$ transition. The neutron excitations take place outside of the nuclear surface reflecting the spatially extended structure of the $2s_{1/2}$ wave function. In $^{30}$Ne, the deformation splitting of the giant resonance is large, and the low-lying resonance is overlapping with the giant resonance.' author: - 'Kenichi Yoshida$^{1,2}$' - 'Nguyen Van Giai$^{2}$' title: 'Low-lying dipole resonance in neutron-rich Ne isotopes ' --- Introduction ============ The study of nuclei far off stability is one of the most active research fields in nuclear physics [@tan01; @hor01; @hag02a], and exploring the collective motions unique in unstable nuclei is one of the main issues experimentally and theoretically [@neu07]. In neutron-rich nuclei, because of the absence of the Coulomb barrier the surface structure is quite different from stable nuclei. One of the unique structures is the neutron skin [@suz95; @miz00]. Since the collective excitations are sensitive to the surface structure, one can expect new kinds of exotic excitation modes associated with the neutron skin to appear in neutron-rich nuclei. One of the examples is the soft dipole excitation [@ike88], which is observed not only in light halo nuclei [@sac93; @shi95; @zin97; @nak06; @nak94; @pal03; @fuk04; @nak99; @pra03; @aum99], but also in heavier systems [@lei01; @try03; @adr05], where an appreciable $E1$ strength is observed above the neutron threshold exhausting several percents of the energy-weighted sum rule (EWSR). The structure of the low-lying dipole state and its collectivity has been studied in the framework of the mean-field calculations by many groups [@cat97; @ham98; @ham99; @gor02; @mat02; @ter06; @col01; @sar04; @vre01a; @vre01b; @paa05; @cao05; @liv07]. A low-lying dipole state in neutron-rich $^{26}$Ne was first predicted by using the relativistic quasiparticle-random-phase approximation (QRPA) in Ref. [@cao05], and recently it was observed at RIKEN around 9 MeV, exhausting about 5% of the Thomas-Reiche-Kuhn (TRK) dipole sum rule [@gib07]. In Ref. [@cao05], the QRPA was solved in the response function formalism. This method can treat the excitations to the continuum exactly by employing the Green’s functions satisfying the out-going-wave boundary conditions, but an additional procedure is required to obtain the microscopic structure of the excitation mode [@kha05]. In the present paper, we investigate the microscopic structure of the low-lying dipole resonance in neutron-rich Ne isotopes, and we discuss the isotopic dependence with special attention to the deformation effects. To this end, we have developed a deformed QRPA code in the matrix formulation based on the coordinate-space Skyrme-Hartree-Fock-Bogoliubov (HFB) theory. The paper is organized as follows: In Sec. \[model\], we explain our method. In Sec. \[check\], we check the results of our new calculation scheme by comparing the existing QRPA results. In Sec. \[results\], we present the results of the deformed QRPA and we discuss the microscopic structure of the low-lying dipole state in $^{26,28,30}$Ne. Finally, we summarize the paper in Sec. \[summary\]. \[model\]Model ============== We briefly summarize here our approach (see Ref. [@yos06] for details). In order to discuss simultaneously effects of nuclear deformation and pairing correlations including the continuum, we solve the HFB equations [@dob84; @bul80] $$\begin{gathered} \begin{pmatrix} h^{\tau}(\boldsymbol{r}\sigma)-\lambda^{\tau} & \tilde{h}^{\tau}(\boldsymbol{r}\sigma) \\ \tilde{h}^{\tau}(\boldsymbol{r}\sigma) & -(h^{\tau}(\boldsymbol{r}\sigma)-\lambda^{\tau}) \end{pmatrix} \begin{pmatrix} \varphi^{\tau}_{1,\alpha}(\boldsymbol{r}\sigma) \\ \varphi^{\tau}_{2,\alpha}(\boldsymbol{r}\sigma) \end{pmatrix} \\ = E_{\alpha} \begin{pmatrix} \varphi^{\tau}_{1,\alpha}(\boldsymbol{r}\sigma) \\ \varphi^{\tau}_{2,\alpha}(\boldsymbol{r}\sigma) \end{pmatrix} \label{eq:HFB1}\end{gathered}$$ directly in the cylindrical coordinates assuming axial and reflection symmetries. Here, $\tau=\nu$ (neutron) and $\pi$ (proton), and $\boldsymbol{r}=(\rho,z,\phi)$. For the mean-field Hamiltonian $h$, we employ the SkM\* interaction [@bar82]. Details for expressing the densities and currents in the cylindrical coordinate representation can be found in Refs. [@ter03; @sto05]. The pairing field is treated by using the density-dependent contact interaction [@ber91; @ter95], $$v_{pp}(\boldsymbol{r},\boldsymbol{r}^{\prime})=V_{0}\dfrac{1-P_{\sigma}}{2} \left[ 1- \left(\dfrac{\varrho^{\mathrm{IS}}(\boldsymbol{r})}{\varrho_{0}}\right)^{\gamma} \right] \delta(\boldsymbol{r}-\boldsymbol{r}^{\prime}). \label{eq:res_pp}$$ with $V_{0}=-390$ MeV $\cdot$fm$^{2}$ and $\varrho_{0}=0.16$ fm$^{-3}$, $\gamma=1$. Here, $\varrho^{\mathrm{IS}}(\boldsymbol{r})$ denotes the isoscalar density and $P_{\sigma}$ the spin exchange operator. The pairing strength $V_{0}$ is determined so as to approximately reproduce the experimental pairing gap of 1.25 MeV in $^{28}$Ne obtained by the three-point formula [@sat98]. Because the time-reversal symmetry and reflection symmetry with respect to the $x-y$ plane are assumed, we have only to solve for positive $\Omega$ and positive $z$. We use the lattice mesh size $\Delta\rho=\Delta z=0.6$ fm and the box boundary condition at $\rho_{\mathrm{max}}=9.9$ fm and $z_{\mathrm{max}}=9.6$ fm. The quasiparticle energy is cut off at 60 MeV and the quasiparticle states up to $\Omega^{\pi}=13/2^{\pm}$ are included. Using the quasiparticle basis obtained by solving the HFB equation (\[eq:HFB1\]), we solve the QRPA equation in the matrix formulation [@row70] $$\sum_{\gamma \delta} \begin{pmatrix} A_{\alpha \beta \gamma \delta} & B_{\alpha \beta \gamma \delta} \\ B_{\alpha \beta \gamma \delta} & A_{\alpha \beta \gamma \delta} \end{pmatrix} \begin{pmatrix} X_{\gamma \delta}^{\lambda} \\ Y_{\gamma \delta}^{\lambda} \end{pmatrix} =\hbar \omega_{\lambda} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} X_{\alpha \beta}^{\lambda} \\ Y_{\alpha \beta}^{\lambda} \end{pmatrix} \label{eq:AB1}.$$ The residual interaction in the particle-particle (p-p) channel appearing in the QRPA matrices $A$ and $B$ is the density-dependent contact interaction (\[eq:res\_pp\]). On the other hand, for the residual interaction in the particle-hole (p-h) channel, we employ the Landau-Migdal (LM) approximation [@bac75] applied to the density-dependent Skyrme forces [@gia81; @gia98], $$\begin{aligned} v_{ph}(\boldsymbol{r},\boldsymbol{r}^{\prime})=& N_{0}^{-1}\{F_{0}+F_{0}^{\prime}\tau\cdot\tau^{\prime} \notag \\ &+(G_{0}+G_{0}^{\prime} \tau\cdot\tau^{\prime})\sigma\cdot\sigma^{\prime} \} \delta(\boldsymbol{r}-\boldsymbol{r}^{\prime}). \label{eq:res_ph}\end{aligned}$$ Here, $N_{0}$ is the density of states and the Landau parameters are deduced from the same Skyrme force which generates the mean field. Because the full self-consistency between the static mean-field calculation and the dynamical QRPA calculation is broken, we have to renormalize the residual interaction in the particle-hole channel by an overall factor $f_{ph}$ to get the spurious $K^{\pi}=0^{-}$ or $1^{-}$ modes (representing the center-of-mass motion) at zero energy ($v_{ph} \rightarrow f_{ph}\cdot v_{ph}$). We cut the two-quasiparticle space at $E_{\alpha}+E_{\beta} \leq 60$ MeV due to the excessively demanding computer memory as well as the calculation time if we used a model space consistent with that adopted in the HFB calculation. Accordingly, we need another factor $f_{pp}$ for the particle-particle channel. We determine this factor such that the spurious $K^{\pi}=0^{+}$ mode associated with the particle number fluctuation appears at zero energy ($v_{pp} \rightarrow f_{pp}\cdot v_{pp}$). \[check\]Check of the calculation scheme ======================================== ![Response function for the isoscalar quadrupole operator in $^{22}$O. The transition strengths are smeared by using a Lorentzian function with a width of $\Gamma=0.5$ MeV. The renormalization factors for the QRPA calculation are $f_{ph}=0.982, f_{pp}=1.18$. The cutoff energy is 60 MeV. []{data-label="22O_response"}](fig1.eps) In this section, we compare our results with those of Ref. [@kha02]. In this reference, the SLy4 interaction [@cha98] for the mean field and the surface-type delta interaction with $\gamma=1.5$ and $V_{0}=-415.73$ MeV$\cdot$fm$^{3}$ for the pairing field were employed for the HFB calculation, and the quasiparticle energy was cut off at 50 MeV. Therefore, we adopt these parameters for the comparisons in this section. The differences between the present calculation and that in Ref. [@kha02] are the mesh size, the boundary condition, the cutoff energy for the QRPA calculation, and the treatment of the spin-dependent interaction ($G_{0}$ and $G_{0}^{\prime}$) in Eq. (\[eq:res\_ph\]). In the present calculation, the spin transition density is treated exactly. ![Cutoff energy dependence of the renormalization factors and the $B(E2\uparrow)$ value for the first $2^{+}$ state in $^{22}$O. []{data-label="22O_dep"}](fig2.eps) In Fig. \[22O\_response\], we show the isoscalar quadrupole response function in $^{22}$O. The first $2^{+}$ state is located at 2.8 MeV with $B(E2\uparrow)=18.9$ $e^{2}$fm$^{4}$. The experimental values are $E(2^{+})_{\mathrm{exp}}=3.2$ MeV and $B(E2)_{\mathrm{exp}}=21\pm 8$ $e^{2}$fm$^{4}$ [@bel01; @thi00; @bec06]. In Ref. [@kha02], the energy and the transition strength are $E(2_{1}^{+})=1.9$ MeV and $B(E2)=22$ $e^{2}$fm$^{4}$. The energy and the transition strength of the low-lying collective state is quite sensitive to the cutoff energy for the RPA calculation [@bla77]. In Fig. \[22O\_dep\], the cutoff energy dependence of the renormalization factors and the $B(E2\uparrow)$ value for the $2_{1}^{+}$ state in $^{22}$O are shown. Even with the cutoff energy of 70 MeV, the transition strength for the low-lying state does not converge yet. In this case, the dimension of the QRPA matrix in Eq. (\[eq:AB1\]) is 11726 for the $K^{\pi}=0^{+}$ channel and the memory size is 13 GB, and the CPU time is about 70,000s per each iteration for determining the renormalization factor $f_{pp}$. If we could perform the QRPA calculation including all quasiparticle states obtained in the HFB calculation, the renormalization factor for the pairing channel $f_{pp}$ would be 1, because the p-p channel is treated self-consistently between the HFB and the QRPA calculations. The peak position of the giant resonance is located slightly higher than in Ref.[@kha02]. The non-collective two-quasiparticle states around 6 and 7 MeV are consistent between the two calculations. The energy-weighted sum ($1.867\times10^{4}$MeV$\cdot$fm$^{4}$) overestimates by about 13.9% the EWSR value ($1.638\times 10^{4}$MeV$\cdot$fm$^{4}$). The overshooting of the EWSR for the isoscalar quadrupole mode in the LM approximation was pointed out in Ref [@miz07].  \[results\]Results and Discussion ================================= $^{26}$Ne $^{28}$Ne $^{30}$Ne ------------------------------------------- ------------ ------------- ------------- $\lambda_{\nu}$ (MeV) $-4.60$ $-3.06$ $-2.90$ $\lambda_{\pi}$ (MeV) $-14.8$ $-17.0$ $-19.9$ $\beta_{2}^{\nu}$ 0.08 0.12 0.32 $\beta_{2}^{\pi}$ 0.14 0.20 0.39 $\langle \Delta_{\nu} \rangle$ (MeV) 0.0 (0.70) 1.27 (1.24) 1.34 (1.30) $\langle \Delta_{\pi} \rangle$ (MeV) 1.04 0.87 0.0 $\sqrt{\langle r^{2} \rangle_{\nu}}$ (fm) 3.20 3.35 3.53 $\sqrt{\langle r^{2} \rangle_{\pi}}$ (fm) 2.93 2.98 3.08 : Ground state properties of $^{26,28,30}$Ne obtained by the deformed HFB calculation with the SkM\* interaction and the surface-type pairing interaction. Chemical potentials, deformations, average pairing gaps and root-mean-square radii for neutrons and protons are listed. []{data-label="GS"} We now discuss the properties of $^{26,28,30}$Ne nuclei calculated with the SkM\* interaction. We summarize in Table \[GS\] the ground state properties of these Ne isotopes obtained by solving Eq. (\[eq:HFB1\]). The ground state is slightly deformed in $^{26}$Ne and $^{28}$Ne, and we obtain a well-deformed ground state for $^{30}$Ne. The values in parentheses are experimental pairing gaps extracted by the three-point mass difference formula [@sat98] using the experimental binding energies taken from Ref. [@aud95]. We define the deformation parameter $\beta_{2}$ and average pairing gap $\langle\Delta\rangle$ [@sau81; @ben00; @dug01; @yam01] as $$\begin{aligned} \beta_{2}^{\tau}&=\dfrac{4\pi}{5}\dfrac{\int d\bold{r} \varrho^{\tau}(\bold{r})r^{2}Y_{20}(\hat{r})} {\int d\bold{r} \varrho^{\tau}(\bold{r})r^{2}},\\ \langle\Delta_{\tau}\rangle&=-\dfrac{\int d\bold{r} \tilde{\varrho}^{\tau}(\bold{r}) \tilde{h}^{\tau}(\bold{r})}{\int d\bold{r} \tilde{\varrho}^{\tau}(\bold{r})},\end{aligned}$$ where $\tilde{\varrho}(\boldsymbol{r})$ is the pairing density. Fig. \[response\] shows the response functions for the isovector dipole mode in neutron-rich Ne isotopes. The isovector dipole operator used in the present calculation is $$\hat{F}_{1K}=e\dfrac{N}{A}\sum_{i}^{Z}r_{i}Y_{1K}(\hat{r}_{i})- e\dfrac{Z}{A}\sum_{i}^{N}r_{i}Y_{1K}(\hat{r}_{i}),$$ and the response functions are calculated as $$S(E)=\sum_{i}\sum_{K} \dfrac{\Gamma/2}{\pi}\dfrac{|\langle i|\hat{F}_{1K}|0\rangle|^{2}} {(E-\hbar \omega_{i})^{2}+\Gamma^{2}/4}.$$ $^{26}$Ne --------- We can clearly see a resonance structure at around the excitation energy of 8-9 MeV, together with the giant resonance at $15-20$ MeV. Because of the small deformation the $K$ splitting is small and smeared out. ![Isovector dipole transition strengths in $^{26}$Ne for the $K^{\pi}=0^{-}$ (the upper) and $K^{\pi}=1^{-}$ (the lower) states. Underlying discrete states are shown together with the smeared response functions. The arrow indicates the neutron emission threshold $E_{\mathrm{th}}=6.58$ MeV. []{data-label="strength"}](fig4.eps) ----- --------------- --------------- ------------------------ -------------------------------------------- ---------------------- $E_{\alpha}+E_{\beta}$ $Q_{10,\alpha\beta}$ $\alpha$ $\beta$ (MeV) $X_{\alpha \beta}^{2}-Y_{\alpha\beta}^{2}$ ($e\cdot$ fm) (a) $\nu[310]1/2$ $\nu[211]1/2$ 8.15 0.670 $-0.309$ (b) $\nu[330]1/2$ $\nu[220]1/2$ 11.4 0.020 $-0.397$ (c) $\nu[312]5/2$ $\nu[202]5/2$ 11.2 0.006 $-0.239$ (d) $\nu[321]3/2$ $\nu[211]3/2$ 11.3 0.006 0.338 (e) $\nu[330]1/2$ $\nu[211]1/2$ 6.54 0.003 $-0.118$ (f) $\nu[312]3/2$ $\nu[211]3/2$ 12.8 0.002 $-0.014$ (g) $\nu[301]1/2$ $\nu[211]1/2$ 9.32 0.002 $-0.117$ (h) $\nu[200]1/2$ $\nu[101]1/2$ 14.0 0.002 $-0.241$ (i) $\nu 1/2^{-}$ $\nu[211]1/2$ 12.6 0.002 $-0.068$ (j) $\pi[220]1/2$ $\pi[101]1/2$ 7.96 0.265 0.0085 (k) $\pi[330]1/2$ $\pi[220]1/2$ 13.4 0.008 $-0.329$ (l) $\pi[220]1/2$ $\pi[110]1/2$ 14.1 0.008 $-0.346$ ----- --------------- --------------- ------------------------ -------------------------------------------- ---------------------- : QRPA amplitudes for the $K^{\pi}=0^{-}$ state in $^{26}$Ne at 8.25 MeV. This mode has the proton strength $B(E1)=2.98 \times10^{-2}~e^{2}$fm$^{2}$, the neutron strength $B(Q^{\nu}1)=2.89 \times10^{-2} e^{2}$fm$^{2}$, and the isovector strength $B(Q^{\mathrm{IV}}1)=1.17 \times10^{-1} e^{2}$fm$^{2}$, and the sum of backward-going amplitude $\sum|Y_{\alpha\beta}|^{2}=4.33\times 10^{-3}$. The single-(quasi)particle levels are labeled with the asymptotic quantum numbers $[Nn_{3}\Lambda]\Omega$. Only components with $X_{\alpha\beta}^{2}-Y_{\alpha\beta}^{2} > 0.001$ are listed. Two-quasiparticle excitation energies are given by $E_{\alpha}+E_{\beta}$ in MeV and two-quasiparticle transition matrix elements $Q_{10,\alpha\beta}$ in $e \cdot$fm. In the row (i), the label $\nu 1/2^{-}$ denotes a non-resonant discretized continuum state of neutron $\Omega^{\pi}=1/2^{-}$ level. []{data-label="26Ne_0-"} ----- --------------- --------------- ------------------------ -------------------------------------------- ---------------------- $E_{\alpha}+E_{\beta}$ $Q_{11,\alpha\beta}$ $\alpha$ $\beta$ (MeV) $X_{\alpha \beta}^{2}-Y_{\alpha\beta}^{2}$ ($e\cdot$ fm) (a) $\nu[312]3/2$ $\nu[211]1/2$ 8.68 0.849 0.339 (b) $\nu[310]1/2$ $\nu[211]1/2$ 8.16 0.040 $-0.131$ (c) $\nu[301]1/2$ $\nu[211]1/2$ 9.32 0.010 0.294 (d) $\nu[321]3/2$ $\nu[220]1/2$ 12.0 0.007 0.250 (e) $\nu[303]7/2$ $\nu[202]5/2$ 12.1 0.006 0.414 (f) $\nu[330]1/2$ $\nu[220]1/2$ 11.4 0.004 $-0.127$ (g) $\nu[312]5/2$ $\nu[211]3/2$ 12.1 0.004 0.348 (h) $\nu[321]3/2$ $\nu[202]5/2$ 10.3 0.001 $-0.010$ (i) $\nu[321]3/2$ $\nu[202]5/2$ 11.8 0.003 $-0.214$ (j) $\nu[330]1/2$ $\nu[211]1/2$ 6.54 0.003 $-0.081$ (k) $\nu[321]3/2$ $\nu[211]1/2$ 7.14 0.001 0.106 (l) $\pi[220]1/2$ $\pi[101]1/2$ 7.96 0.037 0.0095 (m) $\pi[211]3/2$ $\pi[101]1/2$ 7.95 0.015 $-0.011$ (n) $\pi[321]3/2$ $\pi[220]1/2$ 14.0 0.004 0.313 (o) $\pi[312]5/2$ $\pi[211]3/2$ 14.7 0.002 $-0.338$ (p) $\pi[211]3/2$ $\pi[110]1/2$ 14.1 0.002 0.280 (q) $\pi[211]1/2$ $\pi[101]1/2$ 11.5 0.002 $-0.256$ ----- --------------- --------------- ------------------------ -------------------------------------------- ---------------------- : Same as Table \[26Ne\_0-\] but for the $K^{\pi}=1^{-}$ state in $^{26}$Ne at 8.76 MeV. This mode has $B(E1)=1.65 \times10^{-2}~e^{2}$fm$^{2}$, $B(Q^{\nu}1)=3.58 \times10^{-2} e^{2}$fm$^{2}$, $B(Q^{\mathrm{IV}}1)=1.00 \times10^{-1} e^{2}$fm$^{2}$, and $\sum|Y_{\alpha\beta}|^{2}=2.93\times 10^{-3}$. []{data-label="26Ne_1-"} In Fig. \[strength\], we show the transition strengths in the low-energy region. The neutron emission threshold is 6.35 MeV, and the resonance which is composed of several discrete states appears just above the threshold. In contrast to the low-lying quadrupole state in $^{22}$O, the transition strengths for the dipole states in this region converge at the cutoff energy of about 40 MeV. We made a detailed analysis of the QRPA eigenmodes and show in Tables \[26Ne\_0-\],\[26Ne\_1-\] the microscopic structures of the $K^{\pi}=0^{-}$ state at 8.25 MeV and the $K^{\pi}=1^{-}$ state at 8.76 MeV, which have the largest transition strength for each sector. In the Tables, single-(quasi)particle states are labeled with the asymptotic quantum numbers $[Nn_{3}\Lambda]\Omega$ just for convenience. It should be noted that the asymptotic quantum numbers are not good quantum numbers because the deformation is not so large. For the $K^{\pi}=0^{-}$ state at 8.25 MeV, the dominant component is the $\nu[211]1/2 \to \nu[310]1/2$ transition, corresponding to $\nu(2s^{-1}_{1/2} 2p_{3/2})$. Particle-hole excitations of (b), (c) and (d) correspond to $\nu(1d^{-1}_{5/2}1f_{7/2})$ excitation, which have 3.2% contribution in total. The rows (e), (f), (g) and (h) correspond to $\nu(2s^{-1}_{1/2}1f_{7/2})$, $\nu(1d^{-1}_{5/2}2p_{3/2})$, $\nu(2s^{-1}_{1/2}2p_{1/2})$ and $\nu(1p^{-1}_{1/2}1d_{3/2})$ excitations, respectively. Two-quasiparticle proton excitations, furthermore, have an appreciable contribution; the rows (j), (k), and (l) correspond to the hole-hole like $\pi (1p_{1/2} \otimes 1d_{5/2})$ excitation, particle-hole like $\pi (1d^{-1}_{5/2}1f_{7/2})$, and $\pi (1d^{-1}_{5/2}1p_{3/2})$ excitations, respectively. The $K^{\pi}=1^{-}$ state has a similar structure to the $K^{\pi}=0^{-}$ state. The main component is $\nu(2s^{-1}_{1/2} 2p_{3/2})$, which corresponds to (a) and (b) in Table \[26Ne\_1-\]. With a small contribution, many other neutron particle-hole and proton two-quasiparticle excitations build the excitation mode at 8.76 MeV. The resonance is also composed of the $K^{\pi}=1^{-}$ mode appearing at 9.40 MeV. This mode is dominantly (97.6%) generated by the $\nu[211]1/2 \to \nu[301]1/2$ transition corresponding to the $\nu(2s_{1/2}^{-1}2p_{1/2})$ transition. Preliminary calculations of deformed QRPA using the Gogny interaction [@per07], and the relativistic deformed QRPA [@pen07] show that the low-lying dipole state is dominantly constructed by the $\nu(2s^{-1}_{1/2}2p_{3/2})$ configuration. ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- ![Transition densities in $^{26}$Ne for the $K^{\pi}=0^{-}$ state at 8.25 MeV (upper panels), and for the $K^{\pi}=1^{-}$ state at 8.76 MeV (lower panels). Solid and dotted lines indicate positive and negative transition densities, and the contour lines are plotted at intervals of $3 \times 10^{-4}$ fm$^{-3}$. The thick solid lines indicate the neutron and proton half density, 0.058 fm$^{-3}$ and 0.036 fm$^{-3}$, respectively. []{data-label="trans_density"}](fig5-1.eps "fig:") ![Transition densities in $^{26}$Ne for the $K^{\pi}=0^{-}$ state at 8.25 MeV (upper panels), and for the $K^{\pi}=1^{-}$ state at 8.76 MeV (lower panels). Solid and dotted lines indicate positive and negative transition densities, and the contour lines are plotted at intervals of $3 \times 10^{-4}$ fm$^{-3}$. The thick solid lines indicate the neutron and proton half density, 0.058 fm$^{-3}$ and 0.036 fm$^{-3}$, respectively. []{data-label="trans_density"}](fig5-2.eps "fig:") ![Transition densities in $^{26}$Ne for the $K^{\pi}=0^{-}$ state at 8.25 MeV (upper panels), and for the $K^{\pi}=1^{-}$ state at 8.76 MeV (lower panels). Solid and dotted lines indicate positive and negative transition densities, and the contour lines are plotted at intervals of $3 \times 10^{-4}$ fm$^{-3}$. The thick solid lines indicate the neutron and proton half density, 0.058 fm$^{-3}$ and 0.036 fm$^{-3}$, respectively. []{data-label="trans_density"}](fig5-3.eps "fig:") ![Transition densities in $^{26}$Ne for the $K^{\pi}=0^{-}$ state at 8.25 MeV (upper panels), and for the $K^{\pi}=1^{-}$ state at 8.76 MeV (lower panels). Solid and dotted lines indicate positive and negative transition densities, and the contour lines are plotted at intervals of $3 \times 10^{-4}$ fm$^{-3}$. The thick solid lines indicate the neutron and proton half density, 0.058 fm$^{-3}$ and 0.036 fm$^{-3}$, respectively. []{data-label="trans_density"}](fig5-4.eps "fig:") ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- ![Same as Fig. \[trans\_density\], for the unperturbed transition density of $\nu[211]1/2 \to [310]1/2$ excitation.[]{data-label="26Ne_wf"}](fig6.eps) In Fig. \[trans\_density\], we show transition densities for the $K^{\pi}=0^{-}$ state at 8.25 MeV, and for the $K^{\pi}=1^{-}$ state at 8.76 MeV. These transition densities are quite different from the classical picture of the isovector giant resonances. They have an isoscalar character in the surface region of the nucleus. On the other hand, outside of the nucleus, neutrons have an oscillation and the neutron excitation is dominant. Furthermore, the neutron excitations take place in the low-density region around 6 fm, namely, this mode has a unique picture of vibration of the neutron skin. The spatially extended structure of the $\nu[211]1/2$ state is responsible for this tail of the neutron transition density. In order to clearly see the spatial structure of $\nu[211]1/2$, the unperturbed transition density of $\nu[211]1/2 \to [310]1/2$ is shown in Fig. \[26Ne\_wf\]. It is clear that the wave function extends far outside of the nucleus, and the extension is larger than in the dipole state of Fig. \[trans\_density\]. Furthermore, the structure around the surface is different between the unperturbed one and that obtained in the QRPA. These differences are generated by the QRPA correlations; the low-lying dipole state possesses a collective nature, which is small but finite. ![Energy weighted sum of the isovector dipole strength function. The solid and the dotted lines are calculations with the energy cutoff at 60 and 30 MeV. The horizontal lines show the classical TRK, and the RPA sum rule including the enhancement factor, $m_{1}=m_{1}^{\mathrm{cl}}(1+\kappa)$. Here $\kappa=0.32$ for the SkM\* interaction in $^{26}$Ne.[]{data-label="EWSR"}](fig7.eps) Fig. \[EWSR\] shows the energy weighted sum of the isovector dipole strength function together with the sum rule values represented by the horizontal lines. The calculated sum satisfies 89.2% of the EWSR value including the enhancement factor $\kappa$; $m_{1}=m_{1}^{\mathrm{cl}}(1+\kappa)$ [@ter06]. The enhancement factor comes from the momentum dependence of the Skyrme density functionals. The effect of the explicit treatment of the momentum dependence for the EWSR was discussed in the discretized-continuum QRPA [@yam02] and the continuum QRPA [@miz07] for the spherical systems. In the present calculation, we treat the momentum dependence in the LM approximation. Therefore, discrepancy between the calculation and the EWSR value comes from this treatment of the momentum dependence. This point remains to be improved, and it is discussed in Ref. [@yos08]. In the present calculation, the energy-weighted sum up to 10 MeV is 5.51 MeV$\cdot e^{2}$fm$^{2}$, corresponding to 6.0% of the TRK sum-rule value, 4.6% of the EWSR including the enhancement factor and 5.1% of the calculated sum. These values are consistent with the experiment [@gib07]. In Fig. \[EWSR\], we also show the energy-weighted sum calculated with the energy cutoff at 30 MeV (dotted line). In the giant resonance region, two calculations give different results, while they are almost identical in the low-energy region. This is because the collectivity of the low-lying resonance is small, and consequently the transition strength is not very sensitive to the cutoff energy. Before going to the neighboring nuclei, it should be noted that we obtain the collective octupole state at about 5.2 MeV, below the neutron threshold, with $B(E3\uparrow)=2458 e^{2}$fm$^{6}$, which corresponds to about 61 in Weisskopf units and the isoscalar transition strength is $2.60\times10^{4}$fm$^{6}$. The lowest $K^{\pi}=0^{-}$ state is located at 5.03 MeV and the sum of the backward-going amplitudes is 0.099. This state is generated by $\nu[211]1/2\to[330]1/2 (53.2\%)$, $\nu[202]5/2\to[312]5/2 (6.8\%)$, $\nu[211]1/2\to[310]1/2 (6.5\%)$, $\nu[220]1/2\to[330]1/2 (3.7\%)$, and $\pi[101]2/1\otimes[220]1/2 (14.8\%)$, $\pi[101]3/2\otimes[211]3/2 (2.2\%)$, $\pi[220]1/2\otimes[330]1/2 (2.1\%)$, $\pi[101]1/2\otimes[211]1/2 (1.0\%)$. 28Ne and 30Ne ------------- ![Same as Fig. \[strength\] but for $^{28}$Ne and $^{30}$Ne. The arrows indicate the neutron emission threshold $E_{\mathrm{th}}=4.44$ and 4.51 MeV. []{data-label="28Ne_strength"}](fig8-1.eps "fig:") ![Same as Fig. \[strength\] but for $^{28}$Ne and $^{30}$Ne. The arrows indicate the neutron emission threshold $E_{\mathrm{th}}=4.44$ and 4.51 MeV. []{data-label="28Ne_strength"}](fig8-2.eps "fig:") The central panel in Fig. \[response\] shows the response function in $^{28}$Ne. In the low-energy region, we can see a two-bump structure at around 7 and 8 MeV. Because the deformation is small as in $^{26}$Ne, we cannot see a splitting of the giant resonance. In Fig. \[28Ne\_strength\], we show the low-energy part of the strength functions. In the $K^{\pi}=0^{-}$ states, there is a prominent peak at 8.1 MeV with a strength of 0.098 $e^{2}$fm$^{2}$. The strength distribution is fragmented for the $K^{\pi}=1^{-}$ mode, but correspondingly, we can see an eigenmode at 8.9 MeV with the largest transition strength of 0.058 $e^{2}$fm$^{2}$. We show in Table \[28Ne\_0-\] the QRPA amplitude for the $K^{\pi}=0^{-}$ state at 8.14 MeV in $^{28}$Ne. The main component is the neutron two-quasiparticle excitation of $\nu([310]1/2 \otimes [211]1/2)$ corresponding to $\nu(2s^{-1}_{1/2}2p_{3/2})$. Two quasiparticle excitations of (b) and (c) in Table \[28Ne\_0-\] correspond to $\nu(1d^{-1}_{5/2}1f_{7/2})$, and (d): $\nu(2s^{-1}_{1/2}1f_{7/2})$, (e) and (f): $\nu(1d^{-1}_{3/2}2p_{3/2})$, (g): $\nu(1d^{-1}_{3/2}1f_{7/2})$, (h) and (i): $\nu(1d^{-1}_{5/2}2p_{3/2})$, and (j): $\nu(1d^{-1}_{5/2}2p_{1/2})$ excitations, respectively. The proton excitation of $\pi(1p_{1/2}\otimes 1d_{5/2})$ has an appreciable contribution as in $^{26}$Ne. ----- --------------- --------------- ------------------------ -------------------------------------------- ---------------------- $E_{\alpha}+E_{\beta}$ $Q_{10,\alpha\beta}$ $\alpha$ $\beta$ (MeV) $X_{\alpha \beta}^{2}-Y_{\alpha\beta}^{2}$ ($e\cdot$ fm) (a) $\nu[310]1/2$ $\nu[211]1/2$ 8.27 0.569 $-0.303$ (b) $\nu[330]1/2$ $\nu[220]1/2$ 11.2 0.055 $-0.373$ (c) $\nu[321]3/2$ $\nu[211]3/2$ 10.9 0.006 0.323 (d) $\nu[330]1/2$ $\nu[211]1/2$ 6.20 0.036 $-0.096$ (e) $\nu[312]3/2$ $\nu[202]3/2$ 6.82 0.004 $-0.039$ (f) $\nu[310]1/2$ $\nu[200]1/2$ 5.81 0.003 0.026 (g) $\nu[330]1/2$ $\nu[200]1/2$ 3.74 0.004 $-0.006$ (h) $\nu[321]3/2$ $\nu[211]3/2$ 12.9 0.002 $-0.014$ (i) $\nu[310]1/2$ $\nu[220]1/2$ 13.3 0.001 $0.0004$ (j) $\nu[301]1/2$ $\nu[220]1/2$ 14.5 0.001 $-0.022$ (k) $\nu 1/2^{-}$ $\nu[200]1/2$ 10.1 0.009 0.203 (l) $\nu 1/2^{-}$ $\nu[202]3/2$ 10.7 0.002 0.099 (m) $\nu 1/2^{-}$ $\nu[211]1/2$ 12.6 0.004 $-0.071$ (n) $\nu 1/2^{+}$ $\nu[330]1/2$ 14.2 0.003 $-0.114$ (o) $\pi[220]1/2$ $\pi[101]1/2$ 7.63 0.238 0.018 (p) $\pi[330]1/2$ $\pi[220]1/2$ 12.8 0.031 $-0.463$ (q) $\pi[220]1/2$ $\pi[110]1/2$ 13.7 0.007 $-0.441$ (r) $\pi[211]3/2$ $\pi[101]3/2$ 12.3 0.004 $-0.310$ (s) $\pi[211]1/2$ $\pi[101]1/2$ 11.6 0.002 0.288 (t) $\pi 1/2^{-}$ $\pi[220]1/2$ 19.3 0.001 0.004 ----- --------------- --------------- ------------------------ -------------------------------------------- ---------------------- : Same as Table \[26Ne\_0-\] but for the $K^{\pi}=0^{-}$ state at 8.14 MeV in $^{28}$Ne. This mode has $B(E1)=2.62 \times10^{-2}~e^{2}$fm$^{2}$, $B(Q^{\nu}1)=2.29 \times10^{-2} e^{2}$fm$^{2}$, $B(Q^{\mathrm{IV}}1)=9.80 \times10^{-2} e^{2}$fm$^{2}$, and $\sum|Y_{\alpha\beta}|^{2}=9.77\times 10^{-3}$. In the rows (k), (l), (m), (n) and (t), the labels $\nu 1/2^{-}$, $\nu 1/2^{+}$ and $\pi 1/2^{+}$ denote non-resonant discretized continuum states of neutron $\Omega^{\pi}=1/2^{-}$ and $1/2^{+}$ levels and proton $1/2^{+}$ level. []{data-label="28Ne_0-"} The lower energy resonance at around 7 MeV is described by three eigenstates as shown in Fig. \[28Ne\_strength\]. The lower state at 6.70 MeV, which has an isovector strength of 0.028 $e^{2}$fm$^{2}$, is mainly generated by $\nu(1d^{-2}_{3/2}2p_{3/2})$ (87.5%), $\nu(2s^{-1}_{1/2}1f_{7/2})$ (5.7%) and $\nu(1d^{-1}_{3/2}2p_{1/2})$ (2.4%). The state at 6.96 MeV with $B(Q^{\mathrm{IV}}1)=0.044$ $e^{2}$fm$^{2}$ is almost a single p-h excitation of $\nu(1d^{-1}_{3/2}2p_{1/2})$ (90.2%), and the state at 7.19 MeV with 0.021 $e^{2}$fm$^{2}$ is generated dominantly by proton h-h like excitation of $\pi(1p_{1/2}\otimes 1d_{5/2})$ (57.8%), together with $\nu(2s^{-1}_{1/2}1f_{7/2})$ (18.6%), $\nu(1d^{-1}_{3/2}2p_{3/2})$ (6.4%), $\nu(1d^{-1}_{5/2}1f_{7/2})$ (5.2%), $\nu(2s^{-1}_{1/2}2p_{3/2})$ (4.5%) and $\nu(1d^{-1}_{3/2}2p_{1/2})$ (1.8%). Therefore, the higher-energy resonance at 8 MeV has a similar structure to that in $^{26}$Ne; $\nu(2s^{-1}_{1/2}2p_{3/2})$ and $\pi(1p_{1/2}\otimes 1d_{5/2})$ excitations are dominant, and the lower-energy resonance is generated by different eigenmodes. The $K^{\pi}=1^{-}$ state at 8.9 MeV in $^{28}$Ne has a similar structure to that in $^{26}$Ne and $K^{\pi}=0^{-}$ state in $^{28}$Ne, corresponding mainly to the neutron two-quasiparticle excitation of $\nu(2s^{-1}_{1/2}2p_{3/2})$, with 64.0% contribution. In addition to this neutron p-h like excitation, the following excitations have an appreciable contribution; $\nu(1d^{-1}_{5/2}1f_{7/2})$ (13.9%), $\nu(2s^{-1}_{1/2}1f_{7/2})$ (2.8%), $\pi(1p^{-1}_{1/2}1d_{5/2})$ (7.9%) and $\pi(1p_{3/2}\otimes 1d_{5/2})$ (1.4%). ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- ![Same as Fig. \[trans\_density\] but for the $K^{\pi}=0^{-}$ state at 8.1 MeV and for the $K^{\pi}=1^{-}$ state at 6.7 MeV in $^{30}$Ne. []{data-label="30Ne_trans_density"}](fig9-1.eps "fig:") ![Same as Fig. \[trans\_density\] but for the $K^{\pi}=0^{-}$ state at 8.1 MeV and for the $K^{\pi}=1^{-}$ state at 6.7 MeV in $^{30}$Ne. []{data-label="30Ne_trans_density"}](fig9-2.eps "fig:") ![Same as Fig. \[trans\_density\] but for the $K^{\pi}=0^{-}$ state at 8.1 MeV and for the $K^{\pi}=1^{-}$ state at 6.7 MeV in $^{30}$Ne. []{data-label="30Ne_trans_density"}](fig9-3.eps "fig:") ![Same as Fig. \[trans\_density\] but for the $K^{\pi}=0^{-}$ state at 8.1 MeV and for the $K^{\pi}=1^{-}$ state at 6.7 MeV in $^{30}$Ne. []{data-label="30Ne_trans_density"}](fig9-4.eps "fig:") ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- Finally, we discuss the dipole state in $^{30}$Ne. Compared to the response functions in $^{26}$Ne and $^{28}$Ne, that for $^{30}$Ne is quite different, because this nucleus is well deformed as shown in Table \[GS\]. The giant resonance is split into $K^{\pi}=0^{-}$ and $1^{-}$ mode, and the split giant resonance has an overlap with the low-lying resonance below 10 MeV. In the right panel of Fig. \[28Ne\_strength\], we show the strength distribution below 10 MeV in $^{30}$Ne. For the $K^{\pi}=0^{-}$ mode, we can see a prominent peak at 8.1 MeV possessing a large isovector $E1$ strength of 0.48 $e^{2}$fm$^{2}$. This state is mainly generated by $\nu[211]1/2 \to [310]1/2$ (38.4%) and the neutron excitation from $[330]1/2$ to the non-resonance continuum state (37.9%), together with the proton excitation of $\pi[330]1/2 \to [220]1/2$ (6.1%). In Fig. \[30Ne\_trans\_density\], the transition density for the $K^{\pi}=0^{-}$ state is shown. The transition density of protons are quite similar to that in Fig. \[trans\_density\]. For the neutrons, we can easily see the effect of mixing of the excitation into the continuum state; the transition density has large spatial extension. Furthermore, comparing to Fig. \[26Ne\_wf\], this $K^{\pi}=0^{-}$ state still possesses a structure similar to the low-lying dipole state in $^{26}$Ne. ----- --------------- --------------- ------------------------ -------------------------------------------- ---------------------- $E_{\alpha}+E_{\beta}$ $Q_{11,\alpha\beta}$ $\alpha$ $\beta$ (MeV) $X_{\alpha \beta}^{2}-Y_{\alpha\beta}^{2}$ ($e\cdot$ fm) (a) $\nu[312]3/2$ $\nu[200]1/2$ 6.87 0.676 0.207 (b) $\nu[310]1/2$ $\nu[200]1/2$ 7.09 0.089 0.141 (c) $\nu[321]3/2$ $\nu[202]5/2$ 7.52 0.043 $-0.009$ (d) $\nu[321]3/2$ $\nu[211]1/2$ 5.72 0.038 0.026 (e) $\nu[312]5/2$ $\nu[202]3/2$ 6.16 0.025 0.003 (f) $\nu[310]1/2$ $\nu[202]3/2$ 6.13 0.025 0.015 (g) $\nu[330]1/2$ $\nu[211]1/2$ 5.73 0.011 0.040 (h) $\nu1/2^{-}$ $\nu[202]3/2$ 7.57 0.019 0.066 ----- --------------- --------------- ------------------------ -------------------------------------------- ---------------------- : Same as Table \[26Ne\_0-\] but for the $K^{\pi}=1^{-}$ state at 6.69 MeV in $^{30}$Ne, and only components with $X_{\alpha\beta}^{2}-Y_{\alpha\beta}^{2} > 0.01$ are listed. This mode has $B(E1)=1.58 \times10^{-2}~e^{2}$fm$^{2}$, $B(Q^{\nu}1)=3.98 \times10^{-2} e^{2}$fm$^{2}$, $B(Q^{\mathrm{IV}}1)=1.06 \times10^{-1} e^{2}$fm$^{2}$, and $\sum|Y_{\alpha\beta}|^{2}=8.82\times 10^{-3}$. []{data-label="30Ne_1-"} For the $K^{\pi}=1^{-}$ state, we can see a prominent peak at 6.69 MeV possessing an isovector strength of 0.11 $e^{2}$fm$^{2}$. This state has a different structure to the dipole states discussed above. In Table \[30Ne\_1-\], we show its microscopic structure. This state has a collective nature in a sense that a number of two-quasiparticle excitations have an appreciable contribution; in the present case, eight of the neutron excitations have a contribution larger than 1%. In the lower panel of Fig. \[30Ne\_trans\_density\], the transition density of this state is shown. This mode has also a characteristic feature that the neutron and proton contribution have an isoscalar nature around the surface region, and the neutron excitation is dominant outside of the nucleus. It is difficult to link directly with the low-lying dipole states in $^{26}$Ne or $^{28}$Ne, because the deformations are quite different in $^{30}$Ne and in the other two nuclei. The main component of this $K^{\pi}=1^{-}$ state is (a):$\nu[200]1/2 \to [312]3/2$ and (b):$\nu[200]1/2 \to [310]1/2$. These p-h excitations are $\nu(1d^{-1}_{3/2}2p_{3/2})$ in the spherical limit. In this sense, the lower-energy resonance in $^{28}$Ne is connected to this collective $K^{\pi}=1^{-}$ state in $^{30}$Ne. \[summary\]Summary ================== We have investigated a new framework of the deformed QRPA based on the Skyrme density functionals and the Landau-Migdal approximation. With this method, we have made a detailed analysis of the low-lying dipole states in neutron-rich $^{26,28,30}$Ne. In these nuclei, we obtain the excitation mode at $8-8.5$ MeV. The low-lying resonance is composed of several QRPA eigenmodes. In $^{26}$Ne, not only the $\nu(2s_{1/2}^{-1}2p_{3/2})$ transition but also the $\nu(2s_{1/2}^{-1}2p_{1/2})$ transition contribute to generating the resonance. In $^{28}$Ne and $^{30}$Ne, the $\nu[211]1/2 \to [310]1/2$ excitation still plays a major role. Each eigenmode is, however, not purely a single particle-hole excitation, it has a small contribution of the other neutron excitations and proton excitations as well. We have clearly shown the spatially extended structure of the $\nu[211]1/2$ ($2s_{1/2}$) state, and that it is responsible for the oscillation of transition density of neutrons outside of the nucleus. In the well deformed nucleus $^{30}$Ne, the deformation splitting of the giant resonance is large and the low-lying resonance overlaps with the giant resonance. 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--- 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).

--- abstract: 'A classical theory of general relativity in the $4$-dimensional space time is formulated as a Chern–Weil topological theory. An Einstein–Hilbert gravitational action is shown to be invariant with respect to a novel translation (co-translation) operator up to total derivative, and thus a topological invariant of the second Chern class exists due to the Chern–Weil theorem. With a topological insight, fundamental forms are introduced as a principal bundle of a space time manifold. Canonical quantization of the system is performed in a Heisenberg picture using the Nakanishi–Kugo–Ojima formalism. A complete set of the quantum Lagrangian and BRS transformations including auxiliary and ghost fields is given in a self-consistent manner. An appropriate Hilbert space and physical states are introduced into the theory, and the positivity of the physical states and the unitarity of the transition matrix are ensured by the Kugo–Ojima theorem. A non-renormalizability of quantum gravity is reconsidered under the formulation proposed in this study.' author: - Yoshimasa bibliography: - 'ref.bib' title: 'General relativity as the four-dimensional Chern–Weil theory' --- Introduction {#intro} ============ The theory of general relativity is one of the most fundamental theories treating the space time structure of the universe itself, and the correctness of general relativity has been established by many experiments. In particular, new evidence owing to the seminal discovery of gravitational waves was added in 2016[@PhysRevLett.116.241103]. On the other hand, at the microscopic level, nature is described by quantum mechanics. The standard theory of particle physics based on the quantum field theory is shown to be well-established by the discovery of the Higgs boson[@Aad:2012tfa; @Chatrchyan201230]. Thus, our understanding of nature covers a wide range of length scale from the large-scale structure of the universe to the microscopic behavior of sub-atomic elements. However, these two fundamental theories, general relativity and quantum field theory, are not consistent. A construction of a quantum theory of gravity is one of the most fundamental goal of present physics. Here, the “quantum theory of gravity” is understood as: $1$) a theory which can describe the behavior of ($4$-dimensional) space-time in the region where the uncertainty principle becomes essential ($\approx$ the Planck-length), $2$) a theory which is consistent with well-established general relativity at a large scale, and $3$) a theory which can give experimentally measurable predictions. Immediately after an establishment of general relativity and quantum mechanics in 1920’s, construction of quantum gravity was started on 1930’s. (A history of quantization of general relativity is beyond the scope of this report. For detailed history, see [@Rovelli:2000aw; @doi:10.1142qg] and references therein. .) There are three main-streams of the quantization[@Rovelli:2000aw]: 1) Covariant perturbative approach[@Fierz211]: Following the successful method of the QED, a small fluctuation from the flat Minkowski space is treated as a perturbation, then the Feynman rule of gravitational interaction is derived. This method was slowing down after a discovery of non-renormalizability of thees theories[@'tHooft:1974bx], and it becomes active again after appearing the super-string theoretical approach. 2) Canonical quantization of the metric tensor[@DeWitt:1967yk; @DeWitt:1967ub; @DeWitt:1967uc]: The metric tensor is treated as a dynamical variable and interpreted as an operator, and then it is quantized using the canonical method by requiring the commutation relations. The quantum equation of motion is obtained as the deWitte–Wheeler equation[@DeWitt:1967yk]. This approach is also slowing down because the deWitte–Wheeer equation is not mathematically well-defined, and recently it is renovated as loop-gravity[@rovelli2004] and developments are still continuing. 3) Path-integration quantization: When the path-integration method is simply applied to gravity, non-renormalizable infinities appear as the same as the first approach. A spin-network method[@Rovelli:1995ac] can be categorized as this approach. In despite of the decades of intensive efforts on quantization, a widely accepted theory that satisfies above requirements does not yet exist. In contrast to the $4$-dimensional case, it is known that quantum gravity exists in the $(1\hspace{-.1em}+\hspace{-.1em}2)$-dimensional space as a Chern–Simons topological theory[@witten1988], that is renormalizable and does not exhibit dynamics[@Witten198846]. We note that $3$-dimensional general relativity does not have any dynamic degree of freedom even at a classical level. In [@Witten198846], Witten mentioned the following two points: firstly the existence of quantum gravity is due to an accidental feature of the $3$-dimensional case in which the invariant quadratic form exists, and secondly the non-renormalizability of quantum gravity in $4$-dimensions is essentially due to the absence of the invariant quadratic that allows us to understand the short-distance limit of space time as trivial zero-energy solutions. Actuary, it is known that a Chern-Simons action can give a topological invariant only in odd-dimensional spaces[@0264-9381-29-13-133001]. While, at first glance, it seems hopeless to construct quantum general relativity using a Chern-Simons form on the $4$-dimensional space time, we have found a novel symmetry, which is referred to as the co-Poincaré symmetry[@doi:10.1063/1.4990708], allows us to construct general relativity as a Chern–Weil theory on a $4$-dimensional space time manifold. This new symmetry is one extended a translation symmetry, and when it is applied on a pure gravitational Lagrangian without a cosmological term, it induces only a total derivative term. In this study, we show that the invariant quadratic can be defined in the $4$-dimensional space time by introducing a Lie algebra of the co-Poincaré group and the Einstein–Hilbert gravitational Lagrangian can be defined as a second Chern class, when there is no cosmological term. Our approach for quantization is based on the second category “canonical quantization of the metric tensor” in the above list for quantization approaches. In this quantization method, the subject to be quantized is not the space time itself, and thus the space time coordinate $x^\mu$ is not q-number (operator), but c-number[@nakanishi1990covariant; @NakanishiSK2009]. The subject for quantization is a solution of the Einstein equation $g^{(c)}_{\mu\nu}(x)$. In classical general relativity, the geometrical metric tensor $g^{(g)}_{\mu\nu}(x)$ is given by the solution of the classical Einstein equation such as $g^{(g)}_{\mu\nu}(x)=g^{(c)}_{\mu\nu}(x)$, that is nothing other than the Einstein’s equivalent principle. In a quantum level, this relation is not simply fulfilled. The geometrical metric tensor will be given as an expected value of the quantum metric tensor $g^{(g)}_{\mu\nu}(x)=\langle g^{(q)}_{\mu\nu}(x)\rangle$. An outline of our method is as follows; We are starting from the Einstein–Hilbert action of general relativity that has the co-Poincaré symmetry and principal bundle induced by this symmetry. As a result, the fundamental forms can be identified as the spin and surface forms, which will be defined in this article. Based on the fundamental forms, the Nakanishi–Kugo–Ojima covariant quantization is performed, and the complete set of the quantum Lagrangian, equations of motion, BRS transformations and BRS charges for pure gravity is given. As the consequence of quantization, a scattering matrix must fulfill the Kugo–Ojima theorem. This article is organized as follows: In section II, mathematical preliminaries of differential geometry are introduced in order to explain our terminologies and conventions. Standard formalism of a gravitational Lagrangian and geometrical structure of a principal (Poincaré) bundle are also introduced in this section in contrast to our novel topological approach. New translation operator is introduced in section III. It is shown that an Einstein–Hilbert Lagrangian can be recognized as a second Chern class under co-Poincaé symmetry in this section. With a topological insight given here, appropriate fundamental variables (forms) for a Hamiltonian formalism are introduced at the end of this section. An explicit formulation of canonical quantization of the general relativity using the Nakanishi–Kugo–Ojima formalism[@nakanishi1990covariant] is performed in section IV. Section V is devoted for discussions how to construct an appropriate Hilbert space and physical states on it. In consequence, it is shown that the unitarity of the quantum gravitational S-matrix ensured due to the Kugo–Ojima theorem[@kugo1979local; @Kugo1978459]. A renormalizability of our Cher–Weil general relativity is also discussed in section V. At the end, a summary of this study is given in section VI. Preliminaries {#prep} ============= First, standard classical general relativity is geometrically re-formulated in terms of a vierbein formalism according to Ref.[@fre2012gravity; @Kurihara2018]. Differential geometry {#DG} --------------------- A $4$-dimensional pseudo-Riemannian manifold $(\MM,g)$ with $GL(1,3)$ symmetry is considered. On each coordinate patch $U_p\subset\MM$ around $p\in U_p$, orthonormal coordinate vectors are introduced as $x^\mu$. Accordingly, we take standard base vectors on the tangent space $T_p\MM$ as $\partial_\mu$, and those on the cotangent space $T_p^*\MM$ as $dx^\mu$. An abbreviation $\partial_\bullet:=\partial/\partial x^\bullet$ is used throughout this report. Whole manifold can be covered by such coordinate patch, and thus a tangent space $\TMM=\bigcup_pT_p\MM$ and a cotangent space $\TsMM=\bigcup_pT^*_p\MM$ can be defined on $\MM$. We note that these two base vectors are taken to dual each other such as $dx^\mu\partial_\nu=\delta^\mu_\nu$. The tensor product is defined as an ordered pair of two vectors as $\alpha\otimes\beta$ for vectors $\alpha$ and $\beta$ on the tangent space, and it is a rank-$2$ tensor. Higher rank tensors can be constructed as well. A rank-$p$ tensor space on a manifold $\bullet$ is denoted as $V^p(\bullet)$ in this study. The wedge product is defined as an antisymmetrization of a tensor product as $$\begin{aligned} \alpha\wedge\beta:=\alpha\otimes\beta-\alpha\otimes\beta~~&\rm{and}&~~ \aaa\wedge\bbb:=\aaa\otimes\bbb-\bbb\otimes\aaa,\end{aligned}$$ for $\alpha,\beta\in V^1(\TMM)$ and $\aaa,\bbb\in V^1(\TsMM)$. We note that the same symbol “$\wedge$” is used as the antisymmetric tensor-product on both tangent and cotangent spaces. A vector on the cotangent space is named a $1$-form, and wedge products of $p$ cotangent-vectors are named a $p$-form. A scalar function is categorized as a $0$-form. A space of $p$-forms is denoted as $\Omega^p(\TsMM)$. The external derivative, that is a map $d:\Omega^p\rightarrow\Omega^{p+1}$, is introduced as a standard manner, e.g. for a form $\aaa:=a_\mu dx^\mu$, $d\aaa:=(\partial_{\mu_1}a_{\mu_2})dx^{\mu_1}\wedge dx^{\mu_2}$ and $dd\aaa=0$. The Einstein’s convention for repeated indices are used throughout this study. When the external derivative is applied on a $0$-form (function) $f(x^\mu)$, it gives a total derivative of a function such as $df(x^\mu)=(\partial_\nu f(x^\mu))dx^\nu$, and it is a $1$-form. Forms are represented using a Fraktur letters “$\AAA,\aaa,\BBB,\bbb,\CCC,\ccc\cdots$” in this study, as already used above. The “[*dot-product*]{}” between a form and vector is represented using the standard coordinate as $\aaa\cdot{\bm b}=\sum_{\mu_1<\cdots<\mu_p}\alpha_{\mu_1\cdots \mu_p}\beta^{\mu_1\cdots \mu_p}\in\R$, where $$\begin{aligned} \aaa&=&\sum_{\mu_1<\cdots<\mu_p}\alpha_{\mu_1\cdots \mu_p} dx^{\mu_1}\wedge\cdots\wedge dx^{\mu_p}\in\Omega^p\left(\TsMM\right),\\ {\bm b}&=&\sum_{\mu_1<\cdots<\mu_p}\beta^{\mu_1\cdots \mu_p} \partial_{\mu_1}\otimes\cdots\otimes\partial_{\mu_p}\in V^p\left(\TMM\right).\end{aligned}$$ A result of dot-product is independent of choice of the coordinate system, when dual bases are used on $\TMM$ and $\TsMM$. General relativity is constructed on a $4$-dimensional (pseudo) Riemannian manifold $\MM$ with a metric tensor $g_{\mu\nu}$. A line element can be expressed as $$\begin{aligned} ds^2&=&g_{\mu_1\mu_2}(x)dx^{\mu_1}\otimes dx^{\mu_2}.\end{aligned}$$ Because the affine connection $\Gamma^\rho_{~\mu\nu}$ is not a tensor, it is always possible to find a frame in which the affine connection vanishes as $\Gamma^\rho_{~\mu\nu}(p)=0$ at any point $p\in\MM$. A local manifold with a vanishing affine connection has a Poincaré symmetry $ISO(1,3)=SO(1,3)\ltimes T{(4)}$, where $T{(4)}$ is a group of $4$-dimensional translations. This local manifold at $p$ is referred to as the local Lorentz manifold and is denoted by $\M_p$. On $\M_p$, a vanishing gravity, such as $\partial_\rho g_{\mu\nu}=0$, is not required in general. A fiber bundle $\M:=\bigcup_p\M_p$ is regarded as the principal bundle on $\MM$. The metric tensor $diag[{\bm \eta}]=(1,-1,-1,-1)$ is employed in this study. Tangent space $\TM$ and cotangent spaces $\TsM$ can be respectively introduced as well as those on $\MM$. Local coordinate vectors, and standard base-vectors on $\TM$ and $\TsM$, are represented as $x^a$, $\partial_a$ and $\aaa^a$, respectively. A base manifold of vectors and forms, $\MM$ or $\M$, is distinguished by Greek suffices (on $\MM$) or Roman suffices (on $\M$) in our representation. A $SO(1,3)$ invariant coordinate-transformation $\Lambda$ forms an automorphism group referred to as the Lorentz transformation, and it is represented as; $$\begin{aligned} \Lambda:V^p(\TM)\rightarrow V^p(\TM): {\bm \alpha}\mapsto\widetilde{\bm \alpha}={\bm \Lambda}^p{\bm \alpha},&~~~& \left[{\bm \Lambda}^p{\bm \alpha}\right]^{a_1\cdots a_p}:= \left[{\bm \Lambda}\right]^{a_1}_{~b_1}\cdots\left[{\bm \Lambda}\right]^{a_p}_{~b_p} \left[{\bm \alpha}\right]^{b_1\cdots b_p},\end{aligned}$$ where ${\bm \Lambda}$ is the Lorentz transformation matrix. A notation $[\bullet]$ shows a component of a tensor, form, or matrix $\bullet$, and it is used throughout this study. A map $\Varepsilon:\MM\rightarrow\M:{\bm g}\mapsto{\bm \eta}$ is a kind of general coordinate-transformations (diffeomorphism), and is represented using a standard coordinate on any chart of $\MM$ as; $$\begin{aligned} \eta^{ab}&=&\Varepsilon^a_{\mu_1}(x^\mu)\Varepsilon^b_{\mu_2}(x^\mu)g^{\mu_1\mu_2}(x^\mu),\end{aligned}$$ where $\eta^{ab}=[{\bm \eta}]^{ab}$ and $g^{\mu\nu}=[{\bm g}]^{\mu\nu}$ are the metric tensors on $\M$ and $\MM$, respectively. A function $\Varepsilon^a_{\mu}(x^\mu)=[{\bm \Varepsilon}(x^\mu)]^a_{\mu}$ is referred to as the vierbein. We note that the diffeomorphism map $\Varepsilon$ induces isomorphism $\TMM\cong \TM$, and thus a form $\aaa\in\Omega^p(\TsM)$ and its pull-back $\Varepsilon^*[\aaa]\in\Omega^p(\TsMM)$ are identified each other and denoted simply $\aaa\in\Omega^p$ in this study. Orthogonal base vectors can be obtained respectively from those on $\TMM$ and $\TsMM$ as $\partial_a:=\Varepsilon_a^\mu\partial_\mu$ and $\eee^a:=\Varepsilon^a_\mu dx^\mu$, where $\Varepsilon_a^\mu:=[{\bm \Varepsilon}^{-1}]_a^\mu$. They are dual bases each other such as $ \eee^a\cdot\partial_b=\Varepsilon^a_{\mu_1}\Varepsilon_b^{\mu_2}dx^{\mu_1}\partial_{\mu_2} =\Varepsilon^a_{\mu}\Varepsilon_b^{\mu}=\delta^a_b. $ A base vector $\eee^a$ is referred to as the virbein form, which is $\eee\in\Omega^1\cap V^1(\TM)$. (We have to avoid a representation “$dx^a=\Varepsilon^a_\mu dx^\mu$” because it is not closed $d\left(\Varepsilon^a_\mu dx^\mu\right)\neq0$.) A spin connection $\www\in\sss\ooo(1,3)$ can be introduced such as $\www^t\eta+\eta\www=0$, where $\www^t$ is a transpose matrix of $\www$. The spin connection $\www$ is a connection-valued $1$-form on $\TsMM$ and is referred to as the spin form hereafter. The spin form can be represented using a standard coordinate as $\www^{ab}=\omega^{~a}_{\mu~c}\eta^{cb}dx^\mu$ where $\omega^{~a}_{\mu~c}:=[\www]^{~a}_{\mu~c}$, and it is anti-symmetric as $\www^{ab}=-\www^{ba}$. The spin form is not a Lorentz tensor, and its Lorentz transformation is given as; $$\begin{aligned} \widetilde{\www}&=&{\bm \Lambda}^{-1}\www{\bm \Lambda}+{\bm \Lambda}^{-1}d{\bm \Lambda}.\label{wLT}\end{aligned}$$ A $SO(1,3)$ covariant derivative $\td$ can be represented using the spin form as; $$\begin{aligned} \td\aaa^{a_1\cdots a_p}&=& d \aaa^{a_1\cdots a_p}+\www^{a_1}_{~~b}\wedge\aaa^{ba_2\cdots a_p}+\cdots+ \www^{a_p}_{~~b}\wedge\aaa^{a_1\cdots a_{p-1}b}.\end{aligned}$$ A direct calculation can show that $d_\www \aaa^{a_1\cdots a_p}$ is transformed as a rank-$p$ tensor under the $SO(1,3)$ transformation (\[wLT\]). A torsion $2$-form $\TTT^a$ can be defined using the covariant derivative as $\TTT^a:=\td\eee^a=d\eee^a+\www^a_{~b}\wedge\eee^b$. Hereafter, dummy [**Roman**]{}-indices are often abbreviate to a dot (or an asterisk), and this notation is used throughout this study. Dummy [**Greek**]{}-indices are not abbreviated in contrast. In this notation, a curvature $2$-form is written as $\RRR^{ab}:=d\www^{ab}+\www^a_{~\bcdot}\wedge\www^{\bcdot b}$. When multiple dots appear in an expression, pairing must be a left-to-right order at both upper and lower indices, e.g., a $4$-dimensional volume form can be written as $\vvv:=\frac{1}{4!}\epsilon_{\cdott\cdott}\eee^\bcdot\wedge\eee^\bcdot\wedge\eee^\bcdot\wedge\eee^\bcdot=\frac{1}{4!}\epsilon_{abcd}\eee^{a}\wedge\eee^{b}\wedge\eee^{c}\wedge\eee^{d}$, where $\epsilon_{abcd}$ is a completely antisymmetric tensor (the Levi Civita tensor) whose component is $\epsilon_{0123}=1$ on $\TM$. We note that there are two constant-valued tensors on $\TM$, the metric tensor ${\bm \eta}$ and the Levi Civita tensol ${\bm \epsilon}$, and they are not a constant valued on $\TMM$. A $2$-dimensional surface form is defined as $\SSS_{ab}:=\frac{1}{2}\epsilon_{ab\cdott}\eee^\bcdot\wedge\eee^\bcdot$ which is a $2$-dimensional surface perpendicular to both $\eee^a$ and $\eee^b$. While the spin form $\www^{ab}$ is not a $SO(1,3)$ tensor-valued form, the curvature form $\RRR^{ab}$ is a Lorentz tensor-valued $2$-form. More precisely, $\RRR^{ab}$ is a rank-$2$ tensor on $\TM$ with respect to Roman indices, and is $2$-from on $\TsM$ (and also on $\TsMM$), such as $\RRR\in\Omega^2\cap V^2(\TM)$. Using this notation, the Bianchi identities can be expressed as; $$\begin{aligned} \td\TTT^a=\RRR^{a\bcdot}\wedge\eee^{\bcdot}\hspace{.1em}\eta_{\bcdott},&~~~& ~\td\RRR^{ab}=0.\label{bianchi1}\end{aligned}$$ A volume form $\vvv$ which is $GL(1,3)$ (and also $SO(1,3)$) invariant can written as; $$\begin{aligned} \vvv&=&\frac{1}{4!}\epsilon_{\cdott\cdott}\Varepsilon^\bcdot_{\mu_1}\Varepsilon^\bcdot_{\mu_2} \Varepsilon^\bcdot_{\mu_3}\Varepsilon^\bcdot_{\mu_4} dx^{\mu_1}\wedge dx^{\mu_2}\wedge dx^{\mu_3}\wedge dx^{\mu_4} =\sqrt{-det[{\bm g}]}dx^{0}\wedge dx^{1}\wedge dx^{2}\wedge dx^{3}.\end{aligned}$$ We note that $\sqrt{-det[{\bm g}]}=-det[{\bm \Varepsilon}]$. In addition to that, a new operator is introduced for shorter representations. This is a map that transfers a rank-$p$ tensor field to a rank-$(n\hspace{-.1em}-\hspace{-.1em}p)$ tensor field defined on an $n$-dimensional space time manifold such that $$\begin{aligned} \overline{\aaa^{a_1\cdots a_p}}&=&\overline{\aaa}_{a_1\cdots a_{n-p}}=~~ \frac{1}{p!}\epsilon_{b_1\cdots b_p a_1\cdots a_{n-p}} \aaa^{b_1\cdots b_p},\\ \overline{\aaa_{a_1\cdots a_p}}&=&\overline{\aaa}^{a_1\cdots a_{n-p}}=-\frac{1}{p!} \epsilon^{a_1\cdots a_{n-p} b_1\cdots b_p} \aaa_{b_1\cdots b_p},\end{aligned}$$ where $\aaa\in\Omega^p$ and $\epsilon$ is a completely anti-symmetric tensor in $n$-dimensions. It follows the relation $\overline{\overline{\aaa}}=\aaa$, when $\aaa$ is anti-symmetric with respect to all indices. The surface form can be expressed using this expression as $\SSS_{ab}=\overline{\eee^a\wedge\eee^b}$. We note that this operator is totally different from the Hodge-$*$ operator. In contrast with that the Hodge-$*$ operator maps a $p$-form to $(n\hspace{-.2em}-\hspace{-.2em}p)$-form, this operator, which is referred to as the [*bar-dual operator*]{} in this study, maps a rank-$p$ tensor to a rank-$(n\hspace{-.2em}-\hspace{-.2em}p)$ tensor with keeping a rank of a form as $\bar{\bullet}:\Omega^q\cap V^p(\TM)\rightarrow\Omega^q\cap V^{n-p}(\TM)$. We note that the Hodge-$*$ operator is not appropriate for constructing general relativity because the metric tensor is hidden in its definition. Lagrangian formalism {#lagrange} -------------------- The Lagrangian form of gravity that reproduces the Einstein equation without matter fields can be written as $$\begin{aligned} \LLL_G(\vomega,\eee) &=&\frac{1}{2}\SSS_\cdott\wedge\left( \RRR^\cdott-\frac{\Lambda}{3!}\overline{\SSS}^\cdott \right), \label{Lagrangian44}\end{aligned}$$ where $\Lambda$ is the cosmological constant. While this Lagrangian form is invariant under the general coordinate transformation and the local ${SO}(1,3)$ transformation, it is not invariant under a local translation. We will discuss this point in detail later in a next section. On the local Lorentz manifold, number of independent components of the Lagrangian form is ten, of which, six components are from the curvature $2$-form and four components are from the vierbein forms. These degrees of freedom (d.o.f.) correspond to the total number of d.o.f$.$ from the Poincarè group. The Lagrangian form is now treated as a functional with two independent forms, $\{\vomega^{ab}, \eee^c\}$, and is taken to vary independently. This method is commonly known as the Palatini method[@Palatini2008; @ferreris]. The gravitational action is introduced as the integration of the Lagrangian such that $$\begin{aligned} \I_G &=&\frac{c^4}{4\pi G}\int_{\Sigma}\LLL_G.\label{gravact}\end{aligned}$$ The integration region $\Sigma$ is taken to be an entire space of $\MM$, for instance. The equations of motion with respect to the two independent functions (spin and vierbein forms) can be obtained by requiring a stationary condition for the variation of the action for each form separately. From the variation with respect to the spin form, an equation of motion, $$\begin{aligned} \TTT^a&=&d\eee^a+ \vomega^{a}_{~\bcdot}\wedge\eee^\bcdot=0,\label{torsion less}\end{aligned}$$ can be obtained, which is referred to as the torsion less condition. The torsion less condition is obtained from the solution of the equation of motion, rather than being an independent constraint on the manifold. This equation of motion includes six independent equations, which is the same as the number of the independent components of the spin form. Therefore, the spin form can be uniquely determined from (\[torsion less\]) when the vierbein forms are given. Next, taking the variation with respect to the vierbein form, one can obtain an equation of motion as; $$\begin{aligned} \epsilon_{a\cdott\bcdot}\left( \frac{1}{2}\RRR^\cdott\wedge\eee^\bcdot -\frac{\Lambda}{3!}\eee^\bcdot\wedge\eee^\bcdot\wedge\eee^\bcdot \right)&=&0.\label{EoM2}\end{aligned}$$ This is the Einstein equation without any matter fields. The curvature and vierbein forms can be uniquely determined by simultaneously solving the equations of motion (\[torsion less\]) and (\[EoM2\]) as a quotient set of ${GL}(1,3)$ diffeomorphism. The Planck unit ($c=\hbar=4\pi G=1$) is employed hereafter. Form-tensor duality {#TcoT} ------------------- Classical general relativity can be represented simply and clearly in a covariant manner using differential forms as shown in previous section. This representation is referred to as the cotangent representation because differential forms are defined on a cotangent bundle $\TsM$. While the cotangent representation is suitable for a formal discussion due to its simplicity of the expression, it is not convenient for concrete calculations. On the other hand, a tensor representation on a tangent bundle $\TM$ is suitable for concrete calculations, and it is referred to as the tangent representation, hereafter. A relation between these two representations is discussed in this section. A scalar function valued $p$-form $\aaa\in\Omega^p\otimes V^0(\TM)$ is considered here. The dual tensor ${\bm A}$ of the form $\aaa$ is introduced through a map $\V:\Omega^p\rightarrow V^p:\aaa\mapsto \AA$. A form $\aaa$ can be expanded using a standard base on $\TsM$ as; $$\begin{aligned} \aaa &=:&\sum_{a_1<\cdots<a_p} \alpha_{a_1\cdots a_p}\hspace{.1em}\eee^{a_1}\wedge\cdots\wedge\eee^{a_p} =\frac{1}{p!}\alpha_{a_1\cdots a_p}\hspace{.1em}\eee^{a_1}\wedge\cdots\wedge\eee^{a_p}.\end{aligned}$$ We note that $\alpha_{a_1\cdots a_p}$ is completely antisymmetric with respect to its lower suffixes. A rank-$p$ tensor $\AA\in V^p$ can be introduced using these expansion coefficients as; $$\begin{aligned} \AA&:=& \frac{1}{p!}\alpha^{a_1\cdots a_p}\hspace{.1em} \partial_{a_1}\wedge\cdots\wedge\partial_{a_p},\end{aligned}$$ where $\alpha^{a_1\cdots a_p}:=\eta^{a_1b_1}\cdots\eta^{a_pb_p}\alpha_{b_1\cdots b_p}$. We note that symmetric components of the tensor $\AA$ does not appear due to antisymmetry of the coefficient $\alpha^{a_1\cdots a_p}$. This map uniquely gives the dual tensor independently from a choice of coordinate system. A dot product between the form and its dual vector is given as $$\begin{aligned} \aaa\cdot\AA&=&\sum_{a_1<\cdots<a_p}\left(\alpha^{a_1\cdots a_p}\right)^2\in\R,\end{aligned}$$ which is the length of the vector $\AA$, and thus it is $SO(1,3)$ invariant. An external derivative of $\aaa\in\Omega^p\otimes V^0(\TM)$ can be written as; $$\begin{aligned} d\aaa&=&\frac{1}{p!}\left(\partial_{a_0}\alpha_{a_1\cdots a_p}\right)\eee^{a_0} \wedge\eee^{a_1}\wedge\cdots\wedge\eee^{a_p}+\frac{1}{(p-1)!}\alpha_{a_1\cdots a_p}d\eee^{a_1}\wedge\cdots\wedge\eee^{a_p},\\ &=&\frac{1}{(p+1)!}\alpha_{a_0a_1\cdots a_p}\eee^{a_0}\wedge\eee^{a_1}\wedge\cdots\wedge\eee^{a_p},\end{aligned}$$ where $$\begin{aligned} \alpha_{a_0a_1\cdots a_p}&=&(p+1)\left(\partial_{a_0}\alpha_{a_1\cdots a_p}+p\hspace{.1em}\alpha_{ba_2\cdots a_p}(\partial_{a_0}\Varepsilon^{b}_\nu)\Varepsilon^\nu_{a_1}\right), \end{aligned}$$ and thus $d\aaa$ is a $(p\hspace{-.2em}+\hspace{-.2em}1)$-form and its dual is a rank-($p\hspace{-.2em}+\hspace{-.2em}1$) tensor. On the other hand, an external derivative for the dual vector $\AA$ can be obtained as $$\begin{aligned} d\AA&=&\frac{1}{p!}\left(\partial_{a_0}\alpha^{a_1\cdots a_p}\right)\hspace{.1em} \eee^{a_0}\hspace{-.1em}\wedge\partial_{a_1}\wedge\cdots\wedge\partial_{a_p}\\&~&+~ \frac{1}{(p-1)!}\alpha^{ba_2\cdots a_p} \left(\partial_{a_0}\Varepsilon^\nu_b\right)\Varepsilon^{a_1}_\nu\hspace{.1em}\eee^{a_0}\hspace{-.1em}\wedge\partial_{a_1} \wedge\partial_{a_2}\wedge\cdots\wedge\partial_{a_p}.\end{aligned}$$ Therefore, one can confirm a diagram that; $$\begin{aligned} \begin{array}{ccc} \aaa&\longrightarrow^{\hspace{-1.2em}\V}&\AA \\ d\downarrow~&~&d\downarrow~\\ %d\aaa&\longleftarrow^{\hspace{-1.2em}\V^{-1}}&d\AA\label{diag} d\aaa&\longrightarrow^{\hspace{-1.2em}\V }&d\AA\label{diag} \end{array}\end{aligned}$$ This diagram can be straightforwardly extended for a rank-$q$ tensor valued $p$-forms applying it to each tensor component. E.g., the curvature $2$-form $\RRR^{ab}\in\Omega^2\cap V^2(\TM)$ can be expanded as; $$\begin{aligned} \RRR^{ab}&=&\frac{1}{2}R^{ab}_{~~c_1c_2}\eee^{c_1}\wedge\eee^{c_2},\end{aligned}$$ where $R^{ab}_{~~c_1c_2}$ is the equivalent to the standard Riemann curvature tensor. The Einstein equation on the cotangent bundle can be translated into that on the tangent bundle as; $$\begin{aligned} \epsilon_{a\cdott\bcdot}\RRR^\bcdott\wedge\eee^\bcdot=0 &\longleftrightarrow^{\hspace{-1.2em}\V}&~ G^{ab}\hspace{.2em}\partial_a\otimes\partial_b=0,\end{aligned}$$ where $G^{ab}$ is a rank-$2$ tensor referred to as the Einstein tensor defined as; $$\begin{aligned} G^{ab}&:=&R^{ab}-\frac{1}{2}\eta^{ab}R,\end{aligned}$$ and $R^{ab}=\Ri^{\bcdot b}_{~~\bcdot \ast}\eta^{a\ast}$ is a Ricci tensor, and $R=\Ri^{\bcdot\ast}_{~~\bcdot\ast}$ is a scalar curvature. We note that both $R^{ab}$ and $\eta^{ab}$ (and thus the Einstein tensor $G^{ab}$) are symmetric tensor. While two representations looked to have different degrees of freedom (d.o.f$.$) at first glance, it is not the case. A tangent-bundle representation has $4\times(4+1)/2-4=6$ d.o.f$.$ because the Einstein tensor is symmetric and divergenceless under the $SO(1,3)$ covariant derivative. On the other hand, a cotangent-bundle representation of the Einstein equation is given by a vector-valued three-form, and thus it can be written as; $$\begin{aligned} \epsilon_{a\cdott\bcdot}\RRR^\bcdott\wedge\eee^\bcdot=\frac{1}{3!}[R\hspace{-.2em}E_a]_{\bcdot\bcdott} \eee^\bcdot\wedge\eee^\bcdot\wedge\eee^\bcdot,~(a=0,1,2,3),\end{aligned}$$ where $[R\hspace{-.2em}E_a]_{bcd}$ is an expansion coefficient of the form $\epsilon_{a\cdott\bcdot}\RRR^\bcdott\wedge\eee^\bcdot$, and $R\hspace{-.2em}E_a$ is completely anti-symmetric rank-three tensor. There are four such forms according to $a=0,1,2,3$ and they are not independent each other, and it forms a completely antisymmetric rank-$4$ tensor as $[R\hspace{-.2em}E]_{abcd}:=[R\hspace{-.2em}E_a]_{bcd}$. The d.o.f$.$ of $R\hspace{-.2em}E$ is $4\times(4-1)/2=6$, and thus it is the same as that in the tangent-representation. On the other hand, a total d.o.f$.$ of Poincaré group is ten. Other four d.o.f$.$ out of total ten d.o.f$.$ are fixed due to a torsion-less equation, which has four independent constraints on the spin form. A functional derivative with respect to a form can be understand as follows: A 1-forms $\aaa^{ab}:=\alpha^{~ab}_{\bcdot}\eee^\bcdot\in\Omega^1\otimes V^2(\TM)$ and a functional of that $\fff(\aaa^{a_1a_2})$ are considered as an example. On the tangent space $\TM$, a functional derivative with respect to a tensor function can be understand as an operation such as; $$\begin{aligned} \left(\frac{\delta~~~~}{\delta\alpha^{~b_1b_2}_{b_3}}\right)\fff\left(\alpha^{~a_1a_2}_{\bcdot}\eee^\bcdot\right) &=& \fff\left(\delta^{[a_1}_{b_1}\delta^{a_2]}_{b_2}\eee^{b_3}\right),\end{aligned}$$ where $\delta^{[a_1}_{b_1}\delta^{a_2]}_{b_2}=\delta^{a_1}_{b_1}\delta^{a_2}_{b_2}-\delta^{a_2}_{b_1}\delta^{a_1}_{b_2}$. This operator often appears in the quantum field theory, and it is independent of a choice of coordinate systems. A functional derivative on the cotangent space $\TsM$ can be defined as; $$\begin{aligned} \frac{\delta\fff(\aaa^{a_1a_2})}{\delta\aaa^{b_1b_2}}&:=& \left(\iota_{\bcdot}\frac{\delta~~~~}{\delta\alpha^{~b_1b_2}_\bcdot}\right)\fff\left(\alpha^{~a_1a_2}_{\bcdot}\eee^\bcdot\right) = \fff\left(\delta^{[a_1}_{b_1}\delta^{a_2]}_{b_2}\right),\end{aligned}$$ where $\iota_a$ is a contraction with respect to a base vector $\partial_a$ such as $\iota_a\eee^b=\delta^b_a$. This is also a coordinate independent when dual bases are used. It can be extended straightforwardly to more general cases such as $\aaa\in\Omega^p\otimes V^q$. Direct calculations can show that a formal operation, e.g., $$\begin{aligned} \frac{\delta~~~~}{\delta\aaa^{a_1a_2a_3}}\left(\frac{1}{3!}\aaa^{\bcdot\cdott}\wedge\bbb_{\bcdot\cdott}\right)&=& \bbb_{a_1a_2a_3},\end{aligned}$$ gives a consistent result with that in the tangent representation. We note that $\left(\iota_{b_2}\otimes\iota_{b_1}\right)\eee^{a_1}\wedge\eee^{a_2}=\delta^{[a_1}_{b_1}\delta^{a_2]}_{b_2}$. Poincaé bundle {#poincare} -------------- The general relativity described above can be seen as a geometric theory of the manifold with a global ${GL}(1,3)$ symmetry equipped with the local Poincaré bundle as the principal bundle. Let us introduce the generators $P_a$ for $\ttt{(4)}$ and $J_{ab}$ for $\sss\ooo(1,3)$. The commutation relations of the standard Lie algebra of $\iii\sss\ooo(1,3)$ are given as follows; $$\begin{aligned} \left[P_a,P_b\right]&=&0,\label{PJcr1}\\ \left[J_{ab},P_c\right]&=&-\eta_{ac}P_b+\eta_{bc}P_a,\label{PJcr2}\\ \left[J_{ab},J_{cd}\right]&=& -\eta_{ac}J_{bd}+\eta_{bc}J_{ad} -\eta_{bd}J_{ac}+\eta_{ad}J_{bc}.\label{PJcr3}\end{aligned}$$ According to these generators, the principal (Poincaré) connection can be introduced as follows; $$\begin{aligned} \AAA_0&=&\{J_\cdott, P_\bcdot\}\times\{\vomega^\cdott,\eee^\bcdot\} =J_\cdott\vomega^\cdott+ P_\bcdot\eee^\bcdot.\label{PCoF}\end{aligned}$$ A curvature $2$-form based on the principal connection can be expressed as follows; $$\begin{aligned} \FFF_0&=&d\AAA_0+\AAA_0\wedge\AAA_0 = J_\cdott\RRR^\cdott+ P_\bcdot\TTT^\bcdot.\label{PCaF}\end{aligned}$$ New topological formulation {#topol} =========================== So far we introduced a standard formalism of general relativity in terms of differential geometry. While the vierbein form can be considered as a gauge field of a local translation symmetry, general relativity cannot be interpreted as a $ISO(1,3) $ gauge theory, because the Lagrangian does not keep $ISO(1,3)$ symmetry. In this section, we introduce a new translation operator, under which the Lagrangian is invariant up to a total derivative. Under this novel symmetry, an Einstein–Hilbert Lagrangian is interpreted as the topological invariant of the second Chern class. The mathematical details in this section can be found in Ref.[@doi:10.1063/1.4990708]. Co-translation operator {#newsym} ----------------------- As pointed out by Witten[@Witten198846], there is no invariant quadratic-form, and the second Chern class $\mathrm{Tr}[\FFF_0\wedge\FFF_0]$ does not match with the $4$-dimensional Einstein–Hilbert action. The reason why the general relativity on the $(1\hspace{-.2em}+\hspace{-.2em}2)$-dimensional spacetime can be understood as the Chern–Simons theory is that there exists the ${ISO}(1,2)$ non-degenerate invariant, $\epsilon_{\cdott\bcdot}P^\bcdot J^\cdott$. In contrast with the $4$-dimensional case, the Einstein–Hilbert action in ($1\hspace{-.2em}+\hspace{-.2em}2$)-dimension is translation invariant[@Witten198846], and the translation only induces a total derivative on the Lagrangian form[@0264-9381-29-13-133001]. To realize the same trick in the $4$-dimensional theory, we introduce a new operator defined as; $$\begin{aligned} P_{ab}&=&P_a\iota_b,\end{aligned}$$ where $\iota_a$ is a contraction with respect to the vector field $\xi^a\in\Omega^0\cap V^1(\TM)$. More precisely, it is defined as $$\begin{aligned} \iota_a=\iota_{\xi^a},&~~~&\xi^a=\eta^{ab}\Varepsilon^{\mu}_{b}\partial_\mu=.\label{tfv}\end{aligned}$$ This operator is referred to as co-translation hereafter. The Lie algebra associated with the co-translation can be expressed as follows: $$\begin{aligned} \left[P_{ab},P_{cd}\right]&=&0,\label{PJcr1-2}\\ \left[J_{ab},P_{cd}\right]&=&-\eta_{ac}P_{bd}+\eta_{bc}P_{ad},\label{PJcr2-2}\end{aligned}$$ and (\[PJcr3\]). The invariant quadratics corresponding to the Lie algebra are now found to be $$\begin{aligned} %\langle J_{ab},P_{cd} \rangle&=&-\epsilon_{acbd}+\epsilon_{bcad},\label{iq1}\\ \langle J_{ab},P_{cd} \rangle&=&\epsilon_{abcd},\label{iq1}\\ \langle J_{ab},J_{cd}\rangle&=&\langle P_{ab},P_{cd}\rangle~=~0.\label{iq2}\end{aligned}$$ The principal connection can be obtained as; $$\begin{aligned} \AAA&=&\{J_\cdott, P^\cdott\}\times\{\vomega^\cdott,\SSS_\cdott\} = J_\cdott\vomega^\cdott+ P^\cdott\SSS_\cdott,\label{Pcon}\end{aligned}$$ where the lowering and raising of the Roman indices are done using the Lorentz metric. This connection is referred as co-Poincaré connection hereafter. We note that the second term of (\[Pcon\]) is also a one-form because the contraction maps $p$-forms to $(p\hspace{-.2em}-\hspace{-.2em}1)$-forms. The curvature $2$-form due to the co-Poincaré connection is given by $$\begin{aligned} \FFF&=&d\AAA+\AAA\wedge\AAA = J_\cdott\RRR^\cdott+ P^\cdott\td\SSS_\cdott~.\label{c2f}\end{aligned}$$ The second Chern class can be constructed from the invariant quadratics (\[iq1\]), (\[iq2\]), and the curvature $2$-from (\[c2f\]) using a following method: We consider the curvature $2$-form embedded in a $5$-dimensional manifold $\Sigma_5\subset\M_5\in\left(\M_4\otimes\R\right)$, and the integration boundary is $\partial\Sigma_5:=\Sigma_4\subset\Sigma_5$. A manifold $\Sigma_5$ has no boundary other than $\Sigma_4$. An inclusion map $\varphi:\M_4\rightarrow\M_5$ maps the non-singular $(4\hspace{-.2em}\times\hspace{-.2em}4)$-matrix (rank-$2$ tensor) ${\bm T}_4$ to the non-singular $(5\hspace{-.2em}\times\hspace{-.2em}5)$-matrix (rank-$2$ tensor) ${\bm T}_5$ as; $$\begin{aligned} {\bm T}_5&:=& \left[ \begin{array}{cc} {\bm T}_4 & 0\\ 0 & 1 \end{array} \right].\end{aligned}$$ Standard base vectors on $\Sigma_4$ are transformed as $\varphi[(x^0,x^1,x^2,x^3)]=(x^0,x^1,x^2,x^3,1)$ on any trivial chart. A derivative of the map $D\varphi$ acts on ${\bm T}_4$ as; $$\begin{aligned} D\varphi[{\bm T}_4]&=&\left(\varphi[{\bm T}_4+{\bm \tau}]-\varphi[{\bm T}_4]\right){\bm \tau}^{-1}= \left[ \begin{array}{cc} {\bm I}_4 & 0\\ 0 & 0 \end{array} \right],\end{aligned}$$ where ${\bm \tau}$ and ${\bm \tau}^{-1}$ are respectively any non-singular $(4\hspace{-.2em}\times\hspace{-.2em}4)$-matrix and its inverse, and ${\bm I}_4$ is a $(4\hspace{-.2em}\times\hspace{-.2em}4)$-identity matrix. Because the map $D\varphi$ keeps a rank of a matrix, the map $\varphi$ is immersion, and thus $D\varphi$ is a map on the tangent bundle such as $D\varphi:\TM_4\rightarrow\TM_5$. The second Chern class can be express as; $$\begin{aligned} \frac{1}{4}\int_{\Sigma_5}\mathrm{Tr}\left[ \FFF\wedge\FFF \right] &=&\frac{1}{2}\int_{\Sigma_5} \td\SSS_\cdott\wedge\RRR^\cdott =\frac{1}{2}\int_{\Sigma_5}\td\left(\SSS_\cdott\wedge\RRR^\cdott\right) ,\label{csaction1}\end{aligned}$$ where the Bianchi identity (\[bianchi1\]) is used. Now the covariant derivative $\td(\bullet)$ can be replaced by the external derivative $d(\bullet)$ because a form $\SSS_\cdott\wedge\RRR^\cdott$ is a Lorentz scalar. Thus, the action integral can be obtained in the $4$-dimensional manifold $\M_4$ as the second Chern class as; $$\begin{aligned} (\ref{csaction1})&=&\frac{1}{2}\int_{\Sigma_5}d\left(\SSS_\cdott\wedge\RRR^\cdott\right) =\int_{\partial\Sigma_5=\Sigma_4}\LLL_{CW}\label{csaction0},\end{aligned}$$ where $$\begin{aligned} \LLL_{CW}&=&\frac{1}{2}\SSS_\cdott\wedge\RRR^\cdott,\label{csaction}\end{aligned}$$ and it is nothing more than the Einstein action (\[gravact\]) in the $4$-dimensional space time without the cosmological term. This Lagrangian, $\LLL_{CW}$, is referred to as the Chern-Weil Lagrangian in this study because this topological invariant can be understood as the result of a Chern-Weil theory[@zbMATH03077491; @doi:10.1063/1.4990708]. In conclusion, we have confirmed that the Einstein–Hilbert gravitational action in the $4$-dimensional space time is the topological invariant of the second Chern class. Our formalism can be categorized as a topological theory. (Here, the topological theory does not mean the “topological field theory” whose correlation function is independent from the metric tensor. In this report, the “topological theory” means the theory with conserved topological (characteristic) classes. ) Because several theories which have an invariant characteristic class are know and investigated intensively, a brief explanation of the topological theory is summarized in Appendix \[apBF\]. Next the co-translation $\delta_{CT}$ of the Lagrangian is considered. A co-translation operator defined as $\delta_{CT}=\varepsilon^{1/2}\xi^a\delta_T\iota_a$ induces a transformation on the fundamental forms as follows: $$\begin{aligned} \delta_{CT}\left(\eee^a\wedge\eee^b\right)&=&\varepsilon\left\{ \xi^a\wedge\td\xi^b-\xi^b\wedge\td\xi^a\right\},\label{dctS}\\ \delta_{CT} \www^{ab}&=&0,\label{dctW}\end{aligned}$$ where $\varepsilon\in\R^+$ is (positive real) infinitesimal parameter, $\delta_T$ is the translation operator which induces $ \delta_T \eee^a=\varepsilon\td\xi^a. $ Here $\xi^a\in\Omega^0\cap V^1(\TM)$ is a vector introduced in (\[tfv\]). A result of the co-translation operator on the Lagrangian form can be obtained as follows: $$\begin{aligned} \delta_{CT}\LLL_{CW} &=&\frac{1}{4}d\left(\varepsilon\epsilon_{\cdott\cdott} \RRR^\cdott\xi^\bcdot\wedge\xi^\bcdot\right),\label{sft}\end{aligned}$$ which is a total derivative. A relation (\[sft\]) does not depend on a choice of $\xi$’s. In conclusion, it is confirmed that the Einstein–Hilbert Lagrangian without neither the cosmological constant nor matter fields are invariant under the co-translation up to the total derivative. Due to the co-translation symmetry, the $4$-dimensional general relativity can have a topological invariant of a second Chern class when there is no cosmological term. We note that the cosmological term cannot appear in the action under the invariant quadratic of (\[iq2\]), because a term $\vvv=\SSS_\cdott\wedge\overline{\SSS}^\cdott$ has no co-translation symmetry. To implement the cosmological term by means of the second Chern class, we have to use an invariant quadratic such as $\langle P_{ab},P_{cd}\rangle\neq0$. In reality, for orthogonal groups in $4$-dimension, there are two isomorphisms such as ${SO}(4)\simeq{SU}(2)\times{SU}(2)$, and ${SO}(2,2)\simeq{SL}(2,R)\times{SL}(2,L)$. The second splitting induces the invariant quadratic with $\langle P_{ab},P_{cd}\rangle\neq0$. However, one must use a complex representation for the second splitting in the case of the ${SO}(1,3)$ group. It is known that the complexified theory may lead to difficulties in the construction of the quantum Hilbert space[@MenaMarugan:1994sj; @Witten:2003mb; @Oda:2003iu] after the quantization of the theory. On the other hand, the three-dimensional theory has two sets of [*real*]{} invariant-quadratics[@Witten198846]. Therefore, the gravitational theory in the $(1+2)$-dimension can treat both de Sitter and anti-de Sitter spaces as the topological theory. Hamiltonian formalism {#Hamilton} --------------------- For the sake of the quantization of the theory, we rewrite the classical gravitation theory in the Hamiltonian formalism in a completely covariant way. The cosmological constant is set to be zero hereafter, because our interest is in the formulation of general relativity as a Chern–Weil theory. The inclusion of the cosmological constant in the Hamiltonian formalism is straightforward[@Kurihara2018], if a topological aspect is ignored. Based on discussions in a previous section, the covariant Hamiltonian formalism is introduced as follows: In the geometrical quantization method, the principal bundle is identified as the phase space of a dynamical system, and thus $\vomega$ can be identified as the first canonical variable. Then, the second canonical variable $\MMM$ (the canonical momentum) is introduced as; $$\begin{aligned} \MMM_{ab}=\frac{\delta{\LLL}_G}{\delta\left(d\vomega^{ab}\right)}=\SSS_{ab}.\end{aligned}$$ Therefore, the two fundamental forms $\{\vomega,\MMM=\SSS\}$ form a phase space (fundamental forms), consistent with the previous observation. The Hamiltonian density can be obtained using the standard method as; $$\begin{aligned} \HHH_G&=& \frac{1}{2}\MMM_\cdott\wedge d\vomega^\cdott- {\LLL}_G = -\frac{1}{2}\SSS_\cdott\wedge \vomega^{\bcdot}_{~\ast}\wedge\vomega^{\ast\bcdot}.\label{HG}\end{aligned}$$ We note that this Hamiltonian density is $not$ the local ${SO}(1,3)$ invariant because it depends on the spin form itself, which is not the Lorentz tensor. It is natural because the gravitational field can be eliminated at any point of the global manifold due to the Einsteins equivalence principle. Therefore, the energy of the gravitational field depends on the local coordinate. For example, the energy of the gravitational wave is well-defined only in the asymptotically flat space time. A quantity $\iota_a\HHH_G$ can be understood as an energy flow with respect to the direction of the vector $\xi^a$ in the flat asymptotic frame[^1]. The first canonical equations can then be obtained as; $$\begin{aligned} \frac{\delta\HHH_G}{\delta\MMM_{ab}}&=& -\epsilon_{\cdott\cdott}\left(\epsilon_{ab\bcdot\ast}\eee^\ast\right)^{-1}\wedge\eee^\bcdot\wedge\www^\bcdot_{~\ast}\wedge\www^{\ast\bcdot}= d\vomega^{ab},\label{domega}\end{aligned}$$ where $\delta\bullet/\delta\SSS=(\delta\bullet/\delta\eee)(\delta\SSS/\delta\eee)^{-1}$ is used. Thus we can obtain $$\begin{aligned} -\epsilon_{a\cdott\bcdot}\eee^\bcdot\wedge\www^\bcdot_{~\ast}\wedge\www^{\ast\bcdot}&=& \epsilon_{a\cdott\bcdot}\eee^\bcdot\wedge d\vomega^{\cdott}.\label{canonicaleq1}\end{aligned}$$ The second canonical equations can then be obtained as; $$\begin{aligned} \frac{\delta\HHH_G}{\delta\vomega^{ab}}&=&-d\MMM_{ab},\label{canonicaleq2}\end{aligned}$$ The first equation (\[canonicaleq1\]) gives the equation of motion, $ \epsilon_{a\cdott\bcdot}\RRR^\cdott\wedge\eee^\bcdot=0, $ which simply leads to the Einstein equation without matter fields and the cosmological constant, and the second equation (\[canonicaleq2\]) leads to the torsion less condition, such as $\td\SSS_{a\bcdot}\wedge\eee^\bcdot=2\TTT^\bcdot\wedge\SSS_{a\bcdot}=0$, as expected. The Poisson bracket can be introduced in the covariant formalism as; $$\begin{aligned} \left\{\aaa,\bbb\right\}_\mathrm{PB}&=& \frac{\delta\aaa}{\delta\vomega^\cdott}\wedge\frac{\delta\bbb}{\delta\SSS_\cdott}- \frac{\delta\bbb}{\delta\vomega^\cdott}\wedge\frac{\delta\aaa}{\delta\SSS_\cdott}\end{aligned}$$ where $\aaa$ and $\bbb$ are arbitrary forms. The Poisson brackets for the fundamental forms become $$\begin{aligned} \left\{\vomega^{ab},\vomega^{cd}\right\}_\mathrm{PB}&=& \left\{\SSS_{ab},\SSS_{cd}\right\}_\mathrm{PB}~=~0,\label{pb1}\\ \left\{\vomega^{a_1a_2},\SSS_{b_1b_2}\right\}_\mathrm{PB}&=& \delta^{[a_1}_{b_1}\delta^{a_2]}_{b_2}.\label{pb2}\end{aligned}$$ One can confirm easily that $$\begin{aligned} \epsilon_{a\cdott\bcdot}\left\{\vomega^\cdott,\HHH_G\right\}_\mathrm{PB}\wedge\eee^\bcdot&=& -\epsilon_{a\cdott\bcdot}\vomega^\bcdot_{~\ast}\wedge\vomega^{\ast\bcdot}\wedge\eee^\bcdot =\epsilon_{a\cdott\bcdot}d\vomega^\cdott\wedge\eee^\bcdot,\\ % \left\{\SSS_{ab},\HHH_G\right\}_\mathrm{PB}&=& -\left(-\eta_{b\bcdot}\vomega^\cdott\wedge\SSS_{\bcdot a}\right) =d\SSS_{ab},\end{aligned}$$ where the Einstein equation and torsion less condition are used. The Hamiltonian form can be understood as a generator of a total derivative of a given form. Canonical quantization {#canonical} ====================== Quantum field theory of general relativity under the Kugo–Ojima formalism[@Kugo1978459; @kugo1979local] discussed by Nakanishi in a series of papers[@Nakanishi:1977gt; @Nakanishi:1978zx; @Nakanishi:1978np; @Nakanishi:1978rt; @Nakanishi:1979ff; @Nakanishi:1979rr; @Nakanishi:1979rq; @Nakanishi:1979um; @Nakanishi:1980rf; @Nakanishi:1980db; @Nakanishi:1980yf; @Nakanishi:1980hp; @Nakanishi:1980kf; @Nakanishi:1981fj] and summarized in reports[@nakanishi1983manifestly; @nakanishi1990covariant]. It is self consistent, physically and mathematically rigorous, and very beautiful theory. We follow their method to quantize our topological theory in this section. While discussions in this section are highly technical, they are important to understand that our quantum general relativity is mathematically well-defined. BRS-transformation {#6-1} ------------------ The Diracs procedure of the canonical quantization for the field theories with restraint conditions, such as the Yang–Mills gauge theory, require to add supplementary terms to kill a unphysical degree of freedom, e.g. gauge fixing and Faddeev–Popov ghost terms. After killing unphysical [d.o.f$.$]{}, still a symmetry remains, i.e. the BRS-symmetry, in the Lagrangian. A systematic way to perform quantization for such a restraint system has been established by Nakanishi[@Nakanishi01061966], and Kugo and Ojima[@Kugo1978459; @kugo1979local]. The BRS-transformation of the general relativity is introduced according to their recipe[@Nakanishi01061978; @Nakanishi01071978]. Necessary auxiliary and Faddeev–Popov (anti-)ghost fields are introduced as; - auxiliary field:    $\beta_{\mu~b}^{~a}(x)$, - ghost fields:       $\chi_{~b}^a(x)$ and $\chi_\mu(x)$, - anti-ghost field:  $\tilde{\chi}_{\mu~b}^{~a}(x)$, where $x\in\MM$. These fields are assumed to be Hermitian functions (operators). Here $\tilde{\bullet}$ is used to represent an anti-ghost, because a standard representation $\bar{\bullet}$ is already used for another operator. Another reason why $\bar{\bullet}$ is not used for the anti-ghost field here is that a relation between ghost and anti-ghost is not a particle–anti-particle relation, but are independent Hermitian-fields each other. The ghost field is considered to be factorized to two parts, a local Lorentz part $\chi^a_{~b}$ and a global part $\chi_\mu$, but the anti-ghost is not. The b-field and (anti-)ghost fields are assumed to be the global vector with respect to Greek indices, however not local Lorentz tensor with respect to Roman indices. Their Lorentz transformation is assumed to be the same as the spin connection. Roman indices can be brought-up or -down by $\eta^{ab}$ or $\eta_{ab}$, e.g. $\beta_\mu^{~ab}=\beta_{\mu~\bcdot}^{~a}\eta^{\bcdot b}$ and so on. Here, ghost and anti-ghost fields are assumed to anti-commute as in the quantization of the usual field theory. Moreover, those are anti-symmetric under an exchange between two Roman indices. The BRS-transformation, denoted as $\delBRS[\bullet]$ in this report, satisfies four rules stated in Ref.[@Nakanishi01071978]. First the BRS-transformation of the coordinate vector on $\MM$ should respect the general linear-transformation as; $$\begin{aligned} \delBRS\left[x^\mu\right]&=&\chi^\mu.\end{aligned}$$ In addition to that, we require the postulate given in [@Nakanishi01071978] such as; $$\begin{aligned} \delBRS\left[\partial_\mu X\right]&=&\partial_\mu\delBRS\left[X\right] -\left(\partial_\mu\delBRS\left[x^\nu\right]\right)\partial_\nu X,\end{aligned}$$ where $X$ is any field defined on $T\MM$. For differential forms on $T^*\MM$, the BRS-transformation acts as; $$\begin{aligned} \delBRS\left[dx^\mu\right]&=&\left(\partial_\nu\delBRS\left[x^\mu\right]\right)dx^\nu =d\left(\delBRS\left[x^\mu\right]\right) =~d\chi^\mu.\label{deltaBRSd}\end{aligned}$$ Therefore, the BRS-transformation and external derivative are commute each other[@Nakanishi01071978], i.e., $$\begin{aligned} \left[\delBRS,d\right]\bullet&=&\delBRS\left[d\bullet\right]- d\left(\delBRS\left[\bullet\right]\right)=~0.\label{dBRS}\end{aligned}$$ The BRS-transformation of above fields can be defined as; $$\begin{aligned} \left\{ \begin{array}{cl} \delBRS\left[\beta_{\mu~b}^{~a}\right]&=~0\\ \delBRS\left[\chi_\mu\right]&=~- g_{\mu\rho}\left( \partial^\rho\chi^\nu+ \partial^\nu\chi^\rho \right)\chi_\nu\label{BRSchi}\\ \delBRS\left[\chi^{a}_{~b}\right]&=~\chi^a_{~\bcdot}\chi^\bcdot_{~b}\\ \delBRS\left[\tilde{\chi}_{\mu~b}^{~a}\right]&=~i\beta_{\mu~b}^{~a} \end{array} \right.\end{aligned}$$ The BRS transformation is assumed to follow the Leibniz rule as; $$\begin{aligned} \delBRS\left[XY\right]&=&\delBRS\left[X\right]Y+\epsilon_XX\delBRS\left[X\right],\label{Leib}\end{aligned}$$ where the signature $\epsilon_X=-1$ when $X\in\{\chi_{~b}^a(x),\chi_\mu(x), \tilde{\chi}_{\mu~b}^{~a}(x)\}$, and $\epsilon_X=+1$ otherwise. The BRS-transformation of vierbein and spin connection are defined as; $$\begin{aligned} \delBRS\left[\Varepsilon^a_\mu\right]&=& \Varepsilon^{~\bcdot}_\mu~\chi_{~\bcdot}^{a} -\left(\partial_\mu\chi^{\nu}\right)\Varepsilon^a_\nu,\\ \delBRS\left[\omega^{~ab}_{\mu}\right]&=& \omega_{\mu}^{~a\bcdot}~\chi_{~\bcdot}^{b}+ \omega_{\mu}^{~\bcdot b}~\chi_{~\bcdot}^{a}- \partial_\mu\chi^{a}_{~\bcdot}~\eta^{b\bcdot} -\left( \partial_\mu\chi^\nu \right)\omega^{~ab}_{\nu}.\end{aligned}$$ The BRS-transformation of the vierbein inverse $\Varepsilon^\mu_a$ can be obtained from $ \delBRS\left[\Varepsilon^a_\mu~\Varepsilon_a^\nu\right]= \delBRS\left[\delta_\mu^\nu\right]~=~0, $ as; $$\begin{aligned} \delBRS\left[\Varepsilon_a^\mu\right]&=&-\chi^{\bcdot}_{~a}~\Varepsilon_{\bcdot}^{\mu} +\Varepsilon_a^\nu\left(\partial_\nu\chi^{\mu}\right).\end{aligned}$$ Accordingly, the BRS-transformation of the metric tensor and its inverse are obtained from above relations as; $$\begin{aligned} \delBRS\left[g_{\mu\nu}\right]&=&\delBRS\left[ \eta_\cdott \Varepsilon^\bcdot_\mu \Varepsilon^\bcdot_\nu \right]= -g_{\mu\rho}\partial_\nu\chi^\rho -g_{\nu\rho}\partial_\mu\chi^\rho,\label{BRSg}\\ \delBRS\left[g^{\mu\nu}\right]&=&\delBRS\left[ \eta^\cdott \Varepsilon_\bcdot^\mu \Varepsilon_\bcdot^\nu \right]= g^{\mu\rho}\partial_\rho\chi^\nu+ g^{\rho\nu}\partial_\rho\chi^\mu.\label{BRSgi}\end{aligned}$$ Anti-symmetry of the ghost filed $\chi_{ab}=-\chi_{ba}$ and symmetry of the metric tensor are used here. The BRS-transformations of the vierbein and spin forms become $$\begin{aligned} \delBRS\left[\eee^a\right]&=& \chi_{~\bcdot}^{a}\eee^\bcdot,\\ \delBRS\left[\vomega^{ab}\right] &=&\vomega^{a\bcdot}~\chi_{~\bcdot}^{b}+\vomega^{\bcdot b}~\chi_{~\bcdot}^{a}-d\chi^{a}_{~\bcdot}~\eta^{b\bcdot},\end{aligned}$$ follows from above results. In response to above quantities, corresponding one-forms are introduced as; $$\begin{aligned} \left\{ \begin{array}{lll} \bbb^{ab}&=~\beta_{\mu}^{~ab}~\partial_\nu\chi^\mu dx^\nu&=~ \beta_{\mu}^{~ab}~d\chi^\mu,\label{b-form}\\ \ccc^{a}~&=~\chi^a_{~\bcdot}~\eee^\bcdot&=~\chi^a_{~\bcdot}~\Varepsilon^\bcdot_\mu~dx^\mu,\\ \tilde{\ccc}^{ab}&=~\tilde{\chi}_{\mu}^{~ab}\partial_\nu\chi^\mu dx^\nu&=~ \tilde{\chi}_{\mu}^{~ab}~d\chi^\mu,\label{g-form} \end{array} \right.\end{aligned}$$ which are named as “[*b-form*]{}” and “[*$($anti-$)$ghost-form*]{}”, respectively. Note that $\bbb^{ab}$ and $\ccc^{a}$ ($\tilde{\ccc}^{ab}$) are anti-commutable (commutable), respectively. The BRS-transformations of those forms are given by $$\begin{aligned} \left\{ \begin{array}{ll} \delBRS\left[\bbb^{ab}\right]&= %~\delBRS\left[\ccc\right]~= ~\delBRS\left[\ccc^a\right]~=~0,\\ \delBRS\left[\tilde{\ccc}^{ab}\right]&=~i\bbb^{ab}. \end{array} \right.\end{aligned}$$ Moreover, the BRS-transformation of the vierbein form can be represented as; $$\begin{aligned} \delBRS\left[\eee^a\right]&=&\ccc^a.\end{aligned}$$ One can easily confirm that the BRS-transformation is nilpotent for all forms and fields defined above by direct calculations. A proof of nilpotency for all necessary forms are given in Appendix \[ap1\]. Quantum Lagrangian {#6-2} ------------------ We start from the Lagrangian of the classical gravity given in (\[Lagrangian44\]). The quantum Lagrangian can be obtained by adding gauge-fixing and Faddeev–Popov ghost terms to the Lagrangian for the classical gravity, i.e. $$\begin{aligned} \LLL_{QG}&=&\LLL_G+\LLL_{GF}+\LLL_{FP},\label{QLGF}\end{aligned}$$ where $\LLL_{GF}$ is referred to as the gauge-fixing Lagrangian form, and $\LLL_{FP}$ is as the Faddeev–Popov Lagrangian form. The quantum Lagrangian form must satisfy following conditions: 1. invariant under the general linear transformation, 2. invariant under the local Lorentz transformation, 3. invariant under the BRS-transformation, and 4. nilpotent under the BRS-transformation. Requirements $3$ and $4$ imply that the Lagrangian form must be the BRS-null object, i.e., $\delBRS\left[\LLL_{QG}\right]=0$. It can be confirmed that the classical part of the Lagrangian form $\LLL_G$ retains all conditions. The gauge-fixing Lagrangian form is determined to obtain a desired gauge-fixing condition. Here we require the de Donder condition for the gauge fixing, which can be represented using the metric tensor as $\partial_\mu\left(\sqrt{-g}~g^{\mu\nu}\right)=0$. An alternative representation is also possible as $\Gamma^\lambda_{~\mu\nu}~g^{\mu\nu}=0$. Correspondent representations in the vierbein formalism are given as $d\SSS_{ab}=0$, which is identical to $\vomega^a_{~\bcdot}\wedge\eee^\bcdot=0$, if the torsion less condition is assumed. A simplest candidate of the gauge-fixing form, which give the de Donder condition as a Euler-Lagrange equation with respect to variation of the b-form is given as; $$\begin{aligned} \LLL_{GF}&=& -\frac{1}{2}\left(d\bbb^\cdott+\alpha\bbb^\bcdot_{~\ast}\wedge\bbb^{\ast\bcdot} \right)\wedge\SSS_\cdott,\label{LGF}\end{aligned}$$ where $\alpha\in\R$ is a gauge fixing parameter. In a practical sense, the $\alpha$-term is vanished autonomously, i.e., $\alpha\bbb^\bcdot_{~\ast}\wedge\bbb^{\ast\bcdot}\wedge\SSS_\cdott=0$, due to the anti-commutation of the b-form. However, it is left here to discuss how the gauge-fixing term works. The term $d\bbb^{ab}+\alpha\bbb^a_{~\bcdot}\wedge\bbb^{\bcdot b}$ itself (without multiplying the surface form) retains Lorentz invariance, when $\alpha=1$. It can be confirmed as follows: The first two terms $d\bbb^{ab}+\bbb^a_{~\bcdot}\wedge\bbb^{\bcdot b}$ is a Lorentz tensor, because it has the same structure as the curvature form using the b-form instead of the spin from. Therefore, a $4$-form such as $\left(d\bbb^\cdott+\bbb^\bcdot_{~\ast}\wedge\bbb^{\ast\bcdot}\right)\wedge\SSS_\cdott$ is local Lorentz invariant. Searching for the Feddeev–Popov ghost-Lagrangian, we note that the gauge-fixing Lagrangian can be represented as; $$\begin{aligned} \LLL_{GF}&=&\frac{i}{2} \delBRS\left[d\tilde{\ccc}^\cdott+\alpha \tilde{\ccc}^{\bcdot}_{~\ast}\wedge\bbb^{\ast\bcdot} \right]\wedge\SSS_\cdott.\end{aligned}$$ Therefore, if the Faddeev–Popov form is taken as; $$\begin{aligned} \LLL_{FP}&=&-\frac{i}{2}\left( d\tilde{\ccc}^\cdott+ \alpha \tilde{\ccc}^{\bcdot}_{~\ast}\wedge\bbb^{\ast\bcdot} \right)\wedge\delBRS\left[\SSS_\cdott\right],\end{aligned}$$ a sum of gauge-fixing and Faddeev–Popov terms is BRS-null, because anti-ghost and surface forms are nilpotent. As the result, the Faddeev–Popov Lagrangian form can be obtained as; $$\begin{aligned} \LLL_{FP}&=& -\frac{i}{2}\left(d\tilde{\ccc}^\cdott+ \alpha \tilde{\ccc}^{\bcdot}_{~\ast}\wedge\bbb^{\ast\bcdot}\right) \wedge\left(\epsilon_{\cdott c_1c_2}\chi^{c_1}_{~c_3} \eee^{c_3}\wedge\eee^{c_2} \right),\nonumber\\ &=&-\frac{i}{2}\left( d\tilde{\ccc}^\cdott+ \alpha \tilde{\ccc}^{\bcdot}_{~\ast}\wedge\bbb^{\ast\bcdot}\right) \wedge\ccc^\star\wedge\overline{\eee}_{\star\cdott}.\label{LFP}\end{aligned}$$ The Faddeev–Popov Lagrangian form is [*not*]{} Lorentz invariant, even if $\alpha=1$, and the $\alpha$-term in $\LLL_{FP}$ is not vanished after multiplying $\delBRS\left[\SSS_{ab}\right]$. Therefore, the Faddeev–Popov Lagrangian form can fix the Lorentz invariance. Finally the quantum Lagrangian form, (\[QLGF\]), (\[LGF\]), and (\[LFP\]) with any value of $\alpha$, satisfies all four requirements given above. According to the particle-physics terminologies, it is referred to as Landau-gauge when $\alpha=0$, and Feynman-gauge when $\alpha=1$, respectively. An equation of motion can be obtained as the Euler–Lagrange equation under the variational principle. All relevant equations of motion are derived in Appendix \[app2\]. Commutation relations --------------------- From the quantum Lagrangian (\[QLGF\]), the spin form $\vomega^{ab}$, and forms $\bbb^a, \tilde{\ccc}^{ab}$ defined by (\[g-form\]), the formal expressions of the commutation relation can be singled out as $$\begin{aligned} \left[\widehat{\vomega}^{ab}(x),\widehat{\SSS}_{cd}(y)\right]&=&-i\delta^{(4)}(x-y) \delta^{[a}_{c}\delta^{b]}_{d},\label{CR1}\\ \left[\widehat{\bbb}^{ab}(x),\widehat{\SSS}_{cd}(y)\right]&=&-i\delta^{(4)}(x-y) \delta^{[a}_{c}\delta^{b]}_{d},\label{CR2}\\ \left\{\widehat{\tilde\ccc}^{ab}(x),(\widehat{\ccc^\bcdot\wedge\overline{\eee}}_{\bcdot cd})(y)\right\}&=&-i\delta^{(4)}(x-y) \delta^{[a}_{c}\delta^{b]}_{d}.\label{CR3}\end{aligned}$$ where $\widehat{\bullet}$ is denoted the operator corresponding to the field or form $\bullet$, hereafter. We note that $\delta\LLL_{QG}/\delta\bbb^{ab}=\SSS_{ab}$. We dont need to treat the form $\ccc^a$ as the canonical variable. For later discussions, above commutation relations are rewritten by means of original fields instead of forms. First the commutation relations (\[CR1\]) and (\[CR2\]) are given by $$\begin{aligned} \left[\widehat{\omega}_{\mu~\bcdot}^{~a}(x)\eta^{\bcdot b},\widehat{\SSS}_{cd}(y)\right]dx^\mu&=&-i\delta^{(3)}(x-y) \delta^{[a}_{c}\delta^{b]}_{d},\label{fCR1}\\ \left[\widehat{\beta}_{\nu~\bcdot}^{~a}(x)\eta^{\bcdot b}\partial_\mu\widehat{\chi}^\nu, \widehat{\SSS}_{cd}(y)\right]dx^\mu&=&-i\delta^{(3)}(x-y) \delta^{[a}_{c}\delta^{b]}_{d}.\label{fCR2}\end{aligned}$$ They are still formal expressions with omitting a $2$-dimensional integration measure in the right-hand side in order to avoid long expressions. Since these expressions are enough for following discussions, we use these formal expressions. To obtain the field-representation for (\[CR3\]), some manipulations are necessary. The left had side of the (\[CR3\]) can be expressed as $$\begin{aligned} (\ref{CR3})&=& \epsilon_{cde_1e_2}~\widehat{\chi}^{e_1}_{~~\bcdot}~\epsilon^{\bcdot e_2 f_1f_2}\left\{ \widehat{\tilde{\chi}}_{\nu~\ast}^{~a}\eta^{\ast b}\partial_\mu\widehat{\chi}^\nu,\widehat{\SSS}_{f_1f_2} \right\}dx^\mu.\end{aligned}$$ Let us introduce matrix expressions as $$\begin{aligned} \widehat{\bm{\epsilon\chi\bar{\epsilon}}}&:=& \epsilon_{ab e_1e_2}~\widehat{\chi}^{e_2}_{~~\bcdot}~\epsilon^{\bcdot e_1cd},\\ \widehat{\{\tilde{\bm\chi},\bm\SSS\}}&:=& \left\{ \widehat{\tilde{\chi}}_{\nu~\bcdot}^{~a}\eta^{\bcdot b}\partial_\mu\widehat{\chi}^\nu ,\widehat{\SSS}_{cd} \right\}dx^\mu.\end{aligned}$$ Matrices $\widehat{\bm{\epsilon\chi\bar{\epsilon}}}$ and $\widehat{\{\tilde{\bm\chi},\bm\SSS\}}$ are $4\times4$ matrices, whose elements are again $4\times4$ matrices. The commutation relation (\[CR3\]) can be express as $$\begin{aligned} \widehat{\bm{\epsilon\chi\bar{\epsilon}}}\bcdot\widehat{\{\tilde{\bm\chi},\bm\SSS\}}&=& -i\delta^{(3)}(x-y)~\bm{\mathrm I},\label{CR3-2}\end{aligned}$$ where $\bm{\mathrm I}$ is a matrix expression of $\delta^{[a}_c\delta^{b]}_d$ that is also a $4\times4$ matrix of $4\times4$ matrices. If each element of matrix $\widehat{\bm{\epsilon\chi\bar{\epsilon}}}$ is invertible, one can obtain the commutation relation of $\widehat{\{\tilde{\bm\chi},\bm\SSS\}}$ by applying the inverse matrix on (\[CR3-2\]). In reality, the matrix is invertible except a diagonal part of the matrix. While the diagonal part shows $ \left[\widehat{\bm{\epsilon\chi\bar{\epsilon}}}\right]_{aa}=\bm{0} $, where $\bm{0}$ is $4\times4$ zero matrix, the diagonal part is not necessary for the inversion because the diagonal part of $\widehat{\{\tilde{\bm\chi},\bm\SSS\}}$ is also zero-matrix due to the antisymmetric of the surface form. Therefore, one can get a matrix inverse as; $$\begin{aligned} \left[\widehat{\bm{\epsilon\chi\bar{\epsilon}}}^{-1}\right]_{IJ}&=& \left\{ \begin{array}{cl} \left(\left[\widehat{\bm{\epsilon\chi\bar{\epsilon}}}\right]_{JI}\right)^{-1}&(I\neq J) \\ \bm{0}&(I=J), \end{array} \right.\end{aligned}$$ where $I$ and $J$ are indices for each $4\times4$ matrix. Thus the commutation relation can be obtained as; $$\begin{aligned} \widehat{\{\tilde{\bm\chi},\bm\SSS\}}&=& -i\delta^{(3)}(x-y) \widehat{\bm{\epsilon\chi\bar{\epsilon}}}^{-1}, \label{fCR3}\end{aligned}$$ as the matrix representation. Each element of the matrix $\left[\widehat{\bm{\epsilon\chi\bar{\epsilon}}}^{-1}\right]_{IJ}$ is one of $\pm\left(\chi^a_{~b}\right)^{-1}$ or zero. In summary, three non-zero commutation-relations (\[fCR1\]), (\[fCR2\]) and (\[fCR3\]) are obtained. All other commutation relations are zero. BRS-charge {#BRSCharge} ---------- The Noether charge associated with the BRS-transformation can be constructed according to the recipe presented in [@Kurihara2018]. The conserved Noether charge can be given by $$\begin{aligned} \QQQ_\xi&=&\frac{1}{2}\left( \iota_\xi\vomega^\cdott \right)\SSS_\cdott,\end{aligned}$$ where $\xi^\mu$ a vector on $T\MM$. First, taking the auxiliary field as the vector field $\xi^\mu$, we get a conserved charge as; $$\begin{aligned} \QQQ_\bbb&:=& \frac{1}{2}\left(\iota_{\bbb}\www^\cdott\right)\SSS_\cdott= \frac{1}{2}g^{\mu_1\mu_2}{\beta}^{~a_1}_{\mu_3~a_2}\partial_{\mu_1}\chi^{\mu_3}\omega_{\mu_2}^{~a_2a_3}~\SSS_{a_3a_1},\end{aligned}$$ and its BRS-transformation is; $$\begin{aligned} \widehat{\QQQ}_\bbb&:=&\delBRS\left[\QQQ_\bbb\right]= \frac{1}{2}{\beta}^{~a_1}_{\mu_3~a_2}\delBRS\left[g^{\mu_1\mu_2}\partial_{\mu_1}\chi^{\mu_3}\omega_{\mu_2}^{~a_2a_3}~\SSS_{a_3a_1}\right].\end{aligned}$$ All fields in the BRS-transformation, $g$, $\partial\chi$, $\omega$, and $\SSS$, are nilpotent, therefore $\widehat{\QQQ}_\beta$ is BRS-null due to the remark in section \[ap1\]. Next, the anti-ghost form is taken as the vector field $\xi^\mu$. As a consequence, the conserved charge can be obtained as; $$\begin{aligned} \QQQ_{\tilde\ccc}&:=&\frac{1}{2}\left(\iota_{\tilde\ccc}\www^\cdott\right)\SSS_\cdott= \frac{1}{2}g^{\mu_1\mu_2}{\tilde\chi}^{~a_1}_{\mu_3~a_2}\partial_{\mu_1}\chi^{\mu_3}\omega_{\mu_2}^{~a_2a_3}~\SSS_{a_3a_1}.\end{aligned}$$ The BRS-transformation of this charge is obtained as; $$\begin{aligned} \widehat{\QQQ}_{\tilde\ccc}&:=&\frac{i}{2} \delBRS\left[\QQQ_{\tilde\ccc}\right],\\&=& \frac{i}{4}{\tilde\chi}^{~a_1}_{\mu_1~a_2}\delBRS\left[g^{\mu_1\mu_2}\partial_{\mu_1}\chi^{\mu_3}\omega_{\mu_2}^{~a_2a_3}~\SSS_{a_3a_1}\right]\nonumber\\ &~&-\frac{1}{4}{\beta}^{~a_1}_{\mu_1~a_2}\left(g^{\mu_1\mu_2}\partial_{\mu_1}\chi^{\mu_3}\omega_{\mu_2}^{~a_2a_3}~\SSS_{a_3a_1}\right),\end{aligned}$$ where a factor $i/2$ is just a convention. Then again by taking the BRS-transformation of this, one can get $$\begin{aligned} \delBRS\left[\widehat{\QQQ}_{\tilde\ccc}\right]&=&-\frac{1}{2} {\beta}^{~a_1}_{\mu_1~a_2}\delBRS\left[g^{\mu_1\mu_2}\partial_{\mu_1}\chi^{\mu_3}\omega_{\mu_2}^{~a_2a_3}~\SSS_{a_3a_1}\right]= -\widehat{\QQQ}_\bbb,\end{aligned}$$ where nilpotent of $(g\omega\SSS)$ is used. Therefore, the charge $\QQQ_{\tilde\ccc}$ is not nilpotent, but $\widehat{\QQQ}_{\tilde\ccc}$ is. Since charges $\QQQ_\xi$ is conserved[@Kurihara2018] and external derivative and the BRS-transformation are commute each other as (\[dBRS\]), both charges are conserved as $$\begin{aligned} d\widehat{\QQQ}_\xi&=&d\left(\delBRS\left[\QQQ_\xi\right]\right)=~\delBRS\left[d\QQQ_\xi\right]=~0,\end{aligned}$$ where $\xi=\tilde{\ccc},\bbb$. In conclusion, the two conserved charges $\widehat{\QQQ}_\bbb$ and $\widehat{\QQQ}_{\tilde\ccc}$ are obtained as $$\begin{aligned} \widehat{\QQQ}_\bbb&=&\frac{1}{2}\beta^{~\bcdot}_{\mu~\ast}~\delBRS\left[\widehat{\QQQ}_{~\bcdot}^{\mu~\ast}\right],\\ \widehat{\QQQ}_{\tilde\ccc}&=&\frac{i}{4}\tilde\chi^{~\bcdot}_{\mu~\ast}~\delBRS\left[\widehat{\QQQ}_{~\bcdot}^{\mu~\ast}\right]-\frac{1}{4}\beta^{~\bcdot}_{\mu~\ast}~\widehat{\QQQ}_{~\bcdot}^{\mu~\ast},\\\end{aligned}$$ where $$\begin{aligned} \widehat{\QQQ}_{~~b}^{\mu a}&:=&g^{\nu_1\nu_2}\partial_{\nu_1}\chi^{\mu}\omega_{\nu_2}^{~ac}~\SSS_{cb},\end{aligned}$$ which satisfy $$\begin{aligned} \left\{ \begin{array}{cll} d\widehat{\QQQ}_\bbb&=&d\widehat{\QQQ}_{\tilde\ccc}~=~0,\\ \delBRS\left[\widehat{\QQQ}_{\tilde\ccc}\right]&=&-\widehat{\QQQ}_\bbb,\\ \delBRS\left[\widehat{\QQQ}_{\bbb}\right]&=&0. \end{array} \right.\end{aligned}$$ Moreover, the charge $\widehat{\QQQ}_{\bbb}$ is a generator of the BRS-transformation for operators, such as $$\begin{aligned} i\lambda~\delBRS\left[\widehat\Omega\right]&=&\left[\widehat{\Omega},\lambda~\widehat{\QQQ}_\bbb\right],\label{BRSgen0}\end{aligned}$$ where $\widehat\Omega$ is any operator and $\lambda=i(-1)^{\pm\widehat{\QQQ}_{\tilde\ccc}}$[@Nakanishi:1977gt]. Especially by taking $\widehat\Omega=\widehat{\QQQ}_{\tilde\bbb}$ or $\widehat{\QQQ}_{\tilde\ccc}$, one can obtain $$\begin{aligned} \widehat{\QQQ}_{\bbb}^2&=&0,\label{BRSgen1}\\ \left[\widehat{\QQQ}_{\tilde\ccc},\widehat{\QQQ}_\bbb\right]&=&-i\widehat{\QQQ}_\bbb.\label{BRSgen2}\end{aligned}$$ A relation $(\ref{BRSgen1})$ immediately follows from nilpotent of $\widehat{\QQQ}_{\bbb}$ and $\lambda=-1$ in (\[BRSgen0\]). A proof of the relation (\[BRSgen2\]) is given in Appendix \[appC\]. Discussions =========== In a previous section, we performed quantization of our theory according to the canonical method in details. In this section, we discuss physics based on quantum general relativity developed in a previous section. Hilbert space and physical states {#discs} --------------------------------- A tangent representation of the spin and surface forms is considered to introduce a Hilbert space. A set of square integrable real-functions $L^2(\MM)$ is introduced as the Hilbert space $\HH$. Spin forms obtained as a solution of the Einstein equation are assumed to be square integrable, and thus the space of coefficient functions of spin forms $\varpi$ are a subset of the Hilbert space as $\varpi\subset\HH\subset L^2$ The bar-dual space of $\varpi$, which is denoted $\overline{\varpi}$, is simply referred to as the dual space of $\varpi$. The dual space $\overline{\varpi}$ is also square integrable, and thus the Gel’fand triple becomes $\varpi\subseteq\HH\subseteq\overline{\varpi}$. For the spin form $\www^{ab}$ and its dual form $\overline\www^{ab}$, their coefficient functions are written as $\omega^{ab}_{~~\mu}\in\varpi$ and $\frac{1}{2}\epsilon_{ab\cdott}\hspace{.2em}\omega^\cdott_{~\mu}\in\overline\varpi$, respectively. A standard bilinear from $\langle \overline{\www}|\www\rangle$ is defined as; $$\begin{aligned} \langle \overline\www|\www\rangle&:=&\frac{1}{2}\overline\www_\cdott\otimes\www^\cdott= \frac{1}{4}\epsilon_{\bcdott\bcdott}\hspace{.2em} \omega^{~~\cdott}_{\mu_1}\hspace{.2em}\omega^{~~\cdott}_{\mu_2}\hspace{.2em} dx^{\mu_1}\otimes dx^{\mu_2}.\end{aligned}$$ Under this bilinear from, the functional space $\varpi$ can be regarded as the pseudo-Riemannian manifold with a metric tensor of; $$\begin{aligned} g_{\mu\nu}&:=&\left[{\bm g}\left(\overline{\www},\www\right)\right]_{\mu\nu}= \frac{1}{4}\epsilon_{\bcdott\bcdott}\hspace{.2em} \omega^{~~\cdott}_{\mu}\hspace{.2em}\omega^{~~\cdott}_{\nu}.\end{aligned}$$ A norm of the spin form $\|\www\|$ can be defined using the bilinear form as; $$\begin{aligned} \|\www\|^2:=\int_{\Sigma_2}\langle\overline\www|\www\rangle\in\R,\end{aligned}$$ where ${\Sigma_2}$ is an appropriate $2$-dimensional sub-manifold ${\Sigma_2}\subset\M$. A form $d\www^{ab}$ is considered as a $1$-from defined on $\varpi$ whose coefficient functions are $d\omega^{~ab}_\mu=(\partial_\bcdot\omega^{~ab}_\mu)\eee^\bcdot\in\Omega^1(\varpi)$. An external derivative of tensor components can be transformed to that of the corresponding forms as shown in (\[diag\]). A element of the dual space $\overline{\varpi}$ is a linear combination of spin forms $\omega^{ab}_{~~\mu}\in\varpi$, and thus factional spaces $\varpi$ and $\overline{\varpi}$ are equivalent. Therefore, the Hilbert space can be taken as $\HH=\varpi$, because the Gel’fand triple is now $\varpi\subseteq\HH\subseteq \varpi$. The commutation relations corresponding to the Poisson brackets (\[pb1\]) and (\[pb2\]) are $$\begin{aligned} \left[\widehat{\vomega}^{ab},\widehat{\vomega}^{cd}\right]&=& \left[\widehat{\SSS}_{ab},\widehat{\SSS}_{cd}\right]~=~0\label{cr00}\end{aligned}$$ and (\[CR1\]). The above expressions of the commutation relations are formal ones. More precisely, the representation of (\[CR1\]) must be understood as $$\begin{aligned} \left[\widehat{\omega}_{\mu}^{~~ab}(x),\widehat{\Varepsilon_{\nu}^{c}\Varepsilon_{\rho}^{d}}(y)\right] &=&-i \epsilon^{abcd} \epsilon_{\mu\nu\rho\sigma} \delta^{(3)}(x^{\mu,\nu,\rho}-y^{\mu,\nu,\rho}) \int\delta^{(1)}(x^{\sigma}-y^{\sigma})dx^{\sigma}.\end{aligned}$$ Fixing $\sigma=0$ ([*time coordinate*]{}) instead taking a sum and performs the integration with respect to $dx^0$, one can obtain the standard [*equal-time*]{} commutation relation. The operator can be taken as $$\begin{aligned} \widehat{\vomega}^{ab}=\vomega^{ab},&~& \widehat{\SSS}_{ab}=i\frac{\delta~}{\delta\vomega^{ab}},\end{aligned}$$ reproducing the commutation relations (\[CR1\]) and (\[cr00\]). A Schr[" o]{}dinger equation becomes $$\begin{aligned} \widehat{\HHH}_G|\Psi(\omega)\rangle&=&E_G|\Psi(\omega)\rangle,\end{aligned}$$ where $\widehat{\HHH}_G$ is the Hamiltonian operator corresponding to (\[HG\]), $E_G$ is the energy eigenvalue and $|\Psi(\omega)\rangle\in\cal{H}$. While the existence of the norm is ensured, the negative norm states $\langle\vomega | \vomega \rangle<0$ are still included in $\cal{H}$, that corresponds to a negative energy state. For an explicit discussion, a following simple $(1\hspace{-.1em}+\hspace{-.1em}3)$ coordinate decomposition is considered: Assuming that the global space-time manifold is filled with congruence of geodesics whose tangent vector is time-like at any point on the line. A coordinate $x^0$ is taken along the time-like vector on those geodesics, and other three coordinates are taken as the orthonormal base. By using that coordinate system, the three-dimensional boundary is simply taken as a manifold at $x^0=\tau$ (an [*equal-time*]{} boundary) and the state is written as $|\Psi_\tau\rangle$. The expectation value of the operator $\widehat{\cal{O}}$ can be represented as followsWe note that the operator $\widehat{{\cal O}}(\widehat{\vomega},\widehat{\SSS})$ cannot be determined uniquely from the classical function ${\cal O}\left(\vomega,\SSS\right)$ due to the non-commutativity of quantum operators. This problem is known as the operator ordering problem in general. The operator ordering problem in quantum general relativity is discussed in ref.[@2017arXiv171207964K].: $$\begin{aligned} \overline{{\cal O}}&=& \langle\Psi_\tau|\widehat{{\cal O}}\left(\widehat{\vomega},\widehat{\SSS}\right)|\Psi_\tau\rangle= \langle\Psi_\tau|\int{\cal O}\left(\vomega,\SSS\right)|\Psi_\tau\rangle=\int{\cal O}\left(\vomega,\SSS\right).\end{aligned}$$ For instance, the Hamiltonian operator has an expected value given as $E_G=\int_{\partial\Sigma} \iota_{0}\HHH_G$, which is the total energy in the universe on a three-dimensional space at given time defined above. A trivial ground-state of $\www^{ab}=0$ exists with the eigenvalue of $E_G=0$. By also using the torsion less condition, one can obtain $d\eee^a=0$, which means that the ground state corresponds to the flat Lorentz spacetime. This is one of general results for the Chern–Weil theory: the critical point of the Chern-Weil action gives a flat connection. Due to an existence of BRS generators of (\[BRSgen1\]), (\[BRSgen2\]), and nilpotent of the BRS operator on forms, the physical state can be defined according to the Nakanishi–Kugo–Ojima formalism as $$\begin{aligned} \widehat{\QQQ}_\bbb|\mathrm{phys}\rangle&=&\widehat{\QQQ}_{\tilde\ccc}|\mathrm{phys}\rangle=~0,\end{aligned}$$ where $|\mathrm{phys}\rangle\in{\cal H}_{\mathrm{phys}}\subset{\cal H}$. As a consequence, $\langle\mathrm{phys}|\mathrm{phys}\rangle\geq0$ follows due to the Kugo–Ojima theorem. A transition operator $\widehat{{\mathcal S}}$ is introduced such that the matrix element $$\begin{aligned} p\left(\tau_2:\tau_1\right)&=&\langle\Psi_{\tau_2}|\widehat{{\mathcal S}}|\Psi_{\tau_1}\rangle\end{aligned}$$ gives the transition probability from the state $|\Psi_{\tau_1}\rangle$ at $x^0=\tau_1$ to another state $|\Psi_{\tau_2}\rangle$ at $x^0=\tau_2$, where $|\Psi_{\tau_{1,2}}\rangle\in{\cal H}_{\mathrm{phys}}$ and $\widehat{{\mathcal S}}$ is a functional constructed by the spin, surface, auxiliary, and Faddeev-Popov forms. Again the Kugo–Ojima theorem ensures the unitarity of this transition matrix[@nakanishi1990covariant]. In conclusion, we can state from above discussion that a time evolution of the universe should keep the unitarity of the state vector of quantum gravity. Renormalizability {#renorm} ----------------- Renormalizability has not been discussed so far in this study because our quantization is carried out non-perturbatively. However, as is well-known, a simple power-counting shows that quantum general relativity is non-renormalizable. In reality, for the pure gravitational theory including neither matter fields nor a cosmological constant, the following facts are known by direct calculations: ’t Hooft and Veltman confirmed that the on-shell S-matrix element of quantum general relativity is finite at one-loop[@'tHooft:1974bx], however the theory is non-renormalizable, if an interaction with scalar particles is included. On the other hand, Berends and Gastmans reported that the amplitude of the graviton-graviton scattering is not unitary even at that tree-level[@BERENDS197599]. Beyond the one-loop level, Goroff and Sagnotti reported that the theory is non-renormalizable at a $2$-loop[@Goroff198581; @GOROFF1986709]. Even though at first glance, the unitarity violation reported in [@BERENDS197599] appears to contradict our results, this is not the case. First, the metric tensor or vierbein used to be a standard variable of the perturbation expansion. However, the appropriate phase-space variables as the Chern–Weil action in $4$-dimensions are the spin and surface forms. The standard perturbation gravity has been using an improper expansion variable. Moreover, the covariant perturbation gravity, developed by DeWitt[@DeWitt:1967ub; @DeWitt:1967uc] and used in studies to show the non-renormalizability[@'tHooft:1974bx; @BERENDS197599; @Goroff198581; @GOROFF1986709], uses the background field method, in which the metric tensor is separated into the (classical) background metric $g_{\mu\nu}$ and the quantum field $h_{\mu\nu}$ such as $ g_{\mu\nu}\rightarrow g_{\mu\nu}+h_{\mu\nu}. $ On the other hand, the ground state of the quantum Hamiltonian must have a zero-point energy due to the quantum effect. Even though quantum general relativity in the covariant perturbation method is non-renormalizable in previous studies, we cannot conclude that quantum general relativity itself is non-renormalizable[^2]. Llewellyn Smith suggested that the renormalizability and the tree-level unitarity of the theory are equivalent[@LLEWELLYNSMITH1973233], as has been confirmed for some concrete examples, and with no exceptions reported to date. Since our formalism does not employ an interaction picture, but a Heisenberg picture instead, it is nothing to do with perturbative expansion with respect to a coupling constant. A method to solve quantum general relativity in a Heisenberg picture is discussed by Nakanishi and Abe, and summarized in Ref.[@doi:10.1143/PTP.111.301]. There are still possibilities that a renormalizable perturbation of quantum general relativity exists. Summary ======= In this study, we have re-formulated general relativity as a Chern–Weil theory on a $4$-dimensional space-time manifold. In a standard formulation, there is no topological invariance of a Chern-Simons type in $4$-dimensional space time because a translation invariant action cannot be constructed. A key idea to construct a topological invariant in an even-dimensional space is using a co-translation instead of a simple translation. The Einstein–Hilbert action can be identified a second Chern class and is invariant under the co-translation operation up to total derivative, when a cosmological constant does not exist. The theory with topological invariants introduced in this report is different from (so-called) the topological quantum field theory (QTFT)[@witten1988; @Atiyah1988], that is, simply put, the theory whose $2$-point correlation function is given as to independent from the metric. As a consequence, there exhibit no dynamics. Quantum general relativity proposed in this report has the $4$-dimensional Lagrangian given as the second Chern class in a five-dimensional manifold. As a consequence, the theory has topological invariants as the de Rham classes due to the Chern–Weil theory[@zbMATH03077491], For instance, Zanelli shows a natural quantization of the gravitational constant on odd-dimensional spacetime using a Euler classes[@Zanelli:1994ti]. A similar analysis can be performed in a $4$-dimension space time[@Kurihara2018]. This construction of the action integral suggests that a proper set of phase-space variables are the spin and surface forms. A canonical quantization based on those canonical variables is performed in a Heisenberg picture using the Nakanishi–Kugo–Ojima formalism. Since general relativity is a constraint system with global $GL(1,3)$ and local $SO(1,3)$ symmetries, we have to fix unphysical degree of freedom by implementing a gauge fixing and a ghost Lagrangian. After the gauge fixing, still the BRS symmetry is remaining. We gave a complete set of the quantum Lagrangian and BRS transformation operator with auxiliary and ghost fields in a self consistent manner. Next, an appropriate Hilbert space and physical states are introduced into the theory. As a consequence of the Kugo–Ojima theorem, the positivity of the physical states and the unitarity of the transition matrix were shown in this study. These results suggest that the renormalizability of quantum general relativity is worth reconsidering. I appreciate the kind hospitality of all members of the theory group of Nikhef, particularly Prof. J. Vermaseren and Prof. E. Laenen. A major part of this study has been conducted during my stay at Nikhef in 2017. In addition, I would like to thank Dr. Y. Sugiyama and Prof. J. Fujimoto for their continuous encouragement and fruitful discussions. Topological theories {#apBF} ==================== In this report, theories which have topological (characteristic) classes are referred to as the topological theory. We note that the [topological theory]{} is not the same as, so called the “topological field theory”, which has the metric independent correlation functions and does not have any dynamics. Three examples of the [topological theory]{}s are summarized under following setups in this Appendix. On a $n$-dimensional manifold $\M$, a principal bundle $P(\M,G)$ is defined, where $\M$ is the base manifold and $G$ is the structural Lie-group. The connection one-form $\AAA$ and the corresponding curvature $2$-form $\FFF=d\AAA+\AAA\wedge\AAA$ are equipped on the base manifold $\M$. Here $\AAA$ is a Lie-algebra valued one-form $\AAA\in\ggg$, where $\ggg$ is a Lie-algebra of the structural group $G$. Chern–Simons topological theory ------------------------------- In 1988, Witten shows that general relativity in the $(1+2)$-dimensional space can be considered as a Chern–Simons [topological theory]{}[@witten1988]. The base manifold $\M$ is a Riemannian manifold with $(1+2)$ space-time dimension and the structural group is $ISO(1,2)$. The connection one-form and curvature $2$-form can be respectively taken as $\AAA_0$ and $\FFF_0$ as (\[PCoF\]) and (\[PCaF\]) in the three-dimension. The action is given as; $$\begin{aligned} \I_{CS}&=&\frac{1}{4}\int_{\Sigma_4}\FFF_0\wedge\FFF_0 =\frac{1}{2}\int_{\Sigma_4}Tr\left[ \AAA_0\wedge d\AAA_0+\frac{2}{3}\AAA_0\wedge\AAA_0\wedge\AAA_0 \right],\nonumber\\ &=&\frac{1}{2}\int_{\Sigma_4}d\left( \epsilon_{\cdott\bcdot}~\eee^\bcdot\wedge\RRR^\cdott \right)=\frac{1}{2}\int_{\partial\Sigma_4={\Sigma}_3}\left( \epsilon_{\cdott\bcdot}~\eee^\bcdot\wedge\RRR^\cdott \right),\end{aligned}$$ where ${\Sigma}_4$ is an appropriate simply connected and orientable $4$-dimensional manifold. The relation between ${\Sigma}_4$ and ${\Sigma}_3$ is the same as that in (\[csaction0\]). This is nothing other than the Einstein–Hilbert action[^3] in three space-time dimension, and thus it is shown that general relativity can be constructed as the Chern-Simons [topological theory]{}. This coincidence is rather accidental only in the three dimensional space[@0264-9381-29-13-133001; @doi:10.1063/1.4990708]. While the Chern–Simins [topological theory]{} in three-dimension does not have any dynamical degree, this is not due to a topological aspect of the theory. From a simple counting of number of [d.o.f$.$]{}, one can understand that the dynamical [d.o.f$.$]{} in three-dimensional general relativity is zero at the classical level. BF topological theory --------------------- In 1977, the BF [topological theory]{} is introduced at first by Plebański[@doi:10.1063/1.523215], while the term “BF theory” did not exist yet at that time. In 1989, Horowitz first treated general relativity of the BF theory as the topological theory in general $n$ space-time dimension[@Horowitz1989]. Review articles of the BF [topological theory]{} can be found in [@1991PhR...209..129B; @Krasnov2011]. Some additional constraints are necessary to apply on the original BF [topological theory]{} to treat general relativity[@doi:10.1063/1.523215]. This constraint, called the simplicity condition, in four space time dimension is discussed by Gielen and Oritti[@Gielen:2010cu] (linear constraints) and Celada, González and Montesinos[@0264-9381-33-21-213001] (constraints on $\CC$ formalism). A reason why the BF [topological theory]{} requires the constraints is as follows: The base manifold $\M$ is taken as four space time dimensional Riemannian manifold and the structural group is taken as $SO(1,3)$ (or $SO(4)$). The connection one-form and curvature $2$-form can be respectively taken as the spin-form $\www^{ab}$ and curvature $2$-form $\RRR^{ab}$ as introduced in section \[prep\]. In addition to them, new Lie-algebra valued $2$-form $\BBB^{ab}$ is introduced, and then the BF topological action is introduced as; $$\begin{aligned} \tilde{\I}_{BF}&=&\frac{1}{2}\int_{{\Sigma}_4}\epsilon_{\cdott\cdott}~\BBB^\cdott\wedge\RRR^\cdott,\end{aligned}$$ where ${\Sigma}_4$ is an appropriate simply connected and orientable $4$-dimensional manifold. It cannot be simply recognized as the topological action of the structural group $SO(1,3)$ (or $SO(4)$), because the $2$-form $\BBB^{ab}$ does not belong to the principal bundle in general. Instead, the form $\BBB^{ab}$ can be understood as a connection form on a 2-bundle, and it forms principal 2-bundle in the higher-gauge theory[@doi:10.1063/1.1790048]. While the action $\tilde{\I}_{BF}$ can be topological by means of the 2-gauge theory, it is not coincide with the Einstein–Hilbert action. To convert the BF [topological theory]{} to a gravitational theory, additional constraints must be implement as a Lagrange multiplier term[@0264-9381-16-7-303] such as; $$\begin{aligned} {\I}_{BF}&=&\frac{1}{2}\int_{{\Sigma}_4}\left( \epsilon_{\cdott\cdott}~\BBB^\cdott\wedge\RRR^\cdott -\frac{1}{2}\phi_{\cdott\cdott}\BBB^\cdott\wedge\BBB^\cdott \right),\end{aligned}$$ where $\phi_{abcd}$ is a scalar symmetric traceless matrix. The simplicity condition appears as the equation of motion obtained by taking variation with respect to $\phi$. After implementing the constraint term, the action $\I_{BF}$ does not have a topological characteristic class any more. As one of solutions of the simplicity condition, the surface form $\eee^a\wedge\eee^b$ can be taken as the $2$-form $\BBB^{ab}$, and thus a shape of the BF action coincides with the Einstein–Hilbert action. This coincidence of the shape of the action is true only for solutions of the equation of motion (on-shell condition) in the classical level, and the on-shell condition cannot be simply true after quantization of the BF theory. This is the reason why quantization of the BF gravitational theory is complicated[@0264-9381-33-21-213001]. In addition, the spin form (spin connection) is not treated as a fundamental form as a canonical conjugate of $\BBB^{ab}$ in quantization of the BF theory. As a consequence, a quantum BF theory is different from the quantum theory proposed in this article. Chern–Weil topological theory ----------------------------- This is the theory intrduced in this report. The base manifold $\M$ is taken as Riemannian manifold with $(1\hspace{-.2em}+\hspace{-.2em}3)$ space time dimension and the structural group is $(1\hspace{-.2em}+\hspace{-.2em}3)$-dimensional co-Poincaré group introduced in section \[newsym\]. The connection $1$-form and curvature $2$-form can be respectively taken as $\AAA$ given in (\[c2f\]) and $\FFF$ given in (\[Pcon\]), introduced also in section \[newsym\]. The action integral of the Cern–Weil [topological theory]{} can be written as; $$\begin{aligned} \I_{CW}&=&\frac{1}{4}\int_{\widetilde{\M}_5}\mathrm{Tr}\left[ \FFF\wedge\FFF \right]=\frac{1}{2}\int_{\M_4}\SSS_\cdott\wedge\RRR^\cdott,\end{aligned}$$ as given by equations (\[csaction1\]) and (\[csaction0\]). In contrast with the BF gravitational theory, both forms $\SSS_{ab}$ and $\RRR^{ab}(\www)$ are directly obtained from the principal bundle, and thus the Einstein–Hilbert action itself keeps the characteristic class (second Chern-class), which is ensured by the Chern–Weil theory[@zbMATH03077491; @doi:10.1063/1.4990708]. The simplicity condition of the BF gravitational theory corresponds to the definition of the surface form in the Chern–Weil [topological theory]{}, and thus it is exact after quantization. Quantization of the Chern–Weil [topological theory]{} can be performed with a gauge fixing term with respect to the global $GL(1,3)$ and local $ISO(1,3)$ as shown in section \[canonical\]. In contrast with the Chern–Simons theory in a three-dimensional space time, the Chern–Weil theory in a $4$-dimensional space time has dynamical [d.o.f$.$]{} after quantization. Among ten [d.o.f$.$]{} on the symplectic fields of the vierbein and spin connection, two physical degrees are remaining for dynamical [d.o.f$.$]{}, corresponding to two spin states of graviton. Quantization of constrained system is performed using the Kugo–Ojima formalism in section \[6-1\]. At first, the auxiliary, ghost and anti-ghost fields are introduced to fix the gauge and unphysical [d.o.f$.$]{} in the system. Then the BRS transformations are required on all of physical and unphysical fields. At this stage, number of constraints due to the BRS transformations is the same as a total [d.o.f$.$]{} in the system, and thus there is no dynamical degree if all constraints are independent each other. In reality, as shown in section \[BRSCharge\], all constraints are not independent and there are two conserved BRS charges, $\widehat{\QQQ}_\bbb$ and $\widehat{\QQQ}_{\tilde\ccc}$, in the system. Therefore, the system still has two dynamical [d.o.f$.$]{} after quantization. Proof of nilpotency {#ap1} =================== \ The coordinate vectors are one of the most fundamental vectors on $T\MM$. The nilpotent can be confirmed as $$\begin{aligned} &~& \delBRS\left[\delBRS\left[x^{\mu}\right]\right] ~=~\delBRS\left[\chi^\mu\right] ~=~\delBRS\left[g^{\mu\nu}\chi_\nu\right],\\ &~&~=~ g^{\mu\rho}\left(\partial_\rho\chi^\nu\right)\chi_\nu+ g^{\rho\nu}\left(\partial_\rho\chi^\mu\right)\chi_\nu-g^{\mu\nu}g_{\nu\rho}\left( \partial^\rho\chi^\sigma+ \partial^\sigma\chi^\rho \right)\chi_\sigma,\\ &~&~=~ \left(\partial^\mu\chi^\nu\right)\chi_\nu+ \left(\partial^\nu\chi^\mu\right)\chi_\nu-\left(\partial^\mu\chi^\sigma\right)\chi_\sigma- \left(\partial^\sigma\chi^\mu\right)\chi_\sigma=0.\end{aligned}$$\ Starting from the BRS-transformation of the metric tensor (\[BRSg\]), $$\begin{aligned} \delBRS\left[\delBRS\left[g_{\mu\nu}\right]\right]&=& \delBRS\left[ -g_{\mu\rho}\partial_\nu\chi^\rho \right] +\delBRS\left[\mu\leftrightarrow\nu\right],\nonumber\\ &=&\left\{-\delBRS\left[g_{\mu\rho}\right]\partial_\nu\chi^\rho +g_{\mu\rho}\left(\partial_\nu\delBRS\left[x^\sigma\right]\right)\partial_\sigma\chi^\rho\right\} +\left\{\mu\leftrightarrow\nu\right\},\nonumber\\ &=&\left\{ g_{\mu\sigma}\left(\partial_\rho\chi^\sigma\right)\partial_\nu\chi^\rho+ g_{\mu\rho}\left(\partial_\nu\chi^\sigma\right)\partial_\sigma\chi^\rho\right\} +\left\{\mu\leftrightarrow\nu\right\}=0,\end{aligned}$$ where anti-commutativity of the ghost filed is used.\ \ Since the ghost field has two parts, nilpotent is checked separately. First for the $\chi_\mu$, $$\begin{aligned} \delBRS\left[\delBRS\left[\chi_\mu\right]\right] &=& \delBRS\left[\delBRS\left[g_{\mu\nu}\chi^\nu\right]\right] ~=~\delBRS\left[\delBRS\left[g_{\mu\nu}\right]\right]\chi^\nu=0,\end{aligned}$$ where $\delBRS[\chi^\mu]=0$ and nilpotent of the metric tensor are used. Direct calculation from (\[BRSchi\]) gives the same result, too. The second part becomes $$\begin{aligned} \delBRS\left[\delBRS\left[\chi^a_{~b}\right]\right]&=& \delBRS\left[\chi^a_{~c}\chi^{c}_{~b}\right] ~=~ \chi^a_{~c_2}\chi^{c_2}_{~~c_1}\chi^{c_1}_{~~b}-\chi^a_{~c_1}\chi^{c_1}_{~~c_2}\chi^{c_2}_{~~b}=0,\end{aligned}$$ due to anti-commutativity of the ghost field. A tensor $\partial_{\mu}\chi^{\nu}$ is also nilpotent as $$\begin{aligned} \delBRS\left[\delBRS\left[\partial_{\mu}\chi^{\nu}\right]\right]&=& -\delBRS\left[\partial_{\mu}\chi^{\rho}\partial_{\rho}\chi^{\nu}\right]=~ -\partial_{\mu}\chi^{\rho_1}\partial_{\rho_1}\chi^{\rho_2}\partial_{\rho_2}\chi^{\nu} +\partial_{\mu}\chi^{\rho_1}\partial_{\rho_1}\chi^{\rho_2}\partial_{\rho_2}\chi^{\nu}=~0.\end{aligned}$$ $$\begin{aligned} \delBRS\left[\delBRS\left[\eee^a\right]\right]&=& \delBRS\left[\eee^b\chi^a_{~b}\right]=~ \eee^{b_1}\chi^{b_2}_{~~b_1}\chi^a_{~b_2}+\eee^{b_2}\chi^a_{~b_1}\chi^{b_1}_{~~b_2}=0,\end{aligned}$$ due to anti-commutativity of the ghost field.\ \ One can trace the same calculation as a case of the vierbein form due to $\delBRS\left[d\chi^{ab}\right]=0$. Detailed calculations are omitted here.\ \ The volume form is global scalar and their BRS-transformation is expected to vanish, which can be confirmed as $$\begin{aligned} \delBRS\left[\vvv\right]&=&\frac{1}{4!}\epsilon_{\cdott\cdott} \delBRS\left[\eee^\bcdot\wedge\eee^\bcdot\wedge\eee^\bcdot\wedge\eee^\bcdot\right]~=~ \frac{1}{3!}\epsilon_{a_1\bcdot\cdott} \delBRS\left[\chi^{a_1}_{~~a_2}\eee^{a_2}\wedge\eee^\bcdot\wedge\eee^\bcdot\wedge\eee^\bcdot\right]~=~0,\end{aligned}$$ due to $\eee^\bcdot\wedge\eee^\bcdot\wedge\eee^\bcdot\wedge\eee^\bcdot\propto\epsilon^{\cdott\cdott}$ and $\chi^{a_1}_{~~a_2}=0$ when $a_1=a_2.$\ \ The BRS-transformation of the surface form is given by $$\begin{aligned} \delBRS\left[\SSS_{ab}\right]&=&\frac{1}{2}\epsilon_{abc_1c_2}\delBRS\left[\eee^{c_1}\wedge\eee^{c_2}\right] =~\epsilon_{abc_1c_2}\chi^{c_1}_{~c_3}\eee^{c_3}\wedge\eee^{c_2}~ \left(=~\ccc^\bcdot\wedge\overline{\eee}_{ab\bcdot}\right).\end{aligned}$$ Applying the BRS-transformation on it again, one can get $$\begin{aligned} &~&\delBRS\left[\delBRS\left[\SSS_{ab}\right]\right]~=~ \epsilon_{abc_1c_2}\delBRS\left[\chi^{c_1}_{~c_3}\eee^{c_3}\wedge\eee^{c_2}\right],\\ &~&~=~\epsilon_{abc_1c_2}\Bigl\{ \chi^{c_1}_{~c_4}\chi^{c_4}_{~c_3}\eee^{c_3}\wedge\eee^{c_2}- \chi^{c_1}_{~c_3}\chi^{c_3}_{~c_4}\eee^{c_4}\wedge\eee^{c_2}- \chi^{c_1}_{~c_3}\chi^{c_2}_{~c_4}\eee^{c_3}\wedge\eee^{c_4} \Bigr\}=0,\end{aligned}$$ because first term is the same as the second term and the third term is symmetric with $c_1$ and $c_2$ exchange.\ \ The BRS-transformation of $\ccc^a$ is given by $$\begin{aligned} \delBRS\left[\ccc^a\right]&=& \delBRS\left[\chi^a_{~b}~\Varepsilon^b_\mu~dx^\mu\right],\\ &=&\chi^a_{~b_1}\chi^{b_1}_{~b_2}\Varepsilon^{b_2}_\mu dx^\mu- \chi^a_{~b_1}~\Varepsilon^{b_2}_\mu \chi^{b_2}_{~b_1}dx^\mu +\chi^a_{~b}~\left(\partial_\mu\chi^\nu\right)\Varepsilon^b_\nu~dx^\mu -\chi^a_{~b}~\Varepsilon^b_\mu d\chi^\mu=0,\end{aligned}$$ \ Nilpotent of other forms are trivial and the proof is omitted here.\ \ The quantum Lagrangian must be the BRS-null. The gauge-fixing and Fadeef-Popov terms, $\LLL_{GF}+\LLL_{FP}$, will be constructed to satisfy the BRS-null condition. Therefore, the proof for the gravitational Lagrangian is given here by $$\begin{aligned} \delBRS\left[\LLL_G\right]&=& \frac{1}{2}\delBRS\left[\left( d\vomega^\cdott +\vomega^\bcdot_{~\ast}\wedge\vomega^{\ast\bcdot}\right)\wedge{\SSS}_\cdott -\frac{\Lambda}{3!}\vvv\right].\end{aligned}$$ The BRS-transformation for the volume form is vanished by itself. For the derivative term, $$\begin{aligned} \delBRS\left[ d\vomega^\cdott\wedge{\SSS}_\cdott \right]&=& \epsilon_{abc_2c_3}\chi_{~c_1}^{b}d\vomega^{ac_1}\wedge\eee^{c_2}\wedge\eee^{c_3}+ \epsilon_{abc_2c_3}\vomega^{ac_1}\wedge d\chi_{~c_1}^{b}\wedge\eee^{c_2}\wedge\eee^{c_3}\\&~&+ \epsilon_{abc_1c_2}\chi^{c_1}_{~c_3}~d\vomega^{ab}\wedge\eee^{c_3}\wedge\eee^{c_2}\\ &=&2~\vomega^{ac_1}\wedge d\chi^{b}_{~c_1}\wedge\SSS_{ab},\end{aligned}$$ where first- and third-terms are cancelled each other. Remnant term is transformed as $$\begin{aligned} \delBRS\left[\vomega^\bcdot_{~\ast}\wedge\vomega^{\ast\bcdot}\wedge{\SSS}_\cdott\right]&=& \epsilon_{abc_2c_3}\chi^{c_2}_{~c_4}\vomega^{ac_1}\wedge\vomega_{c_1}^{~~b}\wedge\eee^{c_4}\wedge\eee^{c_3}+ \epsilon_{abc_3c_4}\chi^{c_1}_{~c_2}\vomega^{ac_1}\wedge\vomega_{c_2}^{~~b}\wedge\eee^{c_3}\wedge\eee^{c_4}\\&~&+ \epsilon_{abc_3c_4}\chi^{b}_{~c_2}\vomega^{ac_1}\wedge\vomega_{c_1}^{~~c2}\wedge\eee^{c_3}\wedge\eee^{c_4}- 2~\vomega^{ac_1}\wedge d\chi^b_{~c_1}\wedge\SSS_{ab},\\ &=&-2~\vomega^{ac_1}\wedge d\chi^b_{~c_1}\wedge\SSS_{ab}.\end{aligned}$$ In the r.h.s of the first line, the second term is zero as itself, and first- and third-terms are cancelled each other. Therefore one can confirmed $\delBRS\left[\LLL_G\right]=0$ after summing up all terms. If we use a following remake, we can give simpler proofs for above forms.\ \ If both of two fields, $\alpha$ and $\beta$, are nilpotent, $\alpha\beta$ is also nilpotent.\ [*Proof:*]{}\ If a field $X$ is nilpotent, signatures of the Leibniz rule satisfy $\epsilon_{X}=-\epsilon_{\delta X}$ due to $\delBRS[\delBRS[X]]=0$ and (\[Leib\]), where $\epsilon_{X}$ ($\epsilon_{\delta X}$) is a signature of $X$ ($\delBRS[X]$), respectively. Therefore $$\begin{aligned} \delBRS\left[\delBRS\left[\alpha\beta\right]\right]&=& \epsilon_{\alpha}\delBRS\left[\alpha\right]\delBRS\left[\beta\right]+ \epsilon_{\delta\alpha}\delBRS\left[a\right]\delBRS\left[\beta\right] ~=~0.\end{aligned}$$ Equations of motion {#app2} =================== From the classical Lagrangian form, the torsion-less condition and the Einstein equation are obtained as the equations of motion by requiring a stationary condition for a variation of the action. The same procedure can extract Euler-Lagrange equations from the quantum Lagrangian. Here the quantum Lagrangian is summarized as $$\begin{aligned} &~&\LLL_{QG}=\LLL_G+\LLL_{GF}+\LLL_{FP},~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(\ref{QLGF})\\ &~& \left\{ \begin{array}{llc} \LLL_G&=&~~~~~~\frac{1}{2}\left( \RRR^\cdott -\frac{\Lambda}{3!}\overline{\SSS}^\cdott\right)\wedge{\SSS}_\cdott,~~~~~~~~~~~~~~~~~~~~~~(\ref{Lagrangian44})\\ \LLL_{GF}&=& -\frac{1}{2}\left(d\bbb^\cdott+\alpha\bbb^\bcdot_{~\ast}\wedge\bbb^{\ast\bcdot} \right)\wedge\SSS_\cdott,~~~~~~~~~~~~~~~~~~~~(\ref{LGF})\\ \LLL_{FP}&=& -\frac{i}{2}\left( d\tilde{\ccc}^\cdott+ \alpha \tilde{\ccc}^{\bcdot}_{~\ast}\wedge\bbb^{\ast\bcdot}\right) \wedge\ccc^\star\wedge\overline{\eee}_{\star\cdott},~~~~~~~~~~~~~~~(\ref{LFP}) \end{array} \right.\end{aligned}$$ Euler-Lagrange equations are obtained as follows: $$\begin{aligned} \TTT^a&=&d\eee^a+\www^a_{~\bcdot}\wedge\eee^\bcdot~=~0.\end{aligned}$$ This is the torsion-less condition as the the same as a case of the classical Lagrangian.\ $$\begin{aligned} \frac{1}{2}\overline{\left(\RRR^\cdott\wedge\eee^\bcdot\right)}_a-\Lambda\VVV_a -\frac{1}{2}\left(d\bbb^\cdott+\alpha\bbb^\bcdot_{~\ast}\wedge\bbb^{\ast\bcdot}\right)\wedge\overline{\eee}_{a\cdott} -\frac{i}{2}\left(d\tilde{\ccc}^\cdott+\alpha\tilde{\ccc}^\bcdot_{~\ast}\wedge\bbb^{\ast\bcdot}\right)\wedge\overline{\ccc}_{a\cdott} &=&0,\label{ELE2}\end{aligned}$$ where $\VVV_a=\epsilon_{ab_1b_2b_3}\eee^{b_1}\wedge\eee^{b_2}\wedge\eee^{b_3}/3!$. The first two terms give the Einstein equation without matter fields. Third- and fourth-terms newly appeared from the gauge-fixing and Feddeev–Popov Lagrangian forms.\ $$\begin{aligned} d\SSS_{ab}-\alpha\left( \bbb^\bcdot_a\wedge\SSS_{\bcdot b}-i\tilde{\ccc}^\bcdot_a\wedge\ccc^\bcdot\wedge\overline{\eee}_{\cdott b} \right)&=&0.\end{aligned}$$ We note that $\bbb$ and $\delta_\bbb$ are anti-commute each other and the variation operator is applied from the left. When the Landau-gauge is used, the de Donder gauge-fixing condition $d\SSS_{ab}=0$ is obtained.\ $$\begin{aligned} \left(d\tilde{\ccc}^\cdott+ \alpha \tilde{\ccc}^{\bcdot}_{~\ast}\wedge\bbb^{\ast\bcdot}\right)\wedge\overline{\eee}_{a\bcdot\bcdot}&=&0,\label{ELE3}\end{aligned}$$ where the anti-commutation among $\ccc$, $\delta\ccc$ and $\bbb$ is used.\ $$\begin{aligned} \epsilon_{ab\cdott}\left( d\left(\ccc^\bcdot\wedge\eee^\bcdot\right)-\alpha \bbb^\bcdot_{~\ast}\wedge\ccc^\ast\wedge\eee^\bcdot \right)\nonumber&=& \epsilon_{ab\cdott}\left( d\ccc^\bcdot-\alpha \bbb^\bcdot_{~\ast}\wedge\ccc^\ast \right)\wedge\eee^\bcdot ~=~0,\end{aligned}$$ where the de Donder condition is used. The BRS-transformation may give another set of equations, which must be consistent with above equations: $$\begin{aligned} \delBRS\left[\TTT^a\right]&=&\chi^a_{~\bcdot}~d\eee^\bcdot+d\chi^a_{~\bcdot}\wedge\eee^\bcdot +\chi^a_{~\bcdot}~\vomega^{~\bcdot}_{\ast}\wedge\eee^\ast +\chi_{\ast}^{~\bcdot}~\vomega^{a}_{~\bcdot}\wedge\eee^\ast -d\chi^a_{~\bcdot}\wedge\eee^\bcdot +\chi^\bcdot_{~\ast}\vomega^a_{~\bcdot}\wedge\eee^\ast \nonumber\\ &=&\chi^a_{~\bcdot}~\TTT^\bcdot~=~0.\end{aligned}$$ This is consistent with the torsion-less condition. The BRS-transformation for the volume form is vanished and last two terms are cancelled each other such as $$\begin{aligned} \delBRS\left[i\left(d\tilde{\ccc}^\cdott+\alpha\tilde{\ccc}^\bcdot_{~\ast}\wedge\bbb^{\ast\bcdot}\right)\wedge\overline{\ccc}_{a\cdott}\right]\nonumber&=& -\left(d\bbb^\cdott+\alpha\bbb^\bcdot_{~\ast}\wedge\bbb^{\ast\bcdot}\right)\wedge\overline{\eee}_{a\cdott}\end{aligned}$$ Therefor, the BRS-transformation of (\[ELE2\]) is given by $$\begin{aligned} 0&=& \epsilon_{abc\bullet}\delBRS\left[\left(d\www^{ab}+\www^a_{~\bcdot}\wedge\www^{\bcdot b}\right)\wedge\eee^c\right]\\ &=&\epsilon_{abc\bullet}\Bigl\{ \chi^b_{~\bcdot}\left(d\www^{a\bcdot}+\www^a_{~\ast}\wedge\www^{\ast\bcdot}\right)\wedge\eee^c + \left(d\www^{ab}+\www^a_{~\bcdot}\wedge\www^{\bcdot b}\right)\wedge\ccc^c \Bigr\}.\end{aligned}$$ This is consistent with the Einstein equation. $$\begin{aligned} \delBRS\left[d\eee^a-\alpha\left( \bbb^a_{~\bcdot}\wedge\eee^\bcdot-i\tilde{\ccc}_\bcdot^a\wedge\ccc^\bcdot \right)\right]&=&d\ccc^a=0,\end{aligned}$$ where $\alpha$-terms are cancelled each other. The BRS-transformation of (\[ELE3\]) gives an equation of motion for $\bbb$ in (\[ELE2\]). $$\begin{aligned} \epsilon_{ab\cdott}\delBRS\left[ d\left(\ccc^\bcdot\wedge\eee^\bcdot\right) -\alpha \bbb^\bcdot_{~\ast}\wedge\ccc^\ast\wedge\eee^\bcdot \right]&=&0,\end{aligned}$$ where $\epsilon_{ab\cdott}\ccc^\bcdot\wedge\ccc^\bcdot=0$ is used. This is not an equation, but an identity. proof of (\[BRSgen2\]) {#appC} ====================== The proof of (\[BRSgen2\]) can be given as follows: At first, shorthand notations are introduced to omit indices in following calculations for a while such as, $$\begin{aligned} \widehat{\QQQ}_\bbb=\frac{1}{2}\widehat{\bm\beta}\bcdot\delBRS\left[\widehat{\bm\QQQ}\right],&~~& \widehat{\QQQ}_{\tilde\ccc}=\frac{i}{4}\widehat{\bm\chi}\bcdot\delBRS\left[\widehat{\bm\QQQ}\right] -\frac{1}{4}\widehat{\bm\beta}\bcdot\widehat{\bm\QQQ},\\\end{aligned}$$ because the correspondence of indices is clear in them. By using these notions, the commutation relation can be represented as $$\begin{aligned} -8i\left[\widehat{\QQQ}_{\tilde\ccc},\widehat{\QQQ}_\bbb\right]&=& \left[ \widehat{\bm\beta}\bcdot\delBRS\left[\widehat{\bm\QQQ}\right], \widehat{\bm\chi}\bcdot\delBRS\left[\widehat{\bm\QQQ}\right] \right]+i \left[ \widehat{\bm\beta}\bcdot\delBRS\left[\widehat{\bm\QQQ}\right], \widehat{\bm\beta}\bcdot \widehat{\bm\QQQ} \right],\nonumber\\&=& \widehat{\bm\beta}\left[\delBRS\left[\widehat{\bm\QQQ}\right], \widehat{\bm\chi}\right]\delBRS\left[\widehat{\bm\QQQ}\right] +\widehat{\bm\chi} \left[\widehat{\bm\beta},\delBRS\left[\widehat{\bm\QQQ}\right] \right]\delBRS\left[\widehat{\bm\QQQ}\right]\nonumber\\&~& +i~ \widehat{\bm\beta}\left[\delBRS\left[\widehat{\bm\QQQ}\right],\widehat{\bm\beta}\bcdot\widehat{\bm\QQQ}\right]+i \left[ \widehat{\bm\beta}, \widehat{\bm\beta}\bcdot\widehat{\bm\QQQ} \right]\delBRS\left[\widehat{\bm\QQQ}\right].\nonumber\label{DD}\\\end{aligned}$$ where $[\widehat{\bm\beta},\widehat{\bm\chi}]=0$ is used. The first term of (\[DD\]) becomes $$\begin{aligned} \widehat{\bm\beta}\left[\delBRS\left[\widehat{\bm\QQQ}\right], \widehat{\bm\chi}\right]\delBRS\left[\widehat{\bm\QQQ}\right]&=&\widehat{\bm\beta} \left( \delBRS\left[\widehat{\bm\QQQ}\right]\widehat{\bm\chi}- \widehat{\bm\chi}~\delBRS\left[\widehat{\bm\QQQ}\right] \right)\delBRS\left[\widehat{\bm\QQQ}\right], \nonumber\\&=& \widehat{\bm\beta} \left( \delBRS\left[ \left\{\widehat{\bm\QQQ},\widehat{\bm\chi}\right\} \right]+i\left[\widehat{\bm\beta},\widehat{\bm\QQQ}\right] \right)\delBRS\left[\widehat{\bm\QQQ}\right],\\&=& i\widehat{\bm\beta}~ \left[\widehat{\bm\beta},\widehat{\bm\QQQ}\right] \delBRS\left[\widehat{\bm\QQQ}\right]=~4\widehat{\QQQ}_\bbb,\end{aligned}$$ where $\delBRS\left[\left\{\widehat{\bm\QQQ},\widehat{\bm\chi}\right\}\right]=0$ due to (\[fCR3\]), and (\[fCR2\]) are used. Note that $\widehat{\bm\chi}$ and $\widehat{\bm\QQQ}$ have $\epsilon_X=-1$ in (\[Leib\]). The second term of (\[DD\]) is zero, because $$\begin{aligned} \left[\widehat{\bm\beta},\delBRS\left[\widehat{\bm\QQQ}\right] \right]&=& \delBRS\left[ \left[ \widehat{\bm\beta},\widehat{\bm\QQQ} \right] \right]=~0,\end{aligned}$$ due to (\[fCR2\]). The third term is also zero as $$\begin{aligned} i\widehat{\bm\beta}\left[\delBRS\left[\widehat{\bm\QQQ}\right],\widehat{\bm\beta}\bcdot\widehat{\bm\QQQ}\right]&=& i\widehat{\bm\beta}\widehat{\bm\beta} \left[\delBRS\left[\widehat{\bm\QQQ}\right],\widehat{\bm\QQQ}\right]+ i\widehat{\bm\beta}~ \delBRS\left[\left[\widehat{\bm\QQQ},\widehat{\bm\beta}\right]\right]\widehat{\bm\QQQ}=~0,\end{aligned}$$ due to (\[fCR2\]). A relation $\left[\delBRS\left[\widehat{\bm\QQQ}\right],\widehat{\bm\QQQ}\right]=0$ can be confirmed by direct calculations. The last term of (\[DD\]) becomes $$\begin{aligned} i\left[ \widehat{\bm\beta}, \widehat{\bm\beta}\bcdot\widehat{\bm\QQQ} \right]\delBRS\left[\widehat{\bm\QQQ}\right]&=& i\widehat{\bm\beta}~ \left[\widehat{\bm\beta},\widehat{\bm\QQQ}\right] \delBRS\left[\widehat{\bm\QQQ}\right]=~4\widehat{\QQQ}_\bbb.\end{aligned}$$ [^1]: See, for instance, section 7 in Ref.[@fre2012gravity] [^2]: See section 5 in Ref.[@nakanishi1990covariant] [^3]: The cosmological term in the Einstein–Hilbert action is omitted in this Appendix.

--- abstract: 'We study the semidirect product of a Lie algebra with a representation up to homotopy and provide various examples coming from Courant algebroids, string Lie 2-algebras, and omni-Lie algebroids. In the end, we study the semidirect product of a Lie group with a representation up to homotopy and use it to give an integration of a certain string Lie 2-algebra.' author: - | Yunhe Sheng\ Department of Mathematics, Jilin University, Changchun 130012, Jilin, China\ email: ysheng888@gmail.com\ Chenchang Zhu\ Courant Research Centre “Higher Order Structures”, University of Göttingen\ email:zhu@uni-math.gwdg.de bibliography: - '../../bib/bibz.bib' title: ' [Semidirect products of representations up to homotopy [^1] ]{} ' --- Introduction ============ This paper is the first part of our project to integrate representations up to homotopy of Lie algebras (algebroids). Our original motivation is to integrate the standard Courant algebroid $TM\oplus T^*M$ since it is this Courant algebroid that is much used in Hitchin and Gualtieri’s program of generalized complex geometry. Courant algebroids are Lie 2-algebroids in the sense of Roytenberg and Ševera [@royt; @s:funny]. The general procedure to integrate Lie $n$-algebras (algebroids) is already described in [@getzler; @henriques; @s:funny]. We want to pursue some explicit formulas for the special case of the standard Courant algebroid. It turns out that the sections of the Courant algebroid $TM \oplus T^*M$ form a semidirect product of a Lie algebra with a representation up to homotopy. Abad and Crainic [@abad-crainic:rep-homotopy] recently studied the representations up to homotopy of Lie algebras, Lie groups, and even Lie algebroids, Lie groupoids, in general. Just as one can form the semidirect product of a Lie algebra with a representation, one can form the semidirect product with representations up to homotopy too. In our case, the semidirect product coming from the standard Courant algebra is a Lie 2-algebra. But using the fact that it is also a semidirect product, the integration becomes easier. The integration result is related to the semidirect product of Lie groups with its representation up to homotopy, which will be discussed in Section \[sec:gp\]. However it turns out that the concept of representation up to homotopy of Lie groups of Abad and Crainic will not be general enough to cover all the integration results. This we will continue in a forthcoming paper [@sheng-zhu:II]. In this paper we focus on exhibiting more examples of representation up to homotopy and their semidirect products to demonstrate the importance of our integration procedure. The examples are all sorts of variations of Courant algebroids. One is Chen and Liu’s omni-Lie algebroids, which generalizes Weinstein’s omni-Lie algebras. Hence we expect to give an integration to Weinstein’s omni-Lie algebras via Lie 2-algebras in the next paper [@sheng-zhu:II]. Another example comes from the so called string Lie 2-algebra. It is essentially a Courant algebroid over a point (see Section \[sec:string\]), namely a Lie algebra with an adjoint-invariant inner product. This sort of Lie algebra is usually called a [*quadratic Lie algebra*]{}. This concept also appears in the context of Manin triples and double Lie algebras. The example ${\mathbb R}\to {\mathfrak g}\oplus {\mathfrak g}^*$ that we consider in this paper is an analogue of the standard Courant algebroid, and is basically a special case taken from [@lu-weinstein] [^2]. We give an integration of the string Lie 2-algebra ${\mathbb R}\to {\mathfrak g}\oplus {\mathfrak g}^*$ at the end. Usually people require the base Lie algebra of a string Lie algebra to be semisimple and of compact type (see Remark \[rk:semisimple\]). For such usual sort of string Lie 2-algebras, Baez et al [@baez:2gp] have proved a no-go theorem, namely such string Lie 2-algebras can not be integrated to finite dimensional semi-strict Lie 2-groups. Here a [*semi-strict Lie 2-group*]{} is a group object in $\rm DiffCat$, where $\rm DiffCat$ is the 2-category consisting of categories, functors, and natural transformations in the category of differential manifolds, or equivalently $\rm DiffCat$ is a 2-category with Lie groupoids as objects, strict morphisms of Lie groupoids as morphisms, and 2-morphisms of Lie groupoids as 2-morphisms. Our semi-strict Lie 2-group is actually called a Lie 2-group by the authors in [@baez:2gp]. However, we call it a semi-strict Lie 2-group because compared to the Lie 2-group in the sense of Henriques [@henriques], or equivalently the stacky group in the sense of Blohmann [@blohmann], it is stricter. Basically, their Lie 2-group is a group object in the 2-category with objects as Lie groupoids, morphisms as Hilsum-Skandalas bimodules (or generalized morphisms), 2-morphisms as 2-morphisms of Lie groupoids. Schommer-Pries realizes the string 2-group as such a Lie 2-group with a finite dimensional model [@schommer:string-finite-dim] and the integration of a string Lie 2-algebra to such a model is a work in progress [@ccc:integration-string]. It is not needed in the definition of the string Lie 2-algebra for the base Lie algebra to be semisimple of compact type. One only needs a quadratic Lie algebra. As soon as we relax this condition on compactness, we find out that one can integrate ${\mathbb R}\to {\mathfrak g}\oplus {\mathfrak g}^*$ to a finite dimensional semi-strict Lie 2-group in the sense of Baez et al. The integrating object is actually a special Lie 2-group (very close to a strict Lie 2-group) in the sense of Baez et al. Then of course, as we relax the condition, we are in danger that the class corresponding to this Lie 2-algebra in $H^3({\mathfrak g}\oplus {\mathfrak g}^*, {\mathbb R})$ might be trivial, and consequently our Lie 2-algebra might be strict. Then what we have done would not have been a big surprise because a strict Lie 2-algebra corresponds to a crossed module of Lie algebras, and it easily integrates to a strict Lie 2-group by integrating the crossed module. However, we verified that when ${\mathfrak g}$ itself (not ${\mathfrak g}\oplus {\mathfrak g}^*$) is semisimple, this Lie 2-algebra is not strict. [**Acknowledgement:**]{} We give warmest thanks to Zhang-Ju Liu, Jiang-Hua Lu, Giorgio Trentinagia and Marco Zambon for useful comments and discussion. Y. Sheng gives his warmest thanks to Courant Research Centre “Higher Order Structures”, Göttingen University, where this work was done when he visited there. Representations up to homotopy of Lie algebras ============================================== In this section, we first consider the 2-term representation up to homotopy of Lie algebras. We give explicit formulas of the corresponding 2-term $L_\infty$-algebra, which is their semidirect product. Then we give several interesting examples including Courant algebroids and omni-Lie algebroids. Representation up to homotopy of Lie algebras and their semidirect products --------------------------------------------------------------------------- $L_\infty$-algebras, sometimes called strongly homotopy Lie algebras, were introduced by Drinfeld and Stasheff [@stasheff:shla] as a model for “Lie algebras that satisfy Jacobi up to all higher homotopies”. The following convention of $L_\infty$-algebras has the same grading as in [@henriques] and [@rw]. An $L_\infty$-algebra is a graded vector space $L=L_0\oplus L_1\oplus\cdots$ equipped with a system $\{l_k|~1\leq k<\infty\}$ of linear maps $l_k:\wedge^kL\longrightarrow L$ with degree $\deg(l_k)=k-2$, where the exterior powers are interpreted in the graded sense and the following relation with Koszul sign “Ksgn” is satisfied for all $n\geq0$: $$\sum_{i+j=n+1}(-1)^{i(j-1)}\sum_{\sigma}{\mathrm{sgn}}(\sigma){\mathrm{Ksgn}}(\sigma)l_j(l_i(x_{\sigma(1)},\cdots,x_{\sigma(i)}),x_{\sigma(i+1)},\cdots,x_{\sigma(n)})=0,$$ where the summation is taken over all $(i,n-i)$-unshuffles with $i\geq1$. Let $n=1$, we have $$l_1^2=0,\quad l_1:L_{i+1}\longrightarrow L_i,$$ which means that $L$ is a complex and we usually write ${\mathrm{d}}=l_1$. Let $n=2$, we have $${\mathrm{d}}l_2(x,y)=l_2({\mathrm{d}}x,y)+(-1)^pl_2(x,{\mathrm{d}}y),\quad \forall~x\in L_p, y\in L_q,$$ which means that ${\mathrm{d}}$ is a derivation with respect to $l_2$. We usually view $l_2$ as a bracket $[\cdot,\cdot]$. However, it is not a Lie bracket, the obstruction of Jacobi identity is controlled by $l_3$: $$\begin{aligned} \nonumber &&l_2(l_2(x,y),z)+(-1)^{(p+q)r}l_2(l_2(y,z),x)+(-1)^{qr+1}l_2(l_2(x,z),y)\\ &&=-{\mathrm{d}}l_3(x,y,z)-l_3({\mathrm{d}}x,y,z)+(-1)^{pq}l_3({\mathrm{d}}y,x,z)-(-1)^{(p+q)r}l_3({\mathrm{d}}z, x,y),\end{aligned}$$ where $x\in L_p,~ y\in L_q,~z\in L_q$ and $l_3$ also satisfies higher coherence laws. In particular, if the $k$-ary brackets are zero for all $k>2$, we recover the usual notion of [**differential graded Lie algebras**]{} (DGLA). If $L$ is concentrated in degrees $<n$, $L$ is called an [**$n$-term $L_\infty$-algebra**]{}. In this paper, we mainly consider 2-term $L_\infty$-algebras, which are equivalent to Lie 2-algebras given in [@baez:2algebras] by John Baez and Alissa Crans. In this special case, $l_4$ is always zero. Thus restricting the coherence law satisfied by $l_3$ on degree-0, we obtain $$l_3(l_2(x,y),z,w)+c.p.-\big(l_2(l_3(x,y,z),w)+c.p.\big)=0,\quad\forall~x,y,z,w\in L_0.$$ Lie 2-algebras are categorified version of Lie algebras. In a Lie 2-algebra, the Jacobi identity is replaced by an isomorphism which is called the [ **Jacobiator**]{}. The Jacobiator satisfies a certain law of its own. Given a 2-term $L_\infty$-algebra $L_1\stackrel{{\mathrm{d}}}{\longrightarrow}L_0$, the underlying 2-vector space of the corresponding Lie 2-algebra is made up by $L_0$ as the vector space of objects and $L_0\oplus L_1$ as the vector space of morphism. Please see [@baez:2algebras Theorem 4.3.6] for more details. Recall from [@baez:2algebras] that a Lie 2-algebra is skeletal if isomorphic objects are equal. Viewing a Lie 2-algebra as a 2-term $L_\infty$-algebra, explicitly we have \[defi:skeletal L 2\] A 2-term $L_\infty$-algebra $L_1\stackrel{{\mathrm{d}}}{\longrightarrow}L_0$ is called [**skeletal**]{} if ${\mathrm{d}}=0$. They also prove the following theorem [[@baez:2algebras]]{}\[thm:skeletal Lie 2 a\] There is a one-to-one correspondence between 2-term skeletal $L_\infty$-algebra $L_1\stackrel{{\mathrm{d}}}{\longrightarrow}L_0$ and quadruples $({\mathfrak k}_1,{\mathfrak k}_2,\phi,\theta)$ where ${\mathfrak k}_1$ is a Lie algebra, ${\mathfrak k}_2$ is a vector space, $\phi$ is a representation of ${\mathfrak k}_1$ on ${\mathfrak k}_2$ and $\theta$ is a 3-cocycle on ${\mathfrak k}_1$ with values in ${\mathfrak k}_2$. We briefly recall that given a 2-term skeletal $L_\infty$-algebra $L_1\stackrel{{\mathrm{d}}}{\longrightarrow}L_0$, ${\mathfrak k}_1$ is $L_0$, ${\mathfrak k}_2$ is $L_1$, the representation $\phi$ comes from $l_2$ and the 3-cocycle $\theta$ is obtained from $l_3$. Please see [@abad-crainic:rep-homotopy; @abad] for the general theory of representation up to homotopy of Lie algebroids. In this paper we only consider the 2-term representation up to homotopy of Lie algebras. [[@abad-crainic:rep-homotopy]]{} A 2-term representation up to homotopy of a Lie algebra ${\mathfrak g}$ consists of - A 2-term complex of vector spaces $V_1\stackrel{{\mathrm{d}}}{\longrightarrow}V_0$. - Two linear maps $\mu_i: {\mathfrak g}\to End(V_i)$, which are compatible with ${\mathrm{d}}$, i.e. for any $X\in{\mathfrak g},~\xi\in V_1$, we have $$\label{eqn:d mu} {\mathrm{d}}\circ\mu_1(X)(\xi)=\mu_0(X)\circ{\mathrm{d}}(\xi).$$ - A linear map $\nu:\wedge^2{\mathfrak g}\to{\mathrm{Hom}}(V_0,V_1)$ such that $$\begin{aligned} \label{eqn:mu0}\mu_0[X_1,X_2]-[\mu_0(X_1),\mu_0(X_2)]&=&{\mathrm{d}}\circ \nu(X_1,X_2),\\ \label{eqn:mu1}\mu_1[X_1,X_2]-[\mu_1(X_1),\mu_1(X_2)]&=& \nu(X_1,X_2)\circ{\mathrm{d}},\end{aligned}$$ as well as $$\label{eqn:d k} [\mu_0(X_1) + \mu_1(X_1),\nu(X_2,X_3)]+c.p.= \nu([X_1,X_2],X_3)+c.p.,$$ where $c.p.$ stands for cyclic permutation. We usually write $\mu=\mu_0+\mu_1$ and denote a 2-term representation up to homotopy of a Lie algebra ${\mathfrak g}$ by $(V_1\stackrel{{\mathrm{d}}}{\longrightarrow}V_0,\mu,\nu)$. In [@abad-crainic:rep-homotopy Example 3.25], the authors proved that associated to any representation up to homotopy $V_\bullet$ of a Lie algebra ${\mathfrak g}$, one can form a new $L_\infty $-algebra ${\mathfrak g}\ltimes V_\bullet$, which is their semidirect product. Here we make this construction explicit in the 2-term case. Let $(V_1\stackrel{{\mathrm{d}}}{\longrightarrow}V_0,\mu,\nu)$ be a 2-term representation up to homotopy of ${\mathfrak g}$, then we can form a new 2-term complex $$({\mathfrak g}\ltimes V_\bullet,{\mathrm{d}}):V_1\stackrel{{\mathrm{d}}}{\longrightarrow}({\mathfrak g}\oplus V_0).$$ Define $l_2:\wedge^2({\mathfrak g}\ltimes V_\bullet)\longrightarrow{\mathfrak g}\ltimes V_\bullet $ by setting $$\label{2bracket} \left\{\begin{array}{l}l_2(X+\xi,Y+\eta)=[X,Y]+\mu_0(X)(\eta)-\mu_0(Y)(\xi),\\ l_2(X+\xi,f)=\mu_1(X)(f),\\ l_2(f,g)=0,\end{array}\right.$$ for any $X+\xi,Y+\eta\in{\mathfrak g}\oplus V_0$ and $f,~g\in V_1$. One should note that $l_2$ is not a Lie bracket, but we have $$l_2(l_2(X+\xi,Y+\eta),Z+\gamma)+c.p.={\mathrm{d}}(\nu(X,Y)(\gamma))+c.p..$$ Define $l_3:\wedge^3({\mathfrak g}\ltimes V_\bullet)\longrightarrow{\mathfrak g}\ltimes V_\bullet $ by setting: $$\label{3bracket} l_3(X+\xi,Y+\eta,Z+\gamma)=-\nu(X,Y)(\gamma)+c.p.,$$ then we have \[pro:Lie 2\] With the above notation, if $(V_1\stackrel{{\mathrm{d}}}{\longrightarrow}V_0,\mu,\nu)$ is a 2-term representation up to homotopy of a Lie algebra ${\mathfrak g}$, then $(V_1\stackrel{{\mathrm{d}}}{\longrightarrow}({\mathfrak g}\oplus V_0),l_2,l_3)$ is a 2-term $L_\infty$-algebra. Example I: Courant algebroids $TM\oplus T^*M$ --------------------------------------------- Courant algebroids were first introduced in [@lwx] to study the double of Lie bialgebroids. It is a vector bundle $E\longrightarrow M$ equipped with a nondegenerate symmetric bilinear form ${\left\langle \cdot,\cdot\right\rangle}$ on the bundle, an antisymmetric bracket ${\left\llbracket \cdot,\cdot\right\rrbracket }$ on the section space $\Gamma(E)$ and a bundle map $\rho:E\longrightarrow TM$ such that a set of axioms are satisfied. It can be viewed [@royt] as a Lie 2-algebroid with a “degree 2 symplectic form”. The first example is the standard Courant algebroid $({\mathcal{T}}=TM\oplus T^*M,{\left\langle \cdot,\cdot\right\rangle},{\left\llbracket \cdot,\cdot\right\rrbracket },\rho)$ associated to a manifold $M$, where $\rho:{\mathcal{T}}\longrightarrow TM$ is the projection, the canonical pairing ${\left\langle \cdot,\cdot\right\rangle}$ is given by $$\label{eqn:pair} {\left\langle X+\xi,Y+\eta\right\rangle}=\frac{1}{2}\big(\xi(Y)+\eta(X)\big),\quad\forall ~X,Y\in{\mathfrak X}(M),~\xi,\eta\in\Omega^1(M),$$ the antisymmetric bracket ${\left\llbracket \cdot,\cdot\right\rrbracket }$ is given by $$\label{eqn:bracket} {\left\llbracket X+\xi,Y+\eta\right\rrbracket }\triangleq[X,Y]+L_X\eta-L_Y\xi+\frac{1}{2}d(\xi(Y)-\eta(X)),\quad\forall~X+\xi,~Y+\eta\in\Gamma({\mathcal{T}}).$$ It is not a Lie bracket, but we have $$\label{eqn:jacobi} {\left\llbracket {\left\llbracket e_1,e_2\right\rrbracket },e_3\right\rrbracket }+c.p.=d T(e_1,e_2,e_3),\quad \forall~e_1,e_2,e_3\in\Gamma({\mathcal{T}}),$$ where $T(e_1,e_2,e_3) $ is given by $$\label{T} T(e_1,e_2,e_3)=\frac{1}{3}({\left\langle {\left\llbracket e_1,e_2\right\rrbracket },e_3\right\rangle}+c.p.).$$ Now we realize the section space of ${\mathcal{T}}$ as the semidirect product of the Lie algebra ${\mathfrak X}(M)$ of vector fields with the natural 2-term deRham complex $$\label{eq:natural-cx} C^\infty(M)\stackrel{d}{\longrightarrow}\Omega^1(M).$$ For this we need to define a representation up to homotopy of ${\mathfrak X}(M)$ on this complex. For any $X\in{\mathfrak X}(M)$, define linear actions $\mu_0$ and $\mu_1$ by $$\begin{aligned} \label{C 1}\mu_0(X)(\xi)&\triangleq &{\left\llbracket X,\xi\right\rrbracket }=L_X\xi-{\frac{1}{2}}d(\xi(X)),\quad\forall ~\xi\in\Omega^1(M)\\ \label{C 2}\mu_1(X)(f)&\triangleq &{\left\langle X,d f\right\rangle}=\frac{1}{2}X(f),\quad\forall~f\in C^\infty(M).\end{aligned}$$ Define $\nu:\wedge^2{\mathfrak X}(M)\to{\mathrm{Hom}}(\Omega^1(M),C^\infty(M))$ by $$\label{C 3} \nu(X,Y)(\xi)=T(X,Y,\xi),\quad\forall ~X,Y\in{\mathfrak X}(M),~\xi\in\Omega^1(M).$$ \[pro:rep up to homotopy\] With the above notations, $(C^\infty(M)\stackrel{d}{\longrightarrow}\Omega^1(M),\mu=\mu_0+\mu_1,\nu)$ is a representation up to homotopy of the Lie algebra ${\mathfrak X}(M)$. [[**Proof.**]{} ]{}For any $f\in C^\infty(M)$, we have $$\mu_0(X)(d f)=L_X df-\frac{1}{2}d X( f)=\frac{1}{2}d X( f),$$ which implies $\mu_0\circ d=d\circ\mu_1$, i.e. $\mu_i$’s are compatible with the differential $d$. By straightforward computations, we have $$\begin{aligned} \mu_0[X,Y](\xi)-[\mu_0(X),\mu_0(Y)](\xi)&=&{\left\llbracket {\left\llbracket X,Y\right\rrbracket },\xi\right\rrbracket }+c.p.\\ &=&d T(X,Y,\xi) =d \big(\nu(X,Y)(\xi)\big),\\ \mu_1[X,Y](f)-[\mu_1(X),\mu_1(Y)](f)&=&{\frac{1}{2}}[X,Y](f)-\frac{1}{4}\big(X(Y(f))-Y(X(f))\big)\\ &=&\frac{1}{4}[X,Y](f),\\ \nu(X,Y)(d f)&=&T(X,Y,d f)=\frac{1}{3}\big({\frac{1}{2}}[X,Y](f)+\frac{1}{4}\big(X(Y(f))-Y(X(f))\big)\\ &=&\frac{1}{4}[X,Y](f),\end{aligned}$$ which implies (\[eqn:mu0\]) and (\[eqn:mu1\]). At last we need to prove (\[eqn:d k\]), which is obviously equivalent to $$\label{c temp} \mu_1(X)T(Y,Z,\xi)-T(Y,Z,\mu_0(X)(\xi))+c.p.(X,Y,Z)=T([X,Y],Z,\xi)+c.p.(X,Y,Z).$$ Observe that since $\mu_0(X)(\xi)={\left\llbracket X,\xi\right\rrbracket }$, we have $$T(Y,Z,\mu_0(X)(\xi))+c.p.(X,Y,Z)+T([X,Y],Z,\xi)+c.p.(X,Y,Z)=T([X,Y],Z,\xi)+c.p.(X,Y,Z,\xi).$$ Furthermore, since for any $f\in {C^{\infty}(M)}$, $\mu_1(X)(f)={\left\langle X,df\right\rangle}$ and the cotangent bundle $T^*M$ is isotropic under the pairing $(\ref{eqn:pair})$, we have $$\mu_1(X)T(Y,Z,\xi)+c.p.(X,Y,Z)={\left\langle X,dT(Y,Z,\xi)\right\rangle}+c.p.(X,Y,Z,\xi).$$ Thus, (\[c temp\]) is equivalent to $${\left\langle X,d T(Y,Z,\xi)\right\rangle}+c.p.(X, Y, Z, \xi)=T([X,Y],Z,\xi)+c.p.(X, Y, Z, \xi),$$ which holds by Lemma 4.5 in [@rw]. By Proposition \[pro:Lie 2\], we have \[cor:courant\] $\big(C^\infty(M) \xrightarrow{{\mathrm{d}}=0\oplus d }({\mathfrak X}(M)\oplus\Omega^1(M)),l_2,l_3\big)$ is a 2-term $L_\infty$-algebra, where $l_2$ and $l_3$ are given by (\[2bracket\]) and (\[3bracket\]), in which $\mu_0$, $\mu_1$ and $\nu$ are given by (\[C 1\]), (\[C 2\]) and (\[C 3\]) respectively. \[rmk:L infty\] In [@rw] the authors proved that the sections of a Courant algebroid $({{\mathcal{C}}},{\left\langle \cdot,\cdot\right\rangle},{\left\llbracket \cdot,\cdot\right\rrbracket },\rho)$ form an $L_\infty$-algebra. In the case when ${{\mathcal{C}}}={\mathcal{T}}$ the standard Courant algebroid, the 2-term $L_\infty$-algebra is given by $${C^{\infty}(M)}\xrightarrow{{\mathrm{d}}=0\oplus d } {\mathfrak X}(M) \oplus \Omega^1(M)=\Gamma({\mathcal{T}}),$$ with brackets given by $$l_2(e_1,e_2)={\left\llbracket e_1,e_2\right\rrbracket },\quad l_2(e_1,f)={\left\langle e_1,{\mathrm{d}}f\right\rangle},\quad l_3(e_1,e_2,e_3)=-T(e_1,e_2,e_3), \quad l_{i\ge 4} =0,$$ for any $e_1,e_2,e_3\in\Gamma({\mathcal{T}}),~f\in {C^{\infty}(M)}$. Here $T$ is defined by (\[T\]). It is easy to verify that this is the same as our 2-term $L_\infty$-algebra in Corollary \[cor:courant\]. We can also modify our complex to $$\Omega^1(M)\stackrel{{\rm{Id}}}{\longrightarrow}\Omega^1(M).$$ Following the same procedure, we obtain another representation up to homotopy of the Lie algebra ${\mathfrak X}(M)$ and therefore obtain another 2-term $L_\infty$-algebra which is also totally determined by the Courant algebroid $({\mathcal{T}},{\left\langle \cdot,\cdot\right\rangle},{\left\llbracket \cdot,\cdot\right\rrbracket },\rho)$. More precisely, for any $X\in {\mathfrak X}(M),~\xi\in\Omega^1(M)$, $\mu_0=\mu_1$ is given by $$\mu_0(X)(\xi)\triangleq{\left\llbracket X,\xi\right\rrbracket },$$ and $\nu:{\mathfrak X}(M):\longrightarrow\Omega^2({\mathfrak X}^2(M),{\mathrm{End}}(\Omega^1(M),\Omega^1(M)))$ is given by $$\nu(X,Y)(\xi)\triangleq d T(X,Y,\xi).$$ With the above notations, $(\Omega^1(M)\stackrel{{\rm{Id}}}{\longrightarrow}\Omega^1(M),\mu=\mu_0=\mu_1,\nu)$ is a representation up to homotopy of the Lie algebra ${\mathfrak X}(M)$. Example : Courant algebroids over a point and string Lie 2-algebras {#sec:string} ------------------------------------------------------------------- A Courant algebroid over a point is literally a quadratic Lie algebra, namely a Lie algebra ${\mathfrak k}$ together with nondegenerate inner product ${\left\langle \cdot,\cdot\right\rangle}$ which is invariant under the adjoint action. People often think that a Courant algebroid over a point is a string Lie 2-algebra [^3]. Here we justify this thinking. \[defi:string-alg\] The string Lie 2-algebra associated to a quadratic Lie algebra $({\mathfrak k}, {\left\langle \cdot,\cdot\right\rangle})$, is a 2-term $L_\infty$-algebra $\mathbb R\stackrel{0}{\longrightarrow}{\mathfrak k}$, whose degree-$0$ part is ${\mathfrak k}$, degree-$1$ part is $\mathbb R$, and $l_2,~l_3$ are given by $$\begin{aligned} ~l_2(e,c)&=&0,\\ ~l_2((e_1,c_1),(e_2,c_2))&=&([e_1,e_2],0),\\ ~l_3((e_1,c_1),(e_2,c_2),(e_3,c_3))&=&(0,{\left\langle [e_1,e_2],e_3\right\rangle}),\end{aligned}$$ where $e,~e_1,~e_2,~e_3\in{\mathfrak k},~c,~c_1,~c_2,~c_3\in\mathbb R$. The representation $\phi $ of ${\mathfrak k}$ on $\mathbb R$ and the 3-cocycle $\theta:\wedge^3{\mathfrak k}\longrightarrow \mathbb R$ in the corresponding quadruple in Theorem \[thm:skeletal Lie 2 a\] is given by $$\begin{aligned} \nonumber\rho(e)(c)&=&l_2(e,c)\\ \label{3cocycle}\theta(e_1,e_2,e_3)&=&{\left\langle [e_1,e_2],e_3\right\rangle}.\end{aligned}$$ \[rk:semisimple\] In the definition of string Lie 2-algebras [@baez:string; @henriques] the base Lie algebra ${\mathfrak k}$ is usually required to be semisimple and of compact type, such that the Jacobiator gives rise to the generator of $H^3({\mathfrak k}, \mathbb Z)=\mathbb Z$. This is because Witten’s original motivation is to obtain a $3$-connected cover of $Spin(n)$, and $\mathfrak{so}(n)$ is simple and of compact type. However, to write down the structure of the string Lie 2-algebra, we only need a quadratic Lie algebra. Then $H^3({\mathfrak k}, \mathbb Z)$ is not necessarily $\mathbb Z$ for a general quadratic Lie algebra ${\mathfrak k}$. For example, for the abelian Lie algebra ${\mathbb R}$, any inner product is adjoint-invariant, and $H^3({\mathbb R}, {\mathbb Z})=0$. We thus face the danger that sometimes, the string Lie 2-algebra is trivial, that is the Jacobiator corresponds to the trivial element in $H^3({\mathfrak k}, \mathbb Z)$. Then what we have is a strict Lie 2-algebra, which is a crossed module of Lie algebras. Then the integration of a crossed module of Lie algebras is simply a crossed module of Lie groups. However, we will verify that the example we consider is not such a case. The standard Courant algebroid motivates us to consider the case of the direct sum ${\mathfrak k}={\mathfrak g}\oplus{\mathfrak g}^*$ of a Lie algebra ${\mathfrak g}$ and its dual with the semidirect product Lie algebra structure: $$[X+\xi,Y+\eta]=[X,Y]_{\mathfrak g}+{\mathrm{ad}}_X^*\eta-{\mathrm{ad}}_Y^*\xi,$$here $[\cdot,\cdot]_{\mathfrak g}$ is the Lie bracket of ${\mathfrak g}$. The nondegenerate invariant pairing ${\left\langle \cdot,\cdot\right\rangle}$ on ${\mathfrak g}\oplus {\mathfrak g}^*$ is given by $${\left\langle X+\xi,Y+\eta\right\rangle}={\frac{1}{2}}(\eta(X)+\xi(Y)),\quad\forall~X+\xi,Y+\eta\in{\mathfrak g}\oplus{\mathfrak g}^*.$$ With these definitions, $({\mathfrak g}\oplus{\mathfrak g}^*,[\cdot,\cdot],{\left\langle \cdot,\cdot\right\rangle})$ is a quadratic Lie algebra. In fact, we have $$\begin{aligned} {\left\langle [X_1+\xi_1,X_2+\xi_2],X_3+\xi_3\right\rangle}&=&{\left\langle [X_1,X_2]_{\mathfrak g}+{\mathrm{ad}}_X^*\xi_2-{\mathrm{ad}}_Y^*\xi_1,X_3+\xi_3\right\rangle}\\ &=&{\left\langle [X_1,X_2]_{\mathfrak g},\xi_3\right\rangle}+c.p..\end{aligned}$$ Similarly, we have $${\left\langle X_2+\xi_2,[X_1+\xi_1,X_3+\xi_3]\right\rangle}={\left\langle [X_1,X_3]_{\mathfrak g},\xi_2\right\rangle}+c.p.,$$ which implies that ${\left\langle X_2+\xi_2,[X_1+\xi_1,X_3+\xi_3]\right\rangle}+{\left\langle [X_1+\xi_1,X_2+\xi_2],X_3+\xi_3\right\rangle}=0$, i.e. the nondegenerate inner product ${\left\langle \cdot,\cdot\right\rangle}$ is invariant under the adjoint action. This example is a special case of [@lu-weinstein Theorem 1.12] with ${\mathfrak g}^*$ equipped with $0$ Lie bracket. Thus $({\mathfrak g}, {\mathfrak g}^*)$ forms a Lie bialgebra or equivalently $({\mathfrak g}\oplus {\mathfrak g}^*, {\mathfrak g}, {\mathfrak g}^*)$ is a Manin triple. However, honestly we have not found other Lie bialgebras (Manin triples) giving rise to Lie 2-algebras of the form of semidirect products. We denote the corresponding string Lie 2-algebra of $({\mathfrak g}\oplus {\mathfrak g}^*, [\cdot, \cdot], {\left\langle \cdot,\cdot\right\rangle})$ by $$\label{Lie 2 g g dual} \mathbb R\stackrel{0}{\longrightarrow}{\mathfrak g}\oplus {\mathfrak g}^*.$$ The corresponding 3-cocycle (see ), which we denote by $\widetilde{\nu}:\wedge^3({\mathfrak g}\oplus {\mathfrak g}^*)\longrightarrow \mathbb R$ is $$\begin{aligned} \label{eqn:nu3} \widetilde{\nu}(X_1+\xi_1,X_2+\xi_2,X_3+\xi_3)&=&{\left\langle [X_1+\xi_1,X_2+\xi_2],X_3+\xi_3\right\rangle}\\ \nonumber&=&{\left\langle [X_1,X_2]_{\mathfrak g},\xi_3\right\rangle}+c.p.$$ $(\mathbb R\stackrel{0}{\longrightarrow} {\mathfrak g}^*,\mu_1=0, \mu_0={\mathrm{ad}}^*,\nu=[\cdot,\cdot]_{\mathfrak g})$ is a 2-term representation up to homotopy of the Lie algebra ${\mathfrak g}$. Moreover the string Lie 2-algebra $ \mathbb R\stackrel{0}{\longrightarrow}{\mathfrak g}\oplus {\mathfrak g}^* $ is the semidirect product of ${\mathfrak g}$ and the complex $\mathbb R\stackrel{0}{\longrightarrow} {\mathfrak g}^*$. Since ${\mathrm{d}}=0$, we only need to verify that $\mu_0$ and $\mu_1$ are Lie algebra morphisms and . ${\mathrm{ad}}^*: {\mathfrak g}\to {\mathrm{End}}({\mathfrak g}^*)$ and $0$ are both Lie algebra morphisms. follows from Jacobi identity of $[\cdot,\cdot]_{\mathfrak g}$. Then it is not hard to see that the string Lie 2-algebra $\mathbb R\stackrel{0}{\longrightarrow}{\mathfrak g}\oplus {\mathfrak g}^*$ with formulas in Definition \[defi:string-alg\] is exactly the semidirect product of ${\mathfrak g}$ with the complex $(\mathbb R\stackrel{0}{\longrightarrow} {\mathfrak g}^*,\mu_1=0, \mu_0={\mathrm{ad}}^*,\nu=[\cdot,\cdot]_{\mathfrak g})$ with the formulas , . \[pro:nondegenerate\] If the Lie algebra ${\mathfrak g}$ is semisimple, the Lie algebra 3-cocycle $\widetilde{\nu}$ which is given by (\[eqn:nu3\]) is not exact. [[**Proof.**]{} ]{}Let ${\left\langle \cdot,\cdot\right\rangle}_k$ be the Killing form on ${\mathfrak g}$. The proof follows from the fact that the Cartan 3-form ${\left\langle [\cdot,\cdot],\cdot\right\rangle}_k$ on a semisimple Lie algebra is not exact. Since ${\mathfrak g}$ is semisimple, the Killing form ${\left\langle \cdot,\cdot\right\rangle}_k$ is nondegenerate. Identify ${\mathfrak g}^*$ and ${\mathfrak g}$ by using the Killing form ${\left\langle \cdot,\cdot\right\rangle}_k$ and let ${\mathcal{K}}$ be the corresponding isomorphism, $${\left\langle {\mathcal{K}}(\xi),X\right\rangle}_k={\left\langle \xi,X\right\rangle}.$$ Assume that $\widetilde{\nu}=d\phi$ for some $\phi:\wedge^2({\mathfrak g}\oplus{\mathfrak g}^*)\longrightarrow \mathbb R$, define $\varphi:\wedge^2({\mathfrak g}\oplus{\mathfrak g})\longrightarrow \mathbb R$ by $$\phi(X+\xi,Y+\eta)=\varphi(X+{\mathcal{K}}(\xi),Y+{\mathcal{K}}(\eta)),$$then we have $$\begin{aligned} \widetilde{\nu}(X,Y,\xi)&=&d\phi(X,Y,\xi)\\ &=&-\phi([X,Y],\xi)+\phi({\mathrm{ad}}^*_X\xi, Y)-\phi({\mathrm{ad}}^*_Y\xi, X)\\ &=&-\varphi([X,Y],{\mathcal{K}}(\xi))+\varphi([X,{\mathcal{K}}(\xi)], Y)-\varphi([Y,{\mathcal{K}}(\xi)], X)\\ &=&d\varphi(X,Y,{\mathcal{K}}(\xi)).\end{aligned}$$ On the other hand, we have $$\widetilde{\nu}(X,Y,\xi)={\left\langle [X,Y],\xi\right\rangle}={\left\langle [X,Y],{\mathcal{K}}(\xi)\right\rangle}_k,$$ which implies that the Cartan 3-from $${\left\langle [X,Y],{\mathcal{K}}(\xi)\right\rangle}_k=d\varphi(X,Y,{\mathcal{K}}(\xi)),$$ is exact. This is a contradiction. In the last section, we will give the integration of the string Lie 2-algebra $ \mathbb R\stackrel{0}{\longrightarrow}{\mathfrak g}\oplus {\mathfrak g}^* $ by using the semidirect product of a Lie group with its 2-term representation up to homotopy. It turns out that this string Lie 2-algebra can be integrated to a special Lie 2-group with a finite dimensional model. Example III: Omni-Lie algebroids ${\mathfrak{D}}E\oplus {\mathfrak{J}}E$ ------------------------------------------------------------------------ The notion of omni-Lie algebroids, which was introduced by Chen and Liu in [@clomni], is a generalization of Weinstein’s omni-Lie algebras. Just as Dirac structures of an omni-Lie algebra characterize Lie algebra structures on a vector space, Dirac structures of an omni-Lie algebroid characterize Lie algebroid structures on a vector bundle. See [@clsdirac] for more details. In fact, the role of omni-Lie algebroids ${\mathfrak{D}}E\oplus {\mathfrak{J}}E$ in $E$-Courant algebroids, which was introduced in [@clsecourant], is the same as the role of standard Courant algebroids $TM\oplus T^*M$ in Courant algebroids. Now we briefly recall the notion of omni-Lie algebroids and we will see that it gives rise to a 2-term $L_\infty$-algebra which is a semidirect product. In this subsection $E$ is a vector bundle over a smooth manifold $M$, $\Gamma(E)$ is the section space of $E$. For a vector bundle $E$, let ${\mathfrak{D}}E$ be its covariant differential operator bundle. The associated Atiyah sequence is given by $$\label{Seq:DE} \xymatrix@C=0.5cm{0 \ar[r] & {\mathfrak {gl}}(E) \ar[rr]^{{\mathbbm{i}}} && {\mathfrak{D}}{E} \ar[rr]^{a} && TM \ar[r] & 0. }$$ The associated $1$-jet vector bundle ${{\mathfrak{J}}} E$ is defined as follows. For any $m\in M$, $({{\mathfrak{J}}}{E})_m$ is defined as a quotient of local sections of $E$. Two local sections $u_1$ and $u_2$ are equivalent and we denote this by $u_1\sim u_2$ if $$u_1(m)=u_2(m) ~~\mbox{ and }~~ d{\left\langle u_{1 },\xi\right\rangle}_m=d{\left\langle u_{2 },\xi\right\rangle}_m, \quad\forall~ \xi\in\Gamma(E^*).$$ So any $\mu\in ({{\mathfrak{J}}}{E})_m$ has a representative $u\in\Gamma(E)$ such that $\mu=[u]_m$. Let ${\mathbbm{p}}$ be the projection which sends $[u]_m$ to $u(m)$. Then ${\mathrm{Ker}}{\mathbbm{p}}\cong {\mathrm{Hom}}(TM,E)$ and there is a short exact sequence, called the jet sequence of $E$, $$\label{Seq:JetE} \xymatrix@C=0.5cm{0 \ar[r] & {\mathrm{Hom}}(TM,E) \ar[rr]^{~\qquad {\mathbbm{e}}~} && {{\mathfrak{J}}}{E} \ar[rr]^{{\mathbbm{p}}} && E \ar[r] & 0, }$$ from which it is straightforward to see that ${\mathfrak{J}}E$ is a finite dimensional vector bundle. Moreover, $\Gamma({\mathfrak{J}}{E})$ is isomorphic to $\Gamma(E) \oplus \Gamma (T^*M \otimes E)$ as an ${\mathbb R}$-vector space, and any $u\in \Gamma (E)$ has a lift ${\mathbbm{d}}u\in\Gamma ({{\mathfrak{J}}} E)$ by taking its equivalence class, such that $$\label{Eqt:CWMmoduleGammaE} {\mathbbm{d}}(fu)=f {\mathbbm{d}}u+ d f\otimes u, \quad \forall~ f\in {C^{\infty}(M)}.$$ In [@clomni] the authors proved that $$\begin{aligned} {{\mathfrak{J}}E} \cong {\left\{\nu\in {\mathrm{Hom}}({\mathfrak{D}}{E},E)\,|\, \nu(\Phi)=\Phi\circ\nu({\rm{Id}}_{E}),\quad\forall~ ~ ~\Phi\in {\mathfrak {gl}}(E)\right\}}.\end{aligned}$$ Therefore, there is an $E$-pairing between ${\mathfrak{J}}{E}$ and ${\mathfrak{D}}{E}$ by setting: $$\label{Eqt:conpairingE} {\left\langle \mu,{\mathfrak d}\right\rangle }_E~ {\triangleq}{\mathfrak d}(u),\quad\forall~ ~~ \mu\in ({\mathfrak{J}}{E})_m,~{\mathfrak d}\in({\mathfrak{D}}{E})_m,$$ where $u\in \Gamma(E)$ satisfies $\mu=[u]_m$. Particularly, one has $$\begin{aligned} \label{conpairing1} {\left\langle \mu,\Phi\right\rangle }_E &=& \Phi\circ {\mathbbm{p}}(\mu),\quad\forall~ ~~ \Phi\in {\mathfrak {gl}}(E),~\mu\in{\mathfrak{J}}{E};\\ \label{conpairing2} {\left\langle {{\mathfrak y}},{\mathfrak d}\right\rangle }_{E} &=& {{\mathfrak y}}\circ a({\mathfrak d}),\quad\forall~ ~~ {\mathfrak y}\in {\mathrm{Hom}}(TM,E),~{\mathfrak d}\in{\mathfrak{D}}{E}.\end{aligned}$$ Furthermore, we claim that $\Gamma ({\mathfrak{J}}E)$ is an invariant subspace of the Lie derivative ${{\mathfrak L}}_{{\mathfrak d}}$ for any ${\mathfrak d}\in\Gamma({\mathfrak{D}}{E})$, which is defined by the Leibniz rule as follows: $$\begin{aligned} \nonumber{\left\langle {{\mathfrak L}}_{{\mathfrak d}}\mu,{\mathfrak d}{^\prime}\right\rangle }_{E}&{\triangleq}& {\mathfrak d}{\left\langle \mu,{\mathfrak d}{^\prime}\right\rangle }_{E}-{\left\langle \mu,[{\mathfrak d},{\mathfrak d}{^\prime}]_{{\mathfrak{D}}}\right\rangle }_{E}, \quad\forall~ \mu \in \Gamma({\mathfrak{J}}{E}), ~ ~{\mathfrak d}{^\prime}\in\Gamma({\mathfrak{D}}{E}).\end{aligned}$$ Introduce a nondegenerate symmetric $E$-valued $2$-form ${\left ( \cdot,\cdot\right )_E}$ on $ {\mathcal{E}}\triangleq{\mathfrak{D}}{E}\oplus {\mathfrak{J}}{E}$ by: $${\left ( {\mathfrak d}+\mu,{\mathfrak r}+\nu\right )_E}{\triangleq}{\frac{1}{2}}({\left\langle {\mathfrak d},\nu\right\rangle }_E +{\left\langle {\mathfrak r},\mu\right\rangle }_E),\quad\forall~ ~~ {\mathfrak d},{\mathfrak r}\in{\mathfrak{D}}{E},~\mu,\nu\in{\mathfrak{J}}{E}.$$ Furthermore, we define an antisymmetric bracket ${\left\llbracket \cdot,\cdot\right\rrbracket }$ on $\Gamma({\mathcal{E}})$ by: $$\begin{aligned} {\left\llbracket {\mathfrak d}+\mu,{\mathfrak r}+\nu\right\rrbracket }&{\triangleq}& [{\mathfrak d},{\mathfrak r}]_{{\mathfrak{D}}}+{{\mathfrak L}}_{{\mathfrak d}}\nu-{{\mathfrak L}}_{{\mathfrak r}}\mu + {\frac{1}{2}}\big({\mathbbm{d}}{\left\langle \mu,{\mathfrak r}\right\rangle }_E-{\mathbbm{d}}{\left\langle \nu,{\mathfrak d}\right\rangle }_E\big).\end{aligned}$$ In [@clomni] the authors call the quadruple $({\mathcal{E}},{\left\llbracket \cdot,\cdot\right\rrbracket },{\left ( \cdot,\cdot\right )_E},{\rho_{\varepsilon}})$ the [**omni-Lie algebroid**]{} [^4] associated to the vector bundle $E$, where $\rho$ is the projection of ${\mathcal{E}}$ onto ${\mathfrak{D}}{E}$. Even though ${\left\llbracket \cdot,\cdot\right\rrbracket }$ is antisymmetric, it is not a Lie bracket. More precisely, for any $X,Y,Z\in \Gamma({\mathcal{E}})$, we have $${\left\llbracket {\left\llbracket X,Y\right\rrbracket },Z\right\rrbracket }+c.p.={\mathbbm{d}}T(X,Y,Z),$$ where $T:\Gamma(\wedge^3{\mathcal{E}})\longrightarrow \Gamma(E)$ is defined by \[T\] T(X,Y,Z)=([( [X,Y]{},Z)\_E]{}+c.p.). Now let us construct a 2-term $L_\infty$-algebra from the omni-Lie algebroid ${\mathcal{E}}$. Obviously, $\Gamma({\mathfrak{D}}E)$ is a Lie algebra and there is a natural 2-term complex $$\Gamma(E)\stackrel{{\mathrm{d}}=0\oplus{\mathbbm{d}}}{\longrightarrow}\Gamma({\mathfrak{J}}E).$$ For any ${\mathfrak d}\in\Gamma({\mathfrak{D}}E)$, define linear actions $\mu_0$ and $\mu_1$ by $$\label{O 1}\left\{\begin{array}{l} \mu_0({\mathfrak d})(\mu)\triangleq {\left\llbracket {\mathfrak d},\mu\right\rrbracket }={{\mathfrak L}}_{\mathfrak d}\mu-{\mathbbm{d}}{\left ( \mu,{\mathfrak d}\right )_E},\quad\forall~\mu\in \Gamma({\mathfrak{J}}E),\\ \mu_1({\mathfrak d})(u)\triangleq {\left ( {\mathfrak d},{\mathbbm{d}}u\right )_E}=\frac{1}{2}{\mathfrak d}(u),\quad\forall~u\in\Gamma(E).\end{array}\right.$$ Define $\nu:\wedge^2\Gamma({\mathfrak{D}}E)\longrightarrow{\mathrm{Hom}}(\Gamma({\mathfrak{J}}E),\Gamma( E))$ by $$\label{O 2} \nu({\mathfrak d},{\mathfrak t})(\xi)=T({\mathfrak d},{\mathfrak t},\mu),\quad\forall ~{\mathfrak d},{\mathfrak t}\in\Gamma({\mathfrak{D}}E),~\mu\in\Gamma({\mathfrak{J}}E).$$ Similar to Proposition \[pro:rep up to homotopy\], we prove With the above notations, $(\Gamma(E)\stackrel{{\mathrm{d}}=0\oplus{\mathbbm{d}}}{\longrightarrow}\Gamma({\mathfrak{J}}E),\mu=\mu_0+\mu_1,\nu)$ is a representation up to homotopy of the Lie algebra $\Gamma({\mathfrak{D}}E)$. $\big(\Gamma(E)\stackrel{{\mathrm{d}}=0\oplus{\mathbbm{d}}}{\longrightarrow}\Gamma({\mathfrak{D}}E)\oplus\Gamma({\mathfrak{J}}E),l_2,l_3\big)$ is a 2-term $L_\infty$-algebra, where $l_2$ and $l_3$ are given by (\[2bracket\]) and (\[3bracket\]), in which $\mu$ and $\nu$ are given by (\[O 1\]) and (\[O 2\]) respectively. If the base manifold $M$ is a point, i.e. $E$ is a vector space, for which we use a new notation $V$, then ${\mathfrak{D}}V={\mathfrak {gl}}(V),~{\mathfrak{J}}V=V$, and we recover the notion of omni-Lie algebras. The complex $\Gamma(V)\stackrel{0\oplus{\mathbbm{d}}}{\longrightarrow}\Gamma({\mathfrak{J}}V)$ reduces to $V\stackrel{{\rm{Id}}}{\longrightarrow}V$, which is a representation up to homotopy of ${\mathfrak {gl}}(V)$ with $(\mu_0=\mu_1, \nu)$ given by $$\label{omni mu nu} \mu_0(A)(u)={\frac{1}{2}}Au,\quad \nu(A,B)=\frac{1}{4}[A,B],\quad\forall~A,B\in{\mathfrak {gl}}(V),~u\in V.$$ Hence even though an omni-Lie algebra ${\mathfrak {gl}}(V)\oplus V$ is not a Lie algebra, we can extend it to a 2-term $L_\infty$-algebra, of which $L_0={\mathfrak {gl}}(V)\oplus V,~L_1=V$, $l_2$ and $l_3$ are given by (\[2bracket\]) and (\[3bracket\]), in which $\mu$ and $\nu$ are given by (\[omni mu nu\]). This 2-term $L_\infty$-algebra is a semidirect product of ${\mathfrak {gl}}(V)$ with $V\xrightarrow{{\rm{Id}}} V$. We will study the global object of the 2-term $L_\infty$-algebra associated to an omni-Lie algebra in the forthcoming paper [@sheng-zhu:II]. Representation up to homotopy of Lie groups and semidirect product {#sec:gp} ================================================================== The representation up to homotopy of a Lie group was introduced in [@abad]. In this section we define the semidirect product of a Lie group with a 2-term representation up to homotopy and prove that the semidirect product is a Lie 2-group. Thus we first recall some background on Lie 2-groups. A group is a monoid where every element has an inverse. A 2-group is a monoidal category where every object has a weak inverse and every morphism has an inverse. Denote the category of smooth manifolds and smooth maps by $\rm Diff$, a semistrict[^5] Lie 2-group is a 2-group in $\rm DiffCat$, where $\rm DiffCat$ is the 2-category consisting of categories, functors, and natural transformations in $\rm Diff$. For more details, see [@baez:2gp] and here we only recall the expanded definition: [[@baez:2gp]]{} A semistrict Lie 2-group consists of an object $C$ in $\rm DiffCat$, i.e. $$\xymatrix{ C_1 \ar@<1ex>[r]^{s} \ar[r]_{t} & C_0},$$ where $C_1,~C_0$ are objects in $\rm Diff$, $s,~t$ are the source and target maps, and there is a vertical multiplication $\cdot_{\mathrm{v}}:C\times C\longrightarrow C$, together with - a functor (horizontal multiplication) $\cdot_{\mathrm{h}}:C\times C\longrightarrow C$, - an identity object $1$, - a contravariant functor ${\mathrm{inv}}:C\longrightarrow C$ and the following natural isomorphisms: - the [**associator**]{} $$a_{x,y,z}:(x\cdot_{\mathrm{h}}y)\cdot_{\mathrm{h}}z\longrightarrow x\cdot_{\mathrm{h}}(y\cdot_{\mathrm{h}}z),$$ - the [**left**]{} and [**right unit**]{} $$l_x:1\cdot_{\mathrm{h}}x\longrightarrow x,\quad r_x:x\cdot_{\mathrm{h}}1\longrightarrow x,$$ - the [**unit**]{} and [**counit**]{} $$i_x:1\longrightarrow x\cdot_{\mathrm{h}}{\mathrm{inv}}(x),\quad e_x:{\mathrm{inv}}(x)\cdot_{\mathrm{h}}x\longrightarrow 1,$$ such that the following diagrams commute: - the [**pentagon identity** ]{} for the associator $$\xymatrix{ & (w\cdot_{\mathrm{h}}x)\cdot_{\mathrm{h}}(y\cdot_{\mathrm{h}}z)\ar[dr]^{a_{w,x, y\cdot_{\mathrm{h}}z}}& \\ ((w\cdot_{\mathrm{h}}x)\cdot_{\mathrm{h}}y)\cdot_{\mathrm{h}}z\ar[ur]^{a_{(w\cdot_{\mathrm{h}}x),y,z}}\ar[dr]_{a_{w,x,y}\cdot_{\mathrm{h}}1_z}&&w\cdot_{\mathrm{h}}(x\cdot_{\mathrm{h}}(y\cdot_{\mathrm{h}}z))\\ &(w\cdot_{\mathrm{h}}(x\cdot_{\mathrm{h}}y))\cdot_{\mathrm{h}}z\stackrel{a_{w,x\cdot_{\mathrm{h}}y,z}}{\longrightarrow} w\cdot_{\mathrm{h}}((x\cdot_{\mathrm{h}}y)\cdot_{\mathrm{h}}z)\ar[ur]^{1_w\cdot_{\mathrm{h}}a_{x,y,z}}&}$$ - the [**triangle identity**]{} for the left and right unit lows: $$\xymatrix{ ( x\cdot_{\mathrm{h}}1)\cdot_{\mathrm{h}}y\ar[rr]^{a_{x,1,y}}\ar[dr]^{r_x\cdot_{\mathrm{h}}1_y}&&x\cdot_{\mathrm{h}}(1\cdot_{\mathrm{h}}y)\ar[dl]^{1_x\cdot_{\mathrm{h}}l_y}\\ &x\cdot_{\mathrm{h}}y& }$$ - the [**first zig-zag identity**]{}: $$\xymatrix{ &(x\cdot_{\mathrm{h}}{\mathrm{inv}}(x))\cdot_{\mathrm{h}}x\stackrel{a_{x,{\mathrm{inv}}(x),x}}{\longrightarrow}x\cdot_{\mathrm{h}}({\mathrm{inv}}(x)\cdot_{\mathrm{h}}x)\ar[dr]^{1_x\cdot_{\mathrm{h}}e_x}&\\ 1\cdot_{\mathrm{h}}x\ar[dr]^{l_x}\ar[ur]^{i_x\cdot_{\mathrm{h}}1_x}&&x\cdot_{\mathrm{h}}1\\ &x\ar[ur]^{r_x^{-1}}&}$$ - the [**second zig-zag identity**]{}: $$\xymatrix{ &{\mathrm{inv}}(x)\cdot_{\mathrm{h}}(x\cdot_{\mathrm{h}}{\mathrm{inv}}(x))\stackrel{a_{{\mathrm{inv}}(x),x,{\mathrm{inv}}(x)}}{\longrightarrow}({\mathrm{inv}}(x)\cdot_{\mathrm{h}}x)\cdot_{\mathrm{h}}{\mathrm{inv}}(x)\ar[dr]^{e_x\cdot_{\mathrm{h}}1_{{\mathrm{inv}}(x)}}&\\ {\mathrm{inv}}(x)\cdot_{\mathrm{h}}1\ar[dr]^{r_{{\mathrm{inv}}(x)}}\ar[ur]^{1_{{\mathrm{inv}}(x)}\cdot_{\mathrm{h}}i_x}&&1\cdot_{\mathrm{h}}{\mathrm{inv}}(x).\\ &{\mathrm{inv}}(x)\ar[ur]^{l_{{\mathrm{inv}}(x)}^{-1}}&}$$ In the special case where $a_{x,y,z},~l_x,~r_x,~i_x,~e_x$ are all identity isomorphisms, we call such a Lie 2-group a [*strict Lie 2-group*]{}[^6]. In the same reference, they also defined the notion of special 2-groups, which we recall here, \[defi:special 2 G\] A [**special Lie 2-group**]{} is a Lie 2-group of which the source and target coincide and the left unit law $l$, the right unit law $r$, the unit $i$ and the counit $e$ are identity isomorphisms. For classification of special Lie 2-groups, we need the group cohomology with smooth cocycles, that is we consider the cochain complex with smooth morphisms $G^{\times n}\to M$ with $G$ a Lie group, $M$ its module. And the differential is defined as usual for group cohomology. We denote this cohomology by $H_{sm}^\bullet(G, M)$. [[@baez:2gp Theorem 8.3.7]]{}\[thm:special 2 G\] There is a one-to-one correspondence between special Lie 2-groups and quadruples $(K_1,K_2,\Phi,\Theta)$ consisting a Lie group $K_1$, an abelian group $K_2$, an action $\Phi$ of $K_1$ as automorphisms of $K_2$ and a normalized smooth 3-cocycle $\Theta:K_1^3\longrightarrow K_2$. Two special Lie 2-groups are isomorphic if and only if they corresponds to the same[^7] $(K_1, K_2, \Phi)$ and the corresponding 3-cocycles represent the same element in $H^3_{sm}(K_1, K_2)$. We briefly recall that given a quadruple $(K_1,K_2,\Phi,\Theta)$ the corresponding semistrict Lie 2-group has the Lie group $K_1$ as the space of objects and the semidirect product Lie group $K_1 \ltimes_{\Phi} K_2$ as the space of morphisms. The associator is given by $\Theta$. A unital 2-term representation up to homotopy of a Lie group $G$ consists of - A 2-term complex of vector spaces $V_1\stackrel{{\mathrm{d}}}{\longrightarrow}V_0$. - A nonassociative action $F_1$ on $V_0$ and $V_1$ satisfying $${\mathrm{d}}F_1 = F_1 {\mathrm{d}}, \quad F_1(1_G)={\rm{Id}}.$$ - A smooth map $F_2:G\times G\longrightarrow{\mathrm{End}}(V_0,V_1)$ such that $$\label{eqn:F fail} F_1(g_1)\cdot F_1(g_2)-F_1(g_1\cdot g_2)=[{\mathrm{d}},F_2(g_1,g_2)],$$ as well as $$\label{eqn:F closed} F_1(g_1)\circ F_2(g_2,g_3)-F_2(g_1\cdot g_2,g_3)+F_2(g_1, g_2\cdot g_3)-F_2(g_1, g_2)\circ F_1(g_3)=0.$$ We denote this 2-term representation up to homotopy of the Lie group $G$ by $(V_1\stackrel{{\mathrm{d}}}{\longrightarrow}V_0,F_1,F_2)$. One should be careful that even if $F_1$ is a usual associative action, is not equivalent to $F_2$ being a 2-cocycle. This is strangely different from the Lie algebra case (see Section \[sec:int\]). Define $\widetilde{F_2}:(G\ltimes V_0)^3\longrightarrow V_1$ by $$\label{F2tuta} \widetilde{F_2}((g_1,\xi_1),(g_2,\xi_2),(g_3,\xi_3))=F_2(g_1, g_2)(\xi_3).$$ If $F_1$ is a usual associative action, we form $G\ltimes V_0$ the semidirect product. Then $V_1$ is a $G\ltimes V_0$-module with an associated action $\widetilde{F_1}$ of $G\ltimes V_0$ on $V_1$ $$\widetilde{F_1}(g,\xi)(m)=F_1(g)(m),\quad\forall~m\in V_1.$$ \[pro:3cocycle\] If $F_1$ is the usual associative action of the Lie group $G$ on the complex $V_1\stackrel{{\mathrm{d}}}{\longrightarrow}V_0$, then $\widetilde{F_2}$ defined by (\[F2tuta\]) is a group 3-cocycle representing an element in $H^3_{sm}(G\ltimes V_0,V_1)$. [[**Proof.**]{} ]{}By direct computations, we have $$\begin{aligned} &&d\widetilde{F_2}((g_1,\xi_1),(g_2,\xi_2),(g_3,\xi_3),(g_4,\xi_4))\\&=&\widetilde{F_1}(g_1,\xi_1)\widetilde{F_2}((g_2,\xi_2),(g_3,\xi_3),(g_4,\xi_4))\\ &&-\widetilde{F_2}((g_1,\xi_1)\cdot(g_2,\xi_2),(g_3,\xi_3),(g_4,\xi_4))+\widetilde{F_2}((g_1,\xi_1),(g_2,\xi_2)\cdot(g_3,\xi_3),(g_4,\xi_4))\\ &&-\widetilde{F_2}((g_1,\xi_1),(g_2,\xi_2),(g_3,\xi_3)\cdot(g_4,\xi_4))+\widetilde{F_2}((g_1,\xi_1),(g_2,\xi_2),(g_3,\xi_3))\\ &=&F_1(g_1)F_2(g_2,g_3)(\xi_4)-F_2(g_1\cdot g_2,g_3)(\xi_4)+F_2(g_1, g_2\cdot g_3)(\xi_4)\\ &&-F_2(g_1,g_2)(\xi_3+F_1(g_3)(\xi_4))+F_2(g_1,g_2)(\xi_3)\\ &=&\big(F_1(g_1)\circ F_2(g_2,g_3)-F_2(g_1\cdot g_2,g_3)+F_2(g_1, g_2\cdot g_3)-F_2(g_1, g_2)\circ F_1(g_3)\big)(\xi_4).\end{aligned}$$ By (\[eqn:F closed\]), $\widetilde{F_2}$ is a Lie group 3-cocycle. Similar to the fact that associated to any representation of a Lie group, we can form a new Lie group which is their semidirect product, for the 2-term representation up to homotopy of a Lie group, we can form a Lie 2-group. \[thm:main 1\] Given a 2-term representation up to homotopy $(V_1\stackrel{{\mathrm{d}}}{\longrightarrow}V_0,F_1,F_2)$ of a Lie group $G$, its semidirect product with $G$ is defined to be $$\begin{array}{c} G\times V_0\times V_1\\ \vcenter{\rlap{s }}~\Big\downarrow\Big\downarrow\vcenter{\rlap{t }}\\ G\times V_0. \end{array}$$ Then it is a Lie 2-group with the following structure maps: The source and target are given by \[s t\] s(g,,m)=(g,),t(g,,m)=(g,+m). The vertical multiplication $\cdot_{\mathrm{v}}$ is given by $$(h,\eta,n)\cdot_{\mathrm{v}}(g,\xi,m) =(g,\xi,m+n),\quad \mbox{where}~ h=g,\eta=\xi+{\mathrm{d}}m.$$ The horizontal multiplication $\cdot_{\mathrm{h}}$ of objects is given by \[m o\] (g\_1,)\_(g\_2,)=(g\_1g\_2,+F\_1(g\_1)()), the horizontal multiplication $\cdot_{\mathrm{h}}$ of morphisms is given by \[m m\] (g\_1,,m)\_(g\_2,,n)=(g\_1g\_2,+F\_1(g\_1)(),m+F\_1(g\_1)(n)). The inverse map ${\mathrm{inv}}$ is given by (g,)=(g\^[-1]{},-F\_1(g\^[-1]{})()). The identity object is $(1_G,0)$.\ The associator $$a_{(g_1,\xi),(g_2,\eta),(g_3,\gamma)}:\big((g_1,\xi)\cdot_{\mathrm{h}}(g_2,\eta)\big)\cdot_{\mathrm{h}}(g_3,\gamma)\longrightarrow (g_1,\xi)\cdot_{\mathrm{h}}\big((g_2,\eta)\cdot_{\mathrm{h}}(g_3,\gamma)\big)$$ is given by \[associator\] a\_[(g\_1,),(g\_2,),(g\_3,)]{}=(g\_1g\_2g\_3,+F\_1(g\_1)()+F\_1(g\_1g\_2)(),F\_2(g\_1,g\_2)()). The unit $i_{(g,\xi)}:(1_G,0)\longrightarrow (g,\xi)\cdot_{\mathrm{h}}{\mathrm{inv}}(g,\xi)$ is given by \[unit\] i\_[(g,)]{}=(1\_G,0,-F\_2(g,g\^[-1]{})()). All the other natural isomorphisms are identity isomorphisms. [[**Proof.**]{} ]{}By (\[s t\]), (\[m o\]) and (\[m m\]), it is straightforward to see that $$\begin{aligned} s\big((g_1,\xi,m)\cdot_{\mathrm{h}}(g_2,\eta,n)\big)&=&s(g_1,\xi,m)\cdot_{\mathrm{h}}s(g_2,\eta,n),\\ t\big((g_1,\xi,m)\cdot_{\mathrm{h}}(g_2,\eta,n)\big)&=&t(g_1,\xi,m)\cdot_{\mathrm{h}}t(g_2,\eta,n).\end{aligned}$$ Thus the multiplication $\cdot_{\mathrm{h}}$ respects the source and target map. Furthermore, it is not hard to check that the horizontal and vertical multiplications commute, i.e. $$\big((g,\xi+{\mathrm{d}}m,n)\cdot_{\mathrm{h}}(g^\prime,\eta+{\mathrm{d}}p,q)\big)\cdot_{\mathrm{v}}\big((g,\xi,m)\cdot_{\mathrm{h}}(g^\prime,\eta,p)\big)=\big((g,\xi+{\mathrm{d}}m,n)\cdot_{\mathrm{v}}(g,\xi,m)\big)\cdot_{\mathrm{h}}\big((g^\prime,\eta+{\mathrm{d}}p,q)\cdot_{\mathrm{v}}(g^\prime,\eta,p)\big)$$ \[m commute with v\] @C+2em[ & @/\_2pc/\[l\]\_[(g,)]{}\_=“0” |[(g,+m)]{}\^=“1”\_=“1b” @/\^2pc/\[l\]\^[(g,+(m+n))]{}\^=“2” @[=&gt;]{} “0”;“1”\^[m]{} @[=&gt;]{} “1b”;“2”\^[n]{} & @/\_2pc/\[l\]\_[(g\^,)]{}\_=“3” |[(g\^,+p)]{}\^=“4”\_=“4b” @/\^2pc/\[l\]\^[(g\^,+(p+q))]{}\^=“5” @[=&gt;]{} “3”;“4”\^[p]{} @[=&gt;]{} “4b”;“5”\^[q]{} . ]{} It follows from (\[eqn:F fail\]) that the associator $a_{(g_1,\xi),(g_2,\eta),(g_3,\gamma)}$ defined by (\[associator\]) is indeed a morphism from $\big((g_1,\xi)\cdot_{\mathrm{h}}(g_2,\eta)\big)\cdot_{\mathrm{h}}(g_3,\gamma)$ to $(g_1,\xi)\cdot_{\mathrm{h}}\big((g_2,\eta)\cdot_{\mathrm{h}}(g_3,\gamma)\big)$. To see that it is natural, we need to show that \[left\] a\_[(g\_1,+m),(g\_2,+n),(g\_3,+k)]{}\_(((g\_1,,m)\_(g\_2,,n))\_(g\_3,,k)) is equal to \[right\] ((g\_1,,m)\_((g\_2,,n)\_(g\_3,,k)))\_a\_[(g\_1,),(g\_2,),(g\_3,)]{}, i.e. the following diagram commutates: $$\xymatrix{ \big((g_1,\xi)\cdot_{\mathrm{h}}(g_2,\eta)\big)\cdot_{\mathrm{h}}(g_3,\gamma)\ar[d]\ar[r]^{a}&(g_1,\xi)\cdot_{\mathrm{h}}\big((g_2,\eta)\cdot_{\mathrm{h}}(g_3,\gamma)\big)\ar[d]\\ \big((g_1,\xi+{\mathrm{d}}m)\cdot_{\mathrm{h}}(g_2,\eta+{\mathrm{d}}n)\big)\cdot_{\mathrm{h}}(g_3,\gamma+{\mathrm{d}}k)\ar[r]^{a}&(g_1,\xi+{\mathrm{d}}m)\cdot_{\mathrm{h}}\big((g_2,\eta+{\mathrm{d}}n)\cdot_{\mathrm{h}}(g_3,\gamma+{\mathrm{d}}k)\big). }$$ By straightforward computations, we obtain that (\[left\]) is equal to $$\big(g_1\cdot g_2\cdot g_3,\xi+F_1(g_1)(\eta)+F_1(g_1\cdot g_2)(\gamma),m+F_1(g_1)(n)+F_1(g_1\cdot g_2)(k)+F_2(g_1,g_2)(\gamma+{\mathrm{d}}k)\big),$$ and (\[right\]) is equal to $$\big(g_1\cdot g_2\cdot g_3,\xi+F_1(g_1)(\eta)+F_1(g_1\cdot g_2)(\gamma),m+F_1(g_1)(n)+F_1(g_1)\cdot F_1(g_2)(k)+F_2(g_1,g_2)(\gamma)\big).$$ Hence (\[left\]) is equal to (\[right\]) by (\[eqn:F fail\]). This implies that $a_{(g_1,\xi),(g_2,\eta),(g_3,\gamma)}$ defined by (\[associator\]) is a natural isomorphism. By (\[eqn:F fail\]) and the fact that $F_1(1_G)={\rm{Id}}$, the unit given by (\[unit\]) is indeed a morphism from $(1_G,0)$ to $(g,\xi)\cdot_{\mathrm{h}}{\mathrm{inv}}(g,\xi)$. To see that it is natural, we need to prove $$\big((g,\xi,m)\cdot_{\mathrm{h}}{\mathrm{inv}}(g,\xi,m)\big)\cdot_{\mathrm{h}}i_{(g,\xi)}=i_{(g,\xi+{\mathrm{d}}m)},$$ i.e. the following commutative diagram: $$\xymatrix{ & (1_G,0)\ar[dr]^{i_{(g,\xi)}} \ar_{i_{(g,\xi+{\mathrm{d}}m)}}[dl]\\ (g,\xi+{\mathrm{d}}m)\cdot_{\mathrm{h}}{\mathrm{inv}}(g,\xi+{\mathrm{d}}m) &&\ar[ll]_{\quad\qquad\qquad~(g,\xi,m)\cdot_{\mathrm{h}}{\mathrm{inv}}(g,\xi,m)}(g,\xi)\cdot_{\mathrm{h}}{\mathrm{inv}}(g,\xi) }$$ This follows from $$F_2(g,g^{-1})({\mathrm{d}}m)=F_1(g)\cdot F_1(g^{-1})(m)-F_1(g\cdot g^{-1})(m)=F_1(g)\cdot F_1(g^{-1})(m)-m,$$ which is a special case of (\[eqn:F fail\]). Since $F(1_G)={\rm{Id}}$, we have $$(1_G,0)\cdot_{\mathrm{h}}(g,\xi)=(g,\xi),\quad (g,\xi)\cdot_{\mathrm{h}}(1_G,0)= (g,\xi).$$ Hence the left unit and the right unit can also be taken as the identity isomorphism. The counit $e_{(g,\xi)}:{\mathrm{inv}}(g,\xi)\cdot_{\mathrm{h}}(g,\xi)\longrightarrow (1_G,0)$ can be taken as the identity morphism since we have $${\mathrm{inv}}(g,\xi)\cdot_{\mathrm{h}}(g,\xi)=(g^{-1},-F_1(g^{-1})(\xi))\cdot_{\mathrm{h}}(g,\xi)=(1_G,0).$$ At last, we need to show - the pentagon identity for the associator, - the triangle identity for the left and right unit laws, - the first zig-zag identity, - the second zig-zag identity. We only give the proof of the pentagon identity, the others can be proved similarly and we leave them to the readers. In fact, the pentagon identity is equivalent to $$\begin{aligned} &&a_{(g_1,\xi),(g_2,\eta),(g_3,\gamma)\cdot_{\mathrm{h}}(g_4,\theta)}\cdot_{\mathrm{h}}a_{(g_1,\xi)\cdot_{\mathrm{h}}(g_2,\eta),(g_3,\gamma),(g_4,\theta)}=\\ &&\big((g_1,\xi)\cdot_{\mathrm{h}}a_{(g_2,\eta),(g_3,\gamma),(g_4,\theta)}\big)\cdot_{\mathrm{h}}a_{(g_1,\xi),(g_2,\eta)\cdot_{\mathrm{h}}(g_3,\gamma),(g_4,\theta)}\cdot_{\mathrm{h}}\big(a_{(g_1,\xi),(g_2,\eta),(g_3,\gamma)}\cdot_{\mathrm{h}}(g_4,\theta)\big)\end{aligned}$$ By straightforward computations, the left hand side is equal to $$\big(g_1\cdot g_2\cdot g_3\cdot g_4,\xi+F_1(g_1)(\eta)+F_1(g_1\cdot g_2)(\gamma)+F_1(g_1\cdot g_2\cdot g_3)(\theta),F_2(g_1\cdot g_2, g_3)(\theta)+F_2(g_1, g_2)(\gamma+F_1(g_3)(\theta))\big),$$ and the right hand side is equal to $$\begin{aligned} &\big(g_1\cdot g_2\cdot g_3\cdot g_4,\xi+F_1(g_1)(\eta)+F_1(g_1\cdot g_2)(\gamma)+F_1(g_1\cdot g_2\cdot g_3)(\theta),\\&F_2(g_1, g_2)(\gamma)+F_2(g_1, g_2\cdot g_3)(\theta)+F_1(g_1)\circ F_2(g_2, g_3)(\theta)\big).\end{aligned}$$ By (\[eqn:F closed\]), they are equal. Integrating string Lie 2-algebra $\mathbb R\longrightarrow {\mathfrak g}\oplus {\mathfrak g}^*$ {#sec:int} =============================================================================================== As an application of Theorem \[thm:main 1\], we consider the integration of the string Lie 2-algebra $\mathbb R\longrightarrow {\mathfrak g}\oplus {\mathfrak g}^*$ given by (\[Lie 2 g g dual\]). Now we restrict to the case that ${\mathfrak g}$ is finite dimensional. Obviously, given a quadruple $(K_1,K_2,\Phi,\Theta)$ which represents a special Lie 2-group (see Theorem \[thm:special 2 G\]), by differentiation, we obtain a quadruple $({\mathfrak k}_1,{\mathfrak k}_2,\phi,\theta)$, which represents a 2-term skeletal $L_\infty$-algebra. A special Lie 2-group which is represented by $(K_1,K_2,\Phi,\Theta)$ is an integration of a 2-term skeletal $L_\infty$-algebra which is represented by $({\mathfrak k}_1,{\mathfrak k}_2,\phi,\theta)$ if the differentiation of $(K_1,K_2,\Phi,\Theta)$ is $({\mathfrak k}_1,{\mathfrak k}_2,\phi,\theta)$. If the differential ${\mathrm{d}}$ in a 2-term complex $V_1\stackrel{{\mathrm{d}}}{\longrightarrow}V_0$ is 0, a representation up to homotopy of Lie algebra ${\mathfrak g}$ on $V_1\stackrel{0}{\longrightarrow}V_0$ consists of two strict representations $\mu_1$ and $\mu_0$, and a liner map $\nu:{\mathfrak g}\wedge{\mathfrak g}\longrightarrow{\mathrm{Hom}}(V_0,V_1)$ satisfying equation (\[eqn:d k\]). This equation implies that $\nu$ is a Lie algebra 2-cocycle representing an element in $H^2({\mathfrak g},{\mathrm{Hom}}(V_0,V_1))$, with the representation $[\mu(\cdot),\cdot]$ of ${\mathfrak g}$ on ${\mathrm{Hom}}(V_0,V_1)$ defined by $$[\mu(\cdot),\cdot](X)(A)\triangleq[\mu(X), A]=\mu_1(X)\circ A-A\circ \mu_0(X),\quad \forall ~X\in {\mathfrak g},~A\in{\mathrm{Hom}}(V_0,V_1).$$ Define $\widetilde{\nu}:\wedge^3({\mathfrak g}\oplus V_0)\longrightarrow V_1$ by $$\widetilde{\nu}(X_1+\xi_1,X_2+\xi_2,X_3+\xi_3)=\nu(X_1,X_2)(\xi_3)+c.p.$$ then $\nu$ is a 2-cocycle if and only if $\widetilde{\nu}$ is a 3-cocycle where the representation $\widetilde{\mu}$ of ${\mathfrak g}\oplus V_0$ on $V_1$ is given by $$\widetilde{\mu}(X+\xi)(m)=\mu(X)(m).$$ [[**Proof.**]{} ]{}By direct computations, for any $X_i+\xi_i\in{\mathfrak g}\oplus V_0,~i=1,2,3,4,$ we have $$d\widetilde{\nu}(X_1+\xi_1,X_2+\xi_2,X_3+\xi_3,X_4+\xi_4)=d\nu(X_1,X_2,X_3)(\xi_4)+c.p.,$$ which implies the conclusion. The Lie algebra homomorphism $\mu$ from ${\mathfrak g}$ to ${\mathrm{End}}(V_0)\oplus{\mathrm{End}}(V_1)$ integrates to a Lie group homomorphism $F_1$ from the simply connected Lie group $G$ of ${\mathfrak g}$ to $GL(V_0)\oplus GL(V_1)$ with $$\mu(X)=\frac{d}{dt}\Big|_{t=0}F_1(\exp tX),\quad\forall~X\in{\mathfrak g}.$$ Consequently, ${\mathrm{Hom}}(V_0,V_1)$ is a $G$-module with $G$ action $$g\cdot A=F_1(g)\circ A \circ F_1(g)^{-1},\quad\forall ~g\in G,~A\in {\mathrm{Hom}}(V_0,V_1).$$ The Lie algebra 2-cocycle $\nu:{\mathfrak g}\wedge{\mathfrak g}\longrightarrow{\mathrm{Hom}}(V_0,V_1)$ can integrate to a smooth Lie group 2-cocycle $\overline{F_2}:G\times G\longrightarrow {\mathrm{Hom}}(V_0,V_1)$, satisfying $$\label{eqn:F2 closed} F_1(g_1)\circ(\overline{F_2})(g_2,g_3)\circ F_1(g_1)^{-1}-(\overline{F_2})(g_1\cdot g_2,g_3)+(\overline{F_2})(g_1, g_2\cdot g_3)-(\overline{F_2})(g_1,g_2)=0,$$ and $\overline{F_2}(1_G,1_G)=0.$ Let us explain how. The classical theory of cohomology of discrete groups tells us that the equivalence classes of extensions of $G$ by a $G$ module $M$, one to one correspond to the elements in $H^2(G, M)$. In our case, the same theory tells us that $H^2_{sm}(G, {\mathrm{Hom}}(V_0, V_0))$ classifies the equivalence classes of splitting extensions of $G$ by the $G$-module ${\mathrm{Hom}}(V_0, V_1)$, which is a splitting short exact sequence of Lie groups, with ${\mathrm{Hom}}(V_0, V_1)$ endowed with an abelian group structure, $$\label{eq:extension} {\mathrm{Hom}}(V_0, V_1) \to \hat{G} \to G.$$ In a general extension $\hat{G}$ is a principal bundle over $G$, thus it usually does not permit a smooth lift $G \xrightarrow{\sigma} \hat{G}$. It permits such a lift if and only if the sequence splits. However in our case, since the abelian group ${\mathrm{Hom}}(V_0, V_1) $ is a vector space, we have $H^1(X,{\mathrm{Hom}}(V_0, V_1) )=0$ for any manifold $X$. The proof makes use of a partition of unity and similar to the proof showing that $H^1(X, \underline{{\mathbb R}})=0$ for the sheaf cohomology. Hence all $ {\mathrm{Hom}}(V_0, V_1)$ principal bundles are trivial. Therefore always splits. On the other hand it is well-known that when $G$ is simply connected, there is a one-to-one correspondence between extensions of $G$ [@brahic Theorem 4.15] and extensions of its Lie algebra ${\mathfrak g}$, which in turn are classified by the Lie algebra cohomology $H^2({\mathfrak g}, {\mathrm{Hom}}(V_0, V_1))$. Hence in our case the differentiation map $$H^2_{sm} (G, {\mathrm{Hom}}(V_0, V_1)) \to H^2({\mathfrak g}, {\mathrm{Hom}}(V_0, V_1))$$ is an isomorphism. Hence $\nu$ always integrates to a smooth Lie group 2-cocycle unique up to exact 2-cocycles. Then $\overline{F_2}(1_G,1_G)=0$ can be arranged too, because we can always modify the section $\sigma: G \to \hat{G}$ to satisfy $\sigma(1_G)=1_{\hat{G}}$ and the modification of sections results in an exact term. Then combined with (\[eqn:F2 closed\]), it is not hard to see that $$\label{temp1} \overline{F_2}(1_G,g)=\overline{F_2}(g,1_G)=0,\quad\forall~g\in G.$$ Thus $\overline{F_2}$ is a normalized 2-cocycle. \[pro:int mu nu \] For any 2-term representation up to homotopy $(\mu,\nu)$ of a Lie algebra ${\mathfrak g}$ on the complex $V_1\stackrel{0}{\longrightarrow}V_0$, there is an associated representation up to homotopy $(F_1,F_2)$ of the Lie group $G$ on the complex $V_1\stackrel{0}{\longrightarrow}V_0$, where $F_1$ is the integration of $\mu$ and $F_2:G\times G\longrightarrow {\mathrm{End}}(V_0,V_1)$ is defined by $$\label{eqn:F2} F_2(g_1,g_2)=\overline{F_2}(g_1,g_2)\circ F_1(g_1\cdot g_2).$$ [[**Proof.**]{} ]{}Obviously, (\[eqn:F fail\]) is satisfied. To see (\[eqn:F closed\]) is also satisfied, combine (\[eqn:F2\]) with (\[eqn:F2 closed\]). By the fact that $F_1$ is a homomorphism, we obtain $$\begin{aligned} &F_1(g_1)\circ F_2(g_2,g_3)\circ F_1(g_2\cdot g_3)^{-1}\circ F_1(g_1)^{-1}-F_2(g_1\cdot g_2,g_3)\circ F_1(g_1\cdot g_2\cdot g_3)^{-1}\\&+F_2(g_1, g_2\cdot g_3)\circ F_1(g_1\cdot g_2\cdot g_3)^{-1}-F_2(g_1,g_2)\circ F_1(g_1\cdot g_2)^{-1}=0.\end{aligned}$$ Composed with $F_1(g_1\cdot g_2\cdot g_3)$ on the right hand side, we obtain (\[eqn:F closed\]). By Proposition \[pro:int mu nu \] and Theorem \[thm:main 1\], we have \[thm:main 2\] Let $G$ be the simply connected Lie group integrating ${\mathfrak g}$, then the string Lie 2-algebra $\mathbb R\stackrel{0}{\longrightarrow}{\mathfrak g}\oplus{\mathfrak g}^*$ given by (\[Lie 2 g g dual\]) integrates to the following Lie 2-group, $$\label{2 group}\begin{array}{c} G\times {\mathfrak g}^*\times \mathbb R\\ \vcenter{\rlap{s }}~\Big\downarrow\Big\downarrow\vcenter{\rlap{t }}\\ G\times {\mathfrak g}^*, \end{array}$$ in which the source and target are given by s(g,,m)=t(g,,m)=(g,)the vertical multiplication $\cdot_{\mathrm{v}}$ is given by $$(h,\eta,n)\cdot_{\mathrm{v}}(g,\xi,m) =(g,\xi,m+n),\quad \mbox{where}~ h=g,\eta=\xi,$$ the horizontal multiplication $\cdot_{\mathrm{h}}$ of objects is given by $$(g_1,\xi)\cdot_{\mathrm{h}}(g_2,\eta)=(g_1\cdot g_2,\xi+{\mathrm{Ad}}^*_{g_1}\eta),$$ the horizontal multiplication $\cdot_{\mathrm{h}}$ of morphisms is given by $$(g_1,\xi,m)\cdot_{\mathrm{h}}(g_2,\eta,n)=(g_1\cdot g_2,\xi+{\mathrm{Ad}}^*_{g_1}\eta,m+n),$$ the inverse map ${\mathrm{inv}}$ is given by $${\mathrm{inv}}(g,\xi)=(g^{-1},-{\mathrm{Ad}}^*_{g^{-1}}\xi),$$ The identity object is $(1_G,0)$,\ the associator $$a_{(g_1,\xi),(g_2,\eta),(g_3,\gamma)}:\big((g_1,\xi)\cdot_{\mathrm{h}}(g_2,\eta)\big)\cdot_{\mathrm{h}}(g_3,\gamma)\longrightarrow (g_1,\xi)\cdot_{\mathrm{h}}\big((g_2,\eta)\cdot_{\mathrm{h}}(g_3,\gamma)\big)$$ is given by \[associator\] a\_[(g\_1,),(g\_2,),(g\_3,)]{}=(g\_1g\_2g\_3,+\^\*\_[g\_1]{}+\^\*\_[g\_1g\_2]{},F\_2(g\_1,g\_2)()), All the other structures are identity isomorphisms. [[**Proof.**]{} ]{}Since $F_1$ is a usual associative action, we may modify the unit (\[unit\]) given in Theorem \[thm:main 1\] to be the identity natural transformation. It turns out that (\[2 group\]) is a special Lie 2-group and is represented by $(G\ltimes {\mathfrak g}^*,\mathbb R,{\rm{Id}},\widetilde{F_2})$, where $G\ltimes {\mathfrak g}^*$ is the semidirect product with the coadjoint action of $G$ on ${\mathfrak g}^*$, ${\rm{Id}}$ is the constant map $G\ltimes {\mathfrak g}^* \to {\mathrm{Aut}}({\mathbb R})$ by mapping everything to ${\rm{Id}}\in {\mathrm{Aut}}({\mathbb R})$, and $\widetilde{F_2}$ is given by $$\label{eq:tf2} \widetilde{F_2}((g_1,\xi_1),(g_2,\xi_2),(g_3,\xi_3))=F_2(g_1, g_2)(\xi_3)=\overline{F_2}(g_1, g_2)\circ F_1(g_1\cdot g_2)(\xi_3).$$ Since $\overline{F_2}$ is normalized, $\widetilde{F_2}$ is also normalized. The string Lie 2-algebra $\mathbb R\stackrel{0}{\longrightarrow}{\mathfrak g}\oplus{\mathfrak g}^*$ is skeletal and is represented by $({\mathfrak g}\oplus{\mathfrak g}^*,\mathbb R,0,\widetilde{\nu})$, where $\widetilde{\nu}$ is given by (\[eqn:nu3\]). Thus to show that our Lie 2-group is an integration of the string Lie algebra, we only need to show that the differential of the Lie group 3-cocycle $\widetilde{F_2}$ is the Lie algebra 3-cocycle $\widetilde{\nu}$. By direct computations [@brylinski Lemma 7.3.9], we have $$\begin{aligned} &&\frac{\partial^3}{\partial_{t_1}\partial_{t_2}\partial_{t_3}}\Big|_{t_i=0}\sum_{\sigma\in S_3}\epsilon(\sigma)\widetilde{F_2}\big((e^ {t_{\sigma(1)}X_{\sigma(1)}},t_{\sigma(1)}\xi_{\sigma(1)}),(e^ {t_{\sigma(2)}X_{\sigma(2)}},t_{\sigma(2)}\xi_{\sigma(2)}),(e^ {t_{\sigma(3)}X_{\sigma(3)}},t_{\sigma(3)}\xi_{\sigma(3)})\big)\\ &=&\frac{\partial^3}{\partial_{t_1}\partial_{t_2}\partial_{t_3}}\Big|_{t_i=0}\Big(\overline{F_2}(e^{ t_1X_1},e^ {t_2X_2})\circ F_1(e^{ t_1X_1}\cdot e^ {t_2X_2})(t_3\xi_3)\Big)+c.p.\\ &=&\frac{\partial^2}{\partial_{t_1}\partial_{t_2}}\Big|_{t_i=0}\Big(\overline{F_2}(e^{ t_1X_1},e^ {t_2X_2})\circ F_1(e^{ t_1X_1}\cdot e^ {t_2X_2})(\xi_3)\Big)+c.p.\\ &=&\frac{\partial}{\partial_{t_1}}\Big|_{t_1=0}\Big(\frac{\partial}{\partial_{t_2}}\big|_{t_2=0}\overline{F_2}(e^{ t_1X_1},e^ {t_2X_2})\circ F_1(e^{ t_1X_1})(\xi_3)+\overline{F_2}(e^{t_1X_1},1_G)\circ\frac{\partial}{\partial_{t_2}}\big|_{t_2=0}F_1(e^{ t_1X_1}\cdot e^{ t_2X_2})\Big)\\&&+c.p.\\ &=&\frac{\partial}{\partial_{t_1}}\frac{\partial}{\partial_{t_2}}\Big|_{t_i=0}\overline{F_2}(e^{ t_1X_1},e^ {t_2X_2})(\xi_3)+\frac{\partial}{\partial_{t_2}}\big|_{t_2=0}\overline{F_2}(1_G,e^ {t_2X_2})\circ \frac{\partial}{\partial_{t_1}}\Big|_{t_1=0}F_1(e^{ t_1X_1})(\xi_3)\\ &&+c.p.\quad\mbox{by (\ref{temp1})}\\ &=&\nu(X_1,X_2)(\xi_3)+c.p.\\ &=&\widetilde{\nu}(X_1+\xi_1,X_2+\xi_2,X_3+\xi_3) \quad\mbox{by (\ref{eqn:nu3})},\end{aligned}$$ which completes the proof. If Lie algebra ${\mathfrak g}$ is semisimple, the Lie group 3-cocycle $\widetilde{F_2}$ is not exact, i.e. $[\widetilde{F_2}]\neq0$ in $H^3_{sm}(G\ltimes {\mathfrak g}^*,\mathbb R)$. [[**Proof.**]{} ]{}By Theorem \[thm:main 2\], the differentiation of the Lie group 3-cocycle $\widetilde{F_2}$ is the Lie algebra 3-cocycle $\widetilde{\nu}$. We only need to show that when ${\mathfrak g}$ is semisimple, the Lie algebra 3-cocycle $\widetilde{\nu}$ is not exact. This fact is proved in Proposition \[pro:nondegenerate\]. Since $G\ltimes {\mathfrak g}^*$ is a fibration over $G$, the spectral sequence with $E_2^{p,q}=H^p_{sm}(G, H^q_{sm}( {\mathfrak g}^*, {\mathbb R}))$ calculates the group cohomology $H^3_{sm}(G\ltimes {\mathfrak g}^*,\mathbb R)$. Since ${\mathfrak g}^*$ is an abelian group, $H^q_{sm}( {\mathfrak g}^*, {\mathbb R})=\wedge^q{\mathfrak g}^*$. Thus when $G$ is compact, $ H^p_{sm}(G, H^q_{sm}( {\mathfrak g}^*, {\mathbb R})) = (\wedge^q{\mathfrak g}^*)^G$ if $p=0$ and 0 otherwise, where $(\wedge^q{\mathfrak g}^*)^G$ denotes the set of invariant elements of $\wedge^q {\mathfrak g}^*$ under the coadjoint action of $G$. Thus when $G$ is compact, $H^3_{sm}(G\ltimes {\mathfrak g}^*,\mathbb R)=(\wedge^3 g^*)^G \neq 0$ because the Cartan 3-form is an element of $(\wedge^3 g^*)^G$. Notice that our 2-cocycle $\overline{F_2}$ is unique only up to exact terms. Hence by Theorem \[thm:special 2 G\], to verify that our construction is unique up to isomorphism, we need the following lemma, If $\overline{F_2}= {\mathrm{d}}\alpha$ is exact, then $\widetilde{F_2}={\mathrm{d}}\beta$ is also exact with $$\beta((g_1, \xi_1), (g_2, \xi_2))=\alpha(g_1)F_1(g_1)(\xi_2).$$ It is a direct calculation. Since $\overline{F_2}= {\mathrm{d}}\alpha$, we have $$\overline{F_2}(g_1, g_2) = F_1 (g_1) \alpha(g_2) F_1(g_1)^{-1} - \alpha(g_1 g_2) + \alpha(g_1).$$ From the definition of $F_2$, we know that $$F_2(g_1, g_2)= F_1(g_1) \alpha(g_2) F_1(g_1)^{-1} F_1(g_1 g_2) - \alpha(g_1 g_2) F_1(g_1 g_2) + \alpha(g_1) F_1(g_1g_2).$$ By , we have $$\begin{split} \widetilde{F_2}((g_1, \xi_1), (g_2, \xi_2),(g_3, \xi_3))= &F_1(g_1) \alpha(g_2) F_1(g_1)^{-1} F_1(g_1g_2) (\xi_3) - \alpha(g_1g_2) F_1(g_1g_2) (\xi_3) + \alpha(g_1) F_1(g_1g_2) (\xi_3) \\ = &{\mathrm{d}}\beta((g_1, \xi_1), (g_2, \xi_2),(g_3, \xi_3)), \end{split}$$ since $F_1$ is a group homomorphism. Our Lie 2-group as a stacky group has the underlying differential stack $G \times {\mathfrak g}^* \times B{\mathbb R}$. Thus it is 0,1,2-connected (i.e. it has $\pi_0=\pi_1=\pi_2=0$) since $\pi_2(B{\mathbb R})=\pi_1({\mathbb R})=0$ and $\pi_1(B{\mathbb R})=\pi_0({\mathbb R})=0$. Thus it is the unique 0,1,2-connected stacky Lie group integrating the string Lie 2-algebra ${\mathbb R}\xrightarrow{0} {\mathfrak g}\oplus {\mathfrak g}^*$ in the sense of [@z:lie2]. [^1]: Supported by the German Research Foundation (Deutsche Forschungsgemeinschaft (DFG)) through the Institutional Strategy of the University of Göttingen [^2]: private conversation with Jiang-Hua Lu [^3]: private conversation with John Baez and Urs Schreiber [^4]: It is slightly different from the notion given in [@clomni], where the bracket is not skewsymmetric. [^5]: Please see the introduction for the reason why we call it a semistrict Lie 2-group. [^6]: The notion of strict Lie 2-groups is the same as [@baez:2gp]. [^7]: up to isomorphisms of groups of course

--- abstract: 'Galaxy clusters have long been theorised to quench the star-formation of their members. This study uses integral-field unit observations from the $K$-band Multi-Object Spectrograph (KMOS) - Cluster Lensing And Supernova survey with *Hubble* (CLASH) survey (K-CLASH) to search for evidence of quenching in massive galaxy clusters at redshifts $0.3<z<0.6$. We first construct mass-matched samples of exclusively star-forming cluster and field galaxies, then investigate the spatial extent of their H$\alpha$ emission and study their interstellar medium conditions using emission line ratios. The average ratio of [H$\alpha$]{} half-light radius to optical half-light radius ([$r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }$]{}) for all galaxies is $1.14\pm0.06$, showing that star formation is taking place throughout stellar discs at these redshifts. However, on average, cluster galaxies have a smaller [$r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }$]{} ratio than field galaxies: $\langle$[$r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }$]{}$\rangle = 0.96\pm0.09$ compared to $1.22\pm0.08$ (smaller at a 98% credibility level). These values are uncorrected for the wavelength difference between [H$\alpha$]{} emission and $R_c$-band stellar light, but implementing such a correction only reinforces our results. We also show that whilst the cluster and field samples follow indistinguishable mass-metallicity (MZ) relations, the residuals around the MZ relation of cluster members correlate with cluster-centric distance; galaxies residing closer to the cluster centre tend to have enhanced metallicities (significant at the 2.6$\sigma$ level). Finally, in contrast to previous studies, we find no significant differences in electron number density between the cluster and field galaxies. We use simple chemical evolution models to conclude that the effects of disc strangulation and ram-pressure stripping can quantitatively explain our observations.            ' author: - | Sam P. Vaughan,$^{1, 2, 3}$[^1] Alfred L. Tiley,$^{4, 5}$ Roger L. Davies,$^{3}$ Laura J. Prichard,$^{6}$ Scott M. Croom,$^{1,2}$ Martin Bureau,$^3$ John P. Stott,$^{7}$ Andrew Bunker,$^{3}$ Michele Cappellari,$^3$ Behzad Ansarinejad$^{4}$ and Matt J. Jarvis$^{3,8}$\ $^{1}$Sydney Institute for Astronomy, School of Physics, Building A28, The University of Sydney, NSW 2006, Australia\ $^{2}$ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO3D), Australia\ $^{3}$Sub-department of Astrophysics, Department of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK\ $^{4}$International Centre for Radio Astronomy Research, The University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia\ $^{5}$Centre for Extragalactic Astronomy, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK\ $^{6}$Space Telescope Science Institute, 3700 San Martin Drive, Baltimore MD 21218, USA\ $^{7}$Department of Physics, Lancaster University, Bailrigg, Lancaster LA1 4YB, UK\ $^{8}$Department of Physics & Astronomy, University of the Western Cape, Private Bag X17, Bellville, Cape Town, 7535, South Africa\ bibliography: - 'bibliography.bib' date: 'Accepted 2020 June 22. Received 2020 June 21; in original form 2019 November 18.' title: 'K-CLASH: Strangulation and Ram Pressure Stripping in Galaxy Cluster Members at 0.3 &lt; $z$ &lt; 0.6' --- \[firstpage\] galaxies: clusters: general – galaxies: evolution – galaxies: ISM Introduction ============ It is well understood that the environment in which a galaxy resides plays an important role in its formation and evolution. Focusing on the densest environments in particular, we have known for many years that the galaxy population residing in galaxy clusters is markedly different from its counterpart in the field: galaxy cluster members tend to have early-type morphologies [@Dressler:1980; @Dressler:1997], redder optical colours [e.g. @Pimbblet:2002] and spectra free of emission lines [@Gisler:1978]. Current work has extended these observations to much higher redshifts, with studies of protoclusters and overdensities at redshifts between $1.5<z<2.5$ becoming common (e.g. @Muzzin:2013; @Shimakawa:2015; @WangT:2016; @Prichard:2017; @Perez-Martinez+2017; @Boehm:2019; see @Overzier:2016 for a review). The physical processes which cause the differences in galaxy properties can be broadly separated into two categories. On one hand, a number of “external” mechanisms acting on cluster galaxies (involving their interactions with the intracluster medium or other cluster members) have been suggested to quench their star formation and alter their properties. Of these, perhaps the most dramatic is ram-pressure stripping [first proposed in @Gunn:1972]. Galaxy clusters are the largest potential wells in the Universe, and contain vast quantities of hot gas between their members (see e.g. @Sarazin:1986 and @Kravtsov:2012 for reviews). This intracluster medium (ICM) contains an order of magnitude more mass than is in the stars of the galaxies themselves, and is around a thousand times more dense than the inter-galactic medium which surrounds galaxies outside clusters [e.g. @Nicastro:2008; @Zhuravleva:2013]. When a galaxy falls into a cluster, its motion through the ICM creates a pressure which acts on its reservoirs of gas. The force exerted can be strong enough to overcome the disc’s gravitational restoring force, stripping away this reservoir in an occasionally spectacular fashion. Direct observational evidence of gas being stripped from cluster galaxies can be found at local and intermediate redshifts [e.g. @Owers:2012; @Ebeling:2014; @Rawle:2014; @Poggianti:2017; @Boselli:2019], with such objects coming to be known colloquially as “Jellyfish” galaxies following [@Smith:2010]. On the other hand, galaxy clusters are inherently special places, and the initial conditions of galaxies that form within them are different from those of galaxies which form in less dense regions of space. Since the massive clusters of today correspond to the largest overdensities in the early Universe [e.g. @Springel:2005], it has been suggested that these unique initial conditions lead to an “accelerated” evolution of their members [e.g. @Dressler:1980; @Morishita:2017; @Chan:2018]. The question of whether internal or external drivers of galaxy evolution are most important is key to building a complete picture of the way in which galaxies change throughout their lifetimes, and a satisfactory answer has so far remained out of reach. Attempting to answer this question by studying cluster galaxies at $z=0$ is hampered by the fact that so many of them are quiescent, evolved and seemingly at the endpoint of their evolutionary paths. As first discussed in [@Butcher:1978; @Butcher:1984], galaxy clusters at $z\approx0.5$ contain a much higher fraction of star-forming galaxies than today. Furthermore, of those cluster members which are not currently forming stars, some show evidence of recently truncated star formation via the k+a spectral characteristics of post-starburst galaxies [e.g. @Poggianti:2009] or the strong H$\delta$ absorption of “post star-forming” galaxies [e.g. @Couch:1987; @Owers:2019]. These observations imply that intermediate-redshift clusters– which are more likely to be in the process of actively transforming their members– offer a more promising route to address this problem. A number of studies have targeted intermediate-redshift cluster galaxies, often with spatially-unresolved spectroscopy [e.g. recently @Rosati:2014; @Sobral:2015; @Maier:2016; @Morishita:2017]. Whilst these studies have the advantage of targeting large numbers of objects and forming statistically-significant sample sizes, environmental quenching processes are inherently spatially inhomogeneous. Spectroscopic observations which sample multiple positions in the same galaxy at the same time are therefore required to catch these mechanisms to transform galaxies in the act. Our view of intermediate- to high-redshift ($z>1$) star-forming galaxies has been revolutionised in the last decade by integral-field spectroscopic surveys from the ground [e.g. @ForsterSchreiber:2006; @Mancini:2011; @Genzel:2011; @Wisnioski:2015; @Stott:2016; @Beifiori:2017] and deep *Hubble Space Telescope (HST)* grism spectroscopy [e.g. @Atek:2010; @vanDokkum:2011; @Brammer:2012; @Schmidt:2014; @Treu:2015]. These surveys generally study ionised-gas emission, primarily the [H$\alpha$]{} or \[\]$\lambda5007$ lines, resulting in spatially resolved maps of the rate and locations of star formation, two-dimensional maps of the gas kinematics and information on variations in the interstellar medium (ISM) conditions across individual objects. There are few studies of this kind, however, which specifically target star formation in cluster members with spatially-resolved spectroscopy. Pioneering work in this field was carried out by @Kutdemir:2008 [@Kutdemir:2010], who used the Very Large Telescope (VLT) FOcal Reducer/low dispersion Spectrograph 2 (FORS2) with an observing pattern of three adjacent parallel slits to target ionised-gas emission in cluster and field galaxies at $0.1\leq z\leq0.91$. They found a remarkable similarity between the fraction of galaxies with irregular gas kinematics in their field and cluster samples, whilst also finding a correlation between [H$\alpha$]{} luminosity and gas kinematic irregularity which only holds for cluster members. Two further recent examples are [@Vulcani:2015; @Vulcani:2016], who used data from the Grism Lens-Amplified Survey from Space (GLASS) to study the morphologies and star formation rates of 76 [H$\alpha$]{} emitters in 10 clusters from the Cluster Lensing and Supernova survey with *Hubble* [CLASH; @Postman:2012]. They found that [H$\alpha$]{} emitters are observed with a wide variety of morphologies, including a number undergoing ram-pressure stripping, and that their cluster samples follow a mass–star formation rate (SFR) relation similar to that of a matched sample of galaxies in the field. In this work, we perform a similar investigation using observations from the “K-CLASH“ survey (@Tiley:2020; hereafter ). We target 4 clusters from the CLASH sample[^2] using the $K$-band Multi-Object Spectrograph [KMOS; @Sharples:2013], building a sample of 40 star-forming cluster galaxies and 120 star-forming mass-matched ”field" galaxies along the same lines-of-sight. Our goal is to find evidence of environmental quenching mechanisms in action, by comparing the properties of these star-forming cluster and field galaxies which have been observed in a homogeneous way. In Section \[sec:K-CLASH\_survey\], we briefly summarise the K-CLASH survey (introduced fully in ) and define the cluster and field galaxy samples used in this work. In Section \[sec:Halpha\_size\], we discuss our method of creating and characterising our [H$\alpha$]{} surface brightness distributions and the $R_\textrm{c}$-band continuum images. We then measure the half-light radii of the [H$\alpha$]{} and $R_\mathrm{c}$-band images ($r_{\mathrm{e}, \rm{H}\alpha}$ and $r_{\mathrm{e}, R_{\mathrm{c} } }$ respectively) and present distributions of the ratio [$r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }$]{} for the cluster and field galaxies. In Section \[sec:Stacked\_spectra\] we describe our method of measuring emission line fluxes and line ratios from individual galaxies, as well as our spectral stacking methodology, and discuss the results. We place our results into context and discuss the physical implications of our findings in Section \[sec:Discussion\], before drawing our conclusions in Section \[sec:Conclusions\]. We use the program `Stan`[^3] [@Carpenter:2017] a number of times in this work to perform full Bayesian inference of model parameters via dynamic Hamiltonian Monte Carlo (HMC) sampling [@Betancourt:2014; @Betancourt:2017]. In each case, we ensured that the Gelman-Rubin convergence statistic $\hat{R}$ was within acceptable ranges (i.e. below 1.1 for each parameter) and there were no divergent transitions during the sampling. All magnitudes referred to in this work are in the AB system. We assume a Wilkinson Microwave Anisotropy Probe (WMAP) nine-year cosmology [@WMAP9] with Hubble constant $\mathrm{H}_0$ = 69.3[kms$^{-1}$]{}Mpc$^{-1}$, matter density $\Omega_{\mathrm{m}} = 0.287$, spatial curvature density $\Omega_{k}=0$ and cosmological constant $\Omega_{\Lambda}$ = 0.713. The K-CLASH survey {#sec:K-CLASH_survey} ================== The K-CLASH survey design, data reduction procedures and sample properties are introduced and described in . We provide a brief summary here. KMOS is a multi-object near-infrared spectrograph mounted at the Nasmyth focus of Unit Telescope 1 (UT1) at the European Southern Observatory’s (ESO) VLT, Cerro Paranal, Chile. It consists of 24 separate integral-field units (IFUs) on pick-off arms which can be deployed anywhere within a 72 diameter patrol field; each IFU itself has a field of view of 28 $\times$ 28 with spatial sampling of 02 $\times$ 02 per spaxel. K-CLASH observations were conducted with KMOS in the *IZ* band between 2016 and 2018 (proposal IDs 097.A-0397, 098.A-0224, 099.A-0207 and 0100.A-0296). The wavelength coverage in the *IZ* band is from $0.779$ to 1.079 $\mu$m, corresponding to rest-frame [H$\alpha$]{} emission from $z=0.19$ to $z=0.64$. The resolving power varies from $R=2700$ at the bluest wavelength to $R=3700$ at the reddest. The data were reduced with the publicly available <span style="font-variant:small-caps;">EsoRex</span> software [the “ESO Recipe Execution Tool” ; @esorex] and the KMOS instrument pipeline. The pipeline propagates uncertainties in the standard manner, resulting in a “noise” cube for each galaxy to accompany its “data” cube. The target fields were chosen to be the four massive galaxy clusters MACS 2129 ($z=0.589$), MACS 1311 ($z=0.494$), MACS 1931 ($z=0.352$) and MS 2137 ($z=0.313$). These clusters were selected from the full CLASH sample[^4] to be observable from the VLT and to be at redshifts where [H$\alpha$]{} emission from cluster members lies between atmospheric telluric absorption bands and strong night sky emission-line features. A summary of the properties of each cluster is presented in Table \[tab:clashclusters\]. Each cluster was also required to have wide-field optical imaging in multiple bands[^5], from which we select bright galaxies (*V* &lt; 22 for MACS 1931 and *V* &lt; 23 otherwise) with good photometric redshift estimates (measured by @Umetsu:2014) as targets. We preferentially observed galaxies which are blue ($B - V\leq 0.9$ for $z \leq 0.4$ and $V - R_\mathrm{c} \leq 0.9$ for $z > 0.4$) and have photometric redshifts placing them at their respective cluster redshift. Remaining KMOS arms were first placed on blue galaxies at other photometric redshifts, followed by red galaxies at the cluster redshift. During every observing block (OB), one KMOS IFU was allocated to the brightest cluster galaxy (BCG), and at least one arm was placed on a star in the field of view to measure the point-spread function (PSF). Whilst multi-band *HST* photometry is available in each cluster centre, the limited radial extent of these observations means only a small fraction of our K-CLASH targets are covered, and as such we do not make use of this photometry in this work. In total, 282 galaxies were observed across the 4 clusters. We detected stellar continuum and/or ionised-gas emission (from the [H$\alpha$]{} and/or \[\] lines) in 243 galaxies. As discussed in , after integrating each KMOS observation in 06, 12, and 24 diameter apertures, we measured the emission-line signal-to-noise ratio (S/N) by simultaneously fitting the [H$\alpha$]{} line and each of the \[\] doublet lines with a Gaussian component. Following [@Stott:2016], [@Tiley:2016] and [@Tiley:2019], the S/N of the [H$\alpha$]{} emission is then defined as the square root of the difference in $\chi^2$ between that of the best-fitting [H$\alpha$]{} Gaussian component ($\chi^2_{\mathrm{model}}$) and that of a straight line equal to the value of the continuum ($\chi^2_{\mathrm{ continuum}}$); i.e. S/N = $\sqrt{\chi^2_{\mathrm{model}} - \chi^2_{\mathrm{continuum}}}$. [H$\alpha$]{} emission with S/N&gt;5.0 in at least one aperture was found in 191 objects, forming the K-CLASH parent sample. ---------- ------------- ---------------- ---------- ----------------------- ---------------------- -------------- -------------------------- ---------------- -------------- Cluste RA Dec $z$^(a)^ $kT_{X}$$^{\rm{(a)}}$ $R_{200}^{\rm{(b)}}$ KMOS targets [H$\alpha$]{} detections Cluster sample Mass-matched (J2000) (J2000) (keV) (kpc) Field sample **(1)** **(2)** **(3)** **(4)** **(5)** **(6)** **(7)** **(8)** **(9)** **(10)** MACS2129 21:29:26.12 $-$07:41:27.76 0.589 $9.00\pm1.20$ 1904 75 57 12 39 MACS1311 13:11:01.80 $-$03:10:39.68 0.494 $5.90\pm0.40$ 1395 76 44 15 22 MACS1931 19:31:49.63 $-$26:34:32.51 0.352 $6.70\pm0.40$ 1871 63 44 4 30 MS2137 21:40:15.17 $-$23:39:40.33 0.313 $5.90\pm0.30$ 1261 68 46 9 29 ---------- ------------- ---------------- ---------- ----------------------- ---------------------- -------------- -------------------------- ---------------- -------------- \[tab:clashclusters\] Removal of AGN {#sec:AGN_removal} -------------- Since this work concentrates on star-forming galaxies, it is important to distinguish between ionised-gas emission which traces recent star formation and ionising photons from active galactic nuclei (AGN). Unfortunately, we cannot place our objects on many of the common emission-line diagnostic diagrams used to identify AGN contamination [e.g. the BPT diagram: @BPT; @Kauffmann:2003; @Kewley:2006] due to the fact that the KMOS wavelength range does not encompass the H$\beta$ and \[\] emission lines for all of our targets. Instead, we turn to the \[\]/[H$\alpha$]{} ratio as well as a number of sources of ancillary data: the Chandra Advanced CCD Imaging Spectrometer (ACIS) survey of X-ray Point Sources [@Wang:2016], the Wide-field Infrared Survey Explorer [WISE; @WISE; @NeoWISE] “AllWISE” source catalogue[^6], and *Spitzer Space Telescope* Infrared Array Camera [IRAC; @IRAC] observations in the 3.6 and 4.5$\mu$m channels[^7]. Using these sources, we remove from the parent K-CLASH sample: - 6 galaxies with X-ray luminosities between $10^{42}$ and $10^{44}$ erg s$^{-1}$, likely to be powered by an AGN [e.g. @Comastri:2004]. Of these, 5 are detected in [H$\alpha$]{} with S/N&gt;5. - 13 galaxies with WISE colour $W_1-W_2>0.8$ (following the AGN selection criterion of @Stern:2012). Of these, 10 have [H$\alpha$]{} with S/N&gt;5. - 1 galaxy with *Spitzer* colour \[3.6\]-\[4.5\]&gt;1.0, which is not detected in [H$\alpha$]{}. Unfortunately, we were unable to use the common colour-colour cuts from [@Donley:2012], since neither \[5.8\] nor \[8.0\] micron observations of our fields were available. - 13 galaxies with emission line ratios $\log_{10}($\[\]/[H$\alpha$]{})$>-0.1$ (following @Wisnioski:2018). Note that these emission line fluxes are measured in a 12 diameter circular aperture. Five galaxies with [H$\alpha$]{} S/N&gt;5 were classified as containing an AGN using two or more diagnostics. Cluster and field samples {#sec:sample_selection} ------------------------- Next, we differentiate between galaxies that reside in one of the targeted CLASH clusters and those which are simply chance alignments along the same line of sight. Firstly, we calculated the predicted velocity dispersion ($\sigma_{\rm{cluster}}$) of each of the four CLASH clusters using the dispersion-temperature ($\sigma-T$) relation of @Girardi:1996 and the cluster X-ray temperature from [@Postman:2012]. We verified that using the velocity dispersion predicted assuming a hydrostatic isothermal model ($\sigma^2=k_{\rm{B}}T/\mu m_p$[^8]) or the $\sigma$-T relation of [@Wu:1999] made no difference to our sample selection. The cluster redshift ($z_{\rm{cluster}}$) was taken from [@Postman:2012] . For each galaxy with detected [H$\alpha$]{} emission, we then used its spectroscopic redshift ($z_{\rm{member}}$) to calculate its line-of-sight velocity with respect to the rest-frame of the cluster as $v_{\rm{member}}=c(z_{\rm{member}}-z_{\rm{cluster}})/(1+z_{\rm{cluster}})$, where $c$ is the speed of light. Star forming galaxies (SFGs) with a projected radius ($r$) less than twice the radius where the mean interior density is 200 times the critical density of the Universe ($R_{\rm{200}}$) and with $|v_{\rm{member}}|$ less than three times $\sigma_{\rm{cluster}}$ are then classified as cluster members and form the K-CLASH cluster sample. We note that we use updated $R_{\rm{200}}$ measurements of the four CLASH clusters from A. Zitrin (private communication); these values are listed in Table \[tab:clashclusters\]. The widths of these windows were a compromise to account for the possibility that the clusters may not be completely relaxed whilst minimising contamination from non-cluster members. We then selected the following populations: - Galaxies which do not contain an AGN (based on the criteria of Section \[sec:AGN\_removal\]), have [H$\alpha$]{} S/N&gt;5 and satisfy $r < 2 R_{\rm{200}}$ and $|v_{\rm{member}}| < 3 \sigma_{\rm{cluster}}$ form the K-CLASH cluster sample. This selects 40 galaxies. - Galaxies which do not contain an AGN, have [H$\alpha$]{} S/N&gt;5, do not satisfy both $r < 2 R_{\rm{200}}$ and $|v_{\rm{member}}| < 3 \sigma_{\rm{cluster}}$ and have $9.5 < \log_{10}(M_*/\mathrm{M}_{\odot}) < 11.1$ form the K-CLASH field sample. This selects 120 galaxies. A further 8 galaxies have $\log_{10}(M_*/\mathrm{M}_{\odot}) > 11.1$ and form a “high-mass field” sample. Due to their small number, however, we refrain from analysing them further in this work[^9]. A detailed discussion of the properties of these samples can be found in . We note that these criteria are not perfect, and the fact that we have treated each cluster as axisymmetric is unlikely to be strictly correct. It is therefore possible that the field sample contains galaxies which actually reside in the cluster, and vice-versa. This implies that the differences between cluster and field galaxies found in this work are formally only lower limits, as any contamination at all (in either direction) will tend to homogenise both samples and wash out any differences we measure. Measuring H$\alpha$ and stellar continuum sizes {#sec:Halpha_size} =============================================== Utilising the spatially-resolved nature of the KMOS observations of each galaxy, we now measure the extent of the H$\alpha$ emission in each object, a proxy for the spatial extent of star formation. In short, this involves measuring the point-spread function (PSF) of the observations, creating an H$\alpha$ emission line map and then fitting this map with a model light profile which has been convolved with the measured PSF. The KMOS PSF {#sec:KMOS_PSF} ------------ The KMOS PSF represents the response of the instrument to a point-like input signal. Our observing strategy required at least one KMOS arm to target a star in the field of view for each OB, allowing us to measure the PSF. These stellar observations were then reduced in the same way as the science data (see ), including co-adding multiple observations of the same object over separate OBs. We then collapse each reduced cube by summing along the wavelength direction to create an image, which we use during the fitting process (see Section \[sec:Ha\_surface\_brightness\_profiles\]). To characterise variations of the PSF between nights, we fit a two-dimensional Gaussian model to each collapsed PSF image. We found the average FWHM of these images across all K-CLASH fields to be 078, with a standard deviation of 015. In Appendix \[sec:KMOS\_PSF\_variation\], we also investigate changes of the PSF as a function of wavelength and between each of the three KMOS spectrographs, finding the impact of these effects to be minimal for our study. H$\alpha$ line-maps {#sec:Ha_line_maps} ------------------- To construct an [H$\alpha$]{} line-map, we fit a Gaussian emission line profile to the spectrum in each spaxel after subtracting a 4^th^ order polynomial fit to the stellar continuum. We then integrate the best-fitting Gaussian to obtain the [H$\alpha$]{} flux, and assign this value to the corresponding pixel of the [H$\alpha$]{} image. To avoid including flux from the lines on either side, we mask 5 Å regions around 6549.86 and 6585.27 Å (their rest-frame wavelengths) during the fit. To avoid bad pixels, skylines and the [H$\alpha$]{} emission itself biasing the continuum estimate, we iteratively sigma-clip the spectrum during the fitting, discarding pixels with discrepant fluxes and then fitting again to the remaining pixels. We again use a fourth order polynomial, fit with three iterations whilst discarding $\geq2\sigma$ outliers, but reasonable changes to these parameters do not affect our conclusions. We also create a corresponding two-dimensional “noise” image for each galaxy, using the galaxy’s uncertainty cube. For each pixel in the noise image, we add in quadrature the values from the corresponding noise spectrum in a 20 Å window around the [H$\alpha$]{} line. This also allows us to make two dimensional S/N maps for each galaxy, by dividing its [H$\alpha$]{} image its noise image[^10]. An example [H$\alpha$]{} line map is shown in Figure \[fig:spaxel\_example\], with a representative spectrum showing the continuum and [H$\alpha$]{} spectral regions. ![An example [H$\alpha$]{} line map for galaxy MACS 1311: ID 47439. The top panel shows a spectrum corresponding to a single spaxel (highlighted in green in the lower figure) with spectral range chosen to show the region around [H$\alpha$]{} (blue), the emission line fit (green) and fit to the continuum (red).[]{data-label="fig:spaxel_example"}](f1.pdf){width="50.00000%"} An important consideration when measuring the size of [H$\alpha$]{} emission is the minimum surface brightness our observations are sensitive to. We estimate the limit to which we can detect [H$\alpha$]{} emission in each galaxy by taking the median of each wavelength slice in the error cube (thus creating a “median noise spectrum” for each target) and integrating this across a small window centred on the expected wavelength of the [H$\alpha$]{} emission. The average FWHM of [H$\alpha$]{} emission across the K-CLASH star-forming sample is 8.5 Å, so we choose a window of 10 Å; this window size also avoids contribution from \[\] emission on either side. Our 3$\sigma$ detection limit is on the order of $1\times10^{-15}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$, which varies from $4\times10^{-15}$ to $4\times10^{-16}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$. To put this in context, this value is around three orders of magnitude shallower than the recent Multi-Unit Spectroscopic Explorer (MUSE) study of ram-pressure stripping at $z\approx0.7$ by [@Boselli:2019], or the deep stacked [H$\alpha$]{} images from *HST* grism spectroscopy at $z\approx1$ studied in [@Nelson:2016], which both reached a surface brightness limit on the order of $1\times10^{-18}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$ (although we note that the $\approx$100 Å spectral resolution of the grism spectra studied in @Nelson:2016 is much lower than that of our KMOS spectra). We also convert these minimal [H$\alpha$]{} surface brightnesses to minimal SFR surface densities ($\Sigma_{\rm{SFR}}$) using the relation of [@Hao:2011] and [@Murphy:2011] without correcting for dust extinction. The top panel of Figure \[fig:limiting\_SFR\_surface\_density\] shows the $3\sigma$ limiting $\Sigma_{\rm{SFR}}$ of each galaxy in our sample, which is $\approx$0.03-0.1 M$_{\odot}$ yr$^{-1}$ kpc$^{-2}$ for galaxies with an observed [H$\alpha$]{} wavelength $\lambda_{\rm{H}\alpha}>0.9 \mu$m. The bottom panel shows how a representative noise spectrum (averaged spectrally in a rolling window of width 10 Å) varies with wavelength. The decreased sensitivity of the detectors in the KMOS *IZ* band (corresponding to a higher mean noise level shown in the bottom panel of Figure \[fig:limiting\_SFR\_surface\_density\]) implies that our observations of galaxies at $z\approx0.3$ are roughly as shallow in terms of SFR surface density as those at $z\approx0.5$. Furthermore, our best observations occur at redshifts corresponding to gaps between prominent sky emission lines at $\lambda\approx0.92$ Å and $\lambda\approx1.065$ Å. Our $\Sigma_{\rm{SFR}}$ sensitivities are comparable to those of other IFU studies at $z\sim1-2$ [e.g. @Genzel:2011; @Stott:2016], although shallower than *HST* grism studies at intermediate redshifts (e.g. @Vulcani:2015 reach 0.01 M$_{\odot}$ yr$^{-1}$ kpc$^{-2}$). ![Top: The limiting star formation rate surface density ($\Sigma_{\rm{SFR}}$) as a function of the observed [H$\alpha$]{} wavelength of each galaxy in our sample ($\lambda_{\mathrm{H}_{\alpha}}$). Data-points are colour-coded according to the CLASH field the galaxy was observed in. Solid and dashed lines show how a fixed background surface brightness (of $4\times10^{-15}$, $7.5\times10^{-16}$ and $4\times10^{-16}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$) translates to a variable $\Sigma_{\rm{SFR}}$ (in M$_{\odot}$ yr$^{-1}$ kpc$^{-2}$). Note that this plot includes all [H$\alpha$]{} detections, including those with S/N $<5$. Bottom: a representative spectrum showing how the noise level varies as a function of wavelength (due to strong sky emission lines and decreased sensitivity at the blue end of the *IZ* band). To create this plot, we take a representative noise spectrum and find the average noise value (in a rolling 10 Å window) as a function of wavelength. The observations with lowest background noise (around 0.92 and 1.065 $\mu$m) correspond to gaps between skylines. Low-$z$ observations are affected by the higher average noise level at wavelengths $< 0.9\mu$m due to the decreased sensitivity of the KMOS detectors in the *IZ* band. []{data-label="fig:limiting_SFR_surface_density"}](f2.pdf){width="50.00000%"} [H$\alpha$]{} surface brightness profiles {#sec:Ha_surface_brightness_profiles} ----------------------------------------- {width="\textwidth"}\  \ {width="\textwidth"} With high-resolution broad-band and narrow-band imaging, the spatial structure of the continuum light [e.g. @Elmegreen:2005] and [H$\alpha$]{} emission [e.g. @Shapley:2011; @Nelson:2012] in some high-redshift galaxies has been found to be clumpy and disturbed, in contrast to the generally ordered stellar distributions and star formation seen throughout spiral galaxies today. For this reason, a number of studies use a curve-of-growth method to estimate the half-light radius of a galaxy’s [H$\alpha$]{} flux [e.g. @Nelson:2012; @Magdis:2016; @Schaefer:2017]. This approach has some drawbacks, however. Low signal-to-noise [H$\alpha$]{} flux at the outskirts of the galaxy may be missed, and spurious “hot” pixels in the line map are counted as real [H$\alpha$]{} emission if not properly masked. Furthermore, the seeing-limited nature of our observations implies that we are also unable to distinguish distinct clumps of [H$\alpha$]{} emission on scales smaller than the PSF, making irregular systems appear to follow smooth, disc-like surface brightness profiles. We therefore choose instead to fit model light profiles to our [H$\alpha$]{} line maps, in a similar manner to [@Nelson:2016] and [@Wisnioski:2018]. We also note that this choice will have less of an impact on our intermediate-redshift study ($0.2\lessapprox z \lessapprox0.6$) than on work at higher redshift where disturbed morphologies are more common. We fit the [H$\alpha$]{} surface brightness distributions using the publicly available code [<span style="font-variant:small-caps;">imfit</span>]{}[^11] [@Erwin:2015]. [<span style="font-variant:small-caps;">imfit</span>]{} creates two-dimensional surface brightness distributions and fits these to data through a choice of minimisation techniques. In this case, we use [<span style="font-variant:small-caps;">imfit</span>]{} to fit infinitely-thin axisymmetric exponential-disc surface brightness distributions to our [H$\alpha$]{} line maps. Each model is convolved with an observation of the PSF (a two-dimensional image of a bright star in the field, constructed by collapsing the full KMOS data-cube along the wavelength direction; see Section \[sec:KMOS\_PSF\]) during the fitting process. Uncertainties on the fitting parameters are estimated using 1000 bootstrap resamples of the original image. We test the robustness of this fitting method in Appendix \[sec:SN\_tests\] and show it can accurately recover the input disc scale lengths from mock data at various values of signal-to-noise ratio, disc size and disc orientation. The intrinsic (i.e. deconvolved) [H$\alpha$]{} half-light radius of each galaxy was then measured by performing a curve-of-growth analysis in circular apertures on the intrinsic (i.e. before convolution) best-fitting surface brightness profile. Each KMOS IFU has a field of view of 28$\times$28, which corresponds to 9.5$\times$9.5 and 20.3$\times$20.3 kpc at the lowest and highest redshift of our targets, respectively. We note that, in their study of [H$\alpha$]{} emitters at $0.3<z<0.7$, [@Vulcani:2016] found no objects with [H$\alpha$]{} effective radii larger than 10 kpc, implying that a KMOS IFU would encompass at least the half-light radius for all of their targets if observed at the highest redshift of our study. At the lowest redshifts, it is possible that some galaxies would have [H$\alpha$]{} emission more extended than the field of view of an IFU. As we show in Appendix \[sec:SN\_tests\], however, we are able to recover the sizes of mock [H$\alpha$]{} distributions larger than the field of view from high S/N data. We therefore conclude that whilst it is possible the field of view of a KMOS IFU may miss flux from the most spatially extended [H$\alpha$]{} emitters, this is unlikely to significantly affect our conclusions. Continuum imaging {#sec:continuum_imaging} ----------------- Each K-CLASH field has been targeted with deep Subaru Suprime-Cam observations. Suprime-Cam [@Miyazaki:2002] has a 34$^{\prime}\times27^{\prime}$ field of view which was mosaiced over each cluster. The data were reduced and analysed by [@Umetsu:2014], as well as independently by [@vonderLinden:2014], using reduction methods described in [@Nonino:2009] and [@Medezinski:2013]. The images are publicly available from the CLASH archive[^12]. *HST* imaging in a large number of bands is also available in the very centre of each CLASH cluster, but since this imaging covers only a small fraction of K-CLASH galaxies we choose not to use it for any of our targets. We use the Suprime-Cam $R_\mathrm{c}$-band imaging (in the Johnson–Morgan–Cousins system; see @Miyazaki:2002 for details) to measure the surface brightness profile of each K-CLASH galaxy. Images taken in this band are available without having been convolved to the limiting PSF of the other bands (“PSF-matched") before stacking. Instead, the $R_\mathrm{c}$-band images we use were stacked individually at each epoch and camera rotation angle, making them most appropriate for measuring galaxy shapes (e.g. for a weak lensing analysis) and light profiles. At the redshifts of our targets, the $R_\mathrm{c}$ band corresponds to the rest-frame $B$ band. Similarly to the [H$\alpha$]{} line-maps, we used [<span style="font-variant:small-caps;">imfit</span>]{} to fit model profiles to each galaxy. A $6\farcs4\times6\farcs4$ postage-stamp image of each target was extracted from the larger Suprime Cam $R_{\mathrm{c}}$-band image for this purpose. We note that this cutout is larger than a KMOS IFU field of view of $2\farcs8\times2\farcs8$; we found the results of our $R_{\mathrm{c}}$-band fitting to be more robust with this larger cutout size than when matching the KMOS IFUs’ fields of view exactly. A median stack of hundreds of stars in each field was used as the PSF estimate during the fitting process. The seeing varied from 06 to 09 in the four K-CLASH fields. In contrast to the [H$\alpha$]{} spatial modelling, however, we allowed the Sérsic index of the light profile to vary between 1 and 10. We also simultaneously fit foreground and background objects in each postage-stamp cutout with appropriate Sérsic or PSF models to ensure acceptable fits. Uncertainties were again measured using 1000 bootstrap resamples of the input data. Almost all galaxies were well fit with a single Sérsic component, but a small number of disturbed objects required multiple components in order to achieve an adequate fit. The intrinsic $R_{\mathrm{c}}$-band half-light radius of each galaxy was again measured by performing a curve-of-growth analysis on the intrinsic best-fitting model (i.e. the model unconvolved with the PSF) in circular apertures. Signal-to-noise constraints {#sec:SN_constraints} --------------------------- Each galaxy has, up to now, been selected from the K-CLASH parent sample to have an [H$\alpha$]{} S/N greater than 5 in at least one of a 06, 12 or 24 diameter aperture, as well as not being flagged as containing an AGN (see Section \[sec:sample\_selection\]). To ensure our measurements from the image fitting are reliable, we now enforce further constraints. Firstly, we visually inspect each map and remove 2 galaxies from the cluster sample (5%) and 10 from the field sample (8%) where the fit has failed and/or there are problems with the $R_\mathrm{c}$-band imaging (e.g. the galaxy is obscured by the diffraction spike from a bright star). We then require that the reduced $\chi^{2}$ values of the [H$\alpha$]{} and continuum fits should be less than 5[^13]. This removes 12 galaxies from the cluster sample (30%) and 41 from the field sample (34%). Finally, as motivated by our tests in Appendix \[sec:SN\_tests\], we place a constraint on the minimum S/N of the [H$\alpha$]{} images we use for further analysis. We divide each [H$\alpha$]{} image by its associated noise map (described in Section \[sec:Ha\_line\_maps\]) to create a two dimensional S/N map for each galaxy, requiring that the average [H$\alpha$]{} S/N within the best fitting half-light radius is greater than 2. This removes 4 galaxies from the cluster sample (10%) and 21 from the field sample (18%). We are then left with 48 field galaxies and 22 cluster galaxies. Emission line–continuum size ratios {#sec:Ha_Results} ----------------------------------- ![Histograms of the ratio of [H$\alpha$]{} size to continuum size for the cluster and field samples. The average [$r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }$]{} ratio is $0.96\pm0.09$ for the cluster galaxies and $1.22\pm0.08$ for the field galaxies.[]{data-label="fig:size_ratio"}](f5.pdf){width="50.00000%"} Two example fits to the continuum and [H$\alpha$]{} images are shown in Figure \[fig:big\_small\_Ha\]. These objects were chosen to illustrate galaxies with extended (top) and concentrated (bottom) [H$\alpha$]{} emission compared to their $R_\mathrm{c}$-band continuum size. The average ratio of [H$\alpha$]{} effective radius to $R_\mathrm{c}$-band effective radius is 1.14$\pm$0.06, with a range from 0.1 and 2.76. This is in very good agreement with the study of [@Wilman:2020] at higher redshift, who found a median [H$\alpha$]{}-to-continuum half-light radius ratio of 1.19 using the KMOS$^{3\mathrm{D}}$ sample [@Wisnioski:2015; @Wisnioski:2019] at $0.7 < z < 2.7$. It is also in good agreements with the studies of $z\approx1$ galaxies by [@Nelson:2016] ($r_{\mathrm{s,\,H}\alpha}/r_\mathrm{s}[F140W] \approx 1.1$ for star-forming main sequence galaxies) and of “compact” SFGs by [@Wisnioski:2018] at $z\approx0.7-3.7$ ($r_{\mathrm{e,\,H}\alpha}/r_e[F160W] \approx 1.2$). Our measurements also agree well with the mass-(continuum) size relation at intermediate redshifts (@VanDerWel:2014; see ). The average $r_{\rm{e, H}\alpha}/r_{\mathrm{e, } R_{\rm{c}}}$ ratio is $0.96\pm0.09$ for the cluster galaxies and $1.22\pm0.08$ for the field sample, where the quoted uncertainty is the standard error on the mean (i.e. the standard deviation of the sampling distribution). Figure \[fig:size\_ratio\] shows histograms of [$r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }$]{} for the cluster and field samples. To ensure the constraints on $\chi^2$ introduced in Section \[sec:SN\_constraints\] are not biasing our results, we also compute the average [$r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }$]{} ratio for the cluster and field sample without requiring a reduced-$\chi^2$ value of less than 5. In this case, our conclusions are unchanged; we find $\langle$[$r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }$]{}$\rangle=1.03\pm0.09$ for the cluster galaxies (now including 30 galaxies) and $\langle$[$r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }$]{}$\rangle=1.55\pm0.20$ for the field sample (80 galaxies). We also test to see whether our choice of fitting each [H$\alpha$]{} emission-line map with an exponential profile (equivalent to a Sérsic profile with the index fixed at $n=1$) and each stellar continuum map with a Sérsic profile (fitting the index as a free parameter) has impacted our results. We therefore repeat the above analysis using a Sérsic profile for the [H$\alpha$]{} maps, fixing the Sérsic index of each galaxy to be the same as that measured for its continuum light. We find very similar results to before: the average [$r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }$]{} ratio measured from this approach is $1.01+/-0.08$ for the cluster galaxies and $1.26\pm0.09$ for the field galaxies. The average [H$\alpha$]{} half-light radii of the cluster and field galaxies are comparable: $\langle r_{e_{\rm{H}\alpha}}\rangle = 3.4\pm0.4$ kpc for the cluster galaixes compared to $3.9\pm0.3$ kpc for the field galaxies. The two samples also have consistent average $r_{e_{R_{c}}}$ values within the uncertainties: $3.5\pm0.3$ kpc for the cluster sample and $3.7\pm0.3$ kpc for the mass-matched field sample. It is well known that the measured size of an individual galaxy varies as a function of the wavelength it is observed at; using data from the Galaxy And Mass Assembly (GAMA) survey [@Driver:2009; @Driver:2011], for example, [@Vulcani:2014] found that galaxies have smaller $r_{\mathrm{e}}$ when observed with redder photometric filters. Ideally, therefore, we would compare $r_{\mathrm{e}}$ from [H$\alpha$]{} emission-line maps with $r_{\mathrm{e}}$ measured from deep stellar continuum imaging at $\approx 1\mu$m (which matches the [H$\alpha$]{} rest-frame), rather than $r_{\mathrm{e}}$ from the $R_{\mathrm{c}}$-band images (rest-frame $B$ band) as done in this work. A number of studies have discussed empirical methods to convert size measurements carried out at one wavelength to size measurements at another, however [e.g. @Kelvin:2012; @VanDerWel:2014; @Chan:2016]. We therefore use each of these prescriptions to correct our stellar-continuum effective radius measurements to the [H$\alpha$]{} wavelength (6563 Å) in each galaxy’s rest-frame, and then recalculate the size ratios $r_{e, {\rm{H}\alpha}}/r_{e, R_{\mathrm{c}}\mathrm{\,(corrected)} }$ for the cluster and field galaxies. The conversion from [@Chan:2016] applies identically to all galaxies. [@Kelvin:2012] and [@VanDerWel:2014] apply different corrections for disc-dominated/spheroid-dominated galaxies and late-type/early-type galaxies respectively. To use these prescriptions on our sample, we use the Sérsic indices measured in Section \[sec:continuum\_imaging\] to classify galaxies as disc-dominated and late type (Sérsic index $n \leq2$) or spheroid-dominated and early type (Sérsic index $n > 2$). We then recalculate the $r_{e, {\rm{H}\alpha}}/r_{e, R_{\mathrm{c}}\mathrm{\,(corrected)} }$ ratio for each galaxy. The average $r_{e, {\rm{H}\alpha}}/r_{e, R_{\mathrm{c}}\mathrm{\,(corrected)} }$ values are shown in Table \[tab:size\_ratio\_after\_corrections\]. For every correction prescription, the difference between the average size ratio of the cluster and field galaxies is as large or larger than the uncorrected difference, showing that our [$r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }$]{} measurements are likely to be a lower limit on the true difference in size between [H$\alpha$]{} emission and stellar continuum light at 6563Å for cluster and field galaxies. Correction Full sample Cluster sample Field sample ------------------- --------------- ---------------- --------------- Uncorrected $1.14\pm0.06$ $0.96\pm0.09$ $1.22\pm0.08$ [@Chan:2016] $1.28\pm0.07$ $1.09\pm0.10$ $1.37\pm0.09$ [@Kelvin:2012] $1.23\pm0.07$ $1.04\pm0.10$ $1.32\pm0.09$ [@VanDerWel:2014] $1.41\pm0.08$ $1.19\pm0.12$ $1.51\pm0.10$ : Average $r_{e, {\rm{H}\alpha}}/r_{e, R_{\mathrm{c}}\mathrm{\,(corrected)} }$ ratios. Our measurements of stellar-continuum effective radii are from the rest-frame $B$ band of our targets, which is $\approx$ 2000Å bluer than the rest-frame [H$\alpha$]{} emission. Here, we apply a number of different prescriptions to correct our stellar effective radii to the rest-frame [H$\alpha$]{} wavelength (6563Å) and remeasure the size-ratio $r_{e, {\rm{H}\alpha}}/r_{e, \mathrm{continuum} }$. In each case, we find a difference between the average size ratio of the cluster and field galaxies which is as large or larger than that of the uncorrected case. \[tab:size\_ratio\_after\_corrections\] ### Statistical significance We have found that the mean [$r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }$]{} of galaxies in our cluster sample is smaller than the average [$r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }$]{}of galaxies in our field sample. Here, we quantify the significance of this result. We define $\Delta(\langle$[$r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }$]{}$\rangle$) to be: $$\Delta(\langle r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }\rangle) = \langle r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } } \rangle_{\mathrm{cluster}} - \langle r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } } \rangle_{\mathrm{field}}$$ and using the standard rules for addition or subtraction of two Gaussian random variables, our measurements in Section \[sec:Ha\_Results\] imply $\Delta(\langle r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }\rangle) = -0.26\pm0.12$. The 95% credible interval for $\Delta(\langle r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }\rangle)$ therefore excludes zero (-0.49 $\leq\Delta(\langle$[$r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }$]{}$\rangle)\leq$ -0.02), and using the cumulative distribution function of the normal distribution shows that only 1.5% of the probability mass lies above zero. We therefore show that $\langle$[$r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }$]{}$\rangle$ for the cluster galaxies is smaller than $\langle$[$r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }$]{}$\rangle$ for the field galaxies at the 98.5% credibility level. We have also verified this by direct simulation, as well as by fitting the cluster and field [$r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }$]{}distributions with the program `Stan`[^14] and inspecting the posterior probability distribution of $\Delta(\langle$[$r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }$]{}$\rangle$). Finally, we perform a $t$-test (assuming unequal variances) with the null hypothesis that the cluster and field samples have equal means. The test shows we can reject this null hypothesis ($t \mathrm{\,statistic}=-2.12$, $p \mathrm{\,value}=0.038$). ### Comparison with previous work Our study is in very good agreement with the findings of [@Bamford:2007]. Their work measured the radial extent of rest frame $B$-band light and a number of emission lines (\[\]$\lambda3727$, H$\beta$, \[\]$\lambda4959$ and \[\]$\lambda 5007$) in 19 cluster and 50 field galaxies at $0.25<z<1$. They found that the ratio of emission line to stellar scale length was 0.92$\pm$0.07 in cluster galaxies and 1.22$\pm$0.06 in field galaxies, in comparison to $0.96\pm0.09$ and $1.22\pm0.08$ from this sample. We also find agreement with [@Schaefer:2017], who studied the half-light ratio $r_{50, \mathrm{H}\alpha}/r_{50, \mathrm{continuum}}$ in a low-redshift sample ($0.001 < z < 0.1$) of 201 star-forming galaxies in the Sydney-AAO Multi-object Integral-field (SAMI) survey. They found that at larger local environmental densities, the fraction of galaxies with centrally-concentrated [H$\alpha$]{} emission (small [$r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }$]{}) increases. Evidence for truncated [H$\alpha$]{} discs in local galaxy clusters has also been reported by [@Koopmann:2006], using narrow-band [H$\alpha$]{} observations of spiral galaxies in the Virgo cluster. They found $r_{\rm{H}\alpha}/r_\mathrm{R}=0.91\pm0.05$, compared with $1.18\pm0.10$ for a matched sample of isolated spiral galaxies. This is again in excellent agreement with the averages found in this study, and indicates a lack of redshift evolution of the average ratio. [@Koopmann:2004], and a follow up study by [@Crowl:2008], also found that the star formation rates in the centres of truncated spirals in Virgo were comparable to a matched field sample, showing that these galaxies were undergoing stripping of their outskirts rather than experiencing a reduction in star formation at all radii. On the other hand, [@Vulcani:2015; @Vulcani:2016] found a small number of cluster galaxies at redshift $0.3<z<0.7$ with *extended* [H$\alpha$]{} compared to the stellar continuum, using *HST* grism observations and rest-frame UV, optical and infrared *HST* imaging. Of the galaxies with “spiral” morphologies in their sample, 7 out of 25 have [H$\alpha$]{} sizes more than twice their size in the *HST* $F475W$ band. They also concluded that these objects have been ram-pressure stripped, leading to an extended star-forming halo around the stellar component and complicated morphologies. As discussed in Section \[sec:Ha\_line\_maps\], the limiting SFR surface density of our observations is $\approx0.03-0.1$ M$_{\odot}$ yr$^{-1}$ kpc$^{-2}$, shallower than the studies of [@Vulcani:2015; @Vulcani:2016] which reach 0.01 M$_{\odot}$ yr$^{-1}$ kpc$^{-2}$. The reason for the apparent disagreement between the studies, therefore, could be due to the fact that we are insensitive to extended low surface brightness [H$\alpha$]{} emission. What else could lead to small [$r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }$]{} ratios? ------------------------------------------------------------------------------------------------------------ Previously, we have made the association of small [$r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }$]{} ratios with the removal of [H$\alpha$]{}-emitting gas at large galaxy radii. Here, we discuss other factors and processes which could lead to a similar trend whilst also explaining why their influence is expected to be small. The source of [H$\alpha$]{} flux we want to measure is ultimately stars more massive than $\approx20$ [M$_{\odot}$]{} and younger than 5-10 Myrs in individual star-forming regions (via their ionisation of surrounding gas; see e.g. @Kennicutt:1998a and @Calzetti:2013 for reviews). We then want to make the association of [H$\alpha$]{} flux and star formation (assuming a form for the initial mass function; see e.g @Kennicutt:2012). Under the assumption that each region is optically thick to ionising radiation (“Case B”– every energetic photon from a massive star ionises an atom of Hydrogen, which then recombines and produces a cascade of emission lines, with [H$\alpha$]{} and H$\beta$ the most prominent), a simple relation between [H$\alpha$]{} flux and star formation rate exists [@Osterbrock:2006]. However, the effects of dust, AGN and photo-ionisation from evolved stars or shocks can confound this simple picture. Firstly, dust attenuation and extinction will suppress both [H$\alpha$]{} and continuum flux, absorbing photons to be re-emitted at longer wavelengths. Is it possible that a difference in dust properties between cluster and field galaxies is driving the observed trend to small [$r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }$]{} in dense environments? The $R_\mathrm{c}$ band in which we measure our continuum sizes is centred at $\approx6500$ Å, tracing flux from 4000 to 5500 Å in the rest-frame of our sources, 1000 - 2500 Å bluer than rest frame [H$\alpha$]{}. Dust reddening is a function of the emitted wavelength, with stronger extinction at shorter wavelengths [e.g. @Calzetti:2000], naively implying that dust will tend to attenuate the stellar continuum light more than the [H$\alpha$]{} emission. On the other hand, however, a number of studies have reported additional attenuation towards star-forming regions [e.g. @Fanelli:1988; @Calzetti:1994; @Mancini:2011], finding that [H$\alpha$]{} emission is further attenuated by a factor of $\approx2$ compared to the continuum at the same wavelength (A$_{V\mathrm{, H}\alpha}=$ A$_{V\rm{, continuum}}/0.44$). This is due to the fact that regions, where young stars are found, are inherently dustier than the regions surrounding older stellar populations. Using the Calzetti extinction law [@Calzetti:2000] and taking these two effects into account, the [H$\alpha$]{} emission is reddened more than the $R_\mathrm{c}$-band continuum light[^15]. It is therefore possible that a difference in global dust properties between environments could lead to smaller observed size ratios in dense environments, if cluster galaxies are more obscured than their field counterparts. Using the global extinction estimates from our SED fits , however, we find the values of A$_{V}$ to be comparable between the (mass-matched) cluster and field samples, with the average being *lower* for cluster galaxies by 0.1 mag. There is also no significant correlation between A$_{V}$ and size ratio (Pearson’s correlation coefficient $r_{x,y}$=0.035, $p$-value=0.77), implying that galaxies with smaller size ratios are not systematically more attenuated. It should be noted that because the KMOS wavelength range available to us does not cover the H$\beta$ line, it is not possible to make *local* extinction corrections to our [H$\alpha$]{} maps (although differences in the spatial distribution of dust conspiring to suppress the [$r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }$]{} ratio in cluster but not field galaxies is unlikely). We also note that whilst it is well known that dust extinction correlates with stellar mass [e.g. @Reddy:2006; @Momcheva:2013; @Nelson:2016a], we have avoided any associated systematic effect from this correlation by matching our field and cluster samples in mass. Secondly, we consider the [H$\alpha$]{} flux from active galactic nuclei. The effect of AGN would be to add extra flux in the centre of each galaxy, leading to centrally-peaked radial flux profiles and small inferred sizes. Narrow-line AGN in particular could impact the [H$\alpha$]{} flux more than the continuum measurements, and hence bias the inferred [$r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }$]{} to small values. Whilst we have endeavoured to remove all AGN contamination from our sample (see Section \[sec:AGN\_removal\]), without further observations of the H$\beta$ and \[\] emission lines in each galaxy we cannot completely rule out their presence; in particular, weak AGN surrounded by star-forming regions are especially difficult to detect using the methods of Section \[sec:AGN\_removal\]. In the local Universe, the fraction of luminous AGN in high-density environments is lower than in the field [@Kauffmann:2004; @Popesso:2006], although there is strong evolution from $z>1$ to the present day [@Martini:2013]. For weaker AGN, the fraction tends to be comparable [@Best:2005a; @Haggard:2010]. We therefore expect the effect of AGN interlopers which have been missed by our selection cuts to be small, but also– most importantly– comparable for the field and cluster sample. Finally, photo-ionisation from sources such as planetary nebulae and post-AGB stars can contribute to [H$\alpha$]{} emission [e.g. @Binette:1994; @Sarzi:2010]. Regions where such emission is an important fraction of the total ionising photon flux have come to be known as Low-Ionisation Emission Regions or “LIERs” [@Belfiore:2016]. The usual way to identify LIERs is with a BPT diagram, and as such we are unable to remove spectra with LIER-like line ratios from our samples. However, the total fraction of ionising radiation from post-AGB stars is largest for old, quiescent populations, so for the currently star-forming galaxies in our sample their contribution is expected to be small [@Byler:2017]. We therefore take our main result at face value: above the limiting surface brightness of our observations, star-forming galaxies residing in clusters on average have smaller [$r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }$]{} values than similar galaxies in the field. Central surface brightness measurements --------------------------------------- We now explore whether there are differences between the [H$\alpha$]{} surface brightnesses in the cluster and field galaxies. We use the aperture flux measurements from , extracted from a 06 diameter circular aperture centred on each galaxy. To briefly recap the measurement process, we first sum the spectra within the aperture to extract a spectrum and then subtract a sigma-clipped 6th order polynomial fit to the continuum after cleaning any remaining sky emission-line residuals. Fluxes are then measured by fitting a set of Gaussian emission line templates to the [H$\alpha$]{} and \[\] emission lines (See Section 4.1 of for further details). Finally, we calculate the surface brightness of each galaxy within the 06 aperture (which we denote $\mu_{0.6}$) by dividing the integrated flux of the [H$\alpha$]{} line by the area of the aperture. We note that using a fixed aperture size for all galaxies does not account for the fact that the apparent sizes of galaxies in the cluster and field samples are slightly different; the average [H$\alpha$]{} half-light radius of galaxies in the cluster sample is 056 whilst that of galaxies in the field sample is 079. Whilst we would ideally measure the central surface brightness in apertures of e.g. $r_{\mathrm{e}}/4$ or $r_{\mathrm{e}}/8$, the average PSF width of the parent K-CLASH sample is comparable to$\approx$ 06 and we therefore choose not to make measurements using apertures smaller than this. This does mean, of course, that measurements of $\mu_{0.6}$ for our cluster galaixes will include flux originating from slightly larger radii than for field sample galaxies (i.e. including flux from slightly beyond $r_{\mathrm{e}}$ on average for the cluster galaxies, compared to within $r_{\mathrm{e}}$ for the field galaxies). The central surface brightnesses $\mu_{0.6}$ for the cluster galaxies should therefore be taken as upper limits to the true (i.e. deconvolved) surface brightnesses within 06. We find a small difference in the average central surface brightnesses ($\langle \mu_{0.6} \rangle$) between the two samples of $\approx$0.06 dex. We again use `Stan` to fit a Gaussian function to each distribution, measuring the average and standard deviation of each population (incorporating measurement uncertainties during the fit). The field sample is centred at $\log_{10}(\mu_{0.6}/\mathrm{erg} \mathrm{\,s}^{-1}\mathrm{cm}^{-2}\mathrm{arcsec}^{-2}) = -16.29\pm{0.03}$ and the cluster sample at $\log_{10}(\mu_{0.6}/\mathrm{erg} \mathrm{\,s}^{-1}\mathrm{cm}^{-2}\mathrm{arcsec}^{-2}) = -16.35\pm{0.05}$. Histograms of the two distributions is shown in Figure \[fig:surface\_brightness\]. ![Surface brightness measurements within a 06 aperture ($\mu_{0.6}$) for the cluster (red) and field (blue) samples. We find a small difference between the average $\log_{10}(\mu_{0.6}/\mathrm{erg} \mathrm{\,s}^{-1}\mathrm{cm}^{-2}\mathrm{arcsec}^{-2})$ values: $-16.29\pm{0.03}$ for the field sample and $-16.35\pm{0.05}$ for the cluster sample.[]{data-label="fig:surface_brightness"}](f6.pdf){width="50.00000%"} Emission Line Analysis {#sec:Stacked_spectra} ====================== {width="\textwidth"} Having found a difference in the *extent* of star formation between cluster and field galaxies, we now assess whether the *physical conditions* in their star-forming regions differ too. To do this, we investigate similarities and differences between the emission line spectra of galaxies in the cluster and mass-matched field sample. We extract flux from a circular aperture with a diameter of 24 centred on the continuum centre of each object. We measure the fluxes of the [H$\alpha$]{}, \[\]$\lambda6548,\lambda6584$ and \[\]$\lambda6716,\lambda6731$ emission lines. This is accomplished by performing a fit (with a single velocity component) to each spectrum using `pPXF`[^16] [@ppxf; @2017MNRAS.466..798C]. The \[\] doublet is fit with a single template of two Gaussians, fixed at a flux ratio of 3 [@Osterbrock:2006]. The \[\] lines are fit with individual templates, but we use the “`limit_doublets`” keyword in `pPXF` to limit the flux ratio of the two lines to be between 0.44 and 1.44, the values allowed by a physical analysis of the atomic transitions involved [@Osterbrock:2006]. We fail to detect stellar absorption features in our spectra, and as such use a sixth order polynomial to approximate the stellar continuum rather than including a library of stellar templates. We estimate uncertainties by adding random noise (scaled according to each galaxy’s noise spectrum at each wavelength) to the best-fitting model and repeating the fit 1000 times per galaxy. We also investigate a stack of the galaxy spectra in each of the cluster and field samples. Stacking increases the S/N compared to the spectra of individual galaxies, and allows us to make more robust measurements of the relatively faint \[\] doublet. During the stacking procedure, we interpolate each spectrum to be uniformly sampled in $\log\lambda$, fit a Gaussian to find the centroid of the [H$\alpha$]{} emission line, shift the spectrum to its rest frame and divide by the peak [H$\alpha$]{} flux. We remove the stellar continuum by subtracting a 4$^{\rm{th}}$ order polynomial fit and combine all spectra into a median stack. Our conclusions are unchanged if we sigma-clip the spectra before combining. The final stacked spectrum of each sample is shown in Figure \[fig:stacked\_spectra\], where we also show a representative spectrum from an individual galaxy. We perform 10,000 bootstrap resamples to assess the uncertainties in each stack. If $N$ objects contribute to a stack, we randomly draw $N$ spectra from the sample (with replacement) and recombine them. The final error “spectra" are estimated by taking the standard deviation of the bootstrap samples at each wavelength. We then measure the emission lines in the same manner as for individual galaxies (see Section \[sec:Ha\_line\_maps\]), with measurement uncertainties estimated using 10,000 bootstrap resamples of each stacked spectrum. Gas-phase metallicities {#sec:emline_results} ----------------------- Cluster Field -------------------------------------------------- --------------------- ------------------------ [\[\]]{}$\lambda6584/$[H$\alpha$]{} $0.26\pm0.03$ $0.25\pm0.02$ [\[\]]{}$\lambda 6716$ / \[\]$\lambda 6731$ $1.18\pm0.17$ $1.43^{+0.01}_{-0.02}$ [\[\]]{}$\lambda6716,6731/$[H$\alpha$]{} $0.22\pm0.04$ $0.25\pm0.02$ [\[\]]{}$\lambda6584$/[\[\]]{}$\lambda6716,6731$ $1.20\pm0.28$ $1.01\pm0.15$ $12+\log(\rm{O/H})$ $8.57\pm0.02$ $8.56\pm0.02$ $n_e\,(\rm{cm}^{-3})$ $126^{+182}_{-116}$ $<10^{+28}_{-0}$ : Emission line ratios and derived quantities for the stacked spectra. We fix the maximum value of the [\[\]]{}$\lambda 6716$ / [\[\]]{}$\lambda 6731$ ratio to be 1.44 (see Section \[sec:EM\_line\_ratios\]), and as such the upper uncertainty on this ratio in the field stacked spectrum is 0.00. The solar oxygen abundance is $12+\log(\rm{O/H}) = 8.69$ [@Asplund:2009]. \[tab:emission\_line\_ratios\] The stacked spectra of the two samples are shown in Figure \[fig:stacked\_spectra\]. It is clear that the mass-matched field and cluster galaxies show very similar average spectra. This adds to the findings of a number of other studies which show that the environment a galaxy resides in plays only a minor role in setting the conditions of its interstellar medium [e.g. @Mouhcine:2007; @Cooper:2008; @Pilyugin:2017; @Wu:2017]. A number of characteristics of star-forming regions can be investigated using emission-line fluxes and line ratios. Firstly, we measure the gas-phase metallicity, $12+\log(\rm{O/H})$, of the galaxies in our sample. A number of methods exist to convert emission-line measurements to metallicities, although it is well known that large discrepancies exist between metallicities estimated using different methods [e.g. @Pilyugin:2001; @Liang:2007; @Kewley:2008]. Here, we derive the gas-phase metallicity using the ratio \[\]$\lambda6584/$[H$\alpha$]{} and the polynomial conversion of [@Pettini:2004]: $$\label{eqn:mass_metallicity} 12+\log(\rm{O/H})=9.37 + 2.03N +1.26N^{2} + 0.32N^{3},$$ {width="96.00000%"} where $N\equiv\log_{10}($\[\]$\lambda6584/$[H$\alpha$]{}).We recall that we have already removed all galaxies with large \[\]$\lambda6584/$[H$\alpha$]{} from our sample in an effort to remove galaxies containing an AGN. Whilst it is true that studies have shown that galaxies with the same \[\]$\lambda6584/$[H$\alpha$]{} ratio can have different \[\]/Ha ratios (and therefore different metalliicties: e.g. @Maier:2016), with only the [H$\alpha$]{} and emission lines available to us this conversion is the only one we are able to use. It does allow for easy comparison to gas-phase metallicity measurements at high-redshift, however, as many studies also use the \[\]$\lambda6584/$[H$\alpha$]{} ratio [e.g. @Swinbank:2012b; @Stott:2014; @Wuyts:2016; @Magdis:2016]. We measure an \[\]$\lambda$6584/[H$\alpha$]{} ratio of 0.26$\pm$0.03 for the cluster stacked spectrum and 0.25$\pm$0.02 for the mass-matched field stacked spectrum. These results are summarised in Table \[tab:emission\_line\_ratios\], and correspond to $12+\log(\rm{O/H})$= $8.57\pm0.02$ and $8.56\pm0.02$ respectively. For reference, the solar oxygen abundance is $12+\log(\rm{O/H}) = 8.69$ [@Asplund:2009]. We also use Equation \[eqn:mass\_metallicity\] to construct the mass-metallicity (MZ) relation for individual galaxies [see e.g. @Lequeux:1979; @Tremonti:2004]. This is shown in the top panel of Figure \[fig:M\_Z\]. Contours show the local MZ relation derived from 236,114 galaxies from the 12th data release of the Sloan Digital Sky Survey [SDSS; @SDSS_DR12]. Emission line measurements are from [@Thomas:2013], with stellar masses estimated using the technique of [@Maraston:2009] assuming a Kroupa Initial Mass Function [IMF; @Kroupa]. We inferred metallicities for the SDSS galaxies again using Equation \[eqn:mass\_metallicity\]. The average redshift of these objects is 0.06, with galaxies selected to be in the “star-forming” region of the BPT diagram. Following [@Maiolino:2008], we use a relation of the form $$\label{eqtn:M_Z_relation} 12 + \log(\textrm{O/H}) = -0.0864\left(\log \left(\frac{M}{\mathrm{M}_{\odot}}\right)-M_{0}\right)^{2}+K_{0},$$ with free parameters $M_0$ and $K_0$; $K_0$ corresponds to the metallicity of a galaxy with mass $M_0$. We again perform the regression using `Stan`, incorporating uncertainties in the $x$ and $y$ directions and intrinsic scatter around the relation. We place Gaussian priors of $\mathcal{N}(10, 2)$ on $M_0$ and $K_0$ and a “half-normal” prior[^17] of $\mathcal{N}(0, 1)$ on the intrinsic scatter parameter $\sigma$. For the entire K-CLASH sample, we find $M_0 = 12.12\pm0.34$ and $K_0=8.92\pm0.10$, with an intrinsic scatter of $\sigma=0.18\pm0.02$ dex. In the bottom panel of Figure \[fig:M\_Z\], we split our sample by environment and fit a Bayesian hierarchical model to the cluster and field samples. Rather than fitting to the two populations individually, we describe the unknown model parameters of the cluster and field populations as being drawn from shared prior distributions, with these prior distributions themselves described by shared hyper-parameters that are also estimated during the fitting. This allows for inference on the unknown parameters in the cluster and field samples separately, whilst also resulting in tighter constraints on their measurement; both populations can borrow strength from one another by influencing the shared hyper-parameter posterior distributions. In this way, hierarchical modelling is the best compromise between fitting to the cluster and field samples completely independently (resulting in larger uncertainties on $M_0$ and $K_0$ for both populations) and combining all galaxies together to derive single values of $M_0$ and $K_0$ (which prevents us inferring any differences between the two samples). An introduction to hierarchical models can be found in [@BDA3], with some recent discussion and examples of their use in astronomy in e.g. [@Lieu:2017], [@Sharma:2017], [@Thrane:2019], [@Grumitt:2019]. We fully describe our model below. In the following context, the symbol $\sim$ means “is distributed according to”, e.g. $\alpha \sim \mathcal{N}(0, 2)$ means that the parameter $\alpha$ is distributed according to a normal distribution with mean 0 and standard-deviation 2. $$\begin{aligned} \label{eqtn:hierarchical_model} \begin{split} i &= 1...N_{\mathrm{galaxies}}\\ j &= \mathrm{field~or~cluster}\\ \midrule \alpha &\sim \mathcal{N}(0, 2)\\ \beta &\sim \mathrm{Inv-gamma}(2, 0.1)\\ \gamma &\sim \mathcal{N}(0, 2)\\ \delta &\sim \mathrm{Inv-gamma}(2, 0.1)\\ \tau &\sim \mathrm{Inv-gamma}(2, 0.1)\\ \midrule M_{0, j} &\sim \mathcal{N}(\alpha, \beta)\\ K_{0, j} &\sim \mathcal{N}(\gamma, \delta)\\ \sigma_{j} &\sim \mathrm{Half-}\mathcal{N}(0, \tau)\\ \midrule M_{\rm{true}, i} &\sim \mathcal{N}(M_{\rm{obs}, i}, \sigma_{M, i})\\ y_{\rm{true}, i} &\sim \mathcal{N}(y_{\rm{obs}, i}, \sigma_{y, i})\\ \theta_i &= -0.0864\left(\log \frac{M_{\rm{true, i}}}{\mathrm{M}_{\odot}} -\frac{M_{\rm{0, i}}}{\mathrm{M}_{\odot}} \right)^{2}+K_{0, j}\\ y_{\rm{true}, i} &\sim \mathcal{N}(\theta_{i}, \sigma_{j}) \end{split}\end{aligned}$$ This model should be interpreted as follows. The index $i$ labels individual galaxies, and runs from 1 to the total number of objects in our sample. For each galaxy, the quantities we want to relate are its true value of $12 + \log(\mathrm{O/H})$ and its true value of stellar mass. We denote the true gas-phase metallicities of our samples to be $y_{\rm{true}, i}$. This vector $y_{\rm{true}, i}$ is a “latent” variable, in that we do not (and cannot) observe it directly. Instead, we only have uncertain measurements of our galaxies’ gas-phase metallicities, which we denote $y_{\rm{obs}, i}$. We relate $y_{\rm{true}, i}$ to $y_{\rm{obs}, i}$ using a series of Gaussian distributions. These distributions are centred on $y_{\rm{obs}, i}$ and have standard deviations given by the measurement uncertainties on $y_{\rm{obs}, i}$, $\sigma_{y, i}$. The same is true for each galaxy’s stellar mass: we relate our noisy observations ($M_{\rm{obs}, i}$) to each galaxy’s true stellar mass ($M_{\rm{true}, i}$) using a series of Gaussian distributions with standard deviations $ \sigma_{M, i}$. The quantities we wish to infer, $M_0$, $K_0$ and the intrinsic scatter $\sigma$, may take different values for the cluster and field samples. We use the index $j$ to show this; $j$ can take the values “field” or “cluster”, depending on whether galaxy $i$ is in the field or cluster sample. For each galaxy, we use Equation \[eqtn:M\_Z\_relation\] to infer a value of gas-phase metallicity from $M_0$, $K_0$ and $M_{\rm{true}, i}$. We then describe $y_{\rm{true}, i}$ as being distributed as a series of Gaussians centred on these value of gas-phase metallicity, with standard deviation $\sigma_j$. We place Gaussian priors (denoted $\mathcal{N}$) on $M_{0, j}$ and $K_{0, j}$, and a half-Gaussian prior on $\sigma_j$. The parameters $\alpha, \beta, \gamma, \delta$ and $\tau$ are hyper-parameters. We place Gaussian priors on $\alpha$ and $\gamma$ (which describe the “location” of the priors on $M_{0, j}$ and $K_{0, j}$) and inverse gamma priors (denoted $\mathrm{Inv-gamma}$ ) on $\beta, \gamma$ and $\tau$ (which denote the “width” or “scale” of the priors on $M_{0, j}$, $K_{0, j}$ and $\sigma_j$). We choose an inverse gamma prior for these quantities to ensure they remain positive. During the fitting process, we took the standard modelling step of centring our observations around zero by subtracting their mean value. The model was fit using `Stan`, with the “maximum tree-depth” parameter set to 20. We find that the two samples lie on indistinguishable mass-metallicity relations: for the cluster sample $M_0 = 12.07\pm0.34$, $K_0=8.89\pm0.11$ and $\sigma=0.15\pm0.03$ dex; for the field sample $M_0 = 12.07\pm0.34$, $K_0=8.91\pm0.09$ and $\sigma=0.19\pm0.02$ dex. We also note that our conclusions remain unchanged if we perform a standard fit to the field and cluster populations separately, instead of using the hierarchical model outlined above. The fact that the field and cluster MZ relations are the same is in agreement with [@Maier:2016], who found the difference between the MZ relations of field and cluster galaxies (in another CLASH cluster at $z\approx0.4$) to be less than 0.1 dex. Similarly, for local galaxies, [@Mouhcine:2007] found only small differences between the gas-phase metallicity of galaxies with masses greater than $10^{9.5}$ [M$_{\odot}$]{} as a function of environmental density. On the other hand, after removing the trend between environment, colour and luminosity, [@Cooper:2008] found a weak but significant trend between metallicity and environment, with more metal-rich galaxies residing in higher density environments. [@Ellison:2009] also found an elevation of 0.04 dex in metallicity between a sample of 1318 cluster galaxies and a matching sample of field galaxies, but also a stronger trend between *local* density, rather than simply cluster membership, and metallicity. Finally, [@Gupta:2016] studied the MZ relation in two CLASH clusters at $z\sim0.35$, finding that the relation of galaxies residing in RX J1532+30 is consistent with their local comparison sample whilst the relation of galaxies in MACS J1115+01 is enhanced by 0.2 dex. Residuals around the Mass-Metallicity relation {#sec:MZ_residuals} ---------------------------------------------- {width="\textwidth"} Whilst we do not detect a difference between the mass-metallicity relation of galaxies residing in high-density environments (our cluster sample) and lower-density environments (our field sample), a number of studies have documented a correlation between a galaxy’s location in cluster phase space and its metal content. [@Maier:2016] showed that the fraction of accreted galaxies that are metal-rich is higher than their sample of infalling galaxies, and these accreted galaxies have higher metallicities than predicted from models assuming a constant supply of inflowing pristine gas. Similarly, [@Gupta:2016] measured a correlation between the residuals around the MZ relation and cluster-centric distances in one of the two massive CLASH clusters they studied (although they found no correlation in the other). In the local Universe, [@Pilyugin:2017] found that galaxies in the densest environments have a median increase in Oxygen abundance of 0.05 dex with respect to the MZ relation, whilst [@Wu:2017] also showed that the median MZ relation residual is a weak function of environment, with a primary dependence on stellar mass. To investigate these effects in our own samples, we fit a linear model to the residuals around the MZ relation for the cluster and field samples separately. We define $\Delta(\mathrm{O/H})$ for each galaxy to be its gas-phase metallicity measurement minus the metallicity value from the MZ relation at the galaxy’s stellar mass. Our model is of the form $$\Delta(\mathrm{O/H}) = \alpha + \beta_1 (r/R_{200}) + \beta_2 \log(M_{*}/M_{\odot}) + \beta_3 \mathrm{SFR}/(M_{\odot}\mathrm{yr}^{-1})$$ which includes the projected distance from the cluster centre, $r$, (scaled by the $R_{200}$ value of the appropriate cluster), stellar mass and star formation rate as explanatory variables. We again use `Stan` to infer the posterior probability distribution of each coefficient, finding that the $\Delta(\mathrm{O/H})$ has no dependence on $M_*$ or SFR for the cluster or field samples. The only correlation coefficient more than one standard deviation away from zero occurs with projected distance for the cluster sample: $\beta_{1, \mathrm{cluster}} = -0.21 \pm 0.08$, significant at the 2.6$\sigma$ level. As expected, the field sample coefficient is consistent with zero: $\beta_{1, \mathrm{field}} = 0.04 \pm 0.04$. We show the correlation between projected distance and the MZ relation residual in the left-hand panel of Figure \[fig:MZ\_residual\_with\_cluster\_position\]. The right-hand panel of Figure \[fig:MZ\_residual\_with\_cluster\_position\] plots each galaxy in the cluster sample in phase-space, coloured by the galaxy’s residual above or below the MZ relation. We also show regions in phase space from the simulations of [@Rhee:2017], labelled A through E. Region A contains the largest fraction of “first infallers” into the cluster, whilst “recent” infallers (galaxies which have fallen into the cluster 0 - 3 Gyr ago) and intermediate infallers (3 - 6.5 Gyrs ago) tend to be found in regions B and C. Region E, containing “ancient” infallers (accreted &gt; 6.5 Gyrs ago), is underpopulated in our sample, showing that these galaxies are no longer visible in [H$\alpha$]{}. Region D tends to contain a combination of intermediate and ancient infallers, as well as a population of “backsplash” galaxies. We make further comment on these timescales in Section \[sec:Discussion\]. ISM conditions {#sec:EM_line_ratios} -------------- ![Top: A histogram of the electron number density in the mass-matched field (blue) and cluster (red) galaxy samples, estimated from the \[\]$\lambda 6716$/ \[\]$\lambda 6731$ line ratio and the calibration of [@Proxauf:2014]. Bottom: The \[\]$\lambda 6716$/\[\]$\lambda 6731$ ratio against $\log_{10}($\[\]$_{ \lambda6584}/$[H$\alpha$]{}). Each solid point has \[\] and \[\] S/N greater than 3, whilst faded points have S/N less than 3 in at least one line. We limit values of \[\]$\lambda 6716$/ \[\]$\lambda 6731$ to be between 0.44 and 1.44, the maximal values allowed by atomic physics.[]{data-label="fig:electron_density"}](f10.pdf){width="50.00000%"} The ratio \[\]$\lambda 6716$ / \[\]$\lambda 6731$ is a well-known electron number density diagnostic tool [e.g. @Osterbrock:2006; @Proxauf:2014]. The line ratio in the very low ($n_{\mathrm{e}}<10$ cm$^{-3}$) and high ($n_{\mathrm{e}}>10^{4}$ cm$^{-3}$) electron number density limits are 1.44 and 0.44, respectively. Assuming an electron temperature of 10,000 K and using the empirical calibration of [@Proxauf:2014], we find that the average electron number density in the stacked spectra from the cluster sample and mass matched field sample are $n_{e}=126^{+183}_{-116}$ cm$^{-3}$ and $<10^{+28}_{-0}$ cm$^{-3}$ respectively. Figure \[fig:electron\_density\] shows the derived electron number densities and a comparison between the \[\]$\lambda 6716$/\[\]$\lambda 6731$ and \[\]$_{ \lambda6584}$/[H$\alpha$]{} ratios. Among the individual objects, the distribution of electron number densities for the field and cluster galaxies values are similar, although interestingly we do observe proportionally fewer cluster galaxies with a small \[\]$\lambda 6716$/\[\]$\lambda 6731$ ratio compared to the field sample. These comparable electron number densities across both environments are in agreement with [@Kewley:2016], who found no difference between the electron number densities of a sample of 13 galaxies in a $z=2.1$ proto-cluster and a number of $z\approx2$ field galaxies. It should be noted, however, that the general properties of $z\approx2$ galaxies and the galaxy population at $0.3 < z < 0.6$ are very different- as are the environmental conditions in high-redshift proto-clusters and the massive intermediate-redshift clusters studied in this work. On the other hand, some studies have found an environmental dependence of the electron number density. [@Darvish:2015] observed galaxies residing in large-scale filamentary structures at a similar redshift to our sample ($z\approx0.5$), finding significantly smaller electron number densities than those in a similar sample residing in the field. Similarly, [@Sobral:2015] studied a merging cluster at $z\approx0.2$ and found that the cluster galaxies have electron number densities lower than that of field objects with similar properties. Conversely, at higher redshift [@Harshan:2020] found that galaxies in a $z=1.62$ proto-cluster have *higher* electron number densities than a those of a similar field sample, significant at the 2.6$\sigma$ level. Discussion {#sec:Discussion} ========== This work has two main conclusions. Firstly, the average [H$\alpha$]{} to continuum size ratio ([$r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }$]{}) of star-forming cluster galaxies is smaller than that of star-forming field galaxies which are matched in mass (Section \[sec:Ha\_Results\]). Secondly, the emission line ratios of the integrated spectra of cluster and field galaxies lead to identical mass-metallicity (MZ) relations. For the cluster sample, however, the residuals around the MZ relation are correlated with a galaxy’s cluster-centric distance, with galaxies closer to the centre of their cluster preferentially scattering to higher gas-phase metallicities by up to $\sim0.2-0.3$ dex above the relation (Section \[sec:MZ\_residuals\]). We also find a number of secondary conclusions: that galaxies in the cluster sample have an average [H$\alpha$]{} surface brightness within a 06 aperture which is marginally fainter than those in the field sample by 0.06 dex; and that the ISM conditions of the galaxies in the two samples are similar, with comparable electron densities (but tentative evidence that $n_e$ in the cluster galaxies is larger). Environmental effects --------------------- A number of environmental processes have been suggested to quench galaxy cluster members, with the most commonly proposed being “strangulation” and “ram-pressure stripping”. Strangulation occurs when a galaxy’s supply of cold gas residing in its halo is removed. In the absence of a supply of cold gas, galaxies continue to form stars until they run out of the fuel residing in their discs [e.g. @Larson:1980; @Peng:2015]. This leads to an overall reduction of the star-formation rate and [H$\alpha$]{} flux, but also an increase of the gas-phase metallicity (as the ISM is no longer diluted by an inflow of low-metallicity material). [@Maier:2016], for example, found strangulation to be consistent with their measurements of the chemical enrichment of the galaxies in a CLASH cluster at $z\approx0.4$ (and see also @Maier:2019a [@Maier:2019b; @Ciocan:2020] for studies at lower and higher redshift). We use the “bathtub” chemical evolution model of [@Lilly:2013], [@Peng:2014] and [@Peng:2015] to study the evolution of the metallicity and central surface brightness of a galaxy whose halo of cold gas has been removed. Following this model, at a time $t$ after the onset of disc strangulation ($t_q$) the galaxy’s increase in metallicity ($\Delta \log[Z(t)]$) is: $$\Delta \log[Z(t)] = \log\left(1+ \frac{y \varepsilon t}{Z(t_q)}\right)$$ where $y$ is the average metal yield per stellar generation and $\varepsilon$ is the star-formation efficiency. Furthermore, we can model the surface brightness within a 06 diameter aperture. Firstly, we assume that the gas follows an exponential surface brightness distribution with a scale length 1(approximately that found in the cluster and field galaxies). Secondly, we use the stellar masses derived in , assume a gas fraction and use the assumed exponential scale length to find a central gas surface density. Under the assumption that the star formation rate is related to the total gas mass ($\mathrm{SFR}=\varepsilon M_{g}$), we use the conversion of [@Hao:2011] and [@Murphy:2011] to derive an observed [H$\alpha$]{} flux from a star-formation rate (including a dust extinction of $A_V=0.5$ magnitudes, the average value found in the K-CLASH sample; see ). We then integrate the exponential disc profile within the aperture to obtain a surface brightness value. Over time, as the galaxy consumes its gas, the gas mass (and hence the SFR and [H$\alpha$]{} flux) will decrease exponentially according to Equation 15 of @Peng:2014: $\mathrm{SFR} \propto \exp(-\varepsilon(1-R)t)$. Here, $R$ is the fraction of mass of newly-formed stars which is (instantaneously) returned to the ISM via supernovae and stellar winds. Following [@Lilly:2013], we use values of $R=0.4$ and $y = 0.016$ (i.e. $y \approx 9$ in units of $12 + \log(\mathrm{O/H})$). We also use a gas fraction of 1, a star-formation efficiency of 0.1 Gyr$^{-1}$ and a solar metallicity at the start of strangulation ($Z(t_q) = 0.0134$; @Asplund:2009). Finally, based on the simulations of [@Rhee:2017] and the locations of our targets in Figure \[fig:MZ\_residual\_with\_cluster\_position\], we assume an infall time of both 1 and 3 Gyrs (i.e. 1 or 3 Gyrs since strangulation). We then derive the following: simulated surface brightnesses which match those shown in Figure \[fig:surface\_brightness\]; a decrease in central surface brightness after 1 (3) Gyrs of disc strangulation to be $\approx0.05~(0.15)$ dex; and a gas-phase metallicity increase of $\approx 0.1~(0.2)$ dex. These values are in agreement with our findings in this work, although we caution that this analysis is approximate in nature; we want to show that a simple strangulation model can explain our measurements, rather than perform a full quantitative analysis of our results. A simple toy model of strangulation can therefore account for both the $0.06$ dex decrease in central surface brightness as well as the $\approx0.2$ dex scatter to higher metallicities in the cluster sample galaxies, in a timescale which matches the cluster infall times implied by the simulations of [@Rhee:2017]. However, this simple model can *not* account for the decrease in half-light radii measured in Section \[sec:Ha\_Results\]; if gas is being consumed throughout the galaxy disc, to a first approximation the entire disc would become less star-forming but its scale length would remain unchanged [e.g. @Bekki:2002; @Boselli:2006b]. We note that our modelling in Section \[sec:Ha\_Results\] measured the *intrinsic* half-light radii of our targets, rather than the observed half-light radii. As such, even if a star-forming disc appears to become smaller (as its outer regions fell below our detection threshold), we will still recover the same half-light radius. Since we do in fact measure slightly smaller average [$r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }$]{} in the cluster sample galaxies, another process must be at play. We therefore conclude that our galaxy sample is also being affected by ram-pressure stripping. Ram-pressure stripping occurs when the pressure of the intra-cluster medium (ICM) “wind” experienced by a galaxy (due to its motion through the ICM) exceeds the galaxy’s gravitational restoring force and begins to remove material from its outskirts [@Gunn:1972]. It has also been shown to reduce the size of [H$\alpha$]{} discs. Of the many hydrodynamical simulations of ram-pressure stripping available in the literature, the study of [@Bekki:2014] is the most appropriate to compare to our work. They used hydrodynamical simulations of ram pressure to investigate the ratios of [H$\alpha$]{} to optical disc scale lengths of galaxies passing through dense environments. They found that whilst the precise evolution of $r_{e, {\rm{H}\alpha}}/r_{e, \mathrm{optical} }$ for individual star-forming galaxies in clusters depends sensitively on the cluster halo mass and galaxy disc inclination with respect to the cluster core, in general ram-pressure stripping reduces the $r_{e, {\rm{H}\alpha}}/r_{e, \mathrm{optical} }$ ratio in disc galaxies in massive clusters, which matches the findings in this work. They also found that the central star formation of these galaxies can be moderately enhanced (during pericentre passage, primarily for edge-on systems), suppressed or completely quenched (both after pericentre passage). We note that, by definition, the process of ram-pressure stripping also encompasses the effects of disc strangulation, and therefore our previous calculations regarding the increase in metallicity and reduction in surface brightness are still valid. We also note that it is only through the use of integral-field observations, which allow us to make measurements of the extent of [H$\alpha$]{} discs and gas-phase metallicities at the same time, that we have been able to come to this conclusion. We therefore strongly advocate the use of spatially resolved spectroscopy in future studies of environmental quenching processes. Local studies have recently unveiled the complexity of galaxies undergoing gas-stripping processes. The GASP project (GAs Stripping Phenomena in galaxies with MUSE; @Poggianti:2017 and references therein) studies 114 nearby galaxies in group and cluster environments which show evidence of recent stripping by ram pressure or turbulent processes. Truncated gas discs are common in the galaxies published so far, with most observations also showing evidence of spectacular tails of ionised gas [@Poggianti:2017; @Gullieuszik:2017; @Moretti:2018], although these are not ubiquitous [@Fritz:2017]. The 3$\sigma$ limiting surface brightness of the GASP observations is $2.5\times10^{-18}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-1}$ [@Poggianti:2017], with the tidal tails having surface brightnesses of $\lesssim1\times10^{-16}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-1}$ [e.g. @Gullieuszik:2017], below the average K-CLASH limiting surface brightness ($\approx1\times10^{-15}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-1}$; see Section \[sec:Ha\_line\_maps\]). We estimate that we would require nine hours on source to obtain a 3$\sigma$ detection of an [H$\alpha$]{} emission line at 1 $\mu$m with a surface brightness of $1\times10^{-16}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-1}$, 3.6 times longer than the typical K-CLASH galaxy observation. Electron number densities ------------------------- Despite the differences discussed above, the stacked spectra of cluster and field galaxies do not show a significant difference in electron number density measurements (although the uncertainty on these measurements are large). This finding would be in contrast to the work of [@Darvish:2015] and [@Sobral:2015], who find smaller electron densities in higher density environments. Our results could imply that the ram-pressure stripping has not yet directly impacted the ISM in the centres of our targets, where most of the emission line flux originates. As a galaxy moves through the dense ICM, its gaseous halo and disc are compressed towards the cluster centre and stripped on the trailing edge. Physically, one might expect to see a variation of the gas density between the leading and trailing edges of the object, which could be evident in the ratio of the \[\] doublet lines. Spatially resolved maps of the \[\] line ratio have been studied in local AGN, ultra luminous infrared galaxies (ULIRGs) and starburst galaxies [e.g. @Bennert:2006; @Sharp:2010; @Westmoquette:2011; @Kakkad:2018], but not for objects undergoing ram-pressure stripping, for which the GASP project provides an excellent data set. Conclusions {#sec:Conclusions} =========== Using IFU observations from the K-CLASH survey [@Tiley:2020], we have studied the effect of environment on star-forming galaxies in 4 CLASH clusters at $0.3<z<0.6$. We make comparisons to a mass-matched sample of galaxies residing in the field along nearby lines of sight at similar redshifts. We note that we cannot guarantee the purity of our cluster and field samples, as our simple cuts in projected radius and velocity do not account for the undoubtedly complex distribution of mass in each cluster. Our results, therefore, should be viewed as lower limits to the true differences between the cluster and field populations at these redshifts; any contamination (in either direction) will tend to homogenise our samples and reduce the diversity we find. Firstly, we infer the radial extent of ongoing star formation and older stellar populations by fitting exponential disc models to the [H$\alpha$]{} surface brightness distributiomns and Sérsic profiles to $R_\mathrm{c}$-band images. We have ensured that fitting the [H$\alpha$]{} maps with more general Sérsic profiles does not change our results. We then investigate the physical conditions of the ISM of galaxies in our sample by interpreting the emission line ratios measured in the integrated spectrum of each object, as well as stacking these spectra together to improve the signal-to-noise ratio and determine average properties. We summarise our conclusions below. 1. The average ratio of the half-light radius of the [H$\alpha$]{} emission and the $R_\mathrm{c}$-band continuum emission ([$r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }$]{}) across all galaxies is $1.14\pm0.06$, showing that star formation is generally taking place throughout stellar discs at these redshifts. 2. When separating by environment, we find an average [$r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }$]{}=0.96$\pm$0.09 for galaxies in the cluster sample and $1.22\pm0.08$ for galaxies in the field sample. $\langle$[$r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }$]{}$\rangle$ of the cluster galaxies is smaller than $\langle$[$r_{\mathrm{e}, {\rm{H}\alpha}}/r_{\mathrm{e}, R_{\mathrm{c} } }$]{}$\rangle$ of the field galaxies at the 98.5% confidence level. 3. The central surface brightnesses within a 06 diameter aperture are $\approx0.05$ dex fainter for galaxies in the cluster sample than those in the field sample. 4. Using the conversion of [@Pettini:2004], we measure a gas-phase metallicity for each object from the \[\]$\lambda6584$/[H$\alpha$]{} ratio. Both the cluster and field galaxies follow indistinguishable mass-metallicity (MZ) relations. 5. We do, however, see a correlation between a galaxy’s residual around the MZ relation and its projected radius (for galaxies in the cluster sample). Galaxies which are residing closer to the centre of their parent cluster tend to be more metal enriched (by up to $\approx0.2-0.3$ dex more than expected, given their mass). 6. Using the ratio of the \[\]$\lambda6716$/\[\]$\lambda6731$ lines, we infer the electron number density, $n_e$, in each galaxy. The distribution of these values are broadly similar between the field and cluster samples, although we do find a smaller proportion of cluster galaxies with very low $n_e$ compared to the field sample. In contrast to previous studies, the stacked cluster spectrum and the stacked field spectrum do not show a significant difference in electron number density (although the large uncertainties prevent us from drawing strong conclusions from this result). 7. We use the “bathtub” chemical evolution models of [@Lilly:2013] and [@Peng:2014] to show that removal of a galaxy’s halo of cold-gas (i.e. “disc strangulation”) can account for the fainter surface brightnesses and scatter to higher metallicities of galaxies in the cluster sample. However, since strangulation alone cannot explain the measured reduction in the intrinsic size of the [H$\alpha$]{} discs, we conclude that ram-pressure stripping must also be affecting the outskirts of our targets. Acknowledgements {#acknowledgements .unnumbered} ================ We thank the anonymous referee for their comments which greatly improved this work. This paper made use of the `astropy` <span style="font-variant:small-caps;">python</span> package [@astropy], as well as the `matplotlib` plotting software [@matplotlib] and the scientific libraries `numpy` [@numpy], `pandas` [@pandas] and `scipy` [@scipy]. This work was based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere under ESO programme(s) 097.A-0397, 098.A-0224, 099.A-0207 and 0100.A-0296. Parts of this research were supported by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013. SPV acknowledges support from a doctoral studentship from the UK Science and Technology Facilities Council (STFC) grant ST/N504233/1. ALT acknowledges support from a Forrest Research Foundation Fellowship, STFC (ST/L00075X/1 and ST/P000541/1), the ERC advanced Grant DUSTYGAL (321334), and an STFC Studentship. RLD acknowledges travel and computer grants from Christ Church, Oxford, and support from the Oxford Hintze Centre for Astrophysical Surveys, which is funded through generous support from the Hintze Family Charitable Foundation. MB was supported by the consolidated grants ‘Astrophysics at Oxford’ ST/H002456/1 and ST/K00106X/1 from the UK Research Council. Data Availability {#data-availability .unnumbered} ================= The data underlying this article are available in the article and in its online supplementary material. Variations of the KMOS PSF {#sec:KMOS_PSF_variation} ========================== The KMOS instrument comprises three separate spectrographs, which disperse the light from arms 1–8, 9–16 and 17–24 respectively. It has been reported that the PSF can vary slightly between the spectrographs [e.g. @Magdis:2016], and so it is important to investigate whether any systematic differences exist which could impact our results. Firstly, we investigate the wavelength dependence of the KMOS PSF. To do this, we use the reduced data cubes of stars targeted during the K-CLASH observations (see and Section \[sec:K-CLASH\_survey\]). We sum the flux in a window of width 0.05$\mu$m in the wavelength direction and fit a two-dimensional Gaussian function. This is repeated a further five times for increasing central wavelengths. We plot the full-width at half-maximum (FWHM) of the best-fitting Gaussian against wavelength for each spectrograph in Figure \[fig:spectrographs\_PSF\_variation\_with\_lambda\]. We find that the FWHM of the PSF improves by $\approx$01 from the low- to the high-wavelength end of the *IZ* band spectral range, although this effect is negligible for the results of this study. We also find that the PSFs of the three spectrographs are very comparable across the *IZ* band; the largest difference between spectrographs is only 003 at 1.05 $\mu$m. We also compare all of the star observations taken during the K-CLASH survey to investigate the stability of the KMOS PSF over time, as well as any deviation from circularity. To do so, we collapse each datacube along the wavelength direction and fit the resulting image with a two-dimensional Gaussian. The ratio of the best-fitting dispersion values in the $x$ and $y$ directions ($\sigma_x$ and $\sigma_y$) for each star observation is shown in Figure \[fig:spectrographs\_PSF\_variation\], plotted against the FWHM of the best-fitting Gaussian. We find that the PSF is generally circular, with deviations from circularity of at worst $\approx$25%. The average $\frac{\sigma_x}{\sigma_y}$ is 0.96, 1.04 and 1.06 for spectrographs 1, 2 and 3 respectively. Furthermore, we account for these small differences between spectrographs by using the PSF image observed with the same spectrograph as the science data whenever possible. ![The average FWHM of the PSF as a function of wavelength and colour-coded by KMOS spectrograph. A two-dimensional image was created from each star observed during the K-CLASH survey by collapsing the full data-cube in a window of 0.05 $\mu$m in the wavelength direction. We then fit a two-dimensional Gaussian function to derive the FWHM of each observation, and average together all observations in the same spectrograph. This procedure is then repeated a further five times for increasing central wavelength. The FWHM of the PSF varies by around 01 across the *IZ* spectral range, and by 0.03(at most) between spectrographs.[]{data-label="fig:spectrographs_PSF_variation_with_lambda"}](f11.pdf){width="50.00000%"} ![Ratio of the standard deviations ($\sigma_{x}$ and $\sigma_{y}$) of the two-dimensional Gaussian fit to each star observed during the K-CLASH survey, plotted against the FWHM of the two-dimensional Gaussian fit. Points are colour-coded according to the KMOS spectrograph they were observed with. Each PSF is generally circular, with deviation from circularity at worst $\approx25$%.[]{data-label="fig:spectrographs_PSF_variation"}](f12.pdf){width="50.00000%"} [H$\alpha$]{} Line Map Signal-to-Noise tests {#sec:SN_tests} ============================================ To test the robustness of our [H$\alpha$]{} spatial profile measurements, we create mock datacubes with model [H$\alpha$]{} surface brightness distributions and fit them in exactly the same way as real observations. We take the mock radial [H$\alpha$]{} surface brightness profiles to be those of exponential discs, using [<span style="font-variant:small-caps;">imfit</span>]{} to create a two-dimensional surface brightness distributions. The disc models have six free parameters: - the coordinates of the image centre, $x_0$ and $y_0$ - the observed ellipticity, $\epsilon$, defined as $1- \frac{b}{a}$ (where $a$ and $b$ are respectively the semi-major and semi-minor axes of an ellipse) - the disc position angle, PA, measured counter-clockwise from the positive $y$ axis of the image - the central intensity, $I_0$ - the exponential disc scale length, $R_{\rm{disc}}$[^18] We then create a datacube by assigning a mock spectrum to each pixel in the image. We model the continuum light as a second order polynomial, and superimpose a single Gaussian emission line at a wavelength corresponding to [H$\alpha$]{} emission at $z=0.4$. This emission line template has velocity dispersion of 100 [kms$^{-1}$]{} and a peak flux equal to twice the continuum level. The absolute normalisation of each spectrum is defined by the value of the model surface brightness distribution at that spaxel. Random noise is then added. This is accomplished by taking the mean spectrum of the model cube from within its half light radius and dividing by an input S/N value to create a noise “spectrum”. At each wavelength slice, random numbers are then drawn from a Gaussian distribution (centred on zero with width corresponding to the value in the noise spectrum) and added to the cube. This process implies that the *average* S/N value of pixels within the model half-light radius of each datacube is equal to the requested S/N ratio; the S/N at the centre of the image and at the edges will be respectively higher and lower than this average. Finally, the mock cube is convolved with the KMOS PSF and the parameters of [H$\alpha$]{} surface brightness distribution are measured in the manner discussed in Section \[sec:Halpha\_size\]. We note that these simulations do not model the effect of sky-subtraction residuals on our measurement process. ![Recovery of the [H$\alpha$]{} profile parameters of mock observations, at various input S/N ratios of the H$\alpha$ images. Note the image S/N value is the *average* S/N value of pixels within the best-fitting half-light radius. The dashed line represents an average image S/N of 2, and the histogram shows the S/N distribution of our observed [H$\alpha$]{} images described in Section \[sec:Ha\_line\_maps\]. See Appendix \[sec:SN\_tests\] for details.[]{data-label="fig:SN_tests"}](f13.pdf){width="50.00000%"} ![Recovering the [H$\alpha$]{} disc scale length ($R_{\mathrm{disc}}$) of mock observations, for various scale lengths and S/N ratios. $R_{\rm{in}}$ refers to the true value of the model exponential scale length whilst $R_{\rm{out}}$ refers to the measured value. The dashed line represents the size of the KMOS IFU for a source placed in the centre (7 pixels). The top panel shows the difference between the input and output scale lengths, whilst the bottom shows their ratio.[]{data-label="fig:SN_tests_2"}](f14.pdf "fig:"){width="50.00000%"} ![Recovering the [H$\alpha$]{} disc scale length ($R_{\mathrm{disc}}$) of mock observations, for various scale lengths and S/N ratios. $R_{\rm{in}}$ refers to the true value of the model exponential scale length whilst $R_{\rm{out}}$ refers to the measured value. The dashed line represents the size of the KMOS IFU for a source placed in the centre (7 pixels). The top panel shows the difference between the input and output scale lengths, whilst the bottom shows their ratio.[]{data-label="fig:SN_tests_2"}](f15.pdf "fig:"){width="50.00000%"} ![Recovering the [H$\alpha$]{} disc scale length ($R_{\mathrm{disc}}$) of mock observations, for various locations on the IFU and S/N values. The model is a source with $R_{\rm{disc}}=3$ pixels, $\epsilon=0.2$ and position angle 135.[]{data-label="fig:SN_tests_3"}](f16.pdf){width="50.00000%"} The results of our tests are shown in Figures \[fig:SN\_tests\], \[fig:SN\_tests\_2\] and \[fig:SN\_tests\_3\]. Figure \[fig:SN\_tests\] shows that estimating $R_{\rm{disc}}$ requires a larger S/N than simply finding the ($x_0, y_0$) coordinate of the [H$\alpha$]{} flux centre, but is easier than constraining the galaxy ellipticity ($\epsilon$). A histogram of the S/N ratios of our data is shown in orange. We find that to recover $R_{\rm{disc}}$ to an accuracy better than 10%, we require the average of the S/N of the integrated [H$\alpha$]{} flux in all spaxels within the best-fitting half-light radius to be greater than 2. For an exponential surface brightness profile, this implies that the central S/N ratio is $\approx10$. To investigate the effect of the finite size of each KMOS IFU, we placed a mock galaxy in the centre of an IFU and varied the input $R_{\rm{disc}}$, with a range of average S/N ratios. Figure \[fig:SN\_tests\_2\] shows comparisons of the input ($R_{\rm{in}}$) and recovered ($R_{\rm{out}}$) disc scale lengths as a function of $R_{\rm{in}}$ and S/N. For reference, the average half-light radius at $z=0.5$ is approximately 1[4 kpc; @Paulino-Afonso:2017], which corresponds to a disc scale-length of $\approx$06 ($\approx$ 3 pixels). We find that our ability to recover $R_{\rm{disc}}$ is good to better than 10% for high S/N data at all values of $R_{\rm{in}}$ , showing that the limited size of the KMOS field of view does not hinder these measurements. At S/N=2, we recover galaxies almost three times the average disc-scale length (11 pixels) with a 25-50% uncertainty. Finally, we assess the impact of mis-centred [H$\alpha$]{} emission by placing our mock galaxy at various positions across the IFU. We create a model galaxy with $\epsilon=2$, $R_{\rm{disc}}=3$ pixels and position angle 135 and then place it at various $(x, y)$ locations. We measure the best-fitting $R_{\rm{disc}}$ 10 times, and show the average ratio of $R_{\rm{out}}$/$R_{\rm{in}}$ for S/N of 2, 5 and 10 in Figure \[fig:SN\_tests\_3\]. We find that small offsets from the centre have no effect on our ability to recover $R_{\rm{disc}}$. As expected, the largest uncertainties occur when the mis-centring is large (i.e. the object is in a corner of the IFU). We ensured this did not occur for any of our observations. Whilst not exhaustive, these tests show that we can robustly measure $R_{\rm{disc}}$ for a variety of S/N ratios, galaxy sizes and locations on the IFU, and allow us to make an informed decision on the minimum S/N ratio to use in our analyses. It should nevertheless be stressed that these tests are conducted under more favourable conditions than the real observations and analyses, since they are fitting a model which we know to be the true representation of the data, and do not include systematic uncertainties such as sky line residuals in the spectral dimension or “hot” pixels in the [H$\alpha$]{} images. Table of Measurements ===================== We present all measurements used in this work in Table \[tab:all\_KCLASH\_values\]. This table is available online in machine-readable format. \[tab:all\_KCLASH\_values\] ID 41309 41423 44143 45189 45677 46622 ----------------------------------------------------------------------------------- ------------- ------------- ---------- ------------- ---------- ------------- RA **(1**) 197.739 197.781 197.797 197.691 197.795 197.678 Dec **(2**) -3.25943 -3.26086 -3.25042 -3.24392 -3.243 -3.23713 Observation Field **(3**) MACS1311 MACS1311 MACS1311 MACS1311 MACS1311 MACS1311 Spectroscopic Redshift **(4**) 0.271578 0.45068 0.36509 0.435555 0.48838 0.437338 Detected H$\alpha$? **(5**) True False False True False True Cluster sample **(6**) False False False False False False Mass-matched field sample **(7**) True False False True False True AGN flag **(8**) False False False False False False $I_{\mathrm{H}\alpha}$ (D0.6; erg s$^{-1}$ arcsec$^{-1}$) **(9**) 1.7911e-17 4.02604e-18 0 1.37713e-17 0 1.26111e-17 $I_{\mathrm{H}\alpha}$ uncertainty (D0.6; erg s$^{-1}$ arcsec$^{-1}$) **(10**) 5.91279e-18 3.58005e-18 0 2.79143e-18 0 2.24848e-18 $\log_{10} M_{*}$ $(M_{\odot})$ **(11**) 10.0449 10.396 10.1432 10.3793 10.6962 10.173 Cluster-centric velocity ([kms$^{-1}$]{}) **(12**) -44632.1 -8692.74 -25867.5 -11727.8 -1127.82 -11370 Projected radius () **(13**) 301.678 310.833 297.111 337.239 270.619 355.75 Projected radius/$R_{200}$ () **(14**) 1.33089 1.37128 1.31074 1.48777 1.19386 1.56943 H$\alpha$ $r_{\mathrm{e}}$ () **(15**) 1.59826 — — 0.461888 — 0.728819 Stellar $r_{\mathrm{e}}$ () **(16**) 0.582627 0.395013 1.07183 0.511791 0.309265 0.678999 H$\alpha$-to-stellar size ratio **(17**) 2.7432 — — 0.902494 — 1.07337 Reliable size measurement **(18)** True False False True False True $\chi^2$- H$\alpha$ image fitting **(19**) 3.13659 — — 3.1118 — 2.42207 $\chi^2$- stellar image fitting **(20**) 1.20319 1.34134 2.51276 2.38855 3.52322 2.9806 Mean S/N (H$\alpha$ image fitting) **(21**) 2.48416 — — 3.69481 — 4.02895 Sérsic index (stars) **(22**) 0.9406 2.73483 1.17899 1.14667 3.4329 0.56021 Sérsic index uncertainty (stars) **(23**) 0.031186 0.095936 0.017757 0.059112 0.14053 0.01166 H$\alpha$ line S/N (D2.4) **(24**) 12.6412 4.37531 0 0 0 25.2812 `ppxf` [H$\alpha$]{} flux (D2.4; erg s$^{-1}$ arcsec$^{-1}$) **(25**) 1.93485e-16 — — 6.01279e-17 — 2.18856e-16 `ppxf` [H$\alpha$]{} flux uncertainty (D2.4; erg s$^{-1}$ arcsec$^{-1}$) **(26**) 2.10491e-13 — — 0 — 7.85701e-18 `ppxf` flux (D2.4; erg s$^{-1}$ arcsec$^{-1}$) **(27**) 3.39716e-18 — — 8.66973e-18 — 1.22466e-16 `ppxf` flux uncertainty (D2.4; erg s$^{-1}$ arcsec$^{-1}$) **(28**) 2.11108e-13 — — 0 — 1.31339e-17 `ppxf` ratio (D2.4) **(29**) – — — – — 1.43 `ppxf` ratio uncertainty (D2.4) **(30**) – — — – — 0.0967388 `ppxf` flux total (D2.4; erg s$^{-1}$ arcsec$^{-1}$) **(31**) 0 — — 3.00684e-17 — 1.06618e-16 `ppxf` flux total uncertainty (D2.4; erg s$^{-1}$ arcsec$^{-1}$) **(32**) 7.54642e-14 — — 0 — 1.02338e-17 `ppxf` $\chi^2$ (D2.4) **(33**) 1.87915 — — 4.4041 — 5.10002 `ppxf` velocity dispersion (D2.4; [kms$^{-1}$]{}) **(34**) 68.8751 — — 45.3285 — 100.05 $12 + \log_{10} (\mathrm{O}/\mathrm{H})$ (D2.4) **(35**) 7.88033 — — — — 8.76653 $12 + \log_{10} (\mathrm{O}/\mathrm{H})$ uncertainty (D2.4) **(36**) 1.08415 — — — — 0.0601683 \[lastpage\] [^1]: Contact e-mail: <sam.vaughan@sydney.edu.au> (SPV) [^2]: Only one, MACS 2129, is also studied in [@Vulcani:2016] [^3]: <https://mc-stan.org/> [^4]: <https://archive.stsci.edu/prepds/clash/> [^5]: Optical imaging is generally from Suprime-Cam [@Miyazaki:2002], but is supplemented by data from the ESO Wide Field Imager [@Baade:1999] and the Magellan Inamori Magellan Areal Camera (IMACS) in MACS 1311 where only $R_\mathrm{c}$-band Suprime-Cam imaging was available. [^6]: <http://wise2.ipac.caltech.edu/docs/release/allwise/> [^7]: <https://irsa.ipac.caltech.edu/data/SPITZER/CLASH/> [^8]: where $k_{\rm{B}}$ is the Boltzmann constant, $T$ is the gas temperature $m_p$ is the mass of a proton and $\mu=0.6715$ is the mean molecular weight in atomic mass units. [^9]: Note that the “field” sample discussed in encompasses all 128 galaxies. [^10]: Note that this S/N definition is different to the one described in Section \[sec:K-CLASH\_survey\] to measure the [H$\alpha$]{} S/N in an integrated spectrum. [^11]: <http://www.mpe.mpg.de/~erwin/code/imfit/> [^12]: <https://archive.stsci.edu/prepds/clash/> [^13]: To ensure this constraint does not bias our results, we also conduct the same analysis without making this cut; see Section \[sec:Ha\_Results\] [^14]: See the note in the introduction [^15]: For a representative extinction of A$_{\mathrm{V}}$=1 mag and a source at at $z=0.6$, [H$\alpha$]{} is reddened by 1.87 mag compared to 1.36 mag for the continuum [^16]: <https://pypi.org/project/ppxf/> [^17]: defined as a normal distribution for positive values of the dependent variable and zero otherwise. [^18]: We note that in Section \[sec:Halpha\_size\] we have converted all measured values of exponential scale length into half-light radii, $r_{\rm{e, H}\alpha}$, by performing a curve of growth analysis in a circular aperture on the intrinsic (unconvolved) best-fitting model.

--- abstract: 'The charged current antikaon production off nucleons induced by antineutrinos is studied at low and intermediate energies. We extend here our previous calculation on kaon production induced by neutrinos. We have developed a microscopic model that starts from the SU(3) chiral Lagrangians and includes background terms and the resonant mechanisms associated to the lowest lying resonance in the channel, namely, the $\Sigma^*(1385)$. Our results could be of interest for the background estimation of various neutrino oscillation experiments like MiniBooNE and SuperK. They can also be helpful for the planned $\bar \nu-$experiments like MINER$\nu$A, NO$\nu$A and T2K phase II and for beta-beam experiments with antineutrino energies around 1 GeV.' author: - 'M. Rafi' - 'I.' - 'M. Sajjad' - 'M. J.' title: '$\bar\nu$ induced $\bar K$ production off the nucleon' --- Introduction {#Introduction} ============ Weak interaction experiments with neutrino energies around 1 GeV are quite sensitive to the neutrino oscillation parameters and as a consequence many experiments like MiniBooNE, SciBooNE, K2K, T2K, NO$\nu$A, etc. explore this energy range. Although many interesting results can be obtained without a detailed knowledge of the various processes used for the neutrino detection or the neutrino flux, a reliable estimate of the $\nu-$N cross section for various processes is mandatory to carry out a precise analysis of the measurements. Among these processes, strangeness conserving ($\Delta S=0$) weak interactions involving quasielastic production of leptons induced by charged as well as neutral weak currents have been widely studied [@Boyd:2009zz; @Leitner:2006ww; @Leitner:2006sp; @Benhar:2010nx; @Martini:2010ex; @Amaro:2010sd; @Nieves:2011pp]. Much work has also been done to understand one pion production in the weak sector [@AlvarezRuso:1998hi; @Sato:2003rq; @Graczyk:2009qm; @Hernandez:2007qq; @Leitner:2008wx; @Leitner:2010jv; @Hernandez:2010bx; @Lalakulich:2010ss]. There are other inelastic reactions like hyperon and kaon production ($\Delta S=\pm 1$) that could also be measured even at quite low energies. However, very few calculations study these processes [@VicenteSingh; @Mintz:2007zz; @Dewan; @Shrock; @Amer:1977fy; @Mart:2009; @RafiAlam:2010kf]. This is partly justified by their small cross sections due to the Cabibbo suppression. As a result of this situation, the Monte Carlo generators used in the analysis of the current experiments apply models that are not well suited to describe the strangeness production at low energies. NEUT, for example, used by Super-Kamiokande, K2K, SciBooNE and T2K, only considers associated production of kaons within a model based on the excitation and later decay of baryonic resonances and from deep inelastic scattering (DIS) [@Hayato:2009zz]. Similarly, other neutrino event generators like NEUGEN [@Gallagher:2002sf], NUANCE [@Casper:2002sd] (see also discussion in Ref. [@Zeller:2003ey]) and GENIE [@Andreopoulos:2009rq] do not consider single hyperon/kaon production. Recently we have studied single kaon production induced by neutrinos at low and intermediate energies [@RafiAlam:2010kf] using Chiral Perturbation Theory ($\chi$PT). We found that up to E$_{\nu_\mu}\approx 1.2$ GeV, single kaon production dominates over the associated production of kaons along with hyperons which is mainly due to its lower threshold energy. In this work, we extend our model to include weak single antikaon production off nucleons. The theoretical model is necessarily more complicated than for kaons because resonant mechanisms, absent for the kaon case, could be relevant. On the other hand, the threshold for associated antikaon production corresponds to the $K-\bar K$ channel and it is much higher than for the kaon case (KY). This implies that the process we study is the dominant source of antikaons for a wide range of energies. The study may be useful in the analysis of antineutrino experiments at MINER$\nu$A, NO$\nu$A, T2K and others. For instance, MINER$\nu$A has plans to investigate several strange particle production reactions with both neutrino and antineutrino beams [@Solomey:2005rs] with high statistics. Furthermore, the T2K experiment [@Kobayashi:2005] as well as beta beam experiments [@Mezzetto:2010] will work at energies where the single kaon/antikaon production may be important. We introduce the formalism in Sec. \[Formalism\]. In Sec. \[Results and Discussion\], we present the results, discussions and conclusions. Formalism {#Formalism} ========= The basic reaction for antineutrino induced charged current antikaon production is $$\label{reaction} \bar \nu_{l}(k) + N(p) \rightarrow l(k^{\prime}) + N^\prime(p^{\prime}) + \bar K(p_{k}) ,$$ where $l=e^+,\mu^+$ and $ N \& N^\prime $ are nucleons. The expression for the differential cross section in the laboratory frame for the above process is given by $$\begin{aligned} \label{sigma_inelas} d^{9}\sigma &=& \frac{1}{4 M E(2\pi)^{5}} \frac{d{\vec k}^{\prime}}{ (2 E_{l})} \frac{d{\vec p\,}^{\prime}}{ (2 E^{\prime}_{p})} \frac{d{\vec p}_{k}}{ (2 E_{k})} \delta^{4}(k+p-k^{\prime}-p^{\prime}-p_{k})\bar\Sigma\Sigma | \mathcal M |^2,\end{aligned}$$ where $ k( k^\prime) $ is the momentum of the incoming(outgoing) lepton with energy $E( E^\prime)$, $p( p^\prime)$ is the momentum of the incoming(outgoing) nucleon. The kaon 3-momentum is $\vec{p}_k $ having energy $ E_k $, $M$ is the nucleon mass, $ \bar\Sigma\Sigma | \mathcal M |^2 $ is the square of the transition amplitude averaged(summed) over the spins of the initial(final) state. It can be written as $$\label{eq:Gg} \mathcal M = \frac{G_F}{\sqrt{2}}\, j_\mu J^{\mu}=\frac{g}{2\sqrt{2}}j_\mu \frac{1}{M_W^2} \frac{g}{2\sqrt{2}}J^{\mu},$$ where $j_\mu$ and $ J^{\mu}$ are the leptonic and hadronic currents respectively, $G_F=\sqrt{2} \frac{g^2}{8 M^2_W}$ is the Fermi coupling constant, $g$ is the gauge coupling and $M_W$ is the mass of the $W$-boson. The leptonic current can be readily obtained from the standard model Lagrangian coupling the $W$ bosons to the leptons $${\cal L}=-\frac{g}{2\sqrt{2}}\left[j^\mu{ W}^-_\mu+h.c.\right].$$ We construct a model including non resonant terms and the decuplet resonances, that couple strongly to the pseudoscalar mesons. The same approach successfully describes the pion production case (see for example Ref. [@Hernandez:2007qq]). The channels that contribute to the hadronic current are depicted in Fig. \[fg:terms\]. There are s-channels with $\Sigma,\Lambda$(SC) and $\Sigma^*$(SCR) as intermediate states, a kaon pole (KP) term, a contact term (CT), and finally a meson ($\pi$P,$\eta$P) exchange term. For these specific reactions there are no u-channel processes with hyperons in the intermediate state. ![Feynman diagrams for the process $\bar \nu N\rightarrow l N^\prime \bar K$. First row from left to right: s-channel $\Sigma,\Lambda $ propagator (labeled SC in the text), s-channel $\Sigma^*$ Resonance (SCR), second row: kaon pole term (KP); Contact term (CT) and last row: Pion(Eta) in flight ($ \pi P/ \eta P $). []{data-label="fg:terms"}](feynman.eps){width="80.00000%" height=".4\textwidth"} The contribution coming from different terms can be obtained from the $\chi$PT Lagrangian. We follow the conventions of Ref. [@Scherer:2002tk] to write the lowest-order SU(3) chiral Lagrangian describing the interaction of pseudoscalar mesons in the presence of an external current, $$\label{eq:lagM} {\cal L}_M^{(2)}=\frac{f_\pi^2}{4}\mbox{Tr}[D_\mu U (D^\mu U)^\dagger] +\frac{f_\pi^2}{4}\mbox{Tr}(\chi U^\dagger + U\chi^\dagger),$$ where the parameter $f_\pi=92.4$ MeV is the pion decay constant, $U(x)=\exp\left(i\frac{\phi(x)}{f_\pi}\right)$ is the SU(3) representation of the meson fields $\phi(x)$ and $D_\mu U$ is its covariant derivative $$\begin{aligned} D_\mu U&\equiv&\partial_\mu U -i r_\mu U+iU l_\mu\,.\end{aligned}$$ For the charged current case the left and right handed currents $l_\mu$ and $r_\mu$ are given by $$r_\mu=0,\quad l_\mu=-\frac{g}{\sqrt{2}} ({W}^+_\mu T_+ + {W}^-_\mu T_-),$$ with $W^\pm$ the $W$ boson fields and $$T_+=\left(\begin{array}{rrr}0&V_{ud}&V_{us}\\0&0&0\\0&0&0\end{array}\right);\quad T_-=\left(\begin{array}{rrr}0&0&0\\V_{ud}&0&0\\V_{us}&0&0\end{array}\right).$$ Here, $V_{ij}$ are the elements of the Cabibbo-Kobayashi-Maskawa matrix. The second term of the Lagrangian of Eq. \[eq:lagM\], that incorporates the explicit breaking of chiral symmetry coming from the quark masses [@Scherer:2002tk], is not relevant for our study. The lowest-order chiral Lagrangian describing the interaction between baryon-meson octet in the presence of an external weak current can be written in terms of the SU(3) matrix as $$\label{eq:lagB} {\cal L}^{(1)}_{MB}=\mbox{Tr}\left[\bar{B}\left(i\D -M\right)B\right] -\frac{D}{2}\mbox{Tr}\left(\bar{B}\gamma^\mu\gamma_5\{u_\mu,B\}\right) -\frac{F}{2}\mbox{Tr}\left(\bar{B}\gamma^\mu\gamma_5[u_\mu,B]\right),$$ where $M$ denotes the mass of the baryon octet, and the parameters $D=0.804$ and $F=0.463$ can be determined from the baryon semileptonic decays [@Cabibbo:2003cu]. The covariant derivative of $B$ is given by $$D_\mu B=\partial_\mu B +[\Gamma_\mu,B],$$ with $$\Gamma_\mu=\frac{1}{2}\left[u^\dagger(\partial_\mu-ir_\mu)u +u(\partial_\mu-il_\mu)u^\dagger\right],$$ where we have introduced $u^2=U$. Finally, $$\label{Eq:octet-weak_int} u_\mu= i\left[u^\dagger(\partial_\mu-i r_\mu)u-u(\partial_\mu-i l_\mu)u^\dagger\right].$$ The next order meson baryon Lagrangian contains many new terms (see for instance Ref. [@Oller:2006yh]). Their importance for kaon production will be small at low energies and there are some uncertainties in the coupling constants. Nonetheless, for consistency with previous calculations, we will include the contribution to the weak magnetism coming from the pieces $${\cal L}^{(2)}_{MB}= d_5 \mbox{Tr}\left(\bar{B}[f_{\mu\nu}^+,\sigma^{\mu\nu}B]\right)+ d_4 \mbox{Tr}\left(\bar{B}\{f_{\mu\nu}^+,\sigma^{\mu\nu}B\}\right)+\dots,$$ where the tensor $f_{\mu\nu}^+$ can be reduced for our study to $$f_{\mu\nu}^+=\partial_\mu l_\nu-\partial_\nu l_\mu -i [l_\mu,l_\nu].$$ In this case, the coupling constants are fully determined by the proton and neutron anomalous magnetic moments. The same approximation has also been used in calculations of single pion [@Hernandez:2007qq] and kaon production  [@RafiAlam:2010kf] induced by neutrinos. As it is the case for the $\Delta(1232)$ in pion production, we expect that the weak excitation of the $\Sigma^*(1385)$ resonance and its subsequent decay in $NK$ may be important. The lowest order SU(3) Lagrangian coupling the pseudoscalar mesons with decuplet-octet baryons in presence of external weak current is given by $${\cal L}_{dec} = {\cal C} \left( \epsilon^{abc} \bar T^\mu_{ade} u_{\mu,b}^d B_c^e + \, h.c. \right), \label{eq:dec_lag}$$ where $T^\mu$ is the SU(3) representation of the decuplet fields, $a-e$ are flavour indices[^1], $B$ corresponds to the baryon octet and $u_\mu$ is the SU(3) representation of the pseudoscalar mesons interacting with weak left $l_\mu$ and right $r_\mu$ handed currents (See Eq. \[Eq:octet-weak\_int\]). The parameter ${\cal C}\simeq 1$ has been fitted to the $\Delta(1232)$ decay-width. The spin 3/2 propagator for $\Sigma^*$ is given by $$G^{\mu\nu}(P)= \frac{P^{\mu\nu}_{RS}(P)}{P^2-M_{\Sigma^*}^2+ i M_{\Sigma^*} \Gamma_{\Sigma^*}}, \qquad$$ where $P=p+q$ is the momentum carried by the resonance, $q=k-k^\prime$ and $P^{\mu \nu}_{RS}$ is the projection operator $$P^{\mu\nu}_{RS}(P)= \sum_{spins} \psi^{\mu} \bar \psi^{\nu} =- (\slashchar{P} + M_{\Sigma^*}) \left [ g^{\mu\nu}- \frac13 \gamma^\mu\gamma^\nu-\frac23\frac{P^\mu P^\nu}{M_{\Sigma^*}^2}+ \frac13\frac{P^\mu \gamma^\nu-P^\nu \gamma^\mu }{M_{\Sigma^*}}\right], \label{eq:rarita_prop}$$ with $M_{\Sigma^*}$ the resonance mass and $\psi^{\mu}$ the Rarita-Schwinger spinor. The $\Sigma^*$ width obtained using the Lagrangian of Eq. \[eq:dec\_lag\] can be written as $$\begin{aligned} \Gamma_{\Sigma^*}&=&\Gamma_{\Sigma^*\rightarrow \Lambda \pi} + \Gamma_{\Sigma^*\rightarrow \Sigma \pi}+ \Gamma_{\Sigma^*\rightarrow N \bar{K}}\; , \label{eq:width}\end{aligned}$$ where $$\begin{aligned} \Gamma_{\Sigma^* \rightarrow Y,\, meson}&=&\frac{C_Y}{192\pi}\left(\frac{\cal C}{f_\pi}\right)^2 \frac{(W+M_Y)^2-m^2}{W^5}\lambda^{3/2}(W^2,M_Y^2,m^2) \, \Theta(W-M_Y-m).\end{aligned}$$ Here, $m,\, M_Y$ are the masses of the emitted meson and baryon. $\lambda(x,y,z)=(x-y-z)^2-4yz$ and $\Theta$ is the step function. The factor $C_Y$ is 1 for $\Lambda$ and $\frac23$ for $N $ and $\Sigma$. Using symmetry arguments, the most general $W^- N \rightarrow \Sigma^*$ vertex can be written in terms of a vector and an axial-vector part as, $$\begin{aligned} \label{eq:delta_amp} \langle \Sigma^{*}; P= p+q\, | V^\mu | N; p \rangle &=& V_{us} \bar\psi_\alpha(\vec{P} ) \Gamma^{\alpha\mu}_V \left(p,q \right) u(\vec{p}\,), \nonumber \\ \langle \Sigma^{*}; P= p+q\, | A^\mu | N; p \rangle &=& V_{us} \bar \psi_\alpha(\vec{P} ) \Gamma^{\alpha\mu}_A \left(p,q \right) u(\vec{p}\,),\end{aligned}$$ where $$\begin{aligned} \Gamma^{\alpha\mu}_V (p,q) &=& \left [ \frac{C_3^V}{M}\left(g^{\alpha\mu} \slashchar{q}- q^\alpha\gamma^\mu\right) + \frac{C_4^V}{M^2} \left(g^{\alpha\mu} q\cdot P- q^\alpha P^\mu\right) + \frac{C_5^V}{M^2} \left(g^{\alpha\mu} q\cdot p- q^\alpha p^\mu\right) + C_6^V g^{\mu\alpha} \right ]\gamma_5 \nonumber\\ \Gamma^{\alpha\mu}_A (p,q) &=& \left [ \frac{C_3^A}{M}\left(g^{\alpha\mu} \slashchar{q}- q^\alpha\gamma^\mu\right) + \frac{C^A_4}{M^2} \left(g^{\alpha\mu} q\cdot P- q^\alpha P^\mu\right) + C_5^A g^{\alpha\mu} + \frac{C_6^A}{M^2} q^\mu q^\alpha \right ]. \label{eq:del_ffs} \end{aligned}$$ Our knowledge of these form factors is quite limited. The Lagrangian of Eq. \[eq:dec\_lag\] gives us only $C_5^A(0)=-2{\cal{C}}/\sqrt{3}$ (for the $\Sigma^{*-}(1385)$ case). However, using SU(3) symmetry we can relate all other form factors to those of the $\Delta(1232)$ resonance, such that $C_i^{\Sigma*^-}/C_i^{\Delta^{+}}=-1$ and $C_i^{\Sigma*^-}/{C_i^{\Sigma*^0}}=\sqrt{2}$. See Refs. [@AlvarezRuso:1998hi; @Paschos:2005; @Hernandez:2007qq; @Leitner:2008ue; @Hernandez:2010bx] for details of the $W N\Delta$ form-factors. In the $\Delta$ case, the vector form factors are relatively well known from electromagnetic processes and there is some information on the axial ones from the study of pion production. We will use the same set as in Ref. [@Hernandez:2007qq; @Hernandez:2010bx], where pion production induced by neutrinos has been studied, except for $C_5^A(0)$, obtained directly from the Lagrangian and $C_6^A$. These latter two form factors are related by PCAC so that $C_6^A=C_5^A M^2/(m_K^2-q^2)$. In our model, we use an SU(3) symmetric Lagrangian. The only SU(3) breaking comes from the use of physical masses. This is expected to be a good description for the background terms, as it was discussed for the kaon production induced by neutrinos in Ref. [@RafiAlam:2010kf]. Little is known about the SU(3) breaking for the axial couplings of the baryon decuplet, but only a small breaking has been found for their electromagnetic properties [@hep-ph/9211247; @arXiv:0907.0631]. Therefore, we can expect a similarly small uncertainty in the size of the $\Sigma^*(1385)$ contribution. Even from relatively low neutrino energies, other baryonic resonances, beyond the $\Sigma^*(1385)$, could contribute to the cross section, as they are close to the kaon nucleon threshold. However, their weak couplings are basically unknown. Also, the theoretical estimations of these couplings are still quite uncertain. Nonetheless, recent advances on the radiative decays of these resonances, both experimental and theoretical (see, e.g., Refs. [@Doring:2006ub; @Taylor:2005zw]) are very promising and may help to develop a more complete model in the future. Finally, we consider the $q^2$ dependence of the weak current couplings provided by the chiral Lagrangians. In this work, we follow the same procedure as in Ref. [@RafiAlam:2010kf][^2] and adopt a global dipole form factor $ F(q^2)=1/(1-q^2/M_F^2)^2, $ with a mass $M_F\simeq 1$ GeV that multiplies all the hadronic currents, except the resonant one, that has been previously discussed. Its effect, that should be small at low neutrino energies, will give an idea of the uncertainties of the calculation and will be explored in the next section. Detailed expressions of the resulting hadronic currents $J^{\mu}$ containing both background and resonant terms are listed in the appendix \[app:amplitude\]. Results and Discussion {#Results and Discussion} ====================== We consider the following strangeness changing ($| \Delta S | = 1$) charged-current reactions: $$\begin{aligned} \label{processes} \bar \nu_l + p &\rightarrow & l^+ + K^- + p \nonumber\\ \bar \nu_l + p &\rightarrow & l^+ +\bar K^0 + n \nonumber\\ \bar \nu_l + n &\rightarrow & l^+ + K^- + n \,.\end{aligned}$$ ![Cross-section for the processes $\bar\nu_\mu N\rightarrow \mu^+ N^\prime \bar K$ and $\bar\nu_e N\rightarrow e^+ N^\prime \bar K$ as a function of the antineutrino energy[]{data-label="fg:xsec_all_chnl"}](Totalmue.eps){width="60.00000%"} In Fig. \[fg:xsec\_all\_chnl\], we show their total cross section for electronic and muonic antineutrinos as a function of energy. We obtain similar values to the cross sections of kaon production induced by neutrinos of Ref. [@RafiAlam:2010kf], even when there are no resonant contributions. The electronic antineutrino cross sections are slightly larger, but they do not present any other distinguishing feature. For all channels, the cross sections are very small, as compared to other processes induced by antineutrinos at these energies, like pion production, due to the Cabibbo suppression and to the smallness of the available phase space. Nonetheless, the reactions we have studied are the main source of antikaons for a wide range of neutrino energies. In fact, the lowest energy antikaon associate production, ($K\bar{K}$, $| \Delta S | = 0$), has a quite high threshold ($\approx 1.75$ GeV) and thus, it leads to even smaller cross sections in the range of energies we have explored. For instance, at 2 GeV, GENIE predicts antikaon production cross sections at least two orders of magnitude smaller than our calculation[^3]. As it was expected, our results would lead to a very minor signal in past experiments. For instance, we have evaluated the flux averaged cross-section $\langle\sigma\rangle$ for the MiniBooNE antineutrino flux [@AguilarArevalo:2011sz] in the sub GeV energy region. The results are given in Table \[tb:flux\] and compared with the recent measurement of the neutral current $\pi^0$ production per nucleon with the same flux [@AguilarArevalo:2009ww]. . Process $\langle\sigma\rangle$ ($10^{-41}$ cm$^2$) --------------------------------------------------------------- -------------------------------------------- $\bar \nu_\mu + p \rightarrow \mu^+ + K^- + p$ 0.11 $\bar \nu_\mu + p \rightarrow \mu^+ +\bar K^0 + n $ 0.08 $\bar \nu_\mu + n \rightarrow \mu^+ + K^- + n$ 0.04 $\bar \nu_\mu + ^{12}C \rightarrow \bar \nu_\mu +X + \pi^0 $ $14.8\pm 0.5\pm 2.3$ \[tb:flux\] We find that the antikaon production cross section is around two orders of magnitude smaller than the NC $\pi^0$ one at MiniBooNE. Given the number of neutral pions observed for the antineutrino beam we expect that only a few tens of antikaons were produced in this experiment. One should notice here that the average antineutrino energy at MiniBooNE is well below the kaon threshold. Thus, we are only sensitive to the high energy tail of the flux. One could expect a relatively larger signal for the atmospheric neutrino $\bar\nu_e$ and $\bar\nu_\mu$ induced events at SuperK, given the larger neutrino energies. But even there we find a very small background from antikaon events. Taking the antineutrino fluxes from Ref. [@Honda:2006qj] we have calculated the event rates for the 22.5kT water target and a period of 1489 days as in the SuperK analysis of Ref. [@Ashie:2005ik]. We obtain 0.8 $e^+$ and 1.5 $\mu^+$ events. Although the model has large uncertainties at high energies, the rapid fall of the neutrino spectrum implies that the high energy tail contributes very little to the background. We have also estimated the average cross sections for the expected antineutrino fluxes at T2K [@arXiv:1109.3262] and MINER$\nu$A (low energy configuration) [@minflux]. In both cases, we have implemented an energy cut ($E_k+E_l<2$ GeV), that insures that high energy neutrinos, for which our model is less reliable, play a minor role. The results are presented in Table \[tb:fluxes\]. For T2K, we get similar results to the MiniBooNE case whereas the average cross section is much larger at MINER$\nu$A because of the higher neutrino energies. Process $\langle\sigma\rangle$ MINER$\nu$A $\langle\sigma\rangle$ T2K ----------------------------------------------------- ------------------------------------ ---------------------------- $\bar \nu_\mu + p \rightarrow \mu^+ + K^- + p$ 1.1 0.07 $\bar \nu_\mu + p \rightarrow \mu^+ +\bar K^0 + n $ 0.49 0.04 $\bar \nu_\mu + n \rightarrow \mu^+ + K^- + n$ 0.33 0.02 \[tb:fluxes\] Hitherto, our results correspond to relatively low antineutrino energies, where our model is best suited. However, the model could also be used to compare with data obtained at much higher neutrino energies selecting events such that the invariant mass of hadronic part is close to antikaon-nucleon threshold and the transferred momentum $q$ is small. This procedure has been used, for instance, in the analysis of two pion production induced by neutrinos [@Adjei:1981nw; @Kitagaki:1986ct]. In Fig. \[fg:xsec\_mu\_pp\], we show the size of several contributions to the $\bar \nu_\mu p \rightarrow \mu^+ p K^- $ reaction. Obviously, this separation is not an observable and only the full cross section obtained with the sum of the amplitudes has a physical sense. However, it could help us to get some idea of how the uncertainties associated to some of the mechanisms, like the $\Sigma^*(1385)$ one, could affect our results. ![Cross-section for the process $\bar \nu_\mu p \rightarrow \mu^+ p K^- $.[]{data-label="fg:xsec_mu_pp"}](pp.eps){width="60.00000%"} The cross section is clearly dominated by the non–resonant terms, providing the CT term the largest contribution. We see the destructive interference that leads to a total cross section smaller than that predicted by the CT term alone. We could also remark the negligible contribution of the $\Sigma^*(1385)$ channel. This fact is at variance with the strong $\Delta$ dominance for pion production and it can be easily understood because the $\Sigma^*$ mass is below the kaon production threshold. We have also explored, the uncertainties associated with the form factor. The curve labeled as “Full Model” has been calculated with a dipole form factor with a mass of 1 GeV. The band corresponds to a 10 percent variation of this parameter. The effect is similar in the other channels and we will only show the results for the central value of 1 GeV. ![Cross-section for $\bar \nu_\mu n \rightarrow \mu^+ n K^- $.[]{data-label="fg:xsec_mu_nn"}](nn.eps){width="60.00000%"} ![Cross-section for $\bar \nu_\mu p \rightarrow \mu^+ n \bar{K}^0 $.[]{data-label="fg:xsec_mu_pn"}](pn.eps){width="60.00000%"} In Figs. \[fg:xsec\_mu\_nn\] and \[fg:xsec\_mu\_pn\], we show the other two channels. As in the previous case the CT term is very important. We observe, however, that the pion-pole term gives a contribution as large as the CT one for the $\bar \nu_\mu p \rightarrow \mu^+ n K^0 $ process. For the $\bar \nu_\mu n \rightarrow \mu^+ n K^- $ case, we find a substantial contribution of the $\Sigma^*$ resonance, due to the larger value of the couplings (see Table \[tb:currents\]). As in the first case, there is some destructive interference between the different mechanisms participating in these processes. ![$d\sigma/dQ^2$ cross section.[]{data-label="fg:q2"}](dsigdq2.eps){width="60.00000%"} In Fig. \[fg:q2\], we show the $Q^2$ distributions for the three channels at a antineutrino energy $E_{\bar{\nu}} = 2$ GeV. We have checked that the reactions are always forward peaked (for the final lepton),even in the absence of any form factor, favouring relatively small values of the momentum transfer. We should notice however, that the smallness of $Q^2$ does not imply that $q^0$ or $\vec{q}$ are also small. In fact, because of the kaon mass both energy and momentum transfer are always large. Also nucleon laboratory momentum, even at threshold, is quite large ($\sim$ 0.48 GeV). This implies that, for these processes, Pauli blocking in nuclei would be ineffective. In summary, we have developed a microscopical model for single antikaon production off nucleons induced by neutrinos based on the SU(3) chiral Lagrangians, including the lowest lying octet and decuplet baryons. This model is an extension of that of Ref. [@RafiAlam:2010kf], where single kaon production was investigated. The calculation is necessarily more complex for antikaons because resonant mechanisms, absent for the kaon case, could be relevant. On the other hand, the threshold for associated antikaon production corresponds to the $K-\bar K$ channel and it is much higher than for the kaon case (kaon-hyperon). This implies that the process we study is the dominant source of antikaons for a wide range of energies. All parameters of the model involving only octet baryons are well known: Cabibbo’s angle, $f_\pi$, the pion decay constant, the proton and neutron magnetic moments and the axial vector coupling constants D and F. The weak couplings of the $\Sigma^*(1385)$ have been obtained from those of the $\Delta(1232)$ using SU(3) symmetry. Although they contain considerable uncertainties, we find that the resonance contribution is quite small. We obtain for the single antikaon production cross sections similar to those of single kaon production, and around two orders of magnitude smaller that for pion production for antineutrino fluxes such as that from MiniBooNE. Nonetheless, the study may be useful in the analysis of antineutrino experiments at MINER$\nu$A, NO$\nu$A, T2K and others with high statistics and/or higher antineutrino energies. This work is partly supported by DGICYT Contracts No. FIS2006-03438 and FIS2008-01143, the Generalitat Valenciana in the program Prometeo and the EU Integrated Infrastructure Initiative Hadron Physics Project under contract RII3-CT-2004-506078. I.R.S. acknowledges support from the Ministerio de Educación. M.R.A. wishes to acknowledge the financial support from the University of Valencia and Aligarh Muslim University under the academic exchange program and also to the DST, Government of India for the financial support under the grant SR/S2/HEP-0001/2008. Hadronic Currents {#app:amplitude} ================= For consistency with Eq. \[eq:Gg\] the contributions to the hadronic current are $$\begin{aligned} J^\mu \arrowvert_{CT} &=&i A_{CT} V_{us} \frac{ \sqrt{2}}{2 f_\pi} \bar N(p^\prime) \; (\gamma^\mu + B_{CT} \; \gamma^\mu \gamma_5 ) \; N(p) \\ J^\mu \arrowvert_{\Sigma} &=&i A_{\Sigma} (D-F) V_{us} \frac{ \sqrt{2}}{2 f_\pi} \bar N(p^\prime) p_k\hspace{-.9em}/ \; \gamma_5 \frac{ p\hspace{-.5em}/ + q\hspace{-.5em}/ + M_\Sigma} {( p + q)^2 -M_\Sigma^2} \left(\gamma^\mu +i \frac{(\mu_p + 2\mu_n)}{2 M} \sigma^{\mu \nu} q_\nu \right. \\ &+& \left. (D-F) \left\{ \gamma^\mu - \frac{q^\mu}{ q^2-{M_k}^2 } q\hspace{-.5em}/ \right\} \gamma^5 \right) N(p) \\ J^\mu \arrowvert_{\Lambda} &=& i A_{\Lambda} V_{us} (D+3F) \frac{1} {2 \sqrt{2} f_\pi} \bar N(p^\prime) p_k\hspace{-.9em}/ \; \gamma^5 \frac{ p\hspace{-.5em}/ + q\hspace{-.5em}/ +M_\Lambda} {( p + q)^2 -M_\Lambda^2} \left(\gamma^\mu +i \frac{\mu_p}{2 M} \sigma^{\mu \nu} q_\nu \right. \\ &-& \left. \frac{(D + 3 F)}{3} \left\lbrace \gamma^\mu - \frac{q^\mu }{ q^2-{M_k}^2 } q\hspace{-.5em}/ \right\rbrace \gamma^5 \right) N(p) \\ J^\mu \arrowvert_{KP}&=& i A_{KP} V_{us} \frac{\sqrt{2}}{2 f_\pi} \bar N(p^\prime) q\hspace{-.5em}/ \; N(p) \frac{q^\mu}{q^2-M_k^2} \\ J^\mu \arrowvert_{\pi} &=& iA_{\pi } \frac{M\sqrt{2}}{2 f_\pi} V_{us} (D + F)\frac{ 2 {p_k}^\mu -q^\mu}{(q-p_k)^2 - {m_\pi}^2} \bar N(p^\prime) \gamma_5 N(p) \\ J^\mu \arrowvert_{\eta} &=&i A_{\eta } \frac{M\sqrt{2}}{2 f_\pi} V_{us} (D - 3 F)\frac{2 {p_k}^\mu - q^\mu}{(q-p_k)^2 - {m_\eta}^2} \bar N(p^\prime) \gamma_5 N(p) \\ J^\mu \arrowvert_{\Sigma^*} &=&- i A_{\Sigma^*} \frac{\cal C}{ f_\pi } \frac{1}{\sqrt{6}} \; V_{us} \; \frac{p_k^\lambda}{P^2 - M_{\Sigma^*}^2 + i \Gamma_{\Sigma^*} M_{\Sigma^*}}\; \bar N(p^\prime) P_{RS_{\lambda \rho}} ( \Gamma_V^{\rho \mu} +\Gamma_A^{\rho \mu} ) N(p) \end{aligned}$$ In $\Gamma_V^{\rho \mu} +\Gamma_A^{\rho \mu}$, the form factors are taken as for the $\Delta^+$ case. 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[^1]: The physical states of the decuplet are: $T_{111} = \Delta^{++},T_{112} = \frac{\Delta^{+}}{\sqrt3}, T_{122} = \frac{\Delta^{0}}{\sqrt3}, T_{222} = \Delta^{-},T_{113} = \frac{\Sigma^{*+}}{\sqrt3}, T_{123} = \frac{\Sigma^{*0}}{\sqrt6},T_{223} = \frac{\Sigma^{*-}}{\sqrt3},T_{113} = \frac{\Xi^{+}}{\sqrt3}, T_{133} = \frac{\Xi^{0}}{\sqrt3},T_{333} = \Omega^{-} $. [^2]: A more elaborate discussion can be found there. [^3]: This has been obtained with GENIE version 2.7.1 and corresponds to $K\bar{K}$ processes.

--- abstract: | The physical model describing the influence of the electronic subsystem on the properties of one-dimensional chains of metal is presented. It is shown that depending on an interaction potential between atoms in one-dimensional system formation of chains of various length is possible. In case the characteristic depth of the potential well of the interatomic interaction does not exceed a certain magnitude, the chains in 1D system are formed with length of several angstroms, while the increase the depth of the well also leads to the possibility of formation of metal chains of greater length.\ Представлена физическая модель, описывающая влияние электронной подсистемы на свойства одномерных цепочек металлов. Показано, что в зависимости от потенциала взаимодействия между атомами в одномерной системе возможно образование цепочек различной длины. В том случае, если характерная глубина потенциальной ямы межатомного взаимодействия не превышает определенной величины, в 1D-системе образуются цепочки с характерной длиной порядка нескольких ангстрем, в то время как увеличение глубины ямы приводит также к возможности образования цепочек металлов большей длины. author: - | В.Д. Борман, П.В. Борисюк, О.С. Васильев, В.Н. Тронин, И.В. Тронин,\ М.А. Пушкин, В.И. Троян bibliography: - 'biblio1.bib' title: Влияние электронов на стабильность одномерных цепочек металлов --- Правильное понимание свойств наноразмерных контактов имеет решающее значение для многих областей современной нанотехнологии. Уникальные свойства моноатомных цепочек металлов привлекают в настоящее время значительное внимание как экспериментально[@Agrait:2003kr; @Smit:2001gk; @Csonka:2006uu; @Kizuka:2001wj; @Ohnishi:1998tz; @Rodrigues:2001wv; @RubioBollinger:2001us; @Untiedt:2002iy; @Yanson:1998uo], так и теоретически [@Skorodumova:2005dg; @Skorodumova:2003vj; @Skorodumova:2000uc]. Атомарные цепочки могут быть получены в экспериментах по механически контролируемому обрыву цепи с использованием туннельного микроскопа или просвечивающего электронного микроскопа[@Agrait:2003kr]. Структуры, получаемые в данных экспериментах являются одномерными цепочками, состоящие из нескольких атомов металла, находящиеся между двух поверхностей(см рис. \[fig:chain\]). Образование подобных цепочек сильно зависит от материала атомов, из которых состоит цепочка. Было показано, что золотые цепочки могут быть до 2.6 нм в длину[@Untiedt:2002iy; @Yanson:1998uo], тогда как цепочки из атомов серебра в длину не превышают нескольких ангстрем[@Smit:2001gk]. Так, в работах[@Smit:2001gk; @Untiedt:2002iy] с помощью методики механически контролируемого обрыва цепи (mechanically controllable break-junction (MCB)),в серии экспериментов были получены гистрограммы, отображающие частоту обрыва цепочек металлов Ag, Au, Pt, Pd от длины цепочки(см. рис. \[fig:agaupt\]). Таким образом, можно сделать вывод, что металлы Au и Pt, в отличие от Ag и Pd, могут образовывать одномерные цепочки с количеством атомов $N>2$. В связи с отсутствием теоретического описания данного эффекта в настоящее время, необходимо проведение исследования свойств одномерных наноцепочек. Понимание механизмов образования цепочек может помочь улучшить контроль за процессом их получения, что, в свою очередь, может привести к созданию цепочек различных материалов с уникальными свойствами (магнетизм, сверхпроводимость) или созданию цепочек больших размеров. В данной работе представлена физическая модель, описывающая влияние электронной подсистемы на свойства одномерных цепочек металлов. Показано, что в зависимости от потенциала взаимодействия между атомами в одномерной системе возможно образование цепочек различной длины. В том случае, если характерная глубина потенциальной ямы межатомного взаимодействия не превышает определенной величины, в 1D-системе образуются цепочки с характерной длиной порядка нескольких ангстрем, в то время как увеличение глубины ямы приводит также к возможности образования цепочек металлов большей длины. Рассмотрим влияние электронной подсистемы на свойства одномерных цепочек металлов. Предположим, что цепочки металлов, наблюдаемые в экспериментах, являются реализацией возможных состояний одномерной статистической системы. Рассмотрим бесконечную одномерную цепочку. Рассмотрим электрон в поле флуктуаций плотности одномерной системы частиц. Пусть $\overline{n}$ — средняя плотность частиц в рассматриваемой системе, а $n=n(x)=\sum_i \delta (x-x_i)$ — микроскопическая плотность. Тогда $\delta n = \overline{n}-n$ — флуктуация плотности. Гамильтониан электрона в поле флуктуаций плотности будет иметь вид: $$\label{eq:math:H1} H = H[\Psi ,\delta n] + H[\delta n]$$ где первое слагаемое — часть гамильтониана, отвечающая взаимодействию электрона с флуктуацией плотности, а второе слагаемое , отвечающее собственно флуктуации плотности, представляет собой изменение свободной энергии системы частиц: $$\label{eq:math:Hpsin} H[\Psi ,\delta n] = \int \left( - \Psi ^* \frac{{\hbar^2 }}{{2m}}(\Delta \Psi ) + \Psi ^* U\Psi \right)\,dx$$ $$\label{eq:math:Hn} \begin{split} H[\delta n] = F(n + \delta n) - F(n) = \int \left. \frac{\delta F}{\delta n} \right|_{\overline n} \delta n \,dx + \frac{1}{2} \int \frac{\delta^2 F}{\delta n(x) \delta n(x')} \, \delta n(x) \, \delta n(x') \, dx dx' ={} \\ {} = - \frac{1}{2}\int {\beta ^{ - 1} (x,x')\delta n(x)\delta n(x')\,dxdx'} \end{split}$$ где $\beta$ — функция отклика системы, $\beta^{-1}=\frac{\delta^2 F}{\delta n(x) \delta n(x')}$, $F$ — свободная энергия системы. Потенциальная энергия электрона в поле флуктуаций плотности $U=\int{V(x-x')\, dx\, \delta n}$. Будем предполагать одночастичный потенциал $V$ локальным: $V(x-x') = V_0 \delta(x-x')$, тогда $U = V_0 \delta n(x)$. Это приближение, принимаемое при изучении рассеяния медленных частиц, справедливо, когда де-Бройлевская длина волны электрона много больше размера рассеивателя. Это условие выполняется, поскольку длина волны электрона порядка размера кластера, а кластер содержит много атомов($N \gg 1$).\ Для равновесного состояния: $$\label{eq:math:dHdn} \frac{\delta H}{\delta (\delta n)}=0$$ Подставляя в , получим $$\label{eq:math:dHdn1} \begin{split} \int{V_0\Psi^*(x)\Psi(x)\,dx} &-\int{\beta^{-1}(x,x')\delta n'(x')\,dxdx'}=0; \\ {} \delta n'(x')&=\int{\beta(x,x') V_0 \Psi^*(x) \Psi(x)\,dx} \end{split}$$ Таким образом, из , , с учетом получаем: $$\begin{split} H_{eff} [\Psi ] =\left. H[\Psi]\right|_{\delta n'} = \int \left( - \Psi ^* \frac{{h^2 }}{{2m}}\Delta \Psi \right)\,dx + \frac{1}{2}V_0 \int \beta(x,x') \left(\Psi ^*(x) \Psi(x) \right) \times {}\\ {} \times \left(\Psi ^*(x') \Psi(x') \right) \,dxdx' \end{split} \label{eq:math:Heff}$$ Функция отклика $\beta(x,x')$ связана с корреляционной функцией $\nu(x,x')$ флуктуационно-диссипативной теоремой [@balesku1978]: $$\label{eq:math:nu_beta} n^2 \nu(x,x')=-\delta (x-x') n - T \beta(x,x')$$ или, в $k$ — представлении, $$\label{eq:math:nu_beta_k} \beta(k)=-\frac{n}{T}[1+n\nu (k)]$$ Для упрощения воспользуемся локальным приближением (что соответствует длинноволновому приближению($k \rightarrow 0$)) [@Devyatko1990a] : $$\label{eq:math:nu_beta_k1} \begin{split} \beta(k)=&-\frac{n}{T}[1+n\nu(k)+n\nu_{cor}(k)-n\nu_{cor}(k)]=-\frac{n}{T}[1+n\nu_{cor}(k)]- {}\\ {}&- \frac{n}{T}[n\nu(k)-n\nu_{cor}(k)] \approx -\frac{n}{T}[1+n\nu_{cor}(0)]-\frac{n}{T}[n\nu(0)-n\nu_{cor}(0)]\delta(k)\frac{1}{a} \end{split}$$ здесь и далее индекс ${cor}$ — соответствует величинам, вычисленным для системы частиц с взаимодействием типа “твердые шары”, величины без индекса соответствуют вычисленным для системы частиц с полным взаимодействием. Функция отклика $\beta_{k=0}$ связана с сжимаемостью соотношением [@Korneev]: $$\label{eq:math:beta0} \left. \frac{\partial p}{\partial n} \right|_{cor}=n \beta^{-1}_{k=0}=-T [1+n \nu_{cor}(0)]^{-1}$$ в перейдем от плостности $n$ к степени заполнения $\theta$, $\theta=n a$ — степень заполнения в одномерной системе: $$\label{eq:math:beta1} \frac{1}{a} \left. \frac{\partial p}{\partial \theta}\right|_{cor}=-T[1+n\nu_{cor}(0)]^{-1}$$ Сжимаемость $\left. \frac{\partial \theta}{\partial p}\right|_{cor}$, где $p$ — одномерное давление, для взаимодействия типа “твердые шары” может быть получена из уравнения состояния одномерного газа, которое может быть найдено из cоотношений [@fisher1961]: $$\label{eq:math:sost1d} \begin{split} \frac{1}{n} +& T \frac{\partial}{\partial p} \ln(\varphi(p,T))=0 \\ \varphi(p,T) =& \int \limits^{\infty}_0 {dx \,\exp \left(-\frac{px+\Phi(x)}{T}\right)} \end{split}$$ где $\Phi(x)$ — потернциал взаимодействия между атомами. Таким образом, уравнение состояния $p(\theta,T)$ и восприимчивость $\left. \frac{\partial \theta}{\partial p}\right|_{cor}$ одномерного газа частиц для взаимодействия типа “твердые шары” будет иметь вид: $$\label{eq:math:p1d} p = \frac{T}{a}\frac{\theta }{{1 - \theta }}, \quad \left. \frac{\partial \theta}{\partial p}\right|_{cor}=\frac{(1-\theta)^2}{T} a$$ где $a$ — радиус частицы, из и и для $\beta(k)$ получаем: $$\label{eq:math:betaK} \beta (k) = \frac{\theta }{aT}\left( {1 - \theta } \right)^2 - \frac{\theta }{{Ta^3 }}\left( {T\frac{{\partial \theta }}{{\partial p}} - a\left( {1 - \theta } \right)^2 } \right)\delta (k)$$ Восприимчивость системы взаимодействующих частиц $\frac{{\partial \theta }}{{\partial p}}$ найдем в приближении взаимодействия типа “прямоугольная яма”: $$\label{eq:math:yama} \Phi \left( x \right) = \left\{ {\begin{array}{*{20}c} {\infty ,x \le a } \\ { - \varepsilon, a < x \le a + R} \\ {0,x > a + R} \\ \end{array}} \right.$$ Здесь $\varepsilon$ — глубина ямы, $a$ — радиус частицы $R$ — ширина ямы. В этом случае восприимчивость $\frac{{\partial \theta }}{{\partial p}}$ может быть найдена из уравнения состояния для одномерного газа [@herzfeld1932]: $$\label{eq:math:p1d_teta} p_{1D} \sigma \left( {\frac{1}{\theta } - 1} \right) = T - p_{1D} R\left[ {\frac{{\exp \left( {\beta p_{1D} R} \right)}}{{1 - \exp \left( { - \varepsilon/T} \right)}} - 1} \right]^{ - 1}$$ где $p_{1D}=\frac{pa}{T}$ — безразмерное одномерное давление. Таким образом, с учетом , из для эффективного гамильтониана $H_{eff}$ получим: $$\label{eq:math:Heff1} H_{eff} [\Psi ] = \int {\left( { - \Psi ^* \frac{{\hbar ^2 }}{{2m}}\Delta \Psi } \right)dx - \frac{{V_0 ^2 }}{{4\pi }}} \frac{\lambda }{T}\int {\left| \Psi \right|} ^4 dx - \frac{{V_0 ^2 }}{{4\pi }}\frac{\alpha }{T}\left( {\int {\left| \Psi \right|} ^2 dx} \right)^2$$ Здесь $\lambda = \frac{{\theta (1 - \theta )^2 }}{a}$, $\alpha = \frac{\theta }{{a^3 }}\left[ {T\frac{{\partial \theta }}{{\partial p}} - a(1 - \theta )^2 } \right]$ Зная эффективный гамильтониан электрона в поле флуктуации плотности , получим уравнение Шредингера: $$\label{eq:math:nelin_shed} \Delta \Psi = - \frac{m}{{\hbar ^2 }}\frac{{V_0^2 \lambda }}{T}\Psi ^3 - \left( {\frac{m}{{\hbar ^2 }}\frac{{V_0^2 \alpha }}{{2\pi T}} + E\frac{m}{\hbar ^2 }} \right)\Psi$$ Для такого уравнения есть термин “нелинейное уравнение Шредингера”. Показано [@alimenkov2006], что одним из решений подобного нелинейного уравнения будет является решение вида $\Psi(x)=A/{\mathrm{ ch}}(Bx)$, где $A$ и $B$ — константы, $B$ имеет смысл обратного характерного размера солитона. Таким образом, в нашем случае решением будет являться: $$\label{eq:math:psi_resh} \Psi (x) = \frac{V_0}{\hbar}\sqrt{\frac{\sqrt{2\pi}m}{8Ta}\theta (1-\theta)^2}\,\frac{1}{{{\mathrm{ch}}\left( {\frac{\sqrt{2\pi}}{4}\frac{mV_0^2}{\hbar^2 T}\frac{\theta}{a}(1-\theta)^2\,x} \right)}}$$ где амплитуда солитона $A$: $$\label{eq:A} A=\frac{V_0}{\hbar}\sqrt{\frac{\sqrt{2\pi}m}{8Ta}\theta (1-\theta)^2}$$ а обратный характерный размер солитона $B$: $$\label{eq:B} B=\frac{\sqrt{2\pi}}{4}\frac{mV_0^2}{\hbar^2 T}\frac{\theta}{a}(1-\theta)^2;$$ Заметим, что зависимость имеет максимальное значение при $\theta=1/3$, что соответствует минимальной длине цепочки $L_{min}=1/B=\sqrt{2\pi}\frac{mV_0^2}{\hbar^2 Ta}\frac{1}{27}$. Зная волновую функцию электрона, можно найти энергию электрона в зависимости от степени заполнения $\theta$ одномерной системы частиц. Из получим, что энергия электрона будет иметь вид: $$\label{eq:E_sol_f} E=-\sqrt{\frac{\pi}{2}}\frac{V_0^2}{Ta^3}\,\theta\left(T\frac{\partial \theta}{\partial p} - a(1-\theta)^2\right)-\frac{\pi}{16}\frac{mV_0^4}{\hbar^2 T^2a^2}\,\theta^2(1-\theta)^4,$$ Величину $\frac{\partial \theta}{\partial p}$ в приближении взаимодействия типа ”прямоугольная яма“ можно получить из . В таком случае, зависимость энергии электрона $E$ от степени заполнения одномерной системы $\theta$ при различных значениях глубины потенциальной ямы взаимодействия атомов представлена на рис. \[fig:energ(t)\]. Как можно видеть из рис. \[fig:energ(t)\], при малой глубине потенциальной ямы взаимодействия атомов $\varepsilon=5\cdot 10^{-2}$ эВ, зависимость энергии электрона от степени заполнения имеет один минимум при $\theta \simeq 0.22$ (см. рис. \[fig:energ(t)\](a)), что отвечает одному наиболее вероятному состоянию электрона при любом значении величины степени заполнения $\theta$. При увеличении же глубины потенциальной ямы $\varepsilon=9\cdot 10^{-2}$ эВ, в зависимости энергии электрона от степени заполнения появляется второй минимум при $\theta \simeq 0.68$ (см. рис. \[fig:energ(t)\](a)), таким образом при увеличении энергии взаимодействия атомов имеется два наиболее вероятных состояний электрона, соответствующие различным $\theta$. Зная связь характерной длины солитона $1/B$ со степенью заполнения одномерной системы $\theta$ , из можно найти зависимость энергии электрона от характерной длины солитона $E(1/B)=E(L)$ (см. рис. \[fig:energ(l)\]). В связи с громоздкостью, итоговое аналитическое выражение для $E(L)$ не приводится. Как можно видеть из рис. \[fig:energ(l)\](б), при глубине потенциальной ямы взаимодействия атомов $\varepsilon=9\cdot 10^{-2}$ эВ имеются два минимума энергии электрона при $L\simeq 4.9, 8$ Å, соответствующие различным диапазонам $\theta$. Подобное поведение можно объяснить следующим образом. При $\theta=0\div 1/3$ наиболее вероятным является образование цепочек длиной $L \simeq 4.9$ Å. С увеличением же степени заполнения $\theta > 1/3$ система переходит в новое состояние с образованием цепочек длиной $L\simeq 8$ Å. Получим вероятность образования цепочек длинной $L$. Величина $\exp (-E(L)/T)$ — есть вклад в плотность вероятности образования цепочки с характерным размером $L$ за счет взаимодействия электрона с флуктуацией плотности. Для того, чтобы получить вероятность образования цепочки длиной $L$ необходимо дополнительно учесть вероятность образования флуктуации плотности в одномерной системе. Таким образом, плотность вероятности образования цепочки с характерным размером $L$ будет иметь вид: $$\label{eq:f_ver} f=\exp \left(\frac{-E(L)}{T}\right)w_L$$ где $w_L$ — вероятность найти в одномерной системе невзаимодействующих частиц цепочку длиной $L$. $w_L$ определим следующим образом. Известно [@landau5], что вероятность того, что на длине $L$ будет находиться всего $N$ атомов дается формулой Пуассона: $$\label{fpuassona} w_N=\frac{\overline{n}^N \exp{(-\overline{n})}}{N!},$$ где $\overline{n}=\frac{N_0}{L_0}L=\frac{\theta}{2a}L$ — среднее значение числа частиц на длине $L$, $N_0$ — число частиц в системе, $L_0$ — длина системы, $N$ — число частиц в цепочке. Полагая, что в цепочке длиной $L$ содержится $N$ частиц при плотности частиц $\theta'=1$, получим $N=\frac{L}{2a}$. Таким образом, вероятность флуктуационного образования цепочки из $N$ частиц будет выглядеть следующим образом: $$\label{fpuassona_N} w_N=\frac{(\theta N)^N \exp{(-(\theta N))}}{N!}$$ Имея в виду вышеозначенную связь числа частиц с длиной цепочки, получим: $$\label{fpuassona_L} w_L=\frac{\left( \frac{\theta L}{2a}\right)^{\frac{L}{2a}} \exp{\left(- \frac{\theta L}{2a}\right)}}{\left(\frac{L}{2a}\right)!}$$ Таким образом, плотность вероятности образования цепочки длиной $L$ в одномерной системе будет выглядеть следующим образом: $$\label{eq:f_ver_full} f=\exp \left(\frac{-E(L)}{T}\right) \frac{\left( \frac{\theta(L) L}{2a}\right)^{\frac{L}{2a}} \exp{\left(- \frac{\theta(L) L}{2a}\right)}}{\left(\frac{L}{2a}\right)!},$$ где зависимость $\theta(L)=\theta(1/B)$ описывается выражением . На рис. \[fig:frasp(l)\] представлены зависимости плотности вероятности образования цепочек длиной $L$ при глубине потенциальной ямы взаимодействия атомов $\varepsilon=9\cdot 10^{-2}$ эВ (рис. \[fig:frasp(l)\](а), (б)) и $\varepsilon=5\cdot 10^{-2}$(в). Как можно видеть из рис. \[fig:frasp(l)\], при значении глубины потенциальной ямы взаимодействия атомов $\varepsilon=5\cdot 10^{-2}$ эВ, при любой плотности $\theta$ в одномерной системе наиболее вероятным является образование димеров - цепочек, состоящих из двух атомов, тогда как при увеличении глубины ямы появляется дополнительный пик плотности функции распределения, отвечающий образованию цепочек с $N>2$. Таким образом, показано, что в зависимости от потенциала взаимодействия между атомами в одномерной системе возможно образование цепочек различной длины. В том случае, если характерная глубина потенциальной ямы межатомного взаимодействия не превышает определенной величины, в 1D-системе образуются цепочки с характерной длиной порядка нескольких ангстрем, в то время как увеличение глубины ямы приводит к возможности образования цепочек металлов также и большей длины.\ Работа выполнена при финансовой поддержке Федеральной целевой программы “Научные и научно-педагогические кадры инновационной России на 2009 – 2013 годы”.

--- abstract: 'We present near-infrared integral field spectroscopy data obtained with VLT/SINFONI of “the Teacup galaxy”. The nuclear K-band (1.95–2.45 ) spectrum of this radio-quiet type-2 quasar reveals a blueshifted broad component of FWHM$\sim$1600-1800 km s$^{-1}$ in the hydrogen recombination lines (Pa$\alpha$, Br$\delta$, and Br$\gamma$) and also in the coronal line \[Si VI\]$\lambda$1.963 . Thus the data confirm the presence of the nuclear ionized outflow previously detected in the optical and reveal its coronal counterpart. Both the ionized and coronal nuclear outflows are resolved, with seeing-deconvolved full widths at half maximum of 1.1$\pm$0.1 and 0.9$\pm$0.1 kpc along PA$\sim$72–74. This orientation is almost coincident with the radio axis (PA=77), suggesting that the radio jet could have triggered the nuclear outflow. In the case of the H$_2$ lines we do not require a broad component to reproduce the profiles, but the narrow lines are blueshifted by $\sim$50 km s$^{-1}$ on average from the galaxy systemic velocity. This could be an indication of the presence of a nuclear molecular outflow, although the bulk of the H$_2$ emission in the inner $\sim$2 ($\sim$3 kpc) of the galaxy follows a rotation pattern. We find evidence for kinematically disrupted gas (FWHM$>$250 km s$^{-1}$) at up to 5.6 kpc from the AGN, which can be naturally explained by the action of the outflow. The narrow component of \[Si VI\] is redshifted with respect to the systemic velocity, unlike any other emission line in the K-band spectrum. This indicates that the region where the coronal lines are produced is not co-spatial with the narrow line region.' author: - '\' title: 'An infrared view of AGN feedback in a type-2 quasar: the case of the Teacup galaxy' --- \[firstpage\] galaxies: active – galaxies: nuclei – galaxies: evolution – galaxies: individual – galaxies: jets. Introduction {#intro} ============ Cosmological simulations require active galactic nuclei (AGN) feedback to regulate black hole and galaxy growth [@diMatteo05; @Croton06]. This process occurs when the intense radiation produced by the active nucleus sweeps out and/or heats the interstellar gas, quenching star formation and therefore producing a more realistic number of massive galaxies in the simulations (see @Fabian12 for a review). Two major modes of AGN feedback are identified. Radio- or kinetic-mode feedback dominates in galaxy clusters and groups, where jet-driven radio bubbles heat the intra-cluster medium. This type of feedback is generally associated with powerful radio galaxies. Quasar- or radiative-mode feedback consists of AGN-driven winds of ionized, neutral, and molecular gas [@Fabian12; @Fiore17]. However, such a clear distinction between the two modes of feedback can be somewhat misleading. This is because it has been shown, through the detection of nuclear outflows, that radiative-mode feedback also acts in radio-galaxies (see e.g. @Emonts16 and references therein), whilst the presence of jets in quasars that are deemed to be radio-quiet can lead to faster and more turbulent AGN-driven winds [@Mullaney13; @Zakamska14]. Therefore, it is over-simplistic to consider the impact each mode of AGN-feedback has on its host galaxy in isolation. One of the most efficient ways to identify the imprint of outflows in large AGN samples at any redshift is to search for them in the warm ionized phase in the optical range (e.g. \[O III\]$\lambda$5007 Å). Indeed, during recent years it has become clear that ionized outflows are a ubiquitous phenomenon in type-2 quasars (QSO2s) at z$\la$0.7 [@Villar11; @Villar14; @Liu13; @Harrison14; @Karouzos16]. QSO2s are excellent laboratories to search for outflows and study their impact on their host galaxies, as the AGN continuum and the broad components of the permitted lines produced in the broad-line region (BLR) are obscured. Fast motions are often measured in QSO2s, with full-widths at half maximum (FWHM) $>$1000 km s$^{-1}$ and typical velocity shifts (V$_s$) of hundreds km s$^{-1}$. These outflows are likely triggered by AGN-related processes and they originate in the high-density regions (n$_e\ge 10^3~cm^{-3}$) within the central kiloparsecs of the galaxies. Optical integral field spectroscopy (IFS) studies have shown that these outflows can extend up to $\sim$15 kpc from the AGN [@Humphrey10; @Liu13; @Harrison14]. However, these results have been recently questioned, as the reported outflow extents could be overestimated due to seeing smearing effects [@Karouzos16; @Villar16; @Husemann16]. Now that ionized outflows have been identified as a common process in QSO2, the next goal is to investigate their impact on other gaseous phases, such as the molecular and coronal phases. Since H$_2$ is the fuel required to form stars and feed the SMBH, the impact of the outflows in this gaseous phase is what might truly affect how systems evolve. Detecting coronal outflows is also interesting because, due to the high ionization potentials (IP$\ga$100 eV; @Mullaney09 [@Rodriguez11; @Landt15]) of these lines, they are unequivocally associated with nuclear activity. Coronal lines have intermediate widths between those of the broad and the narrow emission lines (FWHM$\sim$500–1500 km s$^{-1}$) and are generally blueshifted and/or more asymmetric than lower-ionization lines [@Penston84; @Appenzeller91; @Rodriguez11]. This indicates that either coronal lines are produced in an intermediate region between the narrow-line region (NLR) and the BLR [@Brotherton94; @Mullaney08; @Denney12], and/or are related to outflows [@Muller06; @Muller11]. The near-infrared (NIR) range, and particularly the K-band, allows us to trace outflow signatures in the molecular, ionized and coronal phases simultaneously. In addition, because ionized outflows in QSO2 are heavily reddened [@Villar14], observing them in the NIR permits us to penetrate through the dust screen and trace the regions closer to the base of the outflow. The rest-frame NIR spectrum of QSO2s at z$<$0.7 has not been fully characterised yet. To the best of our knowledge, this has been done for one QSO2 so far: Mrk477 at z=0.037 [@Villar15]. Additionally, a NIR spectrum of the QSO2 SDSS J1131+1627 at z=0.173 was presented in @Rose11, but only Pa$\alpha$ was detected. Here we explore the NIR spectrum of the QSO2 SDSS J143029.88+133912.0 (J1430+1339; at $z$=0.0852). The Teacup galaxy ----------------- According to its \[O III\] luminosity (5$\times$10$^{42}~erg~s^{-1}$ = 10$^{9.1}L_{\sun}$; @Reyes08), J1430+1339 is a luminous QSO2, and considering its position in the 1.4GHz–\[O III\] luminosity plane [@Lal10] it is classified as radio-quiet (L$_{1.4GHz}=5\times 10^{23}~W Hz^{-1}$; Harrison et al. 2015, hereafter @Harrison15). Nonetheless, it is a factor of 10 above the radio-FIR correlation found for star-forming galaxies [@Villar14; @Harrison14], which makes it a “radio excess source”. The host galaxy shows clear signatures of a past interaction with another galaxy, in the form of shells, tails and chaotic dust lanes [@Keel15]. J1430+1339 was nicknamed “the Teacup galaxy” because of the peculiar appearance of its extended emission-line region (EELR) in SDSS and HST images [@Keel12; @Keel15]. This EELR is dominated by a filamentary bubble to the northeast (NE) with a radial extent of $\sim$12 kpc measured from the nucleus (see Figure \[fig1\]). In the opposite direction there is another knotty emission-line structure resembling a fan extending up to $\sim$7 kpc. The Teacup has been proposed as a fading AGN candidate [@Gagne14]. The NE emission-line bubble coincides with the radio-continuum structure detected in VLA maps @Harrison15. These radio maps also show another radio bubble extending $\sim$10 kpc to the west, as well as two compact radio structures: a brighter one coincident with the AGN position, and a fainter one located $\sim$0.8 kpc northeast from the AGN (PA$\sim$60), identified by @Harrison15 as high-resolution B (HR-B) region. According to the latter authors, this HR-B structure would be co-spatial with the base of the ionized nuclear outflow first reported by @Villar14 and @Harrison14 using the SDSS spectrum and IFU spectroscopy respectively. At the position of HR-B the outflow has an observed velocity of -740 km s$^{-1}$ relative to the narrow component of \[O III\]$\lambda$5007 Å, which @Harrison15 interpreted as gas accelerated by jets or quasar winds at that location. The latter authors also speculated that these jets/winds would be driving the 10–12 kpc radio bubbles. {width="12cm"} Here we study the nuclear and extended NIR emission of the Teacup using seeing-limited SINFONI K-band data. In Section 2 we describe the observations and data reduction, in Section 3 we present the results on the nuclear and extended emission of the galaxy, and in Section 4 we discuss the observations and their implications. Throughout this paper we assume a cosmology with H$_0$=71 km s$^{-1}$ Mpc$^{-1}$, $\Omega_m$ = 0.27, and $\Omega_{\Lambda}$ =0.73. At the redshift of the galaxy (z=0.0852), the spatial scale is 1.591 kpc arcsec$^{-1}$. Observations and Data Reduction {#observations} =============================== We obtained K-band (1.95–2.45 ) observations of the Teacup with SINFONI on the 8 m Very Large Telescope (VLT). The data were taken during the night of 2015 March 7th in service mode (Program ID: 094.B-0189(A)) with a total on-source time of 1800 s and at an airmass of 1.3–1.4. Due to the strong and rapid variation of the IR sky emission, the observations were split into short exposures of 300 s each, following a jittering O-S-S-O pattern for sky and on-source frames. The observing conditions were clear and the seeing variation over the on-source observing period was small according to the DIMM seeing monitor[^1] (median optical seeing FWHM=1.05, standard deviation=0.21, and standard error=0.02). To calculate the seeing FWHM in the K-band we used the photometric standard star observed immediately after the target, which appears slightly elongated along PA=88.3$\pm$0.1, with a maximum FWHM=0.58. Along the minor axis, the FWHM=0.46. Therefore, the seeing error is dominated by the shape of the PSF rather than by the seeing variation during the observations. By averaging the maximum and minimum values of the FWHM measured for the star and by adding the seeing variation and PSF shape errors in quadrature, we get a seeing FWHM=0.52$\pm$0.06 ($\sim$830 pc resolution). We used the 0.125$\times$0.250 pixel$^{-1}$ configuration, which yields a field-of-view (FOV) of 8$\times$8 per single exposure. Due to the jittering process, the effective FOV in the case of our target is $\sim$9$\times$9 ($\sim$14$\times$14 kpc$^2$). The spectral resolution in the K-band is R$\sim$3300 ($\sim$75 km s$^{-1}$) and the instrumental broadening, as measured from the OH sky lines, is 6.0$\pm$0.5 Å with a dispersion of 2.45 Å pixel$^{-1}$. For the reduction of the data, we used the ESO pipeline ESOREX (version 3.8.3) and our own IDL routines for the telluric correction and flux calibration (see @Piqueras12). We applied the usual calibration corrections of dark subtraction, flat fielding, detector linearity, geometrical distortion, wavelength calibration, and subtraction of the sky emission to the individual frames. The individual cubes from each exposures were then combined into a single data cube. To estimate the uncertainty in the wavelength calibration we used the atmospheric OH lines, from which we measured an error of 7.8 km s$^{-1}$. The flux calibration was performed in two steps. First, to obtain the atmospheric transmission curves, we extracted the spectra of the standard stars with an aperture of 5$\sigma$ of the best 2D Gaussian fit of a collapsed image. The spectra were then normalised by a blackbody profile of the appropiate T$_{eff}$, taking the most relevant absorption spectral features of the stars into account. The result is a sensitivity function that accounts for the atmospheric transmission. Second, we flux-calibrated the spectra of the stars using their 2MASS K-band magnitudes. Every individual cube was then divided by the sensitivity function and multiplied by the conversion factor to obtain a fully-calibrated data cube. The uncertainty in the flux calibration is $\sim$15%. We refer the reader to @Piqueras12 [@Piqueras16] for further details on the data reduction. Results ======= Nuclear spectrum ---------------- We extracted two K-band spectra of the nuclear region of the Teacup in two circular apertures of 0.5 and 1.25 diameter ($\sim$0.8 and 2 kpc respectively), centred at the maximum of the Pa$\alpha$ emission (see Section \[nuclear\] for details). The minimum aperture was chosen to match the spatial resolution set by the seeing (FWHM=0.52$\pm$0.06). In the following we will refer to the spectrum extracted in this aperture as the nuclear spectrum. ### Continuum shape {#nuclear} Figure \[fig2\] shows the observed nuclear spectrum of the Teacup with the emission lines labelled. We extracted the spectra centred at the peak of the ionized gas emission (as traced by Pa$\alpha$) because it does not coincide with the maximum of the K-band continuum emission. The peaks of the ionized gas and continuum emission are spatially offset by 0.125 ($\sim$200 pc; i.e. the size of one spaxel) with PA=0. It is the case that 0.125 is 1/4 the seeing size and thus we cannot resolve two spectra spatially offset by 0.125. The continuum slope rises towards the red, showing a maximum at $\sim$2.35 . The red dashed line in the left panel of Figure \[fig2\] corresponds to a blackbody of T=1200 K, which better reproduces the K-band nuclear continuum of the Teacup. This K-band spectral shape is not common in type-2 AGN, which generally show the opposite slope, but it has been reported for a few Seyfert 2 galaxies (e.g. NGC7674 – @Riffel06; Mrk348 – @Ramos09; NGC4472 and NGC7743 – @Burtscher15) and it has been interpreted as emission from AGN-heated nuclear dust near the sublimation temperature. In the case of the Teacup, considering the relatively large area probed by the nuclear spectrum ($\sim$830 pc diameter), we are likely detecting hot polar dust within the ionization cones. The change of the continuum slope with increasing aperture, as shown in the right panel of Figure \[fig2\] is also noteworthy. The red excess is only observed in the nuclear spectrum, it then flattens if we consider intermediate apertures and the slope becomes negative in the case of the large aperture spectrum (1.25 diameter). The latter spectral shape resembles the typical K-band spectrum of Seyfert 2 galaxies [@Riffel06; @Ramos09; @Burtscher15]. This change of slope with increasing aperture is due to the extra contribution from stellar light included in the larger apertures. ### Emission line spectrum {#lines} By far the most prominent emission-line feature in the nuclear spectrum of the galaxy is Pa$\alpha$ (see Figure \[fig2\]), followed by Br$\delta$, He I$\lambda$2.060, Br$\gamma$, and the coronal line \[Si VI\]$\lambda$1.963 (all $\lambda$ given in  unless otherwise specified). We also detect several H$_2$ emission lines, indicative of the presence of a nuclear molecular gas reservoir. We fitted the nuclear emission-line spectrum with Gaussian profiles using the Starlink program [dipso]{}. Since the \[Si VI\] and H$_2$ 1-0S(3) lines are blended (see central panel of Figure \[fig3\]) we fixed the FWHM of H$_2$ 1-0S(3) to match those of the other H$_2$ lines to enable us to obtain a reliable fit. In Table \[tab1\] we report the FWHMs corrected for instrumental broadening, velocity shifts (V$_s$) and fluxes resulting from our fits with [dipso]{} with their correspoding errors. The uncertainties in V$_s$ include the wavelength calibration error (7.8 km s$^{-1}$ as measured from the sky spectrum) and the individual fit uncertainties provided by [dipso]{}. In the case of the fluxes, the errors have been determined by adding quadratically the flux calibration error (15%) and the fit uncertainties. We require two Gaussians to reproduce the hydrogen recombination lines and the \[Si VI\]$\lambda$1.963 line profiles. This includes a narrow component of FWHM$\sim$400-460 km s$^{-1}$ and a broad blueshifted component of FWHM$\sim$1600–1800 km s$^{-1}$. We identify this broad component with the nuclear outflow reported from optical spectroscopy by @Villar14 and @Harrison15. We discard the possibility of a BLR origin because the broad components are significantly blueshifted from the narrow component. Additionally not only are they detected in the permitted lines, but also in the \[Si VI\] coronal line. In Figure \[fig3\] we show the profiles and corresponding fits of the emission lines showing blueshifted broad components in the nuclear spectrum, namely Pa$\alpha$, Br$\delta$, \[Si VI\], and Br$\gamma$. In the case of Pa$\alpha$ we fitted a broad component of FWHM=1800$\pm$90 km s$^{-1}$ with V$_s$=-234$\pm$35 km s$^{-1}$ relative to the central wavelength of the narrow component ($\lambda_c$=20353.39$\pm$0.54 Å, giving $z$=0.08516$\pm$0.00003). For Br$\delta$ and Br$\gamma$ we fitted blueshifted broad components consistent with that of Pa$\alpha$ within the uncertainties (see Table \[tab1\]). The coronal line \[Si VI\] also shows a broad component of FWHM=1600$\pm$120 km s$^{-1}$ blueshifted with respect to the narrow \[Si VI\] emission component. It is noteworthy that the narrow component is redshifted by V$_s$=54$\pm$11 km s$^{-1}$ from the narrow Pa$\alpha$ line, which is not observed in any other emission line in the K-band spectrum of the Teacup. As explained in Section \[intro\], coronal lines are generally blueshifted and slightly broader than lower ionization emission lines (e.g. @Rodriguez11). In the case of the Teacup, we detect the blueshifted broad component associated with the outflow, and a redshifted narrow component whose FWHM is the same as those of the recombination lines. This redshifted narrow component suggests that the coronal region and the region where the narrow core of the hydrogen lines is produced are different. Moreover, the lack of detection of coronal lines in the optical spectrum of the Teacup could be indicating that this coronal region is more reddened than the NLR. Using the SDSS spectrum presented in @Villar14 we measured \[Fe VII\]$\lambda$6087Å/\[O I\]$\lambda$6300Å=0.037$\pm$0.010, which is well below the range of $\sim$0.6-5.7 reported by @Rodriguez06b for Seyfert galaxies with strong coronal lines detected in their NIR spectra. Finally, in the case of the He I$\lambda$2.060 and the molecular lines, single Gaussians with FWHM$\sim$400–480 km s$^{-1}$ were sufficient to reproduce the profiles (see Table \[tab1\]). However, whilst neither the He I line nor the narrow core of the hydrogen recombination lines are shifted from the systemic velocity (as measured from the narrow component of Pa$\alpha$), all the H$_2$ lines are systematically blueshifted, with V$_s$=-51$\pm$32 km s$^{-1}$ on average. This could be a first indication of a molecular outflow in the Teacup (see e.g. @Muller16 and references therein), although deeper observations are required to confirm it. It should also be noted that the use of the narrow component of Pa$\alpha$ as a tracer of the systemic velocity is affected by uncertainties [@Villar14; @Muller16]. Accurate measurements of the systemic velocity are needed to understand the kinematic behaviour of the H$_2$ lines. ### Emission line diagnostics {#emission} To determine the degree of obscuration of the nuclear region of the Teacup we calculated the narrow and broad Pa$\alpha$/Br$\gamma$ ratios in the two apertures considered here. By comparing them (see Table \[tab2\]) with the theoretical value of 12.2 [@Hummer87] we can determine the optical and infrared extinction (A$_V$ and A$_K$ respectively) by using the parametrization A$_{\lambda}\propto\lambda^{-1.75}$ [@Draine89]. For the narrow component, we measure a maximum value of A$_V$=2.8$\pm$1.4 mag in the nucleus of the Teacup ($\sim$830 pc diameter). In the large aperture ($\sim$2 kpc diameter) the level of obscuration decreases to 0.55$\pm$1.14 mag. Following the same procedure, from the ratio of the broad components we measure A$_V$=3.5$\pm$1.9 mag in the nucleus of the Teacup and A$_V$=1.5$\pm$1.7 mag in the large aperture. The latter value of the extinction is consistent with A$_V$=1.9$\pm$0.3 mag reported by @Villar14 for the nuclear outflow as measured from the SDSS spectrum. The ratio H$_2$1-0S(1)/Br$\gamma$ can be used to disentangle the dominant excitation mechanism of the gas (see Table \[tab2\]). In the two apertures considered we measure values in the range 0.6–0.9, consistent with AGN photoinization (this ratio is lower than 0.6 for starburst galaxies and higher for LINERs; @Mazzalay13). Although we observe a tendency in 1-0S(1)/Br$\gamma$ to increase with the aperture, the values are consistent within the errors. ![Molecular line ratios measured in the two apertures considered here (0.5and 1.25; green triangles). Pink circles are the ratios measured from the nuclear spectra of the Seyfert 2 galaxies studied in @Ramos09. Yellow squares correspond to the ratios derived for the broad and narrow components of regions A and B of the luminous IR galaxy (LIRG) and gas-rich merger NGC3256 [@Emonts14]. The dashed line indicates the locus of T$_{vib}$=T$_{rot}$. Vertical dotted lines delimitate the regions of “thermal” and “non-thermal” excitation from @Mouri94.[]{data-label="fig4"}](f4.eps){width="6.5cm"} As can be seen from Figure \[fig2\], we detect several H$_2$ lines in the nuclear spectrum of the Teacup. In AGN, the lowest vibrational levels (v=1) of H$_2$ tend to be thermalized (i.e. excited by shocks and/or X-ray illumination; @Hollenbach89 [@Maloney96]), while higher level transitions are populated due to non-thermal processes such as UV fluorescence [@Black87]. The 1-0S(1)/2-1S(1) line ratio is an excellent discriminator between thermal and non-thermal processes. It is $\le$2 in gas excited by UV fluorescence and $\ge$5 in thermally-dominated gas [@Mouri94]. At the same time, the 1-0S(2)/1-0S(0) line ratio is sensitive to the strength of the incident radiation. In Figure \[fig4\] we show the line ratios that we have measured in the two apertures considered here. For comparison, we also plot the nuclear ratios of the five Seyfert 2 galaxies studied in @Ramos09 and those measured for the broad and narrow component of the lines in regions A and B of the gas-rich merger and luminous IR galaxy NGC3256 [@Emonts14]. We find 1-0S(1)/2-1S(1) ratios $\ge$5, consistent with thermal excitation. Indeed, it can be seen from Figure \[fig4\] that the position of the Teacup ratios in the diagram is very different from those measured in Seyfert 2 galaxies. On the other hand, the lower limits on the 1-0S(1)/2-1S(1) line ratio are consistent with the values reported by @Emonts14 for regions A and B of the LIRG NGC 3256, but the 1-0S(2)/1-0S(0) values are significantly higher. This implies that although the molecular gas in the Teacup and in NGC3256 is thermally excited, the strength of the incident radiation is not the same. This is expected considering that the presence of nuclear activity has not been confirmed yet in NGC3256 [@Emonts14], while the Teacup hosts a very luminous AGN. Using the two line ratios mentioned above we can derive the rotational and vibrational temperatures of the gas following @Reunanen02. Although we only have upper limits, in the case of the nuclear spectrum T$_{vib}\simeq T_{rot}\ga 3000$ K, which is characteristic of thermally excited gas. On the other hand, for the large aperture T$_{vib}<<T_{rot}$, indicating that gas excitation is more complex than local thermal equilibrium (LTE) conditions (T$_{vib}=T_{rot}$). Finally, we can use the H$_2$1-0S(1) luminosity to estimate the amount of molecular gas present in the nucleus of the Teacup. Following @Mazzalay13, the relation between the line flux and the warm molecular gas mass is $$M_{H_2}\simeq 5.0875 \times 10^{13} (\frac{D}{Mpc})^2 (\frac{F_{1-0S(1)}}{erg~s^{-1} cm^{-2}})10^{0.4A_K}, \label{eq1}$$ where D=387 Mpc and A$_K$ is the extinction reported in Table \[tab2\]. In the same Table we show M$_{H_2}$ measured in the two apertures considered, as well as the masses of cold molecular gas (M$_{cold}$) calculated by assuming a cold–to–warm mass ratio M$_{cold}$/M$_{H_2}\simeq0.7\times10^{6}$. This ratio was derived observationally by @Mazzalay13 by comparing values of M$_{cold}$ obtained from CO observations and H$_2$ luminosities for a large number of galaxies covering a wide range of luminosities, morphological types and nuclear activity. A similar ratio was reported by @Dale05 for a large sample of active and star-forming galaxies (M$_{cold}$/M$_{H_2}\simeq10^{5-7}$). In the nucleus of the Teacup we measure M$_{H_2}=(3.0\pm0.8)\times10^3 M_{\sun}$ and M$_{cold}=(2.2\pm0.6)\times10^9 M_{\sun}$. If instead of looking at the inner $\sim$830 pc of the Teacup we measure the molecular gas content in the inner 2 kpc, M$_{H_2}=(1.0\pm0.2)\times10^4 M_{\sun}$ and M$_{cold}=(7.4\pm1.8)\times10^9 M_{\sun}$. These cold gas masses are similar to those reported by @Villar13 for a sample of 10 QSO2s at z$\sim$0.2–0.3 with CO measurements. Ionized, coronal and molecular emission-line maps {#extended} ------------------------------------------------- In the previous sections we studied the spectra of the nuclear region of the Teacup. Here we take advantage of the spatial information provided by SINFONI and study the flux distribution and kinematics of Pa$\alpha$, \[Si VI\] and H$_2$1-0S(1). These three emission lines are the highest S/N representatives of the ionized, coronal and molecular phases of the gas respectively. Pa$\alpha$ is the only line that we detect in the extended emission-line structures (i.e. the NE bubble and the SW fan; see Section \[bubbles\]). We note, however, that the SINFONI FOV (9.2$\times$8.7) does not cover the full extent of the NE bubble, as shown in Figure \[fig1\]. In the case of the nuclear Pa$\alpha$ and \[Si VI\] emission we needed two Gaussians to reproduce the observed line profiles and obtain corresponding flux, velocity and velocity dispersion ($\sigma$) maps. For the H$_2$1-0S(1) emission line (hereafter H$_2$) a single Gaussian was sufficient. Since the \[Si VI\] and H$_2$1-0S(3) lines are blended, we fixed the FWHM of the latter emission line to match that of H$_2$1-0S(1) and get reliable fits for the \[Si VI\]. In Figure \[fig5\] we show the flux, velocity and $\sigma$ maps of the broad Pa$\alpha$, broad and narrow \[Si VI\] components, and H$_2$. These maps correspond to a 4$\times$4 (6.4$\times$6.4 kpc$^2$) FOV. Using the flux maps we can estimate the projected sizes of each emission-line region. In Table \[tab3\] we show the observed and seeing-deconvolved FWHMs along the major axis in each case. We note that the intrinsic, deconvolved sizes for the central distribution of the emission lines considered here are approximate sizes. They have been obtained by applying the standard Gaussian deconvolution method based on the well-known quadrature relation FWHM$_{int}^2$ = FWHM$_{obs}^2$ - FWHM$_{seeing}^2$. All the emission-line maps shown in Figure \[fig5\] are resolved except the narrow \[Si VI\] maps. The broad Pa$\alpha$ and broad \[Si VI\] flux maps show similar morphologies, and are both elongated roughly in the same direction (PA$\sim$72–75). We note that this elongation does not coincide with the orientation of the seeing major axis (PA=88.3$\pm$0.1), as measured from the standard star. In the case of the broad \[Si VI\] flux map, two peaks are observed, separated by $\sim$0.3 with a PA$\sim$70 (although it is difficult to distinguish them in the \[Si VI\] flux map shown in Figure \[fig5\], they are evident when displayed in contours). One of these flux peaks coincides with the AGN position, as traced by the maximum of the Pa$\alpha$ emission, and the other could be the coronal counterpart of the compact HR-B region detected by @Harrison15 in the high-angular resolution VLA radio maps. The HR-B compact region is located 0.5 northeast of the AGN position, with PA$\sim$60. As expected for outflow-related components, the maps of the broad Pa$\alpha$ and broad \[Si VI\] emission are dominated by blueshifted velocity components, reaching maximum values of -250 km s$^{-1}$. The narrow \[Si VI\] emission-line maps (third row of Figure \[fig5\]) are similar to those of the broad component, with the exception of the velocity map. The NE part of the emission is redshifted and the SW blueshifted, with maximum velocities of $\pm$150 km s$^{-1}$. From the narrow \[Si VI\] flux map we can set constraints on the size of the coronal line region (CLR) of the Teacup. The observed FWHM that we measure is consistent with the seeing FWHM within the errors (see Table \[tab3\]). Therefore the size of the CLR is formally unresolved and we can estimate an upper limit as \[(FWHM$_{seeing}$+3$\sigma_{seeing}$)$^2$-FWHM$_{seeing}^2]^{0.5}$ = 0.47$\sim$746 pc. The H$_2$ maps (bottom panels of Figure \[fig5\]) are completely different to those of the ionized and coronal lines. They are elongated almost in the N–S direction (PA=-7.7$\pm0.5\degr$), which is roughly perpendicular to the orientation of the Pa$\alpha$ and \[Si VI\] maps. The seeing-deconvolved FWHM along the major axis is 1.4$\pm$0.2 kpc (see Table \[tab3\]). The global H$_2$ velocity field seems to be dominated by rotation, with maximum velocities of $\pm$250 km s$^{-1}$. Deviations from this pattern are identified at some spatial locations, but they correspond to spaxels where the signal-to-noise is lower. The $\sigma$ values are at a maximum in the central region (FWHM$\sim$450 km s$^{-1}$) and decrease toward the edges (FWHM$\sim$150 km s$^{-1}$). {width="18.0cm"} ### Extent of the nuclear outflow In Figure \[fig6\] we show Pa$\alpha$ flux maps extracted in consecutive velocity intervals of 500 km s$^{-1}$, centred at the maximum of the line profile in the central spaxel (see Figure \[fig7\]). These velocity cuts allow us to characterize the extent and orientation of the Pa$\alpha$ emission in the core and the wings of the line. In the central panel we see the Pa$\alpha$ emission corresponding to the core of the line, which we have assumed as systemic redshift. From this map we can distinguish the nuclear emission (inner $\sim$2) and the extent of the emission-line features covered by the SINFONI FOV (the NE bubble and the SW fan). The right and left middle panels correspond to the velocity bins centred at $\pm$500 km s$^{-1}$, so the Pa$\alpha$ wings dominate the emission. The nuclear emission appears elongated in the two panels and on larger scales, depending on whether we are sampling the blue or the red wing of Pa$\alpha$, we see the SW fan or the NE bubble, respectively. {width="13cm"} ![Pa$\alpha$ line profile as extracted from the central spaxel. The continuum has been normalized (dashed line). The solid-green vertical lines indicate the velocity bins (from -2000 to 2000 km s$^{-1}$ in steps of 500 km s$^{-1}$) used for extracting the nine flux maps shown in Figure \[fig6\].[]{data-label="fig7"}](f7.eps){width="8cm"} The blueshifted broad component that we identified in the nuclear spectrum of the Teacup is sampled by the four negative velocity bins (see Figure \[fig7\]). We fitted a Gaussian model to the nuclear region detected in the corresponding flux maps (middle left and top panels in Figure \[fig6\]) and measured seeing-deconvolved FWHMs ranging from 1 to 1.1 kpc along the major axis, with PA=70–75. We measure a similar extension and orientation of the outflow when we sample the red wing of Pa$\alpha$ (middle right and bottom left panels in Figure \[fig6\]). This indicates that we can trace the nuclear outflow up to 1000 km s$^{-1}$ beyond the core of Pa$\alpha$. It is noteworthy that the elongation of the nuclear Pa$\alpha$ emission that we see at the highest velocities is less obvious at $\Delta$v=0 km s$^{-1}$ (i.e. at the core of the line), confirming that the nuclear outflow is extended. Therefore from the analysis of Figure \[fig6\] we conclude that the nuclear ionized outflow is resolved. The observed maximum extension, which corresponds to the -1000 km s$^{-1}$ velocity bin, is FWHM=1.37$\pm$0.06 along the 73.9$\pm$0.2 axis. As noted in Section \[observations\], the standard star that was used to characterize the seeing is slightly elongated along PA=88.3$\pm$0.1, with a maximum extension of 0.58. Even considering the maximum value of the seeing, the nuclear ionized outflow is resolved, with a seeing-deconvolved FWHM=0.68$\pm$0.09 (1.1$\pm$0.1 kpc). We cannot repeat the same exercise for the \[Si VI\] emission line because its blue wing is blended with H$_2$1-0S(3) and it has lower signal-to-noise than Pa$\alpha$. However, if we fit a Gaussian component to the broad \[Si VI\] flux map shown in Figure \[fig5\], we measure a FWHM=0.75$\pm$0.06, which corresponds to a seeing-deconvolved FWHM=0.54$\pm$0.09 (0.9$\pm$0.1 kpc) along the major axis (PA=72.4$\pm$0.3). Therefore, the coronal outflow is also resolved in the same direction as the ionized outflow. ### Large-scale Pa$\alpha$ emission {#bubbles} In Figure \[fig8\] we show the flux, velocity and dispersion maps of the narrow component of Pa$\alpha$ in a 9.5$\times$9 (15.1$\times$14.3 kpc$^2$) FOV. The flux map resembles the morphology of the \[O III\] and H$\alpha$ HST images [@Keel15]. The bulk of the Pa$\alpha$ emission is dominated by the nuclear component, and beyond we detect part of the NE bubble (radial size of $\sim$9.5 kpc), and the SW fan ($\sim$7 kpc). The velocity map (central panel of Figure \[fig8\]) is similar to the \[O III\] velocity map reported by @Harrison15, although it reveals that the kinematics of the gas in the bubble and fan are different from those of the NLR. Whilst the central $\sim$2 show a smooth velocity field, probably coincident with the rotation pattern of the galaxy, beyond this there is an abrupt change in velocity. The NE bubble is redshifted with maximum velocities of 300 km s$^{-1}$ relative to systemic, and the SW fan is blueshifted with V$_{max}$=-300 km s$^{-1}$. The dispersion map also shows a practically constant line width within the central 2 (FWHM$\sim$450-500 km s$^{-1}$) and smaller values in the bubble and fan (FWHM$\sim$200–300 km s$^{-1}$). We note that the FWHM$>$500 km s$^{-1}$ (i.e. $\sigma\ga 210$ km s$^{-1}$) that we measure around the nuclear region ($\sim$1 radius) corresponds to the transition zone where we no longer need to fit a broad component to reproduce the Pa$\alpha$ profiles. {width="18cm"} To study the properties of the extended Pa$\alpha$ emission of the Teacup in more detail, we extracted spectra from different spatial locations in the bubble and fan. In Figure \[fig1\] we indicate the positions and apertures chosen for extracting the spectra in the galaxy nucleus, bubble (N1, N2, and N3) and fan (S1, S2, and S3). In Table \[tab4\] we report the different apertures chosen in each case and the radial distances from the nucleus. We show the six spectra extracted in the bubble and the fan in Figure \[fig9\]. We used [dipso]{} to fit the emission lines using Gaussian profiles, and in Table \[tab4\] we report the corresponding FWHMs, V$_s$ and line intesities. In the case of the bubble, single Gaussian components of FWHM$\sim$300 km s$^{-1}$ were sufficient to reproduce the narrow line profiles in the N1 and N2 spectra, extracted at $\sim$3 kpc from the nucleus. For the more distant N3 region ($\sim$8 kpc from the nucleus) the FWHM is narrower (FWHM$\sim$200 km s$^{-1}$). We note that, as we already knew from the Pa$\alpha$ velocity map shown in Figure \[fig8\], these narrow Pa$\alpha$ lines are significantly redshifted relative to the systemic velocity (V$_s$$\sim$120–270 km s$^{-1}$; see Figure \[fig9\]). We also extracted three spectra mapping two bright regions within the fan (S1 and S2) and a bright knot at $\sim$3 kpc towards the west (S3). Two Gaussians were necessary to reproduce the narrow Pa$\alpha$ profiles detected in the three spectra (see right panels of Figure \[fig9\]). First, the blueshifted (V$_s\sim$-250 km s$^{-1}$) narrow components with FWHM=100–200 km s$^{-1}$ already shown in the velocity map (see Figure \[fig8\]). Second, in the case of the S1 and S3 spectra we required an additional component of FWHM$\sim$200-300 km s$^{-1}$ redshifted by V$_s\sim$40–120 km s$^{-1}$ from systemic. For the S2 spectrum we fitted a slightly broader component of FWHM$\sim$500 km s$^{-1}$ whose V$_s$ is consistent with that of the narrow component (V$_s\sim$-300 km s$^{-1}$). Finally, we report tentative detection of a very broad Pa$\alpha$ component of FWHM$\sim$3000 km s$^{-1}$ in five of the regions considered here (shown as blue Gaussians in Figure \[fig9\]). These broad components are blueshifted relative to systemic in the bubble and fan, with V$_s$ ranging between -60 and -310 km s$^{-1}$. Unfortunately, the combination of sky residuals and low signal–to–noise of the spectra prevents confirmation of these very broad Pa$\alpha$ components. For this reason the results from the fits reported in Table \[tab4\] do not include the broad components. Discussion ========== The nuclear outflow ------------------- Based on the blueshifted broad lines detected in the nuclear K-band spectrum of the Teacup we confirm the presence of the nuclear ionized outflow previously reported by @Villar14 and @Harrison15 using optical spectra, and we report the existence of its coronal counterpart. Although coronal outflows are commonly detected in Seyfert galaxies (e.g. @Gelbord09 [@Davies14]), this is one of the first detections of coronal outflows in QSO2s. Another example is Mrk477, a QSO2 at z=0.037 with an \[O III\] luminosity of 3.3$\times$10$^{42}~erg~s^{-1}$ for which @Villar15 reported a blueshifted component of FWHM=2460$\pm$340 km s$^{-1}$ in the high-ionization line \[Fe VII\]$\lambda$6087 Å (IP=99.1 eV). The FWHM of this blueshifted component is larger than those measured in the optical for the central 3 of the Teacup using the SDSS spectrum (FWHM$\sim$1000 km s$^{-1}$; @Villar14) and GMOS/Gemini IFU data (maximum FWHM$\sim$1000 km s$^{-1}$; @Harrison14). This is consistent with the outflow being reddened, as first claimed by @Villar14 and confirmed in this work, supporting the advantage of using NIR observations to trace AGN outflows closer to their origin. We measure V$_s$=-234$\pm$35 km s$^{-1}$ for the nuclear broad Pa$\alpha$ component. This value is consistent with the velocities reported by @Harrison14 for the blueshifted component of the \[O III\] line measured from GMOS IFU data. They report a velocity offset of $\Delta$v=-150 km s$^{-1}$ for the broad component in the galaxy-integrated spectrum, which corresponds to the central 5$\times$3.5. However, if we look at the velocity maps (Figure A14 in @Harrison14), there is a strong velocity gradient around the position of the AGN, with a maximum velocity offset of -268 km s$^{-1}$. The broad Pa$\alpha$ velocity map that we report here resembles the \[O III\] velocity field, showing maximum velocities of -250 km s$^{-1}$. On the other hand, @Villar14 reported V$_s$=-70 km s$^{-1}$ for the broad component of \[O III\] relative to the narrow core, but measured in an aperture of 3 diameter ($\sim$5 kpc). The broad Pa$\alpha$ maps, shown in the top middle panel of Figure \[fig5\], demonstrate that the SDSS fiber includes both blueshifted and redshifted components, which explains the smaller V$_s$ reported in @Villar14. @Harrison15 extracted a spectrum at the position of the HR-B region detected in the VLA radio maps and measured a FWHM=720 km s$^{-1}$ with an observed velocity of -740 km s$^{-1}$ relative to the narrow component of \[O III\]. Our SINFONI data have very high S/N in the nuclear region and yet, we do not require such a high velocity component to reproduce any of the profiles (see top middle panel of Figure \[fig5\]). This could be due to the combined effect of reddening variation at different spatial locations, different aperture sizes and kinematic substructure within the outflow region. Using the Pa$\alpha$ and \[Si VI\] flux maps we confirm that both the ionized and coronal nuclear outflows are extended. We derived radial sizes of 1.1$\pm$0.1 kpc and 0.9$\pm$0.1 kpc respectively, with PA=72–74. This PA is almost identical to the radio jet orientation measured from the 1.4 GHz FIRST radio maps (PA$\sim$77; @Harrison14). This could be indicating, as first suggested by @Harrison15, that the interaction between the radio jet and the galaxy interstellar medium may have triggered and accelerated the nuclear outflow. The Teacup is then likely an example of radio jets driving outflows in a radio-quiet AGN. HST studies of Seyfert galaxies with linear radio structures showed strong evidence for interactions between the radio structures and the emission line gas in the NLR occurring on scales of $\la$1 kpc (e.g. @Axon98). In QSO2s, the nuclear outflow in Mrk477, which is thought to be triggered by the triple radio source present in this galaxy, has an estimated size of several hundreds of parsecs [@Heckman97; @Villar15], and recent observations of different samples of luminous QSO2s at z$<$0.6, some of them with relatively high radio luminosities, reveal compact ionized outflow sizes of $<$1–2 kpc [@Villar16; @Karouzos16; @Husemann16]. The case of powerful radio galaxies and quasars is different, with radio-induced outflows that can extend up to several kpc, even outside the galaxy boundaries [@Tadhunter94; @Villar99]. In the case of the H$_2$ lines we do not find evidence for broad components in the nuclear spectrum, and the velocity map shows a dominant rotation pattern. The only possible signatures of a molecular outflow in the Teacup are the nuclear narrow components blueshifted by -50 km s$^{-1}$ on average relative to the systemic velocity. In any case, the bulk of the H$_2$ emission comes from the rotating gas distribution shown in Figure \[fig5\], and only a small percentage would be outflowing. Deeper observations and a more accurate determination of the systemic velocity of the galaxy are needed to confirm the presence of a nuclear molecular outflow. The case of the Teacup is very different from that of its low-luminosity counterpart IC5063. This Seyfert 2 galaxy has a similar redshift and radio-power as the Teacup but its NIR spectrum shows wings in the H$_2$ profiles as broad as those of the Br$\gamma$ line. Furthermore, the H$_2$ emission shows a peak in brightness co-spatial with the radio lobe, which @Tadhunter14 interpreted as being due to gas cooling and forming molecules behind a jet-induced shock. The results presented here suggest a different scenario in the case of the Teacup, demonstrating that we do not always see broad molecular lines in the case of galaxies with strong jet-cloud interactions and ionized outflows. This non-detection of broad H$_2$ components in the Teacup could be explained by a two-stage quasar wind scenario [@Lapi05; @Menci08; @Zubovas12; @Faucher12]. Fast winds accelerated by the AGN interact with the galaxy ISM, reaching temperatures $\ge10^7$ K. Once the gas cools down to $\sim$10$^4$ K, it starts to emit warm ionized and coronal lines such as \[O III\], Pa$\alpha$, and \[Si VI\], but further cooling is necessary for the gas to emit in H$_2$. Emission-line structures ------------------------ The bulk of the Pa$\alpha$ emission is dominated by the nuclear component, and beyond, the NE bubble and SW fan expand in opposite directions with projected velocities $\pm$300 km s$^{-1}$ (see Figure \[fig8\]). This is entirely consistent with the \[O III\] velocity map reported by @Harrison15 in a much larger FOV ($\sim$25$\times$25). We extracted spectra at different locations within the emission line structures and we found double-peaked Pa$\alpha$ profiles in the SW fan with FWHMs=100–500 km s$^{-1}$ which are indicative of disturbed kinematics. In contrast, in the NE bubble only one Gaussian is needed to reproduce the Pa$\alpha$ lines, but we measure FWHMs=300 km s$^{-1}$ at two different positions. For comparison, the extended non-outflowing ionized gas detected in the most dynamically disturbed mergers with nuclear activity show FWHM$<$250 km s$^{-1}$ [@Bellocchi13]. According to this comparison with merger dynamics, the detection of turbulent gas in the Teacup is confirmed in apertures N1, N2, S1, and S2, located at distances between 2.8 and 5.6 kpc from the AGN, which correspond to $\sim$6.5 and $\sim$13.5 times half the seeing size. Furthermore, in both the bubble and the fan we find tentative detection of a a very broad Pa$\alpha$ component of FWHM$\sim$3000 km s$^{-1}$. These broad components are blueshifted relative to systemic, with Vs ranging between -60 and -310 km s$^{-1}$. The velocity shifts that we measure for the very broad components and also for the narrow components (see Table \[tab4\]) are consistent with those recently reported by @Keel17 using GMOS integral field spectroscopy. They claim that asymmetric \[O III\] profiles are present in different locations within the NE bubble, reaching maximum velocities of $\pm$1000 km s$^{-1}$. Unfortunately, @Keel17 did not characterize the lines profiles quantitatively, so we do not know if they are detecting the very broad components that we see in our SINFONI data. Our interpretation is that, if these broad components are real, we would be detecting highly turbulent gas in a huge outflow that has cooled sufficiently to emit hydrogen recombination lines and be observable in the NIR. Nonetheless, independently of whether or not these broad components are real, the large scale gas (up to 5.6 kpc from the nucleus) shows turbulent kinematics that can be explained by the action of the outflow. Finally, the H$_2$ maps presented here reveal a rotation pattern whose axis is misaligned with that of the narrow Pa$\alpha$ emission, suggesting that the molecular gas is not coupled with the velocity distribution shown by the ionized gas. The rotating distribution of molecular gas detected in the Teacup could be the quasar-luminosity counterpart of the 100 pc-scale circumnuclear disks (CNDs) observed in nearby Seyfert galaxies using both NIR [@Hicks13] and sub-mm observations [@Garcia14]. These CNDs rotate with similar velocities to those measured here $\sim$200–300 km s$^{-1}$ (e.g. @Helfer03 [@Garcia05]). Furthermore they are not present in matched samples of quiescent galaxies [@Hicks13], indicating that they constitute a key element in the feeding of active SMBHs. Conclusions =========== We have characterized the K-band emission-line spectrum of the radio-quiet QSO2 known as the Teacup galaxy. Thanks to the IFU capabilities of SINFONI we have not only studied the nuclear region of the galaxy, but also the Pa$\alpha$ emission of its kpc-scale emission-line structures (i.e. the NE bubble and the SW fan) within a $\sim$9$\times$9 FOV. Our major conclusions are as follows: - The nuclear K-band spectrum of the Teacup reveals the presence of a blueshifted component of FWHM$\sim$1600–1800 km s$^{-1}$ in the hydrogen recombination lines and also in the coronal line \[Si VI\]$\lambda$1.963. Therefore, we confirm the presence of the nuclear ionized outflow previously detected from optical spectra, and we reveal its coronal counterpart. - The FWHM of the NIR lines associated with the nuclear ionized outflow are larger than those of their optical counterparts. This is consistent with the idea that, because of the lower extinction in the NIR, we can trace the outflow closer to its origin. - Both the ionized and coronal nuclear outflows are spatially resolved, with seeing-deconvolved radial sizes of 1.1$\pm$0.1 and 0.9$\pm$0.1 kpc along the radio axis (PA=72–74). This suggests that the radio jet could have triggered the nuclear outflow. - We find kinematically disrupted ionized gas (FWHM$>$250 km s$^{-1}$) at up to 5.6 kpc from the AGN, which can be naturally explained by the action of the outflow. - The narrow component of \[Si VI\] is redshifted by V$_s$=54$\pm$11 km s$^{-1}$ with respect to the systemic velocity, which is not the case for any other emission line in the K-band spectrum of the Teacup. This indicates that the coronal region is not co-spatial with the NLR. - In the case of the H$_2$ lines, we do not require a broad component to reproduce the profiles seen in the nuclear spectrum, but the narrow lines are blueshifted by $\sim$50 km s$^{-1}$ on average from the galaxy systemic velocity. This could be an indication of the presence of a molecular outflow, although additional observations are required to confirm this. - The H$_2$ maps reveal a rotating structure oriented roughly perpendicular to the radio jet and the broad Pa$\alpha$ and \[Si VI\] maps. This molecular gas structure could be the quasar-luminosity equivalent of the 100 pc-scale CNDs detected in Seyfert galaxies. - We report tentative detection of very broad Pa$\alpha$ components (FWHM$\sim$3000 km s$^{-1}$) at different locations across the NE bubble and SW fan (at up to 5.6 kpc from the AGN). If confirmed, such extremely turbulent components could be hot shocked gas that has cooled sufficiently to be observable in Pa$\alpha$. Acknowledgments {#acknowledgments .unnumbered} =============== Based on observations made with ESO Telescopes at the Paranal Observatory under programme ID 094.B-0189(A). First, we would like to acknowledge the constructive feedback and suggestions of the referee. CRA acknowledges the Ramón y Cajal Program of the Spanish Ministry of Economy and Competitiveness through project RYC-2014-15779 and the Spanish Plan Nacional de Astronom' ia y Astrofis' ica under grant AYA2016-76682-C3-2-P. JPL acknowledges support from the Science and Technology Facilities Council (STFC) grant ST/N002717/1 and the Spanish Plan Nacional de Astronom' ia y Astrofis' ica under grant AYA2012-39408-C02-01. MVM acknowledges support from the Spanish Ministerio de Econom' ia y Competitividad through the grants AYA2012-32295 and AYA2015-64346-C2-2-P. PSB acknowledges support from FONDECYT through grant 3160374. The authors acknowledge the data analysis facilities provided by the Starlink Project, which is run by CCLRC on behalf of PPARC. We finally acknowledge José Antonio Acosta Pulido, Santiago Garc' ia Burillo, Richard Davies, Erin Hicks, and Clive Tadhunter for useful comments that have substantially contributed to improve this work. Appenzeller, I., Wagner, S. 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\#1 \#1 = harvmac =[ ]{} \#1\#2[\#1\#2==]{}\#1 **Abstract** \#1 =cmr10 =cmr7 =cmr5 =cmmi10 =cmmi7 =cmmi5 =cmsy10 =cmsy7 =cmsy5 =cmti10 =’177 =’177 =’177 =’60 =’60 =’60 \#1[\#1\^]{} \#1[[e]{}\^[\^[\#1]{}]{}]{} \#1\#2[\^[\#1]{}\_[\#2]{}]{} /\#1[/]{} \#1 \#1\#2 \#1[\_[[\#1]{}]{}]{} \#1\#2[\_[[\#1]{}]{}\_[[\#2]{}]{}]{} \#1\#2[[\#1\#2]{}]{} \#1\#2\#3[[\^2 \#1\#2 \#3]{}]{} |\#1 \#1[\#1 ]{} \#1[\#1|]{} \#1[| \#1]{} \#1[| \#1|]{} \#1 \#1[1.5ex-16.5mu \#1]{} \#1\#2 \#1[.3ex]{} \#1\#2\#3[Nucl. Phys. B[\#1]{} (\#2) \#3]{} \#1\#2\#3[Phys. Lett. [\#1]{}B (\#2) \#3]{} \#1\#2\#3[Phys. Rev. Lett. [\#1]{} (\#2) \#3]{} \#1\#2\#3[Phys. Rev. [\#1]{} (\#2) \#3]{} \#1\#2\#3[Ann. Phys. [\#1]{} (\#2) \#3]{} \#1\#2\#3[Phys. Rep. [\#1]{} (\#2) \#3]{} \#1\#2\#3[Rev. Mod. Phys. [\#1]{} (\#2) \#3]{} \#1\#2\#3[Comm. Math. Phys. [\#1]{} (\#2) \#3]{} Zachary Guralnik [*Department of Physics, University of California at San Diego, La Jolla, CA 92093*]{} The divergence of lepton and baryon currents in the Standard Model is independent of the fermion masses. For a single family, the baryon and lepton number anomaly is where $W^{\mu\nu}$ is the $SU(2)$ field strength and $B^{\mu\nu}$ is the $U(1)$ field strength. This differs greatly from the axial current equations of Q.E.D. because in Q.E.D. the production of axial charge depends critically on whether or not the electron is massive. I will begin by reviewing the reasons for this sensitivity. Then I will show why these reasons are not applicable to a spontaneously broken theory with a vector current anomaly, such as the standard model. The results give some insight into the production of baryon number in the standard model by sphalerons, which has been of much recent interest. The divergence of the axial current in Q.E.D. \[\] is In a background gauge field the matrix element of the last term is The remaining terms are higher dimension functions of the gauge fields and vanish in an adiabatic aproximation. If the electron is massive then there is no axial charge violation in an adiabatic approximation because the first and last terms in equation   cancel. This cancellation is obvious from the start if one calculates the anomaly using a Pauli Villars regulator field. Then the regulated axial current satisfies where $\chi$ is the regulator field and $\Lambda$ is its mass. $\chi$ is bosonic, so $\chi$ loops have the opposite sign from $\psi$ loops. Therefore there can be no mass independent terms in the matrix element of $\del_{\mu}J^{5 \mu}_r$ in a background gauge field. This cancellation also has a simple spectral interpretation. An explanation of the Q.E.D. axial anomaly based upon the spectrum of a massless electron in a background magnetic field has been given by Nielson and Ninomiya \[\]. Their arguments are briefly summarized below. Consider a uniform background magnetic field in the z direction. In the massless case, positive and negative chirality fermions decouple, so there are two sets of Landau levels. The positive and negative chirality Landau levels contain zero-modes with $E=-p_z$ and $E=+p_z$ respectively. Suppose one turns on a positive uniform electric field ${\cal E}$ in the $z$ direction. In an adiabatic approximation, solutions flow along spectral lines according to the Lorentz force law ${dp\over dt}=e{\cal E}$. Thus right chiral zero-modes slide out of the Dirac sea while left chiral zero-modes slide deeper into the Dirac sea (). This motion produces a net axial charge but no electric charge. By a careful counting of states one reproduces the global form of the anomaly where V is the volume of space. Now consider the same background fields but suppose the electron is massive. In this case, there are no zero-modes among the Landau levels. In the absence of zero-modes adiabatic evolution just maps the Dirac sea into itself, so axial charge can not be adiabatically generated. The discussion above is not applicable to the standard model because standard model fermions can be given masses without changing the baryon or lepton number violation in fixed gauge field background. Dirac mass terms do not carry vector charge, so they do not effect the divergence of a vector current. Yet in an adiabatic limit it seems that presence or absence of mass terms $\it must$ effect the divergence of a current. In the following, this paradox will be resolved by solving the equations of motion for certain background fields which, according to the anomaly equation, should generate charge. I will demonstrate that spatially uniform backgrounds which generate vector charge have no adiabatic limit. Such backgrounds produce the anomaly by causing hopping between energy levels. On the other hand, localized instanton-like backgrounds do possess an adiabatic limit. Backgrounds of this type will be shown to produce the anomaly via fermionic bound states whose energies traverse the gap between $E=-m$ to $E=m$. This give a better understanding of the mechanism of baryon number production in the standard model by sphalerons. The sphaleron configuration corresponds to the half-way point with a zero energy bound state. Because of the chiral couplings, the standard model Landau levels are quite complicated. To avoid calculating Landau levels in $3+1$ dimensions, I will instead consider a spontaneously broken $U(1)$ axial gauge theory in $1+1$ dimensions. While the details of the computation are different, many of the results obtained in $1+1$ dimensions are expected to hold in $3+1$ dimensions. The lagrangian of this theory is This simplified model possesses the two traits whose consistency I wish to demonstrate; a massive spectrum and a mass independent vector current divergence, For the moment I will not consider the full dynamical theory, but only that given by where $\rho(x) =v$ asymptotically. It should be possible to demonstrate the anomaly by considering the momentum space equations of motion, as was done for massless Q.E.D. by Nielsen and Ninomiya using the Lorentz force law. A few remarks are in order about how to do this. Let the Dirac field in a background be expanded as follows: where $u_{p,i}$ are free massive spinors normalized to 1, and the index $i$ distinguishes between positive and negative frequency solutions when the backgrounds vanish. All the background dependance is contained in the time evolution of $c_{p,i}(t)$ When the backgrounds vanish, where Given a knowledge of which states are occupied at an initial time, one can determine which states are occupied at a final time by looking at the evolution of the coeficients $c_{p,i}$. At this point however, the use of this expansion to determine the vector charge or the particle number is very ambiguous. One can make transformations of $\psi$, corresponding to certain transformations of the background fields, which change the $c_{p,i}$. For example transformations exist which map something that looks like the Dirac sea into something that looks like an excited state with non zero vector charge. An invariant definition of charge is needed. Such a definition must depend on the background fields as well as the Fourier coefficients. In order to make the computation of the charge simple, I will only consider processes in which local gauge invariant functions of the background fields vanish at asymptotic times. This means that the initial and final $\theta$ and $A^{\mu}$ are gauge equivalent to $\theta =0$ and $A^{\mu} =0$. In this case the proper definition of charge at asymptotic times is simple. In Dirac sea language, one subtracts the number of vacant negative frequency states from the number of occupied positive frequency states. The occupation number of a positive or negative frequency state of momentum $p$ is proportional to $|c_{p,\pm}|^2$ in the gauge in which the backgrounds vanish. Equivalently, in second quantized language one can adopt a normal ordered definition of charge at asymptotic times. The change in the charge can then be written in terms of Bogolubov coefficients relating the operators $\hat c_{p,i}$ in the asymptotic past to those in the asymptotic future, where these operators are defined in the gauge in which the backgrounds vanish. Note that at intermediate times the gauge invariant backgrounds do not vanish so a well defined Bogolubov transformation between asymptotic past and intermediate times does not exist. Normal ordering is no longer sensible at intermediate times because solutions can not be classified as positive or negative frequency. However, I will never explicitly calculate the charge at intermediate times [^1]. In the spirit of the anomaly calculations done by Nielsen and Ninomiya, I will first consider a process in which a spatially uniform axial electric field is turned on and then off. I will also choose a uniform (spatially parallel transported) Higgs field background. The particular background to be considered is where ${\cal E}(t)={\cal E}$ for $0<t<T$ and $0$ at all other times. In this gauge, with the initial and final backgrounds vanishing, the coefficients $c_{p,i}$ have an immediate interpretation in terms of particle and charge production. Due to the axial electric field, vector charge generation is expected, and should be evident in the time evolution of these coeficients. The equations of motion for $c_{p,i}(t)$ are complicated at low p, but simplify greatly at large $|p|$. The simplification occurs because, as one would expect, the fermion mass can be neglected at large $|p|$. A straightforward calculation gives the equation describing behavior deep in the Dirac sea: This equation is not complete, but the neglected terms are all supressed by factors of ${m\over |p|}$. The solution is where which is easily recognized as an axial version of the Lorentz force law. Therefore states along the negative frequency spectral lines at large $|p|$ flow inward towards small $|p|$. Because of unitarity and Fermi statistics, solutions can not pile up at small $|p|$. Therefore there must be level hopping at small $|p|$. Positive frequency states must appear at a rate matching the inward flow of negative frequency states across some large $|p|$ cutoff (). I thus arrive at the result that the backgrounds of have no adiabatic limit. Therefore the absence of zeromodes has no effect on charge production. Putting the system on a line of length L with periodic boundary boundary conditions on the Fermi field, one finds that the number of states crossing the cutoff per unit time is ${L\over\pi}gE$. This yields the expected anomaly ${1\over L}{dQ\over dt}= {gE\over \pi}$. There is actually no reason to expect adiabatic behavior with uniform backgrounds. The backgrounds of   have singular time dependence when the electric field is turned on or off. One can make the time dependence of these backgrounds nonsingular either by smoothly switching the electric field on and off, or by going to $A^0=0$ gauge. If one does the former, one can try to make the backgrounds vary slowly in time by having the electric field ${\cal E}(t)$ vary slowly in time. However, no matter how slowly the electric field varies, $A^0$ will vary rapidly at large distances since $A^0=-{\cal E}(t)x$. In $A^0=0$ gauge, the backgrounds of  become One can try to make these backgrounds vary slowly in time by making ${\cal E}$ small. Yet, no matter how small ${\cal E}$ is, the Higgs phase $\theta$ winds wildly with time at large distances. Therefore the non adiabatic nature of uniform charge producing backgrounds is an infinite volume effect. It is actually easy to see the low momentum level hopping explicitly without invoking Fermi statistics. In $1+1$ dimensions $\gamma^{\mu}\gamma^5 =\epsilon^{\mu\nu}\gamma_{\nu}$. One can use this fortuitous fact to solve the equations of motion at all momenta. For $0<t<T$ the background fields of    are equivalent to a a background vector gauge field [^2] with $V^0=0$ and $V^1=-{\cal E}x$. The vector field strength vansishes, so the time evolution of $\psi$ at intermediate times is trivial. $\psi^{\prime}\equiv e^{{i\over 2}{\cal E}x^2}\psi$ evolves as a free field: One only has to transform back from $c^{\prime}$ to $c$ to get $c_{p,i}(t)$ as a function of the initial coefficients $c_{r,l}(0)$. The result is that where The quantity within the brackets can be written where $\phi$ is a massive free scalar field in $1+1$ dimensions, and $(z^0,z^1) \equiv (t,{p-r\over E})$. The “light cone” singularity in  gives the leading term of  : Let us rewrite this in a form which is easier to interpret: where I have used the fact that $\gamma^0\gamma^{\pm}=1 \pm \gamma^5$ in $1+1$ dimensions. The axial Lorentz force law is clearly visible in the delta functions and the associated left or right chiral projectors. The low momentum level hopping is also manifest. The hopping of negative frequency to positive frequency states is described by $T_{p,+:r,-}$. At large $p$ and $r$ of the same sign, the spinors $u_{p,+}$ and $u_{r,-}$ have opposite chirality so that $u^{\dagger}_{p,+} {1\pm\gamma^5\over 2}u_{r,-}$ vanishes. Thus in the limit of large momenta at fixed time, $T_{p,+;r,-}$ vanishes. However at small $|p|$ the spinors have mixed chirality so that  does not vanish when $p-r=\pm {\cal E}t$, and the predicted level hopping occurs. It is interesting to note that factor   is almost the transformation function associated with an axial transformation of $\psi$: is equivalent to where This is not to be confused with an axial $\it gauge$ transformation because the initial and final background fields are the same; $A^{\mu}=0$ and $\theta=0$. An axial gauge transformation does nothing, but an axial transformation which leaves the Higgs and gauge potentials unchanged can produce particles and vector charge. This should be no surprise given the bosonization rules \[\] for an axial gauge theory in $1+1$ dimensions. The vector charge density in bosonized form is where $\chi$ is the bosonic counterpart to $\psi$. An axial transformation corresponds to Therefore an axial transformation of the type  above produces a net vector charge. The uniform backgrounds of  are interesting but perverse because the gauge invariant objects built from the Higgs and gauge fields do not fall off at large spatial distances. Furthermore these configurations can exist only in an infinite volume because they are inconsistent with periodic boundary conditions. Therefore let us instead consider localized, charge producing backgrounds. By localized, I mean that the energy density carried by the backgrounds is at its minimum outside a spacetime disc of finite radius. At fixed $\Delta Q$ one can always make such backgrounds vary arbitrarily slowly in time, so that there is no argument against the existence of an adiabatic limit. We are again confronted with the puzzle of how vector charge can be produced by a weak electric field in a theory with a gap. The clue to the puzzle is that one can not go to unitary $(\theta=0)$ gauge from localized backgrounds which produce charge. For such backgrounds $D_{\mu}\phi=0$ asymptotically. Therefore If $\Delta Q$ is not zero, then $\phi^{\ast}\phi$ must vanish somewhere due to the non vanishing Higgs winding number. In the presence of such a defect there may be a bound state as well as the continuum of “scattering” solutions with $E=\pm\sqrt{p^2+m^2}$. In an adiabatic limit the only way charge can appear is if a bound state traverses the mass gap. As the defect is created and destroyed in a process with $\Delta Q=1$, the bound state energy should change continuously from $-m$ to $m$. I will show that this is indeed the case. The sphaleron corresponds to a bound state at the half-way point and has charge one half \[\]. An example of a localized configuration giving $\Delta Q =1$ is where the phase $\alpha (t)$ rotates by a total angle of $-\pi$ from $\alpha (-\infty)=0$ to $\alpha (\infty)=-\pi$. In an adiabatic limit $\alpha(t)$ varies slowly and the gauge fields can be neglected. The defect at $x=0$ is spatially pointlike for convenience; For a fixed $\alpha$, finding the spectrum is a trivial matching problem. (A less singular version of this background is drawn in ) One finds a set of scattering solutions with $E=\pm\sqrt{p^2+m^2}$, but there is also a bound state solution with $E^2<m^2$. Continuity of the solution across $x=0$ requires This yields a bound state with energy $E=-m\cos\alpha$. As $\alpha$ varies adiabatically from $0$ to $-\pi$, a single bound charge is carried across the gap. Note that this alone does not guarantee the net production of charge. A bound state could travel across the gap and leave a negative energy hole. The axial Lorentz force law causes negative frequency states to slide inwards towards zero momentum, which prevents the appearance of a hole. In an adiabatic approximation, the gauge fields are negligible pertubations on the spectrum, but drive the spectral flows needed to produce the anomaly. For more general localized backgrounds, an index theorem enables one to count the number of time dependent energy eigenvalues which travel across the gap. Consider spinor functions $f(x,\tau)$ anihilated by the operator where by varying the parameter $\tau$ from $-\infty$ to $\infty$ one goes slowly through the same cycle of Dirac hamiltonians $\hat H$ that occur in real time. I will write the energy eigenvalues as $E_n(\tau)$ and the energy eigenfunctions as $\chi_n (x,\tau)$. Since $\hat H (\tau)$ is a slowly varying function of $\tau$, the solutions of equation   can be written as where there is no sum on $n$ and This solution is only normalizable if $E_n(\tau)$ has a negative value at $\tau=-\infty$ and a positive value at $\tau=+\infty$. Now consider the adjoint operator A function $a_n(\tau) \chi_n(x,\tau)$ annihilated by $\hat D^{\dagger}$ is only normalizable if $E_n(\tau)$ has a positive value at $\tau=-\infty$ and a negative value at $\tau=+\infty$. Hence the total charge generated by bound states crossing the gap is equal to the difference in the number of normalizable modes annihilated by $\hat D$ and the number of normalizable modes annihilated by $\hat D^\dagger$. This quantity is known as the index of $\hat D$. The operator whose index I wish to calculate is where asymptotically $A^0$ is absent from $\hat D$ because it is negligible in an adiabatic approximation. One can take the adiabatic limit of a process with fixed $\Delta Q$ by making the following gauge invariant rescaling of the fields: In the large $\lambda$ limit $A^0$ vanishes. $A^1$ is a nonvanishing adiabatic parameter, but one can gauge it to zero. Doing so effects only the eigenfunctions of $\hat H (\tau)$ but not the eigenvalues. A straight-forward method to calulate the index of Dirac operators on $R_n$ has been constructed by Weinberg\[\]. Using these methods, the index of $\hat D$ with $A^1=0$ is found to be [^3] which is gauge invariant. This is just as one expects given equation . The relation of this index theorem to charge production can also be understood in terms of the euclidean path integral using methods due to Fujikawa \[\] and ’t Hooft \[\]. The fermionic portion of the partition function is where Let $\psi$ and $\bar\psi$ be expanded as where and $f_n(x)$ and $g_l(x)$ are normalized to one. There is a one to one mapping between eigenfunctions of $\hat K ^{\dagger} \hat K$ and $\hat K \hat K ^{\dagger}$ provided that the eigenvalue is not zero. $\hat K$ maps eigenfunctions of $\kr$ into eigenfunctions of $\kl$ with the same non zero eigenvalue, while $\hat K ^{\dagger}$ does the inverse mapping. However if $\kr f(x) =0$ or $\kl g(x) =0$, then there is no mapping because $\kl f(x)=0$ implies that $\hat K f(x) =0$, and $\kr g(x)=0$ implies that $\hat K^{\dagger} g(x)=0$. The difference between the number of zeromodes of $\kr$ and $\kl$ is given by the index of $\hat K$. A zeromode of either $\kr$ or $\kl$ contributes nothing to the euclidean action. Therefore the integral over the grassman coefficient of a zeromode will vanish unless the coefficient appears in the expansion of an operator in a Green’s function. It is easy to see from this that the contributions of a given Higgs and gauge field background to a Green’s function vanishes except when the number of $\psi$’s in the Green’s function differs from the number $\bar\psi$ ’s by the index of $\hat K$. For example, if $\kr$ has one zeromode $f_0(x)$ and $\kl$ has no zeromode, then In general the net vector charge produced is given by the index of $\hat K$, which in an adiabatic limit is the same as the index of $\hat D$ because the two operators differ only by a factor of $\gamma^0$. The connection between the spectral and path integral approaches to the anomaly is now clear [^4]. An interesting feature of the index theorem for a spontaneously broken axial theory is that it permits Higgs and gauge field backgrounds to create single fermions and not just pairs. The Euclidean equations of motion possess a symetry $\psi\rightarrow\gamma^0\psi^{\ast}$. In the absence of the Higgs coupling to fermions, $\hat K$ anticommutes with $\gamma^5$, so zeromodes can be chosen to be chiral. Therefore in the massless axial theory zeromodes occur in pairs of opposite chirality which are related by the above symetry. This pairing is a reflection of $Q_5$ conservation. However in the spontaneously broken axial theory, $Q_5$ has a Higgs component as well as a fermionic component, and only the sum is conserved. It is no longer true that $\lbrace\hat K , \gamma^5\rbrace = 0$. Therefore zeromodes can no longer be chosen to be chiral. In fact, in an adiabatic approximation one can prove that the mapping $\psi\rightarrow\gamma^0\psi^{\ast}$ does not yield independent solutions. This is done in appendix B. The production of single fermions by a background is not a violation of gauge or Lorentz invariance. For example a single fermion can not get a vacuum expectation value because the path integral over gauge and Higgs fields in the one instanton sector vanishes, even if the fermionic integral does not. So far it has only been demonstrated how charge violation proceeds independently of the fermion masses in the case of background Higgs and gauge fields. I will now show how this works in the dynamical case. This will be done by demonstrating the consistency of the Ward identities with a massive spectrum. Similiar results should hold for three current correlation functions in $3+1$ dimensions. The current equations are and A simple path integral manipulation relates the current equations to Ward idendities for $\JJ$ . One finds that and If it were not for the last term in  , the two Ward identities   and   would ensure the existence of a massless pole in the current correlator \[\]??[A.D. Dolgov and V.I. Zakharov, Nucl. Phys. B27 (1971) 525Y. Frishman, A. Schwimmer, T. Banks and S. Yankielowicz, Nucl. Phys. B177 (1981) 157S. Coleman and B. Grossman, Nucl. Phys. B203 (1982) 205.]{}. Naively one might expect the last term in  to give at most ${\cal O}(g)$ perturbative corrections to this pole or its residue. We are thus confronted with the same dilemma as before. The massive spectrum of a spontaneously broken U(1) axial gauge theory appears to be inconsistent with its vector current anomaly. The resolution of the puzzle lies in the fact that the gauge boson mass is proportional to $g$. It turns out that the last term in   contains an order zero piece which exactly cancels the first term at small $p^2$. The last term in   can be rewritten as where $\phi=\rho\exp{i\theta\gamma^5}$, $\bra{0}\rho\ket{0}=v$, and terms which do not give a zeroth order contribution have been dropped. In t’Hooft $\xi$-gauge there is no mixing between $\theta$ and $A^{\mu}$, so in momentum space the leading term of   is At small $p^2$ this is just $-{1\over \pi}\epsilon^{\mu\alpha}p_{\mu}$, giving the stated cancellation. An almost identical cancellation occurs in the Schwinger model \[\] with no fermion mass term. This model also has a massive spectrum. Furthermore the Ward identities are like those of the axial Higgs model, except that axial and vector labels are swapped: and In bosonized form \[\] the last term of the latter ward identity can be written as where $\phi$ is a scalar field with mass ${e\over\sqrt\pi}$. At momentum small compared to the coupling $e$, this becomes ${1\over\pi}\epsilon^{\alpha\nu}p_{\nu}$ which cancels against the first (anomalous commutator) term of  . Thus the anomaly equation does not imply a massless pole. The apparent paradox of an anomaly equation which is insensitive to particle masses has been resolved in $1+1$ dimensions. The Higgs mechanism creates a gap, but also provides a means to cross the gap. In the presence of a localized background with Pontryagin number one, there is a bound fermion due to the winding Higgs background. This bound fermion acts as an “elevator” which carries charge across the gap. For uniform charge generating backgrounds, the Higgs degree of freedom prevents the existence of an adiabatic limit. In the dynamical case, the gauge boson becomes massive due to the Higgs. The gauge boson mass alters the anomalous ward identities in such a way that they do not imply the existence of a massless state. I believe the mechanisms described here should extend readily to $3+1$ dimensions and the standard model. [**Acknowledgements**]{} The author would like to thank David Kaplan, Aneesh Manohar and Jan Smit for useful discussions. This work was supported in part by the Department of Energy under grant number DOE-FG03-90ER40546, the Texas National Research Laboratory Commission under grant RGFY93-206, and by the National Science Foundation under grant PHY-8958081. In $1+1$ dimensions $\gamma^{\mu}\gamma^5= \epsilon_{\mu\nu}\gamma^{\nu}$ Therefore the $1+1$ dimensional Dirac equation with an axial gauge field $A^{\mu}$ is equivalent to the Dirac equation with a background vector gauge field $V^{\mu}$ where $V^{\mu}=\epsilon^{\mu\nu}A_{\nu}$. Thus it naively appears that if an axial gauge theory does not conserve vector charge, then neither does a vector gauge theory. Conversly if a vector theory does not conserve axial charge, it seems that an axial theory does not conserve axial charge either. Fortunately both these statements are not true. The reason they are not true in a finite volume is that there is an ambiguity in doing Bogoliubov transformations. This ambiguity is removed by choosing either axial or vector gauge invariance. Consider the massless axial gauge theory in an $S_1\otimes R_1$ space-time, and suppose charge is produced by a field strengh which vanishes at asymptotic times. The change in vector charge is equal to minus the change in the Chern-Simons number: Therefore the gauge can be chosen so that $A^{\mu}$ vanishes in either the asymptotic past or the asymptotic future, but not both. I will call the Fermi field $\psi^{in}$ or $\psi^{out}$ depending on whether $A^{\mu}$ vanishes in the past or future. $\psi^{in}$ can be expanded in terms of spinors which have definite momentum and frequency in the asymptotic past. Similiarly $\psi^{out}$ can be expanded in terms of spinors which have definite momentum and frequency in the asymptotic future. Particle production is then determined from the Bogoliubov transformation relating the two sets of expansion coefficients. Now suppose we were to consider the vector gauge theory with the backgrounds $V^{\mu}=\epsilon^{\mu\nu}A_{\nu}$. Suppose also that both the axial and vector field strenghths vanish at past and future times. If both field strengths vanish then $\epsilon^{\mu\nu}\del_{\mu}A_{\nu}$ and $\del_{\mu}A^{\mu}$ vanish and $A^{\mu}$ must be a constant. Consider a configuration with $A^{\mu}$=0 in that past and $A^{\mu}=a^{\mu}$ in the future. The difference between an axial gauge theory and a vector gauge theory lies in the relation between $\psi^{in}$ and $\psi^{out}$. For the axial theory while for the vector theory In light-cone coordinates, the two $\psi^{out}$ fields are related by the transformation This transformation changes the vector charge by an amount $g(a_+ - a_-){L\over 2\pi}$ and the axial charge by an amount proportional to $g(a_+ + a_-){L\over 2\pi}$, where $L$ is circumference of $S_1$. Thus in a finite volume one finds the desired result that the axial theory produces only vector charge and the vector theory produces only axial charge. The arguments above are not sufficient to show this result in an infinite volume. This is because in an infinite volume one can always find a gauge in which the vector potential vanishes in both the asymptotic past and asymptotic future [^5]. For these gauges there is no difference between the out fields in the axial theory and the out fields in the vector theory: both are equal to the in field. However there is no equivalence between localized gauge invariant backgrounds in the axial theory and localized gauge invariant backgrounds in the vector theory provided that either vector charge or axial charge respectively are produced. If the axial and vector field strenghs are both localized, then $\epsilon^{\mu\nu}\del_{\mu}A^{\nu}$ and $\del_{\mu}A^{\mu}$ vanish outside some finite region of space-time. This means that $A^{\mu}$ must be a constant outside this region. The Pontryagin index for both the axial and the vector theory therefore vanishes. Note also that for the massive axial theory, a winding Higgs background has no Q.E.D. counterpart. The Euclidean equations of motion for the fermions of a spontaneously broken axial gauge theory possess the symetry $\psi\rightarrow\ \gamma^0\psi^*$. In this appendix I show that, in an adiabatic limit, this symetry does not yield independent solutions. To be precise, a solution of $\hat K f_0(x,\tau)=0$ has the property that $\gamma^0f_0^*(x,\tau)=\exp(i\alpha)f_0(x,\tau)$, where the phase $\alpha$ is a constant. The same is true for spinors annihilated by the adjoint operator $\hat K ^{\dagger}$. Recall that the solution of $\hat K f_0(x,\tau)=0$ in an adiabatic limit is where $\chi_0(x,\tau)$ is an eigenfunction of the time dependent Hamiltonian for which the energy $E_0(\tau)$ crosses the gap. The Berry’s phase $\beta(\tau)$ will turn out to be important to prevent pairing of zeromodes. At asymptotic positive $x$ the magnitude of the Higgs field is $v$, and one can always choose the gauge so that the phase of the Higgs field is independent of $x$. With this choice the bound state eigenfunctions of $H(\tau)$ at large $x$ are of the form where $c(\tau)$ is the phase of Higgs, and $a(\tau)$ is an arbitrary phase. Therefore at large positive $x$ It is easy show that the above relation holds at all $x$ without knowing the exact form of the solution. If $\chi$ is an solution of then so is $\gamma^0\chi^{\ast}$, because Furthermore the eigenvalue equation   is linear and first order in $x$. Therefore if the relation   is true at any $x$, then it must be true at all $x$. We thus arrive at the result that It appears that there is a time dependent phase relation, but in fact the product of all the phases above is independent of $\tau$. The Euclidean equations of motion are linear and first order in $\tau$, and possess the symetry $\psi\rightarrow\gamma^0\psi^{\ast}$. Therefore if at some fixed $\tau$ then this relation must hold at all $\tau$. The symetry which gives pairs of zeromodes in the massless theory fails to give pairs in the spontaneously broken theory. [^1]: At intermediate times the charge is defined by axial gauge invariance and charge conjugation symetry. For example one can use an axially gauge invariant point split charge which is odd under charge conjugation. When the gauge fields vanish this is equivalent to the usual normal ordered definition of charge. [^2]: This method of solving the Dirac equation brings up a troubling question. If an axial gauge field background can generate vector charge, then apparently a vector gauge field background can also generate vector charge. I discuss why this last statement is not true in appendix A. [^3]: Weinberg applied his methods to count the number of zero energy modes of a vortex-fermion system in 2 spatial dimensions. This system was previously considered by Jackiw and Rossi \[\] who suggested the existence of an index theorem equating the number of fermion zero energy modes to the vortex number. The index theorem for their model is very similiar to the one considered in this paper. [^4]: This connection is not novel. The relation between modes annihilated by the Euclidean Dirac operator and spectral flows which take states in and out of the Dirac sea was discussed by Nielsen and Ninomiya in the context of massless fermions . [^5]: In a finite volume one is prevented from doing this by the gauge invariance of $\exp(ig\oint dx^1A_1)$

--- abstract: 'We report on the crystal structure, physical properties and electronic structure calculations for the ternary pnictide compound EuCr$_{2}$As$_{2}$. X-ray diffraction studies confirmed that EuCr$_{2}$As$_{2}$ crystalizes in the ThCr$_{2}$Si$_{2}$-type tetragonal structure (space group *I4/mmm*). The Eu-ions are in a stable divalent state in this compound. Eu moments in EuCr$_{2}$As$_{2}$ order magnetically below $T_m$ = 21 K. A sharp increase in the magnetic susceptibility below $T_m$ and the positive value of the paramagnetic Curie temperature obtained from the Curie-Weiss fit suggest dominant ferromagnetic interactions. The heat capacity exhibits a sharp $\lambda$-shape anomaly at $T_m$, confirming the bulk nature of the magnetic transition. The extracted magnetic entropy at the magnetic transition temperature is consistent with the theoretical value $Rln(2S+1)$ for $S$ = 7/2 of the Eu$^{2+}$ ion. The temperature dependence of the electrical resistivity $\rho(T)$ shows metallic behavior along with an anomaly at 21 K. In addition, we observe a reasonably large negative magnetoresistance ($\sim$ -24%) at lower temperature. Electronic structure calculations for EuCr$_{2}$As$_{2}$ reveal a moderately high density of states of Cr-3$d$ orbitals at the Fermi energy, indicating that the nonmagnetic state of Cr is unstable against magnetic order. Our density functional calculations for EuCr$_{2}$As$_{2}$ predict a G-type AFM order in the Cr sublattice. The electronic structure calculations suggest a weak interlayer coupling of the Eu-moments.' author: - 'U. B. Paramanik' - 'R. Prasad' - 'C. Geibel' - 'Z. Hossain' title: 'Itinerant and local-moment magnetism in EuCr$_{2}$As$_{2}$ single crystals' --- INTRODUCTION ============ The layered pnictide intermetallic compounds RT$_{2}$Pn$_{2}$ (R = rare-earth elements, T = transition metal; Pn = pnictide) with ThCr$_{2}$Si$_{2}$-type tetragonal structure (space group *I4/mmm*) exhibit a rich variety of transport and magnetic properties. These compounds consist of alternate ‘T-Pn’ layers and ‘R’ layers stacked along the $c$ axis. Following the exploration of these materials over the last 20 years, recently, the discovery of high temperature superconductivity (SC) in the doped AFe$_{2}$As$_{2}$ (A = divalent alkaline metal or rare-earth metal) has generated a new wave of investigations in search of new compounds in this class, which exhibit interesting magnetic and superconducting properties. The Fe atoms in these materials undergo a spin-density-wave (SDW) antiferromagnetic (AFM) transition below 200 K. Upon doping or under application of external pressure, the Fe AFM ordering weakens and SC emerges.[@Rotter; @Sasmal; @Sefat; @Jeevan; @Miclea] Europium is among the few special rare-earth elements having two stable valence configurations: Eu$^{2+}$ ($J = S$ = 7/2) and Eu$^{3+}$ ($J$ = 0); Eu$^{2+}$ bears a strong magnetic moment ($\sim$7.0 $\mu_{B}$) whereas Eu$^{3+}$ does not carry any moment. In a few cases a mixed-valence state of Eu is also observed, for example, in EuNi$_{2}$P$_{2}$ and EuCu$_{2}$Si$_{2}$ \[6,7,8\]. EuFe$_{2}$As$_{2}$ is a member of the Fe based “122” pnictide family where Eu is divalent. This system undergoes a SDW transition in the Fe sublattice at 190 K accompanied by an AFM ordering of Eu$^{2+}$ moments at 19 K \[9\]. The interplay between SC and Eu$^{2+}$ magnetism in doped EuFe$_{2}$As$_{2}$ has been extensively studied recently.[@Jeevan; @Miclea; @Jeevan1; @Zapf; @Anupam; @Paramanik] Replacing As by P in EuFe$_{2}$P$_{2}$, no Fe moment has been observed in the system and the divalent Eu moments order ferromagnetically at $T_C$ = 30 K as has been detected by neutron diffraction measurements.[@Feng; @Ryan] Incommensurate antiferromagnetic structure of Eu$^{2+}$ moments with $T_N$ = 47 K has been found in EuRh$_{2}$As$_{2}$ \[15\]. While EuCu$_{2}$As$_{2}$ exhibits a delicate balance between FM and AFM ordering,[@Sengupta] EuNi$_{2}$As$_{2}$ and EuCo$_{2}$As$_{2}$ order antiferromagnetically.[@Bauer; @Ballinger] Briefly, the pnictide compounds of this structure class show a variety of novel and interesting behaviors. We synthesized a new isostructural compound, EuCr$_{2}$As$_{2}$. Thia compound crystalizes in the ThCr$_{2}$Si$_{2}$-type tetragonal structure with space group *I4/mmm*. As shown in Fig. 1, alternating Eu layers and CrAs layers are stacked along the $c$ axis where Cr atoms form a square planar lattice in the CrAs layer, similar to the AFe$_{2}$As$_{2}$. Recently, Singh et al. have investigated the closely related compound BaCr$_{2}$As$_{2}$ \[19\]. A combined study of physical properties and electronic structure calculations demonstrate that BaCr$_{2}$As$_{2}$ is a metal with itinerant antiferromagnetism, similarly to the parent phases of Fe-based superconductors but with slightly different magnetic structure. Neutron diffraction measurements on BaFe$_{2-x}$Cr$_{x}$As$_{2}$ crystals reveal that the Cr doping in BaFe$_{2}$As$_{2}$ leads to suppression of the Fe SDW transition but the superconductivity (as usually observed in case of other transition metal doping) is prevented by a new competing magnetic order of G-type antiferromagnetism which becomes the dominant magnetic ground state for $x$ $>$ 0.3.[@Athena; @Marty] BaCr$_{2}$As$_{2}$ shows stronger transition metal-pnictogen covalency than the Fe compounds,[@Singh] and in that respect is more similar to the widely studied compound BaMn$_{2}$As$_{2}$. BaMn$_{2}$As$_{2}$ has been characterized as a small band-gap semiconductor with G-type AFM ordering of Mn moments at $T_N$ = 625 K \[22,23\]. This material becomes metallic by partial substitution of Ba by K or by applied pressure on the parent compound.[@KBaMn; @PBaMn; @PKBaMn] In contrast to BaCr$_{2}$As$_{2}$ and BaMn$_{2}$As$_{2}$, both having tetragonal crystal structure, EuMn$_{2}$As$_{2}$ forms in hexagonal crystal structure[@Ruhel] whereas EuCr$_{2}$As$_{2}$ is found to be tetragonal. Very recently, the closely related compounds LnOCrAs (Ln = La, Ce, Pr,and Nd) possessing similar CrAs layers as in BaCr$_{2}$As$_{2}$ have been synthesized by Park et al.[@Hosono] These compounds are isostructural (ZrCuSiAs-type structure with the space group *P4/nmm*) to that of LnOFeAs, which are the parent compounds of Fe-based high $T_c$ superconductors. Powder neutron diffraction measurements at room temperature reveal that Cr$^{2+}$ ions in LaOCrAs bear a large itinerant moment of 1.57 $\mu_{B}$ pointing along the $c$ axis which undergo a G-type AFM ordering. The Néel temperature $T_N$ has been estimated to be in between 300-550 K. Therefore, the related materials possessing CrAs layers are highly enthralling with regard to the physical properties when the AFM ordering is suppressed by doping. Here we report on the crystal structure, physical properties and electronic structure calculations of EuCr$_{2}$As$_{2}$. Our combined experimental investigations and density functional studies show that Eu-ions are in a divalent state and the Eu$^{2+}$ local moments order magnetically at $T_m$ = 21 K. $M(T)$ and $M(H)$ data suggest competing FM and AFM interactions since the $M(T)$ curves look like that of a ferromagnet while the $M(H)$ curves lack the features typically observed in a ferromagnet. A large negative magnetoresistance is found below $T_m$. Density-functional theory-based calculations indicate that the Cr ions bear itinerant moments and the most stable magnetic state in the Cr sublattice is a G-type AFM order. METHODS ======= The single crystals of EuCr$_{2}$As$_{2}$ were grown using CrAs flux as described by Singh et al.[@Singh] The CrAs binary was presynthesized by reacting the mixture of Cr powder and As pieces at 300 $^\circ$C for 10 h, and then at 600 $^\circ$C for 30 h and finally at 900 $^\circ$C for 24 h. A ratio of Eu : CrAs = 1 : 4 was placed in an alumina crucible, and sealed inside a tantalum tube. The assembly was put into a furnace and heated to 1230 $^\circ$C slowly and held there for 13 hours, and then was cooled to 1120 $^\circ$C at a rate of 2 $^\circ$C/h, finally it was furnace-cooled to room temperature. The shiny plate-like EuCr$_{2}$As$_{2}$ crystals were formed in layers, which were cleaved mechanically from the flux. Several plate like single crystals with typical dimension $4\times4\times0.2$ mm$^3$ were obtained. The polycrystalline samples of EuCr$_{2}$As$_{2}$ were prepared using solid state reaction method similar to that of EuFe$_{2}$As$_{2}$ as described in our earlier reports. [@Jeevan; @Jeevan1; @Anupam] Stoichiometric amounts of the starting elements of Eu chips (99.9%), Cr powder (99.999%), and As chips (99.999%) were used for the reaction. The Single crystals and crushed polycrystalline samples were characterized by x-ray diffraction (XRD) with Cu-$K_\alpha$ radiation to determine the single phase nature and crystal structure. Scanning electron microscope (SEM) equipped with energy dispersive x-ray (EDX) analysis was used to check the homogeneity and composition of the samples. The electrical transport properties were measured by standard four probe technique using close cycle refrigerator (Oxford Instruments) and physical property measurement system (PPMS-Quantum design). The $\chi(T)$ = $M(T)$/$H$ and $M(H)$ isotherms were measured using a commercial SQUID magnetometer (MPMS, Quantum-Design). The specific heat was measured by relaxation method in a PPMS-Quantum design. We have carried out the density-functional band structure calculations using the full potential linear augmented plane wave plus local orbitals (FP-LAPW+lo) method as implemented in the WIEN2k code.[@Blaha] The Perdew-Burke-Ernzerhof (PBE) form of the generalized gradient approximation (GGA) was used to calculate the exchange correlation potential.[@Perdew] Additionally, to correct the on-site strong Coulomb interaction within the Eu-$4f$ orbitals we have included U on a mean-field level using the GGA+U approximation. No spectroscopy data for EuCr$_{2}$As$_{2}$ are available in the literature, therefore, we have used U = 8 eV, the standard value for an Eu$^{2+}$ ion.[@Jeevan; @Jeevan1; @Li] In addition, the spin orbit coupling is included with the second variational method in the Eu-$4f$ shell. The set of plane-wave expansion $K_{MAX}$ was determined as $R_{MT}$$\times$$K_{MAX}$ equals to 7.0 and the $K$-mesh used was $10\times10\times10$. RESULTS AND DISCUSSION ====================== \[ExpDetails\] A. Crystal Structure {#expdetails-a.-crystal-structure .unnumbered} ----------------------------------- ![\[fig:XRD\] (Color online) (a) The powder x-ray diffraction pattern of EuCr$_{2}$As$_{2}$ recorded at room temperature. The Rietveld refinement fit (solid black line), difference profile (lower solid green line) and positions of Bragg peaks (vertical blue bars) are also shown. Inset: x-ray diffraction pattern for EuCr$_{2}$As$_{2}$ plate-like single crystal.](fig1.eps){width="8.7cm"} Fig. 1 shows the powder XRD pattern at room temperature for the crushed polycrystalline sample of EuCr$_{2}$As$_{2}$. All the diffraction peaks could be indexed based on the ThCr$_{2}$Si$_{2}$-type structure (space group *I4/mmm*). The crystallographic lattice parameters are listed in Table I. The $c/a$ ratio for EuCr$_{2}$As$_{2}$ is much larger than that of other Eu based transition metal pnictides. A comparison of the structural parameters is shown in Table III. An increased $c/a$ ratio has also been observed in the homologous compound BaCr$_{2}$As$_{2}$ ($a = 3.96 $ [Å]{}  and  $c = 13.632 $ [Å]{}) \[19\] as compared to other transition metal compounds BaT$_{2}$As$_{2}$. The inset of Fig. 1 shows the x-ray diffraction pattern for a EuCr$_{2}$As$_{2}$ single crystal. Only the (00l) diffraction peaks are observed, confirming that the crystallographic $c$ axis is perpendicular to the plane of the plate-like single crystals. From the EDX analysis, the single phase nature of the sample is manifested with obtained atomic ratio of Eu : Cr : As as 20.8 : 38.3 : 40.9. [llll]{} Structure &\ Space group & *I4/mmm*\ \ $a$ (Å) & 3.893(2)\ $c$ (Å) & 12.872(2)\ $V_{cell}$ (Å$^3$)& 195.08(1)\ Refined Atomic Coordinates\ Atom Wyckoff & x &y  &   z\ Eu  2a & 0 & 0  &   0\ Cr  4d & 0 & 0.5  &   0.25\ As  4e & 0 & 0  &   0.363\ \[ExpDetails\] B. Magnetic susceptibility and isothermal magnetization {#expdetails-b.-magnetic-susceptibility-and-isothermal-magnetization .unnumbered} ---------------------------------------------------------------------- Fig. \[fig:MT\] shows the temperature dependence of the magnetic susceptibility $\chi$$_{ab}(T)$ for EuCr$_{2}$As$_{2}$ with the applied magnetic field H = 1 kOe along the crystallographic ab-plane (H$\parallel$$ab$). There is a sharp increase in $\chi$$_{ab}(T)$ below 21 K which tend to saturate at lower temperature as in the case of a ferromagnetic order. At high temperature $\chi$$_{ab}(T)$ follows the modified Curie-Weiss behavior, $\chi(T) = \chi_{0} + C/(T - \theta_{P})$ where $\chi_{0}$ is the temperature-independent term of the susceptibility, C is the Curie constant and $\theta$$_{P}$ is the Weiss temperature. The fitting of inverse susceptibility data by the Curie-Weiss behavior in the temperature range 50-300 K (shown by the solid line) yields the effective paramagnetic moment $\mu_{eff}$ = 7.95 $\mu_{B}$ and $\theta$$_{P}$ = 19 K. Similar fit for $\chi$$_{c}(T)$ data (not shown here) yields the effective paramagnetic moment $\mu_{eff}$ = 8.27 $\mu_{B}$ and $\theta$$_{P}$ = 22 K. For both $\chi$$_{ab}$ and $\chi$$_{c}$, the effective paramagnetic moments are close to the theoretical value of g$\sqrt{S(S+1)}\mu_{B}$ = 7.94 $\mu_{B}$ for free Eu$^{2+}$ moments ($S$ = 7/2, $L$ = 0). The positive values of the paramagnetic Curie temperature $\theta_{P}$ obtained from the fit suggest predominantly ferromagnetic exchange interactions between the Eu$^{2+}$ moments. ![\[fig:MT\] (Color online) Temperature dependence of magnetic susceptibility $\chi$$_{ab}$ for EuCr$_{2}$As$_{2}$ with the applied magnetic field H = 1 kOe. The solid line represents the fit to the Curie-Weiss behavior.](fig2.eps){width="8.7cm"} ![\[fig:MTH\] (Color online) (a) Temperature dependence of magnetization M of EuCr$_{2}$As$_{2}$ single crystal under applied field of 50 Oe and 500 Oe with H in the ab plane (H$\parallel$$ab$) and parallel to the crystallographic $c$ axis (H$\parallel$$c$). All the data shown are in zero field cooled (ZFC) condition. (b) Isothermal magnetization M of EuCr$_{2}$As$_{2}$ single crystal with H$\parallel$$ab$ and H$\parallel$$c$. M-H data were corrected for the demagnetization effect, taken on a plate-like sample.](fig3a.eps "fig:"){width="4.4cm"} ![\[fig:MTH\] (Color online) (a) Temperature dependence of magnetization M of EuCr$_{2}$As$_{2}$ single crystal under applied field of 50 Oe and 500 Oe with H in the ab plane (H$\parallel$$ab$) and parallel to the crystallographic $c$ axis (H$\parallel$$c$). All the data shown are in zero field cooled (ZFC) condition. (b) Isothermal magnetization M of EuCr$_{2}$As$_{2}$ single crystal with H$\parallel$$ab$ and H$\parallel$$c$. M-H data were corrected for the demagnetization effect, taken on a plate-like sample.](fig3b.eps "fig:"){width="4.1cm"} Fig. 3(a) represents the magnetization M (T) in two different orientations of magnetic field, i.e. H$\parallel$$ab$ and H$\parallel$$c$. At high temperature M(T) is almost isotropic, as normally observed for a stable divalent Eu state. Since it bears a spin only ($J = S$ = 7/2) moment, one expects a negligible anisotropy. However, a significant anisotropy in the magnetization is developed below 25 K (M$_{H\parallel c}$/M$_{H\parallel ab}$ $\approx$ 1.5 at 21 K), suggesting an anisotropic magnetic interaction. The rapid increase of $M(T)$ below 21 K gives an impression that the magnetic order is either ferromagnetic in nature or it has a strong ferromagnetic component. To gain further insight on the nature of the magnetic order we have carried out isothermal magnetization measurements with varying magnetic fields at fixed temperatures \[Fig. 3(b)\]. At temperature $T$ = 2 K, the magnetization for H$\parallel$$c$ saturates more rapidly as the magnetic field is increased from H = 0 to 4.1 kOe. For H$\parallel$$ab$, the magnetization saturates at much higher field (18 kOe). A sizable magnetic field is required to achieve the saturation of magnetization for both H$\parallel$$ab$ and H$\parallel$$c$. We do not observe any hysteresis in the M(H) curve at 2 K. It is known that a good quality single crystal of a ferromagnet with small anisotropy and thus small domain wall energy may not always exhibit remanent magnetization.[@Givord] Thus, one cannot rule out the possibility of a FM state of Eu$^{2+}$ moments in EuCr$_{2}$As$_{2}$. However, combining the experimental results with the electronic structure calculations (to be discussed below) where we get an antiferromagnetic ground state of the interlayer Eu moments with very weak interlayer coupling, it is possible that the dominant nearest-neighbor (i.e. intralayer) Eu-Eu interaction is FM as indicated by a positive value of $\theta$$_{P}$, but there could be a weak or frustrated AFM coupling between the layers. In this connection, we may recall other homologous compounds EuCu$_{2}$As$_{2}$ or EuFe$_{2}$As$_{2}$, wherein the intralayer Eu-Eu interaction has been established to be ferromagnetic, while the interlayer antiferromagnetic coupling is very weak.[@Sengupta; @Kasinathan; @Shuai; @Xiao] If we consider the interlayer Eu-Eu antiferromagnetic coupling to be very weak in EuCr$_{2}$As$_{2}$, then, a small external magnetic field as employed in our magnetic measurements can reorient the spin arrangement along the field direction at the onset of magnetic ordering. The saturated magnetization at 2 K is determined to be $\sim$7.78 $\mu_{B}$/f.u and $\sim$7.66 $\mu_{B}$/f.u for H$\parallel$$ab$ and H$\parallel$$c$ respectively, implying that the system is nearly isotropic. The measured saturated magnetization for both the directions are more than that expected for parallelly aligned Eu$^{2+}$ moments (gS = 7.0 $\mu_{B}$/f.u. with g = 2, $S$ = 7/2), indicating that the Cr ions carry an itinerant moment and are contributing to the total magnetization. The electronic structure calculations on EuCr$_{2}$As$_{2}$ (to be discussed below) also suggest that Cr carries an itinerant moment and the most stable magnetic structure in the Cr sublattice is a G-type AFM order. The homologous compound BaCr$_{2}$As$_{2}$ has been proposed to be a metal with itinerant antiferromagnetism.[@Singh] In fact, neutron diffraction measurements on BaFe$_{2-x}$Cr$_{x}$As$_{2}$ crystals reveal that for $x$$>$0.3, the magnetic ground state is consistent with G-type AFM order.[@Marty] It also suggests that the Cr magnetic ordering could be well above room temperature in the BaCr$_{2}$As$_{2}$ parent compound similar to BaMn$_{2}$As$_{2}$ \[23\] which also exhibits a G-type AFM ordering of Mn moments at $T_N$ = 625 K. Moreover, the closely related compounds LnOCrAs possessing similar CrAs layers, in which, Cr ions bear a large itinerant moment of 1.57 $\mu_{B}$ and undergo a G-type AFM ordering with Néel temperature in between 300–550 K.[@Hosono] Similar magnetic ordering of Cr in EuCr$_{2}$As$_{2}$ is also possible at higher temperature. \[ExpDetails\] C. Specific heat {#expdetails-c.-specific-heat .unnumbered} ------------------------------- Fig. 4 shows the plots of temperature dependence of heat capacity $C_{P}(T)$ of the EuCr$_{2}$As$_{2}$ singe-crystal and that of the reference compound BaCr$_{2}$As$_{2}$ taken from ref. 19. The $C_{P}(T)$ of EuCr$_{2}$As$_{2}$ exhibits a sharp $\lambda$-type anomaly due to the magnetic transition at $T_m$ = 21 K, indicating that the magnetic transition is of second-order. The anomaly in $C_{P}(T)$ remains undisturbed under applied field of 500 Oe but with increasing field up to 5 kOe the anomaly is reduced significantly in height and considerably broadened suggesting a field induced change of the nature of the magnetic transition, presumably a field stabilized ferromagnetic order. The magnetic anomaly in the $C_{P}(T)$ makes it difficult to fit the data at lower temperature to extract the electronic specific-heat coefficient ($\gamma$). The measured value of $C_{P}(T)/T$ at 2 K is $\approx$ 225 mJ/molK$^{2}$, but the estimation of $\gamma$ from this value is not reliable as there are magnon contribution from the nearby magnetic ordering of Eu moments. A large $C_{P}(T)/T$ ($\approx$ 250 mJ/molK$^{2}$) at 2 K was also observed in ferromagnetically ordered EuFe$_{2}$P$_{2}$ \[13\]. ![\[fig:cp\] (Color online) Temperature dependence of the specific heat $C_{P}$ of EuCr$_{2}$As$_{2}$ single crystal and that of the reference compound BaCr$_{2}$As$_{2}$ taken from ref. 19. The lower inset shows the $C_{P}$ of EuCr$_{2}$As$_{2}$ at zero field and under external fields of 500 Oe and 5 kOe.](fig4.eps){width="8.8cm"} ![\[fig:sm\] (Color online) $C_{mag}/T$ vs T of EuCr$_{2}$As$_{2}$ and the calculated magnetic entropy $S_{mag}$ vs $T$ shown by the solid line.](fig5.eps){width="8.5cm"} The magnetic part of heat capacity $C_{mag}(T)$ was deduced by the usual method of subtracting the heat capacity of BaCr$_{2}$As$_{2}$ from that of EuCr$_{2}$As$_{2}$ after adjusting the renormalization due to different atomic masses of Ba and Eu, although the mass difference is small here. Based on the mean-field theory,[@Blanco] the heat-capacity jump at the magnetic transition is calculated for the two possible magnetic structures: (i) the equal moment (EM) structure where the magnetic moments are the same at all sites and (ii) the amplitude modulated (AM) structure where the magnetic-moment amplitude varies periodically from one site to another. For EM structure, the jump in the heat capacity at the ordering temperature is given by[@Blanco] $$\Delta C_{EM} = 5 \frac{J(J+1)} {(2J^2+2J+1)} R \label{eq:C}$$ and for AM structure, $$\Delta C_{AM} = \frac{10} {3} \frac{J(J+1)} {(2J^2+2J+1)} R \label{eq:C}$$ where $J$ is the total angular momentum and $R$ is the gas constant. By using $J = S$ = 7/2 for divalent Eu, $\Delta$C$_{EM}$ and $\Delta$C$_{AM}$ amounts to 20.15 J/molK and 13.4 J/molK respectively. Our estimated $\Delta$$C$ ($\approx$ 20.25 J/molK) at $T_{m}$ suggests that EuCr$_{2}$As$_{2}$ possesses an EM structure. In addition, a hump appears in the specific heat at $T \sim T_m$/3, which arises naturally within the mean-field theory for a (2$J$+1)-fold degenerate multiplet. The hump is seen in experimental $C_{mag}(T)$ for EuCr$_{2}$As$_{2}$, which is more pronounced in the $C_{mag}/T$ versus $T$ plot (Fig. 5). The hump in the ordered state is particularly noticeable in compounds containing Eu$^{2+}$ or Gd$^{3+}$ with $S$ = 7/2, and is not visible for lower $S$, e.g., $S$ = 1/2 \[36, 37, 38\]. The magnetic contribution to the entropy $S_{mag}$ was obtained by integrating the $C_{mag}/T$ versus $T$. The $C_{mag}/T$ data were extrapolated from $T$ = 2 K to $T$ = 0 in order to approximate the missing $C_{mag}/T$ data between 0 and 2  K. As can be seen from Fig. 5, the $S_{mag}$ saturates to the expected theoretical value $R ln(2S+1)$ = 17.3 J/molK, where $S$ = 7/2 for Eu$^{2+}$. The magnetic entropy $S_{mag}$ = 14.6 J/molK at $T_{m}$ is 84% of the theoretical value. \[ExpDetails\] D. Transport Properties {#expdetails-d.-transport-properties .unnumbered} -------------------------------------- The temperature dependence of in-plane electrical resistivity $\rho_{ab}(T)$ of EuCr$_{2}$As$_{2}$ as shown in Fig. \[fig:Resistivity\] exhibits a metallic behavior with residual resistivity $\rho_{ab}$ = 2.0 $\mu\Omega$ cm at 2 K and residual resistivity ratio (RRR) = $\rho_{300\,{K}}/\rho_{2\,{K}} \approx 90$. The high residual resistivity ratio together with a low residual resistivity confirms the high quality of our crystals. Since the compound is metallic, most likely the magnetic coupling between the Eu spins is mainly mediated by the conduction electrons through indirect Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. The resistivity data show a kink at the magnetic transition temperature ($T_m$) followed by a rapid decrease in resistivity below $T_m$ due to reduction of spin disorder scattering. Further, we observe significant reduction of the electrical resistivity adjacent to the magnetic ordering temperature on application of magnetic field, leading to a negative magnetoresistance (MR). The MR reaches its maximum value (-24%) near $T_m$. The MR \[Fig. \[fig:Resistivity\]\] is defined as \[$\rho(H)-\rho(0)]/\rho(0)$, where $\rho(0)$ and $\rho(H)$ are the resistivity measured at zero field and under applied field $H$ = 50 kOe respectively. ![\[fig:Resistivity\] (Color online) Temperature dependence of in plane resistivity $\rho$$_{ab}$ for EuCr$_{2}$As$_{2}$ under zero applied magnetic field. The upper inset shows an enlarge view of $\rho$$_{ab}$ under applied magnetic fields of 0 and 50 kOe parallel to the $c$-axis. The lower inset shows the temperature dependence of magnetoresistance for EuCr$_{2}$As$_{2}$.](fig6.eps){width="8.7cm"} \[ExpDetails\] E. Density-functional calculations {#expdetails-e.-density-functional-calculations .unnumbered} ------------------------------------------------- In order to study the electronic and magnetic properties of the compound we start with the calculation of the density of states (DOS) for EuCr$_{2}$As$_{2}$ in the quenched paramagnetic state, that means no spin polarization is allowed on the Cr ions, but, spin polarization is enabled for the Eu ions. Such a study can provide an intimation of magnetic state for the transition metal ions by analyzing the partial density of states (PDOS) at the Fermi level and we can infer whether the magnetic state is favored or not. A similar approach was adopted in prior calculations for EuFe$_{2}$As$_{2}$ \[12, 31\] and the DOS of which is shown here for comparison purposes. We use the experimental lattice parameters $a = 3.893(2)$ [Å]{}, and $c = 12.872(2)$ [Å]{} of EuCr$_{2}$As$_{2}$ for the calculations. The internal coordinate of As (z$_{As}$ = 0.361) is determined by force minimization, which is very close to the experimental z$_{As}$ for EuCr$_{2}$As$_{2}$. For the reference compound EuFe$_{2}$As$_{2}$, the experimental lattice parameters were taken from ref. 9. ![\[fig:dos\] (Color online) Total and partial Densities of states (DOS) for EuCr$_{2}$As$_{2}$ and EuFe$_{2}$As$_{2}$ in the NM state in Cr/Fe sublattice and FM interaction between the intralayer Eu spins in the Eu sublattice.](fig7.eps){width="8.5cm"} The general shape of our density of states for EuCr$_{2}$As$_{2}$ (Fig. \[fig:dos\]) is similar to that for EuFe$_{2}$As$_{2}$, but, with a shift of 3$d$ orbitals in the binding energy, which is expected as Cr has two $3d$ electrons less as compared to Fe. The calculated DOS for EuFe$_{2}$As$_{2}$ is very similar to that reported by Li et al.[@Li] The Eu $4f$ states for both the compounds are quite localized in between -1.5 to -3 eV, suggesting that the Eu ions are in a stable 2+ valence state. The calculated spin moment for Eu$^{2+}$ is about 6.9 $\mu_{B}$ which is consistent with the experimental value. The rest of the DOS can be divided into two parts: (i) The DOS below -2 eV consists of hybridized Cr $3d$ and As $4p$ orbitals. The $p$-$d$ hybridization between As $4p$ and Cr $3d$ is sizable. (ii) The DOS near the Fermi level ranging from -2 eV to +2 eV is basically composed of the Cr $3d$ orbitals. The Fermi level lies on a steep edge of a peak in the partial density of states of Cr-3$d$ orbitals, resulting in a relatively large DOS at the Fermi energy. The corresponding value of PDOS at the Fermi level for Cr-$3d$ is N($E_F$) = 3.03 states/eV per Cr atom, which is greater than that of Fe-3$d$ states (2.15 states/eV per Fe atom) in EuFe$_{2}$As$_{2}$ \[31\]. According to the Stoner criterion, magnetism may occur if $N(E_{F})*I >$ 1, where $I$ is the Stoner exchange-correlation integral.[@DJ] We use the Stoner exchange-correlation integral $I$= 0.38 eV for Cr-3$d$ from the original work of Janak,[@Janak] which amounts to $N(E_{F})*I$ = 1.15. Therefore, the non magnetic (NM) state of Cr is unstable against the magnetic order in EuCr$_{2}$As$_{2}$. [c c c]{} Cr-ordering & $\Delta$ E(eV) & $m_{Cr}(\mu_{B})/ m_{Eu}(\mu_{B})$\ \ \[0.02ex\] NM & 0 & 0/6.9\ \[1ex\] FM & -0.136 & 1.28/6.9\ \[1ex\] S-AFM & -0.064 & 1.69/6.9\ \[1ex\] G-AFM & -0.417 & 2.10/6.9\ \[0.5ex\] ![\[fig:afm\] (Color online) The top panel shows the G-type (Néel or checkerboard) AFM structure where nearest-neighbor spins are aligned antiparallel. The bottom panel represents stripe-type AFM ordering along with the definitions of the in-plane exchange constants $J_1$ and $J_2$. ](fig8.eps){width="5.5cm"} [c c c c c c]{} Compound & $a$([Å]{}) & $c$([Å]{}) & $T_{N}/T_{C} (K)$ & $\mu_{eff}$($\mu_{B}$/f.u.) & Ref.\ \ \[0.02ex\] EuFe$_{2}$As$_{2}$ &3.907 &12.114 &19 &7.79 &9\ \[1ex\] EuCu$_{2}$As$_{2}$ &4.260 &10.203 &15 &7.90 &16\ \[1ex\] EuNi$_{2}$As$_{2}$ &4.096 &10.029 &14 &7.30 &17,42\ \[1ex\] EuCo$_{2}$As$_{2}$ &3.934 &11.511 &39 &7.40 &18,42\ \[1ex\] EuCr$_{2}$As$_{2}$ &3.893 &12.872 &21 &7.95 &This work\ \[0.5ex\] \[comparison\] To examine the most stable magnetic structure of Cr in EuCr$_{2}$As$_{2}$, we have calculated the total energy for different possible magnetic states, namely, (i) a non-spin polarized calculation (no magnetism on Cr), (ii) FM spin configuration, (iii) stripe-type AFM (similar to that in EuFe$_{2}$As$_{2}$) and (iv) G-type AFM. The corresponding total energies of different magnetic states together with the calculated moment values are listed in Table II. It is shown that a G-type AFM order in the Cr sublattice is the lowest energy state for EuCr$_{2}$As$_{2}$. The large energy differences between different Cr magnetic configurations suggest that the magnetic ordering temperature of Cr moments should be high. BaCr$_{2}$As$_{2}$ also possess a G-type AFM ground state of Cr itinerant moments as has been reported by Singh et al.[@Singh] Neutron diffraction studies on BaFe$_{2-x}$Cr$_{x}$As$_{2}$ show a G-type AFM ground state for $x>$ 0.3 \[21\]. Recent experimental investigation on the closely related compound LaOCrAs reveal a G-type AFM order of Cr itinerant moments of 1.57 $\mu_{B}$ \[28\]. Therefore, our calculation of minimum ground state energy for a G-type AFM order of Cr moments in EuCr$_{2}$As$_{2}$ agrees with the magnetic structure of other related compounds. According to the Heisenberg model with nearest-neighbor ($J_1$) and next-nearest-neighbor ($J_2$) spin interactions, the differences in the ordered energies for several collinear commensurate magnetic structures are given by [@Johnston] $$E_{FM} - E_{G-AFM} = 2NS^2(2J_1)$$ and $$E_{G-AFM} - E_{S-AFM} = 2NS^2(2J_2 - J_1)$$ where $N$ is the number of spins $S$. The in-plane G-type AFM is favored when $J_1 >$ 0 and $J_1 > 2J_2$. Our calculations yield an antiferromagnetic $J_1$ ($> 0$) and a large negative $J_2$ with $J_2$/$J_1$ = -0.77. The large negative value of $J_2$ implies that this model is probably not reliable for the system. This might be expected in a itinerant magnetic system with long range magnetic order, as has been pointed out by Singh et al. for the BaCr$_{2}$As$_{2}$ system.[@Singh] Experimentally we do not observe any signature of Cr moment ordering up to 300 K. This is not surprising considering the large total energy differences between different magnetic Cr moment configurations, which suggest an ordering temperature well above the maximum temperature of our measurements. Magnetic measurements at higher temperature are needed to corroborate the expected AFM ordering of Cr moments. The calculated magnetic structure of Cr moments can be verified experimentally using neutron or x-ray scattering measurements. Finally, we discuss the magnetic order in the Eu sublattice. The calculated total energy for the system is found to be minimum when the interlayer Eu spins are antiferromagnetically coupled. Nevertheless, the difference in total energy is very small ($\sim$ 0.0006 eV) whether the interlayer Eu spins are antiferromagnetically coupled or ferromagnetically coupled, implying a rather weak interlayer coupling in the Eu sublattice. So, it is expected that any small external effect (doping, external pressure or external magnetic field) can easily flip the Eu spin from AFM to FM. The P doped EuFe$_{2}$As$_{2}$ system witnesses a similar weak interlayer coupling (0-6 meV), wherein the antiferromagnetic Eu moments arrangement changes from AFM to FM with slight change in doping concentration.[@Kasinathan] Furthermore, the homologous system EuFe$_{2}$As$_{2}$ with antiferromagnetic ground state experiences a field-induced spin reorientation to the FM state for an applied field of just 1 T in the $ab$-plane and at 2 T along the $c$-axis,[@Shuai; @Xiao] which suggests a weak AFM coupling between the interlayer Eu spins. Taking into consideration the relatively large interlayer distance between the Eu layers along the $c$ axis (6.44 [Å]{} for EuCr$_{2}$As$_{2}$, and 6.057 [Å]{} for EuFe$_{2}$As$_{2}$), the interlayer coupling of Eu spins in EuCr$_{2}$As$_{2}$ is expected to be even lesser. \[Conclusions\] CONCLUSIONS =========================== In summary, we have successfully synthesized single and poly crystals of EuCr$_{2}$As$_{2}$ and characterized them using x-ray diffraction, electrical resistivity $\rho(T)$, magnetization and specific heat $C_p(T)$ measurements. The powder XRD data confirm that this compound crystallizes in the body-centered tetragonal structure (space group *I4/mmm*). The $C_p(T)$ and $\rho(T)$ data show anomalies at a temperature $T_m$ = 21 K. While the susceptibility behavior apparently indicates a ferromagnetic order below 21 K, the magnetization data in the ordered state do not show any hysteresis or spontaneous magnetization. Furthermore, the value of $\theta$$_{P}$ obtained from the Curie-Weiss fit in the paramagnetic state is positive and very close to the magnetic transition temperature. These observations indicate that the dominant nearest-neighbor (i.e. intralayer) Eu-Eu interaction is FM but there could be a weak or frustrated AFM coupling between the layers. Also, we do not rule out the possibility of a FM state of Eu$^{2+}$ moments in EuCr$_{2}$As$_{2}$. The measured saturated magnetization for both H$\parallel$$ab$ and H$\parallel$$c$ are larger than the theoretical value of g$S$ = 7.0 $\mu_{B}$ per Eu atom, suggesting that the Cr moments possibly contribute to the observed saturated magnetization values. The $\rho(T)$ data confirm the metallic state of EuCr$_{2}$As$_{2}$ with a negative magnetoresistance (-24%) around the magnetic transition. The magnetic entropy $S_{mag}(T)$ at $T_m$ is 84% of the theoretical value $Rln(2S+1)$ for $S$ = 7/2 of the Eu$^{2+}$ ion and the remaining 16% is recovered by $\approx$ 34 K. The electronic structure calculations indicate that the Cr ions carry itinerant moment and the most stable magnetic structure in the Cr sublattice is a G-type AFM order. Moreover, the large total energy differences between different magnetic Cr moment configurations suggest an ordering temperature well above the maximum temperature of our measurements. Higher temperature magnetic measurements are needed to observe the expected Cr moment ordering. Density-functional calculations suggest a very weak interlayer coupling between the Eu moments. It would be useful and interesting to determine the magnetic structures of EuCr$_{2}$As$_{2}$ by magnetic neutron or x-ray scattering measurements. 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--- abstract: 'Bilayer graphene can exhibit deformations such that the two graphene sheets are locally detached from each other resulting in a structure consisting of domains with different inter-layer coupling. Here we investigate how the presence of these domains affect the transport properties of bilayer graphene. We derive analytical expressions for the transmission probability, and the corresponding conductance, across walls separating different inter-layer coupling domain. We find that the transmission can exhibit a valley-dependent layer asymmetry and that the domain walls have a considerable effect on the chiral tunnelling properties of the charge carriers. We show that transport measurements allow one to obtain the strength with which the two layers are coupled. We performed numerical calculations for systems with two domain walls and find that the availability of multiple transport channels in bilayer graphene modifies significantly the conductance dependence on inter-layer potential asymmetry.' author: - 'Hasan M. Abdullah' - 'B. Van Duppen' - 'M. Zarenia' - 'H. Bahlouli' - 'F. M. Peeters' title: Quantum transport across van der Waals domain walls in bilayer graphene --- Introduction ============ A decade ago, researchers started investigating graphene and its associated multilayers for use as a basis for next generation of fast and smart electronic logic gates. The absence of a band gap leads to different proposals for gap generation[@14-0; @14-1; @14-2]. For example, by changing the size of the graphene flakes into nanoribbons or quantum dots, one can control the energy gap through size quantization[@15; @15-1; @14]. Important experimental advances were achieved in recent years which enabled the fabrication of graphene based electronic devices at the nano scale[@16; @16-1; @intro-1]. The increasing control over the structure of graphene flakes allowed for new devices that could constitute the building blocks for a fully integrated carbon based electronics. An example of this is deformed bilayer graphene, where the two layers are not aligned due to a mismatch in orientation or stacking order resulting in e.g. twisted bilayer graphene. Its electronic structure is strongly different from normal bilayer graphene and exhibits very peculiar properties such as the appearance of additional Dirac cones[@17; @18; @19; @20; @VanderDonck2016; @VanderDonck2016b]. Recent experiments have shown that epitaxial graphene can form step-like bilayer/single layer (SL/BL) interfaces or that it is possible to create bilayer graphene flakes that are connected to single layer graphene regions[@13-1; @13-2; @20-0]. The appearance of these structures fueled theoretical and experimental investigations on the behavior of massless and massive particles in such junctions. For example, few works have investigated different domain walls that separate, for instance, different type of stacking[@AB-BA-1; @AB-BA-2; @pelc] or even different number of layers[@20-1; @22; @26]. These theoretical investigations showed that the transmission probabilities through SL/BL interfaces exhibits a valley-dependent asymmetry which could be used for valley-based electronic applications[@23; @24; @25]. Other theoretical and experimental works focused on the emergence of Landau levels, edge state properties and peculiar transport properties in such systems[@27-0; @27-1; @27-2; @27; @28; @29; @30; @31; @32; @33; @15]. Bilayer graphene flake sandwiched between two single zigzag or armchair nanoribbons[@15; @34] was also investigated and it was found that the conductance exhibits oscillations for energies larger than the inter-layer coupling. Most of these recent theoretical works considered domain walls separating patches of bilayer graphene with different stacking type or where only a single layer was connected to a bilayer graphene sheet. Very recently, however, a number of new bilayer graphene platforms have been synthesized. These consist of regions where the coupling between the two graphene layers is changed. For example in the case of folded graphene [@Wang2017; @Rode2016] a part of the fold forms a coupled bilayer structure, while another part of it is uncoupled[@Schmitz2017; @Hao2016; @Yan2016]. One has also observed systems with domain walls separating regions of different Bernal stacking [@Yin2017; @Yin2016]. In general, these systems can be modelled as being composed of two single layers of graphene (2SL) which are locally bound by van der Waals interaction into an AA- or AB-stacked bilayer structure. Here, we present a systematic study of electrical transport across domain walls separating regions of different inter-layer coupling. We discuss the dependance on the coupling between the graphene layers, on the distance between subsequent domain walls and on local electrostatic gating. For completeness, we also present all possible combinations of locally detached bilayer systems. Analytical expressions for the transport across a single domain wall are also obtained. These results can serve as a guide for future experiments. From a theoretical point of view, one can wonder how charge carriers will respond to transitions between systems that have completely different transport properties. For example, single layer graphene and AA-stacked bilayer graphene are known to feature Klein tunnelling at normal incidence while AB-stacked bilayer graphene shows anti-Klein tunnelling[@Katsnelson2006; @Stander2009]. It is, therefore, interesting to investigate under which conditions these peculiar chirally-assisted tunnelling properties pertain in combined systems, as well as to investigate how the presence of multiple transport channels changes the transport properties. From our study we obtain useful analytical expressions for the transmission probability across a single domain wall. These results also show that the effect of local gating is to break the symmetry between the two layers and to introduce a valley-dependent angular asymmetry, which could be used for a layer-dependent valley-filtering device. We show that the inter-layer coupling strength and stacking has a characteristic effect on the conductance across a domain wall which can be used to measure structural deformations in bilayer graphene. We find that the presence of multiple conductance channels in bilayer graphene can modify the dependance of the conductance on an applied inter-layer potential difference from constructive to destructive. Finally, we show that transitions in-between AA-stacked and AB-stacked bilayer graphene systems largely conserve the parity of the transport channel. The paper is organized as follows. In Sec. \[Sec:Model\], we discuss the formalism, explain the geometry of the investigated domain walls, and define the possible scattering processes between the different transport modes. In Sec. \[Symmetry\], we give analytical expressions for the transmission probabilities through one domain wall and analyze how the symmetry between the graphene layers can be broken by electrostatic potentials. An overview of the numerical results for more complex set-ups consisting of multiple domain walls and gates is presented in Sec. \[Results\]. Finally, in Sec. \[Concl\] we briefly summarize the main points of this paper and comment on possible experimental signatures of the presence of coupling domain walls in bilayer graphene. Model {#Sec:Model} ===== Single layer graphene consists of two inequivalent sublattices, denoted as $\alpha$ and $\beta$, with interatomic distance $a=0.142$ nm and that are coupled in the tight binding (TB) formalism by $\gamma_{0}=3$ eV[@1]. It has a gapless energy spectrum with band crossings at the so-called Dirac points $K$ and $K'$ that are located at the corners of the Brillouin zone. The energy dispersion around one of these points is depicted in Fig. \[fig01\](a). Bilayer graphene consists of two single layers of graphene which can be stacked in two stable configurations: AB-stacked bilayer graphene (AB-BL) or AA-stacked bilayer graphene (AA-BL). In AB-BL, atom $\alpha_{2}$ is placed directly above atom $\beta_{1}$ with inter-layer coupling $\gamma_1\approx0.4$ eV[@li2009band] as shown in Fig. \[fig01\](b). It has a parabolic dispersion relation with four bands. Two of them touch at zero energy, whereas the other two bands are split away by an energy $\gamma_1$. The skew hopping parameters $\gamma_3$ and $\gamma_4$ between the other two sublattices are negligible since they have insignificant effect on the transmission probabilities and band structure at high energies [@Ben]. In AA-BL two single layers of graphene are placed exactly on top of each other such that the structure becomes mirror-symmetric. Atoms $\alpha_2$ and $\beta_2$ in the top layer are located directly above atoms $\alpha_1$ and $\beta_1$ in the bottom layer, with direct inter-layer coupling $\gamma_1\approx0.2\ {\rm eV}$ [@AA-gamma1], see Fig. \[fig01\](c). AA-BL has a linear energy spectrum with two Dirac cones shifted in energy by an amount of $\pm\gamma_1$ as depicted in Fig. \[fig01\](c) by the full curves. Geometries ---------- We consider four different junctions that can be made of from the building blocks depicted in Fig. \[fig01\]: monolayer, AA- stacked and AB-stacked bilayer graphene. Without loss of generality, we assume that the charge carriers are always propagating from the left to the right hand side. Then we consider three different configurations: ($I$) a structure where the leads on the left ($x<0$) and on the right hand side ($x>d$) consist of two decoupled single layers while in between they are connected into an AB-BL (AA-BL) configuration. This is depicted in Fig. \[intro-fig02\](a). We will refer to such a structure as 2SL-AB-2SL (2SL-AA-2SL). ($II$) A structure where the middle region is made up of two decoupled monolayers and whose leads are AB (AA) stacked bilayer graphene. This is depicted in Figs. \[intro-fig02\](b, d). Such a configuration henceforth will be refereed to as AB-2SL-AB (AA-2SL-AA). ($III$) A structure where a domain wall separates an AB (AA) stacked structure from two decoupled single layers. We will assign the abbreviation 2SL-AB (2SL-AA) to this structure if the charge carriers are incident on one of the two separated layers or AB-2SL (AA-2SL) if the coupled bilayer structure is connected to the source. This is depicted in Fig. \[intro-fig02\](c). (IV) left and right leads are bilayer graphene with AA- and AB stacking, respectively, separated by a domain where the two layers are completely decoupled (AA-2SL-AB), see Fig. \[intro-fig02\](e). To describe transport in the above mentioned structures, we allow for scattering between the layers as well as between the different propagating modes in an AB-BL or between the two Dirac cones in AA-BL. In the next section, we describe the transport modes in 2SL and BL and how charge carriers can be scattered in-between them. \                  Scattering definitions {#Scat} ---------------------- In this section we define the model Hamiltonian that describes the different structures. For this purpose we use a suitable basis defined by $\mathbf{\Psi}=(\Psi_{\alpha 1},\Psi_{\beta 1},\Psi_{\alpha 2},\Psi_{\beta 2})^{T}, $ whose elements refer to the sublattices in each layer. The general form of the Hamiltonian near the K-point reads $$\label{eq01} H=\left( \begin{array}{cccc} V_{1} & v_{F}\pi^{\dag} & \tau\gamma_{1} & 0 \\ v_{F}\pi & V_{1} & \zeta\gamma_{1} & \tau\gamma_{1}\\ \tau\gamma_{1} & \zeta\gamma_{1} & V_{2}& v_{F}\pi^{\dag} \\ 0 & \tau\gamma_{1}& v_{F}\pi &V_{2} \\ \end{array}\right).$$ The coupling between the two graphene layers is controlled by the parameters $\tau $ and $\zeta $ through which we can “*switch on*" or “*switch off*" the inter-layer hopping between specific sublattices. This allows to model different stackings by assigning different values to these parameters. For $\tau = \zeta = 0$, the two layers are decoupled and the Hamiltonian reduces to two independent SL sheets. To achieve AA-stacking we select $\tau =1$ and $\zeta =0$ while for AB-stacking we need $\tau =0$ and $\zeta =1$. In Eq. $v_{F}\approx10^{6}$ m/s is the Fermi velocity[@1] of charge carriers in each graphene layer, $\pi=p_{x}+ip_{y}$ denotes the momentum, $V_1$ and $V_2$ are the potentials on layers 1 and 2. In the present study, we only apply these potentials in the intermediate region. We assume that the domain wall is oriented in the $y$-direction and of infinite length. Therefore, the system is translational invariant and the momentum $p_{y}$ is conserved. This enables us to write the wave function as $\mathbf{\Psi}(x,y)=e^{ik_{y}y} \mathbf{\Phi}(x)$. ### Decoupled graphene layers The eigenfunctions of the 2SL Hamiltonian are those of the isolated graphene sheet, [@1] $$\label{eq02} \mathbf{\Phi}=\left(\begin{array}{cccc} \phi_{1}\ \\ \phi_{2}\ \end{array}\right),\phi_j=\left(\begin{array}{cccc} \mu^{-}_j & -\mu^{+}_j \\ 1 & 1 \end{array}\right)\left(\begin{array}{cccc} e^{ik_jx} \\ e^{-ik_jx} \end{array}\right),$$ where $j=1,2$ is the layer index, $ k_j=\sqrt{(\epsilon+s_{j}\delta)^2-k_y^2} $ with $s_j= $sgn$\left( j-1.5 \right)$, $\mu^\pm_ j=(k_j\pm\ ik_y)/(\epsilon+s_{j}\delta)$, $\epsilon=E-v_0,\ \delta=(V_1-V_2)/2,\ v_0=(V_1+V_2)/2$. Introducing the length scale $l=\hbar v_{F}/\gamma_{1}$, which represents the inter-layer coupling length, allows us to define the following dimensionless quantities:$$\begin{aligned} \epsilon\rightarrow\frac{\epsilon}{\gamma_1},\ v_0\rightarrow\frac{v_0}{\gamma_1},\ \delta\rightarrow\frac{\delta}{\gamma_1},\ k_y\rightarrow lk_y,\ \text{and}\ \vec r\rightarrow\frac{\vec r}{l}. \label{dimensionless}\end{aligned}$$ Notice that for the two stacking configurations, $\gamma_{1}$ was found to be different. For the AB-BL the value is $\gamma_1\approx0.4$ eV while for AA-BL it is $\gamma_1\approx0.2$ eV[@li2009band; @AA-gamma1; @AA-Yuehua2010]. ![(Colour online) Different geometries for bilayer and two decoupled graphene layer interfaces with schematic representation of the transmission probabilities. (a) AA or AB stacking bilayer graphene sandwiched between two SL garaphene layers (2SL-AA(AB)-2SL), (b) AB-BL leads with 2SL as intermediate region (AB-2SL-AB), (c) two single gaphene layers connected to AB-BL(2SL-AB), and (d) similar to (b) but now with AA-BL as the leads with two upper (red)-lower (blue) shifted Dirac cones (AA-2SL-AA). (e) left and right leads are bilayer graphene with different stacking connected to the two decoupled graphene sheets (AA-2SL-AB). The possible transmission processes between the different conduction channel are indicated above the respective junctions. []{data-label="intro-fig02"}](trans-channels1.pdf "fig:"){width="3.6"}\ ![(Colour online) Different geometries for bilayer and two decoupled graphene layer interfaces with schematic representation of the transmission probabilities. (a) AA or AB stacking bilayer graphene sandwiched between two SL garaphene layers (2SL-AA(AB)-2SL), (b) AB-BL leads with 2SL as intermediate region (AB-2SL-AB), (c) two single gaphene layers connected to AB-BL(2SL-AB), and (d) similar to (b) but now with AA-BL as the leads with two upper (red)-lower (blue) shifted Dirac cones (AA-2SL-AA). (e) left and right leads are bilayer graphene with different stacking connected to the two decoupled graphene sheets (AA-2SL-AB). The possible transmission processes between the different conduction channel are indicated above the respective junctions. []{data-label="intro-fig02"}](bands-AA-2SL-AB.pdf "fig:"){width="2.15"} In order to discuss the different scattering modes, we introduce the notation $A_{\rm incoming}^{\rm outgoing}$, where $A$ can stand for transmission ($T$) or reflection ($R$) probabilities and the indexes denote the mode by which the particles are incoming or outgoing. Fig. \[intro-fig02\] depicts all possible transitions that are considered in the present work. Fig. \[intro-fig02\](a) shows all possible transmission processes in a 2SL-BL-2SL system where $t$ denotes the top layer on either side and $b$ the bottom layer. For example, $T^{b}_{t}$ denotes a particle coming through the top layer and exiting on the bottom layer. ### AB-stacking For AB-BL there are two branches corresponding to propagating modes. These branches correspond to the wave vector $k^{\pm}$ given by $$\label{eq03} k^{\pm}=\left[-k^{2}_{y}+\epsilon^{2}+\delta^{2} \pm \sqrt{\epsilon^{2}(1+ 4\delta^{2})-\delta^{2}}\right]^{1/2},$$ The modes presented in Eq. (\[eq03\]) labeled by “$k^{+}$” correspond to eigenstates that are odd under layer inversion, while the “$k^{-}$”modes are even. These modes are shown, respectively, in blue and red in Fig. \[intro-fig02\](b). This means that there are two available channels for transmission at a given energy, and an additional two for the reflection probabilities. Note that for energies $0<E<\gamma_1$, there is only one propagating mode and one transmission and reflection channel. Similarly, the wave function of AB-BL can be written as [@Ben] $$\label{eq04} \Psi(x,y)=G M(x)C e^{ik_{y}y},$$ where $M(x)$ corresponds to a $4 × 4$ diagonal matrix consisting of exponential terms, while the components of the constant vector $C$ depend on the propagating region, and $G$ is given by $$\label{eq05} G=\left(\begin{array}{cccc} \xi^{+}_{-} & -\xi^{+}_{+} & \xi^{-}_{-} & -\xi^{-}_{+} \\ 1 & 1 & 1 & 1 \\ \rho^{+} & \rho^{+} & \rho^{-} & \rho^{-} \\ \zeta^{+}_{+} & -\zeta^{+}_{-} & \zeta^{-}_{+} & -\zeta^{-}_{-} \\ \end{array}\right),$$ where $\xi^{\pm}_{\pm}=(k^{\pm}\pm ik_{y})/E-\delta,\ \rho^{\pm}=(\epsilon-\delta)\left[1-((k^{\pm})^{2}+k^{2}_{y})/(\epsilon-\delta)^{2}\right] $ and $\zeta^{\pm}_{\pm}=(\epsilon-\delta)\rho^{\pm}\xi^{\pm}_{\pm}/(\epsilon+\delta)$. The use of the matrix notation will prove to be very useful to construct the transfer matrix as outlined below. ### AA-stacking In the case of an AA-BL, the corresponding wave function can be written similar to Eq. but now with the matrix $G$ given by $$\label{eq07} G=\left(\begin{array}{cccc} \xi^{-}_{+} & \xi^{+}_{+} & \xi^{-}_{-} & \xi^{+}_{-} \\ 1 & 1 & 1 & 1 \\ \zeta^{-}_{+} & \zeta^{+}_{+} & \zeta^{-}_{-} & \zeta^{+}_{-} \\ \rho^{+} & \rho^{+} & \rho^{-} & \rho^{-} \\ \end{array}\right),$$ where $\rho^{\pm}=\frac{1}{2\epsilon}\left[-(k_y^2+(k^\pm)^2)+(\epsilon^{}-\delta)^2+1)\right] $, $\xi^{\pm}_{\pm}=(\rho^{\pm}+\delta+\epsilon)(ik_y\pm k^\pm)/(\delta^2-\epsilon^2+1) $ and $\zeta^{\pm}_{s}=(\xi^{\pm}_{\pm}-\rho^{\pm}(ik_y \pm k^\pm)/(\epsilon+\delta$). To investigate when scattering between the Dirac cones of AA-BL is allowed or forbidden, one can apply a unitary transformation that forms symmetric and anti-symmetric combinations of the top and bottom layer. This yields a Hamiltonian in the basis $\mathbf{\Psi}=2^{-1/2}(\Psi_{\alpha 2}+\Psi_{\alpha 1},\Psi_{\beta 2}+\Psi_{\beta 1},\Psi_{\alpha 2}-\Psi_{\alpha 1},\Psi_{\beta 2}-\Psi_{\beta 1})^{T}$ of the form: $$\label{eq10} H_{AA}=\left( \begin{array}{cccc} \gamma_1+v_0 & v_{F}\pi^{\dag} & -\delta & 0 \\ v_{F}\pi & \gamma_1+v_0 & 0 & -\delta\\ -\delta & 0 & -\gamma_1+v_0& v_{F}\pi^{\dag} \\ 0 & -\delta& v_{F}\pi & -\gamma_1+v_0 \\ \end{array}\right).$$ For $\delta=0$, this Hamiltonian is block-diagonal and represents two Dirac cones as shown in Fig. \[fig01\](c). The two cones correspond to modes with wave vector $k^{\pm}$ given by $$k^{\pm}=\left[-k^{2}_{y}+\left(\epsilon\pm \sqrt{(1+ \delta^{2})} \right)^2 \right]^{1/2}. \label{k_vector_AA}$$ In Fig. \[fig01\](c) the blue bands corresond to the odd $k^{+}$ modes and red bands, denoting the even modes, are given by the $k^{-}$ wavevector. In these equations, $v_0$ denotes the energy shift of the whole spectrum. This shift can be chosen zero by assigning the same magnitude but different signs to the electrostatic potentials on both layers $V_1=-V_2$. Eq. shows that for zero electric field ($\delta=0$) both cones are decoupled and the scattering between them is strictly forbidden. This was used before in Ref. \[\] to propose AA-BL as a potential candidate for *cone-tronics* based devices. However, this protected cone transport is broken for finite bias ($\delta\neq0$) and hence scattering between the cones is allowed. Furthermore, one might wonder if the charge carriers stay within their cone transport through a domain consisting of two decoupled layers. ### Scattering probability In order to calculate the scattering probability in the reflection and transmission channel, we use the transfer matrix method together with boundary conditions that require the eigenfunctions in each domain to be continious for each sublattice [@Barbier; @Ben2]. To conserve probability current we normalize transmission probabilities $T$ and reflection probabilities $R$ such that $$\sum_{i,j}\left( T_i^j+R_i^j \right)=1,$$ where, the index $i$ refers to the incoming mode while the index $j$ denotes the outgoing mode. For a coupled bilayer the different modes are labelled by “$-$” for the modes that are even under in-plane inversion and by “$+$” for odd modes. For a decoupled 2SL system, we employ the notation $t$ for the top layer and $b$ for the bottom layer. For example, for the system 2SL-AB-2SL and for an incident particle in the top layer of 2SL gives $T_t^t+T_t^b+R_t^t+R_t^b=1$. In Fig. \[intro-fig02\] all possible transition probabilities are shown schematically. ### Conductance To obtain measureable quantities, we finally calculate the zero temperature conductance that can be obtained from the Landauer-Büttiker formula [@Landauer] where we have to sum over all the transmission channels, $$\label{eq08} {G_i^{j}}(E)=G_{0}\frac{L_y}{2 \pi}\int_{-\infty}^{+\infty}dk_{y} T_{i}^j(E,k_y),$$ with $L_y$ the length of the sample in the $y$-direction and $G_0=4\ e^2/h$. The factor $4$ comes from the valley and spin degeneracy in graphene. The total conductance of any configuration is the sum of all available channels $G_T=\sum_{i,j}G_i^j$.     Transmission across a single domain wall {#Symmetry} ======================================== ![(Colour online) The angle-dependent transmission and reflection probabilities through (a, b) 2SL-AA and (c, d) AA-2SL systems. The systems in (b, d) are the same as in (a, c), respectively, but where now the right side of the junction is subjected to an electrostatic potential of strength $ v_0= 1.5\gamma_1.$ In the system 2SL-AA $R_b^{b(t)}=R_t^{t(b)}$ and $T_b^{\pm}=T_t^{\pm}$ while $R_+^{-}=R_-^{+}=0$ and $T_\pm^{b}=T_\pm^{t}$ in AA-2SL system. In all panels $E=1.2\ \gamma_1$.[]{data-label="polar-SL-AA"}](SL-AA-T.pdf "fig:"){width="1.25"} ![(Colour online) The angle-dependent transmission and reflection probabilities through (a, b) 2SL-AA and (c, d) AA-2SL systems. The systems in (b, d) are the same as in (a, c), respectively, but where now the right side of the junction is subjected to an electrostatic potential of strength $ v_0= 1.5\gamma_1.$ In the system 2SL-AA $R_b^{b(t)}=R_t^{t(b)}$ and $T_b^{\pm}=T_t^{\pm}$ while $R_+^{-}=R_-^{+}=0$ and $T_\pm^{b}=T_\pm^{t}$ in AA-2SL system. In all panels $E=1.2\ \gamma_1$.[]{data-label="polar-SL-AA"}](SL-vAA-T.pdf "fig:"){width="1.25"}\ ![(Colour online) The angle-dependent transmission and reflection probabilities through (a, b) 2SL-AA and (c, d) AA-2SL systems. The systems in (b, d) are the same as in (a, c), respectively, but where now the right side of the junction is subjected to an electrostatic potential of strength $ v_0= 1.5\gamma_1.$ In the system 2SL-AA $R_b^{b(t)}=R_t^{t(b)}$ and $T_b^{\pm}=T_t^{\pm}$ while $R_+^{-}=R_-^{+}=0$ and $T_\pm^{b}=T_\pm^{t}$ in AA-2SL system. In all panels $E=1.2\ \gamma_1$.[]{data-label="polar-SL-AA"}](AA-SL-T.pdf "fig:"){width="1.25"} ![(Colour online) The angle-dependent transmission and reflection probabilities through (a, b) 2SL-AA and (c, d) AA-2SL systems. The systems in (b, d) are the same as in (a, c), respectively, but where now the right side of the junction is subjected to an electrostatic potential of strength $ v_0= 1.5\gamma_1.$ In the system 2SL-AA $R_b^{b(t)}=R_t^{t(b)}$ and $T_b^{\pm}=T_t^{\pm}$ while $R_+^{-}=R_-^{+}=0$ and $T_\pm^{b}=T_\pm^{t}$ in AA-2SL system. In all panels $E=1.2\ \gamma_1$.[]{data-label="polar-SL-AA"}](AA-vSL-T.pdf "fig:"){width="1.25"} Here we will present analytical expressions for the transmission probabilities of transport across a single domain wall. These analytical expressions will shed light on the requirements for transport across a domain wall and how local electrostatic gating can affect these transport properties. By doing so, we encounter that curiously, electrostatic gates can break the symmetry between the layers in the transmission probability if there are evanescent modes in the system. The breaking of the layer symmetry results in an asymmetric angular distribution of the transmission probability as will be shown further. We consider a situation where two propagating modes exist in the AB-BL or AA-BL. This requires some caution in defining the incident angle in the calculation of the transmission probabilities. Failing to do so may result in erroneous results such as transmission exceeding unity or unexpected symmetry features[@kumar; @Ben03]. Considering one domain wall, the simplest configuration, separating 2SL and either AA or AB-BL allows to obtain analytic expressions for the transmission probabilities. The incident angle for each propagating mode depends on the type of layer stacking in the incident region. Hence, for charge carriers incident from 2SL we define $$k_j=E\cos\phi,\ k_y=E\sin\phi.$$ On the other hand, when charge carriers are incident from AB-BL we need to define incident angle for each mode separately such that $$k^\pm=\sqrt{E^{2}\pm E}\cos\phi,\ k_y=\sqrt{E^{2}\pm E} \sin\phi.$$ Finally, if charge carriers incident from AA-BL the associated angle is defined as $$k^\pm=(E\pm 1)\cos\phi,\ k_y=(E\pm1) \sin\phi.$$ A straightforward calculation results in the transmission probability for charge carriers incident from 2SL and impinging on AA-BL $$\label{eq15} T_j^\pm=\frac{2(\epsilon+v_0)(\pm1+\epsilon)\textrm{Re}(k^{\pm})}{k_j\left[ \left( \pm1+\epsilon+k^\pm \sec\phi \right)^2+(\mp1+v_0)^{2}\tan^2\phi \right]}\ ,$$ while for the reverse configuration (AA-2SL) it is given by $$\label{eq16} T_\pm^j=\frac{2\epsilon\textrm{Re}(k_{j})}{\cos\phi\left[ \left( \epsilon+k_{j} \sec\phi \right)^2+(\mp1+v_0)^{2}\tan^2\phi \right]}\ .$$ Similar as performed for the AA-BL Hamiltonian, also the AB-BL Hamiltonian can be expressed in terms of symmetric and anti-symmetric combinations of the two layers. This manipulation allows to determine a closed-form expression for the transmission probability of the 2SL-AB structure. The derivation is outlined in Appendix \[Sec:Appendix\] and results in $$\label{eq17} T_j^\pm=4 \textrm{Re}(k^{\pm}) \frac{\eta\left[ \eta^2+\left(\textrm{Im}(k^{\mp})+\kappa_{j} v_0\sin\phi\right)^2\ \right]}{C_{0 }+\sum_{m=1}^4C_m\cos(m\phi)},$$ with $\eta=\epsilon \cos\phi$ and $\kappa_{j}=+1(-1)$ for $j=b( t)$. For the reverse configuration (AB-2SL) the transmission probabilities are $$\label{eq18} T_\pm^j=4 \textrm{Re}(k_j)k^{\pm} \frac{\lambda\left[ \mu^{\pm}+\kappa_{j} v_0 \sin\phi\ \textrm{Im}(k^{\mp}) \right] }{\left\vert Q^{\pm} \right\vert^2},$$ where $ \lambda,\ C_m,\ \mu^\pm$, and$\ Q^\pm$ are functions defined in Appendix \[Sec:Appendix\]. For a domain wall separating 2SL and AA-BL, the transmission probabilities are always symmetric with respect to normal incidence as indicated in Eqs. (\[eq15\],\[eq16\]). In other words, for the 2SL-AA $T_b^\pm(\phi)=T_t^\pm(\phi) $ and similarly $T_\pm^b(\phi)=T_\pm^t(\phi) $ for AA-2SL configuration, and this symmetry still holds when the right side of the junction is gated ($v_0\neq0$). We will refer to this symmetry as *layer symmetry* since it is a consequence of the equivalence of 2SL layers and the symmetric coupling of the AA-BL. Notice that Klein tunnelling for normal incidence in SL and AA-BL is also conserved in the combined structure. For example, in 2SL-AA and for normal incidence ($\phi=0$), the modes become $k_j=\epsilon+v_0,\ k^\pm=\pm1+\epsilon$ and hence Eq. reads $T^\pm_j=1/2.$ Then, for charge carriers propagating in the bottom (top) layer it may be transmitted into $k^+$ or $k^-$ states and thus the total probability is $T_{b(t)}^++T_{b(t)}^-=1/2+1/2=1$. As a result of Klein tunnelling at normal incidence, the corresponding reflection probabilities are zero such that $R_b^{b(t)}=R_t^{t(b)}=0$. In an analogous manner it can be shown that for normal incidence Eq. gives $T_\pm^j=1/2.$ Turning now to the 2SL-AB/AB-2SL case, one can infer from Eqs. (\[eq17\],\[eq18\]) that for $v_0=0$ the layer symmetry holds since the only term carrying asymmetric features is proportional to $v_{0}$. However, for $v_0\neq 0$ it is striking that despite the fact that a homogeneous electrostatic potential does not break any in-plane symmetry in the system, layer symmetry is broken. This leads to an angular asymmetry in the transmission channel, i.e. $T_b^\pm(\phi)=T_t^\pm(-\phi)$ for 2SL-AB and $T_\pm^b(\phi)=T_\pm^t(-\phi)$ for AB-2SL. Upon further analysis of Eqs. (\[eq17\],\[eq18\]), one notices that this asymmetric feature is present in regions in the ($E,k_y$) plane where one of the two modes is propagating while the other is evanescent. In Figs. \[fig003\](a,b) we show a diagram for these different regions associated with 2SL-AB and AB-2SL, respectively. The layer symmetry is broken in the green and pink regions while in the yellow regions layer symmetry holds. The mechanism for breaking the layer symmetry in configurations consisting of AB-BL is attributed only to the evanescent modes. For example, in 2SL-AB (see Fig. \[fig003\]) the transmission probability for charge carriers to be transmitted into $k^+$ from either bottom or top layers of 2SL is $$\label{eq19} T_j^+=4 \textrm{Re}(k^{+}) \frac{\eta\left[ \eta^2+\left(\textrm{Im}(k^{-})+\kappa_{j} v_0 \sin\phi\right)^2 \right]}{C_{0 }+\sum_{m=1}^4C_m\cos(m\phi)},$$ where $\kappa_{b(t)}=1(-1)$. The above equation shows that layer symmetry is broken, $T_b^+(\phi)=T_t^+(-\phi),$ only when $v_0\neq0$ and $\textrm{Im}(k^{-})\neq0$ which is satisfied in the pink and gray regions in Fig. \[fig003\](a). However in the gray region there are no $k^+$ propagating states and consequently the transmission probabilities $T_j^+$ are zero. The same analysis applies also to $T_j^-$ where the asymmetric feature is preserved only when $\textrm{Im}(k^{+})\neq0$ as shown by the green region in Fig. \[fig003\](a). For AB-2SL configuration, the layer asymmetry is only reflected in the $T_+^j$ , see Eq. , since $\textrm{Im}(k^{-})\neq0$ corresponds to the pink region in Fig. \[fig003\](b). While for $T_-^j$, the $k^-$ propagating states are only available for $E>\gamma_1$ (yellow region in Fig. \[fig003\](b)) which coincides with $\textrm{Im}(k^{+})=0$. Thus, the layer symmetry is always conserved in $T_-^j$ as it can be seen in Eq. . Now it is clear why layer symmetry is not broken in the AA-BL configuration; because there are always two propagating modes associated with any energy value. The breaking of angular symmetry in this situation is qualitatively similar to that obtained in AB-BL[@Ben] subject to an inter-layer bias. One can connect this layer asymmetry in the vicinity of the two valleys $K$ and $K'$ through time-reversal symmetry. The Hamiltonian $H_{K'}$ can be related to the Hamiltonian $H_{K}$ through the transformation $$\label{eq20} H_{K'}(\boldsymbol{k})=\Theta H_K(\boldsymbol{-k})\Theta^{-1},$$ where $\Theta$ is the time-reversal symmetry operator. This implies, for example in the $T_{b(t)}^+$ channel, that charge carriers moving from right to left and scattered from the bottom layer to $k^+$ in $K$ valley are equivalent to those scattered from top layer to $k^+$ but moving in the opposite direction in the vicinity of $K'$. If layer symmetry holds in the vicinity of one of the valleys, then the transmission probabilities of charge carriers moving in the opposite directions must be the same. It is worth pointing out here that the layer asymmetry in the $K$ valley is reversed in the $K'$ valley and hence the overall symmetry of the system is restored. Therefore, the macroscopic time reversal symmetry is preserved. Numerical Results {#Results} ================= We first present the results for transmission, and reflection probabilities and for the conductance in the case of domain walls separating 2SL and AA-BL structures. The different regions as defined in Fig. \[fig003\] are superimposed as dashed black and white curves. Moreover, in calculating the transport properties we considered different magnitudes for the electrostatic potential $v_0$ and bias $\delta$ applied to the drain structure. AA-Stacking ----------- ### 2SL-AA/AA-2SL We consider charge carriers tunnelling through 2SL-AA and AA-2SL systems. In Fig. \[polar-SL-AA\](a) we show the transmission and reflection probabilities for charge carriers impinging on pristine AA-BL as a function of incident angle $\phi$. As a result of the layer symmetry, charge carriers incident from bottom/top layer of 2SL and transmitted into the lower Dirac cone ($k^+$) in the AA-BL will have the same transmission probability $T_{b}^+=T_{t}^+$. Similarly, for those charge carriers transmitted into the upper cone, they will also have the same probability $T_{b}^-=T_{t}^-$ regardless which layer they are incident from. This symmetry stems from the fact that the wavefunction in the 2SL are a superposition of two spinors corresponding to the two sublattices while in AA-BL it is a superposition of four. For this reason, charge carriers incident from top or bottom layer of 2SL have the same dynamics and hence share their transmission probability. A partial reflection into the same layer, $R_{b}^{b}=R_{t}^{t}$ is shown in Fig. \[polar-SL-AA\](a), which corresponds to evanescent modes associated with the upper Dirac cone ($k^-$). As in transmission, charge carriers can be back scattered between the layers. However, the absence of the electrostatic potential results in a small scattering current as depicted in Fig. \[polar-SL-AA\](a). In addition, scattering back from top to bottom layer or vice versa occurs also with the same reflection probabilities $R_{b}^{t}=R_{t}^{b}$. Because of chiral decoupling of oppositely propagating waves in AA-BL and in SL, back-scattering is forbidden for normal incidence ($\phi=0$) and thus the reflection probabilities for each channel are zero, i.e. $R_{b}^{b(t)}(0) = R_{t}^{t(b)}(0)=0$. This is associated with perfect tunnelling $T_{b}^+(0) + T_{b}^-(0) = T_{t}^+(0) + T_{t}^-(0) = 1$. The effect holds for all forthcoming structures composed of AA-BL and 2SL. Fig. \[polar-SL-AA\](b) shows the numerical results of the same system, 2SL-AA, but now in the AA region, the potential is increased to $v_0=1.5 \gamma_1$. This shifts the two Dirac cones in energy to $ v_0\pm \gamma_1$. As a result of the presence of the electrostatic potential, a strong scattered reflection $R_{b}^{t}/R_{t}^{b}$ takes place when there are no propagating modes in the AA section. In Figs. \[polar-SL-AA\](c,d), we show the reversed configuration, i.e. an AA-2SL system. The transmission and reflection probabilities for zero ($v_0=0$) and with nonzero ($v_0=1.5 \gamma_1$) electrostatic potentials applied to 2SL are reported in panels (c) and (d) respectively. Similar to the 2SL-AA system, we can note that layer symmetry still holds such that $T_{+}^b = T_{+}^t$ and $T_{-}^b = T_{-}^t$. Furthermore, we find strong non-scattered reflection in the $R_{+}^+$ and $ R_{-}^-$ channels that is associated with evanescent modes on both sides of AA-BL and 2SL whereas the scattered reflection channels $R_{-}^+$ and $ R_{+}^-$ are always zero due to the protected cone transport discussed earlier. ![(Colour online) Density plot of the transmission and reflection probabilities through 2SL-AA-2SL as a function of Fermi energy and transverse wave vector $k_y$ with $v_0= \delta =0$ and width of the AA-BL $d=25$ nm. []{data-label="fig-SL-AA-SL"}](SL-AA-SL-Tbb.pdf "fig:"){width="1.5"} ![(Colour online) Density plot of the transmission and reflection probabilities through 2SL-AA-2SL as a function of Fermi energy and transverse wave vector $k_y$ with $v_0= \delta =0$ and width of the AA-BL $d=25$ nm. []{data-label="fig-SL-AA-SL"}](SL-AA-SL-Rbb.pdf "fig:"){width="1.73"}\ ![(Colour online) Density plot of the transmission and reflection probabilities through 2SL-AA-2SL as a function of Fermi energy and transverse wave vector $k_y$ with $v_0= \delta =0$ and width of the AA-BL $d=25$ nm. []{data-label="fig-SL-AA-SL"}](SL-AA-SL-Tbt.pdf "fig:"){width="1.5"} ![(Colour online) Density plot of the transmission and reflection probabilities through 2SL-AA-2SL as a function of Fermi energy and transverse wave vector $k_y$ with $v_0= \delta =0$ and width of the AA-BL $d=25$ nm. []{data-label="fig-SL-AA-SL"}](SL-AA-SL-Rbt.pdf "fig:"){width="1.73"} ![(Colour online) The same as in Fig. \[fig-SL-AA-SL\], but now with $v_0=1.5 \gamma_1.$ Red and white dashed curves correspond to the lower and upper Dirac cones in AA-BL, respectively, while the black dashed curves are the bands of 2SL. []{data-label="fig-SL-AAv-SL"}](SL-AA-SL-vTbb.pdf "fig:"){width="1.5"} ![(Colour online) The same as in Fig. \[fig-SL-AA-SL\], but now with $v_0=1.5 \gamma_1.$ Red and white dashed curves correspond to the lower and upper Dirac cones in AA-BL, respectively, while the black dashed curves are the bands of 2SL. []{data-label="fig-SL-AAv-SL"}](SL-AA-SL-vRbb.pdf "fig:"){width="1.73"}\ ![(Colour online) The same as in Fig. \[fig-SL-AA-SL\], but now with $v_0=1.5 \gamma_1.$ Red and white dashed curves correspond to the lower and upper Dirac cones in AA-BL, respectively, while the black dashed curves are the bands of 2SL. []{data-label="fig-SL-AAv-SL"}](SL-AA-SL-vTbt.pdf "fig:"){width="1.5"} ![(Colour online) The same as in Fig. \[fig-SL-AA-SL\], but now with $v_0=1.5 \gamma_1.$ Red and white dashed curves correspond to the lower and upper Dirac cones in AA-BL, respectively, while the black dashed curves are the bands of 2SL. []{data-label="fig-SL-AAv-SL"}](SL-AA-SL-vRbt.pdf "fig:"){width="1.73"} ![(Colour online) The same as in Fig. \[fig-SL-AA-SL\], but now with $v_0=1.5\gamma_1$ and $ \delta =0.6\gamma_1$. []{data-label="fig-SL-AAve-SL"}](SL-AA-SL-veTbb.pdf "fig:"){width="1.5"} ![(Colour online) The same as in Fig. \[fig-SL-AA-SL\], but now with $v_0=1.5\gamma_1$ and $ \delta =0.6\gamma_1$. []{data-label="fig-SL-AAve-SL"}](SL-AA-SL-veRbb.pdf "fig:"){width="1.73"}\ ![(Colour online) The same as in Fig. \[fig-SL-AA-SL\], but now with $v_0=1.5\gamma_1$ and $ \delta =0.6\gamma_1$. []{data-label="fig-SL-AAve-SL"}](SL-AA-SL-veTtt.pdf "fig:"){width="1.5"} ![(Colour online) The same as in Fig. \[fig-SL-AA-SL\], but now with $v_0=1.5\gamma_1$ and $ \delta =0.6\gamma_1$. []{data-label="fig-SL-AAve-SL"}](SL-AA-SL-veRtt.pdf "fig:"){width="1.73"}\ ![(Colour online) The same as in Fig. \[fig-SL-AA-SL\], but now with $v_0=1.5\gamma_1$ and $ \delta =0.6\gamma_1$. []{data-label="fig-SL-AAve-SL"}](SL-AA-SL-veTbt.pdf "fig:"){width="1.5"} ![(Colour online) The same as in Fig. \[fig-SL-AA-SL\], but now with $v_0=1.5\gamma_1$ and $ \delta =0.6\gamma_1$. []{data-label="fig-SL-AAve-SL"}](SL-AA-SL-veRbt.pdf "fig:"){width="1.73"} ### 2SL-AA-2SL ![(Colour online) Density plot of the transmission and reflection probabilities through AA-2SL-AA as a function of Fermi energy and transverse wave vector $k_y$ with $v_0= \delta =0$ and width of the 2SL $d=25$ nm. []{data-label="fig-AA-SL-AA"}](AA-SL-AA-Tbb.pdf "fig:"){width="1.5"} ![(Colour online) Density plot of the transmission and reflection probabilities through AA-2SL-AA as a function of Fermi energy and transverse wave vector $k_y$ with $v_0= \delta =0$ and width of the 2SL $d=25$ nm. []{data-label="fig-AA-SL-AA"}](AA-SL-AA-Rbb.pdf "fig:"){width="1.73"}\ ![(Colour online) Density plot of the transmission and reflection probabilities through AA-2SL-AA as a function of Fermi energy and transverse wave vector $k_y$ with $v_0= \delta =0$ and width of the 2SL $d=25$ nm. []{data-label="fig-AA-SL-AA"}](AA-SL-AA-Tbt.pdf "fig:"){width="1.5"} ![(Colour online) Density plot of the transmission and reflection probabilities through AA-2SL-AA as a function of Fermi energy and transverse wave vector $k_y$ with $v_0= \delta =0$ and width of the 2SL $d=25$ nm. []{data-label="fig-AA-SL-AA"}](AA-SL-AA-Rbt.pdf "fig:"){width="1.73"} ![(Colour online) Density plot of the transmission and reflection probabilities through AA-2SL-AA as a function of Fermi energy and transverse wave vector $k_y$ with $v_0= \delta =0$ and width of the 2SL $d=25$ nm. []{data-label="fig-AA-SL-AA"}](AA-SL-AA-Ttt.pdf "fig:"){width="1.5"} ![(Colour online) Density plot of the transmission and reflection probabilities through AA-2SL-AA as a function of Fermi energy and transverse wave vector $k_y$ with $v_0= \delta =0$ and width of the 2SL $d=25$ nm. []{data-label="fig-AA-SL-AA"}](AA-SL-AA-Rtt.pdf "fig:"){width="1.73"}\ ![(Colour online) The same as in Fig. \[fig-AA-SL-AA\], but now with $v_0=1.5\ \gamma_1.$ []{data-label="fig-AA-SLv-AA"}](AA-SL-AA-vTbb.pdf "fig:"){width="1.5"} ![(Colour online) The same as in Fig. \[fig-AA-SL-AA\], but now with $v_0=1.5\ \gamma_1.$ []{data-label="fig-AA-SLv-AA"}](AA-SL-AA-vRbb.pdf "fig:"){width="1.73"}\ ![(Colour online) The same as in Fig. \[fig-AA-SL-AA\], but now with $v_0=1.5\ \gamma_1.$ []{data-label="fig-AA-SLv-AA"}](AA-SL-AA-vTbt.pdf "fig:"){width="1.5"} ![(Colour online) The same as in Fig. \[fig-AA-SL-AA\], but now with $v_0=1.5\ \gamma_1.$ []{data-label="fig-AA-SLv-AA"}](AA-SL-AA-vRbt.pdf "fig:"){width="1.73"}\ ![(Colour online) The same as in Fig. \[fig-AA-SL-AA\], but now with $v_0=1.5\ \gamma_1.$ []{data-label="fig-AA-SLv-AA"}](AA-SL-AA-vTtt.pdf "fig:"){width="1.5"} ![(Colour online) The same as in Fig. \[fig-AA-SL-AA\], but now with $v_0=1.5\ \gamma_1.$ []{data-label="fig-AA-SLv-AA"}](AA-SL-AA-vRtt.pdf "fig:"){width="1.73"}\ In this Section, we show the results of transport across two domain walls forming a system with three regions; where AA-BL is sandwiched between two regions of 2SL, see Fig. \[intro-fig02\](a). Such a system can exhibit a strong layer selectivity when current flows through the intermediate region , i.e. AA-BL. This behaviour has already been investigated in Ref. \[\]. Here, however, we go in much more detail to show how the different transmission and reflection channels are affected by the electrostatic potential or finite bias applied to the intermediate region. In Figs. \[fig-SL-AA-SL\] and \[fig-SL-AAv-SL\] we show the scattered and non-scattered channels for transmission and reflection for pristine AA-BL and with electrostatic potential of strength $v_0=1.5 \ \gamma_1$, respectively. Layer symmetry is preserved in both reflection and transmission channels as clarified in Figs. \[fig-SL-AA-SL\]. and \[fig-SL-AAv-SL\] also show strong scattered transmission, especially for normal incidence which can be altered depending on the width of the AA-BL. When an electrostatic potential is applied to the middle domain, resonances appear in the transmission probabilities for $v_0+\gamma_{1}>E>v_0-\gamma_1$ as shown in Fig. \[fig-SL-AAv-SL\]. This is a consequence of the finite size of the AA-BL and the presence of charge carriers with different chirality in the mentioned range of energies [@AA-cones]. Introducing a finite bias $ \delta =0.6\gamma_1$ on AA-BL breaks the layer symmetry of the system. As a result, $T_{b}^b \neq T_{t}^t$ and $R_{b}^b \neq R_{t}^t$. However, it is still preserved in the scattered channels $T_{t}^b = T_{b}^t$ and $R_{t}^b = R_{b}^t$ ( see Fig. \[fig-SL-AAve-SL\]). It is worth mentioning here that the finite bias does not break the angular symmetry with respect to normal incidence in the transmission and reflection probabilities as it does for normal AB-BL[@Ben]. This is a manifestation of the symmetric inter-layer coupling in AA-BL. ### AA-2SL-AA In this system we interchange the AA-BL and 2SL as shown in Fig. \[intro-fig02\](d). In this case, scattering is defined between the two cones in the AA-BL regions. In Figs. \[fig-AA-SL-AA\] and \[fig-AA-SLv-AA\] we show the transmission and reflection probabilities between the two Dirac cones through the pristine 2SL and in the presence of an electrostatic potential, respectively. The first and the last rows of Figs. (\[fig-AA-SL-AA\]) and (\[fig-AA-SLv-AA\]) show the non-scattered transmission and reflection probabilities corresponding to the lower and upper Dirac cones, respectively. We notice that Klein tunnelling is preserved at normal incidence. This shows that Klein tunnelling in AA-stacked bilayer graphene is a robust feature that is insensitive to local changes in the inter-layer coupling. On the other hand we see that scattering between two different Dirac cones remains strictly forbidden even with a local decoupling of the two layers. Therefore, these devices could be used for conetronics. As a result, in the second row of Figs. \[fig-AA-SL-AA\] and \[fig-AA-SLv-AA\] the scattered transmission and reflection channels are zero $T_{+}^- = T_{-}^+=R_{+}^- = R_{-}^+=0$. In Fig. \[fig-AA-SLve-AA\] we plot the transmission and reflection probabilities for a potential strength $v_0=1.5\ \gamma_1$ and inter-layer bias $\delta = 0.3\ \gamma_1$. The shift in the bands of the top (white) and bottom (red) layer of 2SL is due to the inter-layer bias which couples the two Dirac cones as shown in Eq. . ![(Colour online) The same as in Fig. \[fig-AA-SL-AA\], but now with $v_0=1.5\gamma_1$ and $ \delta =0.3\gamma_1$. Red and white dashed curves correspond to the bands of bottom and top layers of 2SL, respectively, while the black dashed curves are the AA-BL bands.[]{data-label="fig-AA-SLve-AA"}](AA-SL-AA-veTbb.pdf "fig:"){width="1.5"} ![(Colour online) The same as in Fig. \[fig-AA-SL-AA\], but now with $v_0=1.5\gamma_1$ and $ \delta =0.3\gamma_1$. Red and white dashed curves correspond to the bands of bottom and top layers of 2SL, respectively, while the black dashed curves are the AA-BL bands.[]{data-label="fig-AA-SLve-AA"}](AA-SL-AA-veRbb.pdf "fig:"){width="1.73"}\ ![(Colour online) The same as in Fig. \[fig-AA-SL-AA\], but now with $v_0=1.5\gamma_1$ and $ \delta =0.3\gamma_1$. Red and white dashed curves correspond to the bands of bottom and top layers of 2SL, respectively, while the black dashed curves are the AA-BL bands.[]{data-label="fig-AA-SLve-AA"}](AA-SL-AA-veTbt.pdf "fig:"){width="1.5"} ![(Colour online) The same as in Fig. \[fig-AA-SL-AA\], but now with $v_0=1.5\gamma_1$ and $ \delta =0.3\gamma_1$. Red and white dashed curves correspond to the bands of bottom and top layers of 2SL, respectively, while the black dashed curves are the AA-BL bands.[]{data-label="fig-AA-SLve-AA"}](AA-SL-AA-veRbt.pdf "fig:"){width="1.73"} ![(Colour online) The same as in Fig. \[fig-AA-SL-AA\], but now with $v_0=1.5\gamma_1$ and $ \delta =0.3\gamma_1$. Red and white dashed curves correspond to the bands of bottom and top layers of 2SL, respectively, while the black dashed curves are the AA-BL bands.[]{data-label="fig-AA-SLve-AA"}](AA-SL-AA-veTtt.pdf "fig:"){width="1.5"} ![(Colour online) The same as in Fig. \[fig-AA-SL-AA\], but now with $v_0=1.5\gamma_1$ and $ \delta =0.3\gamma_1$. Red and white dashed curves correspond to the bands of bottom and top layers of 2SL, respectively, while the black dashed curves are the AA-BL bands.[]{data-label="fig-AA-SLve-AA"}](AA-SL-AA-veRtt.pdf "fig:"){width="1.73"} Therefore, the suppression of the scattering transmission and reflection probabilities due to the protected cone transport does not hold anymore. It is, therefore, possible that scattering between different cones takes place as clarified in the second row of Fig. \[fig-AA-SLve-AA\]. ### Conductance The conductance of two and three-block systems is shown in Figs. \[SL-AA-G\] and \[SL-AA-SL-G\], respectively. For the two systems 2SL-AA and AA-2SL with pristine AA-BL and 2SL, the conductance for different channels is shown in Figs. \[SL-AA-G\](a, b). It shows that the conductance of these two systems are identical. Referring to Figs. \[polar-SL-AA\](a, c) we notice that the transmission probabilities for pristine 2SL-AA and AA-2SL are quite different. However, the corresponding conductances (see Fig. \[SL-AA-G\]) exhibit time reversal symmetry in spite of the fact that the domain wall separates two different systems. This is a strong point which can be verified experimentally even in the case of zero electrostatic potential. Adding an electrostatic potential to one of the two sides leads to different behavior in the conductance of the above mentioned two systems as depicted in Figs. \[SL-AA-G\](c,d). In Fig. \[SL-AA-G\](c) the charge carriers incident from 2SL and impinging on AA-BL whose bands are shifted by $v_0$. Each conductance channel gives zero at $E=0$ due to the absence of propagating states in the 2SL at this energy, even though there are propagating states available in AA-BL corresponding to two cones. We note also that $G_{b}^\pm = G_{t}^\pm$ are almost zero at upper and lower cones $v_0\pm \gamma_1$ as a result of the absence of states at these points as seen in Fig. \[SL-AA-G\](c). In Fig. \[SL-AA-G\](d) we see that the conductance of different channels is not zero in contrast to the previous case because here at $E=0$ there are propagating states available in both AA-BL and 2SL. Furthermore, all channels have one minimum, due to the lack of states, at $E = v_0$ which corresponds to the Dirac cone in 2SL shifted by $v_0$ while $G_-^{t/b}$ has also another minimum at the upper cone $E=\gamma_1$ as shown in Fig. \[SL-AA-G\](d). Finally, for comparison we add in Figs. \[SL-AA-G\](e, f) the conductance that will be measured in the absence of a domain wall for 2SL-2SL and AA-AA junctions with $v_0=0$ (blue curves). Our results indicate that domain walls are experimentally identifiable channels even in the absence of a gate. As a reference we also calculate the total conductance in the presence of an electrostatic potential ($v_0=1.5\gamma_1$) as shown with black curves in Figs. \[SL-AA-G\](e, f) which corresponds, in this case, to the usual p-n junctions in single-layer graphene and AA-BL, respectively. The conductance of three-block systems is shown in Fig. \[SL-AA-SL-G\] where left and right panels correspond to AA-2SL-AA and 2SL-AA-2SL structure, respectively. Protected cone transport leads to zero conductance in the scattered channels $G_-^+ = G_+^-=0$ as shown in Fig. \[SL-AA-SL-G\](a). A close inspection also reveals that $G_-^- = G_+^+$ at $E=0$ with finite and non-zero values, regardless of the fact that in the 2SL region there are no available propagating states. This is attributed to the evanescent modes in 2SL at $E=0$ which are responsible for ballistic transport in graphene[@Snyman]. We thus also expect that $G_-^-$ (red curve in Fig. \[SL-AA-SL-G\](a)) should be exactly zero at the Dirac cone $E = \gamma_1$ as a result of the absence of propagating states in the leads at this energy. By shifting the bands of 2SL using a local potential with strength $v_0 = 1.5\gamma_1$, a local minimum appears in the conductance $G_T$ at $E = v_0$ which corresponds to the position of the charge-neutrality point in 2SL as shown in Fig. \[SL-AA-SL-G\](c). This minimum can be obtained by aligning the upper cone in AA-BL and the Dirac cone in 2SL such that they are located at the same energy, this can be achieved by choosing $v_0=\gamma_1$. The main difference introduced by applying an inter-layer bias is the broken protected cone transport where now $G_+^-=G_-^+\neq0$ as depicted in Fig. \[SL-AA-SL-G\](e). For completeness, we performed similar calculations but now with 2SL as the leads (2SL-AA-2SL) and the results for the conductance with pristine, gated and biased AA-BL are shown in Figs. \[SL-AA-SL-G\](b, d, f), respectively. Here, all conductance channels are zero at $E=0$ such that $G_t^t=G_b^b$ and $G_t^b=G_b^t$ as shown in Figs. \[SL-AA-SL-G\](b, d). Similarly, the main features in Fig. \[SL-AA-SL-G\](f) are in qualitative agreement with those shown in Figs. \[SL-AA-SL-G\](b, d) but now the tunnelling equivalence through the same channel is broken so that $G_t^t\neq G_b^b$. This is a direct consequence of the perpendicular electric field which leads to the breaking of the inter-layer sublattice equivalence. The peaks appearing in the total conductance are due to the finite size of the AA-BL region. ![(Colour online) Conductance of two-block system for different magnitudes of the applied gate: (a, b) $v_0= \delta =0$, (c, d) $v_0=3\gamma_1/2$, $\delta =0$ . $G_T$ is the total conductance obtained by summation of all possible channels, (e, f) the total conductance for 2SL-2SL and AA-AA junctions, respectively, with $v_0=0$ (blue curves) and $v_0=1.5\gamma_1$ (black curves).[]{data-label="SL-AA-G"}](SL-AA-G.pdf "fig:"){width="1.65"} ![(Colour online) Conductance of two-block system for different magnitudes of the applied gate: (a, b) $v_0= \delta =0$, (c, d) $v_0=3\gamma_1/2$, $\delta =0$ . $G_T$ is the total conductance obtained by summation of all possible channels, (e, f) the total conductance for 2SL-2SL and AA-AA junctions, respectively, with $v_0=0$ (blue curves) and $v_0=1.5\gamma_1$ (black curves).[]{data-label="SL-AA-G"}](AA-SL-G.pdf "fig:"){width="1.65"}\ ![(Colour online) Conductance of two-block system for different magnitudes of the applied gate: (a, b) $v_0= \delta =0$, (c, d) $v_0=3\gamma_1/2$, $\delta =0$ . $G_T$ is the total conductance obtained by summation of all possible channels, (e, f) the total conductance for 2SL-2SL and AA-AA junctions, respectively, with $v_0=0$ (blue curves) and $v_0=1.5\gamma_1$ (black curves).[]{data-label="SL-AA-G"}](SL-AA-vG.pdf "fig:"){width="1.65"} ![(Colour online) Conductance of two-block system for different magnitudes of the applied gate: (a, b) $v_0= \delta =0$, (c, d) $v_0=3\gamma_1/2$, $\delta =0$ . $G_T$ is the total conductance obtained by summation of all possible channels, (e, f) the total conductance for 2SL-2SL and AA-AA junctions, respectively, with $v_0=0$ (blue curves) and $v_0=1.5\gamma_1$ (black curves).[]{data-label="SL-AA-G"}](AA-SL-vG.pdf "fig:"){width="1.65"}\ ![(Colour online) Conductance of two-block system for different magnitudes of the applied gate: (a, b) $v_0= \delta =0$, (c, d) $v_0=3\gamma_1/2$, $\delta =0$ . $G_T$ is the total conductance obtained by summation of all possible channels, (e, f) the total conductance for 2SL-2SL and AA-AA junctions, respectively, with $v_0=0$ (blue curves) and $v_0=1.5\gamma_1$ (black curves).[]{data-label="SL-AA-G"}](2SL-2SL-AA.pdf "fig:"){width="1.65"} ![(Colour online) Conductance of two-block system for different magnitudes of the applied gate: (a, b) $v_0= \delta =0$, (c, d) $v_0=3\gamma_1/2$, $\delta =0$ . $G_T$ is the total conductance obtained by summation of all possible channels, (e, f) the total conductance for 2SL-2SL and AA-AA junctions, respectively, with $v_0=0$ (blue curves) and $v_0=1.5\gamma_1$ (black curves).[]{data-label="SL-AA-G"}](AA-AA.pdf "fig:"){width="1.65"} ![(Colour online) Conductance of three-block system with different magnitudes of the applied gate: (a, b) $v_0=\ \delta =0$, (c, d) $v_0=3\gamma_1/2$, $\delta =0$ and (e, f) $v_0=3\gamma_1/2$, $\delta =0.6\gamma_1$. $G_T$ is the total conductance obtained by summation of all possible channels. []{data-label="SL-AA-SL-G"}](AA-SL-AA-G.pdf "fig:"){width="1.65"} ![(Colour online) Conductance of three-block system with different magnitudes of the applied gate: (a, b) $v_0=\ \delta =0$, (c, d) $v_0=3\gamma_1/2$, $\delta =0$ and (e, f) $v_0=3\gamma_1/2$, $\delta =0.6\gamma_1$. $G_T$ is the total conductance obtained by summation of all possible channels. []{data-label="SL-AA-SL-G"}](SL-AA-SL-G.pdf "fig:"){width="1.65"}\ ![(Colour online) Conductance of three-block system with different magnitudes of the applied gate: (a, b) $v_0=\ \delta =0$, (c, d) $v_0=3\gamma_1/2$, $\delta =0$ and (e, f) $v_0=3\gamma_1/2$, $\delta =0.6\gamma_1$. $G_T$ is the total conductance obtained by summation of all possible channels. []{data-label="SL-AA-SL-G"}](AA-SL-AA-vG.pdf "fig:"){width="1.65"} ![(Colour online) Conductance of three-block system with different magnitudes of the applied gate: (a, b) $v_0=\ \delta =0$, (c, d) $v_0=3\gamma_1/2$, $\delta =0$ and (e, f) $v_0=3\gamma_1/2$, $\delta =0.6\gamma_1$. $G_T$ is the total conductance obtained by summation of all possible channels. []{data-label="SL-AA-SL-G"}](SL-AA-SL-vG.pdf "fig:"){width="1.65"}\ ![(Colour online) Conductance of three-block system with different magnitudes of the applied gate: (a, b) $v_0=\ \delta =0$, (c, d) $v_0=3\gamma_1/2$, $\delta =0$ and (e, f) $v_0=3\gamma_1/2$, $\delta =0.6\gamma_1$. $G_T$ is the total conductance obtained by summation of all possible channels. []{data-label="SL-AA-SL-G"}](AA-SL-AA-veG.pdf "fig:"){width="1.65"} ![(Colour online) Conductance of three-block system with different magnitudes of the applied gate: (a, b) $v_0=\ \delta =0$, (c, d) $v_0=3\gamma_1/2$, $\delta =0$ and (e, f) $v_0=3\gamma_1/2$, $\delta =0.6\gamma_1$. $G_T$ is the total conductance obtained by summation of all possible channels. []{data-label="SL-AA-SL-G"}](SL-AA-SL-veG.pdf "fig:"){width="1.65"}\ AB-Stacking ------------- ### 2SL-AB/AB-2SL In this section, we evaluate how the stacking of the connected region changes the transport properties across a domain wall. The angle-dependent transmission and reflection probabilities for pristine systems 2SL-AB are plotted in Fig. \[polar-SL-AB\](a). The charge carriers can be incident from the two layers in the 2SL structure and impinge on AB-BL where, depending on their energy, they can access only one propagating mode $k^+$ or two $k^\pm$ if the energy is large enough. Scattering from the top or bottom layer of 2SL into one of these modes is equivalent $T_{t}^\pm=T_{b}^\pm$ as well as backscattering $R_{t}^{t(b)}=R_{b}^{b(t)}$ and hence, as before, layer symmetry is preserved (see Fig. \[polar-SL-AB\](a)). In Fig. \[polar-SL-AB\](b) we show results with the AB-BL region subjected to an electrostatic potential of strength $v_0=1.5\gamma_1$. Surprisingly, we see that the layer symmetry is broken and an asymmetric feature with respect to normal incidence shows up in the transmission and non-scattered reflection probabilities, see Appendix \[Sec:Appendix\], such that $[T/R](\phi)=[T/R](-\phi).$ For example, $T_{b}^\pm(\phi)=T_t^\pm(-\phi)$ as well as the non-scattered reflection channels $R_{b}^b(\phi)=R_{t}^t(-\phi)$ as discussed in Sec. \[Symmetry\]. This asymmetric feature can be understood by resorting to the bands on both sides of the junction, where due to the electrostatic potential the band alignment of 2SL and AB-BL is altered. In this case, the center of the AB-BL band is shifted upwards in energy with respect to the crossing of the 2SL[@34] energy bands. The origin of such asymmetry is a direct consequence of the asymmetric coupling in AB-BL which leads to shifting of the bands by $\gamma_1$. Therefore, at low energy $\left\vert E-v_{0} \right\vert< \gamma_1$ there is only one propagating mode $k^{+}$ (i.e one type of charge carrier ) and consequently only $T_{b(t)}^+$ is available. For larger energy, on the other hand, there are two modes available giving rise to four channels $T_{b(t)}^\pm$. ![(Colour online) The angle-dependent transmission and reflection probabilities through (a, b) 2SL-AB and (c, d) AB-2SL junctions. The systems in (b, d) are the same as in (a, c), respectively, but where the right side of the junction is subjected to an electrostatic potential of strength $ v_0= 1.5\gamma_1. $ In (a) $E=1.2\gamma_1$ for all channels while in (b) $E=1.7\gamma_1$ for $T_{b(t)}^+$ and $E=0.6\ \gamma_1$ for the rest of the channels and in (c, d) $E=(0.6, 1.7)\gamma_1$ for $R_+^+/T_+^{b(t)}$ and $R_-^-/T_-^{b(t)}$, respectively. We choose energy values in (b, d) such that they correspond to only one propagating mode in the AB-BL region. []{data-label="polar-SL-AB"}](SL-AB-T.pdf "fig:"){width="1.25"}   ![(Colour online) The angle-dependent transmission and reflection probabilities through (a, b) 2SL-AB and (c, d) AB-2SL junctions. The systems in (b, d) are the same as in (a, c), respectively, but where the right side of the junction is subjected to an electrostatic potential of strength $ v_0= 1.5\gamma_1. $ In (a) $E=1.2\gamma_1$ for all channels while in (b) $E=1.7\gamma_1$ for $T_{b(t)}^+$ and $E=0.6\ \gamma_1$ for the rest of the channels and in (c, d) $E=(0.6, 1.7)\gamma_1$ for $R_+^+/T_+^{b(t)}$ and $R_-^-/T_-^{b(t)}$, respectively. We choose energy values in (b, d) such that they correspond to only one propagating mode in the AB-BL region. []{data-label="polar-SL-AB"}](SL-vAB-T.pdf "fig:"){width="1.25"} ![(Colour online) The angle-dependent transmission and reflection probabilities through (a, b) 2SL-AB and (c, d) AB-2SL junctions. The systems in (b, d) are the same as in (a, c), respectively, but where the right side of the junction is subjected to an electrostatic potential of strength $ v_0= 1.5\gamma_1. $ In (a) $E=1.2\gamma_1$ for all channels while in (b) $E=1.7\gamma_1$ for $T_{b(t)}^+$ and $E=0.6\ \gamma_1$ for the rest of the channels and in (c, d) $E=(0.6, 1.7)\gamma_1$ for $R_+^+/T_+^{b(t)}$ and $R_-^-/T_-^{b(t)}$, respectively. We choose energy values in (b, d) such that they correspond to only one propagating mode in the AB-BL region. []{data-label="polar-SL-AB"}](AB-SL-T.pdf "fig:"){width="1.25"}   ![(Colour online) The angle-dependent transmission and reflection probabilities through (a, b) 2SL-AB and (c, d) AB-2SL junctions. The systems in (b, d) are the same as in (a, c), respectively, but where the right side of the junction is subjected to an electrostatic potential of strength $ v_0= 1.5\gamma_1. $ In (a) $E=1.2\gamma_1$ for all channels while in (b) $E=1.7\gamma_1$ for $T_{b(t)}^+$ and $E=0.6\ \gamma_1$ for the rest of the channels and in (c, d) $E=(0.6, 1.7)\gamma_1$ for $R_+^+/T_+^{b(t)}$ and $R_-^-/T_-^{b(t)}$, respectively. We choose energy values in (b, d) such that they correspond to only one propagating mode in the AB-BL region. []{data-label="polar-SL-AB"}](AB-vSL-T.pdf "fig:"){width="1.25"} ![(Colour online) Density plot of the transmission and reflection probabilities through 2SL-AB-2SL as a function of Fermi energy and transverse wave vector $k_y$ with $v_0=\ \delta =0.$ []{data-label="SL-AB-SL"}](SL-AB-SL-Tbb.pdf "fig:"){width="1.5"} ![(Colour online) Density plot of the transmission and reflection probabilities through 2SL-AB-2SL as a function of Fermi energy and transverse wave vector $k_y$ with $v_0=\ \delta =0.$ []{data-label="SL-AB-SL"}](SL-AB-SL-Rbb.pdf "fig:"){width="1.73"}\ ![(Colour online) Density plot of the transmission and reflection probabilities through 2SL-AB-2SL as a function of Fermi energy and transverse wave vector $k_y$ with $v_0=\ \delta =0.$ []{data-label="SL-AB-SL"}](SL-AB-SL-Tbt.pdf "fig:"){width="1.5"} ![(Colour online) Density plot of the transmission and reflection probabilities through 2SL-AB-2SL as a function of Fermi energy and transverse wave vector $k_y$ with $v_0=\ \delta =0.$ []{data-label="SL-AB-SL"}](SL-AB-SL-Rbt.pdf "fig:"){width="1.73"} The angular asymmetry feature is present only in the region in the $(E,k_y)$-plane where there is only one propagating mode. This can be also understood as a manifestation of the asymmetric amplitude of the wave function in the AB-BL side due to the evanescent modes in this region [@24]. The theory of tunnelling through an interface of monolayer and bilayer was presented earlier[@24] and such asymmetry was noticed as well. Moreover, in our case there are two single layer graphene sheets connected to the bottom and top layers of the bilayer system but the asymmetric feature in Ref. \[\] will be recovered when considering only one propagation channel. For instance, the transmission probabilities $T_{t}^\pm$ and $\ T_{b}^\pm$ presented in Fig. \[polar-SL-AB\](b) show the same asymmetric features discussed in Ref. \[\]. This asymmetry feature is reversed in the other valley, so that the total transmission or reflection averaged over both layers is symmetric as can be seen from Fig. \[polar-SL-AB\](b). However, this valley-dependent angular asymmetry could also be used for the basis of a layer-dependent valley-filtering device as proposed in other works[@Costa2015; @Rycerz2007]. The above analogy, which is discriminating between the presence of one or two modes, applies also to the non-scattered reflection probabilities $R_{b}^b$ and $\ R_{t}^t$. These non-scattered currents are carried by the states localized on the disconnected sublattices $\alpha_2$ and $\beta_1$, as seen in Fig. \[fig01\]. In that case, there is one traveling mode[@pelc] and thus, inherently, a layer asymmetric feature will be present. In contrast, for the scattered channels $R_{b}^t$ and $\ R_{t}^b$ the charge carriers must jump between the layers of AB-BL. This occurs through the localized states on the connected sublattices $\alpha_1$ and $\beta_2$ where there are two travelling modes and, hence, these probabilities exhibit layer symmetry as shown in Fig. \[polar-SL-AB\](b). In the AB-2SL configuration, where charge carriers incident from the AB-BL impinge on the 2SL, we show the angle-dependent transmission and reflection probabilities in Fig. \[polar-SL-AB\](c) for pristine 2SL and AB-BL. Similar to the previous configuration 2SL-AB, the results are symmetric in this case because the Dirac cones of both systems (2SL and AB-BL) are aligned. Furthermore, there is an equivalence in the transmission channels such that $T_{\pm}^t=T_{\pm}^b$ with partial reflection associated with the non-scattered channels $R_{-}^-$ and $R_{+}^+$. While for the scattered channels $R_{+}^-$ and $R_{+}^-$ are almost zero. This is due to efficient transmission resulting from the absence of the electrostatic potential in the 2SL. An electrostatic potential of strength $v_0=1.5 \gamma_1$ induces a scattering between the two modes in the reflection channels so that now $R_{-}^+=R_{+}^-\neq0$ as depicted in Fig. \[polar-SL-AB\](d). In addition, it breaks the band alignment and gives rise to the layer asymmetry feature in the transmission probabilities $T_+^{b(t)}$ where only one travelling mode exists i.e. $E<\gamma_1$. Thus, $T_-^{b(t)}$ always preserves layer symmetry in this case, see Fig. \[polar-SL-AB\](d), because the mode $k^-$ exists for $E>\gamma_1$ where also the mode $k^+$ is available as discussed above. This is also the same reason that configurations consisting of AA-BL always preserve layer symmetry. Indeed, AA-BL does not have a region in the $(E,k_y)$-plane with only one propagating mode, and there are always two travelling modes for all energies. ### 2SL-AB-2SL Different configurations have been proposed to connect a single layer to the AB-stacked bilayer graphene[@25; @34; @15; @Lima]. Now, two SL are connected to the AB-stacked bilayer, see Fig. \[intro-fig02\](a). In Fig. \[SL-AB-SL\] we show the dependence of the transmission and reflection probabilities on the transverse wave vector $k_y$ and the Fermi energy. It appears that all channels are symmetric with respect to normal incidence since the Dirac cones of AB and 2SL are aligned. It also implies that scattered and non-scattered channels of the transmission and reflection are equivalent such that $(T/R)_{b}^t=(T/R)_{t}^b$ and $(T/R)_{t}^t=(T/R)_{b}^b$ (see Fig. \[SL-AB-SL\]). ![(Colour online) The same as in Fig. \[SL-AB-SL\], but now with $v_0=3\gamma_1/2$. []{data-label="SL-AB-SLv"}](SL-AB-SL-vTbb.pdf "fig:"){width="1.5"} ![(Colour online) The same as in Fig. \[SL-AB-SL\], but now with $v_0=3\gamma_1/2$. []{data-label="SL-AB-SLv"}](SL-AB-SL-vRbb.pdf "fig:"){width="1.73"}\ ![(Colour online) The same as in Fig. \[SL-AB-SL\], but now with $v_0=3\gamma_1/2$. []{data-label="SL-AB-SLv"}](SL-AB-SL-vTbt.pdf "fig:"){width="1.5"} ![(Colour online) The same as in Fig. \[SL-AB-SL\], but now with $v_0=3\gamma_1/2$. []{data-label="SL-AB-SLv"}](SL-AB-SL-vRbt.pdf "fig:"){width="1.73"}\ ![(Colour online) The same as in Fig. \[SL-AB-SL\], but now with $v_0=3\gamma_1/2$. []{data-label="SL-AB-SLv"}](SL-AB-SL-vTtb.pdf "fig:"){width="1.5"} ![(Colour online) The same as in Fig. \[SL-AB-SL\], but now with $v_0=3\gamma_1/2$. []{data-label="SL-AB-SLv"}](SL-AB-SL-vRtt.pdf "fig:"){width="1.73"} ![(Colour online) The same as in Fig. \[SL-AB-SL\], but now with $v_0=3\gamma_1/2$, $\delta =0.8\gamma_1.$ New localized states appear inside the *Mexican hat* shape of the low energy bands of AB-BL due to the strong gate potential. []{data-label="SL-AB-SLve"}](SL-AB-SL-veTbb.pdf "fig:"){width="1.5"} ![(Colour online) The same as in Fig. \[SL-AB-SL\], but now with $v_0=3\gamma_1/2$, $\delta =0.8\gamma_1.$ New localized states appear inside the *Mexican hat* shape of the low energy bands of AB-BL due to the strong gate potential. []{data-label="SL-AB-SLve"}](SL-AB-SL-veRbb.pdf "fig:"){width="1.73"}\ ![(Colour online) The same as in Fig. \[SL-AB-SL\], but now with $v_0=3\gamma_1/2$, $\delta =0.8\gamma_1.$ New localized states appear inside the *Mexican hat* shape of the low energy bands of AB-BL due to the strong gate potential. []{data-label="SL-AB-SLve"}](SL-AB-SL-veTbt.pdf "fig:"){width="1.5"} ![(Colour online) The same as in Fig. \[SL-AB-SL\], but now with $v_0=3\gamma_1/2$, $\delta =0.8\gamma_1.$ New localized states appear inside the *Mexican hat* shape of the low energy bands of AB-BL due to the strong gate potential. []{data-label="SL-AB-SLve"}](SL-AB-SL-veRbt.pdf "fig:"){width="1.73"}\ ![(Colour online) The same as in Fig. \[SL-AB-SL\], but now with $v_0=3\gamma_1/2$, $\delta =0.8\gamma_1.$ New localized states appear inside the *Mexican hat* shape of the low energy bands of AB-BL due to the strong gate potential. []{data-label="SL-AB-SLve"}](SL-AB-SL-veTtb.pdf "fig:"){width="1.5"} ![(Colour online) The same as in Fig. \[SL-AB-SL\], but now with $v_0=3\gamma_1/2$, $\delta =0.8\gamma_1.$ New localized states appear inside the *Mexican hat* shape of the low energy bands of AB-BL due to the strong gate potential. []{data-label="SL-AB-SLve"}](SL-AB-SL-veRtb.pdf "fig:"){width="1.73"}\ ![(Colour online) The same as in Fig. \[SL-AB-SL\], but now with $v_0=3\gamma_1/2$, $\delta =0.8\gamma_1.$ New localized states appear inside the *Mexican hat* shape of the low energy bands of AB-BL due to the strong gate potential. []{data-label="SL-AB-SLve"}](SL-AB-SL-veTtt.pdf "fig:"){width="1.5"} ![(Colour online) The same as in Fig. \[SL-AB-SL\], but now with $v_0=3\gamma_1/2$, $\delta =0.8\gamma_1.$ New localized states appear inside the *Mexican hat* shape of the low energy bands of AB-BL due to the strong gate potential. []{data-label="SL-AB-SLve"}](SL-AB-SL-veRtt.pdf "fig:"){width="1.73"}\ ![(Colour online) Transmission probabilities as function of Fermi energy and bias for normal incidence. []{data-label="Tbt-Tbb"}](Tbb.pdf "fig:"){width="1.535"}![(Colour online) Transmission probabilities as function of Fermi energy and bias for normal incidence. []{data-label="Tbt-Tbb"}](Tbt.pdf "fig:"){width="1.775"} Another interesting feature of this configuration is that for $E<\gamma_1$ the scattered and non-scattered transmissions are equal $T_{i}^{j}=T_{i}^{i}$. In this energy regime such device can be used as an electronic beam splitter[@Brand; @Lima]. Fig. \[SL-AB-SLv\] displays the same plot as in Fig. \[SL-AB-SL\] but with an electrostatic potential on the AB-BL region. There is an important difference as compared to the pristine AB-BL case, the layer symmetry is broken such that $T_{t}^{b}(k_{y})= T_{b}^{t}(-k_y)$ as clarified in Fig. \[SL-AB-SLv\]. This can be also understood by pointing out that charge carriers scattered from top to bottom when moving from left to right in the $K$ valley are equivalent to charge carriers scattering from bottom to top when moving oppositely in the second valley $K'$. Introducing a finite bias ($\delta>0$) to the AB-BL region along with an electrostatic potential ($v_0>0$) will shift the bands and opens a gap in the spectrum. As a result of the presence of a strong electric field, the transmission channels are completely suppressed inside the gap due to the absence of traveling modes as seen in Fig. \[SL-AB-SLve\]. Moreover, non-zero asymmetric reflection appears in the gap as well as a violation of the equivalence of non-scattered transmission channels. This is a result of the breaking of inter-layer sublattice equivalence [@Ben]. In addition, some localized states appear inside the “Mexican hat” of the low energy bands where they are pushed by the strong electric field ($\delta=0.8 \gamma_1$), see Fig. \[SL-AB-SLve\]. ![(Colour online) Density plot of the transmission and reflection probabilities through AB-2SL-AB as a function of Fermi energy and transverse wave vector $k_y$ with $v_0= \delta =0$ and $d=25$nm. []{data-label="AB-2SL-AB"}](AB-SL-AB-T11.pdf "fig:"){width="1.5"} ![(Colour online) Density plot of the transmission and reflection probabilities through AB-2SL-AB as a function of Fermi energy and transverse wave vector $k_y$ with $v_0= \delta =0$ and $d=25$nm. []{data-label="AB-2SL-AB"}](AB-SL-AB-R11.pdf "fig:"){width="1.73"}\ ![(Colour online) Density plot of the transmission and reflection probabilities through AB-2SL-AB as a function of Fermi energy and transverse wave vector $k_y$ with $v_0= \delta =0$ and $d=25$nm. []{data-label="AB-2SL-AB"}](AB-SL-AB-T12.pdf "fig:"){width="1.5"} ![(Colour online) Density plot of the transmission and reflection probabilities through AB-2SL-AB as a function of Fermi energy and transverse wave vector $k_y$ with $v_0= \delta =0$ and $d=25$nm. []{data-label="AB-2SL-AB"}](AB-SL-AB-R12.pdf "fig:"){width="1.73"}\ ![(Colour online) Density plot of the transmission and reflection probabilities through AB-2SL-AB as a function of Fermi energy and transverse wave vector $k_y$ with $v_0= \delta =0$ and $d=25$nm. []{data-label="AB-2SL-AB"}](AB-SL-AB-T22.pdf "fig:"){width="1.5"} ![(Colour online) Density plot of the transmission and reflection probabilities through AB-2SL-AB as a function of Fermi energy and transverse wave vector $k_y$ with $v_0= \delta =0$ and $d=25$nm. []{data-label="AB-2SL-AB"}](AB-SL-AB-R22.pdf "fig:"){width="1.73"} ![(Colour online) The same as in Fig. \[AB-2SL-AB\], but here with $v_0=3\gamma_1/2$ []{data-label="AB-SL-ABv"}](AB-SL-AB-vT11.pdf "fig:"){width="1.5"} ![(Colour online) The same as in Fig. \[AB-2SL-AB\], but here with $v_0=3\gamma_1/2$ []{data-label="AB-SL-ABv"}](AB-SL-AB-vR11.pdf "fig:"){width="1.73"}\ ![(Colour online) The same as in Fig. \[AB-2SL-AB\], but here with $v_0=3\gamma_1/2$ []{data-label="AB-SL-ABv"}](AB-SL-AB-vT12.pdf "fig:"){width="1.5"} ![(Colour online) The same as in Fig. \[AB-2SL-AB\], but here with $v_0=3\gamma_1/2$ []{data-label="AB-SL-ABv"}](AB-SL-AB-vR12.pdf "fig:"){width="1.73"}\ ![(Colour online) The same as in Fig. \[AB-2SL-AB\], but here with $v_0=3\gamma_1/2$ []{data-label="AB-SL-ABv"}](AB-SL-AB-vT22.pdf "fig:"){width="1.5"} ![(Colour online) The same as in Fig. \[AB-2SL-AB\], but here with $v_0=3\gamma_1/2$ []{data-label="AB-SL-ABv"}](AB-SL-AB-vR22.pdf "fig:"){width="1.73"} There is a link between the transmission probabilities of our system 2SL-AB-2SL and those investigated by González *et al.* [@34]. The channels $T_b^b$ and $T_b^t$ are qualitatively equivalent to those obtained in Ref. \[\]. For example, $T_b^t$ shows electron-hole ($e-h$) and $\delta\rightarrow-\delta$ symmetry whereas $T_b^b$ exhibits another symmetry which can be obtained under the exchange $(e,\delta)\leftrightarrow\ (h,-\delta)$. The results in Fig. \[Tbt-Tbb\] are in good agreement with those of Ref. \[\] where we fix $v_0=0$ and $d=25 $ nm. ### AB-2SL-AB {#AB-2SL-AB-T} For leads composed of AB-BL where the intermediate region is pristine 2SL, we show the results in Fig. \[AB-2SL-AB\] for the transmission and reflection probabilities. Now charge carriers will scatter between the different modes of the AB-BL on the left and right leads as shown in Fig. \[intro-fig02\](b). As expected, all channels are symmetric and as a result of the finite size of the 2SL region, resonances appear in $T$ as shown in Fig. \[AB-2SL-AB\]. These so-called Fabry-Pérot resonances appear at quantized energy levels[@masir2010] $$\label{eq019} E^n_{SL}(k_y)=\sqrt{k_y^{2}+\left( \frac{n \pi \ }{d}\right)^{2}}.$$ This is the dispersion relation for modes confined in the 2SL region with width $d$. The results presented in Fig. \[AB-2SL-AB\] reveal no scattering between the two modes $k^+$ and $k^-$ and charge carriers are only transmitted or reflected through the same channel from which they come from. Unexpectedly, introducing an electrostatic potential induces a strong scattering in the reflection channels ($R_+^-=R_-^+\neq0$) and very weak scattering in the transmission channels ($T_+^-=T_-^+\neq0$), as seen in Fig. \[AB-SL-ABv\]. When the 2SL are biased, the Dirac cones at bottom and top layers will be shifted up (white dashed lines) and down (red dashed lines) in energy, respectively (see Fig. \[AB-SL-ABve\]). This bias will strengthen the coupling between the two modes resulting in a strong scattering between them. In addition, the inversion symmetry is broken due to the bias leading to an asymmetry with respect to normal incidence. ### Conductance ![(Colour online) The same as in Fig. \[AB-2SL-AB\], but here with $v_0=3\gamma_1/2$, $\delta =0.8\gamma_1$. Red and white dashed curves correspond to the bands of bottom and top layers of 2SL while the black dashed curves are the AB-BL bands. []{data-label="AB-SL-ABve"}](AB-SL-AB-veT11.pdf "fig:"){width="1.5"} ![(Colour online) The same as in Fig. \[AB-2SL-AB\], but here with $v_0=3\gamma_1/2$, $\delta =0.8\gamma_1$. Red and white dashed curves correspond to the bands of bottom and top layers of 2SL while the black dashed curves are the AB-BL bands. []{data-label="AB-SL-ABve"}](AB-SL-AB-veR11.pdf "fig:"){width="1.73"}\ ![(Colour online) The same as in Fig. \[AB-2SL-AB\], but here with $v_0=3\gamma_1/2$, $\delta =0.8\gamma_1$. Red and white dashed curves correspond to the bands of bottom and top layers of 2SL while the black dashed curves are the AB-BL bands. []{data-label="AB-SL-ABve"}](AB-SL-AB-veT12.pdf "fig:"){width="1.5"} ![(Colour online) The same as in Fig. \[AB-2SL-AB\], but here with $v_0=3\gamma_1/2$, $\delta =0.8\gamma_1$. Red and white dashed curves correspond to the bands of bottom and top layers of 2SL while the black dashed curves are the AB-BL bands. []{data-label="AB-SL-ABve"}](AB-SL-AB-veR12.pdf "fig:"){width="1.73"}\ ![(Colour online) The same as in Fig. \[AB-2SL-AB\], but here with $v_0=3\gamma_1/2$, $\delta =0.8\gamma_1$. Red and white dashed curves correspond to the bands of bottom and top layers of 2SL while the black dashed curves are the AB-BL bands. []{data-label="AB-SL-ABve"}](AB-SL-AB-veT22.pdf "fig:"){width="1.5"} ![(Colour online) The same as in Fig. \[AB-2SL-AB\], but here with $v_0=3\gamma_1/2$, $\delta =0.8\gamma_1$. Red and white dashed curves correspond to the bands of bottom and top layers of 2SL while the black dashed curves are the AB-BL bands. []{data-label="AB-SL-ABve"}](AB-SL-AB-veR22.pdf "fig:"){width="1.73"}\ ![(Colour online) The same as in Fig. \[AB-2SL-AB\], but here with $v_0=3\gamma_1/2$, $\delta =0.8\gamma_1$. Red and white dashed curves correspond to the bands of bottom and top layers of 2SL while the black dashed curves are the AB-BL bands. []{data-label="AB-SL-ABve"}](AB-SL-AB-veT21.pdf "fig:"){width="1.5"} ![(Colour online) The same as in Fig. \[AB-2SL-AB\], but here with $v_0=3\gamma_1/2$, $\delta =0.8\gamma_1$. Red and white dashed curves correspond to the bands of bottom and top layers of 2SL while the black dashed curves are the AB-BL bands. []{data-label="AB-SL-ABve"}](AB-SL-AB-veR21.pdf "fig:"){width="1.73"} The conductance of the two-block system consisting of 2SL and BA-BL is shown in Fig. \[AB-SL-G\] for different values of the applied gate voltage. Figs. \[AB-SL-G\](a,b) reveal that the system where charge carriers are incident from the 2SL and impinge on AB-BL and vice versa are equivalent to the case when both 2SL and AB-BL are at the same potential. As seen in Figs. \[AB-SL-G\](a,b), $G_+^{t(b)}=G_{t(b)}^+$ are contributing to the total conductance $G_T$ starting from $E=0$ where the $k^+$ mode exists. On the contrary, $G_-^{t(b)}=G_{t(b)}^-$ only contributes when $E>\gamma_1$ where $k^-$ states are available and this appears as a sharp increase in $G_T$ at $E=\gamma_1$. On the other hand, considering an applied electrostatic potential on the right side of the two-block system will break this equivalence as seen in Figs. \[AB-SL-G\](c,d). In addition, as a result of the shift of the Dirac cone in AB-BL (see Fig. \[AB-SL-G\](c)) or 2SL (see Fig. \[AB-SL-G\](d)) due to the electrostatic potential, all conductance channels are zero at $E=v_0$. Similar to the AA-BL case, the conductances of the pristine systems 2SL-AB/AB-2SL (see Figs. \[AB-SL-G\](a, b)) preserve the time reversal symmetry. Even though, both systems have different transmission probabilities as can be seen from Figs. \[polar-SL-AB\](a, c). We also show in Figs. \[AB-SL-G\](e, f) the total conductance in the absence of domain wall in 2SL-SL and AB-AB systems, respectively, for $v_0=0$ (blue curves) and $v_0=1.5\gamma_1$ (black curves). This shows that transport channels in the presence of domain walls are experimentally recognisable. ![(Colour online) Conductance of different junctions for different magnitudes of the applied gate: (a, b) $v_0= \delta =0$, (c, d) $v_0=3\gamma_1/2$, $\delta =0$, (e, f) the total conductance for 2SL-2SL and AB-AB junctions, respectively, with $v_0=0$(blue curves) and $v_0=1.5\gamma_1$(black curves). []{data-label="AB-SL-G"}](2SLG-BLG-G0.pdf "fig:"){width="1.65"} ![(Colour online) Conductance of different junctions for different magnitudes of the applied gate: (a, b) $v_0= \delta =0$, (c, d) $v_0=3\gamma_1/2$, $\delta =0$, (e, f) the total conductance for 2SL-2SL and AB-AB junctions, respectively, with $v_0=0$(blue curves) and $v_0=1.5\gamma_1$(black curves). []{data-label="AB-SL-G"}](AB-2SLG-G0.pdf "fig:"){width="1.65"}\ ![(Colour online) Conductance of different junctions for different magnitudes of the applied gate: (a, b) $v_0= \delta =0$, (c, d) $v_0=3\gamma_1/2$, $\delta =0$, (e, f) the total conductance for 2SL-2SL and AB-AB junctions, respectively, with $v_0=0$(blue curves) and $v_0=1.5\gamma_1$(black curves). []{data-label="AB-SL-G"}](2SLG-BLG-Gv.pdf "fig:"){width="1.65"} ![(Colour online) Conductance of different junctions for different magnitudes of the applied gate: (a, b) $v_0= \delta =0$, (c, d) $v_0=3\gamma_1/2$, $\delta =0$, (e, f) the total conductance for 2SL-2SL and AB-AB junctions, respectively, with $v_0=0$(blue curves) and $v_0=1.5\gamma_1$(black curves). []{data-label="AB-SL-G"}](AB-2SLG-Gv.pdf "fig:"){width="1.65"}\ ![(Colour online) Conductance of different junctions for different magnitudes of the applied gate: (a, b) $v_0= \delta =0$, (c, d) $v_0=3\gamma_1/2$, $\delta =0$, (e, f) the total conductance for 2SL-2SL and AB-AB junctions, respectively, with $v_0=0$(blue curves) and $v_0=1.5\gamma_1$(black curves). []{data-label="AB-SL-G"}](2SL-2SL-AB.pdf "fig:"){width="1.65"} ![(Colour online) Conductance of different junctions for different magnitudes of the applied gate: (a, b) $v_0= \delta =0$, (c, d) $v_0=3\gamma_1/2$, $\delta =0$, (e, f) the total conductance for 2SL-2SL and AB-AB junctions, respectively, with $v_0=0$(blue curves) and $v_0=1.5\gamma_1$(black curves). []{data-label="AB-SL-G"}](AB-AB.pdf "fig:"){width="1.65"} ![(Colour online) Conductance across the 2SL-AB system as a function of the bias on the AB-BL with $v_0=0.$ (a) and (b) correspond to the single and double modes regime with $E=0.3 \gamma_1$ and $E=1.15 \gamma_1$, respectively. With $G^{\pm}_T=G_t^\pm+G_b^\pm.$ []{data-label="SL-eAB-G"}](2SLG-eBLG-G.pdf "fig:"){width="1.65"}![(Colour online) Conductance across the 2SL-AB system as a function of the bias on the AB-BL with $v_0=0.$ (a) and (b) correspond to the single and double modes regime with $E=0.3 \gamma_1$ and $E=1.15 \gamma_1$, respectively. With $G^{\pm}_T=G_t^\pm+G_b^\pm.$ []{data-label="SL-eAB-G"}](2SLG-eBLG-G2.pdf "fig:"){width="1.62"} ![(Colour online) Conductance of different junctions for different magnitudes of the applied gate: (a, b) $v_0=\ \delta =0$, (c, d) $v_0=3\gamma_1/2$, $\delta =0$ and (e, f) $v_0=3\gamma_1/2$, $\delta =0.8\gamma_1$. []{data-label="AB-SL-AB-G"}](AB-SL-AB-G0.pdf "fig:"){width="1.65"} ![(Colour online) Conductance of different junctions for different magnitudes of the applied gate: (a, b) $v_0=\ \delta =0$, (c, d) $v_0=3\gamma_1/2$, $\delta =0$ and (e, f) $v_0=3\gamma_1/2$, $\delta =0.8\gamma_1$. []{data-label="AB-SL-AB-G"}](SL-AB-SL-G0.pdf "fig:"){width="1.65"}\ ![(Colour online) Conductance of different junctions for different magnitudes of the applied gate: (a, b) $v_0=\ \delta =0$, (c, d) $v_0=3\gamma_1/2$, $\delta =0$ and (e, f) $v_0=3\gamma_1/2$, $\delta =0.8\gamma_1$. []{data-label="AB-SL-AB-G"}](AB-SL-AB-Gv.pdf "fig:"){width="1.65"} ![(Colour online) Conductance of different junctions for different magnitudes of the applied gate: (a, b) $v_0=\ \delta =0$, (c, d) $v_0=3\gamma_1/2$, $\delta =0$ and (e, f) $v_0=3\gamma_1/2$, $\delta =0.8\gamma_1$. []{data-label="AB-SL-AB-G"}](SL-AB-SL-Gv.pdf "fig:"){width="1.65"}\ ![(Colour online) Conductance of different junctions for different magnitudes of the applied gate: (a, b) $v_0=\ \delta =0$, (c, d) $v_0=3\gamma_1/2$, $\delta =0$ and (e, f) $v_0=3\gamma_1/2$, $\delta =0.8\gamma_1$. []{data-label="AB-SL-AB-G"}](AB-SL-AB-Gve.pdf "fig:"){width="1.65"} ![(Colour online) Conductance of different junctions for different magnitudes of the applied gate: (a, b) $v_0=\ \delta =0$, (c, d) $v_0=3\gamma_1/2$, $\delta =0$ and (e, f) $v_0=3\gamma_1/2$, $\delta =0.8\gamma_1$. []{data-label="AB-SL-AB-G"}](SL-AB-SL-Gve.pdf "fig:"){width="1.65"} In Fig. \[SL-eAB-G\] we show the conductance in a 2SL-AB system as a function of the bias for transport using a single Fig. \[SL-eAB-G\](a) or a double Fig. \[SL-eAB-G\](b) mode. The results show that the contribution from the top and bottom layers to the conductances have opposite behaviours as a function of the inter-layer bias. The total conductance $G_{T}$, however, has a convex form, increasing with the application of an inter-layer bias. From Fig. \[SL-eAB-G\](b), on the other hand, we see that when a second mode is available, four channels contribute to the conductance and the total conductance assumes a concave form, i.e. decreasing with increasing inter-layer bias. This is a characteristic experimental feature that can signal the presence of a second mode of propagation. For the three-block system we show the conductance of the configuration AB-2SL-AB and 2SL-AB-2SL in the left and right columns of Fig. \[AB-SL-AB-G\], respectively. The resulting conductance of the first configuration shows only two non-zero channels $G_+^+$ and $G_-^-$, while the scattered ones $G_+^{-}=G_-^{+} =0$ since $T_+^{-}=T_-^{+} =0$ (see Fig. \[AB-SL-AB-G\](a)). Furthermore, for low energy $G_T=G_+^+$ since the mode $k^{-}$ is not available in this regime but it starts conducting when $E>\gamma_1$. The applied electrostatic potential on the 2SL keeps the scattered conductance channels at zero and a minimum in the conductance appears around the shifted Dirac cone $E=v_0$ of the 2SL as depicted in Fig. \[AB-SL-AB-G\](c). As pointed out before, if the Fermi energy approaches the strength of the electrostatic potential, a non-zero minimum is present in the conductance because charge carriers can be transmitted through a width $d$ of 2SL via evanescent modes[@Snyman]. In Fig. \[AB-SL-AB-G\](f) this minimum disappears and the conductance dramatically increases at $E=\gamma_1$. This is because the bias will couple the two modes and two additional scattered channels $G_+^-$ and $G_-^+$ start conducting. The resonant peaks resulting in the conductance, see Figs. \[AB-SL-AB-G\](a,c,e), are due to the finite size of the intermediate region and hence strictly depend on its width $d$. ![(Colour online)Density plot of the transmission and reflection probabilities through AA-2SL-AB junction as a function of Fermi energy and transverse wave vector $k_y$ with $v_0=1.5\gamma_1,\ \delta =0$ and $d=25$nm. The superimposed dashed curves represent the bands of AB-BL(black), AA-BL(green) and 2SL (white), with $\gamma_1$ being the inter-layer coupling of AB-BL. []{data-label="T-AA-SL-AB"}](T11-AA-SL-AB.pdf "fig:"){width="1.5"} ![(Colour online)Density plot of the transmission and reflection probabilities through AA-2SL-AB junction as a function of Fermi energy and transverse wave vector $k_y$ with $v_0=1.5\gamma_1,\ \delta =0$ and $d=25$nm. The superimposed dashed curves represent the bands of AB-BL(black), AA-BL(green) and 2SL (white), with $\gamma_1$ being the inter-layer coupling of AB-BL. []{data-label="T-AA-SL-AB"}](R11-AA-SL-AB.pdf "fig:"){width="1.73"}\ ![(Colour online)Density plot of the transmission and reflection probabilities through AA-2SL-AB junction as a function of Fermi energy and transverse wave vector $k_y$ with $v_0=1.5\gamma_1,\ \delta =0$ and $d=25$nm. The superimposed dashed curves represent the bands of AB-BL(black), AA-BL(green) and 2SL (white), with $\gamma_1$ being the inter-layer coupling of AB-BL. []{data-label="T-AA-SL-AB"}](T12-AA-SL-AB.pdf "fig:"){width="1.5"} ![(Colour online)Density plot of the transmission and reflection probabilities through AA-2SL-AB junction as a function of Fermi energy and transverse wave vector $k_y$ with $v_0=1.5\gamma_1,\ \delta =0$ and $d=25$nm. The superimposed dashed curves represent the bands of AB-BL(black), AA-BL(green) and 2SL (white), with $\gamma_1$ being the inter-layer coupling of AB-BL. []{data-label="T-AA-SL-AB"}](R12-AA-SL-AB.pdf "fig:"){width="1.73"}\ ![(Colour online)Density plot of the transmission and reflection probabilities through AA-2SL-AB junction as a function of Fermi energy and transverse wave vector $k_y$ with $v_0=1.5\gamma_1,\ \delta =0$ and $d=25$nm. The superimposed dashed curves represent the bands of AB-BL(black), AA-BL(green) and 2SL (white), with $\gamma_1$ being the inter-layer coupling of AB-BL. []{data-label="T-AA-SL-AB"}](T22-AA-SL-AB.pdf "fig:"){width="1.5"} ![(Colour online)Density plot of the transmission and reflection probabilities through AA-2SL-AB junction as a function of Fermi energy and transverse wave vector $k_y$ with $v_0=1.5\gamma_1,\ \delta =0$ and $d=25$nm. The superimposed dashed curves represent the bands of AB-BL(black), AA-BL(green) and 2SL (white), with $\gamma_1$ being the inter-layer coupling of AB-BL. []{data-label="T-AA-SL-AB"}](R22-AA-SL-AB.pdf "fig:"){width="1.73"}\ ![(Colour online)Density plot of the transmission and reflection probabilities through AA-2SL-AB junction as a function of Fermi energy and transverse wave vector $k_y$ with $v_0=1.5\gamma_1,\ \delta =0$ and $d=25$nm. The superimposed dashed curves represent the bands of AB-BL(black), AA-BL(green) and 2SL (white), with $\gamma_1$ being the inter-layer coupling of AB-BL. []{data-label="T-AA-SL-AB"}](T21-AA-SL-AB.pdf "fig:"){width="1.5"} ![(Colour online)Density plot of the transmission and reflection probabilities through AA-2SL-AB junction as a function of Fermi energy and transverse wave vector $k_y$ with $v_0=1.5\gamma_1,\ \delta =0$ and $d=25$nm. The superimposed dashed curves represent the bands of AB-BL(black), AA-BL(green) and 2SL (white), with $\gamma_1$ being the inter-layer coupling of AB-BL. []{data-label="T-AA-SL-AB"}](R21-AA-SL-AB.pdf "fig:"){width="1.73"} ![(Colour online) (a) Transmission and reflection probabilities for normal incidence for $v_0=3\gamma_1/2,\ \delta=0$. (b) Transmission probabilities with normal incidence for AA-BL (AB-BL) n-p-n junction, green (black) curves. Blue (red) curves are the non-zero channels $T_+^+$ ($T_-^-$) in AA-2SL-AB. All energies are considered to be less than the electrostatic potential strength. Conductance of AA-2SL-AB junction for different magnitudes of the applied gate: (c) $v_0= \delta =0$, (d) $v_0=3\gamma_1/2$, $\delta =0$, (e) $v_0=3\gamma_1/2$, $\delta =0.6 \gamma_1$, with $\gamma_1$ being the inter-layer coupling of AB-BL. []{data-label="cond-AA-SL-AB"}](T-norm.pdf "fig:"){width="1.65"} ![(Colour online) (a) Transmission and reflection probabilities for normal incidence for $v_0=3\gamma_1/2,\ \delta=0$. (b) Transmission probabilities with normal incidence for AA-BL (AB-BL) n-p-n junction, green (black) curves. Blue (red) curves are the non-zero channels $T_+^+$ ($T_-^-$) in AA-2SL-AB. All energies are considered to be less than the electrostatic potential strength. Conductance of AA-2SL-AB junction for different magnitudes of the applied gate: (c) $v_0= \delta =0$, (d) $v_0=3\gamma_1/2$, $\delta =0$, (e) $v_0=3\gamma_1/2$, $\delta =0.6 \gamma_1$, with $\gamma_1$ being the inter-layer coupling of AB-BL. []{data-label="cond-AA-SL-AB"}](T-vs-d.pdf "fig:"){width="1.65"}\ ![(Colour online) (a) Transmission and reflection probabilities for normal incidence for $v_0=3\gamma_1/2,\ \delta=0$. (b) Transmission probabilities with normal incidence for AA-BL (AB-BL) n-p-n junction, green (black) curves. Blue (red) curves are the non-zero channels $T_+^+$ ($T_-^-$) in AA-2SL-AB. All energies are considered to be less than the electrostatic potential strength. Conductance of AA-2SL-AB junction for different magnitudes of the applied gate: (c) $v_0= \delta =0$, (d) $v_0=3\gamma_1/2$, $\delta =0$, (e) $v_0=3\gamma_1/2$, $\delta =0.6 \gamma_1$, with $\gamma_1$ being the inter-layer coupling of AB-BL. []{data-label="cond-AA-SL-AB"}](cond-AA-SL-AB.pdf "fig:"){width="1.65"} ![(Colour online) (a) Transmission and reflection probabilities for normal incidence for $v_0=3\gamma_1/2,\ \delta=0$. (b) Transmission probabilities with normal incidence for AA-BL (AB-BL) n-p-n junction, green (black) curves. Blue (red) curves are the non-zero channels $T_+^+$ ($T_-^-$) in AA-2SL-AB. All energies are considered to be less than the electrostatic potential strength. Conductance of AA-2SL-AB junction for different magnitudes of the applied gate: (c) $v_0= \delta =0$, (d) $v_0=3\gamma_1/2$, $\delta =0$, (e) $v_0=3\gamma_1/2$, $\delta =0.6 \gamma_1$, with $\gamma_1$ being the inter-layer coupling of AB-BL. []{data-label="cond-AA-SL-AB"}](condv-AA-SL-AB.pdf "fig:"){width="1.65"}\ ![(Colour online) (a) Transmission and reflection probabilities for normal incidence for $v_0=3\gamma_1/2,\ \delta=0$. (b) Transmission probabilities with normal incidence for AA-BL (AB-BL) n-p-n junction, green (black) curves. Blue (red) curves are the non-zero channels $T_+^+$ ($T_-^-$) in AA-2SL-AB. All energies are considered to be less than the electrostatic potential strength. Conductance of AA-2SL-AB junction for different magnitudes of the applied gate: (c) $v_0= \delta =0$, (d) $v_0=3\gamma_1/2$, $\delta =0$, (e) $v_0=3\gamma_1/2$, $\delta =0.6 \gamma_1$, with $\gamma_1$ being the inter-layer coupling of AB-BL. []{data-label="cond-AA-SL-AB"}](condve-AA-SL-AB.pdf "fig:"){width="1.65"} On the other hand, the conductance of the configuration 2SL-AB-2SL has different features. In Fig. \[AB-SL-AB-G\](b) the four channels, in contrast to the previous configuration, start conducting from $E=0$. This possess layer symmetry such that $G_t^t=G_b^b$ and $G_b^t=G_t^b$. Of particular importance is the equivalence of the four channels for $E<\gamma_1$ while for $E>\gamma_1$ charge carriers strongly scatter between the layers ($i.e.\ G_i^j>G_i^i$) as shown in Fig. \[AB-SL-AB-G\](b). This equivalence of the four channels in the regime $E<\gamma_1$ vanishes when an electrostatic potential is applied ($v_0>0$) to the intermediate region as seen in Fig. \[AB-SL-AB-G\](d). However, the scattered and non-scattered conducting channels are still equivalent in this case where $G_t^{t(b)}=G_b^{b(t)}$ with $G_i^j>G_i^i$ for all energy ranges, see Fig. \[AB-SL-AB-G\](d). As discussed before, the most characteristic feature of the inter-layer bias in the AB-BL is the opening of a gap in the energy spectrum between $v_0\pm\delta$ which is reflected in the conductance as seen in Fig. \[AB-SL-AB-G\](f). The resonant sharp peaks in the conductance near the edges of the gap result from the localized states inside the Mexican hat of the low energy bands. Another consequence of the inter-layer bias is the breaking of the equivalence in the non-scattered conducting channels where now $G_t^{t}\neq G_b^{b}$ as seen in Fig. \[AB-SL-AB-G\](f). AA-2SL-AB {#Concl} --------- Here we consider the case where the leads consist of BL with different stackings separated by two uncoupled graphene sheets. Such a structure can be formed if in the decoupled region one of the graphene sheets has larger lattice constant, e.g. due to strain, leading to an inter-layer shift when the two layers couple. Notice that the inter-layer coupling strength $\gamma_{1}$ differs for the two bilayer structures. Their ratio is $\gamma_1^{AA}/\gamma_1^{AB}\approx1/2$ [@li2009band; @AA-gamma1; @AA-Yuehua2010]. To account for this difference the energy is normalized to $\gamma_1^{AB}$ such that the upper Dirac cone of pristine AA-BL is now located at $E=1/2$ instead of $E=1$ as in the previous sections. In the junction AA-2SL-AB the charge carriers incident from AA-BL and transmitted through 2SL into AB-BL. The results for the transmission and reflection probabilities of this junction are shown in Fig. \[T-AA-SL-AB\] for $v_0=1.5 \gamma_1,\ \delta=0$ and $ d=25$ nm. The carriers incident from lower($k^+$)/upper($k^-$) Dirac cones in AA-BL can be transmitted into one of the modes ($k^+$ or $k^-$) in the AB-BL, see Fig. \[intro-fig02\](e). On the other hand, the reflection process occurs between the intra- or inter-cone in the AA-BL. Remarkably, Fig. \[T-AA-SL-AB\] shows that the scattered transmission probabilities are very small and that almost all transmission is carried by the non-scattered channels. This is not immediately expected since a priori the $k^{+}$-mode in AA-BL is not related to the $k^{+}$-mode in AB-BL. However, both modes have the same parity under in-plane inversion, showing that this feature is robust against variations in the inter-layer coupling. In contrast to the AA-2SL-AA junction where the scattering between lower and upper cones is forbidden in case of zero bias, here the two cones are coupled even without bias. This results in non-zero reflection in the scattered channels $R_+^-$ and $R_-^+$. For normal incidence, the scattered transmission ($T_+^-$ and $T_-^+$) and the non-scattered reflection ($R_+^+$ and $R_-^-$) channels are zero (see Fig. \[T-AA-SL-AB\]) because in that case both the AA and AB Hamiltonian are block diagonal in the even and odd modes basis. Now, we can investigate Klein tunnelling when transitioning in-between the two types of stacking. For this, we show the non-zero channels of transmission and reflection for normal incidence in Fig. \[cond-AA-SL-AB\](a). We find that in contrast with the AA-2SL-AA case, perfect Klein tunnelling does not occur in the junction AA-2SL-AB. However, as shown in Fig. \[cond-AA-SL-AB\](b), we do find that the transmission probability does not depend on the length or even presence of the 2SL region, in contrast to the previous cases with two domain walls. For $\delta\neq0$ the coupling between the different modes is strengthened and, hence, strong scattering in the transmission and reflection channels occurs. Furthermore, the symmetry with respect to normal incidence in the reflection and transmission channels is broken. The conductance for the discussed structure is shown in Figs. \[cond-AA-SL-AB\](c, d, e) for $(v_0=\delta=0),\ (v_0=1.5\gamma_1,\delta=0)$ and ($v_0=1.5\gamma_1, \delta=0.6\gamma_1$), respectively. For pristine 2SL, the dominant channels are $G_+^+$ and $G_-^-$ . Notice that the latter one starts conducting only when $E>\gamma_1$ and this shows up as a rapid increase in the total conductance $G_T$ at $E=\gamma_1$. The scattered channels $G_+^-$ and $G_-^+$ are only weakly contributing to the total conductance as a result of weak coupling of the modes. In contrast to the junctions AA(AB)-2SL-AA(AB), in this case the scattered channels of the conductance are not equivalent $G_+^-\neq G_-^+$, see Fig. \[cond-AA-SL-AB\](c,d). This is because the scattering occurs between modes in bilayer graphene of different stackings.  The electrostatic potential introduces a minimum at $E=v_0$ in the total conductance due to the absence of propagating states at this energy in the 2SL, see Fig. \[cond-AA-SL-AB\](d). Biasing the intermediate region (2SL) of the junction AA-2SL-AB provides propagating states at $E=v_0$, and hence removing the minima in $G_T$ as shown in Fig. \[cond-AA-SL-AB\](e). In addition, the contribution of the scattered channels $G_+^-$ and $ G_-^+$ becomes more pronounced as a result of the strong coupling between the modes induced by the bias. Finally, notice that the counterpart junction AB-2SL-AA, represents the time-reversal case of the system discussed above. We have verified that the transmission channels are equivalent in the absence of a bias. In the presence of a bias, the angular symmetry is broken and, consequently, the reversed junction features the opposite angular asymmetry, preserving time-reversal invariance. summary and conclusion {#Concl} ====================== Using the four-band model we obtained the conductance, transmission and reflection probabilities through single and double domain walls separating two single layers and AA/AB-stacked bilayer graphene. We discussed in detail the scattering mechanism from detached layers to bilayer graphene and presented compact analytical formulae for the transmission probabilities. These results showed that one can find the inter-layer coupling strength solely through measuring the conductance. We found that an electrostatic potential applied to AB-BL, in an 2SL-AB junction, breaks the layer symmetry in the single-valley transmission probability channels. Such asymmetry originates from the asymmetric coupling in AB-BL and arises as a consequence of the mismatch in energy between the 2SL and AB-BL Dirac cones caused by the electrostatic potentials applied to the AB-BL region. Layer asymmetry exists when only one propagating mode is present and hence is not seen in configurations consisting of AA-BL where the entire energy range is associated with two transport channels. We have also evaluated the robustness of chirality-induced properties, such as Klein tunnelling and anti-Klein tunnelling, to scattering on domains without inter-layer coupling. We found that in domain walls separating 2SL and AA-BL, Klein tunnelling is still preserved. On the other hand, for domain walls separating 2SL and AB-BL, the well known anti-Klein tunnelling in AB-BL is not preserved any more, but neither is Klein tunnelling itself. Moreover, in two domain walls separating three regions whose interlayer coupling is all different, i.e. the AA-2SL-AB case, we find that although perfect Klein tunnelling does not hold, the tunnelling does not depend on the thickness of the 2SL region either. This remarkable effect is attributed to a conservation of parity of the modes. Furthermore, we have found that a strong gate potential difference allows some states to be localized inside the Mexican hat of the low energy bands in the AB-BL. Those states contribute to the conductance and appear as sharp peaks at the two edges of the gap. We showed that scattering between these modes, in the transmission channels, is not allowed in the configuration (AA/AB)-2SL-(AA/AB). However, such scattering can be induced by applying an inter-layer bias on the 2SL which in addition to shifting the bands of the top and bottom layers of 2SL, also couples the modes. In contrast, we showed that the two modes of AA-BL are coupled even without biasing the system in the junction AA-2SL-AB and revealed that the latter junction is equivalent to the AB-2SL-AA. In order to limit the number of parameters, through this article we only considered abrupt domain walls, however, the results are robust against smoothness of the domain walls.[@Hasan1] Our study reveals that the presence of the local domain wall in bilayer graphene samples change the transport properties significantly. Our results may shed light on the design of electronic devices based on bilayer graphene. Finally, we showed that for a given sample with unknown sizes of local stacking domains, the average inter-layer coupling can be estimated through quantum transport measurements. Acknowledgments {#acknowledgments .unnumbered} =============== HMA and HB acknowledge the Saudi Center for Theoretical Physics (SCTP) for their generous support and the support of KFUPM under physics research group projects RG1502-1 and RG1502-2. This work is supported by the Flemish Science Foundation (FWO-Vl) by a post-doctoral fellowship (BVD). Functions definitions {#Sec:Appendix} ===================== The transmission probabilities are calculated by applying appropriate boundary conditions at the 2SL-BL interfaces together with the transfer matrix. After some cumbersome algebra, we obtain for 2SL-AB $$\begin{aligned} T_j^\pm=4 \textrm{Re}(k^{\pm}) \frac{\eta\left[ \eta^2+\left(\textrm{Im}(k^{\mp})+\kappa_{j}\ v_0\ \sin\phi\right)^2\ \right]}{C_{0 }+\sum_{m=1}^4C_m\ \cos(m\phi)},\end{aligned}$$ where\ $C_0=2\left( \textrm{Im}(k^\mp)\textrm{Re}(k^\pm) \right)^2+\epsilon^2\left( \textrm{Im}^{2}(k^\mp)+\textrm{Re}^{2}(k^\pm) \right) +\Gamma_{1}$,\ \ $\Gamma_{1}=2v_{0}^{4}-4v_0^3E+5v_0^{2}E^{2}-3v_0E^{3}+\frac{3}{4}E^4$,\ \ $C_1=-\epsilon \textrm{Re}(k^\pm)\left[ 4\left( v_0^2+\textrm{Im}^2(k^\mp) \right)-6v_0E+3E^2 \right],$\ \ $C_2=\epsilon^2\left( \textrm{Im}^{2}(k^\mp)+\textrm{Re}^{2}(k^\pm) \right)+\Gamma _{2}$,\ \ $\Gamma _{2}=E\left( -4v_0^3+6v_0^2E-4v_{0}E^2+E^3\right)$,\ \ $C_3=\textrm{Re}(k^\pm)E\left( 2v_0^2-3v_0E+E^2 \right)$,\ \ $C_4=\frac{1}{4}E^2(E-2v_0)^2$.\ Similarly, the transmission probabilities for the AB-2SL system are obtained as $$\begin{aligned} T_\pm^j=4 \textrm{Re}(k_j)k^{\pm} \frac{\lambda\left[ \mu^{\pm}+\kappa_{j} v_0 \sin\phi\ \textrm{Im}(k^{\mp}) \right] }{\left\vert Q^{\pm} \right\vert^2},\end{aligned}$$ $$\begin{aligned} \mu^{\pm}=\frac{\epsilon\left( \textrm{Im}^{2}(k^{^{\mp}})+E^{2} \right)- E(\pm1+E)(E+v_0)\sin^{2}\phi }{2\sqrt{E(\pm1+E)}},\\\end{aligned}$$ $ \lambda=E\sqrt{E(\pm1+E)},$\ $Q^{\pm}=\frac{1}{2}[z_{0}-z_{1}\left( k^{\pm}+i\textrm{Im}(k^\mp) \right)+z_{2}k^{\pm}\textrm{Im}(k^\mp)],$\ with\ $z_0=2i\left[ v_0\alpha\ -ik_jE \right]\left[ \alpha \left( -ik_{j} +\alpha\right)+\epsilon E\right]$,\ \ $z_1=E\left[ \left( ik_{j} +\alpha\right)^{2}-\epsilon^{2}\right]$,\ and finally\ $z_2=2\epsilon \left[ik_{j} +\alpha\right]$,\ where $\alpha=\sqrt{E^2\pm E}\sin\phi.$\ \ \ [99]{} T. 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--- abstract: 'A capacity bounded grammar is a grammar whose derivations are restricted by assigning a bound to the number of every nonterminal symbol in the sentential forms. In the paper the generative power and closure properties of capacity bounded grammars and their Petri net controlled counterparts are investigated.' author: - Ralf Stiebe - Sherzod Turaev bibliography: - 'stiebe.bib' title: Capacity Bounded Grammars and Petri Nets --- Introduction {#sec:introduction} ============ The close relationship between Petri nets and language theory has been extensively studied for a long time [@cre:man; @das:pau]. Results from the theory of Petri nets have been applied successfully to provide elegant solutions to complicated problems from language theory [@esp; @hau:jan]. A context-free grammar can be associated with a context-free (communica-tion-free) Petri net, whose places and transitions, correspond to the nonterminals and the rules of the grammar, respectively, and whose arcs and weights reflect the change in the number of nonterminals when applying a rule. In some recent papers, context-free Petri nets enriched by additional components have been used to define regulation mechanisms for the defining grammar [@das:tur; @tur]. Our paper continues the research in this direction by restricting the (context-free or extended) Petri nets with place capacity. Quite obviously, a context-free Petri net with place capacity regulates the defining grammar by permitting only those derivations where the number of each nonterminal in each sentential form is bounded by its capacity. A similar mechanism was discussed in [@gin:spa1] where the total number of nonterminals in each sentential form is bounded by a fixed integer. There it was shown that grammars regulated in this way generate the family of context-free languages of finite index, even if arbitrary nonterminal strings are allowed as left-hand sides. The main result of this paper is that, somewhat surprisingly, grammars with capacity bounds have a greater generative power. This paper is organized as follows. Section \[sec:def\] contains some necessary definitions and notations from language and Petri net theory. The concepts of grammars with capacities and grammars controlled by Petri nets with place capacities are introduced in section \[sec:capacities\]. The generative power and closure properties of capacity-bounded grammars are investigated in sections \[sec:power-gs\] and \[sec:nb-cfg\]. Results on grammars controlled by Petri nets with place capacities are given in section \[sec:PNC\]. Preliminaries {#sec:def} ============= Throughout the paper, we assume that the reader is familiar with basic concepts of formal language theory and Petri net theory; for details we refer to [@das:pau; @han; @rei:roz]. The set of natural numbers is denoted by ${\mathbb{N}}$, the power set of a set S by ${\mathcal{P}({S})}$. We use the symbol $\subseteq$ for inclusion and $\subset$ for proper inclusion. The *length* of a string $w \in X^*$ is denoted by $|w|$, the number of occurrences of a symbol $a$ in $w$ by $|w|_a$ and the number of occurrences of symbols from $Y\subseteq X$ in $w$ by $|w|_Y$. The *empty* string is denoted by ${\lambda}$. A *phrase structure grammar* (due to Ginsburg and Spanier [@gin:spa1]) is a quadruple $G=(V, \Sigma, S, R)$ where $V$ and $\Sigma$ are two finite disjoint alphabets of *nonterminal* and *terminal* symbols, respectively, $S\in V$ is the *start symbol* and is a finite set of *rules*. A string $x\in (V\cup \Sigma)^*$ *directly derives* a string $y\in (V\cup \Sigma)^*$ in $G$, written as $x{\Rightarrow}y$, if and only if there is a rule $u\to v\in R$ such that $x=x_1ux_2$ and $y=x_1vx_2$ for some $x_1, x_2\in (V\cup \Sigma)^*$. The reflexive and transitive closure of the relation ${\Rightarrow}$ is denoted by ${\Rightarrow}^*$. A derivation using the sequence of rules $\pi=r_1r_2\cdots r_k$, $r_i\in R$, $1\leq i\leq k$, is denoted by $\xRightarrow{\pi}$ or $\xRightarrow{r_1r_2\cdots r_k}$. The *language* generated by $G$, denoted by $L(G)$, is defined by $L(G)=\{w\in \Sigma^*{:}S{\Rightarrow}^* w\}.$ A phrase structure grammar $G=(V, \Sigma, S, R)$ is called *context-free* if each rule $u\to v\in R$ has $u\in V$. The family of context-free languages is denoted by $\mathbf{CF}$. A *matrix grammar* is a quadruple $G=(V, \Sigma, S, M)$ where $V, \Sigma, S$ are defined as for a context-free grammar, $M$ is a finite set of *matrices* which are finite strings (or finite sequences) over a set of context-free rules. The language generated by the grammar $G$ consists of all strings $w\in \Sigma^*$ such that there is a derivation $S\xRightarrow{r_1r_2\cdots r_n}w$ where $r_1r_2\cdots r_n$ is a concatenation of some matrices $m_{i_1}, m_{i_2}, \ldots, m_{i_k}\in M$, $k\geq 1$. The family of languages generated by matrix grammars without erasing rules (with erasing rules, respectively) is denoted by $\mathbf{MAT}$ (by $\mathbf{MAT}^{{\lambda}}$, respectively). A *vector grammar* is defined like a matrix grammar, but the derivation sequence $r_1r_2\cdots r_n$ has to be a shuffle of some matrices $m_{i_1}, m_{i_2}, \ldots, m_{i_k}\in M$, $k\geq 1$. A *semi-matrix grammar* is defined like a matrix grammar, but the derivation sequence $r_1r_2\cdots r_n$ has to be the semi-shuffle of some matrices $m_{i_1}, m_{i_2}, \ldots, m_{i_k}\in M$, $k\geq 1$, i.e., from the shuffle of sequences from $\bigcup_{i=1}^t m_i^*$ where $$M=\{m_1,\ldots,m_t\}.$$ The language families generated by vector and semi-matrix grammars are denoted by ${{\bf V}}^{[{\lambda}]}$ and ${{\bf sMAT}}^{[{\lambda}]}$. A *Petri net* (PN) is a construct $N = (P, T, F, \phi)$ where $P$ and $T$ are disjoint finite sets of *places* and *transitions*, respectively, $F \subseteq (P\times T) \cup (T\times P)$ is the set of *directed arcs*, $$\varphi: (P\times T) \cup (T\times P) \rightarrow \{0, 1, 2, \dots\}$$ is a *weight function*, where $\varphi(x,y)=0$ for all $(x,y)\in ((P\times T) \cup (T\times P))-F$. A mapping $$\mu: P \rightarrow \{0,1,2, \ldots\}$$ is called a *marking*. For each place $p\in P$, $\mu(p)$ gives the number of *tokens* in $p$. $^{\bullet}x=\{y{:}\, (y,x)\in F\}$ and $x^{\bullet}=\{y{:}\, (x,y)\in F\}$ are called the sets of *input* and *output* elements of $x\in P\cup T$, respectively. A sequence of places and transitions $\rho=x_1x_2\cdots x_n$ is called a *path* if and only if no place or transition except $x_1$ and $x_n$ appears more than once, and $x_{i+1}\in x^\bullet_{i}$ for all $1\leq i\leq n-1$. We denote by $P_\rho, T_\rho, F_\rho$ the sets of places, transitions and arcs of $\rho$. Two paths $\rho_1$, $\rho_2$ are called *disjoint* if $P_{\rho_1}\cap P_{\rho_2}=\emptyset$ and $T_{\rho_1}\cap T_{\rho_2}=\emptyset$. A path $\rho=t_{1}p_{1}t_{2}p_{2}\cdots p_{k-1}t_{k}$ ($\rho=p_{1}t_{1}p_{1}t_{2}\cdots t_{k}p_{1}$) is called a *chain* (*cycle*). A transition $t \in T$ is *enabled* by marking $\mu$ iff $\mu(p)\geq \phi(p,t)$ for all $p\in P$. In this case $t$ can *occur*. Its occurrence transforms the marking $\mu$ into the marking $\mu'$ defined for each place $p \in P$ by $\mu'(p)=\mu(p)-\phi(p,t)+\phi(t,p)$. This transformation is denoted by $\mu\xrightarrow{t}\mu'$. A finite sequence $t_1t_2\cdots t_k$ of transitions is called *an occurrence sequence* enabled at a marking $\mu$ if there are markings $\mu_1, \mu_2, \ldots, \mu_k$ such that $\mu \xrightarrow{t_1} \mu_1 \xrightarrow{t_2} \ldots \xrightarrow{t_k} \mu_k$. For each $1\leq i\leq k$, marking $\mu_i$ is called *reachable* from marking $\mu$. $\mathcal{R}(N, \mu)$ denotes the set of all reachable markings from a marking $\mu$. A *marked* Petri net is a system $N=(P, T, F, \phi, \iota)$ where $(P, T, F, \phi)$ is a Petri net, $\iota$ is the *initial marking*. Let $M$ be a set of markings, which will be called *final* markings. An occurrence sequence $\nu$ of transitions is called *successful* for $M$ if it is enabled at the initial marking $\iota$ and finished at a final marking $\tau$ of $M$. A Petri net $N$ is said to be $k$-*bounded* if the number of tokens in each place does not exceed a finite number $k$ for any marking reachable from the initial marking $\iota$, i.e., $\mu(p)\leq k$ for all $p\in P$ and for all $\mu\in \mathcal{R}(N, \iota)$. A Petri net is called *bounded* if it is $k$-bounded for some $k\geq 1$. A Petri net with *place capacity* is a system $N=(P, T, F, \phi, \iota,\kappa)$ where $(P, T, F, \phi,\iota)$ is a marked Petri net and $\kappa:P \to {\mathbb{N}}$ is a function assigning to each place a number of maximal admissible tokens. A marking $\mu$ of $N$ is valid if $\mu(p)\leq \kappa(p)$, for each place $p\in P$. A transition $t \in T$ is *enabled* by a marking $\mu$ if additionally the successor marking is valid. A *cf Petri net* with respect to a context-free grammar $G=(V,\Sigma, S, R)$ is a system $$N=(P, T, F, \phi, \beta, \gamma, \iota)$$ where - labeling functions $\beta:P\rightarrow V$ and $\gamma:T\rightarrow R$ are bijections; - $(p,t)\in F$ iff $\gamma(t)=A\rightarrow \alpha$ and $\beta(p)=A$ and the weight of the arc $(p,t)$ is 1; - $(t,p)\in F$ iff $\gamma(t)=A\rightarrow \alpha$, $\beta(p)=x$ where $|\alpha|_x>0$ and the weight of the arc $(t,p)$ is $|\alpha|_x$; - the initial marking $\iota$ is defined by $\iota(\beta^{-1}(S))= 1$ and $\iota(p) = 0$ for all $p\in P-\beta^{-1}(S)$. Further we recall the definitions of extended cf Petri nets, and grammars controlled by these Petri nets (for details, see [@das:tur; @tur]). Let $G=(V, \Sigma, S, R)$ be a context-free grammar with its corresponding cf Petri net $$N=(P, T, F, \phi, \beta, \gamma, \iota).$$ Let $T_1, T_2, \ldots, T_n$ be a partition of $T$. 1\. Let $\Pi=\{\rho_1, \rho_2, \ldots, \rho_n\}$ be the set of disjoint chains such that $T_{\rho_i}=T_i$, $1\leq i\leq n$, and $$\bigcup_{\rho\in\Pi}P_\rho\cap P=\emptyset.$$ An *$h$-Petri net* is a system $N_h=(P\cup Q, T, F\cup E, \varphi, \zeta, \gamma, \mu_0, \tau)$ where and $E=\bigcup_{\rho\in\Pi}F_\rho$; the weight function $\varphi$ is defined by $\varphi(x,y)=\phi(x,y)$ if $(x,y)\in F$ and $\varphi(x,y)=1$ if $(x,y)\in E$; the labeling function $\zeta:P\cup Q\rightarrow V\cup\{\lambda\}$ is defined by $\zeta(p)=\beta(p)$ if $p\in P$ and $\zeta(p)=\lambda$ if $p\in Q$; the initial marking $\mu_0$ is defined by $\mu_0(p)=\iota(p)$ if $p\in P$ and $\mu_0(p)=0$ if $p\in Q$; $\tau$ is the final marking where $\tau(p)=0$ for all $p\in P\cup Q$. 2\. Let $\Pi=\{\rho_1, \rho_2, \ldots, \rho_n\}$ be the set of disjoint cycles such that $T_{\rho_i}=T_i$, $1\leq i\leq n$, and $$\bigcup_{\rho\in\Pi}P_\rho\cap P=\emptyset.$$ A *$c$-Petri net* is a system $N_c=(P\cup Q, T, F\cup E, \varphi, \zeta, \gamma, \mu_0, \tau)$ where $Q=\bigcup_{\rho\in\Pi}P_\rho$ and $E=\bigcup_{\rho\in\Pi}F_\rho$; the weight function $\varphi$ is defined by $\varphi(x,y)=\phi(x,y)$ if and $\varphi(x,y)=1$ if $(x,y)\in E$; the labeling function $\zeta:P\cup Q\rightarrow V\cup\{\lambda\}$ is defined by $\zeta(p)=\beta(p)$ if $p\in P$ and $\zeta(p)=\lambda$ if $p\in Q$; the initial marking $\mu_0$ is defined by $\mu_0(p)=\iota(p)$ if $p\in P$, and $\mu_0(p_{i,1})=1$, $\mu_0(p_{i,j})=0$ where $p_{i,j}\in P_i$, $1\leq i\leq n$, $2\leq j\leq k_i$; $\tau$ is the final marking where $\tau(p)=0$ if $p\in P$, and $\tau(p_{i,1})=1$, $\tau(p_{i,j})=0$ where $p_{i,j}\in P_i$, $1\leq i\leq n$, $2\leq j\leq k_i$. 3\. Let $\Pi=\{\rho_1, \rho_2, \ldots, \rho_n\}$ be the set of cycles such that $T_{\rho_i}\!=T_i$, $1\leq i\leq n$, and $$\bigcup_{\rho\in\Pi}P_\rho\cap P=\emptyset.$$ An *$s$-Petri net* is a system $N_s=(P\cup Q, T, F\cup E, \varphi, \zeta, \gamma, \mu_0, \tau)$ where $Q=\bigcup_{\rho\in\Pi}P_\rho, E=\bigcup_{\rho\in\Pi}F_\rho$; the weight function $\varphi$ is defined by $\varphi(x,y)=\phi(x,y)$ if $(x,y)\in F$ and $\varphi(x,y)=1$ if $(x,y)\in E$; the labeling function $\zeta:P\cup Q\rightarrow V\cup\{\lambda\}$ is defined by $\zeta(p)=\beta(p)$ if $p\in P$ and $\zeta(p)=\lambda$ if $p\in Q$; $\mu_0$ is the initial marking where $\mu_0(p_0)=1$ and $\mu_0(p)=\iota(p)$ if $p\in (P\cup Q)-\{p_0\}$; $\tau$ is the final marking where $\tau(p_0)=1$ and $\tau(p)=0$ if $p\in (P\cup Q)-\{p_0\}$. Figure \[fig:xPNs\] depicts extended cf Petri nets which are constructed with respect to the context-free grammar $G'=(\{S, A, B\}, \Sigma, S, R)$ where $R$ consists of $r_0: S\to AB$, $r_1: A\to \lambda$, $ r_3: A\rightarrow aA$, $r_5: A\to bA$, $r_2: B\to \lambda$, $r_4: B\to aB$, $r_6: B\to bB$.$\diamond$ A *$z$-PN controlled grammar* is a system $G=(V, \Sigma, S, R, N_z)$ where is a context-free grammar and $N_z$ is $z$-Petri net with respect to the context-free grammar $G'$ where $z\in\{h, c, s\}$. The *language* generated by a $z$-Petri net controlled grammar $G$ consists of all strings $w\in \Sigma^*$ such that there is a derivation $S\xRightarrow{r_1r_2\cdots r_k}w\in \Sigma^*$ and a successful occurrence sequence of transitions $\nu=t_1t_2\cdots t_k$ of $N_z$ such that $r_1r_2\cdots r_k=\gamma(t_1t_2\cdots t_k)$. Grammars and Petri nets with capacities {#sec:capacities} ======================================= We will now introduce grammars with capacities and show some relations to similar concepts known from the literature. A *capacity-bounded* grammar is a quintuple $G=(V,\Sigma,S,R,\kappa)$ where $G'=(V,\Sigma,S,R)$ is a grammar and $\kappa: V \to {\mathbb{N}}$ is a capacity function. The language of $G$ contains all words $w \in L(G')$ that have a derivation $S{\Rightarrow}^* w$ such that $|\beta|_A\leq \kappa(A)$ for all $A\in V$ and each sentential form $\beta$ of the derivation. The families of languages generated by arbitrary capacity-bounded grammars (due to Ginsburg and Spanier) and by context-free capacity-bounded grammars are denoted by $\mathbf{GS}_{{\mathit{cb}}}$ and $\mathbf{CF}_{{\mathit{cb}}}$, respectively. The capacity function mapping each nonterminal to $1$ is denoted by $\mathbf{1}$. Capacity bounded grammars are closely related to nonterminal-bounded, deri-vation-bounded and finite index grammars. A grammar $G=(V,\Sigma,S,R)$ is *nonterminal bounded* if $|\beta|_V\leq k$ for some fixed $k \in {\mathbb{N}}$ and all sentential forms $\beta$ derivable in $G$. The *index* of a derivation in $G$ is the maximal number of nonterminal symbols in its sentential forms. $G$ is of *finite index* if every word in $L(G)$ has a derivation of index at most $k$ for some fixed $k\in {\mathbb{N}}$. The family of context-free languages of finite index is denoted by $\mathbf{CF}_{{\mathit{fin}}}$. A *derivation-bounded* grammar is a quintuple $G=(V,\Sigma,S,R,k)$ where $G'=(V,\Sigma,S,R)$ is a grammar and $k \in {\mathbb{N}}$ is a bound on the number of allowed nonterminals. The language of $G$ contains all words $w \in L(G')$ that have a derivation $S{\Rightarrow}^* w$ such that $|\beta|_V\leq k$, for each sentential form $\beta$ of the derivation. It is well-known that the family of derivation bounded languages is equal to $\mathbf{CF}_{{\mathit{fin}}}$, even if arbitrary grammars due to Ginsburg and Spanier are permitted [@gin:spa2]. \[exa:NBLnotCF1\] Let $G=(\{S, A, B, C, D, E, F\}, \{a, b, c\}, S, R,\mathbf{1})$ be the capacity-bounded grammar where $R$ consists of the rules: $$\begin{array}{llll} r_1: S\to ABCD, & r_2: AB\to aEFb, & r_3: CD\to cAD, &r_4: EF\to EC,\\ r_5: EF\to FC, & r_6: AD\to FD, & r_7: AD\to ED, & r_8: EC\to AB,\\ r_9: FD\to CD, & r_{10}: FC\to AF,& r_{11}: AF\to {\lambda}, & r_{12}: ED\to {\lambda}. \end{array}$$ The possible derivations are exactly those of the form $$\begin{array}{ll} S &\xRightarrow{r_1}ABCD\xRightarrow{(r_2r_3r_4r_6r_8r_9)^n}a^nABb^nc^nCD \xRightarrow{r_2r_3}a^{n+1}EFb^{n+1}c^{n+1}AD \\ & \xRightarrow{r_5r_7}a^{n+1}FCb^{n+1}c^{n+1}ED\xRightarrow{r_{10}r_{11}r_{12}}a^nb^nc^n \end{array}$$ (in the last phase, the sequences $r_{10}r_{12}r_{11}$ and $r_{12}r_{10}r_{11}$ could also be applied with the same result). Therefore, $L(G)=\{a^nb^nc^n{:}n\geq 1\}$.$\diamond$ \[exa:NBLnotCF2\] Let $G=(\{S,A,B,C\},\{a,b,c\},S,R,\mathbf{1})$ be the context-free capa-city-bounded grammar where $R$ consists of the rules $r_1: S\to aBbaAb$, $r_2: A\to aBb$, $r_3: B\to C$, $r_4: C\to A$, $r_5: A\to BC$, $r_6: A\to c$, and let $M$ be the regular set $M=\{a^*ccb^*a^*cb^*\}$. The derivations in $G$ generating words from $M$ are exactly those of the form $$\begin{array}{ll} S &\xRightarrow{r_1}aBbaAb\xRightarrow{(r_3r_2r_4r_3r_2r_4)^n}a^nBb^na^nAb^n \xRightarrow{r_6r_3r_4}a^nAb^na^ncb^n\\ &\xRightarrow{(r_2r_3r_4)^m}a^{n+m}Ab^{n+m}a^ncb^n \xRightarrow{r_5r_4r_3r_6r_4r_6}a^{n+m}ccb^{n+m}a^ncb^n \end{array}$$ (one can also apply $r_3r_6r_4$ in the third phase and $r_5r_4r_6r_3r_4r_6$ in the last phase with the same result). Hence, $ L(G)\cap M=\{a^nccb^na^mcb^m{:}n\geq m\geq 1\}\not\in \mathbf{CF}, $ implying that $L(G)$ is not context-free.$\diamond$ The above examples show that capacity-bounded grammars – in contrast to derivation bounded grammars – can generate non-context-free languages. The generative power of capacity-bounded grammars will be studied in detail in the following two sections. The notions of finite index and bounded capacities can be extended to matrix, vector and semi-matrix grammars. The corresponding language families are denoted by ${{\bf MAT}}^{[{\lambda}]}_{{\mathit{fin}}}$, ${{\bf V}}^{[{\lambda}]}_{{\mathit{fin}}}$, ${{\bf sMAT}}^{[{\lambda}]}_{{\mathit{fin}}}$, ${{\bf MAT}}^{[{\lambda}]}_{cb}$, ${{\bf V}}^{[{\lambda}]}_{cb}$, ${{\bf sMAT}}^{[{\lambda}]}_{cb}$. Also control by Petri nets can in a natural way be extended to Petri nets with place capacities. Since an extended cf Petri net $N_z$, $z\in\{h, c, s\}$, has two kinds of places, i.e., places labeled by nonterminal symbols and *control* places, it is interesting to consider two types of place capacities in the Petri net: first, we demand that only the places labeled by nonterminal symbols are with capacities (*weak capacity*), and second, all places of the net are with capacities (*strong capacity*). A $z$-Petri net $N_z=(P\cup Q, T, F\cup E, \varphi, \zeta, \gamma, \mu_0, \tau)$ is with *weak capacity* if the corresponding cf Petri net $(P, T, F, \phi, \iota)$ is with place capacity, and *strong capacity* if the Petri net $(P\cup Q, T, F\cup E, \varphi, \mu_0)$ is with place capacity. A grammar controlled by a $z$-Petri net with *weak* (*strong*) *capacity* is a $z$-Petri net controlled grammar $G = (V, \Sigma, S, R, N_z)$ where $N_z$ is with weak (strong) place capacity. We denote the families of languages generated by grammars (with erasing rules) controlled by $z$-Petri nets with weak and strong place capacities by $\mathbf{wPN}_{cz}$, $\mathbf{sPN}_{cz}$ ($\mathbf{wPN}^{\lambda}_{cz}$, $\mathbf{sPN}^{\lambda}_{cz}$), respectively, where $z\in\{h, c, s\}$. The power of arbitrary grammars with capacities {#sec:power-gs} =============================================== It will be shown in this section that arbitrary grammars (due to Ginsburg and Spanier) with capacity generate exactly the family of matrix languages of finite index. This is in contrast to derivation bounded grammars which generate only context-free languages of finite index. First we show that we can restrict to grammars with capacities bounded by $1$. Let $\mathbf{CF}_{{\mathit{cb}}}^{1}$ and $\mathbf{GS}_{{\mathit{cb}}}^{1}$ be the language families generated by context-free and arbitrary grammars with capacity function $\mathbf{1}$. $\mathbf{CF}_{{\mathit{cb}}}=\mathbf{CF}_{{\mathit{cb}}}^{1}$ and $\mathbf{GS}_{{\mathit{cb}}}=\mathbf{GS}_{{\mathit{cb}}}^{1}$. Let $G=(V,\Sigma,S,R, \kappa)$ be a capacity-bounded phrase structure grammar. We construct the grammar $G'=(V',\Sigma,(S,1),R')$ with capacity function $\mathbf{1}$ and $$\begin{aligned} V'&=& \{(A,i) {:}A \in V, 1\leq i\leq \kappa(A)\},\\ R'&=& \{\alpha' \to \beta' {:}\alpha' \in h(\alpha), \beta' \in h(\beta), \mbox{ for some } \alpha \to \beta \in R\}, \end{aligned}$$ where $h:(V\cup \Sigma)^* \to (V' \cup \Sigma)^*$ is the finite substitution defined by $h(a)=\{a\}$, for $a \in \Sigma$, and , for $A \in V$. It can be shown by induction on the number of derivation steps that $S \!{\Rightarrow}^*_{G,\kappa}\! \alpha$ holds iff , for some $\alpha' \in h(\alpha)$. \[lem:GScbSubsetMATfin\] $\mathbf{GS}_{{\mathit{cb}}}\subseteq \mathbf{MAT}_{{\mathit{fin}}}$. Consider some language $L\in \mathbf{GS}_{{\mathit{cb}}}$ and let $G=(V,\Sigma,S,R,\mathbf{1})$ be a capacity-bounded phrase structure grammar (due to Ginsburg and Spanier) such that $L=L(G)$. A word $\alpha\in (V\cup \Sigma)^*$ can be uniquely decomposed as $$\alpha=x_1 \beta_1 x_2 \beta_2 \cdots x_n \beta_n x_{n+1}, x_1,x_{n+1} \in \Sigma^*, x_2,\ldots,x_n \in \Sigma^+, \beta_1,\ldots, \beta_n\in V^+.$$ The subwords $\beta_i$ are referred to as the *maximal nonterminal blocks* of $\alpha$. Note that the length of a maximal block in any sentential form of a derivation in $G$ is bounded by $|V|$. We will first construct a capacity-bounded grammar $G'$ with $L(G')=L$ such that all words of $L$ can be derived in $G'$ by rewriting a maximal nonterminal block in every step. Let $G'=(V,\Sigma,S,R',\mathbf{1})$ where $$\begin{aligned} R'&=& \{\alpha_1 \alpha \alpha_2 \to \alpha_1 \beta \alpha_2 {:}\alpha \to \beta \in R, \alpha_1,\alpha_2 \in V^*, |\alpha_1 \alpha \alpha_2|_A \leq 1, \mbox{ for all } A\in V\}. \end{aligned}$$ The inclusion $L(G) \subseteq L(G')$ is obvious since $R\subseteq R'$. On the other hand, any derivation step in $G'$ can be written as $\gamma_1 \underline{\alpha_1 \alpha \alpha_2} \gamma_2 {\Rightarrow}_{G'} \gamma_1 \underline{\alpha_1 \beta \alpha_2} \gamma_2$, where $\alpha \to \beta \in R$, implying that the same step can be performed in $G$ as $\gamma_1 \alpha_1 \underline{\alpha} \alpha_2 \gamma_2 {\Rightarrow}_{G,1} \gamma_1 \alpha_1 \underline{\beta} \alpha_2 \gamma_2.$ Thus $L(G')\subseteq L(G)$ holds as well. Moreover, any derivation step in $G$, $\gamma_1 \alpha_1 \underline{\alpha} \alpha_2 \gamma_2 {\Rightarrow}_{G,1} \gamma_1 \alpha_1 \underline{\beta} \alpha_2 \gamma_2$, $\alpha_1\alpha\alpha_2$ being a maximal nonterminal block, can be performed in $G'$ replacing the maximal nonterminal block $\alpha_1\alpha\alpha_2$ by $\alpha_1\beta\alpha_2$. In the second step we construct a context-free matrix grammar $H$ which simulates exactly those derivations in $G'$ that replace a maximal nonterminal block in each step. We introduce two alphabets $$\begin{aligned} [V]&=&\{[\alpha] {:}\alpha \in V^+, |\alpha|_A\leq 1, \mbox{ for all } A \in V\}\mbox{ and } \overline{V}=\{\overline{A} {:}A \in V\}. \end{aligned}$$ The symbols of $[V]$ are used to encode each maximal nonterminal block as single symbols, while $\overline{V}$ is a disjoint copy of $V$. Any word $$\alpha=x_1 \beta_1 x_2 \beta_2 \cdots x_n \beta_n x_{n+1}, x_1,x_{n+1} \in \Sigma^*, x_2,\ldots,x_n \in \Sigma^+, \beta_1,\ldots \beta_n\in V^+$$ such that $|\alpha|_A\leq 1$, for all $A \in V$, can be represented by the word $[\alpha]=x_1 [\beta_1] x_2 [\beta_2] \cdots x_n [\beta_n] x_{n+1}$, where the maximal nonterminal blocks in $\alpha$ are replaced by the corresponding symbols from $[V]$. The desired matrix grammar is obtained as $H=(V_H,\Sigma,S',M)$, with $V_H=[V]\cup V \cup \overline{V} \cup \{S'\}$ and the set of matrices defined as follows. For any rule $r=\alpha\to \beta$ in $R'$, $M$ contains the matrix $m_r$ consisting of the rules - $[\alpha] \to [\beta]$ (note that $\alpha \in [V]$, but $\beta\in ([V]\cup\Sigma)^*$), - $A \to \overline{A}$, for all $A \in V$ such that $|\alpha|_A=1$ and $|\beta|_A=0$, - $\overline{A}\to A$, for all $A \in V$ such that $|\alpha|_A=0$ and $|\beta|_A=1$. (The order of the rules in $m_r$ is arbitrary). Additionally, $M$ contains the starting and the terminating matrices $$(S'\to [S] S \overline{A_1} \cdots \overline{A_m}) \mbox{ and } (\overline{S} \to {\lambda},\overline{A_1} \to {\lambda}, \ldots,\overline{A_m} \to {\lambda}),$$ where $V=\{S,A_1,\ldots,A_m\}$. Intuitively, $H$ generates sentential forms of the shape $[\beta] \gamma$ where$[\beta] \in ([V] \cup \Sigma)^*$ encodes a sentential form $\beta$ derivable in $G'$ and $\gamma \in (V\cup \overline{V})$ gives a count of the nonterminal symbols in $\beta$ as follows: $|\gamma|_A+|\gamma|_{\overline{A}}=1$ and $|\gamma|_A=|\beta|_A$. Formally, it can be shown by induction that a sentential form over $V_H \cup \Sigma$ can be generated after applying $k\geq 1$ matrices (except for the terminating) iff it has the form $[\beta] \gamma$ where - $\beta \in (V\cup\Sigma)^*$ can be derived in $G'$ in $k-1$ steps, - $\gamma \in \{S,\overline{S}\} \{A_1,\overline{A}_1\} \cdots\{A_m,\overline{A}_m\}$ and $|\gamma|_A=1$ iff $|\beta|_A=1$. We can also show that the inverse inclusion also holds. \[MATfinInCScb\] $\mathbf{MAT}_{{\mathit{fin}}}\subseteq \mathbf{GS}_{{\mathit{cb}}}$. Capacity-bounded context-free grammars {#sec:nb-cfg} ====================================== In this section, we investigate capacity-bounded context-free grammars. It turns out that they are strictly between context-free languages of finite index and matrix languages of finite index. Closure properties of capacity bounded languages with respect to AFL operations are shortly discussed at the end of the section. As a first result we show that the family of context-free languages with finite index is properly included in ${{\bf CF}}_{{\mathit{cb}}}$. \[thm:hierarchyCapacityBounded1\] ${{\bf CF}}_{{\mathit{fin}}} \subset {{\bf CF}}_{{\mathit{cb}}}$. Any context-free language generated by a grammar $G$ of index $k$ is also generated by the capacity-bounded grammar $(G,\kappa)$ where $\kappa$ is the capacity function constantly $k$. The properness of the inclusion follows from Example \[exa:NBLnotCF2\]. An upper bound for ${{\bf CF}}_{{\mathit{cb}}}$ is given by the inclusion ${{\bf CF}}_{{\mathit{cb}}}\subseteq {{\bf GS}}_{{\mathit{cb}}}={{\bf MAT}}_{{\mathit{fin}}}$. We can prove the properness of the inclusion by presenting a language from ${{\bf MAT}}_{{\mathit{fin}}} \setminus {{\bf CF}}_{{\mathit{cb}}}$. $L=\{a^n b^n c^n {:}n\geq 1\} \notin \mathbf{CF}_{{\mathit{cb}}}$. Consider a capacity-bounded context-free grammar $G=(V,\Sigma,S,R,\mathbf{1})$ such that $L \subseteq L(G)$. For $A \in V$, let $G_A=(V,\Sigma,R,A,\mathbf{1})$. The following holds obviously for any derivation in $G$ involving $A$: If $\alpha A \beta {\Rightarrow}^*_{G} xyz$, where $\alpha,\beta \in (V\cup \Sigma)^*$, $x,y,z \in \Sigma^*$ and $y$ is the yield of $A$, then $y \in L(G_A)$. On the other hand, for all $x,y,z \in \Sigma^*$ such that $y\in L(G_A)$, the relation $xAz {\Rightarrow}^*_{G} xyz$ holds. The nonterminal set $V$ can be decomposed as $V=V_{{\mathit{inf}}} \cup V_{{\mathit{fin}}}$, where $$\begin{aligned} V_{{\mathit{inf}}} &=& \{A \in V {:}L(G_A) \mbox{ is infinite}\}\mbox{ and } V_{{\mathit{fin}}} = \{A \in V {:}L(G_A) \mbox{ is finite}\}. \end{aligned}$$ Let $K$ be a number such that $|w|<K$, for all $w \in \bigcup_{A \in V_{{\mathit{fin}}}} L(G_A)$. Consider the word $w=a^{rK} b^{rK} c^{rK}$, where $r$ is the longest length of a right side in a rule of $R$. There is a derivation $S {\Rightarrow}^*_{G} w$. Consider the last sentential form $\alpha$ in this derivation that contains a symbol from $V_{{\mathit{inf}}}$. Let this symbol be $A$. All other nonterminals in $\alpha$ are from $V_{{\mathit{fin}}}$, and none of them generates a subword containing $A$ in the further derivation process. We get thus another derivation of $w$ in $G$ by postponing the rewriting of $A$ until all other nonterminals have vanished by applying on them the derivation sequence of the original derivation. This new derivation has the form $S{\Rightarrow}^*_{G} \alpha {\Rightarrow}^*_{G} xAz {\Rightarrow}^*_{G} xyz=w.$ The length of $y$ can be estimated by $|y|\leq rK$, as $A$ is in the first step replaced by a word over $(\Sigma \cup V_{{\mathit{fin}}})$ of length at most $r$. By the remarks in the beginning of the proof, any word $xy'z$ with $y' \in L(G_A)$ can be derived in $G$. A case analysis shows that $xy'z$ is not in $L$, for any $y'\neq y$. Hence $L(G) \neq L$. The results can be summarized as follows: \[thm:hierarchyCapacityBounded\] $\mathbf{CF}_{{\mathit{fin}}}\subset \mathbf{CF}_{{\mathit{cb}}} \subset \mathbf{GS}_{{\mathit{cb}}}=\mathbf{MAT}_{{\mathit{fin}}}.$ As regards closure properties, we remark that the constructions showing the closure of $\mathbf{CF}$ under homomorphisms, union, concatenation and Kleene closure can be easily extended to the case of capacity bounded languages. \[thm:closureCapacityBounded\] $\mathbf{CF}_{{\mathit{cb}}}$ is closed under homomorphisms, union, concatenation and Kleene closure. We give here a proof only for the Kleene closure and leave the other cases to the reader. Let $L\in \mathbf{CF}_{{\mathit{cb}}}$ and let $G=(V,\Sigma,S,R,\mathbf{1})$ be a context-free grammar such that $L=L(G)$. We construct $G'=(V\cup\{S'\},\Sigma,S',R\cup \{S'\to SS',S'\to {\lambda}\},\mathbf{1}).$ Any terminating derivation in $G'$ that applies the rule $S'\to SS'$ $k$ times generates a word, where $w_i$ is the yield of the $i$-th symbol $S$ introduced by $S'\to SS'$. The subderivation from $S$ to $w_i$ only uses rules from $R$. Moreover, any sentential form $\beta_i$ in this subderivation is the subword of some sentential form $\beta$ in the derivation of $w$ in $G'$. Hence, $|\beta_i|_A \leq |\beta|_A\leq 1$, for all $1\leq i \leq k$ and all $A \in V$. Consequently, $w_i\in L(G)=L$ and $w \in L^*$. Conversely, any word $w=w_1 w_2 \cdots w_k$ with $w_i\in L$, for $1\leq i\leq k$, can be obtained in $G'$ by the derivation $$S'{\Rightarrow}SS' {\Rightarrow}^* w_1 S' {\Rightarrow}w_1SS' {\Rightarrow}^* w_1w_2S' {\Rightarrow}^* w_1w_2 \cdots w_kS' {\Rightarrow}w_1w_2\cdots w_k$$ where the subwords $w_i$ are derived from $S$ as in $G$. As regards closure under intersection with regular sets and under inverse homomorphisms, the constructions to show closure of $\mathbf{CF}$ cannot be extended, since they do not keep the capacity bound. We suspect that $\mathbf{CF}_{{\mathit{cb}}}$ is not closed under any of these operations. Control by Petri nets with place capacities {#sec:PNC} =========================================== We will first establish the connection between context-free Petri nets with place capacities and capacity-bounded grammars. Later we will investigate the generative power of various extended context-free Petri nets with place capacities. The proof for the equivalence between context-free grammars and grammars controlled by cf Petri nets can be immediately transferred to context-free grammars and Petri nets with capacities: \[thm:CapacityPetriNetGrammar\] Grammars controlled by context-free Petri nets with place capacity functions generate the family of capacity-bounded context-free languages. Let us now turn to grammars controlled by extended cf Petri nets with capacities. We will first study the generative power of capacity-bounded matrix and vector grammars, which are closely related to these Petri net grammars. \[thm:matrixGrammarBounds\] ${{\bf MAT}}_{{\mathit{fin}}}={{\bf V}}^{[{\lambda}]}_{{\mathit{cb}}}={{\bf MAT}}^{[{\lambda}]}_{{\mathit{cb}}}={{\bf sMAT}}^{[{\lambda}]}_{{\mathit{cb}}}$. We give the proof of ${{\bf MAT}}_{{\mathit{fin}}}={{\bf V}}^{{\lambda}}_{{\mathit{cb}}}$. The other equalities can be shown in an analogous way. Since ${{\bf MAT}}_{{\mathit{fin}}}={{\bf V}}_{{\mathit{fin}}}={{\bf V}}^{{\lambda}}_{{\mathit{fin}}}$, it suffices to prove ${{\bf V}}_{{\mathit{fin}}}\subseteq {{\bf V}}^{{\lambda}}_{{\mathit{cb}}}$ and ${{\bf V}}^{{\lambda}}_{{\mathit{cb}}}\subseteq {{\bf V}}^{{\lambda}}_{{\mathit{fin}}}$. The first inclusion is obvious because any vector grammar of finite index $k$ is equivalent to the same vector grammar with capacity function constantly $k$. To show ${{\bf V}}^{{\lambda}}_{{\mathit{cb}}}\subseteq {{\bf V}}^{{\lambda}}_{{\mathit{fin}}}$, consider a capacity-bounded vector grammar $$G=(\{A_0,A_1,\ldots,A_m\},\Sigma,A_0,M,\mathbf{1}).$$ (The proof that it suffices to consider the capacity function $\mathbf{1}$ is like for usual grammars.) To construct an equivalent vector grammar of finite index, we introduce the new nonterminal symbols $B_i,B'_i$, $0\leq i\leq m$, $C$, $C'$. For any rule $r: A\to \alpha$, we define the matrix $\mu(r)=(C\to C',s_0,s_1,\ldots,s_m,r,C'\to C)$ such that $s_i=B_i \to B'_i$ if $A=A_i$ and $|\alpha|_A=0$, $s_i=B'_i \to B_i$ if $A\neq A_i$ and $|\alpha|_{A_i}=1$, and $s_i$ is empty, otherwise. Now we can construct $G'=(V',\Sigma,S',M')$ where $M'$ contains - for any matrix $m=(r_1,r_2, \ldots, r_k)$, the matrix $m'=(\mu(r_1), \ldots, \mu(r_k))$, - the start matrix $(S'\to A_0 B_0 B'_1 \cdots B'_m C)$, - the terminating matrix $(C\to {\lambda}, B'_0\to {\lambda},B'_1\to{\lambda}, \ldots, B'_m\to {\lambda})$, and $V'=V\cup \{B_i,B'_i{:}0\leq i\leq m\} \cup \{S',C,C'\}$. The construction of $G'$ allows only derivation sequences where complete submatrices $\mu(r)$ are applied: when the sequence $\mu(r)$ has been started, there is no symbol $C$ before $\mu(r)$ is finished, and no other submatrix can be started. It is easy to see that $G'$ can generate after applying complete submatrices exactly those words $\beta \gamma C$ such that $\beta \in (V\cup \Sigma)^*$, such that $\beta$ can be derived in $G$ and $|\gamma|_{B_i}=1$ iff $|\beta|_{A_i}=1$. Moreover, $G'$ is of index $2 |V|+1$. By constructions similar to those in [@tur] and Theorem \[thm:matrixGrammarBounds\] we can show with respect to weak capacities: \[lem:VfinInwPNch\] For $z\in \{h,c,s\}$, ${{\bf MAT}}_{{\mathit{fin}}}=\mathbf{wPN}^{[{\lambda}]}_{cz}$. We give only the proof for $z=h$. The other equations can be shown using analogous arguments. By Theorem \[thm:matrixGrammarBounds\] it is sufficient to show the inclusions ${{\bf V}}_{fin}\subseteq\mathbf{wPN}_{ch}$ and $\mathbf{wPN}^{{\lambda}}_{ch}\subseteq {{\bf V}}^{{\lambda}}_{cb}$. As regards the first inclusion, let $L$ be a vector language of finite index (with or without erasing rules), and let $ind(L)=k$, $k\geq 1$. Then, there is a vector grammar $G=(V, \Sigma, S, M)$ such that $L=L(G)$ and $ind(G)\leq k$. Without loss of generality we assume that $G$ is without repetitions. Let $R$ be the set of the rules of $M$. By Theorem 16 in [@tur], we can construct an $h$-Petri net controlled grammar $G'=(V, \Sigma, S, R, N_h)$, $N_h=(P\cup Q, T, F\cup E, \varphi, \zeta, \gamma, \mu_0, \tau)$, which is equivalent to the grammar $G$. By definition, for every sentential form $w\in (V\cup\Sigma)^*$ in the grammar $G$, $|w|_V\leq k$. It follows that $|w|_A\leq k$ for all $A\in V$. By bijection $\zeta:P\cup Q\to V\cup\{{\lambda}\}$ we have $\mu(p)=\mu(\zeta^{-1}(A))\leq k$ for all $p\in P$ and $\mu \in \mathcal{R}(N_h, \mu_0)$, i.e., the corresponding cf Petri net $(P, T, F, \phi, \beta, \gamma, \iota)$ is with $k$-place capacity. Therefore $G'$ is with weak place capacity. On the other hand, the construction of an equivalent vector grammar for an $h$-Petri net controlled grammar, can be extended to the case of weak capacities just by assigning the capacities of the corresponding places to the nonterminal symbols of the grammar. As regards strong capacities, there is no difference between weak and strong capacities for grammars controlled by $c$- and $s$-Petri nets because the number of tokens in every circle is limited by $1$. This yields: \[lem:wPNx=sPNx\] For $z\in \{c,s\}$, ${{\bf MAT}}_{{\mathit{fin}}}=\mathbf{sPN}^{[{\lambda}]}_{cz}$. The only families not characterized yet are $\mathbf{sPN}^{[{\lambda}]}_{ch}$. We conjecture that they are also equal to ${{\bf MAT}}_{{\mathit{fin}}}$. Conclusions {#sec:conclusions} =========== We have introduced grammars with capacity bounds and their Petri net controlled counterparts. In particular, we have shown that their generative power lies strictly between the context-free languages of finite index and the matrix languages of finite index. Moreover, we studied extended context-free Petri nets with place capacities. A possible extension of the concept is to use capacity functions that allow an unbounded number of some nonterminals. The investigation shows that for every grammar controlled by a cf Petri net with $k$-place capacity, $k\geq 1$, there exists an equivalent grammar controlled by a cf Petri net with 1-place capacity, i.e., the families of languages generated by cf Petri nets with place capacities do not form a hierarchy with respect to the place capacities.

--- abstract: | We consider the problem of maintaining a dynamic set of integers and answering queries of the form: report a point (equivalently, all points) in a given interval. Range searching is a natural and fundamental variant of integer search, and can be solved using predecessor search. However, for a RAM with $w$-bit words, we show how to perform updates in $O(\lg w)$ time and answer queries in $O(\lg\lg w)$ time. The update time is identical to the van Emde Boas structure, but the query time is exponentially faster. Existing lower bounds show that achieving our query time for predecessor search requires doubly-exponentially slower updates. We present some arguments supporting the conjecture that our solution is optimal. Our solution is based on a new and interesting recursion idea which is “more extreme” that the van Emde Boas recursion. Whereas van Emde Boas uses a simple recursion (repeated halving) on each path in a trie, we use a nontrivial, van Emde Boas-like recursion on every such path. Despite this, our algorithm is quite clean when seen from the right angle. To achieve linear space for our data structure, we solve a problem which is of independent interest. We develop the first scheme for dynamic perfect hashing requiring sublinear space. This gives a dynamic Bloomier filter (an approximate storage scheme for sparse vectors) which uses low space. We strengthen previous lower bounds to show that these results are optimal. author: - | Christian Worm Mortensen[^1]\ IT U. Copenhagen\ `cworm@itu.dk` - | Rasmus Pagh\ IT U. Copenhagen\ `pagh@itu.dk` - | Mihai Pǎtraşcu\ MIT\ `mip@mit.edu` bibliography: - '../general.bib' title: On Dynamic Range Reporting in One Dimension --- Introduction ============ Our problem is to maintain a set $S$ under insertions and deletions of values, and a range reporting query. The query ${\texttt{findany}}(a,b)$ should return an arbitrary value in $S \cap [a,b]$, or report that $S \cap [a,b] = \emptyset$. This is a form of existential range query. In fact, since we only consider update times above the predecessor bound, updates can maintain a linked list of the values in $S$ in increasing order. Given a value $x \in S \cap [a,b]$, one can traverse this list in both directions starting from $x$ and list all values in the interval $[a,b]$ in constant time per value. Thus, the ${\texttt{findany}}$ query is equivalent to one-dimensional range reporting. The model in which we study this problem is the word RAM. We assume the elements of $S$ are integers that fit in a word, and let $w$ be the number of bits in a word (thus, the “universe size” is $u = 2^w$). We let $n = |S|$. Our data structure will use Las Vegas randomization (through hashing), and the bounds stated will hold with high probability in $n$. Range reporting is a very natural problem, and its higher-dimensional versions have been studied for decades. In one dimension, the problem is easily solved using predecessor search. The predecessor problem has also been studied intensively, and the known bounds are now tight in almost all cases [@beame02predecessor]. Another well-studied problem related to ours is the lookup problem (usually solved by hashing), which asks to find a key in a set of values. Our problem is more general than the lookup problem, and less general than the predecessor problem. While these two problems are often dubbed “the integer search problems”, we feel range reporting is an equally natural and fundamental incarnation of this idea, and deserves similar attention. The first to ask whether or not range reporting is as hard as finding predecessors were Miltersen et al in STOC’95 [@miltersen99asymmetric]. For the static case, they gave a data structure with space $O(nw)$ and constant query time, which cannot be achieved for the predecessor problem with polynomial space. An even more surprising result from STOC’01 is due to Alstrup, Brodal and Rauhe [@alstrup01range], who gave an optimal solution for the static case, achieving linear space and constant query time. In the dynamic case, however, no solution better than the predecessor problem was known. For this problem, the fastest known solution in terms of $w$ is the classic van Emde Boas structure [@veb77predecessor], which achieves $O(\lg w)$ time per operation. For the range reporting problem, we show how to perform updates in $O(\lg w)$ time, while supporting queries in $O(\lg\lg w)$ time. The space usage is optimal, i.e. $O(n)$ words. The update time is identical to the one given by the van Emde Boas structure, but the query time is exponentially faster. In contrast, Beame and Fich [@beame02predecessor Theorem 3.7] show that achieving any query time that is $o(\lg w / \lg\lg w)$ for the predecessor problem requires update time $\Omega(2^{w^{1 - \epsilon}})$, which is doubly-exponentially slower than our update time. We also give an interesting tradeoff between update and query times; see theorem \[thm:range\] below. Our solution incorporates some basic ideas from the previous solutions to static range reporting in one dimension [@miltersen99asymmetric; @alstrup01range]. However, it brings two important technical contributions. First, we develop a new and interesting recursion idea which is more advanced than van Emde Boas recursion (but, nonetheless, not technically involved). We describe this idea by first considering a simpler problem, the bit-probe complexity of the greater-than function. Then, the solution for dynamic range reporting is obtained by using the recursion for this simpler problem, on *every path* of a binary trie of depth $w$. This should be contrasted to the van Emde Boas structure, which uses a very simple recursion idea (repeated halving) on every root-to-leaf path of the trie. The van Emde Boas recursion is fundamental in the modern world of data structures, and has found many unrelated applications (e.g. exponential trees, integer sorting, cache-oblivious layouts, interpolation search trees). It will be interesting to see if our recursion scheme has a similar impact. The second important contribution of this paper is needed to achieve linear space for our data structure. We develop a scheme for dynamic perfect hashing, which requires sublinear space. This can be used to store a sparse vector in small space, if we are only interested in obtaining correct results when querying non-null positions (the Bloomier filter problem). We also prove that our solution is optimal. To our knowledge, this solves the last important theoretical problem connected to Bloom filters. The stringent space requirements that our data structure can meet are important in data-stream algorithms and database systems. We mention one application below, but believe others exist as well. Data-Stream Perfect Hashing and Bloomier Filters ------------------------------------------------ The Bloom filter is a classic data structure for testing membership in a set. If a constant rate of false-positives is allowed, the space *in bits* can be made essentially linear in the size of the set. Optimal bounds for this problem are obtained in [@pagh05bloom]. Bloomier filters, an extension of the classical Bloom filter with a catchy name, were defined and analyzed in the static case by Chazelle et al [@chazelle04bloom]. The problem is to represent a vector $V[0..u-1]$ with elements from $\{ 0, \dots, 2^r - 1\}$ which is nonzero in only $n$ places (assume $n \ll u$, so the vector is sparse). Thus, we have a sparse set as before, but with values associated to the elements. The information theoretic lower bound for representing such a vector is $\Omega(n\cdot r + \lg \binom{u}{n}) \approx \Omega(n (r + \lg u))$ bits. However, if we only want correct answers when $V[x] \ne 0$, we can obtain a space usage of roughly $O(nr)$ bits in the static case. For the dynamic problem, where the values of $V$ can change arbitrarily at any point, achieving such low space is impossible regardless of the query and update times. Chazelle et al. [@chazelle04bloom] proved that $\Omega(n(r + \min(\lg\lg \frac{u}{n^3}, \lg n)))$ bits are needed. No non-trivial upper bound was known. We give matching lower and upper bounds: \[thm:bloomlb\] The randomized space complexity of maintaining a dynamic Bloomier filter for $r\geq 2$ is $\Theta(n(r + \lg\lg \frac{u}{n}))$ bits in expectation. The upper bound is achieved by a RAM data structure that allows access to elements of the vector in worst-case constant time, and supports updates in amortized expected $O(1)$ time. To detect whether $V[x] = 0$ with probability of correctness at least $1-\epsilon$, one can use a Bloom filter on top. This requires space $\Theta(n\lg( 1/\epsilon ))$, and also works in the dynamic case [@pagh05bloom]. Note that even for $\epsilon = 1$, randomization is essential, since any deterministic solution must use $\Omega(n \lg(u/n))$ bits of space, i.e. it must essentially store the set of nonzero entries in the vector. With marginally more space, $O(n(r + \lg\lg u))$, we can make the space and update bounds hold with high probability. To do that, we analyze a harder problem, namely maintaining a perfect hash function dynamically using low space. The problem is to maintain a set $S$ of keys from $\{0, \dots, u-1\}$ under insertions and deletions, and be able to evaluate a perfect hash function (i.e. a one-to-one function) from $S$ to a small range. An element needs to maintain the same hash value while it is in $S$. However, if an element is deleted and subsequently reinserted, its hash value may change. \[thm:hash\] We can maintain a perfect hash function from a set $S \subset \{ 0, \dots, u-1 \}$ with $|S| \leq n$ to a range of size $n + o(n)$, under $n^{O(1)}$ insertions and deletions, using $O(n\lg\lg u)$ bits of space w.h.p., plus a constant number of machine words. The function can be evaluated in worst-case constant time, and updates take constant time w.h.p. This is the first dynamic perfect hash function that uses less space than needed to store $S$ ($\lg \binom{u}{n}$ bits). Our space usage is close to optimal, since the problem is harder than dynamic Bloomier filtering. These operating conditions are typical of data-stream computation, where one needs to support a stream of updates and queries, but does not have space to hold the entire state of the data structure. Quite remarkably, our solution can achieve this goal without introducing errors (we use only Las Vegas randomization). We mention an independent application of Theorem \[thm:hash\]. In a database we can maintain an index of a relation under insertions of tuples, using internal memory per tuple which is logarithmic in the length of the key for the tuple. If tuples have fixed length, they can be placed directly in the hash table, and need only be moved if the capacity of the hash table is exceeded. Tradeoffs and the scheme of things {#scheme} ---------------------------------- We begin with a discussion of the greater-than problem. Consider an infinite memory of bits, initialized to zero. Our problem has two stages. In the update stage, the algorithm is given a number $a \in [0..n-1]$. After seeing $a$, the algorithm is allowed to flip $O(T_u)$ bits in the memory. In the query stage, the algorithm is given a number $b \in [0..n-1]$. Now the algorithm may inspect $O(T_q)$ bits, and must decide whether or not $b > a$. The problem was previously studied by Fredman [@fredman82sums], who showed that $\max(T_u, T_q) = \Omega(\lg n / \lg\lg n)$. It is quite tempting to believe that one cannot improve past the trivial upper bound $T_u = T_q = O(\lg n)$, since, in some sense, this is the complexity of “writing down” $a$. However, as we show in this paper, Fredman’s bound is optimal, in the sense that it is a point on our tradeoff curve. We give upper and lower bounds that completely characterize the possible asymptotic tradeoffs: \[thm:bitgt\] The bit-probe complexity of the greater-than function satisfies the tight tradeoffs: $$\begin{aligned} T_q \geq \lg\lg n,\ T_u \leq \lg n &:& T_u = \Theta(\lg_{T_q} n) \\ T_q \leq \lg\lg n,\ T_u \geq \lg n &:& 2^{T_q} = \Theta(\lg_{T_u} n) \\ \end{aligned}$$ As mentioned already, we use the same recursion idea as in the previous algorithm for dynamic range reporting, except that we apply this recursion to every root-to-leaf path of a binary trie of depth $w$. Quite remarkably, these structures can be made to overlap in-as-much as the paths overlap, so only one update suffices for all paths going through a node. Due to this close relation, we view the lower bounds for the greater-than function as giving an indication that our range reporting data structure is likewise optimal. In any case, the lower bounds show that markedly different ideas would be necessary to improve our solution for range reporting. Let $T_{pred}$ be the time needed by one update and one query in the dynamic predecessor problem. The following theorem summarizes our results for dynamic range reporting: \[thm:range\] There is a data structure for the dynamic range reporting problem, which uses $O(n)$ space and supports updates in time $O(T_u)$, and queries in time $O(T_q)$, $(\forall) T_u, T_q$ satisfying: $$\begin{aligned} T_q \geq \lg\lg w,\ \frac{\lg w}{\lg\lg w} \leq T_u \leq \lg w &:& T_u = O(\lg_{T_q} w) + T_{pred} \\ T_q \leq \lg\lg w,\phantom{\ \ \frac{\lg w}{\lg\lg w} \leq} T_u \geq \lg w &:& 2^{T_q} = O(\lg_{T_u} w) \\ \end{aligned}$$ Notice that the most appealing point of the tradeoff is the cross-over of the two curves: $T_u = O(\lg w)$ and $T_q = O(\lg\lg w)$ (and indeed, this has been the focus of our discussion). Another interesting point is at constant query time. In this case, our data structure needs $O(w^{\epsilon})$ update time. Thus, our data structure can be used as an optimal static data structure, which is constructed in time $O(n w^{\epsilon})$, improving on the construction time of $O(n \sqrt{w})$ given by Alstrup et al [@alstrup01range]. The first branch of our tradeoff is not interesting with $T_{pred} = \Theta(\lg w)$. However, it is generally believed that one can achieve $T_{pred} = \Theta( \lg w / \lg\lg w)$, matching the optimal bound for the static case. If this is true, the $T_{pred}$ term can be ignored. In this case, we can remark a very interesting relation between our problem and the predecessor problem. When $T_u = T_q$, the bounds we achieve are identical to the ones for the predecessor problem, i.e. $T_u = T_q = O(\lg w / \lg\lg w)$. However, if we are interested in the possible tradeoffs, the gap between range reporting and the predecessor problem quickly becomes huge. The same situation appears to be true for deterministic dictionaries with linear space, though the known tradeoffs are not as general as ours. We set forth the bold conjecture (the proof of which requires many missing pieces) that all three search problems are united by an optimal time of $\Theta(\lg w / \lg\lg w)$ in this point of their tradeoff curves. We can achieve bounds in terms of $n$, rather than $w$, by the classic trick of using our structure for small $w$ and a fusion tree structure [@fredman93fusion] for large $w$. In particular, we can achieve $T_q = O(\lg\lg n)$ and $T_u = O\left( \frac{\lg n}{\lg\lg n} \right)$. Compared with the optimal bound for the predecessor problem of $\Theta\left( \sqrt{\frac{\lg n}{\lg\lg n}} \right)$, our data structure improves the query time exponentially by sacrificing the update time quadratically. Data-Stream Perfect Hashing =========================== We denote by $S$ be the set of values that we need to hash at present time. Our data structure has the following parts: - A hash function $\rho: \{0,\dots,u-1 \} \rightarrow \{0,1\}^{v}$, where $v = O(\lg n)$, from a family of universal hash functions with small representations (for example, the one from [@dietzfel96universal]). - A hash function $\phi: \{0,1\}^{v} \rightarrow \{1,\dots,r\}$, where $r=\lceil n/\lg^2 n \rceil$, taken from Siegel’s class of highly independent hash functions [@siegel04hash]. - An array of hash functions $h_1,\dots,h_r: \{0,1\}^v \rightarrow \{0,1\}^s$, where $s=\lceil (6+2c)\lg\lg u \rceil$, chosen independently from a family of universal hash functions; $c$ is a constant specified below. - A high performance dictionary [@dietzfel90highperf] for a subset $S'$ of the keys in $S$. The dictionary should have a capacity of $O(\lceil n/\lg u \rceil)$ keys (but might expand further). Along with the dictionary we store a linked list of length $O(\lceil n/\lg u \rceil)$, specifying certain vacant positions in the hash table. - An array of dictionaries $D_1,\dots,D_r$, where $D_i$ is a dictionary that holds $h_i(\rho(k))$ for each key $k\in S \setminus S'$ with $\phi(\rho(k))=i$. A unique value in $\{0,\dots,j-1\}$, where $j=(1+o(1))\lg^2 n$, is associated with each key in $D_i$. A bit vector of $j$ bits and an additional string of $\lg n$ bits is used to keep track of which associated values are in use. We will return to the exact choice of $j$ and the implementation of the dictionaries. The main idea is that all dictionaries in the construction assign to each of their keys a unique value within a subinterval of $[1 .. m]$. Each of the dictionaries $D_1, \dots, D_r$ is responsible for an interval of size $j$, and the high performance dictionary is responsible for an interval of size $O(n/\lg u) = o(n)$. The hash function $\rho$ is used to reduce the key length to $v$. The constant in $v = O(\lg n)$ can be chosen such that with high probability, over a polynomially bounded sequence of updates, $\rho$ will never map two elements of $S$ to the same value (the conflicts, if they occur, end up in $S'$ and are handled by the high performance dictionary). When inserting a new value $k$, the new key is included in $S'$ if either: - There are $j$ keys in $D_i$, where $i=\phi(\rho(k))$, or - There exists a key $k'\in S$ where $\phi(\rho(k))=\phi(\rho(k'))=i$ and $h_i(\rho(k))=h_i(\rho(k'))$. Otherwise $k$ is associated with the key $h_i(\rho(k))$ in $D_i$. Deletion of a key $k$ is done in $S'$ if $k\in S'$, and otherwise the associated key in the appropriate $D_i$ is deleted. To evaluate the perfect hash function on a key $k$ we first see whether $k$ is in the high performance dictionary. If so, we return the value associated with $k$. Otherwise we compute $i=\phi(\rho(k))$ and look up the value $\Delta$ associated with the key $h_i(\rho(k))$ in $D_i$. Then we return $(i-1)j+\Delta$, i.e., position $\Delta$ within the $i$-th interval. Since $D_1,\dots,D_r$ store keys and associated values of $O(\lg\lg u)$ bits, they can be efficiently implemented as constant depth search trees of degree $w^{\Omega(1)}$, where each internal node resides in a single machine word. This yields constant time for dictionary insertions and lookups, with an optimal space usage of $O(\lg^2 n\lg\lg u)$ bits for each dictionary. We do not go into details of the implementation as they are standard; refer to [@hagerup98ram] for explanation of the required word-level parallelism techniques. What remains to describe is how the dictionaries keep track of vacant positions in the hash table in constant time per insertion and deletion. The high performance dictionary simply keeps a linked list of all vacant positions in its interval. Each of $D_1,\dots,D_r$ maintain a bit vector indicating vacant positions, and additional $O(\lg n)$ summary bits, each taking the or of an interval of size $O(\lg n)$. This can be maintained in constant time per operation, employing standard techniques. Only $o(n)$ preprocessing is necessary for the data structure (essentially to build tables needed for the word-level parallelism). The major part of the data structure is initialized lazily. Analysis -------- Since evaluation of all involved hash functions and lookup in the dictionaries takes constant time, evaluation of the perfect hash function is done in constant time. As we will see below, the high performance dictionary is empty with high probability unless $n/\lg u > \sqrt{n}$. This means that it always uses constant time per update with high probability in $n$. All other operations done for update are easily seen to require constant time w.h.p. We now consider the space usage of our scheme. The function $\rho$ can be represented in $O(w)$ bits. Siegel’s highly independent hash function uses $o(n)$ bits of space. The hash functions $h_1,\dots,h_r$ use $O(\lg n + \lg\lg u)$ bits each, and $o(n\lg\lg u)$ bits in total. The main space bottleneck is the space for $D_1,\dots,D_r$, which sums to $O(n\lg\lg u)$. Finally, we show that the space used by the high performance dictionary is $O(n)$ bits w.h.p. This is done by showing that each of the following hold with high probability throughout a polynomial sequence of operations: The function $\rho$ is one-to-one on $S$. There is no $i$ such that $S_i = \{ k \in S \mid \phi(\rho(k))=i \}$ has more than $j$ elements. The set $S'$ has $O(\lceil n/\lg u \rceil)$ elements. That 1. holds with high probability is well known. To show 2. we use the fact that, with high probability, Siegel’s hash function is independent on every set of $n^{\Omega(1)}$ keys. We may thus employ Chernoff bounds for random variables with limited independence to bound the probability that any $i$ has $|S_i| > j$, conditioned on the fact that 1. holds. Specifically, we can use [@schmidt95chernoff Theorem 5.I.b] to argue that for any $l$, the probability that $|S_{i}| > j$ for $j = \lceil \lg^2 n + \lg^{5/3} n \rceil$ is $n^{-\omega(1)}$, which is negligible. On the assumption that 1. and 2. hold, we finally consider 3. We note that every key $k'\in S'$ is involved in an $h_i$-collision in $S_i$ for $i=\phi(\rho(k'))$, i.e. there exists $k''\in S_i \setminus \{k'\}$ where $h_i(k')=h_i(k'')$. By universality, for any $i$ the expected number of $h_i$-collisions in $S_i$ is $O(\lg^4 n / (\lg u)^{6+2c}) = O((\lg u)^{-(2+2c)})$. Thus the probability of one or more collisions is $O((\lg u)^{-(2+2c)})$. For $\lg u \geq \sqrt{n}$ this means that there are no keys in $S'$ with high probability. Specifically, $c$ may be chosen as the sum of the constants in the exponents of the length of the operation sequence and the desired high probability bound. For the case $\lg u < \sqrt{n}$ we note that the expected number of elements in $S'$ is certainly $O(n/\lg u)$. To see than this also holds with high probability, note that the event that one or more keys from $S_i$ end up in $S'$ is independent among the $i$’s. Thus we can use Chernoff bounds to get that the deviation from the expectation is small with high probability. Lower Bound for Bloomier Filters ================================ For the purpose of the lower bound, we consider the following two-set distinction problem, following [@chazelle04bloom]. The problem has the following stages: 1. a random string $R$ is drawn, which will be available to the data structure throughout its operation. This is equivalent to drawing a deterministic algorithm from a given distribution, and is more general than assuming each stage has its own random coins (we are giving the data structure free storage for its random bits). 2. the data structure is given $A \subset [u], |A| \le n$. It must produce a representation $f_R(A)$, which for any $A$ has size at most $S$ bits, in expectation over all choices of $R$. Here $S$ is a function of $n$ and $u$, which is the target of our lower bound. 3. the data structure is given $B \subset [u]$, such that $|B| \le n, A \cap B = \emptyset$. Based on the old state $f_R(A)$, it must produce $g_R(B, f_R(A))$ with expected size at most $S$ bits. 4. the data structure is given $x \in [u]$ and its previously generated state, i.e. $f_R(A)$ and $g_R(B, f_R(A))$. Now it must answer as follows with no error allowed: if $x \in A$, it must answer “A”; if $x \in B$, it must answer “B”; if $x \notin A \cup B$, it can answer either “A” or “B”. Let $h_R(x,f,g)$ be the answer computed by the data structure, when the previous state is $f$ and $g$. It is easy to see that a solution for dynamic Bloomier filters supporting ternary associated data, using expected space $o(n\lg\lg \frac{u}{n})$, yields a solution to the two-set distinction problem with $S = o(n\lg\lg \frac{u}{n})$. We will prove such a solution does not exist. Since a solution to the distinction problem is not allowed to make an error we can assume w.l.o.g. that step 3 is implemented as follows. If there exist appropriate $A, B \subset [u]$, with $x \in A$ such that $f_R(A) = f_0$ and $g_R(B, f_0) = g_0$, then $h_R(x, f_0, g_0)$ must be “A”. Similarly, if there exists a plausible scenario with $x \in B$, the answer must be “B”. Otherwise, the answer can be arbitrary. Assume that the inputs $A \times B$ are drawn from a given distribution. We argue that if the expected sizes of $f$ and $g$ are allowed to be at most $2S$, the data structure need not be randomized. This uses a bicriteria minimax principle. We have $E_{R,A,B}\left[ \frac{|f|}{S} + \frac{|g|}{S} \right] \leq 2$, where $|f|, |g|$ denote the length of the representations. Then, there exists a random string $R_0$ such that $E_{A,B} \left[ \frac{|f|}{S} + \frac{|g|}{S} \right] \leq 2$. Since $|f|, |g| \geq 0$, this implies $E_{A,B}[|f|] \leq 2S, E_{A,B}[|g|] \leq S$. The data structure can simply use the deterministic sequence $R_0$ as its random bits; we drop the subscript from $f_R, g_R$ when talking about this deterministic data structure. Lower Bound for Two-Set Distinction ----------------------------------- Assume $u = \omega(n)$, since a lower bound of $\Omega(n)$ is trivial for universe $u \ge 2n$. Break the universe into $n$ equal parts $U_1, \dots, U_n$; w.l.o.g. assume $n$ divides $u$, so $|U_i| = \frac{n}{u}$. The hard input distribution chooses $A$ uniformly at random from $U_1 \times \dots \times U_n$. We write $A = \{ a_1, \dots, a_n \}$, where $a_i$ is a random variable drawn from $U_i$. Then, $B'$ is chosen uniformly at random from the same product space; again $B' = \{b_1, \dots, b_n\}, b_i \gets U_i$. We let $B = B' \setminus A$. Note that $E[|B|] = n \cdot \Pr[A_1 \ne B_1] = (1 - \frac{n}{u}) \cdot n = (1 - o(1)) \cdot n$. Let $A_i^p$ be the plausible values of $A_i$ after we see $f(A)$; that is, $A_i^p$ contains all $a \in U_i$ for which there exists a valid $A'$ with $a \in A'$ and $f(A') = f(A)$. Intuitively speaking, if $f(A)$ has expected size $o(n \lg\lg \frac{u}{n})$, it contains on average $o(\lg\lg \frac{u}{n})$ bits of information about each $a_i$. This is much smaller than the range of $a_i$, which is $\frac{u}{n}$, so we would expect that the average $|A_i^p|$ is quite large, around $\frac{u}{n} / (\lg \frac{u}{n})^{o(1)}$. This intuition is formalized in the following lemma: With probability at least a half over a uniform choice of $A$ and $i$, we have $|A_i^p| \geq \frac{u/n}{2^{O(S/n)}}$. The Kolmogorov complexity of $A$ is $n\lg \frac{u}{n} - O(1)$; no encoding for $A$ can have an expected size less than this quantity. We propose an encoding for $A$ consisting of two parts: first, we include $f(A)$; second, for each $i$ we include the index of $a_i$ in $|A_i^p|$, using $\lceil \lg|A_i^p| \rceil$ bits. This is easily decodable. We first generate all possible $A'$ with $f(A') = f(A)$, and thus obtain the sets $A_i^p$. Then, we extract from each plausible set the element with the given index. The expected size of the encoding is $2S + \sum_i E_{A}[\lg |A_i^p|] + O(n)$, which must be $\ge n\lg \frac{u}{n} - O(1)$. This implies $\lg \frac{u}{n} - E_{i,A}[\lg |A_i^p|] \le \frac{2S}{n} + O(1)$. By Markov’s inequality, with probability at least a half over $i$ and $A$, $\lg \frac{u}{n} - \lg |A_i^p| \le \frac{4S}{n} + O(1)$, so $\lg |A_i^p| \ge \lg \frac{u}{n} - O(\frac{S}{n})$. We now make a crucial observation which justifies our interest in $A_i^p$. Assume that $b_i \in A_i^p$. In this case, the data structure must be able to determine $b_i$ from $f(A)$ and $g(B,f(A))$. Indeed, suppose we compute $h(x,f,g)$ for all $x \in |A_i^p|$. If that data strucuture does not answer “B” when $x = b_i$, it is obviously incorrent. On the other hand, if it answers “B” for both $x = b_i$ and some other $x' \in A_i^p$, it also makes an error. Since $x'$ is plausible, there exist $A'$ with $x' \in A'$ such that $f(A') = f(A)$. Then, we can run the data structure with $A'$ as the first set and $B$ as the second set. Since $f(A') = f(A)$, the data structure will behave exactly the same, and will incorrectly answer “B” for $x'$. To draw our conclusion, we consider another encoding argument, this time in connection to the set $B'$. The Kolmogorov complexity of $B'$ is $n \lg \frac{u}{n} - O(1)$. Consider a randomized encoding, depending on a set $A$ drawn at random. First, we encode an $n$-bit vector specifying which indices $i$ have $a_i = b_i$. It remains to encode $B' \setminus A = B$. We encode another $n$-bit vector, specifying for which positions $i$ we have $b_i \in A_i^p$. For each $b_i \notin A_i^p$, we simply encode $B_i$ using $\lceil \lg \frac{u}{n} \rceil$ bits. Finally, we include in the encoding $g(B, f(A))$. As explained already, this is enough to recover all $b_i$ which are in $A_i^p$. Note that we do not need to encode $f(A)$, since this depends only on our random coins, and the decoding algorithm can reconstruct it. The expected size of this encoding will be $O(n + S) + n\cdot \Pr_{A,B',i} [b_i \notin A_i^p] \cdot \lg \frac{u}{n}$. We know that with probability a half over $A$ and $i$, we have $|A_i^p| \geq \frac{u/n}{2^{O(S/n)}}$. Thus, $\Pr_{A,B',i} [b_i \in A_i^p] \geq \frac{1}{2} \cdot 2^{-O(S/n)}$. Thus, the expected size of the encoding is at most $O(n + S) + (1 - 2^{-O(S/n)}) \cdot n \lg \frac{u}{n}$. Note that by the minimax principle, randomness in the encoding is unessential and we can always fix $A$ guaranteeing the same encoding size, in expectation over $B$. We now get the bound: $$\begin{aligned} & & O(n + S) + (1 - 2^{-O(S/n)}) \cdot n \lg \frac{u}{n} \geq n \lg \frac{u}{n} - O(1) \\ & \Rightarrow & O\left( \frac{S}{n} \right) \geq 2^{-O(S/n)} \lg \frac{u}{n} - O(1) \Rightarrow 2^{O(S/n)} O(S / n) \geq \lg \frac{u}{n} \Rightarrow \frac{S}{n} = \Omega \left( \lg\lg \frac{u}{n} \right)\end{aligned}$$ A Space-Optimal Bloomier Filter =============================== It was shown in [@carter78bloom] that the approximate membership problem (i.e., the problem solved by Bloom filters) can be solved optimally using a reduction to the exact membership problem. The reduction uses universal hashing. In this section we extend this idea to achieve optimal dynamic Bloomier filters. Recall that Bloomier filters encode sparse vectors with entries from $\{0,\dots,2^r - 1\}$. Let $S\subseteq [u]$ be the set of at most $n$ indexes of nonzero entries in the vector $V$. The data structure must encode a vector $V'$ that agrees with $V$ on indexes in $S$, and such that for any $x\not\in S$, $\Pr[V'[x]\neq 0]\leq \epsilon$, where $\epsilon > 0$ is the error probability of the Bloomier filter. Updates to $V$ are done using the following operations: - [Insert($x$, $a$)]{}. Set $V[x]:=a$, where $a\neq 0$. - [Delete($x$)]{}. Set $V[x]:=0$. The data structure assumes that only valid updates are performed, i.e. that inserts are done only in situations where $V[x]=0$ and deletions are done only when $V[x]\neq 0$. \[thm:filter\] Let positive integers $n$ and $r$, and $\epsilon > 0$ be given. On a RAM with word length $w$ we can maintain a Bloomier filter $V'$ for a vector $V$ of length $u=2^{O(w)}$ with at most $n$ nonzero entries from $\{0,\dots,2^r - 1\}$, such that: - [Insert]{} and [Delete]{} can be done in amortized expected constant time. The data structure assumes all updates are valid. - Computing $V'[x]$ on input $x$ takes worst case constant time. If $V[x]\neq 0$ the answer is always ’V\[x\]’. If $V[x]=0$ the answer is ’0’ with probability at least $1-\epsilon$. - The expected space usage is $O(n(\lg\lg(u/n) + \lg(1/\epsilon) + r))$ bits. The Data Structure ------------------ Assume without loss of generality that $u\geq 2n$ and that $\epsilon\geq u/n$. Let $v=\max(n \log(u/n), n/\epsilon)$, and choose $h: \{0,\dots,u-1\} \rightarrow \{0,\dots,v-1\}$ as a random function from a universal class of hash functions. The data structure maintains information about a minimal set $S'$ such that $h$ is 1-1 on $S \setminus S'$. Specifically, it consists of two parts: 1. A dictionary for the set $S'$, with corresponding values of $V$ as associated information. 2. A dictionary for the set $h(S\backslash S')$, where the element $h(x)$, $x\in S\backslash S'$, has $V[x]$ as associated information. Both dictionaries should succinct, i.e., use space close to the information theoretic lower bound. Raman and Rao [@raman03succinct] have described such a dictionary using space that is $1+o(1)$ times the minimum possible, while supporting lookups in $O(1)$ time and updates in expected amortized $O(1)$ time. To compute $V'[x]$ we first check whether $x\in S'$, in which case $V'[x]$ is stored in the first dictionary. If this is not the case, we check whether $h(x)\in h(S\backslash S')$. If this is the case we return the information associated with $h(x)$ in the second dictionary. Otherwise, we return ’0’. [Insert($x$, $a$)]{}. First determine whether $h(x)\in h(S\backslash S')$, in which case we add $x$ to the set $S'$, inserting $x$ in the first dictionary. Otherwise we add $h(x)$ to the second dictionary. In both cases, we associate $a$ with the inserted element. [Delete($x$)]{} proceeds by deleting $x$ from the first dictionary if $x\in S'$, and otherwise deleting $h(x)$ from the second dictionary. Analysis -------- It is easy to see that the data structure always return correct function values on elements in $S$, given that all updates are valid. When computing $V'[x]$ for $x\not\in S$ we get a nonzero result if and only if there exists $x'\in S$ such that $h(x)=h(x')$. Since $h$ was chosen from a universal family, this happens with probability at most $n/v \leq \epsilon$. It remains to analyze the space usage. Using once again that $h$ was chosen from a universal family, the expected size of $S'$ is $O(n/\log(u/n))$. This implies that the expected number of bits necessary to store the set $S'$ is $\log\binom{u}{O(n/\log(u/n))} = O(n)$, using convexity of the function $x\mapsto \binom{u}{x}$ in the interval $0\dots u/2$. In particular, the first dictionary achieves an expected space usage of $O(n)$ bits. The information theoretical minimum space for the set $h(S\backslash S')$ is bounded by $\log\binom{r}{n} = O(n \log(r/n)) = O(n \log\log(u/n) + n\log(1/\epsilon))$ bits, matching the lower bound. We disregarded is the space for the universal hash function, which is $O(\log u)$ bits. However, this can be reduced to $O(\log n + \log\log u)$ bits, which is vanishing, by using slightly weaker universal functions and doubling the size $r$ of the range. Specifically, $2$-universal functions suffice; see [@pagh00dispers] for a construction. Using such a family requires preprocessing time $(\log u)^{O(1)}$, expected. Upper Bounds for the Greater-Than Problem ========================================= We start with a simple upper bound of $T_u = O(\lg n), T_q = O(\lg\lg n)$. Our upper bound uses a trie structure. We consider a balanced tree with branching factor 2, and with $n$ leaves. Every possible value of the update parameter $a$ is represented by a root-to-leaf path. In the update stage, we mark this root-to-leaf path, taking time $O(\lg n)$. In the query stage, we want to find the point where $b$’s path in the trie would diverge from $a$’s path. This uses binary search on the $\lg n$ levels, as follows. To test if the paths diverge on a level, we examine the node on that level on $b$’s path. If the node is marked, the paths diverge below; otherwise they diverge above. Once we have found the divergence point, we know that the larger of $a$ and $b$ is the one following the right child of the lowest common ancestor. For the full tradeoff, we consider a balanced tree with branching factor $B$. In the update stage, we need to mark a root-to-leaf path, taking time $\lg_B n$. In the query stage, we first find the point where $b$’s path in the trie would diverge from $a$’s path. This uses binary search on the $\lg_B n$ levels, so it takes time $O(\lg\lg_B n)$. Now we know the level where the paths of $a$ and $b$ diverge. The nodes on that level from the paths of $a$ and $b$ must be siblings in the tree. To test whether $b > a$, we must find the relative order of the two sibling nodes. There are two strategies for this, giving the two branches of the tradeoff curve. To achieve small update time, we can do all work at query time. We simply test all siblings to the left of $b$’s path on the level of divergence. If we find a marked one, then $a$’s path goes to the left of $b$’s path, so $a < b$; otherwise $a > b$. This stragegy gives $T_u = O(\lg_B n)$ and $T_q = O(\lg(\lg_B n) + B)$, for any $B \geq 2$. For $T_q > \Omega(\lg\lg n)$, we have $T_q = \Theta(B)$, so we have achieved the tradeoff $T_u = O(\lg_{T_q} n)$. The second strategy is to do all work at update time. For every node on $a$’s path, we mark all left siblings of the node as such. Then to determine if $b$’s path is to the left or to the right of $a$’s path, we can simply query the node on $b$’s path just below the divergence point, and see if it is marked as a left sibling. This strategy gives $T_u = O(B \lg_B n)$ and $T_q = O(\lg(\lg_B n))$. For small enough $B$ (say $B = O(\lg n)$), this strategy gives $T_q = O(\lg\lg n)$ regardless of $B$ and $T_u$. For $B = \Omega(\lg n)$, we have $\lg B = \Theta(\lg T_u)$. Therefore, we can express our tradeoff as: $2^{T_q} = O(\lg_{T_u} n)$. Dynamic Range Reporting ======================= We begin with the case $T_u = O(\lg w), T_q = O(\lg\lg w)$. Let $S$ be the current set of values stored by the data structure. Without loss of generality, assume $w$ is a power of two. For an arbitrary $t \in [0, \lg w]$, we define the trie of order $t$, denoted $T_t$, to be the trie of depth $w / 2^t$ and alphabet of $2^t$ *bits*, which represents all numbers in $S$. We call $T_0$ the *primary trie* (this is the classic binary trie with elements from $S$). Observe that we can assign distinct names of $O(w)$ bits to all nodes in all tries. We call *active paths* the paths in the tries which correspond to elements of $S$. A node $v$ from $T_t$ corresponds to a subtree of depth $2^t$ in the primary trie; we denote the root of this subtree by $r_0(v)$. A node from $T_t$ corresponds to a 2-level subtree in $T_{t-1}$; we call such a subtree a *natural subtree*. Alternatively, a 2-level subtree of any trie is natural iff its root is at an even depth. A root-to-leaf path in the primary trie is seen as the leaves of the tree used for the greater-than problem. The paths from the primary trie are broken into chunks of length $2^t$ in the trie of order $t$. So $T_t$ is similar to the $t$-th level (counted bottom-up) of the greater-than tree. Indeed, every node on the $t$-th level of that tree held information about a subtree with $2^t$ leaves; here one edge in $T_t$ summarizes a segment of length $2^t$ bits. Also, a natural subtree corresponds to two siblings in the greater-than structure. On the next level, the two siblings are contracted into a node; in the trie of higher order, a natural subtree is also contracted into a node. It will be very useful for the reader to hold these parallels in mind, and realize that the data structure from this section is implementing the old recursion idea *on every path*. The root-to-leaf paths corresponding to the values in $S$ determine at most $n-1$ branching nodes in any trie. By convention, we always consider roots to be branching nodes. For every branching node from $T_0$, we consider the extreme points of the interval spanned by the node’s subtree. By doubling the universe size, we can assume these are never elements of $S$ (alternatively, such extreme points are formal rationals like $x + \frac{1}{2}$). We define $\overline{S}$ to be the union of $S$ and the two special values for each branching node in the primary trie; observe that $|\overline{S}| = O(n)$. We are interested in holding $\overline{S}$ for navigation purposes: it gives a way to find in constant time the maximum and minimum element from $S$ that fits under a branching node (because these two values should be the elements from $S$ closest to the special values for the branching node). Our data structure has the following components: - a linked list with all elements of $S$ in increasing order, and a predecessor structure for $S$. - a linked list with all elements of $\overline{S}$ in increasing order, accompanied by a navigation structure which enables us to find in constant time the largest value from $S$ smaller than a given value from $\overline{S} \setminus S$. We also hold a predecessor structure for $\overline{S}$. - every branching node from the primary trie holds pointers to its lowest branching ancestor, and the two branching descendants (the highest branching nodes from the left and right subtrees; we consider leaves associated with elements from $S$ as branching descendants). We also hold pointers to the two extreme values associated with the node in the list in item 2. Finally, we hold a hash table with these branching nodes. - for each $t$, and every node $v$ in $T_t$, which is either a branching node or a child of a branching node on an active path, we hold the depth of the lowest branching ancestor of $r_0(v)$, using a Bloomier filter. We begin by showing that this data structure takes linear space. Items 1-3 handle $O(n)$ elements, and have constant overhead per element. We show below that the navigation structure from 2. can be implemented in linear space. The predecessor structure should also use linear space; for van Emde Boas, this can be achieved through hashing [@willard83predecessor]. In item 4., there are $O(n)$ branching nodes per trie. In addition, there are $O(n)$ children of branching nodes which are on active paths. Thus, we consider $O(n\lg w)$ nodes in total, and hold $O(\lg w)$ bits of information for each (a depth). Using our solution for the Bloomier filter, this takes $O(n(\lg w)^2 + w)$ bits, which is $o(n)$ words. Note that storing the depth of the branching ancestor is just a trick to reduce space. Once we have a node in $T_0$ and we know the depth of its branching ancestor, we can calculate the ancestor in $O(1)$ time (just ignore the bits below the depth of the ancestor). So in essence these are “compressed pointers” to the ancestors. We now sketch the navigation structure from item 2. Observe that the longest run in the list of elements from $\overline{S} \setminus S$ can have length at most $2w$. Indeed, the leftmost and rightmost extreme values for the branching nodes form a parenthesis structure; the maximum depth is $w$, corresponding to the maximum depth in the trie. Between an open and a closed parenthesis, there must be at least one element from $S$, so the longest uninterrupted sequence of parenthesis can be $w$ closed parenthesis and $w$ open parenthesis. The implementation of the navigation structure uses classic ideas. We bucket $\Theta(\sqrt{w})$ consecutive elements from the list, and then we bucket $\Theta(\sqrt{w})$ buckets. Each bucket holds a summary word, with a bit for each element indicating whether it is in $S$ or not; second-order buckets hold bits saying whether first order buckets contain at least one element from $S$ or not. There is also an array with pointers to the elements or first order buckets. By shifting, we can always insert another summary bit in constant time when something is added. However, we cannot insert something in the array in constant time; to fix that, we insert elements in the array on the next available position, and hold the correct permutation packed in a word (using $O(\sqrt{w} \lg w)$ bits). To find an element from $S$, we need to walk $O(1)$ buckets. The time is $O(1)$ per traversed bucket, since we can use the classic constant-time subroutine for finding the most significant bit [@fredman93fusion]. We also describe a useful subroutine, ${\texttt{test-branching}}(v)$, which tests whether a node $v$ from some $T_t$ is a branching node. To do that, we query the structure in item 4. to find the lowest branching ancestor of $r_0(v)$. This value is defined if $v$ is a branching node, but the Bloomier filter may return an arbitrary result otherwise. We look up the purported ancestor in the structure of item 3. If the node is not a branching node, the value in the Bloom filter for $v$ was bogus, so $v$ is not a branching node. Otherwise, we inspect the two branching descendants of this node. If $v$ is a branching node, one of these two descendants must be mapped to $v$ in the trie of order $t$, which can be tested easily. Implementation of Updates ------------------------- We only discuss insertions; deletions follow parallel steps uneventfully. We first insert the new element in $S$ and $\overline{S}$ using the predecessor structures. Inserting a new element creates exactly one branching node $v$ in the primary trie. This node can be determined by examining the predecessor and successor in $S$. Indeed, the lowest common ancestor in the primary trie can be determined by taking an xor of the two values, finding the most significant bit, and them masking everything below that bit from the original values [@alstrup01range]. We calculate the extreme values for the new branching node $v$, and insert them in $\overline{S}$ using the predecessor structure. Finding the branching ancestor of $v$ is equivalent to finding the enclosing parentheses for the pair of parentheses which was just inserted. But $\overline{S}$ has a special structure: a pair of parentheses always encloses two subexpressions, which are either values from $S$, or a parenthesized expression (i.e., the branching nodes from $T_0$ form a binary tree structure). So one of the enclosing parentheses is either immediately to the left, or immediately to the right of the new pair. We can traverse a link from there to find the branching ancestor. Once we have this ancestor, it is easy to update the local structure of the branching nodes from item 3. Until now, the time is dominated by the predecessor structure. It remains to update the structure in item 4. For each $t > 0$, we can either create a new branching node in $T_t$, or the branching node existed already (this is possible for $t > 0$ because nodes have many children). We first test whether the branching node existed or not (using the ${\texttt{test-branching}}$ subroutine). If we need to introduce a branching node, we simply add a new new entry in the Bloomier filter with the depth of the branching ancestor of $v$. It remains to consider active children of branching nodes, for which we must store the depth of $v$. If we have just introduced a branching node, it has exactly two active children (if there exist more than two children on active paths, the node was a branching node before). These children are determined by looking at the branching descendants of $v$; these give the two active paths going into $v$. Both descendants are mapped to active children of the new branching node from $T_t$. If the branching node already existed, we must add one active child, which is simply the child that the path to the newly inserted value follows. Thus, to update item 4., we spend constant time per $T_t$. In total, the running time of an update is $T_{pred} + O(\lg w) = O(\lg w)$. Implementation of Queries ------------------------- Remember that a query receives an interval $[a,b]$ and must return a value in $S \cap [a,b]$, if one exists. We begin by finding the node $v$ which is the lowest common ancestor of $a$ and $b$ in the primary trie; this takes constant time [@alstrup01range]. Note that $v$ spans an interval which includes $[a,b]$. The easiest case is when $v$ is a branching node; this can be recognized by a lookup in the hash table from item 3. If so, we find the two branching descendants of $v$; call the left one $v_L$ and the right one $v_R$. Then, if $S \cap [a,b] \ne \emptyset$, either the rightmost value from $S$ that fits under $v_L$ or the leftmost value from $S$ that in fits under $v_R$ must be in the interval $[a,b]$. This is so because $[a,b]$ straddles the middle point of the interval spanned by $v$. The two values mentioned above are the two values from $S$ closest (on both sides) to this middle point, so if $[a,b]$ is non-empty, it must contain one of these two. To find these two values, we follow a pointer from $v_L$ to its left extreme point in $\overline{S}$. Then, we use the navigation structure from item 2., and find the predecessor from $S$ of this value in constant time. The rightmost value under $v_R$ is the next element from $S$. Altogether, the case when $v$ is a branching node takes constant time. Now we must handle the case when $v$ is not a branching node. If $S \cap [a,b] \ne \emptyset$, it must be the case that $v$ is on an active path. Below we describe how to find the lowest branching ancestor of $v$, *assuming that $v$ is on an active path*. If this assumption is violated, the value returned can be arbitrary. Once we have the branching ancestor of $v$, we find the branching descendant $w$ which is in $v$’s subtree. Now it is easy to see, by the same reasoning as above, that if $[a,b] \cap S \ne \emptyset$ either the leftmost or the rightmost value from $S$ which is under $w$ must be in $[a,b]$. These two values are found in constant time using the navigation structure from item 2., as described above. So if $[a,b] \cap S \ne \emptyset$, we can find an element inside $[a,b]$. If none of these two elements were in $[a,b]$ it must be the case that $[a,b]$ was empty, because the algorithm works correctly when $[a,b] \cap S \ne \emptyset$. It remains to show how to find $v$’s branching ancestor, assuming $v$ is on an active path, but is not a branching node. If for some $t > 0$, $v$ is mapped to a branching node in $T_t$, it will also be mapped to a branching node in tries of higher order. We are interested in the smallest $t$ for which this happens. We find this $t$ by binary search, taking time $O(\lg\lg w)$. For some proposed $t$, we check whether the node to which $v$ is mapped in $T_t$ is a branching node (using the ${\texttt{test-branching}}$ subroutine). If it is, we continue searching below; otherwise, we continue above. Suppose we found the smallest $t$ for which $v$ is mapped to a branching node. In $T_{t-1}$, $v$ is mapped to some $z$ which is *not* a branching node. Finding the lowest branching ancestor of $v$ is identical to finding the lowest branching ancestor of $r_0(z)$ in the primary trie (since $z$ is a not a branching node, there is no branching node in the primary trie in the subtree corresponding to $z$). Since in $T_t$ $z$ gets mapped to a branching node, its natural subtree in $T_{t-1}$ must contain at least one branching node. We have two cases: either $z$ is the root or a leaf of the natural subtree (remember that a natural subtree has two levels). These can be distinguished based on the parity of $z$’s depth. If $z$ is a leaf, the root must be a branching node (because there is at least another active leaf). But then $z$ is an active child of a branching node, so item 4. tells us the branching ancestor of $r_0(z)$. Now consider the case when $z$ is the root of the natural subtree. Then $z$ is above any branching node in its natural subtree, so to find the branching ancestor of $r_0(z)$ we can find the branching ancestor of the node from $T_t$ to which the natural subtree is mapped. But this is a branching node, so the structure in item 4. gives the desired branching ancestor. To summarize, the only super-constant cost is the binary search for $t$, which takes $O(\lg\lg w)$ time. Tradeoffs from Dynamic Range Reporting ====================================== Fix a value $B \in [2,\sqrt{w}]$; varying $B$ will give our tradeoff curve. For an arbitrary $t \in [0, \lg_B w]$, we define the trie of order $t$ to be the trie of depth $w / B^t$ and alphabet of $B^t$ bits, which represents all numbers in $S$. We call the trie for $t = 0$ the primary trie. A node $v$ in a trie of order $t$ is represented by a subtree of depth $B^t$ in the primary trie; we say that the root of this subtree “corresponds to” the node $v$. A node from a trie of order $t$ is represented by a subtree of depth $B$ in the trie of order $t-1$; we call such a subtree a “natural depth-$B$ subtree”. Alternatively, a depth-$B$ subtree is natural if it starts at a depth divisible by $B$. The root-to-leaf paths from the primary trie are boken into chunks of length $B^t$ in the trie of order $t$. A trie of order $t$ is similar to the $t$-th level (counted bottom-up) of the tree used for the greater-than problem, since a path in the primary trie is seen as the leaves of that tree. Indeed, every node on the $t$-th level of that tree held information about a subtree with $B^t$ leaves; here one edge in a trie of order $t$ summarizes a segment of length $B^t$ bits. Also, a natural depth-$B$ subtree corresponds to $B$ siblings in the old structure. On the next level, the $B$ siblings are contracted into a node; in the trie of higher order, a natural depth-$B$ subtree is also contracted into a node. Our data structure has the following new components: - choose this for the first branch of the tradeoff (faster updates, slower queries): hold the same information as in 4. for each $t$, and every node $v$ in the trie of order $t$ which is not a branching node, is on an active path, and is the child of a branching node in the trie of order $t$. - choose this for the second branch of the tradeoff: hold the same information as above for each $t$, and every node $v$ which is not a branching node, is on an active path, and has a branching ancestor in the same natural depth-$B$ subtree. In item 5A., notice that for every $t$ there are at most $2n - 2$ children of branching nodes which are on active paths. We store $O(\lg w)$ bits for each, and there are $O(\lg_B w)$ values of $t$, so we can store this in a Bloomier filter with $o(n)$ words of space. In item 5B., the number of interesting nodes blows up by at most $B$ compared to 5A., and since $B \leq \sqrt{w}$, we are still using $o(n)$ words of space. #### Updates. For each $t > 0$, we can either create a new branching node in the trie of order $t$, or the branching node existed already. We first test whether the branching node existed or not. If we just introduced a branching node, it has at most two children which are not branching nodes and are on active paths (if more than two such children exist, the node was a branching node before). If the branching node was old, we might have added one such child. These children are determined by looking at the branching descendents of $v$ (these give the two active paths going into $v$, one or both of which are new active paths going into the node in the subtrie of order $t$). For such children, we add the depth of $v$ in the structure from item 5A. If we are in case 5B, we follow both paths either until we find a branching node, or the border of the natural depth-$B$ subtree. For of these $O(B)$ positions, we add the depth of $v$ in item $5B$. To summarize, the running time is $O(T_{pred} + \lg_B w)$ if we need to update 5A., and $O(T_{pred} + B \lg_B w)$ is we need to update 5B. #### Queries. We need to show how to find $v$’s branching ancestor, assuming $v$ is on an active path, but is not a branching node. For some $t > 0$, and all $t$’s above that value, $v$ will be mapped in the trie of order $t$ to some branching node. That is the smallest $t$ such that the depth-$B^t$ natural subtree containing $v$ contains some branching node. We find this $t$ by binary search, taking time $O(\lg(\lg_B w))$. For some proposed $t$, we check if the node to which $v$ is mapped is a branching node in the trie of order $t$ (using the subroutine described above). If it is, we continue searching below; otherwise, we continue above. Say we found the smallest $t$ for which $v$ is mapped to a branching node. In the trie of order $t-1$, $v$ is mapped to some $w$ which is not a branching node. Finding the lowest branching ancestor of $v$ is identical to finding the lowest branching ancestor of the node corresponding to $w$ in the primary trie (since $w$ is a not a branching node, there is no branching node in the primary trie in the subtree represented by $w$). In the trie of order $t$, $w$ gets mapped to a branching node, so the natural depth-$B$ subtree of $w$ contains at least one branching node. The either: (1) there is some branching node above $w$ in its natural depth-$B$ subtree, or (2) $w$ is on the active path going to the root of this natural subtree (it is above any branching node). We first deal with case (2). If $w$ is above any branching node in its natural subtree, to find $w$’s branching ancestor we can find the branching ancestor of the node from the trie of order $t$, to which this subtree is mapped. But this is a branching node, so the structure in item 4. gives the branching ancestor $z$. We can test that we are indeed in case (2), and not case (1), by looking at the two branching descendents of $z$, and checking that one of them is strictly under $v$. Now we deal with case (1). If we have the structure 5B., this is trivial. Because $w$ is on an active path and has a branching ancestor in its natural depth-$B$ subtree, it records the depth of the branching ancestor of the node corresponding to $w$ in the primary trie. So in this case, the only super-constant cost is the binary search for $t$, which is $O(\lg(\lg_B w))$. If we only have the structure 5A., we need to walk up the trie of order $t-1$ starting from $w$. When we reach the child of the branching node above $w$, the branching node from the primary trie is recorded in item 5A. Since the branching node is in the same natural depth-$B$ subtree as $w$, we reach this point after $O(B)$ steps. One last detail is that we do not actually know when we have reached the child of a branching node (because the Bloomier filter from item 5A. can return arbitrary results for nodes not satisfying this property). To cope with this, at each level we hope that we have reached the destination, we query the structure in item 5A., we find the purported branching ancestor, and check that it really is the lowest branching acestor of $v$. This takes constant time; if the result is wrong, we continue walking up the trie. Overall, with the structure of 5A. we need query time $O(\lg(\lg_B w) + B)$. We have shown how to achieve the same running times (as functions of $B$) as in the case of the greater-than function. The same calculation establishes our tradeoff curve. Lower Bounds for the Greater-Than Problem ========================================= A lower bound for the first branch of the tradeoff can be obtained based on Fredman’s proof idea [@fredman82sums]. We ommit the details for now. To get a lower bound for the second case ($T_q < O(\lg\lg n)$), we use the sunflower lemma of Erdős and Rado. A sunflower is collection of sets (called petals) such that the intersection of any two of the sets is equal to the intersection of all the sets. Consider a collection of $n$ sets, of cardinalities at most $s$. If $n > (p-1)^{s+1} s!$, the collection contains as a subcollection a sunflower with $p$ petals. For every query parameter in $[0,n-1]$, the algorithm performs at most $T_q$ probes to the memory. Thus, there are $2^{T_q}$ possible execution paths, and at most $2^{T_q} - 1$ bit cells are probed on at least some execution path. This gives $n$ sets of cells of sizes at most $s = O(2^{T_q})$; we call these sets query schemes. By the sunflower lemma, we can find a sunflower with $p$ petals, if $p$ satisfies: $n > (p-1)^{s+1} s! \Rightarrow \lg n > \Theta(s (\lg p + \lg s))$. If $T_q < (1-\epsilon) \lg\lg n$, we have $s\lg s = o(\lg n)$, so our condition becomes $\lg n > \Theta(s \lg p)$. So we can find a sunflower with $p$ petals such that $\lg p = \Omega((\lg n)/s)$. Let $P$ be the set of query parameters whose query schemes are these $p$ petals. The center of the sunflower (the intersection of all sets) obviously has size at most $s$. Now consider the update schemes for the numbers in $P$. We can always find $T \subset P$ such that $|T| \geq |P| / 2^s$ and the update schemes for all numbers in $T$ look identical if we only inspect the center of the sunflower. Thus $\lg |T| = \lg |P| - s = \Omega(\frac{\lg n}{s} - s)$. If $T_u < (\frac{1}{2} - \epsilon) \lg\lg n$, we have $s = o(\frac{\lg n}{s})$, so we obtain $\lg |T| = \Omega(\frac{\lg n}{s})$. Now we restrict our attention to numbers in $T$ for both the update and query value. The cells in the center of the sunflower are thus fixed. Define the natural result of a certain query to be the result (greater than vs. not greater than) of the query if all bit cells read by the query outside the center of the sunflower are zero. Now pick a random $x \in T$. For some $y$ in the middle third of $T$ (when considering the elements of $T$ in increasing order), we have $\Pr[y \leq x] \geq \frac{1}{3}, \Pr[y > x] \geq \frac{1}{3}$, so no matter what the natural result of querying $y$ is, it is wrong with probability at least $\frac{1}{3}$. So for a random $x$, at least a fraction of $\frac{1}{9}$ of the natural results are wrong. Consider an explicit $x$ with this property. The update scheme for $x$ must set sufficiently many cells to change these natural results. But these cells can only be in the petals of the queries whose natural results are wrong, and the petals are disjoint except for the center, which is fixed. So the update scheme must set at least one cell for every natural result which is wrong. Hence $T_u \geq |T|/9 \Rightarrow \lg T_u = \Omega(\lg |T|) = \Omega(\frac{\lg n}{s}) = \Omega(\frac{\lg n}{2^{T_q}}) \Rightarrow 2^{T_q} = \Omega(\lg_{T_u} n)$. #### Acknowledgement. {#acknowledgement. .unnumbered} The authors would like to thank Gerth Brodal for discussions in the early stages of this work, in particular on how the results could be extended to dynamic range counting. [^1]: Part of this work was done while the author was visiting the Max-Planck-Institut für Informatik, Saarbrücken, as a Marie Curie doctoral fellow.

[**[Old Galaxies in the Young Universe]{}**]{} [**[A. Cimatti$^1$, E. Daddi$^2$, A. Renzini$^2$, P. Cassata$^3$, E. Vanzella$^{3}$, L. Pozzetti$^4$, S. Cristiani$^5$, A. Fontana$^6$, G. Rodighiero$^3$, M. Mignoli$^4$, G. Zamorani$^4$ ]{}**]{}\ $^1$ INAF - Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125, Firenze, Italy\ $^2$ European Southern Observatory, Karl-Schwarzschild-Str. 2, D-85748, Garching, Germany\ $^3$ Dipartimento di Astronomia, Università di Padova, Vicolo dell’Osservatorio, 2, I-35122 Padova, Italy\ $^4$ INAF - Osservatorio Astronomico di Bologna, via Ranzani 1, I-40127, Bologna, Italy\ $^5$ INAF - Osservatorio Astronomico di Trieste, Via Tiepolo 11, I-34131 Trieste, Italy\ $^6$ INAF - Osservatorio Astronomico di Roma, via dell’Osservatorio 2, Monteporzio, Italy [ **More than half of all stars in the local Universe are found in massive spheroidal galaxies$^{1}$, which are characterized by old stellar populations$^{2,3}$ with little or no current star formation. In present models, such galaxies appear rather late as the culmination of a hierarchical merging process, in which larger galaxies are assembled through mergers of smaller precursor galaxies. But observations have not yet established how, or even when, the massive spheroidals formed$^{2,3}$, nor if their seemingly sudden appearance when the Universe was about half its present age (at redshift $z \approx 1$) results from a real evolutionary effect (such as a peak of mergers) or from the observational difficulty of identifying them at earlier epochs. Here we report the spectroscopic and morphological identification of four old, fully assembled, massive ($>10^{11}$ solar masses) spheroidal galaxies at $1.6<z<1.9$, the most distant such objects currently known. The existence of such systems when the Universe was only one-quarter of its present age, shows that the build-up of massive early-type galaxies was much faster in the early Universe than has been expected from theoretical simulations$^{4}$.** ]{} In the $\Lambda$CDM scenario$^5$, galaxies are thought to build-up their present-day mass through a continuous assembly driven by the hierarchical merging of dark matter halos, with the most massive galaxies being the last to form. However, the formation and evolution of massive spheroidal early-type galaxies is still an open question. Recent results indicate that early-type galaxies are found up to $z\sim1$ with a number density comparable to that of local luminous E/S0 galaxies$^{6,7}$, suggesting a slow evolution of their stellar mass density from $z\sim1$ to the present epoch. The critical question is whether these galaxies do exist in substantial number$^{8,9}$ at earlier epochs, or if they were assembled later$^{10,11}$ as favored by most renditions of the hierarchical galaxy formation scenario$^{4}$. The problem is complicated also by the difficulty of identifying such galaxies due to their faintness and, for $z>1.3$, the lack of strong spectral features in optical spectra, placing them among the most difficult targets even for the largest optical telescopes. For example, while star-forming galaxies are now routinely found up to $z\sim6.6$$^{12}$, the most distant spectroscopically confirmed old spheroid is still a radio–selected object at $z=1.552$ discovered almost a decade ago$^{13,14}$. One way of addressing the critical question of massive galaxy formation is to search for the farthest and oldest galaxies with masses comparable to the most massive galaxies in the present-day universe ($10^{11-12}$ M$_{\odot}$), and to use them as the “fossil” tracers of the most remote events of galaxy formation. As the rest-frame optical – near-infrared luminosity traces the galaxy mass$^{15}$, the $K_s$-band ($\lambda \sim 2.2\,\mu$m in the observer frame) allows a fair selection of galaxies according to their mass up to $z\sim 2$. Following this approach, we recently conducted the K20 survey$^{16}$ with the Very Large Telescope (VLT) of the European Southern Observatory (ESO). Deep optical spectroscopy was obtained for a sample of 546 objects with $K_s<20$ (Vega photometric scale) and extracted from an area of 52 arcmin$^2$, including 32 arcmin$^2$ within the GOODS–South field $^{17}$ (hereafter the GOODS/K20 field). The spectroscopic redshift ($z_{spec}$) completeness of the K20 survey is 92%, while the available multi-band photometry ($BVRIzJHK_s$) allowed us to derive the spectral energy distribution (SED) and photometric redshift ($z_{phot}$) of each galaxy. The K20 survey spectroscopy was complemented with the ESO/GOODS public spectroscopy (Supplementary Table 1). The available spectra within the GOODS/K20 field were then used to search for old, massive galaxies at $z>1.5$. We spectroscopically identified four galaxies with $18 \lesssim K_s \lesssim 19$ and $1.6 \lesssim z_{spec} \lesssim 1.9$ which have rest-frame mid-UV spectra with shapes and continuum breaks compatible with being dominated by old stars and $R-K_s \gtrsim 6$ (the colour expected at $z>1.5$ for old passively evolving galaxies due to the combination of old stellar populations and k-correction effects$^{9}$). The Supplementary Table 1 lists the main galaxy information. The spectrum of each individual object allows a fairly precise determination of the redshift based on absorption features and on the overall spectral shape (Fig. 1). The co-added average spectrum of the four galaxies (Fig. 2–3) shows a near-UV continuum shape, breaks and absorption lines that are intermediate between those of a F2 V and a F5 V star$^{18}$, and typical of about 1-2 Gyr old synthetic stellar populations$^{19,20}$. It is also very similar to the average spectrum of $z\sim1$ old Extremely Red Objects$^7$ (EROs), and slightly bluer than that of the $z\sim0.5$ SDSS red luminous galaxies$^{21}$ and of the $z=1.55$ old galaxy LBDS 53w091$^{13}$. However, it is different in shape and slope from the average spectrum of $z\sim1$ dusty star-forming EROs$^7$. The multi-band photometric SED of each galaxy was successfully fitted without the need for dust extinction, and using a library of simple stellar population (SSP) models$^{19}$ with a wide range of ages, $Z=Z_{\odot}$ and Salpeter IMF. This procedure yielded best-fitting ages of 1.0-1.7 Gyr, the mass-to-light ratios and hence the stellar mass of each galaxy, which results in the range of 1–3$\times 10^{11}$ $h_{70}^{-2}$ M$_{\odot}$. $H_0=70$ km s$^{-1}$ Mpc$^{-1}$ (with $h_{70} \equiv H_0/70$), $\Omega_{\rm m}=0.3$ and $\Omega_{\Lambda}=0.7$ are adopted. In addition to spectroscopy, the nature of these galaxies was investigated with the fundamental complement of [*Hubble Space Telescope*]{}+ ACS ([*Advanced Camera for Surveys*]{}) imaging from the GOODS public [*Treasury Program*]{}$^{17}$. The analysis of the ACS high-resolution images reveals that the surface brightness distribution of these galaxies is typical of elliptical/early-type galaxies (Fig. 4). Besides pushing to $z\sim1.9$ the identification of the highest redshift elliptical galaxy, these objects are very relevant to understand the evolution of galaxies in general for three main reasons: their old age, their high mass, and their substantial number density. Indeed, an average age of about 1-2 Gyr ($Z=Z_{\odot}$) at $<\! z\!>\sim 1.7$ implies that the onset of the star formation occurred not later than at $z\sim 2.5-3.4$ ($z\sim 2-2.5$ for $Z=2.5Z_{\odot}$). These are strict lower limits because they follow from assuming instantaneous bursts, whereas a more realistic, prolonged star formation activity would push the bulk of their star formation to an earlier cosmic epoch. As an illustrative example, the photometric SED of ID 646 ($z=1.903$) can be reproduced (without dust) with either a $\sim$1 Gyr old instantaneous burst occurred at $z \sim 2.7$, or with a $\sim$2 Gyr old stellar population with a star formation rate declining with $exp(-t/ \tau)$ ($\tau=0.3$ Gyr). In the latter case, the star formation onset would be pushed to $z \sim 4$ and half of the stars would be formed at $z \sim 3.6$. In addition, with stellar masses $M_*>10^{11} h_{70}^{-2} M_\odot$, these systems would rank among the most massive galaxies in the present-day universe, suggesting that they were fully assembled already at this early epoch. Finally, their number density is considerably high. Within the comoving volume relative to 32 arcmin$^2$ and $1.5<z<1.9$ (40,000 $h_{70}^{-3}$ Mpc$^3$), the comoving density of such galaxies is about $10^{-4}$ $h_{70}^{3}$ Mpc$^{-3}$, corresponding to a stellar mass density of about $2 \times 10^{7}$ $h_{70}$ M$_{\odot}$Mpc$^{-3}$, i.e. about 10% of the local ($z=0$) value$^{22}$ for masses greater than $10^{11}$ M$_{\odot}$. This mass density is comparable to that of star-forming $M_*>10^{11}M_\odot$ galaxies at $z\sim 2$ $^{23}$, suggesting that while the most massive galaxies in the local universe are now old objects with no or weak star formation, by $z\sim 2$ passive and active star-forming massive galaxies coexist in nearly equal number. Although more successful than previous models, the most recent realizations of semi-analytic hierarchical merging simulations still severely underpredict the density of such old galaxies: just one old galaxy with $K_s<20$, $R-K_s>6$, and $z>1.5$ is present in the mock catalog$^{4}$ for the whole five times wider GOODS/CDFS area. As expected for early-type galaxies$^{9,24}$, the three galaxies at $z\sim 1.61$ may trace the underlying large scale structure. In this case, our estimated number density may be somewhat biased toward a high value. On the other hand, the number of such galaxies in our sample is likely to be a lower limit due to the spectroscopic redshift incompleteness. There are indeed up to three more candidate old galaxies in the GOODS/K20 sample with $18.5 \lesssim K_s \lesssim 19.5$, $1.5 \lesssim z_{phot} \lesssim 2.0$, $5.6 \lesssim R-K_s \lesssim 6.8$ and compact HST morphology. Thus, in the GOODS/K20 sample the fraction of old galaxies among the whole $z>1.5$ galaxy population is 15$\pm$8% (spectroscopic redshifts only), or up to 25$\pm$11% if also all the 3 additional candidates are counted. It is generally thought that the so-called “redshift desert” (i.e. around $1.4<z<2.5$) represents the cosmic epoch when most star formation activity and galaxy mass assembly took place$^{25}$. Our results show that, in addition to actively star forming galaxies$^{26}$, also a substantial number of “fossil” systems already populate this redshift range, and hence remain undetected in surveys biased towards star-forming systems. The luminous star-forming galaxies found at $z>2$ in sub-mm$^{27}$ and near-infrared$^{23,28}$ surveys may represent the progenitors of these old and massive systems. 1\. Fukugita, M., Hogan, C.J., Peebles, P.J.E. The Cosmic Baryon Budget. Astrophys. J. 503, 518-530 (1998).\ 2. Renzini, A. Origin of Bulges. In “The formation of galactic bulges”, ed. C.M. Carollo, H.C. Ferguson, R.F.G. Wyse, Cambridge University Press, p.9-26 (1999).\ 3. Peebles, P.J.E. When did the Large Elliptical Galaxies Form? In “A New Era in Cosmology”, ASP Conference Proceedings, Vol. 283. ed. N. Metcalfe and T. Shanks, Astronomical Society of the Pacific, 2002., p.351-361 (2002).\ 4. Somerville, R.S. et al. The Redshift Distribution of Near-Infrared-Selected Galaxies in the Great Observatories Origins Deep Survey as a Test of Galaxy Formation Scenarios. Astrophys. J., 600, L135-139 (2004).\ 5. Freedman, W.L. & Turner, M.S. Colloquium: Measuring and understanding the universe. Reviews of Modern Physics, 75, 1433-1447 (2003).\ 6. Im, M. et al. The DEEP Groth Strip Survey. X. Number Density and Luminosity Function of Field E/S0 Galaxies at $z<1$. Astrophys. J. 571, 136-171 (2002).\ 7. Cimatti, A. et al. The K20 survey. I. Disentangling old and dusty star-forming galaxies in the ERO population. Astron. Astrophys. 381, L68-73 (2002).\ 8. Benitez, N. et al. Detection of Evolved High-Redshift Galaxies in Deep NICMOS/VLT Images. Astrophys. J. 515, L65-69 (1999).\ 9. Daddi, E. et al. Detection of strong clustering of extremely red objects: implications for the density of $z>1$ ellipticals. Astron. Astrophys. 361, 535-549 (2000).\ 10. Zepf, S.E. Formation of elliptical galaxies at moderate redshifts. Nature 390, 377-380 (1997).\ 11. Rodighiero G., Franceschini A., Fasano G. Deep Hubble Space Telescope imaging surveys and the formation of spheroidal galaxies. Mon. Not. R. Astron. Soc. 324, 491-497 (2001).\ 12. Taniguchi, Y. et al. Lyman$\alpha$ Emitters beyond Redshift 5: The Dawn of Galaxy Formation. Journal of the Korean Astronomical Society 36, no.3, 123-144 (2003).\ 13. Dunlop, J.S. et al. A 3.5-Gyr-old galaxy at redshift 1.55. Nature, 381, 581-584 (1996).\ 14. Spinrad, H., Dey, A., Stern, D., Dunlop, J., Peacock, J., Jimenez, R., Windhorst, R. LBDS 53W091: an Old, Red Galaxy at z=1.552. Astrophys. J., 484, 581-601 (1997).\ 15. Gavazzi, G., Pierini, D., Boselli, A., The phenomenology of disk galaxies. Astron. Astrophys. 312, 397-408 (1996).\ 16. Cimatti, A. et al. The K20 survey. III. Photometric and spectroscopic properties of the sample. Astron. Astrophys. 392, 395-406 (2002).\ 17. Giavalisco, M. et al., The Great Observatories Origins Deep Survey: Initial Results from Optical and Near-Infrared Imaging. Astrophys. J. 600, L93-98 (2004).\ 18. Pickles, A.J. A Stellar Spectral Flux Library: 1150-25000 Å. PASP 110, 863-878 (1998).\ 19. Bruzual, G. & Charlot, S. Stellar population synthesis at the resolution of 2003. Mon. Not. R. Astron. Soc. 344, 1000-1028 (2003).\ 20. Jimenez, R. et al. Synthetic stellar populations: single stellar populations, stellar interior models and primordial proto-galaxies, Mon. Not. R. Astron. Soc. 349, 240-254 (2004).\ 21. Eisenstein, D.J. et al. Average Spectra of Massive Galaxies in the Sloan Digital Sky Survey. Astrophys. J. 585, 694-713 (2003).\ 22. Cole, S. et al. The 2dF galaxy redshift survey: near-infrared galaxy luminosity functions. Mon. Not. R. Astron. Soc., 326, 255-273 (2001).\ 23. Daddi, E. et al., Near-Infrared Bright Galaxies at $z\sim2$. Entering the Spheroid Formation Epoch ? Astrophys. J., 600, L127-131 (2004).\ 24. Davis, M., Geller, M.J. Galaxy Correlations as a Function of Morphological Type. Astrophys. J., 208, 13-19 (1976).\ 25. Dickinson, M., Papovich, C., Ferguson, H.C., Budavari, T. The Evolution of the Global Stellar Mass Density at $0<z<3$. Astrophys. J., 587, 25-40 (2003).\ 26. Steidel, C.C. et al. A Survey of Star-Forming Galaxies in the z=1.4-2.5 ‘Redshift Desert’: Overview. Astrophys. J. 604, 534-550 (2004).\ 27. Genzel, R., Baker, A.J., Tacconi, L.J., Lutz, D., Cox, P.; Guilloteau, S., Omont, A. Spatially Resolved Millimeter Interferometry of SMM J02399-0136: A Very Massive Galaxy at $z=2.8$. Astrophys. J. 584, 633-642 (2003).\ 28. Franx, M. et al. A Significant Population of Red, Near-Infrared-selected High-Redshift Galaxies. Astrophys. J., 587, L79-L83 (2003).\ 29. Pignatelli, E. & Fasano, G. GASPHOT: A Tool for Automated Surface Photometry of Galaxies. Astrophys. Sp. Sci. 269, 657-658 (1999).\ 30. Peng, C.Y., Ho, L.C., Impey, C.D., Rix, H.-W., Detailed Structural Decomposition of Galaxy Images. Astron. J., 124, 266-293 (2002).\ Correspondence and requests for material should be sent to Andrea Cimatti (cimatti@arcetri.astro.it). This work is based on observations made at the European Southern Observatory, Paranal, Chile, and with the NASA/ESA Hubble Space Telescope obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy (AURA). We thank Rachel Somerville for information on the GOODS/CDFS mock catalog. We are grateful to the GOODS Team for obtaining and releasing the HST and FORS2 data. [**Figure 1**]{} [*The individual and average spectra of the detected galaxies.*]{} From bottom to top: the individual spectra smoothed to a 16 Å  boxcar (26 Å  for ID 237) and the average spectrum of the four old galaxies ($z_{average}=1.68$). The red line is the spectrum of the old galaxy LBDS 53w091 ($z=1.55$) used to search for spectra with a similar continuum shape. Weak features in individual spectra (e.g. MgII$\lambda$2800 and the 2640 Å  continuum break, B2640) become clearly visible in the average spectrum. The object ID 235 has also a weak \[OII\]$\lambda$3727 emission (not shown here). The spectra were obtained with ESO VLT+FORS2, grisms 200I (R($1^{\prime\prime})\sim$400) (ID 237) and 300I (R($1^{\prime\prime})\sim$600) (IDs 235,270,646), 1.0$^{\prime\prime}$ wide slit and $\lesssim 1^{\prime\prime}$ seeing conditions. The integrations times were 3 hours for ID 237, 7.8 hours for IDs 235 and 270, and 15.8 hours for ID 646. For ID 646, the ESO/GOODS public spectrum was co-added to our K20 spectrum (see Supplementary Tab. 1). “Dithering” of the targets along the slits was applied to remove efficiently the CCD fringing pattern and the strong OH sky lines in the red. The data reduction was done with the IRAF software package (see$^{16}$). The spectrophotometric calibration of all spectra was achieved and verified by observing several standard stars. The average spectrum, corresponding to 34.4 hours integration time, was obtained by co-adding the individual spectra convolved to the same resolution, scaled to the same arbitrary flux (i.e. with each spectrum having the same weight in the co-addition), and assigning wavelength–dependent weights which take into account the noise in the individual spectra due to the OH emission sky lines. ![[]{data-label="fig1"}](cimatti_fig1.ps){width="15cm"} [**Figure 2**]{} [*The detailed average spectrum of the detected galaxies.*]{} A zoom on the average spectrum (blue) compared with the synthetic spectrum$^{19}$ of a 1.1 Gyr old simple stellar population (SSP) with solar metallicity ($Z=Z_{\odot}$) and Salpeter IMF (red). The observed average spectrum was compared to a library of synthetic SSP template spectra$^{19,20}$ with a range of ages of 0.1-3.0 Gyr with a step of 0.1 Gyr, and with assumed metallicities $Z$=0.4$\times$, 1.0$\times$, and 2.5$\times Z_{\odot}$. The best fit age for each set of synthetic templates was derived through a $\chi^2$ minimization over the rest-frame wavelength range 2300–3400 Å. The rms as a function of wavelength used in the $\chi^2$ procedure was estimated from the average spectrum computing a running mean rms with a step of 1 Å  and a box size of 20 Å, corresponding to about three times the resolution of the observed average spectrum. The median signal-to-noise ratio is $\sim$20 per resolution element in the 2300–3400 Å  range. The wavelength ranges including the strongest real features (i.e. absorptions and continuum breaks) were not used in the estimate of the rms. The resulting reduced $\chi^2$ is of the order of unity for the best fit models. In the case of solar metallicity, the ranges of ages acceptable at 95% confidence level are $1.0^{+0.5}_{-0.1}$ Gyr and $1.4^{+0.5}_{-0.4}$ Gyr for SSP models of$^{19}$ and$^{20}$ respectively (see also Fig. 3, top panel). Ages $\sim 50\%$ younger or older are also acceptable for $Z=2.5Z_{\odot}$ or $Z=0.4Z_{\odot}$ respectively. The 2640 Å  and 2900 Å  continuum break$^{13}$ amplitudes measured on the average spectrum are B2640=1.8$\pm$0.1 and B2900=1.2$\pm$0.1. These values are consistent with the ones expected in SSP models$^{19-20}$ for ages around 1–1.5 Gyr and solar metallicity. For instance, the SSP model spectrum shown here has B2640=1.84 and B2900=1.27. ![[]{data-label="fig2"}](cimatti_fig2.ps){width="15cm"} [**Figure 3**]{} [*The comparison between the average spectrum and a set of spectral templates.*]{} The average spectrum (blue) compared to a set of template spectra. From bottom: F2 V (green) and F5 V (red) stellar spectra$^{18}$ with $Z=Z_{\odot}$, the composite spectrum (red) of 726 luminous red galaxies at $0.47<z<0.55$ selected from the SDSS$^{21}$ (available only for $\lambda>2600$ Å), the average spectra of $z\sim1$ old (red) and dusty star-forming (green) EROs$^7$, SSP synthetic spectra$^{19}$ ($Z=Z_{\odot}$, Salpeter IMF) with ages of 0.5 Gyr (magenta), 1.1 Gyr (green) and 3.0 Gyr (red). ![[]{data-label="fig3"}](cimatti_fig3.ps){width="15cm"} [**Figure 4**]{} [*The morphological properties of the detected galaxies.*]{} Images of the four galaxies taken with the [*Hubble Space Telescope*]{} +ACS through the F850LP filter (from GOODS data$^{17}$) which samples the rest-frame $\sim$3000-3500 Å  for $1.6<z<2$. The images are in logarithmic grey–scale and their size is $2^{\prime\prime} \times 2^{\prime\prime}$, corresponding to $\sim 17 \times 17$ kpc for the average redshift $z=1.7$ and the adopted cosmology. At a visual inspection, the galaxies show rather compact morphologies with most of the flux coming from the central regions. A fit of their surface brightness profiles was performed with a “Sersic law” ($\propto r^{1/n}$) convolved with the average point spread function extracted from the stars in the ACS field and using the GASPHOT$^{29}$ and GALFIT$^{30}$ software packages. Objects ID 237 and ID 646 have profiles with acceptable values of $n$ in the range of $4<n<6$, i.e., typical of elliptical galaxies, object ID 270 is better reproduced by a flatter profile ($1<n<2$), whereas a more ambiguos result is found for the object showing some evidence of irregularities in the morphology (ID 235, $1<n<3$). These latter objects may be bulge-dominated spirals but no bulge/disk decomposition was attempted. Ground-based near-infrared images taken under 0.5$^{\prime\prime}$ seeing conditions with the ESO VLT+ISAAC through the $K_s$ filter (rest-frame $\sim$6000-8000 Å) show very compact morphologies, but no surface brightness fitting was done. ![[]{data-label="fig4"}](cimatti_fig4.ps){width="15cm"} [**SUPPLEMENTARY TABLE 1**]{}\ \ [cccccccl]{}\ IAU & K20 & R.A. (J2000) & Dec (J2000) & $K_s$ & $R-K_s$ & $z$ & Spectrum\ ID & ID & h m s & $\circ$ $\prime$ $\prime\prime$& & & &\ \ J033210.79-274627.8 & 235 & 03 32 10.776 & -27 46 27.73 & 17.98$\pm$0.04 & 6.47$\pm$0.10&1.610&K20\ J033210.52-274628.9 & 237 & 03 32 10.507 & -27 46 28.84 & 19.05$\pm$0.05 & 6.83$\pm$0.28&1.615&K20\ J033212.53-274629.2 & 270 & 03 32 12.525 & -27 46 29.16 & 18.74$\pm$0.05 & 5.99$\pm$0.10&1.605&K20\ J033233.85-274600.2 & 646 & 03 32 33.847 & -27 46 00.24 & 19.07$\pm$0.07 & 5.99$\pm$0.10&1.903&K20+GOODS\ \ [**Supplementary Table 1**]{}\ \ IAU ID: official identification number in the GOODS–South catalog (z-band)\ (http://www.stsci.edu/science/goods/catalogs).\ \ K20 ID: identification number in the K20 survey catalog (http://www.arcetri.astro.it/$\sim$k20/).\ \ R.A., Dec: Right Ascension and Declination at equinox J2000 based on the public ESO/GOODS $K_s$-band VLT+ISAAC image.\ \ $K_s$: K20 survey total magnitude in the $K_s$-band (Vega scale).\ \ $R-K_s$ color (Vega scale) in 2$^{\prime\prime}$ diameter aperture.\ \ $z$: spectroscopic redshift.\ \ Spectrum: K20: K20 survey, GOODS: public ESO/GOODS VLT+FORS2 spectroscopy (Vanzella et al., in preparation; http://www.eso.org/science/goods).

--- 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>

--- abstract: 'Synthetic ladders realized with one-dimensional alkaline-earth(-like) fermionic gases and subject to a gauge field represent a promising environment for the investigation of quantum Hall physics with ultracold atoms. Using density-matrix renormalization group calculations, we study how the quantum Hall-like chiral edge currents are affected by repulsive atom-atom interactions. We relate the properties of such currents to the asymmetry of the spin resolved momentum distribution function, a quantity which is easily addressable in state-of-art experiments. We show that repulsive interactions significantly stabilize the quantum Hall-like helical region and enhance the chiral currents. Our numerical simulations are performed for atoms with two and three internal spin states.' address: - '$^{1}$NEST, Scuola Normale Superiore & Istituto Nanoscienze-CNR, I-56126 Pisa, Italy' - '$^{2}$Scuola Normale Superiore, I-56126 Pisa, Italy' - '$^{3}$CNR - Istituto Nazionale di Ottica, UOS di Firenze LENS, I-50019 Sesto Fiorentino, Italy' - '$^{4}$The Abdus Salam International Centre for Theoretical Physics (ICTP), I-34151 Trieste, Italy' author: - | Simone Barbarino$^{1}$, Luca Taddia$^{2,3}$, Davide Rossini$^{1}$,\ Leonardo Mazza$^{1}$, and Rosario Fazio$^{4,1}$ title: 'Synthetic gauge fields in synthetic dimensions: Interactions and chiral edge modes' --- Introduction ============ One of the most noticeable hallmarks of topological insulators is the presence of robust [*gapless edge modes*]{} [@topins]. Their first experimental observation goes back to the discovery of the quantum Hall effect [@qhe], where the existence of chiral edge states is responsible for the striking transport properties of the Hall bars. The physics of edge states has recently peeked out also in the arena of ultracold gases [@Atala_2014; @Mancini_2015; @Stuhl_2015], triggered by the new exciting developments in the implementation of topological models and synthetic gauge potentials for neutral cold atoms [@Dalibard_2011; @Struck_2012; @Hauke_2012; @Goldman_2013; @Goldman_2014]. Synthetic gauge potentials in cold atomic systems have already led to the experimental study of Bose-Einstein condensates coupled to a magnetic field [@Lin_2009] or with an effective spin-orbit coupling [@Lin_2011], and more recently to lattice models with non-zero Chern numbers [@Aidelsburger_2013; @Miyake_2013; @Jotzu_2014; @Aidelsburger_2014] and frustrated ladders [@Atala_2014]. In a cold-gas experiment, the transverse dimension of a two-dimensional setup does not need to be a *physical* dimension, i.e. a dimension in real space: an extra *synthetic* dimension on a given *d*-dimensional lattice can be engineered taking advantage of the internal atomic degrees of freedom [@Boada_2012]. The crucial requirement is that each of them has to be coupled to two other states in a sequential way through, for example, proper Raman transitions induced by two laser beams. In this situation, it is even possible to generate gauge fields in synthetic lattices [@Celi_2014]. In this work we focus on one-dimensional systems with a finite synthetic dimension coupled to a synthetic gauge field, i.e. *frustrated ladders*. The study of such ladders traces back to more than thirty years ago, when frustration and commensurate-incommensurate transitions have been addressed in Josephson networks [@kardar1; @kardar2]. Thanks to the experimental advances with optical lattices, these systems are now reviving a boost of activity. Both bosonic (see, e.g., Refs. [@dhar; @petrescu; @grudsdt; @piraud; @tokuno]) and fermionic (see, e.g., Refs. [@roux; @sun; @Barbarino_2015; @Zeng_2015; @Cornfeld_2015; @Mazza_2015; @Budich_2015; @Lacki_2015]) systems have been considered. The emerging phenomenology is amazingly rich, ranging from new phases with chiral order [@dhar] to vortex phases [@piraud] or fractional Hall-like phases in fermionic systems [@Barbarino_2015; @Cornfeld_2015], just to give some examples. Very recently, two experimental groups [@Mancini_2015; @Stuhl_2015] have observed persistent spin currents in one dimensional gases of $^{173}$Yb (fermions) and $^{87}$Rb (bosons) determined by the presence of such gauge field. Within the framework of the synthetic dimension, such *helical* spin currents can be regarded as the *chiral* edge states of a two-dimensional system and are reminiscent of the edge modes of the Hall effect. Up to now, the study of edge currents in optical lattices has mainly focused on aspects related to the single-particle physics and a systematic investigation of the interaction effects is missing. [Repulsive]{} interactions considerably affect the properties of the edge modes: this is well known in condensed matter, where the fractional quantum Hall regime [@fqhe] can be reached for proper particle fillings and for sufficiently strong Coulomb interactions. In view of the new aforementioned experiments in bosonic [@Stuhl_2015] and fermionic [@Mancini_2015] atomic gases, a deeper understanding of the role of repulsive interactions in these setups is of the uttermost importance. Here we model the experiment on the frustrated $n$-leg ladder performed in Ref. [@Mancini_2015] and analyze, by means of density-matrix renormalization group (DMRG) simulations, how atom-atom repulsive interactions modify the edge physics of the system (in this article we disregard the effects of an harmonic confinement and of the temperature). We concentrate on the momentum distribution function, which has been used in the experiment to indirectly probe the existence of the edge currents. The purpose of this article is twofold. First, we want to present numerical evidence that helical modes, reminiscent of the chiral currents of the integer quantum Hall effect, can be stabilized by repulsive interactions. Second, we want to discuss the influence of interactions on experimentally measurable quantities that witness the chirality of the modes. In this context the words “chiral” and “helical” can be interchanged, depending whether one considers a truly one-dimensional system with an internal degree of freedom or a synthetic ladder. There is an additional important point to be stressed when dealing with synthetic ladders in the presence of interactions. The many-body physics of alkaline-earth(-like) atoms (like Ytterbium) with nuclear spin $I$ larger than $1/2$ is characterized by a SU($2I+1$) symmetry [@Gorshkov_2010; @Cazalilla_2014]. When they are viewed as ($2I+1$)-leg ladders, the interaction is strongly anisotropic, i.e. it is short-range in the physical dimension and long-range in the synthetic dimension. This situation is remarkably different from [the typical]{} condensed-matter systems and may lead to quantitative differences especially when considering narrow ladders, as in Ref. [@Mancini_2015]. The paper is organized as follows. In the next section we introduce the model describing a one-dimensional gas of earth-alkaline(-like) atoms with nuclear spin $I\geq 1/2$. In order to make a clear connection with the experiment of Ref. [@Mancini_2015], we briefly explain how this system can be viewed as a ($2I+1$)-leg ladder. Moreover, we present a discussion of the single-particle spectrum to understand the main properties of the edge currents in the non-interacting regime and to identify the regimes where the effects of repulsive interactions are most prominent. Then, in Sec. \[obs-sec\] we introduce two quantities, evaluated by means of the DMRG algorithm, that characterize the edge currents: the (spin-resolved) momentum distribution function and the average current derived from it. In Sec. \[results\] we present and comment our results; we conclude with a summary in Sec. \[conclusions\]. Synthetic gauge fields in synthetic dimensions {#model} ============================================== The model --------- We consider a one-dimensional gas of fermionic earth-alkaline-(like) neutral atoms characterized by a large and tunable nuclear spin $I$, see Fig. \[ladder\](a). Based on the predictions of Ref. [@Gorshkov_2010], Pagano [*et al.*]{} have experimentally showed that, by conveniently choosing the populations of the nuclear-spin states, the number of atomic species can be reduced at will to $2\mathcal{I}+1$, giving rise to an effective atomic spin $\mathcal{I}\leq I$ [@Pagano_2014]. We stress that $I$ has to be an half-integer to enforce the fermionic statistics, while $\mathcal{I}$ can also be an integer, see Fig. \[ladder\](b). Moreover, as extensively discussed in Refs. [@Boada_2012; @Celi_2014], the system under consideration can be both viewed as a mere one-dimensional gas with $2\mathcal{I}+1$ spin states or as a ($2\mathcal{I}+1$)-leg ladder, see Fig. \[ladder\](c). When loaded into an optical lattice, the Hamiltonian can be written as [@Gorshkov_2010]: $$\hat{H}_0 =- t \sum_j \sum_{m=-\mathcal{I}}^\mathcal{I} \left( \hat{c}^\dagger_{j, m} \hat{c}_{j+1, m} + \mathrm{H.c.} \right) + U \sum_j \sum_{m < m'} \hat{n}_{j,m} \hat{n}_{j,m'} \,, \label{Hubbard}$$ where $\hat{c}_{j,m}$ ($\hat{c}^\dagger_{j,m}$) annihilates (creates) a spin-$m$ fermion ($m=-\mathcal{I}, \ldots, \mathcal{I}$) at site $j=1,\ldots,L$ and $\hat{n}_{j,m}=\hat{c}^\dagger_{j,m} \hat{c}_{j,m}$; $t$ is the hopping amplitude, while $U$ is the strength of the SU($2\mathcal{I} + 1$)-invariant interaction; the first sum in the hopping term runs over $j=1, \ldots, L-1$ if open boundary conditions (OBC) in the real dimension are considered, or over $j=1, \ldots, L$ if periodic boundaries (PBC) are assumed. Hereafter we set $\hbar=1$. The Hamiltonian (\[Hubbard\]), also known as the SU($2\mathcal{I}+1$) Hubbard model, has attracted considerable attention in the last few decades, see e.g. Refs. [@Assaraf_1999; @Szirmai_2005; @Buchta_2007; @Manmana_2011]. ![Implementation of $\hat{H} = \hat{H}_0 + \hat{H}_1$ in a cold-atom system. (a) Sketch of a one-dimensional atomic gas with nuclear spin $I=5/2$, e.g. $^{173}$Yb. (b) Definition of the effective spins $\mathcal I=1$ and $\mathcal I=1/2$ as in the experimental implementation with $^{173}$Yb of Ref. [@Mancini_2015]. (c) Graphical representation of the non-interacting Hamiltonian in the synthetic-dimension picture, for the case ${\mathcal I}=1$. []{data-label="ladder"}](Figure1.pdf){width="\linewidth"} The presence of two additional laser beams can induce a coupling between spin-states with $\Delta m = \pm 1$ of amplitude $\Omega_m$ endowed with a running complex phase factor $e^{i\gamma j}$. For simplicity, in the following we assume that $\Omega_m$ does not depend on $m$ and set $\Omega_m=\Omega$. The coupling $\Omega$ is related to the amplitude of the laser beams, while the phase $\gamma$ [depends on]{} their wavelength and relative propagation angle. Explicitly, the Hamiltonian gets a contribution of the form $$\hat{H}_1 = \sum_j \sum_{m=-\mathcal{I}}^{\mathcal{I}-1} \; \Omega_m \left( e^{-i\gamma j} \hat{c}^\dagger_{j, m} \hat{c}_{j, m+1}+ \mathrm{H.c.} \right) \, .$$ As already mentioned, the system characterized by the Hamiltonian $\hat{H} \equiv \hat{H}_0+\hat{H}_1$ is equivalent to a $(2\mathcal{I}+1)$-leg ladder [where the]{} coordinate in the transverse direction is given by the effective-spin index $m=-\mathcal{I}, \dots, \mathcal{I}$. For all purposes, such direction can be regarded as a synthetic dimension with sharp edges; in this framework, the Hamiltonian $\hat{{H}}_1$ describes the hopping in the synthetic dimension and introduces a constant magnetic field perpendicular to the ladder with dimensionless magnetic flux $+\gamma$ per plaquette. The peculiarity of our synthetic ladder resides in the interaction term, which is $SU(2\mathcal{I}+1)$ invariant: it therefore describes an on-site interaction in the real dimension and a long-range interaction in the synthetic one. Since the Hamiltonian $\hat H$ is not translationally invariant, for later convenience, we perform the unitary transformation $\hat{d}_{j,m}=\hat{\mathcal{U}}\hat{c}_{j,m}\hat{\mathcal{U}}^\dagger=e^{-im \gamma j }\hat{c}_{j,m}$ such that $\hat{\mathcal{U}}(\hat{H}_0+\hat{H}_1)\hat{\mathcal{U}}^\dagger = \hat{\mathcal{H}}_0+\hat{\mathcal{H}}_1 = \hat{\mathcal{H}}$ reads $$\begin{aligned} \hat{\mathcal{H}}_0 & = & -t \sum_j \sum_{m=-\mathcal{I}}^\mathcal{I} \left(e^{i\gamma m} \hat{d}^\dagger_{j, m} \hat{d}_{j+1, m} + \mathrm{H.c.} \right) + U \sum_j \sum_{m < m'} \hat{\nu}_{j,m} \hat{\nu}_{j,m'} \,, \\ \hat{\mathcal{H}}_1 & = & \sum_j \sum_{m=-\mathcal{I}}^{\mathcal{I}-1} \left( \Omega_m \; \hat{d}^\dagger_{j, m} \hat{d}_{j, m+1} + \mathrm{H.c.} \right) \,,\end{aligned}$$ where $\hat{\nu}_{j,m}= \hat{d}^\dagger_{j,m} \hat{d}_{j,m}$. Assuming PBC in the real dimension, the quadratic part of $\hat{\mathcal{H}}$ can be diagonalized in Fourier space, in terms of the operators $\hat{{d}}_{p,m}=L^{-1/2} \sum_{j=1}^L e^{ik_pj} \hat{d}_{j,m}$, with $k_p={2\pi p}/{L}$ and $p \in \{-L/2, \ldots, L/2-1\}$. Non-interacting helical liquid ------------------------------ In order to discuss the helical properties of this system, a good starting point is the analysis of the non-interacting physics for the $\mathcal{I}=1/2$ case. The single-particle spectrum of the Hamiltonian $\hat{\mathcal H}$ has two branches with the following dispersion relations: $$\epsilon_\pm(k_p) = -2t \cos \frac \gamma2 \, \cos k_p \pm \sqrt{4t^2 \sin^2 \frac \gamma2 \, \sin^2 k_p + \Omega^2} \, . \label{spectrum_formula}$$ When the condition $\Omega < 2t \sin \frac \gamma2 \tan \frac \gamma2$ is satisfied, the lower branch displays two minima at $k_p \approx \pm\gamma/2$ and a local maximum at $k_p=0$, see Fig. \[spectra\_SU2\](a): this case will be referred to as the weak-Raman-coupling (WRC) regime. In the opposite case, dubbed strong-Raman-coupling (SRC) regime, the lower branch has one single minimum at $k_p=0$ without any special feature at $k_p\neq0$, see Fig. \[spectra\_SU2\](c). ![Spectral properties of $\hat{\mathcal{H}}$ in the non-interacting case. Left panels: energy spectra; right panels: spin polarization along the $z$ axis of the quasi-momentum single-particle eigenstates for several cases (lines with the same colors are in correspondence). Panels (a)-(b): $\mathcal{I}=1/2$ and WRC regime ($\Omega/t=0.3$). Panels (c)-(d): $\mathcal{I}=1/2$ and SRC regime ($\Omega/t=1.8$). Panels (e)-(f): $\mathcal{I}=1$ and WRC regime ($\Omega/t=0.1$). In all the situations, we assumed $\gamma=0.37\pi$, PBC and $L\rightarrow\infty$. In panels (a) and (e), the orange, violet and green lines describe, respectively, the low-, intermediate- and high-filling situations considered in the text.[]{data-label="spectra_SU2"}](Figure2b.pdf){width="\linewidth"} The study of the spin polarization $S^z$ (related to the operator $\sum_{j,m} \hspace{-0.1cm} m \, \hat{\nu}_{j,m} \,$) of each eigenmode highlights an important difference between the SRC and the WRC regimes, see Figs. \[spectra\_SU2\](b) and \[spectra\_SU2\](d). In the WRC case, for most of the values of $k_p$, the eigenstates are prevalently polarized along the $z$ direction, while in the SRC regime this is not true (the dominating polarization is along the $x$ direction, not shown here). Figure \[spectra\_SU2\](a) also shows that in the WRC regime depending on the filling, the low-energy excitation may have very different properties. For low (e.g. the orange line) or high (e.g. the green line) fillings, there are four low-energy excitations. However, when the chemical potential (here we consider zero temperature) lies between $-2t \cos(\gamma/2)-\Omega$ and $-2t \cos (\gamma/2)+\Omega$ (e.g. the violet line), there are two gapless excitations which have definite quasi-momentum and definite spin in the $z$ direction. In the non-interacting case, this is an [*helical liquid*]{} which, once interpreted as a ladder, features two chiral edge modes. Similar considerations about the single-particle spectrum hold for the $\mathcal{I}=1$ case, even though the analytic form of the eigenenergies is more involved. In Fig. \[spectra\_SU2\](e) we show the single-particle energy spectrum of the eigenstates in the WRC regime. Low, intermediate and high fillings can be identified also in this case, and are indicated by the three different horizontal lines. The intermediate filling (violet line) corresponds to the regime where the helical liquid appears; indeed the spin polarization $S^z$ shown in Fig. \[spectra\_SU2\](f) exhibits almost full polarization of the eigenstates close to the considered Fermi energy. Here, in the synthetic-dimension representation, the three-leg ladder displays chiral modes. In the interacting case, the spectral properties of the Hamiltonian are not trivially computable. In the following section we define the physical quantities used to properly characterize the helical modes, which can be calculated by means of the DMRG algorithm. In the remainder of this paper we carefully analyze such quantities. Observables {#obs-sec} =========== The study of the momentum distribution function, both spin-resolved and non-spin-resolved, can provide, as we shall see, information about the helical/chiral nature of the interacting liquid under consideration. The spin-resolved momentum distribution function is defined as $$n_{p,m} = \langle \hat c^\dagger_{p,m} \hat c_{p,m}\rangle = \frac{1}{L} \, \sum_{j,l} \, e^{-i\frac{2\pi p}{L}(j-l)} \langle\hat{c}_{j,m}^\dagger \hat{c}_{l,m} \rangle \,, \label{eq:n_k}$$ where expectation values are taken over the ground state. Since $p$ is not a good quantum number for $\hat{H}$, we will conveniently consider Hamiltonian $\hat{\mathcal{H}}$ and the momentum distribution function $\nu_{p,m} = \langle \hat d^\dagger_{p,m} \hat d_{p,m} \rangle$, for which it easy to verify that $\nu_{p,m}=n_{p-m\gamma,m}$. Accordingly, the total momentum distribution is given by $n_p = \sum_{m=-\mathcal{I}}^{\mathcal{I}} n_{p,m}$. Based on these definitions, we introduce two chirality witnesses, i.e. two quantities which diagnose and identify the edge currents determined by the presence of the gauge field $\gamma \neq0$, even in the presence of repulsive interactions. To this aim, we first solve the continuity equation for the Hamiltonian $\hat{{H}}$ and define the ground-state average chiral current $$\mathcal{J}_{j,m} = -i \, t \, \langle \hat{c}^\dagger_{j,m} \hat{c}_{j+1,m} \rangle + \text{H.c.}\;. \label{cur}$$ Assuming PBC in the real dimension and using Eq. (\[eq:n\_k\]), its spatial average can be re-expressed as $$Q_m = \frac{1}{L} \sum_j \mathcal{J}_{j,m} = - \frac{2t}{L}\sum_{p>0}\sin k_p\left(n_{p,m}-n_{-p,m}\right) \,, \label{QMM}$$ with $k_p = 2\pi p / L$. The latter relation allows to indirectly probe the existence of chiral currents using a quantity, namely $n_{p,m}$, which can be experimentally observed in the state-of-art laboratories using a band-mapping technique [@Kohl_2005] followed by a Stern-Gerlach time-of-flight imaging [@Mancini_2015; @Stuhl_2015]. The quantity $Q_m$ is the first chirality witness to be employed in the following. The second chirality witness is the quantity $$J_m = -\sum_{p>0}\left(n_{p,m}-n_{-p,m}\right) \,, \label{JM}$$ defined in Ref. [@Mancini_2015], which is more directly related to the asymmetry of the spin-resolved momentum distribution function. Both $J_m$ and $Q_m$ give information about the circulating currents and, as we shall see below, display the same qualitative behavior (they only differ for a cut-off at low wavelength). Results ======= Equipped with the definitions given in the previous sections, we now discuss how atom-atom repulsive interactions affect the momentum distribution functions $n_p$ and $n_{p,m}$ and the chirality witnesses $Q_m$ and $J_m$. The results for the non-interacting cases, here used as a reference, are computed by means of exact diagonalization, while for $U/t\neq0$ the DMRG algorithm is used [@White_1992; @Schollwock_2011]. We only address the ground-state properties, i.e. rigorously work at zero temperature. In the finite-size sweeping procedure, up to 250 eigenstates of the reduced density matrix are kept, in order to achieve a truncation error of the order of $10^{-6}$ (in the worst cases) and a precision, for the computed correlations, at the fourth digit. The resulting inaccuracy is negligible on the scale of all the figures shown hereafter. Unless differently specified, in the $\mathcal{I}=1/2$ case we consider $L=96$ and $\Omega/t=0.3$, while in the $\mathcal{I}=1$ case we set $L=48$ and $\Omega/t=0.1$ (the ratio $\Omega/t$ is chosen in order to be in the WRC regime); $\gamma=0.37\pi$ coincides with the experimental value of Ref. [@Mancini_2015]. As shown in Figs. \[spectra\_SU2\](a) and \[spectra\_SU2\](e), in the non-interacting regime we can outline three inequivalent classes of fillings that we dub low, intermediate and high. In the specific, we consider $N/L=3/16$, $3/8$ and $7/12$ for $\mathcal{I}=1/2$, and $N/L=1/4$, $13/24$ and $5/6$ for $\mathcal{I}=1$ corresponding to the low-, intermediate-, and high-filling cases respectively. OBC in the real dimension have been adopted. Momentum distribution functions ------------------------------- Let us first focus on the $\mathcal{I}=1/2$ case. In Figs. \[nkTOT\](a-c) we plot the momentum distribution function $\nu_p$ for the three fillings listed above. For $U/t=0$, the behavior of $\nu_p$ can be easily predicted by looking at the single-particle spectrum: in the low and high-filling cases peaks arise in correspondence of the partially occupied energy wells, while in the intermediate-filling case a more homogeneous momentum distribution function emerges. ![Momentum distribution functions $\nu_p$ for different values of the interaction coefficient. First row: $\mathcal{I}=1/2$; second row: $\mathcal{I}=1$. First column: low-filling case ($\eta=1$); second column: intermediate-filling case ($\eta=1$); last column: high-filling case ($\eta=2$). The various colors denote different $U/t$ values: 0 (black circles), 3 (brown squares), 5 (red diamonds), 8 (green triangles up), 20 (blue triangles down), $ U/t\to \infty$ (orange stars). []{data-label="nkTOT"}](Figure3.pdf){width="\linewidth"} The presence of repulsive atom-atom interactions significantly modifies the momentum distribution functions in the low- and high-filling cases: when $U/t$ is increased, they drive the distribution towards a more homogeneous shape with enhanced tails, a typical effect of interactions [@Giamarchi_2003]. On the contrary, in the intermediate-filling case the homogeneous behavior is unmodified, apart from the mentioned tails. Such a phenomenology is well explained using bosonization and renormalization-group techniques, as discussed in Ref. [@Braunecker_2010]. Interactions lead to an effective enhancement of the energy of the two gapped modes, whose presence characterizes the helical liquid. Effectively, the interacting system behaves as if $\Omega/t$ were renormalized and increased, thus enhancing the filling regimes for which an helical liquid can be expected. Furthermore, this is in agreement with the fact that the non-interacting helical liquid is essentially left unchanged by the interactions. Thus, provided the interaction is sufficiently strong, even low- and high-filling setups can be driven into an helical liquid. This is the first important result of our analysis: repulsive interactions enhance the gap protecting the helical liquid. The momentum distribution functions for $\mathcal{I}=1$ at the three cited fillings display the same qualitative behavior, see Figs. \[nkTOT\](d-f). Again, the underlying physics can be explained in terms of an effective enhancement of $\Omega/t$, due to the presence of interactions. Contrary to the previous case, for values of $\mathcal I$ larger than $1/2$, no analytical prediction is available, but it seems reasonable to believe that a similar behavior should occur. It is important to note that in the SRC regime on-site interactions are not expected to significantly modify the momentum distribution function of the non-interacting system. The occupied single-particle states belong only to the lowest band and are almost polarized in the same direction, $x$: the gas is thus quasi-spinless and an on-site interaction should only weakly alter the ground state because of Pauli exclusion principle. Additional numerical investigations may help in clarifying this issue. ![Spin-resolved momentum distribution functions $\nu_{p,m}$ for different values of $U/t$ in the WRC regime. First row: $\mathcal{I}=1/2$ (note that $\nu_{p,-1/2}=\nu_{-p,1/2}$); second and third row: $\mathcal{I}=1$ (note that $\nu_{p,-1}=\nu_{-p,1}$). Panels (a), (d) and (g): low-filling case; panels (b), (e) and (h): intermediate-filling case; panels (c), (f) and (i): high-filling case. For the color code, see the caption of Fig. \[nkTOT\]. []{data-label="srnk"}](Figure4b.pdf){width="\linewidth"} Further information about the system can be revealed by the spin-resolved momentum distribution functions $\nu_{p,m}$. In Figs. \[srnk\](a-c) we plot such functions in the WRC regime for the spin species $m=1/2$ and $\mathcal I=1/2$. Such profiles are clearly asymmetric with respect to $k_p=0$, indicating the helical nature of the ground state. Note that the asymmetry is enhanced by the interactions. A similar behavior is observed for $m = \pm 1$ and $\mathcal I=1$, see Figs. \[srnk\](d-f). On the other hand, for symmetry reasons, the momentum distribution function $\nu_{p,m=0}$ is symmetric with respect to $k_p=0$, although it is modified by the interactions, see Figs. \[srnk\](g-i). Chirality witnesses {#currents-sec} ------------------- ![Dependence of $Q_{m=\mathcal I}$ on the interaction strength. Panel (a): $Q_{1/2}$ for $\mathcal{I}=1/2$ as a function of the interaction strength $U/t$; dashed lines are the values of $Q_{1/2}$ in the limit $U/t\rightarrow\infty$. Panel (b): $Q_{1/2}$ for $\mathcal{I}=1/2$ in the non-interacting case ($U/t=0$) for different values of $\Omega/t$. Panels (c) and (d): same analysis for $\mathcal{I}=1$ and $m=1$. The various curves denote the different regimes of low (orange circles), intermediate (violet squares) and high (green diamonds) filling.[]{data-label="Q-fig"}](Figure5.pdf){width="\linewidth"} In this paragraph we discuss the properties of the chirality witnesses $Q_m$ and $J_m$ for an interacting system. Even though a preliminary analysis of these quantities has been carried out in Ref. [@Barbarino_2015], a systematic study of the effects of repulsive atom-atom interactions in a relevant experimental setup [@Mancini_2015] is still lacking. In Figs. \[Q-fig\](a) and \[Q-fig\](c) we display the behavior of $Q_{m=\mathcal I}$ as a function of $U$ for the cases $\mathcal{I}=1/2$ and $\mathcal I=1$; we focus again on the three fillings outlined above. In \[app:currents\] we show that, although the system has OBC and it is not homogeneous, averaging over many lattice sites yields a value related to the bulk current. A first striking observation is that one can observe different trends, also displaying non-monotonic features. The role of interactions in protecting the helical liquid here encounters a first naive confirmation: in all cases, the value of $|Q_m|$ in the $U/t \to \infty$ limit exceeds that of the non-interacting system. . Panel (b): same analysis for $\mathcal{I}=1$ and $m=1$.[]{data-label="Q-fig-nint"}](Figure6.pdf){width="\linewidth"} In order to understand the dependence of $Q_m$ on $U/t$, we employ an effective model. We have already noticed that the most prominent effect of the interactions on $\nu_p$ is that of letting the system behave as if it were non-interacting but with a renormalized value of $\Omega$. Here we test this observation by studying the dependence of $Q_m$ on $\Omega$ in the absence of interactions. Results displayed in Figs. \[Q-fig\](b) and \[Q-fig\](d) show that this simple model offers a good qualitative understanding of the interacting system. For example, in both the $\mathcal{I}=1/2$ and $\mathcal{I}=1$ cases, $Q_{m=\mathcal I}$ displays the same (quasi-)monotonic increasing behavior with $U/t$ and with $\Omega/t$, for the low and intermediate fillings. In the high-filling case, $Q_{m=\mathcal I}$ exhibits a strongly non-monotonic behavior as a function of $U$; in particular the plot points out a change in sign which is *a priori* unexpected because in the classical case the magnetic field determines unambiguously the direction of the circulating currents. To further elucidate this problem, in Fig. \[Q-fig-nint\] we plot the dependence of $Q_m$ on the filling $N/L$ for a fixed value of $\Omega/t$ and $U/t=0$. The plot shows that at low fillings the value of $Q_{m=\mathcal I}$ increases gently, but experiences an abrupt decrease once the helical region is entered, marked by the violet line (intermediate fillings). For higher fillings (even outside the helical region) and for small $\Omega$, the value of $Q_{m=\mathcal{I}}$ is negative and thus the current changes sign; however, by increasing $\Omega$, $Q_{m=\mathcal{I}}$ also increases, crossing $0$ and becoming positive and finite. It thus follows that in this system the chiral currents are not strictly speaking chiral and states with opposite current flow occur at accessible energies. ![Dependence of $J_{m=\mathcal I}$ on the interaction strength. Panel (a): $Q_{1/2}$ for $\mathcal{I}=1/2$ at low (orange circles), intermediate (violet squares) and high filling (green diamonds) as a function of the interaction strength $U/t$; dashed lines denote the values of $J_{1/2}$ in the limit $U/t\rightarrow\infty$. Panel (b): same analysis for $\mathcal{I}=1$ and $m=1$.[]{data-label="J-fig"}](Figure7.pdf){width="\linewidth"} The chirality witness $J_{m=\mathcal I}$ shares many similarities with $Q_{m=\mathcal I}$. In Fig. \[J-fig\] we plot $J_{m = \mathcal I}$ as a function of $U$, to be compared with Figs. \[Q-fig\](a) and \[Q-fig\](c) for $Q_{m=\mathcal I}$. Again, in the low- and intermediate-filling regimes $J_{m = \mathcal I}$ is almost monotonous, whereas monotonicity is significantly broken for high fillings. The explanation of this behavior can again be sought in the peculiar dependence of the current carried by the eigenmodes of the system. Conclusions =========== By means of DMRG simulations, we have studied the impact of atom-atom repulsive interactions on the quantum-Hall-like chiral currents recently detected in Refs. [@Mancini_2015; @Stuhl_2015]. We have modeled the experimental setup of Ref. [@Mancini_2015] and characterized the behavior of the edge currents through the asymmetry of the momentum distribution function. We have considered different particle fillings and identified the filling range where a chiral/helical liquid appears (in the text dubbed as “intermediate”). When the filling is slightly higher or lower, in the presence of repulsive interactions, the system starts behaving as the non-interacting chiral/helical liquid. This leads to the first conclusion that interactions stabilize such phase. To better assess its nature, we have introduced two chirality witnesses, which are displayed in Figs. \[Q-fig\] and \[J-fig\], where the chirality of the currents is studied as a function of the interaction strength $U/t$. As highlighted in the plots, the role of the interaction is non-trivial, and in the strongly-repulsive limit leads to the enhancement of the persistent currents. In the analysis presented here we have neglected the role of an harmonic trapping confinement as well as finite-temperature effects. Their interplay with interactions and the edge physics highlighted so far is left for a future work. The edge currents studied here do not have a topological origin. However, these synthetic ladders may support fractional quantum Hall-like states [@Barbarino_2015; @Cornfeld_2015], and it would be very interesting to understand how to explore this regime by means of the quantities discussed in the present paper. In particular it would be important to develop a complete characterization of how fractional quantization may emerge in a cold atomic setup. Acknowledgements {#acknowledgements .unnumbered} ================ We thank Leonardo Fallani and Guido Pagano for enlightening discussions, and S. Sinigardi for technical support. We acknowledge INFN-CNAF for providing us computational resources and support, and D. Cesini in particular. We acknowledge financial support from the EU integrated projects SIQS and QUIC, from Italian MIUR via PRIN Project 2010LLKJBX and FIRB project RBFR12NLNA. R.F acknowledges the Oxford Martin School for support and the Clarendon Laboratory for hospitality during the completion of the work. Currents {#app:currents} ======== The chirality witness $Q_m$ is the space-average value of the expectation value of the current operator over the ground state of the system, $\mathcal J_{j,m}$. Whereas in a homogeneous system with PBC this value coincides with the expectation value of the current on every site, the effects of the boundaries in a system with OBC might play an important role. ![Spatial profile of the spin-resolved currents $\mathcal J_{j,m}$. Panel (a): $\mathcal{I}=1/2$ (blue: $m=-1/2$; red: $m=1/2$). Panel (b): $\mathcal{I}=1$ (blue: $m=-1$; red: $m=0$; orange: $m=1$). In both cases, intermediate filling and $U/t=5$ were chosen. The color code refers to Fig. \[ladder\]. The other parameters of the simulations are set as in Sec. \[results\]. []{data-label="cur-fig"}](FigureA1.pdf){width="\linewidth"} In Fig. \[cur-fig\] we plot $\mathcal J_{j,m}$ both for a system with $\mathcal I=1/2$ \[panel (a)\] and with $\mathcal{I}=1$ \[panel (b)\]. 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--- abstract: 'We present two new constructions of quantum hash functions: the first based on expander graphs and the second based on extractor functions and estimate the amount of randomness that is needed to construct them. We also propose a keyed quantum hash function based on extractor function that can be used in quantum message authentication codes and assess its security in a limited attacker model.' author: - 'M. Ziatdinov' date: 'May 28, 2016' title: From Graphs to Keyed Quantum Hash Functions --- Introduction {#sec:introduction} ============ Quantum hash functions are similar to classical (cryptographic) hash functions and their security is guaranteed by physical laws. However, their construction and applications are not fully understood. Quantum hash functions were first implicitly introduced in @Buhrman2001 as quantum fingerprinting. Then @Gavinsky2010 noticed that quantum fingerprinting can be used as cryptoprimitive. However, binary quantum hash function are not very suitable if we need group operations (and group is not ${\mathbb{Z}}_{2^k}$. For example, several classical hash functions were proposed that use groups, e.g. by @Charles2009 and by @Tillich1994. @Ablayev2015 gave a definition and construction of non-binary quantum hash functions. @Ziatdinov2016 showed how to generalize quantum hashing to arbitrary finite groups. Recently, @Vasiliev2016 showed how quantum hash functions are connected with $\epsilon$-biased sets. Quantum hash functions map a classical message into a Hilbert space. Such space should be as small as possible, so eavesdropper can’t read a lot of information about classical message (this is guaranteed by physical laws as Holevo-Nayak’s theorem states). But images of different messages should be as far apart as possible, so recipient can check that hash differ or not with high probability. We measure this distance using an absolute value of scalar product of hashes of different messages. Informally speaking, to define a quantum hash function we need some random data. Then our input is mixed with this random data. Quantum parallelism allows us to do it in different subspaces simultaneously, so resulting hash is small. For example, random subsets suffice (for ${\mathbb{Z}}_m$) [@Ablayev2008], random codes suffice (for ${\mathbb{Z}}_2^n$) [@Buhrman2001], random automorphisms suffice (for any finite group) [@Ziatdinov2016]. @Vasiliev2015 used some heuristics to find best subsets of ${\mathbb{Z}}_m$. However, typically the amount of randomness that is needed to construct such quantum hash functions is large (about $O(\log^2 |G|)$). We reduce amount of randomness needed to define quantum hash function to $O(\log |G| \log \log |G|)$ in expander-based quantum hash function. Extractor-based quantum hash function allows us to introduce a notion of keyed quantum hash function. It can be used, for example, in quantum message authentication codes. Unlike [@Barnum2001] and [@Barnum2002] we use classical keys and authenticate classical messages. Unlike [@Curty2001] we authenticate whole messages, not single bits. However, our security analysis has only limited attacker. It is known that walk on expander graph gives results very similar to random sampling. We show that walks on expander graphs give a quantum hash functions in section \[sec:expander-qhf\]. Structure of these quantum hash functions is somewhat different from previous versions. Extractor is a generalization of expander graph. In the section \[sec:keyed-qhf\] we propose a keyed quantum hash function based on extractors and assess its security against limited attacker. #### Acknowledgements. I thank Farid Ablayev, Alexander Vasiliev and Marco Carmosino for helpful discussions. A part of this research was done while attending a Special Semester Program on Computational and Proof Complexity (April-June 2016) organized by Chebyshev Laboratory of St.Petersburg State University in cooperation with Skolkovo Institute of Science and Technology and Steklov Institute of Mathematics at St.Petersburg. Partially supported by Russian Foundation for Basic Research, Grants 14-07-00557, 15-37-21160. The work is performed according to the Russian Government Program of Competitive Growth of Kazan Federal University. Definitions =========== Let us recall some basic definitions. Statistics ---------- We use a standard definition of the statistical distance. We say that two distributions $F$ and $G$ are $\epsilon$-close, if for every event $A$, $|\Pr[F \in A] - \Pr[G \in A]| \le \epsilon$. The support of a distribution $X$ is ${\mathrm{Supp}}(X) = \{ x : \Pr[X = x] > 0 \}$. The uniform distribution over ${\{0,1\}}^m$ is denoted by $U_m$ and we say that $X$ is $\epsilon$-close to uniform if it is $\epsilon$-close to $U_m$. We denote that distribution $F$ is $\epsilon$-close to distribution $G$ by $F {\overset{\epsilon}{\approx}}G$. We also use a standard definition of the min-entropy. Let $X$ be a distribution. The min-entropy of $X$ is $H_\infty(X) = \min_{x \in {\mathrm{Supp}}(X)} \log \frac 1 {\Pr[X=x]}$. Quantum model of computation ---------------------------- We use the following model of computation. Recall that a qubit ${\left| \Psi \right\rangle}$ is a superposition of basis states ${\left| 0 \right\rangle}$ and ${\left| 1 \right\rangle}$, i.e. ${\left| \Psi \right\rangle} = \alpha{\left| 0 \right\rangle} + \beta{\left| 1 \right\rangle}$, where $\alpha, \beta \in {\mathbf{C}}$ and $|\alpha|^2 + |\beta|^2 = 1$. So, qubit ${\left| \Psi \right\rangle} \in {\mathcal{H}^2}$, where ${\mathcal{H}^2}$ is a two-dimensional Hilbert complex space. Let $s \ge 1$. We denote $2^s$-dimensional Hilbert complex space by ${({\mathcal{H}^2})^{\otimes s}}$: $${({\mathcal{H}^2})^{\otimes s}} = {\mathcal{H}^2}\otimes {\mathcal{H}^2}\otimes \ldots \otimes {\mathcal{H}^2}= \mathcal{H}^{2^s}$$ We denote a state ${\left| a_1 \right\rangle}{\left| a_2 \right\rangle}\ldots{\left| a_n \right\rangle}$, each $a_i \in {\{0,1\}}$, by ${\left| i \right\rangle}$, where $i$ is $\overline{a_1a_2\ldots a_n}$ in binary. For example, we denote ${\left| 1 \right\rangle}{\left| 1 \right\rangle}{\left| 0 \right\rangle}$ by ${\left| 6 \right\rangle}$. Usually it is clear, which space this state belongs to. Computation is done by multiplying a state by a unitary matrix: ${\left| \Psi_1 \right\rangle} = U {\left| \Psi_0 \right\rangle}$, where $U$ is a unitary matrix: $U^\dagger U = I$, $U^\dagger$ is the conjugate matrix and $I$ is the identity matrix. The density matrix of a mixed state $\{p_i, {\left| \psi_i \right\rangle}\}$ is a matrix $\rho = \sum_i p_i {\left| \psi_i \right\rangle}{\left\langle \psi_i \right|}$. A density matrix belongs to ${\mathrm{Hom}({({\mathcal{H}^2})^{\otimes s}},{({\mathcal{H}^2})^{\otimes s}})}$, the set of linear transformations from ${({\mathcal{H}^2})^{\otimes s}}$ to ${({\mathcal{H}^2})^{\otimes s}}$. At the end of computation state is measured by POVM (Positive Operator Valued Measure). A POVM on a ${({\mathcal{H}^2})^{\otimes s}}$ is a collection $\{E_i\}$ of positive semi-definite operators $E_i : {\mathrm{Hom}({({\mathcal{H}^2})^{\otimes m}},{({\mathcal{H}^2})^{\otimes m}})} \to {\mathrm{Hom}({({\mathcal{H}^2})^{\otimes m}},{({\mathcal{H}^2})^{\otimes m}})}$ that sums up to the identity transformation, i.e. $E_i \succeq 0$ and $\sum_i E_i = I$. Applying a POVM $\{E_i\}$ on a density matrix $\rho$ results in answer $i$ with probability $\operatorname{Tr}(E_i \rho)$. Character theory ---------------- Let $G$ be a group with unity $e$ and operation $\circ$. The character $\chi: G \to {\mathbb{C}}$ of the group $G$ is a homomorphism of $G$ to ${\mathbb{C}}$: for any $g, g' \in G$ it holds that $\chi(g \circ g') = \chi(g) \chi(g')$. Graphs ------ Let the graph $\Gamma = (V,E)$ with set of vertices $V$ and set of edges $E$ be fixed. Self-loops and multiple edges are allowed. Graph $\Gamma$ is the $d$-regular graph if all vertices have the same degree $d$; i.e. each vertex is incident to exactly $d$ edges. Adjacency matrix of the graph $A = A(\Gamma)$ is an $n \times n$ matrix whose $(u,v)$ entry is the number of edges between vertex $u$ and vertex $v$. Let $\lambda_1 \ge \lambda_2 \ge \ldots \ge \lambda_n$ be eigenvalues of matrix $A = A(\Gamma)$, i.e. for some $v_i$ it holds that $A v_i = \lambda_i v_i$. We refer to the eigenvalues of $A(\Gamma)$ as the spectrum of the graph $\Gamma$. Given a $d$-regular graph $\Gamma$ with $n$ vertices and spectrum $\lambda_1 \ge \lambda_2 \ge \ldots \ge \lambda_n$ we denote $\lambda(\Gamma) = \max\{ |\lambda_2|, |\lambda_n| \}$. We call the graph $\Gamma$ a $(d,\lambda)$-expander graph if $\Gamma$ is $d$-regular and has $\lambda(\Gamma) = \lambda$. Every expander graph can be converted to a bipartite expander graph. One can just take two copies of vertex sets and change original edges to go from one copy to another. Generalization of these bipartite expander graphs is extractor graphs. The extractor graph is a bipartite graph where size of components can be different. An extractor can also be defined in terms of function that maps pair of first component vertex and edge to second component vertex. A function $E: {\{0,1\}}^n \times {\{0,1\}}^d \to {\{0,1\}}^m$ is a $(k,\epsilon)$-extractor if for every distribution $X$ over ${\{0,1\}}^n$ with $H_\infty(X) \ge k$, $E(X,Y)$ is $\epsilon$-close to uniform (where $Y$ is distributed like $U_d$ and is independent of $X$). Sometimes we use extractor functions that map one (arbitrary) set to other: $E: G \times {\{0,1\}}^d \to H$. These functions can be thought of as bipartite graphs with vertices $(G,H)$. In this case we denote uniform distribution on $H$ by $U_H$. We also use extractors against quantum storage. Informally, their output is $\epsilon$-close to uniform and no quantum circuit operating on $b$ qubits can distinguish output from uniform. An $(n,b)$ quantum encoding is a collection $\{\rho(x)\}_{x \in {\{0,1\}}^n}$ of density matrices $\rho(x) \in {({\mathcal{H}^2})^{\otimes b}}$. A boolean test $T$ $\epsilon$-distinguishes a distribution $D_1$ from a distribution $D_2$ if $|\Pr_{x_1 \in D_1}[T(x_1) = 1] - \Pr_{x_2 \in D_2}[T(x_2) = 1]| \ge \epsilon$. We say $D_1$ is $\epsilon$-indistinguishable from $D_2$ if no boolean POVM can $\epsilon$-distinguish $D_1$ from $D_2$. A function $X: {\{0,1\}}^n \times {\{0,1\}}^d \to {\{0,1\}}^m$ is a $(k,b,\epsilon)$ strong extractor against quantum storage, if for any distribution $X \subseteq {\{0,1\}}^n$ with $H_\infty(X) \ge k$ and every $(n,b)$ quantum encoding $\{\rho(x)\}$, $U_t \circ E(X,U_t) \circ \rho(X)$ is $\epsilon$-indistinguishable from $U_{t+m} \circ \rho(X)$. Quantum hash functions {#sec:qhf} ====================== Informally, quantum hash function is a function that maps [*large*]{} classical input to a [*small*]{} quantum (hash) state such that two requirements are satisfied: (1) it is hard to restore input given the hash state and (2) it is easy to check with high probability that inputs for two quantum hash states are equal or different. It is easy to meet the first requirement for a constant hash size. One can simply take a qubit ${\left| \Psi(w) \right\rangle} = \alpha(w){\left| 0 \right\rangle} + \beta{\left| 1 \right\rangle}$ and encode the input in a fractional part of $\alpha$. But then the second requirement is not satisfied. It is easy to meet the second requirement for a hash size that is logarithmic in input size. One can simply map the input to the corresponding base state: ${\left| \Psi(i) \right\rangle} = {\left| i \right\rangle}$. However, then the first requirement is not satisfied. Let us give the formal definition. For $\delta \in (0, 1/2)$ we call a function $\psi : X \to {({\mathcal{H}^2})^{\otimes s}}$ a $\delta$-resistant function if for any pair $w,w'$ of different elements of $X$ their images are almost orthogonal: $$\label{eq:qhf-resistance} |{\langle \psi(w) | \psi(w') \rangle}| \le \delta.$$ We call a map $\psi: X \to {({\mathcal{H}^2})^{\otimes s}}$ an $\delta$-resistant $(K;s)$ quantum hash function if $\psi$ is a $\delta$-resistant function, and $\log|X| = K$. Quantum hash function maps inputs of length $K$ to (quantum) outputs of length $s$. If $K \gg s$ any attacker can’t get a lot of information by Holevo-Nayak theorem [@Nayak1999]. The equality of two hashes can be checked using, for example, well-known SWAP-test [@Gottesman2001]. All our hash functions have the following form: $$\label{eq:qhf-structure} {\left| \psi(g) \right\rangle} = \sum_{i=1}^t \chi(k_i(g)) {\left| i \right\rangle},$$ where $g$ is an element of some group $G$, $\{k_i, i=1,\ldots,t\}$, $k_i: G \to H$ is a set of mappings from group $G$ with operation $\circ$ to group $H$ with operation $\bullet$ and $\chi: H \to {\mathbb{C}}$ is a character of the group $H$. For example, the group $G$ can be thought of as $Z_{2^n}$ with group operation $+$, then elements of $G$ can be encoded as binary strings ${\{0,1\}}^n$ of length $n$ Why groups? {#sec:why-group} ----------- We use groups in quantum hash functions of form (\[eq:qhf-structure\]), not just arbitrary sets, because groups have nice structure. We can combine elements of group and we can inverse them. Several classical cryptoprimitives were proposed that use groups, e.g. by @Charles2009 and by @Tillich1994. Expanders for Quantum Hashing {#sec:expander-qhf} ============================= As noted in Section \[sec:introduction\], randomly chosen parameters with high probability lead to a quantum hash function. We replace this process with random walk on expander graph that is known to be close to uniform sampling. In this section we fix a group $G$ with group operation $\odot$ and unity $e$. Let $\Gamma = (V,E)$ be an extractor - i.e. $d$-regular graph with spectral gap $\lambda$. We label vertices $V$ of graph $\Gamma$ with messages (i.e. elements of group $G$). Let us randomly choose one vertex and perform a random walk of length $t$ starting from it. Denote vertices that occured in this walk by $s_j$. Parameter $t$ depend on security parameter $\epsilon$ of quantum hash function and we derive its value in theorem \[thm:t-expander\]. It is easy to note that such construction requires only $t d + \log |G|$ bits of randomness. Let us define the expander quantum hash function. The expander quantum hash function ${{\Psi}_{\Gamma,t}}(g)$ maps elements of $G$ to unitary transformations in $m$-dimensional Hilbert space ${({\mathcal{H}^2})^{\otimes m}}$: $${{\Psi}_{\Gamma,t}}(g) = \sum_{k=1}^t \chi(g \odot s_k) {\left| k \right\rangle}.$$ If we choose $\Gamma$ and $t$ appropriately, ${{\Psi}_{\Gamma,t}}$ is a quantum hash function. \[thm:t-expander\] For any $\delta \in (0; \frac 1 2)$ the function ${{\Psi}_{\Gamma,t}}$ is a $\delta$-resistant $(\log |G|;\log t)$ quantum hash function if $t > O(\frac{\log |G|}{\delta})$. Let us fix some $t$. $${\langle {\Psi}^\dagger(g) | {{\Psi}_{\Gamma,t}}(g') \rangle} = \sum_{k=1}^t {\langle \chi^*(g \odot s_k) | \chi(g' \odot s_k) \rangle} = |\sum_{k=1}^t \chi(s_k^{-1} \odot g^{-1} \odot g' \odot s_k)|.$$ Denoting $g'' = g^{-1} \odot g'$, we get $${\langle {\Psi}^\dagger(g) | {{\Psi}_{\Gamma,t}}(g') \rangle} = |\sum_{k=1}^t \chi(s_k^{-1} \odot g'' \odot s_k)|,$$ and $x_k = s_k^{-1} \odot g'' \odot s_k$ is also some random walk on graph $\Gamma$. Let $G$ be a weighted graph with eigenvalue gap $\epsilon = 1 - \lambda$ and non-uniformity $\nu$. Let random walk on $G$ starts in distribution $q$ and has stationary distribution $\pi$. Then Chernoff bound for expander graphs [@Gillman1993] states that for any positive integer $n$ and for any $\gamma > 0$: $$\label{eq:gillman-chernoff} \Pr\left[ \bigg| \sum_{i=1}^n f(x_i) - n {\mathbf{E}}_\pi f \bigg| \ge \gamma \right] \le 4 N_q \exp\left[-\bigg(\frac{\gamma}{||f||_\infty}\bigg)^2 \frac{\epsilon}{20n} \right].$$ Here we have graph weights $w_{ij} = \frac 1 d$ for all $i,j$ and $w_x=1$, thus $\nu = 1$ and $\pi(x) = \frac 1 V$. Initial distribution $q$ is uniform distribution over $G$, therefore $N_q = 1$. Function $f(x) = \chi(x)$ obviously has $||f||_\infty \le 1$. We also bound (\[eq:gillman-chernoff\]) with some small probability, e.g. $\frac 1 {|G|}$. Then (\[eq:gillman-chernoff\]) becomes $$\Pr \bigg[ \big| \sum_{i=1}^t f(x_i) - t {\mathbf{E}}_\pi f \big| \ge \gamma \bigg] \le 4\exp \bigg[ - \frac {\gamma^2 \epsilon} {20t} \bigg] \le \frac 1 {|G|}.$$ Solving with respect to $t$ gives us: $$t \ge \frac {20}{(1-\lambda) \delta} \ln(4 |G|) = O(\log |G|).$$ If we make a random walk of length $t = O(\log |G|)$, we will get a quantum hash function with high probability. So, construction of this quantum hash function requires only $O(\log |G|)$ bits of randomness if underlying expander graph is chosen carefully. For all $n$ and $\delta \in (0;\frac 1 2)$ there exist a $\delta$-resistant $(\log n;\log t + 1)$ quantum hash function with $t \ge \frac{160 \sqrt 2}{3 \delta} \ln(4 n)$. We use Margulis construction [@Hoory2006] of $(8;\frac{5 \sqrt 2}{8})$ expander graph with $n^2$ vertices and character of group ${\mathbb{Z}}_n^2$. Extractors for Quantum Hashing {#sec:extractor-qhf} ============================== Let ${\mathrm{Ext}}: G \times {\{0,1\}}^d \to H$ be a $(k;\epsilon)$ extractor function. Let $t$ and $s_i \in G, i \in \{1,\ldots,t\}$ be parameters. We choose them in Theorem \[thm:extractor-qhf\]. Denote $S = \{ s_i \}$. We define a quantum hash function ${\Psi}$ based on extractor ${\mathrm{Ext}}$ as follows. $${{\Psi}_{{\mathrm{Ext}},t,S}}(g) = \sum_{i=1}^t \sum_{j=1}^{2^d} \chi({\mathrm{Ext}}(g \circ s_i,j)) {\left| j \right\rangle} {\left| i \right\rangle}.$$ Intuitively, we start from several vertices and move along all incident edges simultaneously. Parameters $t$, $s_i$ depend on security parameter $\epsilon$. Let us choose it. \[thm:extractor-qhf\] If ${\mathrm{Ext}}$ is a $(k,\epsilon)$ extractor, parameter $t > \frac{\log |H| + 1}{2\epsilon^2} ||\chi||_\infty$ and $s_i$ are chosen according to distribution $X$ with $H_\infty(X) \ge k$, then ${\Psi}_{{\mathrm{Ext}}}$ is an $\epsilon$-resistant $(n;d + \log t)$ quantum hash function. It is sufficient to prove that for any $g' \neq g$ $$\begin{aligned} \bigg| {\langle {{\Psi}_{{\mathrm{Ext}},t,S}}(g) | {{\Psi}_{{\mathrm{Ext}},t,S}}(g') \rangle} \bigg| &= \bigg| \sum_{i=1}^t \sum_{j=1}^{2^d} \chi({\mathrm{Ext}}(g \circ s_i, j)^{-1} \bullet {\mathrm{Ext}}(g' \circ s_i, j) ) \bigg| \le \\ & \le \sum_{i=1}^t \sum_{j=1}^{2^d} | \chi({\mathrm{Ext}}(g \circ s_i, j)^{-1} \bullet {\mathrm{Ext}}(g' \circ s_i, j) ) | < \epsilon. \end{aligned}$$ Define $X_i$ to be a distribution of (random variable) $s_i$. Let $Y_i$ be a random variable ${\mathbf{E}}_{U_d}[|\chi({\mathrm{Ext}}(X_i,U_d))|]$. It is easy to see that $Y_i \le ||\chi||_\infty = 1$. Then by Hoeffding’s inequality: $$Pr \Bigg[ \bigg| \frac 1 t \sum_{i=1}^t Y_i - {\mathbf{E}}\Big[ \frac 1 t \sum_{i=1}^t Y_i \Big] \bigg| \ge \epsilon \Bigg] \le 2 \exp \bigg( - 2t\epsilon^2 \bigg).$$ Bounding this probability by $\frac 1 {|H|}$ and solving with respect to $t$ gives $$t \ge \frac{\log|H| + 1}{2\epsilon^2}.$$ Note that selecting parameters $S$ requires $O(\log |G| \times \log |H|)$ random bits. For every $\epsilon > 0$, $\alpha > 0$ and all positive integers $n, k$ there exist an $\epsilon$-resistant $(n; \log t + d + 1)$ quantum hash function, where $t \ge \frac{m+1}{2\epsilon^2}$, $d = O(\log n + \log (1 / \epsilon))$ and $m \ge (1-\alpha) k$. @Guruswami2009 proved that for every $\alpha > 0$ and all positive integers $n,k$ and all $\epsilon > 0$ there is an explicit construction of a $(k;\epsilon)$ extractor $E: {\{0,1\}}^n \times {\{0,1\}}^d \to {\{0,1\}}^m$ with $d = O(\log n + \log (1 / \epsilon))$ and $m \ge (1-\alpha) k$. Quantum hash function ${\Psi}_{E,t}$ is the required function. Keyed quantum hash functions {#sec:keyed-qhf} ============================ Classical message authentication codes (MAC) have wide range of applications. They are defined as a triple of algorithms: $G$ that generates a key, $S$ that uses the key and the message to generate a tag of the message, and $V$ that uses the key, the message and the tag to verify message integrity. Formally, $G : 1^n \to K$, where $n$ is a a security parameter and $K$ is a set of all possible keys, $S: K \times X \to T$, where $X$ is a set of messages and $T$ is a set of tags and $V: K \times X \times T \to \{\mathrm{Acc}, \mathrm{Rej}\}$. We require the following property for MAC to be a sound system: $$\label{eq:mac-sound} \forall n, \forall x \in X: k = G(1^n), V \big( k, x, S(k, x) \big) = \mathrm{Acc},$$ i.e. that verifier always accepts a generated tag. We also require that MAC is a secure system and for any adversary $A$ that can query MAC: $$\label{eq:mac-secure} \forall n, k \notin \mathrm{Query}(A), (x,t) \gets A(S), \Pr \big[ V(k, x, t) = \mathrm{Acc} \big] \le \mathrm{negl}(n),$$ i.e. any adversary that can query MAC outputs correct tag for some key that was not queried and some message with negligible probability. One classical construction of MAC is hash-based MAC (also known as keyed hash functions). Basically, keyed hash function is a function $H(k,x)$, such that $H(k,\cdot)$ is a cryptographic hash function for every $k$. It is easy to see that such function can be used as MAC. With the same considerations as in Section \[sec:qhf\], we define these algorithms to be the following. An $(\epsilon,\delta)$ keyed quantum hash function is a quantum function $S$, such that A function $S$ accepts a key $k \in K$ and a message $x \in X$ and outputs a quantum tag for $x$: $S: K \times X \to T = {({\mathcal{H}^2})^{\otimes t}}$. We require soundness, i.e. tags should be different for different messages under the same key. $$\forall k \in K, \forall x \in X, \forall y \neq x: {\langle S(k,x) | S(k,y) \rangle} < \epsilon.$$ For $x = y$ we get ${\langle S(k,x) | S(k,x) \rangle} = 1$. We also require unforgeability: $$\forall k \in K, k \notin \mathrm{Query}(A), (x,t) \gets A(S), \Pr \big[ {\langle t | S(k,x) \rangle} \ge \epsilon) \big] \le \delta,$$ where $A$ is arbitrary attacker that can query $S$ and $\mathrm(Query)(A)$ is a set of queries made. Informally, keyed quantum hash function outputs a tag for a message. If someone changes a message, then the verification step fails with high probability. If an attacker Eve can query a keyed quantum hash function, access to a function doesn’t help her to forge a tag for some message with some (unqueried) key. \[thm:extractor-keyed-qhf\] Let us define an extractor-based keyed quantum hash function as follows. Let ${\mathrm{Ext}}: {\{0,1\}}^n \times {\{0,1\}}^d \to {\{0,1\}}^m$ be a $(k,b,\epsilon)$ extractor against $b$ quantum storage and $b > r (d + \log t)$ . Then a function $${{\Psi}_{{\mathrm{Ext}}}}({\mathsf{key}},g) = \sum_{i=1}^t \sum_{j=1}^{2^d} \chi({\mathrm{Ext}}(g \circ {\mathsf{key}}\circ s_i,j)) {\left| j \right\rangle}{\left| i \right\rangle}$$ is a $(\epsilon; \epsilon + \epsilon^{2^s+1})$ keyed quantum hash function secure against an attacker $A$ with access to $r$ queries to ${\Psi}_{{\mathrm{Ext}}}$. We have to prove two claims. First, for any $k,x$ and $x' \neq x$, it holds that ${\langle {{\Psi}_{{\mathrm{Ext}}}}(k,x) | {{\Psi}_{{\mathrm{Ext}}}}(k,x') \rangle} < \epsilon$. Second, for any attacker $A$ and any $k \notin \mathrm{Query}(A)$ attacker output $x,t$ such that ${\langle t | {{\Psi}_{{\mathrm{Ext}}}}(k,x) \rangle} \ge \epsilon$ with negligible probability. The first claim is implied by Theorem \[thm:extractor-qhf\]. To prove the second claim we note that access to hash function doesn’t help attacker to output correct tag. Proof by contradiction. Suppose $A$ to be such attacker. Then we can distinguish between ${\mathrm{Ext}}(X,U_d)$ and $U_m$ using a $r (\log t + d)$ qubits. But $r (\log t + d) < b$ that contradicts the fact that ${\mathrm{Ext}}$ is an extractor against $b$ quantum storage. Then attacker should output the tag without access to hash function. This is equal to outputting a state that is close to correct tag. Then the probability of correct guessing $p$ is a ratio of the volume of sphere with radius $\epsilon$ to the volume of the whole space: $$p = \frac {c \epsilon^{2^s + 1}} {c (1 + \epsilon)^{2^s+1}} \le \epsilon^{2^s+1}.$$ For all positive integers $k,n$ and all $c > 0$ there exist a $(N^{-c}; 2 N^{-c})$ keyed quantum hash function. @De2009 proved that for every $\alpha, c > 0$ there exist an explicit $(\alpha N,b,N^{-c})$ extractor $E: {\{0,1\}}^N \times {\{0,1\}}^d \to {\{0,1\}}^m$ against $b$ quantum storage with $d = O(\log^4 n)$ and $m = \Omega (\alpha N - b)$. Open problems ============= Groups that we considered here and all constructions known to us use finite groups or sets and hash input strings of finite lengths. Can quantum hash functions be constructed for infinite groups? On the one hand, even one qubit can store arbitrary length binary string. On the other hand, the measurement of one qubit can’t result in more than one classical bit of information. And “dual” question about infinite strings. Can quantum hash functions work on infinite input strings (i.e. ${\{0,1\}}^*$)? This problem seems to be easier, but it probably requires careful analysis. Another interesting line of research would be improving keyed quantum hash function. Can keyed quantum hash function be secure against an attacker with unlimited number of queries?

--- abstract: 'We study how unique features of non-Hermitian lattice systems can be harnessed to improve Hamiltonian parameter estimation in a fully quantum setting. While the so-called non-Hermitian skin effect does not provide any distinct advantage, alternate effects yield dramatic enhancements. We show that certain asymmetric non-Hermitian tight-binding models with a $\mathbb{Z}_2$ symmetry yield a pronounced sensing advantage: the quantum Fisher information per photon increases exponentially with system size. We find that these advantages persist in regimes where non-Markovian and non-perturbative effects become important. Our setup is directly compatible with a variety of quantum optical and superconducting circuit platforms, and already yields strong enhancements with as few as three lattice sites.' author: - 'A. McDonald$^{1,2}$ and A. A. Clerk$^1$' bibliography: - 'NonHermitianSensing\_Bib.bib' title: 'Exponentially-enhanced quantum sensing with non-Hermitian lattice dynamics' --- Introduction ============ Quantum metrology and sensing aim to improve measurement precision over classical devices by exploiting uniquely quantum phenomena such as entanglement and squeezing [@Giovannetti2011; @DegenRMP; @RMP_Spins]. It is interesting to ask whether distinct effects associated with non-Hermitian dynamics can also be used to improve sensors operating in quantum regimes [@Langbein_2018; @Kero_Nat_Comm; @Liang_2019; @Liu2019; @Murch_2019]. In purely classical settings, mode degeneracies specific to non-Hermitian systems (so-called exceptional points) have been suggested as a means for enhanced parametric sensing [@Wiersig_2014]. Evidence for enhancement has been demonstrated in several classical-domain experiments involving small coupled mode systems (see e.g. Refs. ). Theory suggests that particular kinds of non-Hermitian effects could also be useful in truly quantum settings [@Kero_Nat_Comm]. To date, both theory and experiment have focused on non-Hermitian sensing schemes that utilize at most a few coupled modes. It is however well known that unusual new phenomena appear when considering genuinely multi-mode non-Hermitian dynamics. The paradigmatic example is the so-called “non-Hermitian skin effect" [@Zhong_PRL_2018_1; @Lee_2016; @Alexander_2018], which occurs in several non-Hermitian tight-binding models [@Xiong_2017; @Thomale_2019; @Udea_PRX_2019]. In these systems, all eigenvalues and wavefunctions of the Hamiltonian exhibit a dramatic sensitivity to a change of boundary conditions. This extreme sensitivity would seem to be a potentially powerful resource for parametric sensing [@Schomerus_2020]. ![ (a) Basic lattice sensor: two $N$-site non-Hermitian tight binding chains, each with opposite chirality. Each chain has asymmetric hopping: for the top (bottom) chain, hopping to the right is a factor of $e^{2A}$ larger (smaller) than hopping to the left. The two lattices are only coupled via a weak symmetry breaking perturbation $\epsilon$ on the rightmost site; the goal is to estimate $\epsilon$. A signal entering the top X chain induces an exponentially large output in the bottom P chain, but only if $\epsilon \neq 0$. (b) An array of bosonic cavities coupled via nearest neighbour hopping $w$ and coherent two-photon drive $\Delta$ with a small detuning $\epsilon$ on the last site. This provides a dissipation-free realization of the setup in (a), where the canonical quadratures $\hx$ and $\hp$ play the role of the top and bottom chains respectively. This system yields an exponentially enhanced SNR even when quantum noise effects are included. []{data-label="fig:Schematic"}](Model_3.pdf){width="45.00000%"} In this work, we show that non-Hermitian lattice dynamics does indeed provide a unique means for constructing enhanced sensors; moreover, this advantage persists even when operating in truly quantum regimes. We study in detail Hamiltonian parameter estimation using a one-dimensional lattice model with asymmetric tunneling (akin to the well-studied Hatano-Nelson model [@Hatano_Nelson]). We find, somewhat surprisingly, that the non-Hermitian skin effect does not provide any advantage over more traditional sensing protocols. Rather, we find another distinct non-Hermitian mechanism that enables a dramatic enhancement of measurement sensitivity: the quantum Fisher information per photon exhibits an exponential scaling with system size. As we discuss, the underlying mechanism makes use of both non-reciprocity and an unusual kind of symmetry breaking. While our ideas are general, our analysis focuses on a system that uses parametric driving to realize non-Hermitian dynamics; this has the strong advantage of not requiring any external dissipation or post-selection [@Alexander_2018; @Yuxin_2019]. Further, we ultimately focus on dispersive sensing, where the parameter of interest shifts the frequency of a resonant mode. This is a ubiquitous sensing strategy, with applications ranging from superconducting qubit measurement [@Circuit_QED_PRA] to virus detection [@Vollmer2008]. Our proposal is also compatible with a number of different experimental platforms in superconducting quantum circuits and quantum optics, and ultimately requires one to make a standard homodyne measurement. We also consider physics that goes beyond the usual limit of strictly infinitesimal parameter sensing. We find that the exponential enhancement of measurement sensitivity persists even when considering limitations associated with the finite propagation time of a large lattice. Even for parameters large enough to invalidate a full linear response analysis, we find that our scheme provides a strong advantage: it achieves a square-root enhancement of the sensitivity (including noise effects). This is similar to what is found in exceptional point sensors in the absence of noise [@Wiersig_2014]. Finally, while our discussion focuses on large lattices, the results we present are already interesting in a small system consisting of just three coupled resonators. Ingredients for a non-Hermitian lattice sensor ============================================== Amplified non-reciprocal response in the Hatano-Nelson model ------------------------------------------------------------ A key feature that we will exploit in our new sensor is the dramatically large and uni-directional response exhibited by certain non-Hermitian lattice models: perturbing a single lattice site induces a large change at one end of the chain, but not the other (see e.g. [@Schomerus_2020; @Nunnenkamp_2019]). We start by providing a physically-transparent explanation of this effect, based on interpreting non-Hermitian asymmetry in tight-binding matrix elements as directional gain and loss. The simplest relevant system is the well-known Hatano-Nelson model [@Hatano_Nelson; @Hatano_Nelson_2]. This is a 1D tight-binding chain with asymmetric nearest-neighbour hoppings, $\hat{H} = i J \sum_n \left( e^A \ketbra{n+1}{n} - e^{-A} \ketbra{n}{n+1} \right) $, where $J, A$ are real and $\ket{n}$ is a position eigenket. The corresponding single-particle Schrödinger equation is ($\hbar = 1$ throughout) $$\begin{aligned} \label{eq:Hatano-Nelson} \dot{\psi}_n = J e^{A} \psi_{n-1} - J e^{-A} \psi_{n+1}, \end{aligned}$$ where $\psi_n = \braket{n}{\psi}$. While $A$ formally plays the role of an imaginary vector potential, it is more usefully thought of as an amplification factor. Assuming $A$ is positive for definiteness, Eq. (\[eq:Hatano-Nelson\]) describes a system where a wavefunction’s amplitude grows by $e^{A}$ every time a particle hops one site to the right, and decays an equal amount $e^{-A}$ as it travels to the left, regardless of its energy. With this picture in mind, the form of the real-space susceptibility (i.e. single particle Green’s function) $\chi(n,m;t)$ for a finite open chain has an intuitive form. Letting $\ket{m(t)} = e^{-i \hat{H} t} \ket{m}$, a simple calculation yields (see App. \[app:chi\_quadrature\]): $$\begin{aligned} \chi(n,m;t) & \equiv \braket{n}{m(t)} \label{eq:Susceptibility} = e^{A(n-m)} \chi_0(n,m;t). \end{aligned}$$ Here, $\chi_0(n,m;t)$ is the susceptibility matrix when $A=0$, i.e. the Green’s function of a Hermitian tight-binding chain. This quantity is reciprocal, in the sense that $\chi_0(n,m;t) = (-1)^{m-n}\chi_0(m,n;t)$ (i.e. apart from a phase, there is no asymmetry in rightwards versus leftwards propagation). The Green’s function $\chi_0(n,m;t)$ both describes how particles propagate in the lattice, and also the response properties of the system (i.e. if you perturb site $m$ at $t=0$, how does site $n$ respond at some later time?). The simple factorization in Eq. (\[eq:Susceptibility\]) makes it clear that there are two basic processes determining the response. The first is a distance and direction-dependent amplification / deamplification factor, whereas the second encodes the dynamics of the underlying ($A=0$) Hermitian tight-binding model. We thus have a simple intuitive picture for the susceptibility, without having to make recourse to other seemingly more complicated non-Hermitian features, such as exceptional points, the non-Hermitian skin effect, or the Petermann factor [@Petermann_1979; @Grangier_1998]. Note that Eq. (\[eq:Susceptibility\]) can be easily derived via a similarity transformation, which is analogous to the gauge transformation one would make if $A$ were imaginary (and hence a real synthetic gauge field) [@Hatano_Nelson; @Hatano_Nelson_2]. $\mathbb{Z}_2$ symmetry in non-Hermitian lattice models ------------------------------------------------------- The second basic ingredient we will exploit in constructing our sensor is symmetry breaking. The Hatano-Nelson chain breaks reciprocity for any $A \neq 0$; formally, it picks a preferred amplification direction, and does not remain invariant (up to a local gauge change) under a spatial inversion operation $\ket{n} \rightarrow \ket{-n}$. We can trivially restore this symmetry by considering a system with [*two*]{} uncoupled Hatano-Nelson chains indexed by $\sigma = \uparrow, \downarrow$ with amplification factors $A_{\uparrow}, A_{\downarrow}$. If we pick $A_{\uparrow} = -A_{\downarrow}$, then the composite system restores some of the lost symmetry. Formally, the two-chain system is invariant up to a local gauge change under the combined operations $\ket{n} \rightarrow \ket{-n}$ (spatial inversion) and $\sigma \rightarrow \bar{\sigma} $ (pseudospin inversion). While this may seem trivial, this kind of discrete symmetry can persist even for certain forms of interchain coupling, and has recently been interpreted as a formal $\mathbb{Z}_2$ symmetry class with its own distinct non-Hermitian topological phenomena [@Sato2020]. We discuss this symmetry more formally in Appendix \[app:Symmetry\]. For our purposes, the interesting feature here will be to consider [*breaking*]{} this symmetry with an external perturbation whose magnitude we wish to estimate. As we will see, the response to this symmetry breaking can be exponentially large in system size, enabling a new kind of sensor. Model and Measurement Protocol ============================== With the motivation of the previous section, we now consider a sensor comprised of two Hatano-Nelson chains with an opposite chirality (see Fig. \[fig:Schematic\](a)). There are a variety of means for such realizing non-Hermitian directional tight-binding models using dissipation [@Metelmann2015; @Metelmann2018]; approaches based on feedback control are also possible and have been recently implemented [@Coulais2019]. However, for optimal sensing properties in quantum settings, methods that are both autonomous and avoid the noise associated with dissipation are desirable. We thus focus on a dissipation-free method for realizing non-Hermitian dynamics based on parametric driving [@Alexander_2018; @Yuxin_2019]. We stress that the response properties of our sensor will be independent of how the non-Hermitian dynamics is implemented, and hence apply equally well to dissipative and feedback based strategies. We consider an $N$-site chain of driven, coupled bosonic modes described by the fully Hermitian Hamiltonian $$\begin{aligned} \label{eq:HBKC} \hH_B = \sum_{n =1 }^{N-1} \left( i w \ha^{\dagger}_{n+1} \ha_n + i \Delta \ha^{\dagger}_{n+1} \ha_{n}^\dagger + h.c. \right). \end{aligned}$$ Here $\hat{a}_j$ is the photon annihilation operator on site $j$, $w$ is the nearest-neighbour hopping term, $\Delta$ is the nearest-neighbour two-photon drive, and we consider open boundary conditions. We take both $w$ and $\Delta$ to be positive and $w > \Delta$. This model describes a 1D cavity array subject to parametric drives on each bond (described in a rotating frame set by the external pump frequency). As discussed extensively in Ref. , this system could be realized in both quantum superconducting circuits or nonlinear quantum optical systems. Note the lack of any on-site terms corresponds to the parametric driving frequency matching the resonance frequency of each isolated cavity. Although not immediately obvious, the dynamics generated by $\hH_B$ corresponds to two copies of the Hatano-Nelson model. In the basis of local canonical quadrature operators $\hx_j$ and $\hp_j$, defined via $\ha_j = (\hx_j+i\hp_j)/\sqrt{2}$, the Hamiltonian reads $$\begin{aligned} \label{eq:H_B_xp} \hH_B = \sum_{n=1}^{N-1} \left( -(w-\Delta) \hx_{n+1} \hp_n + (w+\Delta) \hp_{n+1} \hx_n \right). \end{aligned}$$ This then yields the Heisenberg equations of motion $$\begin{aligned} \label{eq:X_EOM} \dot{\hx}_n & = J e^{A} \hx_{n-1} - J e^{-A} \hx_{n+1},\\ \label{eq:P_EOM} \dot{\hp}_n & = J e^{-A} \hp_{n-1} - J e^{A} \hp_{n+1}, \end{aligned}$$ where the effective hopping amplitude $J$ and imaginary vector potential $A$ are related to $w$ and $\Delta$ by $$\begin{aligned} \label{eq:DefJ} &J = \sqrt{w^2-\Delta^2}, \\ \label{eq:DefA} &e^{2A} = \frac{w+\Delta}{w-\Delta}. \end{aligned}$$ Comparing against Eq. (\[eq:Hatano-Nelson\]), we see that the dynamics of each canonical quadrature corresponds to that of a Hatano Nelson model, with opposite chiralities for $\hat{x}$ and $\hat{p}$ (Fig. \[fig:Schematic\]). These orthogonal quadratures correspond to different phases of photonic excitations, and hence the system exhibits phase-dependent non-reciprocal amplification [@Alexander_2018]. Note that there is a constraint on our mapping: the complex wavefunction amplitudes in the Hatano-Nelson model have been replaced by Hermitian quadrature operators in our system. This will play no role in what follows. We now demonstrate how this setup can be used for Hamiltonian parameter estimation. We add a Hermitian perturbation $\epsilon \hat{V}$ to our Hamiltonian where $\hat{V}$ is some system operator; the goal is to estimate $\epsilon$. We also couple the first site of our lattice to an input-output waveguide as a means to probe its properties. The simplest protocol is to use this waveguide to drive the system with a classical tone (i.e. a coherent state), and then measure the outgoing light in the waveguide (see Fig. \[fig:Schematic\](b)). The full Hamiltonian becomes $$\begin{aligned} \hH[\epsilon] = \hH_B + \epsilon \hat{V} + \hH_\kappa -i\sqrt{\kappa} \left( \ha^\dagger_1 \beta - h.c. \right) \end{aligned}$$ $\hH_\kappa$ describes damping of the first site at a rate $\kappa$, due to coupling to the modes of the waveguide which we treat using standard input-output theory [@RMP_Clerk]. The last term corresponds to a classical drive with amplitude $\beta = |\beta| e^{i \theta}$. Note that we take the drive frequency to match the resonance frequency of the isolated cavities; this frequency is zero in our rotating frame. Using the standard input-output boundary condition, the output field in the waveguide is given by $$\begin{aligned} \label{eq:In_Out} \hB^{\rm (out)}(t) = \left( \beta + \hB^{\rm (in)}(t) \right) +\sqrt{\kappa} \ha_1(t) \end{aligned}$$ where $\hB^{\rm (in)}$, the operator equivalent of Gaussian white noise, describes the noise entering the lattice through the waveguide. Our goal is to estimate $\epsilon$ by making an optimal measurement of the output field. In what follows, we take $\epsilon$ to have units of frequency and $\hat{V}$ to be dimensionless. We further specialize to the usual case where $\epsilon$ is so small that it can only be estimated by integrating the output field over a long timescale $\tau$. If we turn on the drive tone at $t=0$, the relevant temporal mode of the output field to consider is $$\begin{aligned} \hfancyB_\tau(N) = \frac{1}{\sqrt{\tau}} \int_0^{\tau} dt \hB^{\rm (out)} (t) \end{aligned}$$ Note that this is normalized to be a canonical bosonic lowering operator, satisfying $[\hfancyB_\tau(N),\hfancyB_\tau^\dagger(N)] = 1$. We write an explicit dependence on the chain size $N$, as we will be interested in understanding how things scale as $N$ is increased. The maximum amount of information available in $\hfancyB_\tau(N)$ on $\epsilon$ is quantified by the quantum Fisher information (QFI). The QFI provides a lower bound on the root mean square error of any (unbiased) estimate of $\epsilon$ regardless of how $\hfancyB_{\tau}(N)$ is measured [@Giovannetti2011]. Calculation of the QFI unfortunately does not in general tell one the form of the optimal measurement. However, in our linear Gaussian system, things are much simpler: for large $| \beta|$, the optimal measurement will always correspond to a standard homodyne measurement [@Pirandola_2015; @Braun_2013]. The relevant Hermitian measurement operator has the form $$\begin{aligned} \label{eq:Meas_Op} \hM_\tau(N) = \frac{1}{\sqrt{2}} \left( e^{-i \phi} \hfancyB_\tau(N) + e^{i \phi} \hfancyB_\tau^\dagger(N) \right), \end{aligned}$$ i.e. a quadrature of the output operator $\hfancyB_\tau(N)$ along a direction in phase space determined by the angle $\phi$. We will focus throughout on the large-drive limit, and will be interested in characterizing the QFI to leading order in $|\beta|$. In this limit, QFI is determined by the statistics of $\hM_\tau(N)$ via [@Kero_Nat_Comm; @Pirandola_2015] $$\begin{aligned} \label{eq:QFI} \mathrm{QFI}_{\tau}(N) &= \max_{\phi} \left[ \lim_{\epsilon \to 0} \left( \frac{1}{\epsilon} \frac { \mathcal{S}_\tau(N, \epsilon) } { \mathcal{N}_{\tau}(N, \epsilon) } \right)^2 \right], \end{aligned}$$ where $$\begin{aligned} & \mathcal{S}_{\tau}(N, \epsilon) = | \langle \hM_\tau(N) \rangle_{\epsilon} - \langle \hM_\tau(N) \rangle_{0} |, \label{eq:SignalDef} \\ & \mathcal{N}_{\tau}(N, \epsilon) = \sqrt{ \langle \hM^2_\tau(N) \rangle_\epsilon - \langle \hM_\tau(N) \rangle^2_\epsilon, } \end{aligned}$$ are the signal and the noise respectively associated with the measurement. Here, $\langle \cdot \rangle_z$ means an average with respect to a state whose dynamics are governed by $\hH[z]$. This expression for the QFI coincides with the SNR of an optimal homodyne measurement, and scales as $| \beta |^2$; the next-leading order term is independent of $|\beta|$. Note that the QFI only depends on the noise $\mathcal{N}_{\tau}(N, \epsilon)$ calculated to zeroth order in $\epsilon$. We stress that the expression for the QFI still depends on the drive phase $\theta$ as well as the form of the operator $\hat{V}$; in what follows, we will be interested in optimizing these as well. Given its role as a fundamental performance metric, it is tempting to declare that a better sensor has been built if it increases the QFI. Different measurement strategies however use resources differently, and one must carefully consider which to constrain when making comparisons. In our case, we wish to distinguish a true sensing enhancement from a more trivial effect, where a different protocol simply results in there being more photons in the system available to interact with the perturbation $\hat{V}$ (as occurs with standard exceptional-point based sensing schemes [@Kero_Nat_Comm; @Liang_2019]). For this reason, we will take as the relevant metric the QFI scaled by the [*total*]{} average photon number $\bar{n}_{\rm tot}$ [@Kero_Nat_Comm]: $$\begin{aligned} \bar{n}_{\rm tot} \equiv \sum_n \langle \ha_n^\dagger \ha_n \rangle_0 \simeq \sum_n \langle \ha_n^\dagger \rangle_0 \langle \ha_n \rangle_0 \propto |\beta|^2/\kappa. \end{aligned}$$ As we consider throughout the large-drive limit, we only keep the leading-order-in-$\beta$ contribution to $\bar{n}_{\rm tot}$. This is simply the photon number associated with the drive-induced displacement of each cavity annihilation operator. The additional contribution to $\bar{n}_{\rm tot}$ due to amplification of vacuum fluctuations is $\beta$-independent, hence plays no role in the large-drive limit we consider (see Appendix \[app:n\_bar\]). {width="100.00000%"} Exponential SNR and QFI enhancement {#sec:ExpEnhancement} =================================== We now focus on computing the optimal SNR of the measurement operator $\hM_\tau(N)$ for our $N$ site chain in the $\epsilon \rightarrow 0$ limit; via Eq. (\[eq:QFI\]), this directly yields the QFI. In this limit, a SNR $\sim 1$ will only be achieved for $\tau$ much longer than any internal dynamical timescale. We thus consider the long-$\tau$ limit, effectively ignoring any transient behaviour and assuming the system is in its steady state. Note that our system is dynamically stable as long as $w > \Delta$ and $\kappa>0$, ensuring that a steady state exists. From Eqs. (\[eq:SignalDef\]),(\[eq:Meas\_Op\]) and (\[eq:In\_Out\]), the first order in $\epsilon$ in this limit reads $$\begin{aligned} \label{eq:Signal_First_Order} \mathcal{S}_\tau(N,\epsilon) & = \sqrt{2\kappa \tau} \big\lvert \Re [ e^{-i \phi} \delta \langle \ha_1 \rangle^{\rm ss} ] \big\rvert \end{aligned}$$ where $$\delta \langle \ha_1 \rangle^{\rm ss} \equiv \epsilon \lim_{\epsilon \rightarrow 0} \left( \frac{\langle \ha_1 \rangle^{\rm ss}_\epsilon - \langle \ha_1 \rangle^{\rm ss}_0} {\epsilon} \right)$$ is the steady state linear response of the site-$1$ average amplitude to a non-zero $\epsilon$. This response will be determined by the zero-frequency susceptibilities (Green’s functions) of the unperturbed system. It will be convenient to split up $\delta \langle \ha_1 \rangle^{\rm ss}_\epsilon$ into its real and imaginary parts, or equivalently to think of the dynamics in the quadrature picture. There are then four different types of susceptibilities: $\chi^{\alpha \beta}[n,m;\omega]$ is the response of the $\alpha$ quadrature on site $n$ to a force which directly drives the $\beta$ quadrature on site $m$. From Eqs. (\[eq:X\_EOM\])-(\[eq:P\_EOM\]) and Eq.(\[eq:Susceptibility\]), we find that the $\epsilon=0$ susceptibilities are $$\begin{aligned} \chi^{xx}[n,m;\omega] \label{eq:chi_x} &= e^{A(n-m)} \tilde{\chi}^{xx}[n,m;\omega], \\ \label{eq:chi_p} \chi^{pp}[n,m;\omega] &= e^{-A(n-m)} \tilde{\chi}^{pp}[n,m;\omega], \\ \label{eq:Off_Diag} \chi^{xp}[n,m;\omega] &= \chi^{px}[n,m;\omega] = 0. \end{aligned}$$ Here $\tilde{\chi}^{\alpha \beta}[n,m;\omega]$ the susceptibility of a Hermitian $N$ site tight-binding chain with hopping $i J$ and amplitude decay rate $\kappa/2$ on the first site (see Appendix \[app:chi\_particle\]). The above structure reflects the fact that the dynamics of the $\hx$ and $\hp$ quadratures correspond to two uncoupled copies of the Hatano-Nelson chain with opposite signed imaginary vector potential $A$. Hence, $\hx$ quadrature signals are amplified as they propagate to the right, and deamplified as they traverse to the left, while the opposite is true for $\hp$ quadrature signals. Note that if we started with two explicit Hatano-Nelson chains, the discussion here would be identical; $x$ and $p$ would then just index the two different chains. To proceed, we need to specify the form of the perturbation Hamiltonian $\hat{V}$. Our system exhibits the the non-Hermitian skin effect (NHSE), implying a strong sensitivity to changes in boundary conditions. As the unperturbed system is an open chain, this suggests that an optimal $\hat{V}$ would induce tunneling between the first and last site, i.e. $$\begin{aligned} \hat{V}_{\rm NHSE} = e^{i \varphi}\ha^\dagger_1 \ha_N + e^{-i \varphi }\ha^\dagger_N \ha_1, \end{aligned}$$ with $\varphi$ an arbitrary phase. As we show in Appendix \[app:SNR\], this choice of $\hat{V}$ does not result in an enhanced sensitivity if one uses the proper metric of QFI/$\bar{n}_{\rm tot}$ (or equivalently SNR/$\sqrt{\bar{n}_{\rm tot}}$). While the signal produced by $\hat{V}_{\rm NHSE}$ is large, this is simply because our system is an amplifier with a large end-to-end gain. The number of photons on the last site (and hence $\bar{n}_{\rm tot}$) will be amplified equally by this gain. As a result, QFI/$\bar{n}_{\rm tot}$ does not show any enhancement as one increases the system size $N$, nor any enhancement over a conventional, single-cavity dispersive detector. We are thus left with a depressing conclusion: the non-Hermitian skin effect does not provide any true advantage in sensing. Note also that $\hat{V}_{\rm NHSE}$ does not break the $\mathbb{Z}_2$ symmetry of the unperturbed system (see App. \[app:Symmetry\]). Luckily, this is not the end of the story. Enhanced sensing is possible with our system, if we chose a $\hat{V}$ that fully exploits the opposite chiralities of our two (effective) Hatano-Nelson chains. Consider the innocuous-looking purely local perturbation $$\begin{aligned} \hat{V}_{N} = \ha_N^\dagger \ha_N, \end{aligned}$$ so that $\epsilon$ now corresponds to a small change in the resonance frequency of the last site. This perturbation does indeed break the $\mathbb{Z}_2$ symmetry of the unperturbed system. To understand how $\hat{V}_{N}$ affects the dynamics of the lattice, it is best to re-examine the equations of motion in the $\hx$ and $\hp$ basis. They remain the same everywhere except the last site $N$, where they now read $$\begin{aligned} \label{eq:XPerturbation} &\dot{\hx}_N = J e^{A} \hx_{N-1} + \epsilon \hp_{N}, \\ \label{eq:PPerturbation} &\dot{\hp}_N = Je^{-A} \hp_{N-1}-\epsilon \hx_N. \end{aligned}$$ Recall that without the perturbation present, the dynamics of the $\hx$ and $\hp$ quadratures are completely independent (c.f. Eqs. (\[eq:X\_EOM\]) and (\[eq:P\_EOM\])). The dispersive shift $\epsilon$ on site $N$ now effectively couples the two non-Hermitian chains, thereby breaking phase-dependent non-reciprocity (see Fig. \[fig:Schematic\]). While the intuitive picture of directional amplification remains unchanged in the rest of the lattice, a wavepacket with a well defined global phase can now scatter off of the perturbation $\epsilon$ and change its phase in the process. The role of $\epsilon$ is reminiscent to that of a magnetic impurity in the quantum spin Hall effect: in both cases the propagation direction of a particle is determined by some internal degree of freedom, which the impurity can change [@Spin_Hall_Review]. We next judiciously choose the phase of the drive $\beta$ to be real and the measurement angle $\phi = \pi/2$. Equivalently, we apply a driving force $-\sqrt{2\kappa} |\beta|$ to $\hx_1$ and consider the corresponding response of its canonically conjugate quadrature $\hp_1$. When $\epsilon = 0$, this off-diagonal susceptibility vanishes, see Eq.(\[eq:Off\_Diag\]). To first order in $\epsilon$, it becomes non-zero. We further take $N$ to be odd in what follows, as this guarantees (via the chiral symmetry of our unperturbed system) that the lattice will have a resonant mode at zero frequency. This then provides a further resonant enhancement of our system’s zero frequency response properties. Note that for an even $N$, we would still have the same exponential enhancement quoted in Eqs. (\[eq:chi\_x\])-(\[eq:chi\_p\]); in this case however, there is no resonant mode at zero frequency, causing a suppression of susceptibilities by a multiplicative factor of $\kappa/(2J)$ (see Eqs.(\[eq:chi\_n1\])-(\[eq:chi\_1n\])). With these optimized choices, first order perturbation theory yields: $$\begin{aligned} \nonumber \mathcal{S}_\tau(N,\epsilon) & = \sqrt{\kappa \tau} |\sqrt{2 \kappa} \beta| \Big( |\delta \chi^{px}[1,1;0]| \Big) \\ \nonumber & = \sqrt{2\kappa \tau}\sqrt{\kappa}|\beta| \Big( |\chi^{pp}[1,N;0]\epsilon \chi^{xx}[N,1;0]| \Big) \\ \label{eq:Large_SNR0} & = \sqrt{8\kappa \tau \bar{n}_{N}} \left| \frac{\epsilon}{\kappa} \right| e^{A(N-1)}. \end{aligned}$$ Here $\bar{n}_N$ denotes the leading-order-in-$\beta$ average photon number of the last site in the lattice, and is given by: $$\bar{n}_N = |\langle \ha_N \rangle^{ss}_0|^2 = \kappa|\beta|^2 |\chi^{xx}[N,1,0]|^2 \propto e^{2A(N-1)}$$ For large $A$, the average photon number on site $N$ is exponentially larger than that on other sites. Writing $\bar{n}_N = Z(A) \bar{n}_{\rm tot}$ we have $ Z(A) = 1 - \mathcal{O}(e^{-4A})$ (see App. \[app:n\_bar\]). We thus obtain: $$\mathcal{S}_\tau(N,\epsilon) = \sqrt{8 Z(A) \kappa \tau \bar{n}_{\rm tot}} \left| \frac{\epsilon}{\kappa} \right| e^{A(N-1)}. \label{eq:Large_SNR}$$ Eq. (\[eq:Large\_SNR\]) is a central result of this work: it shows that even when the total photon number $\bar{n}_{\rm tot}$ is held fixed, our system exhibits a signal power that grows exponentially with system size. For an intuitive picture, consider the propagation of $x$-quadrature photons injected from the waveguide into site $1$, as depicted in Fig. \[fig:Protocol\]. These photons will propagate to the last site $N$, with an amplitude $\chi^{xx}[N,1;\omega] \propto e^{A(N-1)}$. Photons that then scatter off the perturbation $\epsilon \hat{V}_N$ will change phase, so that they now correspond to the $p$ quadrature (c.f. Eq. (\[eq:PPerturbation\])). They can then propagate back to the first lattice site with an amplitude $-\epsilon\chi^{pp}[1,N;\omega] \propto e^{A(N-1)}$. This simple scattering process (involving both $x$ and $p$ quadrature propagation) leads to a parametrically large signal in $\hp_1$. The above heuristic picture also explains why the signal is amplified more than the average photon number $\bar{n}_{\rm tot}$: the average photon number only involves amplification along one traversal of the chain, whereas the signal magnitude involves two traversals (forward and back). This directly explains the extra large factor of $e^{A(N-1)}$ in Eq. (\[eq:Large\_SNR\]). We stress that this exponential signal enhancement would also occur in dissipative realizations of our doubled Hatano-Nelson chain. The final step in characterizing our sensor is to examine its noise properties. Naively, one might expect that the same dynamics responsible for our signal enhancement would also exponentially amplify fluctuations in the output field. This is not the case: as already discussed, calculating the QFI only requires computing the noise to zeroth order in $\epsilon$, see Eq.(\[eq:QFI\]). Without the perturbation, the two effective Hatano-Nelson chains are completely decoupled. Thus, any noise entering through the waveguide will undergo equal amounts of amplification and deamplification before exiting the lattice. For the ideal case of zero internal loss, this means that the noise temperature of the output field will be identical to that of the input field. As a result, the noise in the homodyne current is $$\begin{aligned} \label{eq:Noise} \mathcal{N}_{\tau}(N, 0) = \sqrt{\bar{n}_{\rm th}+\frac{1}{2}} \end{aligned}$$ with $\bar{n}_{\rm th}$ representing the number of thermal quanta in the input field. Combining these two results, our signal-to-noise ratio is $$\begin{aligned} \nonumber \text{SNR}_\tau(N,\epsilon) &= 4 \sqrt{\frac{Z(A) \bar{n}_{\rm tot}\kappa \tau}{2\bar{n}_{\rm th}+1}} |\frac{\epsilon}{\kappa}| e^{A(N-1)} \\ \label{eq:LargeSNR} &= \sqrt{Z(A)} e^{A(N-1)} \, \text{SNR}_\tau(1,\epsilon), \end{aligned}$$ where $\text{SNR}_{\tau}(1,\epsilon)$ is the signal-to-noise ratio of a ubiquitous single-mode dispersive detector [@RMP_Clerk; @Ben_PRX]. As we have stressed, $\text{SNR}_\tau(N,\epsilon)$ also represents the QFI of our system. We see that the SNR and QFI can be exponentially enhanced by either increasing system size $N$ or amplification factor $A$, while all the while maintaining a [*fixed*]{} total photon number $\bar{n}_{\rm tot}$. This is the central result of our work. The crucial ingredients here are the inherent chiral amplification present in a Hatano-Nelson chain, the effective symmetry breaking that occurs when coupling the two opposite-chirality chains in our sensor, and the lack of any amplified output noise in the unperturbed system. Several comments are in order. First, note that the large SNR achieved here is not contingent on approaching a parametric instability: our system is dynamically stable for any value of $\epsilon$ and $A$ (see Appendix \[app:Non-Pert\]). Second, the mechanism we discuss here is useful even in small systems, as the fixed photon number QFI has an exponential dependence on $A$; an arbitrarily large QFI can thus be achieved with only three lattice sites. We further emphasize that the spatially-dependent amplification is a crucial aspect of our scheme. Indeed, the signal-to-noise ratio for a single-mode cavity amplifier can never achieve this sort of sensing enhancement, since the signal and noise are amplified in a similar manner [@Ben_PRX]. Finally, we stress that this enhanced QFI in no way requires or is even related to the existence of an exceptional point in our dynamical matrix. It is also worth stressing that our mechanism is completely distinct from other recently introduced methods that use parametric amplifiers to enhance dispersive sensing [@Terhal_2014; @Ben_PRX; @Florian_PRL_2015]. These works exploit noise squeezing as the basic mechanism for enhancing the SNR and QFI. Unfortunately, in many practical settings this squeezing is difficult to exploit, as one becomes extremely sensitive to the added noise of amplification stages that follow the primary measurement (i.e. one needs following amplifiers to be quantum limited). In contrast, our scheme does not rely on squeezing the measurement noise, but instead effectively amplifies the signal power at fixed total photon number. The output noise has the same magnitude as the input noise, and hence taking advantage of our enhanced QFI does not need following amplification stages to be quantum limited. This represents a significant practical advantage. We end this section by pointing out that the $N$ dependence of the QFI in Eq. (\[eq:LargeSNR\]) does not violate standard Heisenberg-limit constraints [@Giovannetti2011], as the setting here is different. The usual Heisenberg limit applies to $N$ sensor systems which each interact independently with the parameter of interest; the QFI here scales as best as $\propto N$, a result which requires entanglement. In contrast, each of the $N$ modes in our system is not an independent sensor interacting independently with the dispersive perturbation, as the sites are coupled. The enhanced scaling we find is not the result of entanglement: we stress that the input light to our system is just a coherent state. Instead, the enhancement is a consequence of our system’s unusual mechanism for non-reciprocal amplification. Non-Markovian Effects ===================== We now relax the assumption that the parameter $\epsilon$ is infinitely weak. For concreteness, we assume the sensing target is to distinguish the case $\epsilon = 0$ from the case $\epsilon = \epsilon_0 \neq 0$. This kind of discrimination is relevant in many practical situations, for example the dispersive measurement of the state of a qubit [@RMP_Clerk]. We assume that $\epsilon_0$ is small enough such that linear response is still valid, but not so small that measurement will be infinitely long compared to internal system timescales. We thus need to understand the finite-frequency response and noise properties of our non-Hermitian lattice sensor. In this section, we will characterize our sensor by its measurement time $\tau_M$: what is the minimum integration time to to achieve a SNR of unity? Heuristically, $\tau_M$ is the minimum amount of time required to distinguish between $\epsilon = 0$ and $\epsilon = \epsilon_0$. In the limit $\epsilon_0 \rightarrow 0$, $\tau_M$ will be much longer than any internal sensor timescale, and we can use the long-time limit SNR expression derived in the previous section (c.f. Eq. (\[eq:LargeSNR\])). We define $\tau^{*}_{M}(N)$ to be this $\epsilon_0 \rightarrow 0$ expression for the measurement time. Assuming that the input field has only vacuum noise, we find: $$\begin{aligned} \label{eq:T_meas_infty} \tau^{*}_{M}(N) = \frac{1}{16 Z(A) \Bar{n}_{\rm tot} \kappa} \left(\frac{\kappa}{\epsilon_0} \right)^2 e^{-2A(N-1)}. \end{aligned}$$ The obviously attractive feature here is the exponential reduction of $\tau_M$ with increasing lattice size $N$ (but at fixed total photon number). As $N$ or $\epsilon_0$ is increased, $\tau^*_M(N)$ will become increasing smaller, and at some point will become comparable to internal system timescales. At this point, the long-time limit assumption used to derive this expression becomes invalid. There are two distinct relevant timescales that govern the dynamics of our sensor. The first $t_{rt}(N)$ determines the ballistic propagation time to traverse the lattice end to end: $$\begin{aligned} t_{rt}(N) = \frac{N}{J}, \end{aligned}$$ The second $t_{esc}(N)$ involves the coupling to the waveguide: how quickly does a particle that is delocalized in the lattice leak out to the waveguide. A simple Fermi’s Golden Rule estimate yields the scale: $$\begin{aligned} t_{esc}(N) = \frac{N+1}{\kappa} \end{aligned}$$ Both these timescales increase with system size. As a result, non-Markovian effects associated with internal dynamics become increasingly important with increasing $N$. The crucial question is how this physics modified or places a limit on the exponential-in-$N$ measurement enhancement predicted by Eqs. (\[eq:LargeSNR\]) and (\[eq:T\_meas\_infty\]). For large enough $N$ the measurement will be so fast that these internal timescales matter. Do they simply put a bound on the measurement time, or does performance continue to increase with increasing $N$? We first consider the limit $J \gg \kappa$; the only relevant dynamical timescale is then $t_{esc}(N)$, the time it takes a delocalized photon to escape the lattice. In this regime, the level spacing of lattice resonances is much larger than their widths. We can thus accurately approximate the relevant low-frequency behaviour of lattice susceptibilities by the contribution from the zero-frequency resonance (whose width is $1/t_{esc}(N)$). Assuming as always that $N$ is odd, we have: $$\begin{aligned} & \chi^{xx}[N,1;\omega] \approx \frac{2i^{N}}{N+1} \frac{e^{A(N-1)}} {\omega+i \frac{\kappa}{N+1}} \\ & \chi^{pp}[1,N;\omega] \approx \frac{-2i^{-N}}{N+1} \frac{e^{A(N-1)}} {\omega+i \frac{\kappa}{N+1}}.\end{aligned}$$ Note crucially that the residue at the poles are exponentially large in system size; this directly reflects the amplification physics we have discussed previously. Because of these factors, the above response functions are not simply equivalent to those of a single mode system with a very small linewidth. With this approximation, we find that the SNR is given by (see Appendix \[app:Single\_Pole\] for details) $$\begin{aligned} \label{eq:SNR_Approx} &\text{SNR}_\tau(N, \epsilon,J \rightarrow \infty) = \\ & \sqrt{ \frac{\tau}{\tau^*_{M}(N) }} \left( 1+e^{-\frac{\tau}{t_{esc}(N)}} -\frac{2 t_{esc}(N)}{ \tau } (1-e^{-\frac{\tau}{t_{esc}(N)}}) \right) \nonumber\end{aligned}$$ The bracketed factor represents the non-Markovian correction to the long-time limit expression. Note that the correction is only to the magnitude of the signal. As we continue to use linear response, we only need to compute the homodyne current noise to zeroth order in $\epsilon_0$. This noise is thus always vacuum noise regardless of the choice of integration time $\tau$. Using the above expression, we can then directly compute the measurement time $\tau_M$ in the $J \to \infty$ limit. While finding the measurement time analytically is unfeasible, we can describe its asymptotic behavior in the strong and weak measurement limit (see App. \[app:Single\_Pole\]) $$\begin{aligned} \nonumber &\tau_M^{J = \infty}(N) = \begin{cases} \tau_M^*(N), & \tau_M^*(N) \gg t_{esc}(N) \\ \sqrt{6}t_{esc}(N) \sqrt[\leftroot{-2}\uproot{2}5]{\frac{\tau_M^*(N)}{\sqrt{6}t_{esc}(N)}}, & \tau_M^*(N) \ll t_{esc}(N) \end{cases} \\ & \propto \begin{cases} e^{-2A(N-1)}, & \tau_M^*(N) \gg t_{esc}(N) \\ (N+1)^{4/5}e^{-2A(N-1)/5}, & \tau_M^*(N) \ll t_{esc}(N). \end{cases} \end{aligned}$$ We thus find a surprising result: even for fast measurements where the escape time from the lattice plays a role, the measurement time continues to improve exponentially with lattice size $N$. Intuitively, this is because the deleterious effects of increasing the escape time $t_{esc} = (N+1)/\kappa$ with increasing $N$ is more than offset by the exponentially large number of photons $e^{2A(N-1)}$ that exit through the waveguide when $\epsilon = \epsilon_0$. We next consider the case where the hopping amplitude $J$ is not infinitely larger than all other scales. In this case, we must also take into account the finite propagation speed $v \propto J$ of particles the lattice. Because an injected wavepacket must make a round trip before acquiring any information about the perturbation $\epsilon$, for times less than $2N/v = N/J = t_{rt}(N)$ we expect the signal to be approximately zero. After this first round trip, the limiting factor in obtaining a large signal is once again the escape rate. Including the effects of a finite $J$, we find that the SNR is well approximated by simply adding a cutoff to the $J \rightarrow \infty$ result in Eq. (\[eq:SNR\_Approx\]): $$\begin{aligned} \label{eq:SNR_Approx_J_Finite} \text{SNR}_\tau(N,\epsilon, J) & = \Theta \left(\tau - t_{rt}\right ) \text{SNR}_\tau(N,\epsilon, J\rightarrow \infty) \end{aligned}$$ where $\Theta(t)$ is the Heaviside step function. This form reflects the basic intuition that it is impossible to make a measurement faster than the propagation time. Combining these results, we finally find that including the effects of both internal timescales $t_{rt}(N)$ and $t_{esc}(N)$, the measurement time (to good approximation) is given by $$\begin{aligned} \label{eq:meas_time} \tau_M(N) = \max ( \tau_M^{J = \infty}(N), t_{rt}(N) ) \end{aligned}$$ where $\tau_M^{J = \infty}(N)$ is given in Eq. (\[eq:T\_meas\_infty\]). Thus, as a function of increasing system size $N$, the measurement time first decreases exponentially until it reaches the round-trip time in the lattice, after which it increases with $N$. The upshot of our analysis is that increasing the lattice size still provides an exponential sensing advantage when including non-Markovian effects. This continues to be true until the measurement time is reduced to being on par with the round-trip propagation time $t_{rt}(N) = N/J$. In Fig. \[fig:Measurement\_Time\], we plot the numerically-calculated measurement time $\tau_M(N)$ versus lattice size $N$ for a fixed total photon number $\bar{n}_{\rm tot}$ and perturbation size $\epsilon_0 / \kappa$; different curves correspond to different values of the hopping $J / \kappa$. We find an excellent agreement with the analytic approximation given in Eq. (\[eq:meas\_time\]). The measurement time follows $\tau_M^{J = \infty}(N)$ (dark solid line) until it reaches the round-trip time $t_{rt}(N)$ (faint dashed lines), after which it increases linearly with $N$. ![Measurement time $\tau_{M}(N)$ versus lattice size $N$, for different choices of the hopping amplitudes $J$. The solid black line is the measurement time in the $J \rightarrow \infty$ limit, $\tau^{J=\infty}_M(N)$. Faint dashed lines are the round trip propagation timescale $t_{rt}(N) \equiv N/J$. The measurement time decays exponentially with increasing $N$, up until $\tau^{J = \infty}_{M}(N) \approx t_{rt} (N)$. Further increases of $N$ cause the measurement time to scale with $t_{rt}(N)$, implying that it increases linearly with $N$. We take $\epsilon_0 = 10^{-8} \kappa$, $\Bar{n}_{\rm tot} = 5\times 10^{9}$ and $A = 0.2$. We also plot results for odd values of $N$ only, as this guarantees the existence of a zero-frequency lattice eigenstate and thus an additional resonant enhancement of our measurement (c.f. main text before Eq. (\[eq:Large\_SNR0\])). []{data-label="fig:Measurement_Time"}](Measurement_Time_Corrected_2.pdf){width="45.00000%"} Beyond linear response {#sec:Beyond_Linear_Response} ====================== In this final section, we again consider the sensing problem of distinguishing $\epsilon = 0$ from $\epsilon = \epsilon_0$; now however, we analyze the regime where (due to amplification effects) $\epsilon_0$ is too large for a linear response analysis to be valid. This is in contrast to the previous section, where $\epsilon_0$ was small enough that linear response was still valid, but large enough that non-Markovian detector effects were important. For any $\epsilon_0$ the output state of the light leaving the waveguide will be Gaussian, and the statistics of the measured homodyne current will be Gaussian. We can thus again quantify our sensor’s performance by calculating the signal-to-noise ratio. We now however need to account for the fact that the homodyne current noise will also depend on $\epsilon_0$. The definition of the signal-to-noise ratio becomes: $$\begin{aligned} \label{eq:SNR_Average} \mathrm{SNR}_\tau(N,\epsilon_0) \equiv \frac { | \langle \hM_\tau(N) \rangle_{\epsilon_0} - \langle \hM_\tau(N) \rangle_{0} | } {\sqrt{ \frac{\mathcal{N}^2_{\tau}(N,0)+\mathcal{N}^2_{\tau}(N, \epsilon_0)} {2}}} \end{aligned}$$ This SNR quantifies the distinguishability between the Gaussian homodyne current distributions obtained for $\epsilon = 0$ versus $\epsilon = \epsilon_0$ (see e.g. [@RMP_Clerk; @Laflamme_PRL]). As might be expected, the nonlinear dependence of SNR on $\epsilon_0$ will prevent one from indefinitely improving the measurement with increasing $N$. The key issue is that beyond linear response, noise amplification will also play a role. We show in what follows that even with this complication, our system yields a strong advantage, allowing one to fundamentally change the scaling of the SNR with $\epsilon_0$. We will focus on the most interesting situation where $\epsilon_0/\kappa \ll 1$, but where linear response breaks down because of a large amplification factor (i.e. $e^{A(N-1)} \epsilon_0/\kappa $ is not necessarily small). Further, we take the round-trip time $t_{rt}(N) = N/J$ to be small enough that we can ignore the transient dynamics and consider only the steady-state response. Formally, we now need to calculate the output field leaving the waveguide to all orders in $\epsilon_0$. We thus expand the zero frequency quadratures of the output field as a power series in $\epsilon_0 / \kappa$: $$\begin{aligned} \hat{X}^{(\rm out)}[0] & \equiv \sum_{k=0}^{\infty} \left( \frac{\epsilon_0}{ \kappa} \right)^k \hat{X}^{(\rm out)}_k \\ \hat{P}^{(\rm out)}[0] & \equiv \sum_{k=0}^{\infty} \left( \frac{\epsilon_0}{ \kappa} \right)^k \hat{P}^{(\rm out)}_k \end{aligned}$$ To zeroth order in $\epsilon_0$, there is no mixing of quadratures, and input signals are reflected with no net amplification (but just a trivial sign change): $$\begin{aligned} \hat{X}^{(\rm out)}_0 = - \hat{X}^{(\rm in)}[0], \,\,\, \hat{P}^{(\rm out)}_0 = - \hat{P}^{(\rm in)}[0], \end{aligned}$$ Note that throughout this section, we associate the coherent drive tone amplitude $\beta$ with the average value of $ \hat{X}^{(\rm in)}[0]$. In contrast, the first order contributions correspond to a process where input fields scatter once off the “impurity" before returning to the waveguide. This scattering converts one canonical quadrature to the other, and also results in a net amplification or deamplification $$\begin{aligned} \hat{X}^{(\rm out)}_1 & = 4e^{-2A(N-1)} \hP^{\rm(in)}[0] \\ \hat{P}^{(\rm out)}_1 & = -4 e^{2A(N-1)} \hX^{\rm (in)}[0] \end{aligned}$$ The amplification of $\hat{X}^{(\rm in)}$ is exactly the process we discussed in Sec. \[sec:ExpEnhancement\] that is responsible for the exponentially-enhanced signal. The attenuation of $\hat{P}^{(\rm in)}$ at this order can be understood analogously. What about the second order in $\epsilon_0$ contributions? Heuristically, these correspond to input fields scattering off the impurity twice. While we expect such a process to preserve the identity of each canonical quadrature, it also has a more surprising feature: it results in no net amplification or deamplification: $$\begin{aligned} \hat{X}^{(\rm out)}_2 = 8 \hX^{\rm(in)}[0], \,\,\,\,\,\, \hat{P}^{(\rm out)}_2 = 8 \hP^{\rm (in)}[0] \end{aligned}$$ This unexpected result can again be traced by to the chiral and quadrature-dependent nature of gain and loss in our system. Interacting with the impurity twice implies that an input signal has performed at least two round-trip traversals of the lattice (partially as an $X$, partially as a $P$). The gain and attenuation for each of these roundtrips necessarily cancel. This pattern continues to higher order, and provides a simple explanation for the full expression we find for the output field: the net amplification / deamplification factor for each kind of quadrature to quadrature scattering process is independent of $\epsilon_0$. We find $$\begin{aligned} &\hX^{\rm (out)}[0] = R(\epsilon_0)\hX^{\rm (in)}[0] -T(\epsilon_0) e^{-2A(N-1)} \hP^{\rm (in)}[0] \\ &\hP^{\rm (out)}[0] = T(\epsilon_0) e^{2 A(N-1)} \hX^{\rm (in)}[0] + R(\epsilon_0) \hP^{\rm (in)}[0] \end{aligned}$$ where $$\begin{aligned} & R(\epsilon_0) = - \frac {(\frac{\kappa}{2})^2-\epsilon_0^2} {(\frac{\kappa}{2})^2+\epsilon_0^2} \\ & T(\epsilon_0) = \frac{\kappa \epsilon_0} {(\frac{\kappa}{2})^2+\epsilon_0^2} \end{aligned}$$ are elements of an orthogonal scattering matrix describing the conversion of quadratures (see Appendix \[app:Non-Pert\] for details). We see that quadrature-preserving scattering processes never come with amplification factors, whereas the amplification factors for quadrature-changing scattering are independent of $\epsilon_0$. Crucially, there are no amplification factors in denominators in this expression. This result can be derived via a canonical squeezing transformation which eliminates the anomalous terms in Eq. (\[eq:HBKC\]); it also reflects the fact that our system is dynamically stable regardless of the strength of $\epsilon_0$. From these input-output relations, we can readily compute the SNR. Taking the noise of the input field to be vacuum, we have: $$\begin{aligned} \nonumber \mathrm{SNR}_\tau(N,\epsilon_0) = & \frac{ \sqrt{8 \tau}|\beta||T(\epsilon_0)|e^{2A(N-1)} } { \sqrt{1+R^2(\epsilon_0)+T^2(\epsilon_0)e^{4A(N-1)}} } \\ \label{eq:Full_SNR} = & \frac{ \sqrt{2 Q(A,\epsilon_0) \bar{n}_{\rm tot}\kappa \tau }|T(\epsilon_0)|e^{A(N-1)} } { \sqrt{1+R^2(\epsilon_0)+T^2(\epsilon_0)e^{4A(N-1)}} } \end{aligned}$$ where $\bar{n}_{\rm tot} = (\bar{n}_{\rm tot}(0) + \bar{n}_{\rm tot}(\epsilon_0))/2$. We see that now, the denominator in Eq. (\[eq:Full\_SNR\]) also depends on the amplification factor $A$, which corresponds to the amplification of noise. Because of this, increasing $A$ and/or $N$ indefinitely is no longer optimal. There remains nonetheless an advantage in using a carefully chosen amount of amplification. Ignoring $Q(A, \epsilon_0)$ and maximizing the SNR Eq. (\[eq:Full\_SNR\]) with respect to the amplification, we see that the optimal choice corresponds to amplification that simply doubles the output noise over pure vacuum noise. In the $\epsilon_0 \ll \kappa$ limit of interest, the condition is: $$\begin{aligned} \label{eq:A_Opt} e^{4A^*(N-1)} \equiv \frac{1+R^{2}(\epsilon_0)}{T^2(\epsilon_0)} \approx \frac{\kappa^2}{8 \epsilon_0^2} \end{aligned}$$ With this optimized choice of $A$, the SNR written in terms of $\epsilon_0$ is then $$\begin{aligned} \mathrm{SNR}_\tau(N,\epsilon_0) = 8^{1/4} \sqrt{Q(A^*,\epsilon_0)\bar{n}_{\rm tot} \kappa \tau} \sqrt{\frac{\epsilon_0}{\kappa} }. \end{aligned}$$ We show in Appendix \[app:Non-Pert\] that $Q(A^*,\epsilon_0) = 1-\mathcal{O}((\frac{8 \epsilon_0^2}{\kappa^2})^{\frac{1}{N-1}})$. Comparing against Eq. (\[eq:Large\_SNR0\]), we see that the optimized amplification has changed the fundamental scaling of the long-time SNR from being linear in the small parameter $\epsilon_0/\kappa$ to a square-root dependence. Thus, by extending our analysis beyond a simple linear-response treatment, we see that the exponential enhancement of the SNR predicted in Eq. (\[eq:Large\_SNR0\]) cannot extend indefinitely: the best one can do is to enhance the SNR (over a conventional dispersive measurement) by a large factor $\sqrt{\kappa / \epsilon_0}$. This behaviour is plotted in Fig. (\[fig:Scatter\_Twice\]). We again note that this predicted measurement enhancement does not require a large number of lattice sites; just three is already enough. The enhanced square-root dependence of the SNR on $\epsilon_0$ is superficially reminiscent of the behaviour found in non-Hermitian exceptional point (EP) sensors [@Wiersig_2014]. We stress that these phenomena are completely distinct. For EP sensors, it is the frequency of a resonance that exhibits a square root dependence, and not the SNR of a specific measurement (or other metric that also quantifies fluctuations). Further, EP sensing is based on operating near a point where the system’s dynamical matrix becomes defective and normal modes coalesce. In contrast, our system is not operating near such a special operating point. As we have stressed, the mechanism for enhanced SNR in our system is based on its directional amplification, and its ability to amplify signals and noise differently. ![Non-perturbative signal-to-noise ratio in the long time limit $\text{SNR}_{\tau}(N, \epsilon_0)/\text{SNR}_{\tau}(1, \epsilon_0)$, as a function of lattice size $N$. The SNR initially increases exponentially with $N$, as predicted by our linear-response analysis in Sec. \[sec:ExpEnhancement\]. For sufficiently large $N$, linear response breaks down due to the amplification of noise; this causes the SNR to decrease with $N$ for large $N$. A non-trivial maximum is thus reached for an intermediate value of $N$ given by Eq. (\[eq:A\_Opt\]). For this optimal $N$ and a weak perturbation $\epsilon_0$, the SNR scales like $\sqrt{\epsilon_0/\kappa}$ (as opposed to the more standard scaling $\epsilon_0/\kappa$) . The parameters here are $A = 0.05$, $\epsilon_0 = 10^{-7} \kappa$ and $\bar{n}_{\rm tot} = 5\times 10^{9}$. We only plot the results for odd values of $N$, which ensures an resonant enhancement of the zero-frequency response (c.f discussion preceding Eq. (\[eq:Large\_SNR0\]))[]{data-label="fig:Scatter_Twice"}](Non_Pert_SNR_N.pdf){width="45.00000%"} Conclusion ========== In this work, we have shown how the unique features of non-Hermitian lattice dynamics can be used for highly enhanced Hamiltonian parameter estimation and parametric sensing. We analyzed a concrete setup involving two copies of the Hatano-Nelson model and a symmetry breaking perturbation. The response to the perturbation grows exponentially with system size, even when the total system photon number is kept fixed. Our analysis focused on a specific realization of this idea using a chain of parametrically driven cavities and a standard dispersing coupling to the parameter of interest. Here, even in the presence of quantum noise effects, the SNR and quantum Fisher information both grow exponentially with system size (all the while keeping photon number fixed). The system we described could be achieved in a variety of superconducting circuit and quantum optical platforms, and only requires one to make a homodyne measurement of the output field leaving the sensor. We also analyzed effects that go beyond standard linear-response and Markovian assumptions. Even including higher-order effects, we show that our scheme allows one to dramatically enhance the SNR so that it depends on the square root of the sensing parameter. Our work highlights the usefulness of multi-mode non-Hermitian features that go beyond the mere existence of exceptional points. An open question is whether other unique features attributed to non-Hermiticity, such as exotic topological phases or chiral mode switching, are also advantageous to quantum sensing problems. Acknowledgements {#acknowledgements .unnumbered} ================ We thank Kero Lau for a careful reading of the manuscript. This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-19-1-0362. $\mathbb{Z}_2$ symmetry of a non-Hermitian tight-binding model {#app:Symmetry} ============================================================== We discuss in more detail the $\mathbb{Z}_2$ symmetry that we wish to break to order to obtain an exponentially large response. We consider two finite, $N$ site open Hatano-Nelson lattices with opposite chiralities (i.e. oppositely signed imaginary vector potentials $A$). The time-dependent Schrödinger equation reads $$\begin{aligned} \dot{\psi}^{\uparrow}_n = J e^{A} \psi^{\uparrow}_{n-1} - J e^{-A} \psi^{\uparrow}_{n+1} \\ \dot{\psi}^{\downarrow}_n = J e^{-A} \psi^{\downarrow}_{n-1} - J e^{A} \psi^{\downarrow}_{n+1} \end{aligned}$$ where $\sigma$ indexes the two chains. These equations of motion are invariant under a combination of time reversal $\dot{\psi}^{\sigma}_n \to - \dot{\psi}^{\sigma}_n $, spatial inversion $\psi^{\sigma}_n \to \psi^{\sigma}_{N+1-n}$ and pseudospin inversion $\sigma \to \bar{\sigma}$. While this is a seemingly trivial symmetry, we note that the Heisenberg equations of motion of our dissipation-free realization of the same model $$\begin{aligned} \label{eq:Heisenberg_x_App} \dot{\hx}_n & = J e^{A} \hx_{n-1} - J e^{-A} \hx_{n+1}, \\ \label{eq:Heisenberg_p_App} \dot{\hp}_n & = J e^{-A} \hp_{n-1} - J e^{A} \hp_{n+1}, \end{aligned}$$ are invariant under the same set of symmetries, where $\hx_n$ and $\hp_n$ play the role of pseudospin. To describe these symmetries requires using operators acting on the bosonic Hilbert space. To this end, we consider the antiunitary time reversal operator $\mathcal{T}$, a unitary rotation operator $\mathcal{R}$ and the unitary spatial inversion operator $\mathcal{S}$ whose action on the quadratures reads $$\begin{aligned} \mathcal{T} \hx_n \mathcal{T}^{-1} = \hx_n, & \hspace{0.25cm} \mathcal{T} \hp_n \mathcal{T}^{-1} = -\hp_n \\ \mathcal{R} \hx_n \mathcal{R}^{-1} = \hp_n, & \hspace{0.25cm} \mathcal{R} \hp_n \mathcal{R}^{-1} = -\hx_n \\ \mathcal{S} \hx_n \mathcal{S}^{-1} = \hx_{N+1-n}, & \hspace{0.25cm} \mathcal{S} \hp_n \mathcal{S}^{-1} = \hp_{N+1-n}.\end{aligned}$$ Equivalently, we have $$\begin{aligned} &\mathcal{T} \ha_n \mathcal{T}^{-1} = \ha_n \\ &\mathcal{R} \ha_n\mathcal{R}^{-1} = -i\ha_n \\ & \mathcal{S} \ha_n\mathcal{S}^{-1} = \ha_{N+1-n}\end{aligned}$$ With these definitions in hand, it is easy to verify that the Hermitian Hamiltonian which gives the equations of motion Eqs.(\[eq:Heisenberg\_x\_App\]) and (\[eq:Heisenberg\_p\_App\]) $$\begin{aligned} \hH_B = J \sum_{n=1}^{N-1} \left( -e^{-A}\hx_{n+1} \hp_n + e^{A} \hp_{n+1} \hx_n \right). \end{aligned}$$ is invariant under the combination of time-reversal, rotation, and spatial inversion. Similarly, the non-local perturbation considered in Section \[sec:ExpEnhancement\] $$\begin{aligned} \hat{V}_{\rm NHSE} = e^{i \varphi} \ha_1^\dagger \ha_N + e^{-i\varphi} \ha_N^\dagger \ha_1\end{aligned}$$ is invariant under the same combination of symmetries. Time-reversal changes the phase $\varphi \to -\varphi$, $\hat{V}_{\rm NHSE}$ commutes with $\mathcal{R}$ and spatial inversion sends $\ha^\dagger_1 \ha_N \to \ha_N^\dagger \ha_1$. Quadrature susceptibility matrices {#app:chi_quadrature} ================================== We first compute the susceptibilities for the $\hx_n$ and $\hp_n$ quadratures, defined as $\ha_n = (\hx_n+i\hp_n )/\sqrt{2}$. The Heisenberg-Langevin equations of motion in this basis are $$\begin{aligned} \label{eq:HL_x} \dot{\hx}_n & = -i[\hx_n, \hat{H}_B] - \delta_{n1} \left( \frac{\kappa}{2}\hx_n + \sqrt{\kappa} \left( \sqrt{2}\beta + \hX^{\rm(in)} \right) \right) \\ \label{eq:HL_p} \dot{\hp}_n & = -i[\hp_n, \hat{H}_B] - \delta_{n1} \left( \frac{\kappa}{2}\hp_n + \sqrt{\kappa} \hP^{\rm(in)} \right), \end{aligned}$$ where $\hX^{\rm (in)}$ and $\hP^{\rm (in)}$ are the operator equivalent of Gaussian white noise. They average to zero, and their second moment is $$\begin{aligned} & \langle \hX^{\rm (in)}(t) \hX^{\rm (in)} (t') \rangle = \left( \bar{n}_{\rm th} + \frac{1}{2} \right) \delta(t-t') \\ & \langle \hP^{\rm (in)}(t) \hP^{\rm (in)} (t') \rangle = \left( \bar{n}_{\rm th} + \frac{1}{2} \right) \delta(t-t') \\ & \frac{1}{2} \langle \{ \hX^{\rm (in)}(t), \hP^{\rm (in)}(t') \} \rangle = 0 \end{aligned}$$ where $\bar{n}_{\rm th}$ is the number of thermal quanta in the input field. We now focus on the case where $\bar{n}_{\rm th} = 0$, with generalizations to finite-temperature inputs being straightforward. An immense simplification arises by making a local Bogoliubov (squeezing) transformation, such that the Hamiltonian preserves the total number of these new quasiparticles. The dynamical matrix of in this new basis is then explicitly Hermitian. Defining new canonically conjugate quadrature operators $\htx_n$ and $\htp_n$ by $$\begin{aligned} \label{eq:Squeeze_X} & \hx_n = e^{A(n-n_0)} \htx_n \\ \label{eq:Squeeze_P} & \hp_n = e^{-A(n-n_0)} \htp_n \end{aligned}$$ with $n_0$ an arbitrary real number, we have $$\begin{aligned} \hat{H}_B & = J \sum_{n=1}^{N-1} \left( - \htx_{n+1} \htp_j + \htp_{n+1} \htx_j \right) \\ & = iJ \sum_{n=1}^{N-1} \left( \hta_{n+1}^\dagger\hta_n - h.c. \right) \end{aligned}$$ with $\hta_{n} = (\htx_n+i\htp_n)/\sqrt{2}$ a transformed canonical annihilation opreator. The parameter $n_0$ does not enter the Hamiltonian since $\hH_B$ is invariant under a uniform local squeezing operation that doesn’t mix quadratures $\hx_n \to e^{-An_0} \hx_n, \hp_n \to e^{An_0} \hp_n $. In this section, it will be convenient to set $n_0 = 1$, so that the annihilation operators on the first stie remain unchanged $\hta_1 = \ha_1$. The Heisenberg-Langevin equations of motion in this new basis read $$\begin{aligned} \dot{\htx}_n & = -i[\htx_n, \hat{H}_B] - \delta_{n1} \left( \frac{\kappa}{2}\htx_n + \sqrt{\kappa} \left( \sqrt{2}\beta + \hX^{\rm(in)} \right) \right), \\ \dot{\htp}_n & = -i[\htp_n, \hat{H}_B] - \delta_{n1} \left( \frac{\kappa}{2}\htp_n + \sqrt{\kappa} \hP^{\rm(in)} \right). \end{aligned}$$ As expected, the response properties of $\htx_n$ and $\htp_n$ are then determined by a completely Hermitian matrix (other than the waveguide-induced decay on the first site). Using the squeezing transformations Eqs. (\[eq:Squeeze\_X\])-(\[eq:Squeeze\_P\]) and the fact that the dynamics of the $\hx$ and $\hp$ quadratures are uncoupled, the relevant quadrature-quadrature susceptibilities read $$\begin{aligned} \label{eq:chi_xx} & \chi^{xx}(n,m;t) = -i\langle[\hx_n(t), \hp_m(0)]\rangle = e^{A(n-m)} \tilde{\chi}^{xx}(n,m;t) \\ \label{eq:chi_pp} & \chi^{pp}(n,m;t) = i\langle [\hp_n(t), \hx_m(0)] \rangle = e^{-A(n-m)} \tilde{\chi}^{pp}(n,m;t) \\\label{eq:chi_xp} & \chi^{xp}(n,m;t) = i\langle [\hx_n(t), \hx_m(0)] \rangle = 0 \\ \label{eq:chi_px} & \chi^{px}(n,m;t) = -i\langle [\hp_n(t), \hp_m(0)] \rangle = 0 \end{aligned}$$ where $\tilde{\chi}^{\alpha \beta}(n,m;t)$ are quadrature response functions of a regular (i.e. reciprocal particle-conserving) tight-binding chain with a waveguide attached to the first site. Note that our convention differs from that used in the condensed matter community, where $\chi^{\alpha \beta}(n,m;t)$ is the response of quadrature $\alpha$ to a force which couples to $\beta$ in the Hamiltonian. Computing the quadrature-quadrature susceptibilities $\chi^{\alpha \beta}(n,m;t)$ of our non-reciprocal system is then no more complicated than finding the susceptibilities of a reciprocal tight-binding chain $\tilde{\chi}^{xx }(n,m;t)$ and $\tilde{\chi}^{pp}(n,m;t)$. The susceptibilities of the Hatano-Nelson model Eq. (\[eq:Hatano-Nelson\]) are computed in a similar manner. There, instead of a local squeezing transformation, one makes a so called imaginary gauge transformation $\ket{n} \to e^{A (n-j_0)} \ket{n}$ and $\bra{n} \to e^{-A(n-j_0)} \bra{n}$. In this new gauge, the Hamiltonian is Hermitian and completely independent of $A$. The factorization of Eq. (\[eq:Susceptibility\]) as $\chi(n,m;t) = e^{A(n-m)}\tilde{\chi}(n,m;t)$ immediately follows. Particle-conserving susceptibilities {#app:chi_particle} ==================================== Although so far we’ve only considered quadrature-quadrature response functions, the fact that we can map our Hamiltonian onto a particle conserving one makes it so that it is much simpler to keep track of the dynamics of the single squeezed mode $\hta_n$. Indeed, we have $$\begin{aligned} \label{eq:quad_diag} &\tilde{\chi}^{xx}(n,m;t) = \tilde{\chi}^{pp}(n,m;t) = \text{Re} \: \tilde{\chi}(n,m;t) \\ \label{eq:Suscept_Zero} & \tilde{\chi}^{px}(n,m;t) = - \tilde{\chi}^{xp}(n,m;t) = \text{Im} \: \tilde{\chi}(n,m;t)\end{aligned}$$ where $$\begin{aligned} \tilde{\chi}(n,m;t) = \langle [\hta_n(t), \hta_m^\dagger(0)] \rangle.\end{aligned}$$ Because our Hamiltonian is quadratic in boson opeators and conserves total quasiparticle number, we can readily use the single-particle formalism to find the relevant susceptibilities. If we let $\ket{n}$ denote a position eigenket, we then have $$\begin{aligned} \tilde{\chi}(n,m;t) = \bra{n} e^{-i t \left( \boldsymbol{\tilde{H}} -i \frac{\boldsymbol{\kappa}}{2} \right) } \ket{m}\end{aligned}$$ with $$\begin{aligned} &\boldsymbol{\tilde{H}} = i J \left( \sum_{n=1}^{N-1} \ket{n+1}\bra{n} - h.c. \right) \\ & \boldsymbol{\kappa} = \kappa \ket{1}\bra{1}\end{aligned}$$ It is more convenient to write the susceptibilities in the frequency domain: $$\begin{aligned} \tilde{\chi}[n,m;\omega] & = \int_0^\infty dt \chi(n,m;t) e^{i\omega t} \\ & = \bra{n} \frac{i}{\omega \boldsymbol{1} -\boldsymbol{\tilde{H}} + i \frac{\boldsymbol{\kappa}}{2}} \ket{m}\end{aligned}$$ We’ll first compute the susceptibilities without the effects of $\epsilon$ or $\kappa$, that is $$\begin{aligned} \tilde{\chi}_0[n,m;\omega] = \bra{n} \frac{i}{\omega \boldsymbol{1} -\boldsymbol{\tilde{H}} } \ket{m}\end{aligned}$$ Written out explicitly, the matrix elements of the susceptibility for a finite open chain then satisfy the difference equation $$\begin{aligned} \label{eq:Diff_Eq} i \tilde{\chi}_0[n-1,m;\omega] - \frac{\omega}{J} \tilde{\chi}_0[n,m;\omega] - i \tilde{\chi}_0[n+1,m;\omega] = -\frac{i \delta_{nm}}{J}\end{aligned}$$ with boundary conditions $\tilde{\chi}_0[0, m; \omega] = \tilde{\chi}_0[N+1,m;\omega] = 0$. The exact form of the susceptibility matrix is known; here for the sake of completeness we quickly sketch how to obtain it. First, we note that Eq.(\[eq:Diff\_Eq\]) has the form of a translationally invariant Green’s function problem in the index space $n$, with $-i \delta_{nm}/J$ acting as a source term. The general solution will then consist of a linear combination of the source free solution and a convolution (in the index space $n$) of the source with the homogeneous solution. The source free solution, which satisfies $$\begin{aligned} i \tilde{\chi}^{\rm sf}_0[n-1,m;\omega] - \frac{\omega}{J} \tilde{\chi}^{\rm sf}_0[n,m;\omega] - i \tilde{\chi}_0^{\rm sf}[n+1,m;\omega] = 0\end{aligned}$$ is precisely (up to a factor of $i$) the recursion relation that defines $T_n(\omega/2J)$ and $U_n(\omega/2J)$, the Chebyshev polynomials of the first and second kind respectively. Since $U_{-1}(\omega/2J) = 0$, and given our boundary condition $\tilde{\chi}_0[0,m;\omega] = 0$, we conclude that the source free solution is $$\begin{aligned} \tilde{\chi}_0^{\rm sf}[n,m; \omega] = c_mi^n U_{n-1}(\frac{\omega}{2J})\end{aligned}$$ with $c_m$ a constant that will be used to satisfy the second boundary condition. The full solution to Eq. (\[eq:Diff\_Eq\]) is then $$\begin{aligned} \tilde{\chi}_0[n,m\omega] = c_mi^n U_{n-1}(\frac{\omega}{2J}) -\frac{i}{J}i^{n-m} U_{n-m-1}(\frac{\omega}{2J})\Theta(n-m)\end{aligned}$$ with $\Theta(n-m)$ the Heaviside step function (where $\Theta(0) = 0$). Enforcing the second boundary condition $\tilde{\chi}_0[N+1,m;\omega] = 0$ yields $$\begin{aligned} \label{eq:Bare_Suscept} \tilde{\chi}_0[n, m; \omega] = i^{1+n-m} \frac{U_{\min(n,m)-1}(\frac{\omega}{2J}) U_{N-\max(n,m)}(\frac{\omega}{2 J})}{J U_{N}(\frac{\omega}{2J})}\end{aligned}$$ We now turn our attention to computing the response functions in the presence of the waveguide on the first site. Formally, this introduces a local term $-\kappa/2 \delta_{n,1} \delta_{m,1}$ to the the dynamical matrix. The full susceptibilities $\tilde{\chi}[n,m;\omega]$ can then readily be solved algebraically using Dyson’s equation $$\begin{aligned} \label{eq:Dyson} \tilde{\chi}[n,m;\omega] & = \tilde{\chi}_0[n,m;\omega] - \frac{\kappa}{2} \tilde{\chi}_0[n,1;\omega] \tilde{\chi}[1,m;\omega] \\ \nonumber & = \tilde{\chi}_0[n,m;\omega] - \frac{ \frac{\kappa}{2} \tilde{\chi}_0[n,1;\omega] \tilde{\chi}_0[1,m;\omega] } { 1+\frac{\kappa}{2}\tilde{\chi}_0[1,1;\omega] }.\end{aligned}$$ Since there is only a driving force on the first site and we are only interested in the response on the first site, we must only compute $\tilde{\chi}[n,1;\omega]$ and $\tilde{\chi}[1,m;\omega]$: $$\begin{aligned} \label{eq:chi_n1} \tilde{\chi}[n,1;\omega] = i^{n} \frac{U_{N-n}(\frac{\omega}{2J})} {J U_N(\frac{\omega}{2J})+i \frac{\kappa}{2}U_{N-1}(\frac{\omega}{2J})} \\ \label{eq:chi_1n} \tilde{\chi}[1,m;\omega] = -i^{-m} \frac{U_{N-m}(\frac{\omega}{2J})} {J U_N(\frac{\omega}{2J})+i \frac{\kappa}{2}U_{N-1}(\frac{\omega}{2J})}\end{aligned}$$ Because $\tilde{\chi}^{px}(n,m;t) = - \tilde{\chi}^{xp}(n,m;t) = 0$, from Eq. (\[eq:Suscept\_Zero\]) we conclude that $\tilde{\chi}[n,m;\omega] = \tilde{\chi}^{xx}[n,m;\omega] = \tilde{\chi}^{pp}[n,m;\omega]$. With this result and Eqs.(\[eq:chi\_xx\])-(\[eq:chi\_pp\]), we now have all the relevant quadrature-quadrature susceptibilities. Total photon number {#app:n_bar} =================== Let us compute the total steady-state intracavity photon number on each site to zeoreth order in $\epsilon$. To do so, we must solve the Heisenberg-Langevin equations for the cavity annihilation operators $\ha_n$. Recall that we were able to define new squeezed annihilation and creation operators $$\begin{aligned} \ha_n = \cosh(A(n-1)) \hta_n + \sinh(A(n-1)) \hta_n^\dagger\end{aligned}$$ where the Hamiltonian $\hH_B$ conserved the total number of quasiparticles. Thus, the total number of photons on site $n$ reads $$\begin{aligned} \nonumber \langle \ha_n^\dagger \ha_n \rangle & = \cosh(2A(n-1)) \langle \hta_n^\dagger \hta_n\rangle \\ \nonumber & + \sinh(2A(n-1))\Re(\langle \hta_n \hta_n \rangle) \\ &+ \sinh^2(A(n-1))\end{aligned}$$ The last term is due to noise that enter the port on site 1 and is turned into real photons by the parametric amplifier-type interactions. We can readily solve the Heisenberg-Langevin equations for the squeezed modes $\hta_n$: $$\begin{aligned} \label{eq:HL_Solved} \nonumber \hta_n(t) & = \tilde{\chi}(n,m;t) \hta_m(t) \\ \nonumber & - \sqrt{\kappa}\beta \int_0^{t} dt' \tilde{\chi}[n,1;t-t'] \\ & -\sqrt{\kappa} \int_0^{t} dt' \tilde{\chi}[n,1;t-t'] \ha^{\rm(in)}(t')\end{aligned}$$ where $\ha^{\rm (in)}(t) = (\hX^{\rm(in)}(t) + i \hP^{\rm (in)}(t))/\sqrt{2}$ is the operator equivalent of Gaussian white noise. Note that we’re using Einstein summation notation. Assuming a zero temperature environment we have in the steady-state $$\begin{aligned} \langle \hta_n^\dagger \hta_n\rangle = \langle \hta_n \hta_n\rangle = \kappa \beta^2 |\tilde{\chi}[n,1; \omega =0]|^2.\end{aligned}$$ Using Eq. (\[eq:chi\_n1\]). we obtain $$\begin{aligned} \label{eq:photon_number} \langle \ha^\dagger_n \ha_n \rangle & = \frac{4\beta^2}{\kappa} e^{2A(n-1)} \sin^2 \frac{\pi}{2}n \\ \nonumber & + \sinh^2(A(n-1))\end{aligned}$$ where we’ve assumed (and will do so throughout) that $N$ is odd. Summing Eq. (\[eq:photon\_number\]) over all lattice sites gives $$\begin{aligned} \label{eq:Total_n} \bar{n}_{\rm tot} &= \frac{4 |\beta|^2}{\kappa} \frac{e^{2A(N+1)}-1}{e^{4A}-1} \\ \nonumber & + \frac{1}{4} \left( \frac{\sinh(A(2N-1))}{\sinh(A)} - (2N-1) \right) \\ \nonumber & = \bar{n}_N \frac{1-e^{-2A(N+1)}}{1-e^{-4A}}+ \bar{n}_{\rm vac}\end{aligned}$$ with $\bar{n}_{\rm vac}$ the photons that are present due to amplified vacuum fluctuations. Thus, the ratio of the average photon number on the last site to the total number of photons $Z(A)$ is $$\begin{aligned} Z(A) = \left( \frac{1-e^{-2A(N+1)}}{1-e^{-4A}} + \frac{\bar{n}_{\rm vac}}{\bar{n}_N} \right)^{-1}\end{aligned}$$ In the limit where $|\beta|^2/\kappa$ is large, the coherent photons dominate $\bar{n}_{\rm vac}$, which we can ignore. We then have $$\begin{aligned} Z(A) = \frac{1-e^{-4A}}{1-e^{-2A(N+1)}} = 1-\mathcal{O}(e^{-4A})\end{aligned}$$ as in the main text. QFI for $\hat{V}_{NHSE}$ {#app:SNR} ======================== We are now in a position to compute $\text{QFI}_{\tau}(N)/\bar{n}_{\rm tot}$ for any choice of perturbation $\hat{V}$. Recall that that in the large $\beta$ limit of interest, the QFI coincides with SNR squared, optimizing over the homodyne angle $\phi$ ( see Eqs.\[eq:QFI\]). As is written in the main text, see Eq.(\[eq:Signal\_First\_Order\]), the steady-state signal takes the form $$\begin{aligned} \mathcal{S}_{\tau}(N,\epsilon) = \sqrt{\kappa \tau} | \Re [ e^{-i \phi} ( \delta \langle \hx_1 \rangle^{\rm ss} + i \delta \langle \hp_1 \rangle^{\rm ss} ) ] |\end{aligned}$$ with $\delta \langle \hx_1 \rangle^{\rm ss} $ and $\delta \langle \hp_1 \rangle^{\rm ss} $ the steady state linear response of the site-1 average quadrature amplitude to a non-zero $\epsilon$. The signal will depend on the phase of the coherent drive $\beta$ which we take to be real, as in the main text. Our conclusion that $\hat{V}_{NHSE}$ does not have an exponentially large QFI/$\bar{n}_{\rm tot}$ is independent of the phase of $\beta$, as will become evident. This choice of phase is equivalent to driving the $\hx_1$ quadrature with a force $-\sqrt{2 \kappa}\beta$, so that the signal is $$\begin{aligned} \mathcal{S}_{\tau}(N,\epsilon) = \kappa \beta \sqrt{2\tau} | \Re [ e^{-i \phi} ( \delta \chi^{xx}[1,1;0] + i \delta \chi^{px}[1,1;\omega] ) ] |\end{aligned}$$ The form of the responses $\delta \chi^{xx}[1,1;0]$ and $\delta \chi^{px}[1,1;0]$ will depend on $\hat{V}$. For the non-local hopping perturbation $$\begin{aligned} \hat{V}_{\rm NHSE} = e^{i \varphi}\ha^\dagger_1 \ha_N + e^{-i \varphi }\ha^\dagger_N \ha_1\end{aligned}$$ the change to the equations of motion induced by $\epsilon$ to the quadratures on the first site read: $$\begin{aligned} \label{eq:Couple_1} &\delta \dot{\hx}_1 = \epsilon \left( \sin \varphi \: \hx_N + \cos \varphi \: \hp_N \right) \\ \label{eq:Couple_2} & \delta \dot{\hp}_1 = \epsilon \left( -\cos \varphi \: \hx_N+\sin \varphi \: \hp_N \right).\end{aligned}$$ First order perturbation theory then yields $$\begin{aligned} &\delta \chi^{xx}[1,1;0] = \chi^{xx}[1,1;0] (\epsilon \sin \varphi) \chi^{xx}[N,1;0] \\ & \delta \chi^{px}[1,1;0] = \chi^{pp}[1,1;0] (-\epsilon \cos \varphi) \chi^{xx}[N,1;0]\end{aligned}$$ where $\chi^{\alpha \alpha}[n,m;\omega]$ the susceptibilities of the unperturbed system, which we computed in Appendix \[app:chi\_quadrature\] and Appendix \[app:chi\_particle\]. The salient feature is that $\chi^{xx}[n,m;\omega] \propto e^{A(n-m)}$ and $\chi^{pp}[n,m;\omega] \propto e^{-A(n-m)}$ due to the phase-dependent chiral propagation. With the factor of $\chi^{xx}[N,;0]$, it would appear that we have we do in fact have an exponentially large response. Yet it precisely this terms which controls the number of coherent photons on site $N$, since $\bar{n}_N = \kappa |\beta|^2|\chi^{xx}[N,1,0]|^2$. Expressing the signal in terms of $\bar{n}_N$ gives $$\begin{aligned} \label{eq:Boring_SNR} \mathcal{S}_{\tau}(N,\epsilon) = \sqrt{8\tau \kappa \bar{n}_N} |\frac{\epsilon}{\kappa}| | \sin (\varphi-\phi) |\end{aligned}$$ where we’ve used $\chi^{xx}[1,1;0] = \chi^{pp}[1,1;0] = 2/\kappa$ for a chain with a odd number of sites. The form of Eq. (\[eq:Boring\_SNR\]) makes it evident that SNR$/\sqrt{\bar{n}_{\rm tot}}$ doesn’t scale exponentially with system size, and therefore neither does QFI$/\bar{n}_{\rm tot}$. Despite the perturbation having coupled the two effective Hatano-Nelson chains with an amplitude of $\epsilon \cos \varphi $ (see Eqs. (\[eq:Couple\_1\] and \[eq:Couple\_2\])), this is not enough to ensure a large SNR$/\sqrt{\bar{n}_{\rm tot}}$. The non-local form of $\hat{V}_{\rm NHSE}$ implies that a wavepacket only experiences unidirectional amplification before exiting the waveguide. In contrast, the perturbation $\hat{V}_N = \ha^\dagger_N \ha_N$ studied throughout the main text allows for amplification before and after interacting with $\epsilon$. Single-Pole Approximation {#app:Single_Pole} ========================= As mentioned in the main text, we need to understand finite-time dynamics of our non-Hermitian lattice sensor. While we have the exact frequency-space susceptibilities through Eqs.(\[eq:chi\_xx\]-\[eq:chi\_px\]) and Eqs. (\[eq:chi\_n1\])-(\[eq:chi\_1n\]) to zeroth-order in $\epsilon$, Fourier transforming to the time-domain becomes an intractable problem. Note that this is only true of the signal: to zeroeth order in $\epsilon$, the noise is always vacuum. There is however an exact form of the SNR in the limit where the hopping is infinite $J \to\infty$. In this limit the susceptibilities Eqs. (\[eq:chi\_n1\])-(\[eq:chi\_1n\]) take the form $$\begin{aligned} \label{eq:Single_Pole_N1} \tilde{\chi}[N,1;\omega] = \frac{2i^{N}}{N+1} \frac{1} {\omega+i \frac{\kappa}{N+1}} \\ \label{eq:Single_Pole_1N} \tilde{\chi}[1,N;\omega] = \frac{-2i^{-N}}{N+1} \frac{1} {\omega+i \frac{\kappa}{N+1}}\end{aligned}$$ such that the width of the zero mode is $\kappa/(N+1)$. The Fourier transform of each susceptibility (and their product, which is what determines linear response) is then easily computed. The change to the cavity quadrature amplitude $\hp_1$ at a time $t$ in response to the perturbation $\epsilon$ can be found using Eq. (\[eq:HL\_Solved\]) and first order perturbation theory $$\begin{aligned} \nonumber \langle \hp_1(t) \rangle &= -\sqrt{2\kappa}\beta \int_0^t dT \delta \chi^{px}(1,1;T) \\ &= \sqrt{2 \kappa} \epsilon \beta \int_0^t dT \int_0^T dT' \chi^{pp}(1,N:T-T') \chi^{xx}(N,1;T')\end{aligned}$$ Using $\chi^{xx}(n,m;t) = e^{A(n-m)} \tilde{\chi}(n,m;t)$, $\chi^{pp}(n,m;t) = e^{-A(n-m)} \tilde{\chi}(n,m;t)$, Eqs. (\[eq:Single\_Pole\_N1\]) and (\[eq:Single\_Pole\_1N\]) we get $$\begin{aligned} \langle \hp_1(t) \rangle = -\epsilon \sqrt{2 \kappa}\beta(\frac{2}{N+1})^2 e^{2A(N-1)} \int_0^{t}dT T e^{-\frac{\kappa T}{N+1}}\end{aligned}$$ From which we obtain the signal $$\begin{aligned} &\mathcal{S}_{\tau}(N,\epsilon, J \rightarrow \infty) = \\ \nonumber &\frac{\sqrt{2} \kappa |\beta||\epsilon|}{\sqrt{\tau}} (\frac{2}{N+1})^2 e^{2A(N-1)} \int_0^\tau dt \int_0^{t} dT T e^{-\frac{\kappa T}{N+1}}\end{aligned}$$ whereas the noise is always just $\mathcal{N}_\tau(N,\epsilon) = 1/\sqrt{2}$. The SNR for finite $\tau$ is then $$\begin{aligned} &\text{SNR}_\tau(N, \epsilon,J \rightarrow \infty) = \\ & \sqrt{ \frac{\tau}{\tau^*_{M}(N) }} \left( 1+e^{-\frac{\tau}{t_{esc}(N)}} -\frac{2 t_{esc}(N)}{ \tau } (1-e^{-\frac{\tau}{t_{esc}(N)}}) \right) \nonumber\end{aligned}$$ where recall $$\begin{aligned} \tau^{*}_{M}(N) = \frac{1}{16 Z(A) \Bar{n}_{\rm tot} \kappa} \left(\frac{\kappa}{\epsilon_0} \right)^2 e^{-2A(N-1)}\end{aligned}$$ is the measurement time when the steady-state expression holds. We now want to find the measurement time $\tau_M^{J = \infty}(N)$ where in both the weak and strong measurement limit. In the weak measurement limit $\tau^{*}_{M}(N) \gg t_{esc}(N)$, we recover the steady state result $\tau_M^{J = \infty}(N) = \tau_M^*(N)$. In the strong measurement limit $ \tau_M^{*}(N) \ll t_{esc}(N)$, we seek the leading order contribution to the measurement time. To that end, let us define $$\begin{aligned} \gamma = \frac{\tau^{J = \infty}_M(N)}{\tau_M^*(N) } \end{aligned}$$ so that $$\begin{aligned} \left[ \sqrt{\gamma} \left( 1+e^{-\gamma \frac{\tau^*_M(N)}{t_{esc}(N)}} \right) - \frac{2 t_{esc}(N)}{\tau_M^*(N) \sqrt{\gamma}} \left( 1-e^{-\gamma \frac{\tau_M^*(N)}{t_{esc}(N)}} \right) \right]^2 = 1\end{aligned}$$ Assuming that $\gamma \tau_M^*(N)/t_{esc}(N)$ is small (which can be verified to be self-consistent after solving for $\gamma$), then we can Taylor expand the exponential to third order and obtain $$\begin{aligned} \gamma^5 = \left( \frac{\sqrt{6}t_{esc}(N)}{ \tau_M^*(N)} \right)^4\end{aligned}$$ from which $$\begin{aligned} \tau^{J=\infty}_M(N) = \sqrt{6}t_{esc}(N)\sqrt[\leftroot{-2}\uproot{2}5]{\frac{ \tau_M^*(N)}{\sqrt{6}t_{esc}(N)}}\end{aligned}$$ as in the main text. Non-perturbative effects of $\epsilon_0$ to total photon number and output field {#app:Non-Pert} ================================================================================ We now want to consider the full effect of $\epsilon_0$ on the output field. To do so, we must compute the susceptibilities $\chi^{\alpha \beta}_{\epsilon_0}[1,1;\omega]$ to all orders in $\epsilon_0$. The full Heisenberg-Langevin equations are $$\begin{aligned} \label{eq:HL_x_2} \dot{\hx}_n & = -i[\hx_n, \hat{H}_B+\epsilon_0 \ha^\dagger_N \ha_N] - \delta_{n1} \left( \frac{\kappa}{2}\hx_n + \sqrt{\kappa} \hX^{\rm(in)} \right) \\ \label{eq:HL_p_2} \dot{\hp}_n & = -i[\hp_n, \hat{H}_B+\epsilon_0 \ha^\dagger_N \ha_N] - \delta_{n1} \left( \frac{\kappa}{2}\hp_n + \sqrt{\kappa} \hP^{\rm(in)} \right), \end{aligned}$$ where as in the main text we’ve incorporated the drive tone amplitude in the definition of the input operators $\langle \hX^{\rm (in)}\rangle = \beta$ and $\langle \hP^{\rm (in)}\rangle = 0$. Our strategy for solving the Heisenberg-Langevin equations will be nearly identical to that presented in Appendix \[app:chi\_quadrature\]. The key difference is that our squeezing transformation is now defined as $$\begin{aligned} \label{eq:Squeeze_X_2} & \hx_n = e^{A(n-N)} \htx_n \\ \label{eq:Squeeze_P_2} & \hp_n = e^{-A(n-N)} \htp_n.\end{aligned}$$ In this new frame, the Hamiltonian $\hH_B + \epsilon_0 \hta_N^\dagger \hta_N$ preserves total quasiparticle number $\hat{\tilde{N}}$. The Heisenberg-Langevin equations are then $$\begin{aligned} \dot{\htx}_n & = -i[\htx_n, \hat{H}_B+\epsilon_0 \hta_N^\dagger \hta_N] - \delta_{n1} \left( \frac{\kappa}{2}\htx_n + e^{A(N-1)} \sqrt{\kappa} \hX^{\rm(in)} \right) \\ \dot{\htp}_n & = -i[\htp_n, \hat{H}_B+\epsilon_0 \hta_N^\dagger \hta_N] - \delta_{n1} \left( \frac{\kappa}{2}\htp_n + e^{-A(N-1)} \sqrt{\kappa} \hP^{\rm(in)} \right) \end{aligned}$$ Crucially, we can immediately conclude that our chain is dynamically stable for any value of $\epsilon_0$ and $A$: the spectrum is determined by the particle conserving Hamiltonian $\hH_B+\epsilon_0 \hta_N^\dagger \hta_N $ and dissipation $\kappa/2$ on the first site. Using these squeezing transformations in conjunction with Eqs. (\[eq:quad\_diag\]) and (\[eq:Suscept\_Zero\]), we obtain the full form of the susceptibilities: $$\begin{aligned} \label{eq:chi_xx_full} \chi^{xx}_{\epsilon_0}(n,m;t) &= e^{A(n-m)} \Re \tilde{\chi}_{\epsilon_0}(n,m;t) \\ \chi^{pp}_{\epsilon_0}(n,m;t) &= e^{-A(n-m)} \Re \tilde{\chi}_{\epsilon_0}(n,m;t) \\ \label{eq:chi_xp_full_1} \chi^{xp}_{\epsilon_0}(n,m;t) &= - e^{-A(2N-n-m)} \Im \tilde{\chi}_{\epsilon_0}(n,m;t) \\ \label{eq:chi_xp_full} \chi^{px}_{\epsilon_0}(n,m;t) &= e^{A(2N-n-m)} \Im \tilde{\chi}_{\epsilon_0}(n,m;t)\end{aligned}$$ where $\tilde{\chi}_{\epsilon_0}(n,m;t)$ is the susceptibility matrix of the complex modes $\hta_n$. We already have the susceptibilities $\tilde{\chi}[n,m;\omega]$ of our tight-binding chain which incorporate the full effects of the the waveguide via Eq. (\[eq:Dyson\]). The frequency shift on the last site adds a term $-i\epsilon_0 \delta_{n,N} \delta_{m,N}$ to the dynamical matrix. Dyson’s equation in frequency space the gives: $$\begin{aligned} \tilde{\chi}_{\epsilon_0}[n,m;\omega] &= \tilde{\chi}[n,m;\omega] - i\epsilon_0 \tilde{\chi}[n,N;\omega] \tilde{\chi}_{\epsilon_0}[N,m;\omega] \\ &= \tilde{\chi}[n,m;\omega] - \frac{i \epsilon_0 \tilde{\chi}[n,N;\omega] \tilde{\chi}[N,m;\omega] }{1+i \epsilon_0 \tilde{\chi}[N,N;\omega]}.\end{aligned}$$ Since there is a driving force only on the first site, we just need to find the susceptibilities to a force on the first site: $$\begin{aligned} \label{eq:chi_full} \tilde{\chi}_{\epsilon_0}[n,1;\omega] = i^{n} \frac{U_{N-n}(\frac{\omega}{2J})-\frac{\epsilon_0}{J} U_{N-1-n}(\frac{\omega}{2J})} {J U_N(\frac{\omega}{2J})+(i\frac{\kappa}{2}-\epsilon_0)U_{N-1}(\frac{\omega}{2J})-i\frac{\epsilon_0}{J} \frac{\kappa}{2}U_{N-2}(\frac{\omega}{2J})}\end{aligned}$$ We now compute the steady state total photon number $\bar{n}_{\rm tot}(\epsilon_0)$ when $\epsilon_0 \neq 0$. Recall we are interested in the regime where $\epsilon_0/\kappa \ll 1$ but $e^{A(N-1)} \epsilon_0/\kappa$ is not a priori small. The form of our susceptibilities Eqs.(\[eq:chi\_xx\_full\])-(\[eq:chi\_xp\_full\]) implies that $A$ doesn’t effect the spectrum, but just the residue of the poles as expected from our previous discussion. A non-zero value of $\epsilon_0$ changes both the coherent drive-induced photon number, in addition to drive-independent photons generated from input vacuum fluctuations. The leading order correction to the total photon number when $\epsilon_0 \neq 0$ is therefore $$\begin{aligned} \bar{n}_{\rm tot}(\epsilon_0) = \bar{n}_{\rm tot}(0) + ( c \frac{\beta^2}{\kappa} + d ) e^{4A(N-1)} (\frac{\epsilon_0}{\kappa})^2 + \mathcal{O}( e^{4A(N-2)} (\frac{\epsilon_0}{\kappa})^2 )\end{aligned}$$ where $c$ and $d$ are constants of order unity. Thus, Eq. (\[eq:Full\_SNR\]) gives $$\begin{aligned} Q(A, \epsilon_0) = \frac{4|\beta|^2 e^{2A(N-1)}/\kappa} { \bar{n}_{\rm tot}(0) + \frac{1}{2} (c \frac{\beta^2}{\kappa}+d)e^{4A(N-1)}(\frac{\epsilon_0}{\kappa})^2 + \mathcal{O}(e^{4A(N-2)}(\frac{\epsilon_0}{\kappa})^2) }.\end{aligned}$$ With the optimal amplification factor $A^*$ $$\begin{aligned} e^{4A^*(N-1)} = \frac{\kappa^2}{8 \epsilon_0^2}\end{aligned}$$ in conjunction with Eq. (\[eq:Total\_n\]), we get $$\begin{aligned} Q(A^*, \epsilon_0) & = \left( \frac{1-e^{-2A^*(N+1)}}{1-e^{-4A^*}} + \mathcal{O}(e^{-2A^*(N-1)}) \right)^{-1} \\ & = 1-\mathcal{O}((\frac{8 \epsilon_0^2}{\kappa^2})^{\frac{1}{N-1}})\end{aligned}$$ As in the main text. Note that we’ve taken the relevant limit $\beta^2/\kappa \gg 1$ such that we can ignore the amplified vacuum fluctuations to the total photon number With the susceptibilities Eqs. (\[eq:chi\_xx\_full\])-\[eq:chi\_xp\_full\], we can also compute the quadrature-quadrature scattering matrix. If we first define $$\begin{aligned} s[\omega] &= 1- \kappa \tilde{\chi}_{\epsilon_0}[1,1;\omega] \\ &= \frac {a[\omega]+i b[\omega]} {a[\omega]-ib[\omega]}\end{aligned}$$ with $$\begin{aligned} &a[\omega] = J U_N(\frac{\omega}{2J})-U_{N-1}(\frac{\omega}{2J}) \epsilon_0 \\ &b[\omega] = \frac{\kappa}{2}\left(\frac{\epsilon_0}{J}U_{N-2}(\frac{\omega}{2J})-U_{N-1}(\frac{\omega}{2J})\right)\end{aligned}$$ then using the input-output boundary conditions Eq. (\[eq:In\_Out\]) we find that the scattering matrix is $$\begin{aligned} \boldsymbol{s}[\omega] = \begin{pmatrix} R[\omega] & -T[\omega]e^{-2A(N-1)} \\ T[\omega]e^{2A(N-1)} & R[\omega] \end{pmatrix}\end{aligned}$$ with $$\begin{aligned} & R[\omega] = \frac{1}{2} \left( s[\omega] + s^*[-\omega] \right) \\ & T[\omega] = \frac{1}{2i} \left( s[\omega] - s^*[-\omega] \right).\end{aligned}$$ Note that $|s[\omega]|^2 = 1$, which implies $|R[\omega]|^2+|T[\omega]|^2 = 1$. The zero-frequency component of $R(\epsilon_0)$ and $T(\epsilon_0)$ are then: $$\begin{aligned} R(\epsilon_0) = -\frac{(\frac{\kappa}{2})^2-\epsilon_0^2}{(\frac{\kappa}{2})^2+\epsilon_0^2} \\ T(\epsilon_0) = \frac{\kappa \epsilon_0}{(\frac{\kappa}{2})^2+\epsilon_0^2}\end{aligned}$$ as in the main text.

--- abstract: 'A *binary frame template* is a device for creating binary matroids from graphic or cographic matroids. Such matroids are said to *conform* or *coconform* to the template. We introduce a preorder on these templates and determine the nontrivial templates that are minimal with respect to this order. As an application of our main result, we determine the eventual growth rates of certain minor-closed classes of binary matroids, including the class of binary matroids with no minor isomorphic to $PG(3,2)$. Our main result applies to all highly-connected matroids in a class, not just those of maximum size. As a second application, we characterize the highly-connected 1-flowing matroids.' address: - | Department of Mathematics\ Louisiana State University\ Baton Rouge, Louisiana - | Department of Mathematics\ Louisiana State University\ Baton Rouge, Louisiana author: - Kevin Grace - 'Stefan H. M. van Zwam' title: Templates for Binary Matroids --- [^1] Introduction ============ Geelen, Gerards, and Whittle  [@ggw15] recently announced a structure theorem describing the highly connected members of any proper minor-closed class of matroids representable over a given finite field. In this paper we study some consequences of their result. To state a first, rough version of their result, we need the following definitions. A matroid $M$ is *vertically $k$-connected* if, for each partition $(X,Y)$ of the ground set of $M$ with $r(X)+r(Y)-r(M)<k-1$, either $X$ or $Y$ is spanning. We denote the unique prime subfield of $\mathbb{F}$ by $\mathbb{F}_{\textnormal{prime}}$. We say that a matroid $M_2$ is a *rank-$(\leq t)$ perturbation* of a matroid $M_1$ if there exist matrices $A_1$ and $A_2$ over $\mathbb{F}$ such that $r(M(A_1-A_2))\leq t$ and such that $M_1\cong M(A_1)$ and $M_2\cong M(A_2)$. We now restate  [@ggw15 Theorem 3.3]. Its proof is forthcoming in a future paper by Geelen, Gerards, and Whittle. \[ggw3.3\] Let $\mathbb{F}$ be a finite field and let $m_0$ be a positive integer. Then there exist $k,n,t\in\mathbb{Z}_+$ such that, if $M$ is a matroid representable over $\mathbb{F}$ such that $M$ or $M^*$ is vertically $k$-connected and such that $M$ has an $M(K_n)$-minor but no $PG(m_0-1,\mathbb{F}_{\textnormal{prime}})$-minor, then $M$ is a rank-$(\leq t)$ perturbation of a frame matroid representable over $\mathbb{F}$. Let us consider a very simple example of a rank-1 perturbation. Let $A_1$ be the binary matrix $$\begin{bmatrix} 1&0&0&0&1&1&1&0&0&0\\ 0&1&0&0&1&0&0&1&1&0\\ 0&0&1&0&0&1&0&1&0&1\\ 0&0&0&1&0&0&1&0&1&1\\ \end{bmatrix},$$ and let $A_2$ be the binary matrix $$\begin{bmatrix} 0&1&1&1&1&1&1&0&0&0\\ 1&0&1&1&1&0&0&1&1&0\\ 0&0&1&0&0&1&0&1&0&1\\ 0&0&0&1&0&0&1&0&1&1\\ \end{bmatrix}.$$ Note that $A_2$ is the result of adding the rank-1 matrix $$\begin{bmatrix} 1&1&1&1&0&0&0&0&0&0\\ 1&1&1&1&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ \end{bmatrix}$$ to $A_1$. Therefore, the vector matroid $M(A_2)$ is a rank-1 perturbation of $M(A_1)$. Theorem  \[ggw3.3\] is essentially a simplified version of a much more complex structure theorem  [@ggw15 Theorem 4.2]. Geelen, Gerards, and Whittle introduced the concept of a *template* as a tool to capture much of this complexity. Our focus in this paper is on the binary case. Roughly speaking, a binary frame template can be thought of as a recipe for constructing a representable matroid from a graphic or cographic matroid. A matroid constructed in this way is said to *conform* or *coconform* to the template. In the example above, we may think of $M(A_2)$ as the matroid obtained from the vector matroid of the following matrix by contracting the element indexing the final column. Note that the large submatrix on the bottom left is $A_1$: $$\left[ \begin{array}{@{}cccccccccc|c@{}} 1&1&1&1&0&0&0&0&0&0&1\\ \hline 1&0&0&0&1&1&1&0&0&0&1\\ 0&1&0&0&1&0&0&1&1&0&1\\ 0&0&1&0&0&1&0&1&0&1&0\\ 0&0&0&1&0&0&1&0&1&1&0\\ \end{array} \right]$$ In fact, for any matrix $A$ of the following form, where $v$ and $w$ are arbitrary binary vectors, the matroid $M(A)/c$ conforms to the template $\Phi_{CX}$, which we will define in Section  \[Reducing a Template\]: ----------------------------- ----- $v$ 1 incidence matrix of a graph $w$ ----------------------------- ----- Let $\mathcal{M}(\Phi)$ denote the set of matroids representable over a field $\mathbb{F}$ that conform to a frame template $\Phi$. Theorem  \[ggwframe\] below is a slight modification of  [@ggw15 Theorem 4.2]. The modification is explained in Section  \[Preliminaries\]. \[ggwframe\] Let $\mathbb{F}$ be a finite field, let $m$ be a positive integer, and let $\mathcal{M}$ be a minor-closed class of matroids representable over $\mathbb{F}$. Then there exist $k,l\in \mathbb{Z}_+$ and frame templates $\Phi_1,\dots, \Phi_s, \Psi_1,\dots, \Psi_t$ such that - $\mathcal{M}$ contains each of the classes $\mathcal{M}(\Phi_1),\dots,\mathcal{M}(\Phi_s)$, - $\mathcal{M}$ contains the duals of the matroids in each of the classes $\mathcal{M}(\Psi_1),\dots,\mathcal{M}(\Psi_t)$, and - if $M$ is a simple vertically $k$-connected member of $\mathcal{M}$ with at least $l$ elements and with no $PG(m-1,\mathbb{F}_{\textnormal{prime}})$ minor, then either $M$ is a member of at least one of the classes $\mathcal{M}(\Phi_1),\dots,\mathcal{M}(\Phi_s)$, or $M^*$ is a member of at least one of the classes $\mathcal{M}(\Psi_1),\dots,\mathcal{M}(\Psi_t)$. Our contribution is to shed some light on how these templates are related to each other. We define a preorder on the set of frame templates. Our main result, Theorem  \[minimal\], is a list of nontrivial binary frame templates that are minimal with respect to this preorder. One application of this result involves growth rates of minor-closed classes of binary matroids. The *growth rate function* of a minor-closed class $\mathcal{M}$ is the function whose value at an integer $r\geq0$ is given by the maximum number of elements in a simple matroid in $\mathcal{M}$ of rank at most $r$. We prove that a minor-closed class of binary matroids has a growth rate that is eventually equal to the growth rate of the class of graphic matroids if and only if it contains all graphic matroids but does not contain the class of matroids conforming to a certain template. The class of matroids conforming to this template is exactly the class of matroids having an even-cycle representation with a blocking pair. Geelen and Nelson also proved this result in  [@gn15]. We also prove the following theorem. Here, $\mathcal{EX}(F)$ denotes the class of binary matroids with no $F$-minor. If $f$ and $g$ are functions, we write $f(r)\approx g(r)$ if $f(r)=g(r)$ for all but finitely many $r$. \[EXPG32\] The growth rate function for $\mathcal{EX}(PG(3,2))$ is $$h_{\mathcal{EX}(PG(3,2))}\approx r^2-r+1.$$ Note that $r^2-r+1$ is the growth rate of the class of even-cycle matroids. Our main result goes beyond growth rates because it gives information about all highly-connected matroids in a minor-closed class, not just the maximum-sized matroids. This is illustrated by our second application, involving 1-flowing matroids. The 1-flowing property is a generalization of the max-flow min-cut property of graphs. We prove the following. \[1flowing\] There exist $k,l\in\mathbb{Z}_+$ such that every simple, vertically $k$-connected, 1-flowing matroid with at least $l$ elements is either graphic or cographic. We use templates to study a minor-closed class $\mathcal{M}$ by describing the highly-connected matroids in the class. This analysis follows a certain pattern: 1. Find a matroid $N$ not in $\mathcal{M}$. 2. Find all templates such that $N$ is not a minor of any matroid conforming to that template. 3. If all matroids conforming to these templates are in $\mathcal{M}$, then the analysis is complete. 4. Otherwise, repeat Step (1). From the definition of conforming to a template, which we will give in Section  \[Preliminaries\], it will not be difficult to see that for each binary frame template $\Phi$, there are integers $t_1$ and $t_2$ such that every matroid conforming to $\Phi$ is a rank-$(\leq t_1)$ perturbation of a graphic matroid and every matroid coconforming to $\Phi$ is a rank-$(\leq t_2)$ perturbation of a cographic matroid. Thus, by Theorem  \[ggwframe\], the highly connected matroids in a minor-closed class of binary matroids are “close” to being graphic or cographic. In this regard, the work regarding templates resembles work done by Robertson and Seymour concerning minor-closed classes of graphs. In Theorem 1.3 of  [@rs03], Robertson and Seymour showed that highly-connected graphs in a minor-closed class are in some sense “close” to being embeddable in some surface. Section  \[Preliminaries\] of this paper repeats the necessary definitions found in  [@ggw15]. In Section  \[Reducing a Template\], we prove our main result, as well as giving some machinery leading up to it. Section  \[Growth Rates\] applies our result to growth rates of minor-closed classes of binary matroids, and in Section  \[1-flowing Matroids\], we prove Theorem  \[1flowing\]. Preliminaries {#Preliminaries} ============= We repeat here several definitions concerning highly connected matroids which can be found in Geelen, Gerards, and Whittle  [@ggw15]. Although the results found in  [@ggw15] are technically about matrices rather than matroids, it suffices for our purposes to state the results in terms of their immediate matroid consequences. Let $A$ be a matrix over a field $\mathbb{F}$. Then $A$ is a *frame matrix* if each column of $A$ has at most two nonzero entries. We let $\mathbb{F}^{\times}$ denote the multiplicative group of $\mathbb{F}$. Let $\Gamma$ be a subgroup of $\mathbb{F}^{\times}$. A $\Gamma$-frame matrix is a frame matrix $A$ such that: - Each column of $A$ with a nonzero entry contains a 1. - If a column of $A$ has a second nonzero entry, then that entry is $-\gamma$ for some $\gamma\in\Gamma$. In the case where $\Gamma$ is the multiplicative group of one element, a matrix is a $\Gamma$-frame matrix if and only if it is the signed incidence matrix of a graph, with possibly a row removed. In particular, a binary matroid is graphic if and only if it can be represented by a matrix over $\mathrm{GF}(2)$ in which no column has more than two nonzero entries. To facilitate the description of their structure theorem, Geelen, Gerards, and Whittle capture capture much of the complexity with the concept of a “template.” Let $\mathbb{F}$ be a finite field. A *frame template* over $\mathbb{F}$ is a tuple $\Phi=(\Gamma,C,X,Y_0,Y_1,A_1,\Delta,\Lambda)$ such that the following hold[^2]: - $\Gamma$ is a subgroup of $\mathbb{F}^{\times}$. - $C$, $X$, $Y_0$ and $Y_1$ are disjoint finite sets. - $A_1\in \mathbb{F}^{X\times (C\cup Y_0\cup Y_1)}$. - $\Lambda$ is a subgroup of the additive group of $\mathbb{F}^X$ and is closed under scaling by elements of $\Gamma$. - $\Delta$ is a subgroup of the additive group of $\mathbb{F}^{C\cup Y_0 \cup Y_1}$ and is closed under scaling by elements of $\Gamma$. Let $\Phi=(\Gamma,C,X,Y_0,Y_1,A_1,\Delta,\Lambda)$ be a frame template. Let $B$ and $E$ be finite sets, and let $A'\in\mathbb{F}^{B\times E}$. We say that $A'$ *respects* $\Phi$ if the following hold: - $X\subseteq B$ and $C, Y_0, Y_1\subseteq E$. - $A'[X, C\cup Y_0\cup Y_1]=A_1$. - There exists a set $Z\subseteq E-(C\cup Y_0\cup Y_1)$ such that $A'[X,Z]=0$, each column of $A'[B-X,Z]$ is a unit vector, and $A'[B-X, E-(C\cup Y_0\cup Y_1\cup Z)]$ is a $\Gamma$-frame matrix. - Each column of $A'[X,E-(C\cup Y_0\cup Y_1\cup Z)]$ is contained in $\Lambda$. - Each row of $A'[B-X, C\cup Y_0\cup Y_1]$ is contained in $\Delta$. Figure \[fig:A’\] shows the structure of $A'$. [ r|c|c|ccc| ]{} &&&&\ &&&&&\ $X$&columns from $\Lambda$&$0$&&$A_1$&\ &&&&&\ &&&\ &&&&&\ &&&&&\ &&&&&\ &&&&&\ Suppose that $A'$ respects $\Phi$ and that $Z$ satisfies (iii) above. Now suppose that $A\in \mathbb{F}^{B\times E}$ satisfies the following conditions: - $A[B,E-Z]=A'[B,E-Z]$ - For each $i\in Z$ there exists $j\in Y_1$ such that the $i$-th column of $A$ is the sum of the $i$-th and the $j$-th columns of $A'$. We say that any such matrix *conforms* to $\Phi$. Let $M$ be a matroid representable over $\mathbb{F}$. We say that $M$ *conforms* to $\Phi$ if there is a matrix $A$ that conforms to $\Phi$ such that $M$ is isomorphic to $M(A)/C\backslash Y_1$. Let $\mathcal{M}(\Phi)$ denote the set of matroids representable over $\mathbb{F}$ that conform to $\Phi$. Recall that a matroid $M$ is *vertically $k$-connected* if, for each partition $(X,Y)$ of the ground set of $M$ with $r(X)+r(Y)-r(M)<k-1$, either $X$ or $Y$ is spanning. We denote the unique prime subfield of $\mathbb{F}$ by $\mathbb{F}_{\textnormal{prime}}$. Geelen, Gerards, and Whittle will prove Theorem  \[ggwframe\] in a future paper. This theorem is actually a slight modification of the theorem found in  [@ggw15]. In that paper, there is no mention of the requirement that a matroid have size at least $l$. However, Geelen (personal communication) has stated that this is necessary to ensure that adding a finite number of matroids to the class $\mathcal{M}$ does not add any templates to the list $\Phi_1,\dots, \Phi_s, \Psi_1,\dots, \Psi_t$. Although the term *coconform* does not appear in  [@ggw15], we define it in the following obvious way. A matroid $M$ *coconforms* to a template $\Phi$ if its dual $M^*$ conforms to $\Phi$. To simplify the proofs in this paper, it will be helpful to expand the concept of conforming slightly. \[virtual\] Let $A'$ be a matrix that respects $\Phi$, as defined above, except that we allow columns of $A'[B-X,Z]$ to be either unit columns or zero columns. Let $A$ be a matrix that is constructed from $A'$ as described above. Thus, $A[B,E-Z]=A'[B,E-Z]$, and for each $i\in Z$ there exists $j\in Y_1$ such that the $i$-th column of $A$ is the sum of the $i$-th and the $j$-th columns of $A'$. Let $M$ be isomorphic to $M(A)/C\backslash Y_1$. We say that $A$ and $M$ *virtually conform* to $\Phi$ and that $A'$ *virtually respects* $\Phi$. If $M^*$ virtually conforms to $\Phi$, we say that $M$ *virtually coconforms* to $\Phi$. We will denote the set of matroids representable over $\mathbb{F}$ that virtually conform to $\Phi$ by $\mathcal{M}_v(\Phi)$ and the set of matroids representable over $\mathbb{F}$ that virtually coconform to $\Phi$ by $\mathcal{M}^*_v(\Phi)$. The following notation will be used throughout this paper. We denote an empty matrix by $[\emptyset]$. We denote a group of one element by $\{0\}$ or $\{1\}$, if it is an additive or multiplicative group, respectively. If $S'$ is a subset of a set $S$ and $G$ is a subgroup of the additive group $\mathbb{F}^S$, we denote by $G|S'$ the projection of $G$ into $\mathbb{F}^{S'}$. Similarly, if $\bar{x}\in G$, we denote the projection of $\bar{x}$ into $\mathbb{F}^{S'}$ by $\bar{x}|S'$. Unexplained notation and terminology will generally follow Oxley  [@o11]. One exception is that we denote the vector matroid of a matrix $A$ by $M(A)$, rather than $M[A]$. Reducing a Template {#Reducing a Template} =================== In this section, we will introduce reductions and show that every template reduces to one of several basic templates. Since templates are used to study minor-closed classes of matroids, a natural question to ask is whether the set of matroids conforming to a particular template is minor-closed. The answer is no, in general. For example, if $|Y_0|=1$, then a matroid conforms to the following binary frame template if and only if it is a graphic matroid with a loop: $$(\{1\},\emptyset,\emptyset,Y_0,\emptyset,[\emptyset],\{0\},\{0\}).$$ Clearly, this is not a minor-closed class. Another question to ask is whether there might be some sort of minor relationship between a pair of templates, where every matroid conforming to one template is a minor of a matroid conforming to the other. These questions motivate the following discussion. A *reduction* is an operation on a frame template $\Phi$ that produces a frame template $\Phi'$ such that $\mathcal{M}(\Phi')\subseteq \mathcal{M}(\Phi)$. \[reductions\] The following operations are reductions on a frame template $\Phi$: - Replace $\Gamma$ with a proper subgroup. - Replace $\Lambda$ with a proper subgroup closed under multiplication by elements from $\Gamma$. - Replace $\Delta$ with a proper subgroup closed under multiplication by elements from $\Gamma$. - Remove an element $y$ from $Y_1$. (More precisely, replace $A_1$ with $A_1[X, Y_0\cup (Y_1-y)\cup C]$ and replace $\Delta$ with $\Delta|(Y_0\cup (Y_1-y)\cup C)$. - For all matrices $A'$ respecting $\Phi$, perform an elementary row operation on $A'[X, E]$. (Note that this alters the matrix $A_1$ and performs a change of basis on $\Lambda$.) - If there is some element $x\in X$ such that, for every matrix $A'$ respecting $\Phi$, we have that $A'[\{x\},E]$ is a zero row vector, remove $x$ from $X$. (This simply has the effect of removing a zero row from every matrix conforming to $\Phi$.) - Let $c\in C$ be such that $A_1[X,\{c\}]$ is a unit column whose nonzero entry is in the row indexed by $x\in X$, and let either $\lambda_x=0$ for each $\lambda\in\Lambda$ or $\delta_c=0$ for each $\delta\in\Delta$. Let $\Delta'$ be the result of adding $-\delta_cA_1[\{x\},Y_0\cup Y_1\cup C]$ to each element $\delta\in\Delta$. Replace $\Delta$ with $\Delta'$, and then remove $c$ from $C$ and $d$ from $D$. (More precisely, replace $A_1$ with $A_1[X-x, Y_0\cup Y_1\cup (C-c)]$, replace $\Lambda$ with $\Lambda|(X-x)$, and replace $\Delta$ with $\Delta'|(Y_0\cup Y_1\cup (C-c))$.) - Let $c\in C$ be such that $A_1[X,\{c\}]$ is a zero column and $\delta_c=0$ for all $\delta\in\Delta$. Then remove $c$ from $C$. (More precisely, replace $A_1$ with $A_1[X, Y_0\cup Y_1\cup (C-c)]$, and replace $\Delta$ with $\Delta|(Y_0\cup Y_1\cup (C-c))$.) Let $\Phi'$ be the template that results from performing one of operations (1)-(8) on $\Phi$. For (1)-(3), every matrix $A'$ respecting $\Phi'$ also respects $\Phi$. For (4), let $A'$ be a matrix respecting $\Phi'$, and let $M$ be the matroid $M(A)/C\backslash Y_1$, where $A$ is a matrix conforming to $\Phi'$ that has been constructed from $A'$ respecting $\Phi'$ as described in Section  \[Preliminaries\]. Since $Y_1$ is deleted to produce $M$, the only effect of $Y_1$ on $M$ is that for each $i\in Z$ there exists $j\in Y_1$ such that the $i$-th column of $A$ is the sum of the $i$-th and the $j$-th columns of $A'$. But each $j\in Y_1$ in the template $\Phi'$ is also contained in $Y_1$ in the template $\Phi$. Therefore, $A$ conforms to $\Phi$, as does $M$. For (5) and (6), elementary row operations and removing zero rows produce isomorphic matroids. Operations (7) and (8) have the effect of contracting $c$ from $M(A)\backslash Y_1$ for every matrix $A$ conforming to $\Phi$. Since all of $C$ is contracted to produce a matroid $M$ conforming to $\Phi$, the matroids we produce by performing either of these operations still conform to $\Phi$. For $i\in\{1,\dots, 8\}$, we call operation $(i)$ above a *reduction of type $i$*. The operations listed in the definition below are not reductions as defined above, but we continue the numbering from Proposition  \[reductions\] for ease of reference. \[weaklyconforming\] A template $\Phi'$ is a *template minor* of $\Phi$ if $\Phi'$ is obtained from $\Phi$ by repeatedly performing the following operations: - Performing a reduction of type 1-8 on $\Phi$. - Removing an element $y$ from $Y_0$, replacing $A_1$ with $A_1[X,(Y_0-y)\cup Y_1\cup C]$, and replacing $\Delta$ with $\Delta|((Y_0-y)\cup Y_1\cup C)$. (This has the effect of deleting $y$ from every matroid conforming to $\Phi$.) - Let $x\in X$ with $\lambda_x=0$ for every $\lambda\in\Lambda$, and let $y\in Y_0$ be such that $(A_1)_{x,y}\neq0$. Then contract $y$ from every matroid conforming to $\Phi$. (More precisely, perform row operations on $A_1$ so that $A_1[X, \{y\}]$ is a unit column with $(A_1)_{x,y}=1$. Then replace every element $\delta\in\Delta$ with the row vector $-\delta_y A_1[\{x\}, Y_0\cup Y_1\cup C]+\delta$. This induces a group homomorphism $\Delta\rightarrow\Delta'$, where $\Delta'$ is also a subgroup of the additive group of $\mathbb{F}^{C\cup Y_0 \cup Y_1}$ and is closed under scaling by elements of $\Gamma$. Finally, replace $A_1$ with $A_1[X-x,(Y_0-y)\cup Y_1\cup C]$, project $\Lambda$ into $\mathbb{F}^{X-x}$, and project $\Delta'$ into $\mathbb{F}^{(Y_0-y)\cup Y_1\cup C}$. The resulting groups play the roles of $\Lambda$ and $\Delta$, respectively in $\Phi'$.) - Let $y\in Y_0$ with $\delta_y=0$ for every $\delta\in\Delta$. Then contract $y$ from every matroid conforming to $\Phi$. (More precisely, if $A_1[X, \{y\}]$ is a zero vector, this is the same as simply removing $y$ from $Y_0$. Otherwise, choose some $x\in X$ such that $(A_1)_{x,y}\neq0$. Then for every matrix $A'$ that respects $\Phi$, perform row operations so that $A_1[X,\{y\}]$ is a unit column with $(A_1)_{x,y}=1$. This induces a group isomorphism $\Lambda\rightarrow\Lambda'$ where $\Lambda'$ is also a subgroup of the additive group of $\mathbb{F}^X$ and is closed under scaling by elements of $\Gamma$. Finally, replace $A_1$ with $A_1[X-x,(Y_0-y)\cup Y_1\cup C]$, project $\Lambda'$ into $\mathbb{F}^{X-x}$, and project $\Delta$ into $\mathbb{F}^{(Y_0-y)\cup Y_1\cup C}$. The resulting groups play the roles of $\Lambda$ and $\Delta$, respectively in $\Phi'$.) Let $\Phi'$ be a template minor of $\Phi$, and let $A'$ be a matrix that virtually respects $\Phi'$. Let $A$ be a matrix that virtually conforms to $\Phi'$, and let $M$ be a matroid that virtually conforms to $\Phi'$. We say that $A'$ *weakly respects* $\Phi$ and that $A$ and $M$ *weakly conform* to $\Phi$. Let $\mathcal{M}_w(\Phi)$ denote the set of matroids representable over $\mathbb{F}$ that weakly conform to $\Phi$, and let $\mathcal{M}^*_w(\Phi)$ denote the set of matroids representable over $\mathbb{F}$ whose duals weakly conform to $\Phi$. If $M\in\mathcal{M}^*_w(\Phi)$, we say that $M$ *weakly coconforms* to $\Phi$. \[minor\] If a matroid $M$ weakly conforms to a template $\Phi$, then $M$ is a minor of a matroid that conforms to $\Phi$. Let $\Phi'$ be a template minor of $\Phi$. As we can see from Definition  \[weaklyconforming\], every matroid $M$ weakly conforming to $\Phi'$ is a minor of a matroid virtually conforming to $\Phi$. It remains to analyze the case where $M$ virtually conforms to $\Phi$; so $M$ is isomorphic to $M(K)/C\backslash Y_1$, where $K$ is built from a matrix $K'$ that virtually respects $\Phi$. Consider the following matrix $A'$ obtained from $K'$ by adding a row $r$ and a column $c$. [ r|c|c|c|c|ccc| ]{} &&&&&&\ &&&&&&\ $X$&0&columns from $\Lambda$&&&$A_1$&\ &&&&&&\ &&&&&\ &&&&&&&\ &&&&&&&\ &&&&&&&\ &&&&&&&\ $r$&1&0&$1\cdots1$&0&\ From $A'$, we can obtain a matrix $A$ conforming to $\Phi$ such that $M$ is isomorphic to $M(A)/C\backslash Y_1/c$. So $M$ is a minor of a matroid conforming to $\Phi$. An easy consequence of Lemma  \[minor\] is that Theorem  \[ggwframe\], which deals with minor-closed classes, can be restated in terms of weak conforming. \[weakframe\] Let $\mathbb{F}$ be a finite field, let $m$ be a positive integer, and let $\mathcal{M}$ be a minor-closed class of matroids representable over $\mathbb{F}$. Then there exist $k,l\in \mathbb{Z}_+$ and frame templates $\Phi_1,\dots, \Phi_s, \Psi_1,\dots, \Psi_t$ such that - $\mathcal{M}$ contains each of the classes $\mathcal{M}_w(\Phi_1),\dots,\mathcal{M}_w(\Phi_s)$, - $\mathcal{M}$ contains the duals of the matroids in each of the classes $\mathcal{M}_w(\Psi_1)$,$\dots$,$\mathcal{M}_w(\Psi_t)$, and - if $M$ is a simple vertically $k$-connected member of $\mathcal{M}$ with at least $l$ elements and with no $PG(m-1,\mathbb{F}_{\textnormal{prime}})$ minor, then either $M$ is a member of at least one of the classes $\mathcal{M}_v(\Phi_1),\dots,\mathcal{M}_v(\Phi_s)$ or $M^*$ is a member of at least one of the classes $\mathcal{M}_v(\Psi_1),\dots,\mathcal{M}_v(\Psi_t)$. Let $\Phi_1,\dots, \Phi_s, \Psi_1,\dots, \Psi_t$ be the templates whose existence is implied by Theorem  \[ggwframe\]. For $\Phi\in\{\Phi_1,\dots, \Phi_s\}$, Lemma  \[minor\] implies that any matroid $M\in \mathcal{M}_w(\Phi)$ is a minor of a matroid $N\in\mathcal{M}(\Phi)$. Since $\mathcal{M}$ contains $\mathcal{M}(\Phi)$ and is minor-closed, $\mathcal{M}$ contains $\mathcal{M}_w(\Phi)$ as well. Similarly, $\mathcal{M}$ contains the duals of the matroids in each of the classes $\mathcal{M}_w(\Psi_1),\dots,\mathcal{M}_w(\Psi_t)$. The third condition above holds since every matroid conforming to a template also virtually conforms to it. If $\mathcal{M}_{w}(\Phi)=\mathcal{M}_{w}(\Phi')$, we say that $\Phi$ is *equivalent* to $\Phi'$ and write $\Phi\sim\Phi'$. It is clear that $\sim$ is indeed an equivalence relation. Let $T_{\mathbb{F}}$ be the set of all frame templates over $\mathbb{F}$. We define a preorder $\preceq$ on $T_{\mathbb{F}}$ as follows. We say $\Phi\preceq\Phi'$ if $\mathcal{M}_w(\Phi)\subseteq\mathcal{M}_w(\Phi')$. This is indeed a preorder since reflexivity and transitivity follow from the subset relation. We may obtain a partial order by considering equivalence classes of templates, with equivalence as defined above. However, the templates themselves, rather than equivalence classes, are the objects we work with in this paper. Let $\Phi_0$ be the frame template with all groups trivial and all sets empty. We call this template the *trivial template*. In general, we say that a template $\Phi$ is *trivial* if $\Phi\preceq\Phi_0$. It is easy to see that for any template $\Phi$, we have $\Phi_0\preceq \Phi$. Therefore, if $\Phi\preceq\Phi_0$, then actually $\Phi\sim\Phi_0$. Our desire is to find a collection of minimal nontrivial templates. For the remainder of this paper, we restrict our attention to binary frame templates: those frame templates where $\mathbb{F}=\mathrm{GF}(2)$ and $\Gamma$ is the group of one element. - Let $\Phi_C$ be the template with all groups trivial and all sets empty except that $|C|=1$ and $\Delta\cong\mathbb{Z}/2\mathbb{Z}$. - Let $\Phi_X$ be the template with all groups trivial and all sets empty except that $|X|=1$ and $\Lambda\cong\mathbb{Z}/2\mathbb{Z}$. - Let $\Phi_{Y_0}$ be the template with all groups trivial and all sets empty except that $|Y_0|=1$ and $\Delta\cong\mathbb{Z}/2\mathbb{Z}$. - Let $\Phi_{CX}$ be the template with $Y_0=Y_1=\emptyset$, with $|C|=|X|=1$, with $\Delta\cong\Lambda\cong\mathbb{Z}/2\mathbb{Z}$, with $\Gamma$ trivial, and with $A_1=[1]$. - Let $\Phi_{Y_1}$ be the template with all groups trivial, with $C=Y_0=\emptyset$, with $|Y_1|=3$ and $|X|=2$, and with $A_1= \begin{bmatrix} 1& 0 &1\\ 0& 1 & 1 \end{bmatrix}$. It is not too difficult to see that the Fano matroid $F_7$ virtually conforms to each of $\Phi_C$, $\Phi_X$, $\Phi_{CX}$, $\Phi_{Y_0}$, and $\Phi_{Y_1}$. Therefore, these templates are nontrivial. In fact, one can see that $\mathcal{M}(\Phi_{Y_0})$ is the set of graft matroids, that $\mathcal{M}(\Phi_C)$ is the class of matroids obtained by closing the set of graft matroids under minors, and that $\mathcal{M}(\Phi_X)$ is the class of even-cycle matroids. In Lemma  \[Y1minors\], we will show that $\mathcal{M}_v(\Phi_{Y_1})$ is the class of matroids having an even-cycle representation with a blocking pair. Our goal in defining reductions and weak conforming was essentially to perform operations on matrices while leaving the $\Gamma$-frame submatrix intact. The following lemma does not contribute to that goal; so we will only make occasional use of it. \[YCD\] The following relations hold: - $\Phi_{Y_1}\preceq\Phi_X$ - $\Phi_{Y_1}\preceq\Phi_C$ - $\Phi_{Y_0}\preceq\Phi_C$ - $\Phi_C\preceq\Phi_{CX}$ - $\Phi_X\preceq\Phi_{CX}$ For (1), note that any simple matroid $M$ of rank $r$ virtually conforming to $\Phi_{Y_1}$ is a restriction of the vector matroid of a matrix $A$ of the following form: ----------------------- --- ----- ----- ------------ ------------ ------------ 1 0 1 $1\cdots1$ $0\cdots0$ $1\cdots1$ 0 1 1 $0\cdots0$ $1\cdots1$ $1\cdots1$ $\Gamma$-frame matrix $I$ $I$ $I$ ----------------------- --- ----- ----- ------------ ------------ ------------ If we label the sets of rows and columns of $A$ as $B$ and $E$ respectively, and the first row as $x$, then we see that $A[B-x,E]$ is a $\Gamma$-frame matrix. If we let $X=\{x\}$, then we see that $M$ conforms to $\Phi_X$. For (2), consider the matrix $A$ above. Note that it is obtained by contracting $c$ in the following matrix: ----------------------- --- ----- ----- ------------ ------------ ------------ --- 0 0 1 0$\cdots$0 0$\cdots$0 1$\cdots1$ 1 1 0 0 $1\cdots1$ $0\cdots0$ $0\cdots0$ 1 0 1 0 $0\cdots0$ $1\cdots1$ $0\cdots0$ 1 $\Gamma$-frame matrix $I$ $I$ $I$ 0 ----------------------- --- ----- ----- ------------ ------------ ------------ --- Removing $c$ from this matrix, we obtain a $\Gamma$-frame matrix. Therefore, $M$ conforms to $\Phi_C$. For (3), any matroid $M$ conforming to $\Phi_{Y_0}$ is the vector matroid of a matrix of the following form, where $v$ is an arbitrary column vector: ----------------------- ----- $\Gamma$-frame matrix $v$ ----------------------- ----- Let $A$ be the matrix below. Label its sets of rows and columns as $B$ and $E$ respectively, and let $c$ be the last column, with $C=\{c\}$. 0 1 1 ----------------------- --- ----- $\Gamma$-frame matrix 0 $v$ Note that $M$ is isomorphic to $M(A)/C$. Since $A[B,E-C]$ is a $\Gamma$-frame matrix, we see that $M$ conforms to $\Phi_C$. For (4), let $A$ be a matrix conforming to $\Phi_C$ and let $M=M(A)/C$ be the corresponding matroid conforming to $\Phi_C$. If the column of $A$ indexed by $C$ is a zero column, then construct the matrix $\bar{A}$ by adding a unit row, indexed by $X$, whose nonzero entry is in the column indexed by $C$. One readily sees that $\bar{A}$ conforms to $\Phi_{CX}$ and that the corresponding matroid $M(\bar{A})/C$ is equal to $M$. Otherwise, if the column of $A$ indexed by $C$ has a nonzero entry, then one readily sees that $A$ conforms to $\Phi_{CX}$ by considering the row containing the nonzero entry to be indexed by $X$. For (5), any matroid $M$ conforming to $\Phi_D$ is the vector matroid of a matrix of the following form, where $v$ is an arbitrary row vector: [|c|]{} $v$\ \ $\Gamma$-frame matrix\ \ Consider the following matrix $A$, whose last column is indexed by $\{c\}=C$: $v$ 1 ----------------------- --- $0$ 1 $\Gamma$-frame matrix 0 The matroid $M$ is isomorphic to $M(A)/c$, which conforms to $\Phi_{CX}$. \[yshift\] Let $\Phi$ be a template with $y\in Y_1$. Let $\Phi'$ be the template obtained from $\Phi$ by removing $y$ from $Y_1$ and placing it in $Y_0$. Then $\Phi'\preceq \Phi$. Any matrix respecting $\Phi'$ virtually respects $\Phi$ by adding column $y$ only to the zero $Z$ column. Thus, any matroid conforming to $\Phi'$ weakly conforms to $\Phi$. We call the operation described in Lemma  \[yshift\] a *$y$-shift*. Let $\Phi=(\Gamma,C,X,Y_0,Y_1,A_1,\Delta,\Lambda)$ be a frame template over a finite field $\mathbb{F}$. We say that $\Phi$ is in *standard form* if there are disjoint sets $C_0,C_1,X_0,$ and $X_1$ such that $C=C_0\cup C_1$, such that $X=X_0\cup X_1$, such that $A_1[X_0,C_0]$ is an identity matrix, and such that $A_1[X_1,C]$ is a zero matrix. Figure  \[fig:A’ standard\], with the stars representing arbitrary matrices, shows a matrix that virtually respects a template in standard form. Note that if $\Phi$ is in standard form, $|C_0|=|X_0|$. Also note that any of $C_0,C_1,X_0,$ or $X_1$ may be empty. Finally, note that we have defined standard form for frame templates over any finite field, not just binary frame templates. [ r|c|c|cccc| ]{} &&&&&\ $X_0$&columns from $\Lambda|X_0$&0&&&$*$\ $X_1$&columns from $\Lambda|X_1$&0&&&\ &&&\ &&&&&&\ &&&&&&\ &&&&&&\ &&&&&&\ \[standard\] Every frame template $\Phi=(\Gamma,C,X,Y_0,Y_1,A_1,\Delta,\Lambda)$ is equivalent to a frame template in standard form. Choose a basis $C_0$ for $M(A_1[X,C])$, and let $C_1=C-C_0$. Repeatedly perform operation (5) to obtain a template $\Phi'$ where $A_1[X,C_0]$ consists of an identity matrix on top of a zero matrix. Each use of operation (5) results in an equivalent template; therefore, $\Phi\sim\Phi'$. Let $X_0\subseteq X$ index the rows of the identity matrix, and let $X_1\subseteq X$ index the rows of the zero matrix. Since $C_0$ is a basis for $M(A_1[X,C])$, the matrix $A_1[X,C_1]$ must be a zero matrix as well. Thus, $\Phi'$ is in standard form. Throughout the rest of this paper, we will implicitly use Lemma  \[standard\] to assume that all templates are in standard form. Also, the operations (1)-(12) to which we will refer throughout the rest of this paper are the operations (1)-(8) from Proposition  \[reductions\] and (9)-(12) from Definition  \[weaklyconforming\]. \[PhiD\] If $\Phi=(\{1\},C,X,Y_0,Y_1,A_1,\Delta,\Lambda)$ is a binary frame template with $\Lambda|X_1$ nontrivial, then $\Phi_X\preceq\Phi$. Perform operations (2) and (3) on $\Phi$ to obtain the following template, where $\lambda$ is an element of $\Lambda$ with $\lambda_x\neq0$ for some $x\in X_1$: $$(\{1\},C,X,Y_0,Y_1,A_1,\{0\},\{\mathbf{0}, \lambda\}).$$ On this template, repeatedly perform operation (7), then (8), then (4), and then (10) until the following template is obtained: $$(\{1\},\emptyset,X_1,\emptyset,\emptyset,[\emptyset],\{0\},\{\mathbf{0}, \lambda\}).$$ On this template, repeatedly perform operation (5) to obtain a template that is identical to the previous one except that the support of $\lambda$ contains only one element of $X_1$. On this template, repeatedly perform operation (6) to obtain the following template, where $x\in X_1$: $$(\{1\},\emptyset,\{x\},\emptyset,\emptyset,[\emptyset],\{0\},\mathbb{Z}/2\mathbb{Z}).$$ This template is $\Phi_X$. \[PhiC\] If $\Phi=(\{1\},C,X,Y_0,Y_1,A_1,\Delta,\Lambda)$ is a binary frame template, then either $\Phi_C\preceq\Phi$ or $\Phi$ is equivalent to a template with $C_1=\emptyset$. Suppose there is an element $\delta\in\Delta|C$ that is not in the row space of $A_1[X,C]$. Repeatedly perform operations (4) and (10) on $\Phi$ until the following template is obtained: $$(\{1\},C,X,\emptyset,\emptyset,A_1[X,C],\Delta|C,\Lambda).$$ On this template, perform operations (2) and (3) to obtain the following template: $$(\{1\},C,X,\emptyset,\emptyset,A_1[X,C],\{\mathbf{0}, \delta\},\{0\}).$$ Every matrix virtually respecting this template is row equivalent to a matrix virtually respecting a template that is identical to the previous template except that there is the additional condition that $\delta|C_0$ is a zero vector. Note that $\delta|C_1$ is nonzero since, in the previous template, $\delta$ was not in the row space of $A_1[X,C]$. Now, on the current template, repeatedly perform operation (7) and then operation (6) to obtain the following template: $$\Phi'=(\{1\},C_1,\emptyset,\emptyset,\emptyset,[\emptyset],\{\mathbf{0}, \delta|C_1\},\{0\}).$$ Now, any matroid $M$ conforming to $\Phi'$ is obtained by contracting $C_1$ from $M(A)$, where $A$ is a matrix conforming to $\Phi'$. By contracting any single element $c\in C_1$, where $\delta_c=1$, we turn the rest of the elements of $C_1$ into loops. So $C_1-c$ is deleted to obtain $M$. Thus, $M$ conforms to the template $$(\{1\},\{c\},\emptyset,\emptyset,\emptyset,[\emptyset],\mathbb{Z}/2\mathbb{Z},\{0\}),$$ which is $\Phi_C$. Similarly, the converse is true that any matroid conforming to $\Phi_C$ conforms to $\Phi'$. Thus, $\Phi_C\sim\Phi'\preceq\Phi$. Now suppose that every element of $\Delta|C$ is in the row space of $A_1[X,C]$. Thus, contraction of $C_0$ turns the elements of $C_1$ into loops, and contraction of $C_1$ is the same as deletion of $C_1$. By deleting $C_1$ from every matrix virtually conforming to $\Phi$, we see that $\Phi$ is equivalent to a template with $C_1=\emptyset$. \[PhiCD\] If $\Phi=(\{1\},C,X,Y_0,Y_1,A_1,\Delta,\Lambda)$ is a binary frame template, then one of the following is true: - $\Phi_C\preceq\Phi$ - $\Phi$ is equivalent to a template with $\Lambda|X_1$ nontrivial and $\Phi_X\preceq\Phi$ - $\Phi$ is equivalent to a template with $\Lambda|X_0$ nontrivial and $\Phi_{CX}\preceq\Phi$ - $\Phi$ is equivalent to a template with $\Lambda$ trivial and $C=\emptyset$. By Lemmas  \[PhiD\] and  \[PhiC\], we may assume that $\Lambda|X_1$ is trivial and that $C_1=\emptyset$. First, suppose there exist elements $\delta\in\Delta|C_0$ and $\lambda\in\Lambda|X_0$ such that there are an odd number of natural numbers $i$ with $\delta_i=\lambda_i=1$. Thus, $\Lambda|X_0$ is nontrivial. Repeatedly perform operations (4) and (10) on $\Phi$ until the following template is obtained: $$(\{1\},C_0,X,\emptyset,\emptyset,A_1[X,C_0],\Delta|C_0,\Lambda).$$ On this template, repeatedly perform operation (6) to obtain the following template: $$\Phi'=(\{1\},C_0,X_0,\emptyset,\emptyset,A_1[X_0,C_0],\Delta|C_0,\Lambda|X_0).$$ Perform operations (2) and (3) on $\Phi'$ to obtain the following template: $$(\{1\},C_0,X_0,\emptyset,\emptyset,A_1[X_0,C_0],\{\mathbf{0}, \delta\},\{\mathbf{0}, \lambda\}).$$ Any matroid conforming to this template is obtained by contracting $C_0$. If $\delta$ is in the row labeled by $r$ and $\lambda$ is in the column labeled by $c$, then when $C_0$ is contracted, 1 is added to the entry of the $\Gamma$-frame matrix in row $r$ and column $c$. Otherwise, the entry remains unchanged when $C$ is contracted. We see then that this template is equivalent to $\Phi_{CX}$, where 1s are used to replace $\delta$ and $\lambda$. Thus, we may assume that for every element $\delta\in\Delta|C_0$ and $\lambda\in\Lambda|X_0$, there are an even number of natural numbers $i$ such that $\delta_i=\lambda_i=1$. This implies that contraction of $C$ has no effect on the $\Gamma$-frame matrix. So $\Phi$ is equivalent to a template with $\Lambda|X_0$ trivial. Therefore, since $\Lambda|X_1$ is trivial, we see that $\Lambda$ is trivial. Note that operation (7) is a reduction that produces an equivalent template, since $C$ must be contracted to produce a matroid that conforms to a template. By repeatedly performing operation (7), we obtain a template equivalent to $\Phi$ with $C=\emptyset$. \[PhiY0\] If $\Phi=(\{1\},C,X,Y_0,Y_1,A_1,\Delta,\Lambda)$ is a binary frame template with $\Lambda$ trivial and with $C=\emptyset$, then either $\Phi_{Y_0}\preceq\Phi$ or $\Phi$ is equivalent to a template with $\Delta$ trivial. First, suppose there is an element $\delta\in\Delta$ that is not in the row space of $A_1=A_1[X_1,(Y_0\cup Y_1)]$. Recall that a $y$-shift is the operation described in Lemma  \[yshift\]. Repeatedly perform $y$-shifts to obtain the following template, where $Y'_0=Y_0\cup Y_1$: $$(\{1\},\emptyset,X,Y'_0,\emptyset,A_1,\Delta,\{0\}).$$ On this template, perform operation (3) to obtain the following template: $$(\{1\},\emptyset,X,Y'_0,\emptyset,A_1,\{\mathbf{0}, \delta\},\{0\}).$$ Choose a basis $B'$ for $M(A_1)$. By performing elementary row operations on every matrix virtually respecting $\Phi$, we may assume that $A_1[X,B']$ consists of an identity matrix with zero rows below it and that $\delta|B'$ is the zero vector. By assumption, there is some element $y\in (Y'_0-B')$ such that $\delta_y$ is nonzero. Thus, we can repeatedly perform operation (10) to obtain the following template: $$(\{1\},\emptyset,X,B'\cup y,\emptyset,A_1[X,B'\cup y],\{\mathbf{0}, \delta|(B'\cup y)\},\{0\}).$$ Now, we can repeatedly perform operation (6) and then operation (12) to obtain the following template: $$(\{1\},\emptyset,\emptyset,\{y\},\emptyset,[\emptyset],\mathbb{Z}/2\mathbb{Z},\{0\}),$$ which is $\Phi_{Y_0}$. Now suppose that every element $\delta\in\Delta$ is in the row space of $A_1=A_1[X,(Y_0\cup Y_1)]$. Since $\Lambda$ is trivial, by performing elementary row operations on every matrix virtually respecting $\Phi$, we obtain a template equivalent to $\Phi$ with $\Delta$ trivial. \[PhiY1\] Let $\Phi=(\{1\},C,X,Y_0,Y_1,A_1,\Delta,\Lambda)$ be a binary frame template with $\Lambda$ and $\Delta$ trivial. If $M(A_1[X_1,(Y_0\cup Y_1)])$ has a circuit $Y'$ with $|Y'\cap Y_1|\geq3$, then $\Phi_{Y_1}\preceq\Phi$. Any matroid conforming to $\Phi$ is obtained by contracting $C$. Since $\Lambda$ and $\Delta$ are trivial, we may assume that $C=X_0=\emptyset$ and therefore that $X=X_1$. Repeatedly perform operation (4) and then operation (10) on $\Phi$ to obtain the following template: $$(\{1\},\emptyset,X,Y_0\cap Y',Y_1\cap Y',A_1[X,Y'],\{0\},\{0\}).$$ Choose any 3-element subset of $Y'\cap Y_1$ and call it $Y''$. Repeatedly perform $y$-shifts to obtain the following template: $$(\{1\},\emptyset,X,Y'-Y'',Y'',A_1[X,Y'],\{0\},\{0\}).$$ On this template, repeatedly perform operation (11) to obtain the following template: $$(\{1\},\emptyset,X',\emptyset,Y'',A_1[X',Y''],\{0\},\{0\}),$$ where $X'$ is the subset of $X$ that remains after $Y'-Y''$ is contracted. On this template, repeatedly perform operations (5) and (6) to obtain the following template, where $X''$ is a 2-element subset of $X'$: $$(\{1\},\emptyset,X'',\emptyset,Y'',\begin{bmatrix} 1& 0 &1\\ 0& 1 & 1 \end{bmatrix},\{0\},\{0\}).$$ This template is $\Phi_{Y_1}$. \[simpleY1\] If $\Phi$ is a frame template with $\Delta$ trivial, then $\Phi$ is equivalent to a template $\Phi'$ where $A_1[X,Y_1]$ is a matrix with every column nonzero and where no column is a copy of another. Moreover, if $\Phi$ is a binary frame template, then $M(A_1[X,Y_1])$ is simple. Let $A$ be a matrix that virtually conforms to $\Phi$. Since $\Delta$ is trivial, the columns of $A$ indexed by elements of $Z$ are formed by placing a column of $A_1[X,Y_1]$ on top of a unit column or a zero column. These columns can be made using any copy of the same column of $A_1[X,Y_1]$; so only one copy is needed. If any column of $A_1[X,Y_1]$ is a zero column, then any column indexed by an element of $Z$ that is made with this zero column can also be made as a column indexed by an element of $E-(Z\cup Y_0\cup Y_1\cup C)$ and choosing for the element of $\Lambda$ the zero vector. Thus, no zero columns of $A_1[X,Y_1]$ are needed. In the binary case, $M(A_1[X,Y_1])$ has no parallel elements because any such elements index copies of the same column. Also, $M(A_1[X,Y_1])$ has no loops because every column of $A_1[X,Y_1]$ is nonzero. Therefore, $M(A_1[X,Y_1])$ is simple. \[HL\] Let $\Phi$ be a binary frame template. Then at least one of the following is true: - $\Phi_0\sim\Phi$ - $\Phi'\preceq \Phi$ for some $\Phi'\in\{\Phi_X,\Phi_C,\Phi_{CX},\Phi_{Y_0},\Phi_{Y_1}\}$ - $\Phi$ is equivalent to a template where $C=\emptyset$, where $\Lambda$ and $\Delta$ are trivial, and where $A_1$ is of the following form, with $Y_0=V_0\cup V_1$, with $L$ an arbitrary binary matrix, and with each column of $H$ containing at most two nonzero entries: ----- ----- ----- $I$ 0 $H$ 0 $I$ $L$ ----- ----- ----- . Suppose neither (i) nor (ii) holds. By Lemma  \[PhiCD\], we may assume that $\Lambda$ is trivial and $C=\emptyset$. By Lemma  \[PhiY0\], we may assume that $\Delta$ is trivial. By Lemma  \[PhiY1\], every dependent set of $M(A)=M(A_1[X_1,(Y_0\cup Y_1)])$ has an intersection with $Y_1$ with size at most 2. So by elementary row operations, we may assume that $A_1$ is of the following form, where $Y_0=V_0\cup V_1$, where $L$ is an arbitrary binary matrix, where $K$ consists of unit and zero columns, and where each column of $H$ contains at most two nonzero entries: ----- ----- ----- ----- $I$ $K$ 0 $H$ 0 0 $I$ $L$ ----- ----- ----- ----- . However, by Lemma  \[simpleY1\], we may assume that $K$ is an empty matrix. Thus, (iii) holds. \[minimal\] Let $\Phi$ be a binary frame template. Then at least one of the following is true: - $\Phi_0\sim\Phi$ - $\Phi'\preceq \Phi$ for some $\Phi'\in\{\Phi_X,\Phi_C,\Phi_{CX},\Phi_{Y_0},\Phi_{Y_1}\}$ - There exist $k,l\in\mathbb{Z}_+$ such that no simple, vertically $k$-connected matroid with at least $l$ elements either virtually conforms or virtually coconforms to $\Phi$. Suppose for contradiction that none of outcomes (i)-(iii) hold for $\Phi$. By Lemma  \[HL\], outcome (iii) of that lemma holds. Note that any simple matroid $N$ virtually conforming to $\Phi$ is a restriction of a matroid $M$ represented by a matrix of the following form, where $Z=Z_0\cup Z_1$, where $Y_0=V_0\cup V_1$, and where the $\Gamma$-frame matrix has $n$ rows and has a vector matroid isomorphic to the cycle matroid of the graph $K_{n+1}$: [ r|c|cccc|c|c|c| ]{} &&&&\ &&$1\cdots1$&&&&&&\ &&&$1\cdots1$&&&&&\ &&&&$\cdots$&&&&\ &&&&&$1\cdots1$&&&\ &&&0&$I$&$L$\ &$\Gamma$-frame matrix&$I$&$I$&$\cdots$&$I$&0&0&0\ Also recall from the definition of conforming to a template that $Y_0\subseteq E(N)$. We see that $$\begin{aligned} \lambda_N(Y_0\cup (Z_1\cap E(N))) &\leq \lambda_M(Y_0\cup Z_1)\\ &= r_M(Y_0\cup Z_1)+r_M(E-(Y_0\cup Z_1))-r(M)\\ &= |V_0|+|Y_1| + |Y_1|+n-(|Y_1|+|V_0|+n)\\ &= |Y_1|.\end{aligned}$$ Note that each column of the above matrix, except possibly those columns indexed by $V_1$, has at most two nonzero entries. Thus, $M$ is graphic and $\Phi$ is trivial if $V_0=\emptyset$. Since (i) does not hold, $\Phi$ is nontrivial. Therefore, $V_0\neq\emptyset$, and $E(N)-(Y_0\cup Z_1)$ is not spanning. Thus, if $k>|Y_1|+1$, then $N$ is not vertically $k$-connected unless $Y_0\cup(Z_1\cap E(N)) $ is spanning in $N$. This implies that $n=0$; in that case, $N$ is only simple if the $\Gamma$-frame matrix is a $0\times0$ matrix. This implies that $|E(N)|\leq |Y_0\cup Y_1|$. So if $l>|Y_0\cup Y_1|$, then no simple, vertically $k$-connected matroid with at least $l$ elements virtually conforms to $\Phi$. Now, consider a simple matroid $N^*$ which virtually coconforms to $\Phi$. Then $N$ is a restriction of $M$ with $Y_0\subseteq E(N)$. Since a matroid and its dual have the same connectivity function, we have $\lambda_{N^*}(Y_0\cup (Z_1\cap E(N))\leq |Y_1|$. So if $k>|Y_1|+1$, then $N^*$ is not vertically $k$-connected unless either $Y_0\cup (Z_1\cap E(N))$ or $E(N)-(Z_1\cup Y_0)$ is spanning in $N^*$, implying that either $E(N)-(Z_1\cup Y_0)$ or $Y_0\cup (Z_1\cap E(N))$ is independent in $N$. If $E(N)-(Z_1\cup Y_0)$ is independent in $N$, then $$\begin{aligned} |E(N)-(Z_1\cup Y_0)|&=r_N(E(N)-(Z_1\cup Y_0))\\ &\leq r_M(E(M)-(Z_1\cup Y_0))\\ &=|Y_1|+n.\end{aligned}$$ By the formula for corank, we have $$\begin{aligned} r_{N^*}(E(N)-(Z_1\cup Y_0))&\leq r_{M^*}(E(N)-(Z_1\cup Y_0))\\ &= |E(N)-(Z_1\cup Y_0)|+r_M(Z_1\cup Y_0)-r(M)\\ &\leq |Y_1|+n+|Y_1|+|V_0|-(|Y_1|+|V_0|+n)\\ &=|Y_1|.\end{aligned}$$ Since $N^*$ is simple and binary, we have $|E(N)-(Z_1\cup Y_0)|\leq 2^{|Y_1|}-1$. This implies that $|E(N)|\leq 2^{|Y_1|}-1+|Y_1|+|Y_0|$. Thus, if we set $l$ greater than this value, then no simple, vertically $k$-connected matroid with at least $l$ elements virtually coconforms to $\Phi$ unless $Y_0\cup (Z_1\cap E(N))$ is independent in $N$. Since (iii) does not hold, this must be true for some matroid $N$. In particular, $Y_0=V_0\cup V_1$ is independent in $N$, implying that $H$ is a linearly independent matrix. Let $P$ denote the matrix $$P= \left[ \begin{array}{cc} 1& 0\\ 0& 1\\ 0&1\\ \hline 1&1\\ \end{array} \right].$$ Suppose $A_1[X,V_1]$ has $P$ as a submatrix, with the first three rows of $P$ contained in $H$ and the last row of $P$ contained in $L$. Then $A_1$ contains the following submatrix, with the first three columns contained in $A_1[X,Y_1]$ and the last two contained in $A_1[X,V_1]$: $$\left[ \begin{array}{ccc|cc} 1&0&0&1&0\\ 0&1&0&0&1\\ 0&0&1&0&1\\ \hline 0&0&0&1&1\\ \end{array} \right].$$ After contracting all other elements of $Y_1$ by repeatedly performing $y$-shifts and operation (12), the columns of this submatrix form a circuit in $M(A_1)$ whose intersection with $Y_1$ has size 3. However, we have already deduced by Lemma  \[PhiY1\] that this is impossible. Therefore, $A_1$ does not contain $P$ as a submatrix, with the first three rows of $P$ contained in $H$ and the last row of $P$ contained in $L$. We will refer to this fact by saying that $A_1$ has no *$P$-configuration*. Let $\{1,2,\dots, m\}$ be the rows of $L$. (So $|V_0|=m$.) Let $S_i$ be the submatrix of $H$ obtained by restricting $H$ to the columns $j$ such that $L_{i,j}=1$. Recall that $H$, and therefore $S_i$, contain at most two nonzero entries per column. Also, since $H$ is linearly independent, each column has at least one nonzero entry, and no column is a copy of another. Suppose a column $e$ of $S_i$ contains exactly two nonzero entries. Since $A_1$ has no $P$-configuration, all other columns of $S_i$ must contain a nonzero entry in exactly one of the same rows as $e$. Suppose that there are columns $f$ and $g$ in $S_i$ such that $f$ contains a nonzero entry in one of the same rows as $e$, but $g$ contains a nonzero entry in the other row. Then $S_i$ contains the following submatrix: $$\begin{blockarray}{ccc} e & f & g \\ \begin{block}{[ccc]} 1 & 1 & 0 \\ 1 & 0 & 1 \\ \end{block} \end{blockarray}.$$ Since $H$ is a linearly independent matrix, $f$ or $g$ (say $f$) must have an additional nonzero entry in $H$. To avoid $f$ and $g$ forming a $P$-configuration, $g$ must have an additional nonzero entry in the same row as $f$. Therefore, $S_i$ contains the following submatrix: $$\begin{blockarray}{ccc} e & f & g \\ \begin{block}{[ccc]} 1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1\\ \end{block} \end{blockarray}.$$ Since each column of $H$ contains at most two nonzero entries, $\{e,f,g\}$ is a dependent set of columns, contradicting the assumption that $H$ is linearly independent. Therefore, we deduce that each $S_i$ either consists entirely of unit columns or contains a row $s_i$ consisting entirely of 1s. Note that each $S_i$ is the incidence matrix of a star, with possibly one row removed. We will call $s_i$ the *star center* of row $i$. If $S_i$ consists entirely of unit columns, then we define its star center to be $s_i=\emptyset$. If the sets of columns of all the $S_i$ are pairwise disjoint, then by adding each row $i$ to its star center $s_i$, we see that every matroid virtually conforming to $\Phi$ can be represented by a matrix with at most two nonzero entries per column. Thus, $\Phi$ is trivial, contradicting the assumption that (i) does not hold. Also, if $i$ and $j$ are distinct rows of $L$ with distinct star centers $s_i$ and $s_j$, then $S_i$ and $S_j$ can have at most one column in common because otherwise, the columns they have in common form a linearly dependent set in $H$. Now suppose there are $S_i$ and $S_j$ with $s_i=s_j$. Also, suppose that neither $S_i$ nor $S_j$ is a submatrix of the other. Then $A_1$ contains the following submatrix. In fact, after repeatedly performing $y$-shifts, operation (11), and operation (10), we may assume that $A_1$ is the following matrix, with the first three columns indexed by $Y_1$, the next two indexed by $V_0$, and the last three by $V_1$: $$\left[ \begin{array}{ccc|cc|ccc} 1&0&0&0&0&1&1&1\\ 0&1&0&0&0&0&1&0\\ 0&0&1&0&0&0&0&1\\ \hline 0&0&0&1&0&1&1&0\\ 0&0&0&0&1&0&1&1\\ \end{array} \right].$$ Add the fourth row to the first, and swap the fourth and sixth columns to obtain the following matrix: $$\left[ \begin{array}{ccc|cc|ccc} 1&0&0&0&0&1&0&1\\ 0&1&0&0&0&0&1&0\\ 0&0&1&0&0&0&0&1\\ \hline 0&0&0&1&0&1&1&0\\ 0&0&0&0&1&0&1&1\\ \end{array} \right].$$ The last two columns of this matrix contain a $P$-configuration. Now suppose there are matrices $S_i$ and $S_j$ so that $S_j$ is a submatrix of $S_i$. Then $A_1$ contains a submatrix obtained by deleting columns from a matrix of the following form, where the left portion comes from the set $V_0$, the upper-right portion comes from the matrix $H$, the lower-left portion comes from the matrix $L$, and $x$ is 1 or 0 depending on whether or not the last column is contained in $S_j$: $$\left[ \begin{array}{cc|ccccccc} 0&0&1&\cdots&1&1&\cdots&1&1\\ 0&0&1&&&&&&0\\ \vdots&\vdots&&\ddots&&&&&\vdots\\ 0&0&&&1&&&&0\\ 0&0&&&&1&&&0\\ \vdots&\vdots&&&&&\ddots&&\vdots\\ 0&0&&&&&&1&0\\ \hline 1&0&1&\cdots&1&1&\cdots&1&1\\ 0&1&0&\cdots&0&1&\cdots&1&x \end{array} \right].$$ Choose any column contained in $S_j$ and perform row operations so that this column becomes a unit column with nonzero entry in $L$. Then we obtain the following matrix: $$\left[ \begin{array}{cc|cccccccc} 0&1&1&\cdots&1&0&0&\cdots&0&x+1\\ 0&0&1&&&&&&&0\\ \vdots&\vdots&&\ddots&&&&&&\vdots\\ 0&0&&&1&&&&&0\\ 0&1&&&&0&1&\cdots&1&x\\ 0&0&&&&&1&&&0\\ \vdots&\vdots&&&&&&\ddots&&\vdots\\ 0&0&&&&&&&1&0\\ \hline 1&1&1&\cdots&1&0&\cdots&\cdots&0&x+1\\ 0&1&0&\cdots&0&1&\cdots&\cdots&1&x \end{array} \right].$$ Now, by swapping the appropriate columns, we obtain the following: $$\left[ \begin{array}{cc|cccccccc} 0&0&1&\cdots&1&1&0&\cdots&0&x+1\\ 0&0&1&&&&&&&0\\ \vdots&\vdots&&\ddots&&&&&&\vdots\\ 0&0&&&1&&&&&0\\ 0&0&&&&1&1&\cdots&1&x\\ 0&0&&&&&1&&&0\\ \vdots&\vdots&&&&&&\ddots&&\vdots\\ 0&0&&&&&&&1&0\\ \hline 1&0&1&\cdots&1&1&0&\cdots&0&x+1\\ 0&1&0&\cdots&0&1&\cdots&\cdots&1&x \end{array} \right].$$ We see that in this new matrix, $S_i$ and $S_j$ have only one column in common and $s_i\neq s_j$. The last column is in $S_i$ if $x=0$ and $S_j$ if $x=1$. Thus, this case reduces to the final case that remains to be checked: for all $i$ and $j$, we have $s_i\neq s_j$ and $S_i$ and $S_j$ have at most one column in common. Since each column of $H$ contains at most two nonzero entries, and since all $S_i$ have distinct star centers, we see that a column of $H$ can be contained in at most two $S_i$. By adding each row $i$ to its star center $s_i$, one can see that every matrix virtually conforming to $\Phi$ can be rewritten so that every column contains at most two nonzero entries. Therefore, $\Phi$ is trivial, and (i) holds. This completes the contradiction and proves the result. Outcome (iii) of Theorem  \[minimal\] only occurs in very specific situations. In fact, due to connectivity considerations, it is not needed in order to use Corollary  \[weakframe\]. \[describes\] Let $\mathcal{M}$ be a minor-closed class of binary matroids, and suppose there exist $k,l,m\in \mathbb{Z}_+$ and a set $\mathcal{T}_{\mathcal{M}}=\{\Phi_1,\dots, \Phi_s, \Psi_1,\dots, \Psi_t\}$ of binary frame templates such that - $\mathcal{M}$ contains each of the classes $\mathcal{M}_w(\Phi_1),\dots,\mathcal{M}_w(\Phi_s)$, - $\mathcal{M}$ contains the duals of the matroids in each of the classes $\mathcal{M}_w(\Psi_1)$,$\dots$,$\mathcal{M}_w(\Psi_t)$, - if $M$ is a simple vertically $k$-connected member of $\mathcal{M}$ with at least $l$ elements and with no $PG(m-1,2)$ minor, then either $M$ is a member of at least one of the classes $\mathcal{M}_v(\Phi_1),\dots,\mathcal{M}_v(\Phi_s)$ or $M^*$ is a member of at least one of the classes $\mathcal{M}_v(\Psi_1),\dots,\mathcal{M}_v(\Psi_t)$, and - for each template $\Phi\in\mathcal{T}_{\mathcal{M}}$, either $\Phi$ is trivial or $\Phi'\preceq \Phi$ for some $\Phi'\in\{\Phi_X,\Phi_C,\Phi_{CX},\Phi_{Y_0},\Phi_{Y_1}\}$. We say that $\mathcal{T}_{\mathcal{M}}$ *describes* $\mathcal{M}$. By combining Corollary  \[weakframe\] with Theorem  \[minimal\], one can observe that every proper minor-closed class $\mathcal{M}$ of binary matroids can be described by a set of templates. Moreover, that set is nonempty if and only if $\mathcal{M}$ contains all graphic matroids or all cographic matroids. \[Y0Y1\] Let $\mathcal{M}$ be a minor-closed class of binary matroids, and let $\{\Phi_1,\dots, \Phi_s, \Psi_1,\dots, \Psi_t\}$ be a set of templates describing $\mathcal{M}$. If any of these templates is nontrivial, then $\mathcal{M}$ contains $\mathcal{M}(\Phi_{Y_0})$, $\mathcal{M}(\Phi_{Y_1})$, $\mathcal{M}^*(\Phi_{Y_0})$, or $\mathcal{M}^*(\Phi_{Y_1})$. Let $\Phi$ be a nontrivial template in the set $\{\Phi_1,\dots, \Phi_s\}$. By Definition  \[describes\] and Lemma  \[YCD\], either $\Phi_{Y_0}\preceq\Phi$ or $\Phi_{Y_1}\preceq\Phi$. If $\Phi_{Y_0}\preceq\Phi$, then $$\mathcal{M}(\Phi_{Y_0})\subseteq\mathcal{M}_v(\Phi_{Y_0})\subseteq\mathcal{M}_v(\Phi)\subseteq\mathcal{M},$$ where the first containment holds because every matroid conforming to a template also virtually conforms to it, the second containment holds by definition of $\preceq$, and the third containment holds by Definition  \[describes\]. In the case where $\Phi_{Y_1}\preceq\Phi$, a similar argument shows that $\mathcal{M}(\Phi_{Y_1})\subseteq\mathcal{M}$. If $\Psi$ is a nontrivial template in the set $\{\Psi_1,\dots, \Psi_s\}$, a similar argument shows that either $\mathcal{M}^*(\Phi_{Y_0})\subseteq\mathcal{M}$, or $\mathcal{M}^*(\Phi_{Y_1})\subseteq\mathcal{M}$. Growth Rates {#Growth Rates} ============ Let $\mathcal{M}$ be a minor-closed class of matroids. Let $h_{\mathcal{M}}(r)$ denote the *growth rate function* of $\mathcal{M}$: the function whose value at an integer $r\geq0$ is given by the maximum number of elements in a simple matroid in $\mathcal{M}$ of rank at most $r$. For a matroid $M$, we denote by $\varepsilon(M)$ the size of the simplification of $M$, that is the number of rank-1 flats of $M$. By combining the main result in  [@gkw09] with earlier results of Geelen and Whittle  [@gw03] and Geelen and Kabell  [@gk09], Geelen, Kung, and Whittle proved the following: \[growthrate\] If $\mathcal{M}$ is a nonempty minor-closed class of matroids, then there exists $c\in\mathbb{R}$ such that either: - $h_{\mathcal{M}}(r)\leq cr$ for all $r$, - $\binom{r+1}{2}\leq h_{\mathcal{M}}(r)\leq cr^2$ for all $r$ and $\mathcal{M}$ contains all graphic matroids, - there is a prime-power $q$ such that $\frac{q^r-1}{q-1}\leq h_{\mathcal{M}}(r)\leq cq^r$ for all $r$ and $\mathcal{M}$ contains all $\mathrm{GF}(q)$-representable matroids, or - $h_{\mathcal{M}}$ is infinite and $\mathcal{M}$ contains all simple rank-2 matroids. If outcome (2) of the Growth Rate Theorem holds for a minor-closed class $\mathcal{M}$, then $\mathcal{M}$ is said to be *quadratically dense*. In this section, we will consider growth rates of some quadratically dense classes of binary matroids. Let $\mathcal{EX}(F)$ denote the class of binary matroids with no $F$-minor. If $f$ and $g$ are functions, we write $f(r)\approx g(r)$ if $f(r)=g(r)$ for all but finitely many $r$. Since the growth rate function for the class of graphic matroids is $\binom{r+1}{2}$, the Growth Rate Theorem implies that, if $F$ is a nongraphic binary matroid, $$h_{\mathcal{EX}(F)}(r)\geq\binom{r+1}{2}.$$ Kung et. al.  [@kmpr14] pose the following question: For which nongraphic binary matroids $F$ of rank 4 does equality hold above for all but finitely many $r$? Geelen and Nelson answer this question in  [@gn15]. Let $N_{12}$ be the matroid formed by deleting a three-element independent set from $PG(3,2)$. The nongraphic binary matroids $F$ of rank 4 for which $h_{\mathcal{EX}(F)}(r)\approx\binom{r+1}{2}$ are exactly the nongraphic restrictions of $N_{12}$. We present here an alternate proof. Both proofs allow us to answer the question when $F$ is a matroid of any rank, not just rank 4. We will prove the following theorem after proving several lemmas. \[quadgrowth\] Let $\mathcal{M}$ be a minor-closed class of binary matroids. Then $h_{\mathcal{M}}(r)\approx\binom{r+1}{2}$ if and only if $\mathcal{M}$ contains all graphic matroids but does not contain $\mathcal{M}_v(\Phi_{Y_1})$. Our proof of Theorem  \[quadgrowth\] will depend on the following theorem, proved by Geelen and Nelson in  [@gn15]: \[gn51\] Let $\mathcal{M}$ be a quadratically dense minor-closed class of matroids and let $p(x)$ be a real quadratic polynomial with positive leading coefficient. If $h_{\mathcal{M}}(n)>p(n)$ for infinitely many $n\in\mathbb{Z}^+$, then for all integers $r,s\geq1$ there exists a vertically $s$-connected matroid $M\in\mathcal{M}$ satisfying $\varepsilon(M)>p(r(M))$ and $r(M)\geq r$. An *even-cycle matroid* is a binary matroid of the form $M=M\binom{w}{D}$, where $D\in\mathrm{GF}(2)^{V\times E}$ is the vertex-edge incidence matrix of a graph $G=(V,E)$ and $w\in\mathrm{GF}(2)^E$ is the characteristic vector of a set $W\subseteq E$. The pair $(G,W)$ is an *even-cycle representation* of $M$. The edges in $W$ are called *odd* edges, and the other edges are *even* edges. An *odd cycle* of $(G,W)$ is a cycle of $G$ with an odd number of odd edges. A *blocking pair* of $(G,W)$ is a pair of vertices $u,v$ of $G$ so that every odd cycle passes through at least one of these vertices. *Resigning* at a vertex $u$ of $G$ occurs when all the edges incident with $u$ are changed from even to odd and vice-versa. It is easy to see that this corresponds to adding the row of the matrix corresponding to $u$ to the characteristic vector of $W$. Therefore, resigning at a vertex does not change an even-cycle matroid. It is also easy to see that if an even-cycle representation has a blocking pair, then we can resign so that every odd edge is incident with at least one vertex in the blocking pair. For our purposes, it will be convenient to think of a blocking pair in this way. For $r\geq2$, let $A_r$ be the following binary matrix, where we choose for the $\Gamma$-frame matrix the matrix representation of $M(K_{r-1})$, so that the identity matrices are $(r-2)\times(r-2)$ matrices. ----------------------- --- ----- ----- ------------ ------------ ------------ 1 0 1 $1\cdots1$ $0\cdots0$ $1\cdots1$ 0 1 1 $0\cdots0$ $1\cdots1$ $1\cdots1$ $\Gamma$-frame matrix $I$ $I$ $I$ ----------------------- --- ----- ----- ------------ ------------ ------------ Note that $M(A_r)$ is the largest simple matroid of rank $r$ that virtually conforms to $\Phi_{Y_1}$. Let $X_r$ be the largest simple matroid of rank $r$ that virtually conforms to $\Phi_{Y_1}$. Equivalently, $X_1=U_{1,1}$, and for $r\geq2$, we have $X_r=M(A_r)$. \[Y1minors\] The class $\mathcal{M}_v(\Phi_{Y_1})$ is the class of matroids having an even-cycle representation with a blocking pair. This class is minor-closed. Any simple matroid $M$ virtually conforming to $\Phi_{Y_1}$ is a restriction of $X_r$ for some $r$. Label the rows of $A_r$ as $1,\dots,r$. Add to the matrix row $r+1$, which is the sum of rows $2,\dots, r$. This does not change the matroid $X_r$. We see that $X_r$ is an even-cycle matroid $(G,W)$, where row 1 is the characteristic vector of $W$ and rows $2,\dots, r+1$ form the incidence matrix of $G$. Moreover, every edge in $W$ is incident with the vertex corresponding to either row 2 or row $r+1$. Thus, every matroid virtually conforming to $\Phi_{Y_1}$ has an even-cycle representation with a blocking pair. Conversely, every matroid that has an even-cycle representation with a blocking pair $\{u,v\}$ virtually conforms to $\Phi_{Y_1}$, by making $u$ correspond to the second row and making $v$ correspond to row $r+1$, which can be removed without changing the matroid. By resigning whenever we wish to contract an element represented by an odd edge, it is not difficult to see that the class of matroids having an even-cycle representation with a blocking pair is minor-closed. \[restriction\] Any simple, rank-$r$ matroid $M$ that is a minor of a matroid virtually conforming to $\Phi_{Y_1}$ is a restriction of $X_r$. From the preceding lemma, $M$ is a restriction of some $X_{r'}$. So $M$ has an even-cycle representation $(G,W)$ with a blocking pair $\{u,v\}$. Let $w$ be the characteristic vector of $W$. There are $r'-r$ rows in the matrix $A_{r'}[(V\cup w)-{v}, E(M)]$ whose deletion does not alter the matroid $M$. After these rows are deleted, the resulting matrix is a submatrix of $A_r$. \[Y0minorY1\] Every matroid virtually conforming to $\Phi_{Y_1}$ is a minor of a matroid conforming to $\Phi_{Y_0}$. By Lemma  \[YCD\], we have $\Phi_{Y_1}\preceq\Phi_C$. Every matroid conforming to $\Phi_C$ is obtained by contracting an element from a matroid conforming to $\Phi_{Y_0}$. \[graphicvscographic\] Let $k$ be a positive integer. Then there are at most finitely many integers $r$ such that the complete graphic matroid $M(K_{r+1})$ is a rank-($\leq k$) perturbation of a cographic matroid. Let $N$ be a cographic matroid. Observe that adding a rank-1 matrix to a matrix representation of a binary matroid $N$ changes $\varepsilon(N)$ by a factor of at most 2. This occurs when, in every rank-1 flat of $N$, there is at least one nonloop element indexing a column that is changed by adding the rank-1 matrix and at least one nonloop element indexing a column that remains unchanged when the rank-1 matrix is added. Thus, if $M$ is a rank-$(\leq t)$ perturbation of $N$, we have $\varepsilon(M)\leq2^t\varepsilon(N)$. Let $r=r(M)$. Recall that a cographic matroid $N$ has $\varepsilon(N)\leq3r(N)-3$. Therefore, $\varepsilon(M)\leq2^t(3r(N)-3)\leq2^t(3(r+t)-3)$. For fixed $t$ and sufficiently large $r$, this expression is less than $\binom{r+1}{2}=\varepsilon(M(K_{r+1}))$. \[conformonly\] Let $\mathcal{M}$ be a quadratically dense minor-closed class of matroids representable over a given field $\mathbb{F}$. Let $\{\Phi_1,\dots,\Phi_s,\Psi_1,\dots,\Psi_t\}$ be a set of templates describing $\mathcal{M}$. For sufficiently large $r$, the growth rate $h_{\mathcal{M}}(r)$ is equal to the size of the largest simple matroid of rank $r$ that virtually conforms to any template in $\{\Phi_1,\dots,\Phi_s\}$. Let $h'_{\mathcal{M}}(r)$ denote the size of the largest simple matroid of rank $r$ that virtually conforms to any template in $\{\Phi_1,\dots,\Phi_s\}$. So $h_{\mathcal{M}}(r)\geq h'_{\mathcal{M}}(r)$. The size of the largest simple matroid of rank $r$ that virtually conforms to any particular template is a quadratic polynomial in $r$. Thus, for sufficiently large $r$, the function $h'_{\mathcal{M}}(r)$ is a quadratic polynomial as well. By Definition  \[describes\], there exist $k,l\in \mathbb{Z}_+$ so that every simple vertically $k$-connected member of $\mathcal{M}$ with at least $l$ elements either weakly conforms to a template in $\{\Phi_1,\dots,\Phi_s\}$ or weakly coconforms to some template in $\{\Psi_1,\dots,\Psi_t\}$. Suppose, for contradiction, that $h_{\mathcal{M}}(r)>h'_{\mathcal{M}}(r)$ for infinitely many $r$. Theorem  \[gn51\], with $h'_{\mathcal{M}}(r)$ playing the role of $p(r)$, implies that there is a sequence $M_1, M_2,\dots$ of vertically $k$-connected matroids in $\mathcal{M}$ such that $\varepsilon(M_i)>h'_{\mathcal{M}}(i)$ and $r(M_i)\geq i$. Thus, in this sequence, there are infinitely many matroids that are vertically $k$-connected and have size at least $l$. Since these matroids are too large to virtually conform to any template in $\{\Phi_1,\dots,\Phi_t\}$, there is at least one nontrivial template $\Psi\in\{\Psi_1,\dots,\Psi_t\}$ such that infinitely many vertically $k$-connected matroids in $\mathcal{M}$ coconform to $\Psi$. However, since $\mathcal{M}$ contains all graphic matroids and since every complete graphic matroid has infinite vertical connectivity (hence vertical $k$-connectivity), we have that infinitely many complete graphic matroids coconform to $\Psi$. For some $t$ depending on $\Psi$, every matroid coconforming to $\Psi$ is a rank-$(\leq t)$ perturbation of a cographic matroid. This contradicts Lemma  \[graphicvscographic\]. By contradiction, the result holds. \[proof\] First, suppose $h_{\mathcal{M}}(r)\approx\binom{r+1}{2}$. By the Growth Rate Theorem, $\mathcal{M}$ contains all graphic matroids. For $r\geq1$, we have $|X_r|=\binom{r-1}{2}+3r-3$, which for $r>2$ is greater than $\binom{r+1}{2}$. Thus, $\mathcal{M}$ does not contain $\mathcal{M}_v(\Phi_{Y_1})$. Now, suppose $\mathcal{M}$ contains all graphic matroids but does not contain $\mathcal{M}_v(\Phi_{Y_1})$. Since $\mathcal{M}$ contains all graphic matroids, there is a nonempty set $\{\Phi_1,\dots,\Phi_s,\Psi_1,\dots,\Psi_t\}$ of binary frame templates describing $\mathcal{M}$. By Lemma  \[conformonly\], $h_{\mathcal{M}}(r)$ is equal to the size of the largest simple matroid of rank $r$ that conforms to any template in $\{\Phi_1,\dots,\Phi_s\}$. Suppose $\Phi$ is a nontrivial template in $\{\Phi_1,\dots,\Phi_s\}$. By Corollary  \[Y0Y1\], either $\Phi_{Y_0}\preceq\Phi$ or $\Phi_{Y_1}\preceq\Phi$. Since $\mathcal{M}$ does not contain $\mathcal{M}_v(\Phi_{Y_1})$, we must have $\Phi_{Y_0}\preceq\Phi$. However, by Lemma  \[Y0minorY1\], this implies $\mathcal{M}_v(\Phi_{Y_1})\subseteq\mathcal{M}$. Therefore, we conclude that $h_{\mathcal{M}}(r)\approx\binom{r+1}{2}$, completing the proof. \[EXF\] Let $F$ be a simple, binary matroid of rank $r$. Then $h_{\mathcal{EX}(F)}\approx\binom{r+1}{2}$ if and only if $F$ is a nongraphic restriction of $X_r$. By Theorem  \[quadgrowth\], $h_{\mathcal{EX}(F)}\approx\binom{r+1}{2}$ if and only if $\mathcal{EX}(F)$ contains all graphic matroids but does not contain $\mathcal{M}_v(\Phi_{Y_1})$. The condition that $\mathcal{EX}(F)$ contains all graphic matroids is equivalent to the condition that $F$ is nongraphic. By Lemma  \[restriction\], the condition that $\mathcal{EX}(F)$ does not contain $\mathcal{M}_v(\Phi_{Y_1})$ is equivalent to the condition that $F$ is a restriction of $X_r$. Note that $X_4=N_{12}$; so this answers the question posed in  [@kmpr14]. We now consider the growth rate of $\mathcal{EX}(PG(3,2))$. We will prove Theorem  \[EXPG32\], which we restate below. The growth rate function for $\mathcal{EX}(PG(3,2))$ is $$h_{\mathcal{EX}(PG(3,2))}\approx r^2-r+1.$$ We will use the following. \[PG32Phi\] Let $\mathcal{T}_{\mathcal{EX}(PG(3,2))}=\{\Phi_1,\dots\Phi_s,\Psi_1,\dots,\Psi_t\}$. If $\Phi\in\{\Phi_1,\dots\Phi_s\}$, then either $\Phi=\Phi_X$ or $\Phi$ is a template with $C=\emptyset$ and with $\Lambda$ and $\Delta$ trivial. The class of matroids conforming to $\Phi_X$ is exactly the class of even-cycle matroids. This class is minor-closed. The largest simple, even-cycle matroid of rank $r$ has an even-cycle representation obtained from the graph $K_r$ by adding to each even edge an odd edge in parallel as well as adding one odd loop to the graph. Therefore, the class of even-cycle matroids has growth rate $2\binom{r}{2}+1=r^2-r+1$. So the largest simple, even-cycle matroid of rank 4 has size 13. Since $PG(3,2)$ has size $15$, we have $\mathcal{M}(\Phi_X)\subseteq\mathcal{EX}(PG(3,2))$. Therefore, we may assume that $\Phi_X\in\mathcal{T}_{\mathcal{EX}(PG(3,2))}$. Since $\Phi_0\preceq\Phi_X$, we may assume that $\Phi_0\notin\{\Phi_1,\dots\Phi_s\}$. Let $$\Phi=(\{1\},C,X,Y_0,Y_1,A_1,\Delta,\Lambda)$$ be a nontrivial template such that $\Phi\neq\Phi_X$ and $\Phi\in\{\Phi_1,\dots\Phi_s\}$. Consider the graft matroid $M(K_6,V(K_6))$. A straightforward computation shows that, by contracting the nongraphic element, we obtain $PG(3,2)$. Therefore, $\Phi_{Y_0}\npreceq\Phi$. By Lemma  \[YCD\], we also have $\Phi_C\npreceq\Phi$ and $\Phi_{CX}\npreceq\Phi$. Now, we may assume that $\Phi$ is in standard form. Since $\Phi_C\npreceq\Phi$, by Lemma  \[PhiC\] we may assume that $C_1=\emptyset$. Also, by Lemma  \[PhiCD\], since $\Phi_{CX}\npreceq\Phi$ and $\Phi_C\npreceq\Phi$, either $\Lambda|X_1$ is nontrivial and $\Phi_X\preceq\Phi$ or $\Lambda$ is trivial and $C=\emptyset$. First, suppose that $\Lambda$ is trivial and $C=\emptyset$. Since $\Phi_{Y_0}\npreceq\Phi$, Lemma  \[PhiY0\] implies that $\Phi$ is equivalent to a template with $\Delta$ trivial. So we may assume $$\Phi=(\{1\},\emptyset,X,Y_0,Y_1,A_1,\{0\},\{0\}),$$ which is one of the possible conclusions of the lemma. Thus, we may assume that $\Lambda|X_1$ is nontrivial and $\Phi_X\preceq\Phi$. Suppose $|\Lambda|X_1|>2$. On the template $$\Phi=(\{1\},C_0,Y_0,Y_1,A_1,\Delta,\Lambda),$$ perform operation (3) and then repeatedly perform operations (4) and (10) to obtain the template $$(\{1\},C_0,X,\emptyset,\emptyset,A_1[X,C_0],\{0\},\Lambda).$$ Then repeatedly perform operation (7) to obtain $$(\{1\},\emptyset,X_1,\emptyset,\emptyset,[\emptyset],\{0\},\Lambda|X_1).$$ Since $\Lambda|X_1$ has characteristic 2 and size greater than 2, it contains a subgroup $\Lambda'$ isomorphic to $(\mathbb{Z}/2\mathbb{Z})\times(\mathbb{Z}/2\mathbb{Z})$. Perform operation (2) to obtain the template $$(\{1\},\emptyset,X_1,\emptyset,\emptyset,[\emptyset],\{0\},\Lambda');$$ then repeatedly perform operations (5) and (6) to obtain $$(\{1\},\emptyset,X',\emptyset,\emptyset,[\emptyset],\{0\},\Lambda''),$$ where $|X'|=2$ and $\Lambda''$ is the additive group generated by $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ and $\begin{bmatrix} 0 \\ 1 \end{bmatrix}$. One readily sees that $PG(3,2)$ conforms to this template. Therefore, $|\Lambda|=2$. We may perform row operations so that $\Lambda$ is generated by $[1,0\ldots,0]^T$. Let $\Sigma$ be the element of $X$ such that $\Lambda|\{\Sigma\}$ is nonzero. Now, suppose there is an element $\bar{x}\in\Delta$ that is not in the row space of $A_1$. Perform operations (2) and (3) on $\Phi$ to obtain $$(\{1\},C_0,X,Y_0,Y_1,A_1,\{0,\bar{x}\},\{0\}).$$ Now, by a similar argument to the one used in the proof of Lemma  \[PhiY0\], we have $\Phi_{Y_0}\preceq\Phi$. Since we already know this is not the case, we deduce that every element of $\Delta$ is in the row space of $A_1$. Let $\bar{x}\in\Delta|C_0$ and $\bar{y}\in\Lambda$ be such that there are an odd number of natural numbers $i$ such that $\bar{x}_i=\bar{y}_i=1$. Then we call the ordered pair $(\bar{x},\bar{y})$ a *pair of odd type*. Otherwise, $(\bar{x},\bar{y})$ is a *pair of even type*. Suppose $(\bar{x},\bar{y})$ is a pair of odd type with $\bar{y}|X_1$ a zero vector. By performing operations (2) and (3) and repeatedly performing operations (4) and (10), we obtain $$(\{1\},C_0,X,\emptyset,\emptyset,A_1[X,C],\{0,\bar{x}\},\{0,\bar{y}\}),$$ which is equivalent to $\Phi_{CX}$. We already know this is not the case. Therefore, for every pair $(\bar{x},\bar{y})$ of odd type, $\bar{y}|X_1=[1,0,\dots,0]^T$. Suppose $\bar{x}\in\Delta|C$ and $\bar{y}_1,\bar{y}_2\in\Lambda$ are such that $\bar{y}_1|X_1=\bar{y}_2|X_1=[1,0,\dots,0]^T$, such that $(\bar{x},\bar{y}_1)$ is a pair of odd type, and such that $(\bar{x},\bar{y}_2)$ is a pair of even type. Then $(\bar{y}_1+\bar{y}_2)|X_1$ is a zero vector, and $(\bar{x},\bar{y}_1+\bar{y}_2)$ is a pair of odd type. Therefore, either all pairs $(\bar{x},\bar{y})\in\Delta|C\times\Lambda$ are of even type, in which case $\Phi$ is equivalent to a template with $\Lambda|X_0$ trivial and $C=\emptyset$, or if $(\bar{x},\bar{y})$ is a pair of odd type, then $(\bar{x},\bar{z})$ is of odd type for every $\bar{z}\in\Lambda$ with $\bar{z}|X_1$ nonzero. In this case, consider any matrix virtually conforming to $\Phi$. After contracting $C$, we can restore the $\Gamma$-frame matrix by adding $\Sigma$ to each row where the $\Gamma$-frame matrix has been altered. Therefore, $\Phi$ is equivalent to a template with $\Lambda|X_0$ trivial and $C=\emptyset$. So we now have that $$\Phi=(\{1\},\emptyset,X,Y_0,Y_1,A_1,\Delta,\Lambda),$$ with $\Lambda$ generated by $[1,0\ldots,0]^T$ and with every element of $\Delta$ in the row space of $A_1$. We will now show that, in fact, $\Phi$ is equivalent to a template with $\Delta$ trivial. On $\Phi$, perform $y$-shifts to obtain the following template, where $Y'_0=Y_0\cup Y_1$: $$\Phi'=(\{1\},\emptyset,X,Y'_0,\emptyset,A_1,\Delta,\Lambda).$$ By repeatedly performing operation (5) and then operation (6) on this template, we may assume that $A_1$ has the following form, with the star representing an arbitrary binary matrix and $\bar{v}$ representing an arbitrary row vector: $$\left[ \begin{array}{c|c} 0\cdots0&\bar{v}\\ \hline I_{|X|-1}&* \end{array} \right].$$ Also, since $\Lambda|(D-\{\Sigma\})$ is trivial, we may perform row operations on every matrix conforming to $\Phi'$ to obtain a template $$\Phi''=(\{1\},\emptyset,X,Y'_0,\emptyset,A_1,\Delta'',\Lambda),$$ so that every element of $\Delta''$ has 0 for its first $|X|-1$ entries. Since every element of $\Delta$ was in the row space of $A_1$, the only possible nonzero element of $\Delta''$ is the row vector with 0 for its first $|X|-1$ entries and whose last $|Y'_0|-|X|+1$ entries form the row vector $\bar{v}$. Note that operations (5) and (6) and the row operations we performed on every matrix conforming to $\Phi'$ each changes a template to an equivalent template. Thus, we may assume that $\bar{v}$ is nonzero and that $\Delta''=\{\bf{0},$$\bar{v}\}$ because otherwise, $\Phi$ is equivalent to a template with $\Delta$ trivial. So, for some $y\in Y'_0$, we have $\bar{v}_y=1$. On the template $\Phi''$, repeatedly perform operation (11) and then operation (10) to obtain the following template: $$\Phi'''=(\{1\},\emptyset,\{\Sigma\},\{y\},\emptyset,[1],\mathbb{Z}/2\mathbb{Z},\mathbb{Z}/2\mathbb{Z}).$$ The following matrix conforms to $\Phi'''$: $$\left[ \begin{array}{ccccccccccccccc|c} 0&0&0&0&0&0&0&0&0&0&1&1&1&1&1&1\\ \hline 1&0&0&0&1&1&1&0&0&0&0&0&0&1&0&1\\ 0&1&0&0&1&0&0&1&1&0&0&0&1&0&0&1\\ 0&0&1&0&0&1&0&1&0&1&0&1&0&0&0&1\\ 0&0&0&1&0&0&1&0&1&1&1&0&0&0&0&1 \end{array} \right].$$ By contracting $y$, we obtain $PG(3,2)$. Thus, we have shown that $\Phi$ must be equivalent to a template with $\Delta$ trivial. So we may assume $$\Phi=(\{1\},\emptyset,X,Y_0,Y_1,A_1,\{0\},\Lambda),$$ with $\Lambda$ generated by $[1,0,\ldots,0]^T$. Now, let us consider the structure of the matrix $A_1$. By repeated use of operation (5), we may assume that $A_1$ is of the following form, with the top row indexed by $\Sigma$, with $*$ representing an arbitrary row vector, with $Y_0=V_0\cup V_1$, and with each $L_i$ representing an arbitrary binary matrix: ------------ ------------ ------------ ------------ ------- $0\cdots0$ $0\cdots0$ $1\cdots1$ $0\cdots0$ $*$ $I$ $L_0$ $L_1$ 0 $L_2$ 0 0 0 $I$ $L_3$ ------------ ------------ ------------ ------------ ------- Suppose either $L_0$ or $L_1$ has a column with two or more nonzero entries. Let $y$ be the element of $Y_1$ that indexes that column, and let $Y'$ be the union of $\{y\}$ with the subset of $Y_1$ that indexes the columns of the identity submatrix of $A_1[X,Y_1]$. Repeatedly perform operations (4) and (10) on $\Phi$ to obtain $$(\{1\},\emptyset,X,\emptyset,Y',A_1,\{0\},\Lambda).$$ On this template, repeatedly perform $y$-shifts, operation (11), and operation (6) to obtain $$(\{1\},\emptyset,X',\emptyset,Y'',\begin{bmatrix} 0&0&x\\ 1& 0 &1\\ 0& 1 & 1 \end{bmatrix},\{0\},\Lambda),$$ where $x=i$ if $y$ indexes a column of $L_i$ and where $X'$ and $Y''$ index the set of rows and columns, respectively, of the matrix $\begin{bmatrix} 0&0&x\\ 1& 0 &1\\ 0& 1 & 1 \end{bmatrix}$. The following matrix conforms to this template. By contracting the columns printed in bold, we obtain $PG(3,2)$. $$\left[ \begin{array}{cccccccccccccc|cccc} 0&0&0&0&0&0&0&0&0&0&1&1&1&\bf{1}&\bf{0}&\bf{0}&x&x\\ 0&0&0&0&0&0&0&0&0&0&0&0&0&\bf{0}&\bf{1}&\bf{0}&1&1\\ 0&0&0&0&0&0&0&0&0&0&0&0&0&\bf{0}&\bf{0}&\bf{1}&1&1\\ \hline 1&0&0&0&1&1&1&0&0&0&0&0&0&\bf{1}&\bf{0}&\bf{0}&1&0\\ 0&1&0&0&1&0&0&1&1&0&0&0&0&\bf{1}&\bf{0}&\bf{0}&0&1\\ 0&0&1&0&0&1&0&1&0&1&1&0&1&\bf{0}&\bf{1}&\bf{0}&0&0\\ 0&0&0&1&0&0&1&0&1&1&0&1&1&\bf{0}&\bf{0}&\bf{1}&0&0\\ \end{array} \right].$$ This shows that $L_0$ and $L_1$ consist entirely of unit and zero columns. Thus, by Lemma  \[simpleY1\], $L_0$ is an empty matrix and $L_1$ consists entirely of distinct unit columns. Therefore, $A_1$ is of the following form: ------------ ------------ ------------ ------------ ------- $0\cdots0$ $0\cdots0$ $1\cdots1$ $0\cdots0$ $*$ $I$ 0 $I$ 0 $Q_1$ 0 $I$ 0 0 $Q_2$ 0 0 0 $I$ $Q_3$ ------------ ------------ ------------ ------------ ------- with each $Q_i$ representing an arbitrary binary matrix. Let $M$ be any matroid conforming to $\Phi$ with rank and connectivity functions $r$ and $\lambda$, respectively. Let $r'$ be the rank of the submatrix of $A_1$ consisting of $Q_1$, $Q_2$, and the row vector we have denoted with a star. Then $r(Y_0)=|V_0|+r'$ and $r(E(M)-Y_0)=r(M)-|V_0|$. Thus, $\lambda(Y_0)=r'$. So if $k>r'+1$, then $M$ is not vertically $k$-connected unless $Y_0$ or $E(M)-Y_0$ is spanning. If $Y_0$ is spanning in $M$, then the $\Gamma$-frame matrix used to construct $M$ has 0 rows. Thus, $M$ is not simple unless $|E(M)|\leq |Y_0|+|Y_1|+1$, with the 1 coming from the element $[1,0\cdots,0]^T$ of $\Lambda$. Thus, if we set $l>|Y_0|+|Y_1|+1$, then no simple, vertically $k$-connected matroid with at least $l$ elements conforms to $\Phi$ unless $E(M)-V_0$ is spanning in $M$. Therefore, we have $V_0=\emptyset$. Let $Q$ be the submatrix of $A_1$ consisting of $Q_1$ and $Q_2$. If every column of $Q$ has at most two nonzero entries, then $\Phi\preceq\Phi_X$, and as we deduced above, we may assume $\Phi=\Phi_D$. Therefore, we assume that $Q$ has a column $c$, indexed by the element $y\in Y_0$ with three or more nonzero entries. Repeatedly perform operation (10) on $\Phi$ to obtain the template $$\Phi'=(\{1\},\emptyset,X,\{y\},Y_1,A_1[D,Y_1\cup\{y\}],\{0\},\Lambda).$$ Let $c=\left[\begin{array}{c} c_1\\ \hline c_2 \end{array}\right]$, with $c_1$ a column of $Q_1$ and $c_2$ a column of $Q_2$. Consider the following cases: 1. The vector $c_1$ has three nonzero entries. 2. The vector $c_1$ has two nonzero entries, and $c_2$ has one nonzero entry. 3. The vector $c_1$ has one nonzero entry, and $c_2$ has two nonzero entries. 4. The vector $c_2$ has three nonzero entries. In Case $i$, repeatedly perform $y$-shifts and operation (11) to obtain the template $$\Phi''_i=(\{1\},\emptyset,X',\{y\},Y'_1,A_{1,i},\{0\},\Lambda),$$ where $A_{1,i}$ is the matrix defined below with rows indexed by $X'$ and columns indexed by $Y'_1\cup\{y\}$. In each case, the last column is indexed by $y$, and it turns out that the value of $x$ does not matter. $$A_{1,1}=\left[ \begin{array}{ccc|ccc|c} 0&0&0&1&1&1&x\\ \hline 1&0&0&1&0&0&1\\ 0&1&0&0&1&0&1\\ 0&0&1&0&0&1&1\\ \end{array} \right] A_{1,2}=\left[ \begin{array}{ccc|cc|c} 0&0&0&1&1&x\\ \hline 1&0&0&1&0&1\\ 0&1&0&0&1&1\\ 0&0&1&0&0&1\\ \end{array} \right]$$ $$A_{1,3}=\left[ \begin{array}{ccc|c|c} 0&0&0&1&x\\ \hline 1&0&0&1&1\\ 0&1&0&0&1\\ 0&0&1&0&1\\ \end{array} \right] A_{1,4}=\left[ \begin{array}{ccc|c} 0&0&0&x\\ \hline 1&0&0&1\\ 0&1&0&1\\ 0&0&1&1\\ \end{array} \right]$$ In Case $i$, the matrix below virtually conforms to $\Phi''_i$. By contracting the columns printed in bold, we obtain $PG(3,2)$. 1. $$\left[ \begin{array}{ccc|cccccccccccc|c} 1&1&0&0&0&0&1&1&1&0&0&0&1&1&1&\textbf{\textit{x}}\\ 0&0&0&1&0&0&1&0&0&1&0&0&1&0&0&\bf{1}\\ 0&0&0&0&1&0&0&1&0&0&1&0&0&1&0&\bf{1}\\ 0&0&0&0&0&1&0&0&1&0&0&1&0&0&1&\bf{1}\\ \hline 1&0&1&1&1&1&1&1&1&0&0&0&0&0&0&\bf{0}\\ \end{array} \right]$$ 2. $$\left[ \begin{array}{ccccccc|ccccccccc|c} 1&0&0&0&1&1&1&0&0&\bf{0}&1&1&0&0&1&1&\textbf{\textit{x}}\\ 0&0&0&0&0&0&0&1&0&\bf{0}&1&0&1&0&1&0&\bf{1}\\ 0&0&0&0&0&0&0&0&1&\bf{0}&0&1&0&1&0&1&\bf{1}\\ 0&0&0&0&0&0&0&0&0&\bf{1}&0&0&0&0&0&0&\bf{1}\\ \hline 0&1&0&1&1&0&1&1&1&\bf{1}&1&1&0&0&0&0&\bf{0}\\ 0&0&1&1&0&1&1&0&0&\bf{0}&0&0&1&1&1&1&\bf{0}\\ \end{array} \right]$$ 3. $$\left[ \begin{array}{ccccc|cccccccccccc|c} 0&0&0&\bf{1}&\bf{1}&0&0&0&0&0&0&1&0&0&0&0&0&\textbf{\textit{x}}\\ 0&0&0&\bf{0}&\bf{0}&1&0&0&1&0&0&1&0&0&1&0&0&\bf{1}\\ 0&0&0&\bf{0}&\bf{0}&0&1&0&0&1&0&0&1&0&0&1&0&\bf{1}\\ 0&0&0&\bf{0}&\bf{0}&0&0&1&0&0&1&0&0&1&0&0&1&\bf{1}\\ \hline 1&0&1&\bf{1}&\bf{0}&1&1&1&0&0&0&0&0&0&0&0&0&\bf{0}\\ 0&1&1&\bf{0}&\bf{1}&0&0&0&1&1&1&1&0&0&0&0&0&\bf{0}\\ 0&0&0&\bf{0}&\bf{1}&0&0&0&0&0&0&0&1&1&0&0&0&\bf{0}\\ \end{array} \right]$$ 4. $$\left[ \begin{array}{ccccc|cccccccccccc|c} 1&0&\bf{1}&1&\bf{1}&0&0&0&0&0&0&0&0&0&0&0&0&\textbf{\textit{x}}\\ 0&0&\bf{0}&0&\bf{0}&1&0&0&1&0&0&1&0&0&1&0&0&\bf{1}\\ 0&0&\bf{0}&0&\bf{0}&0&1&0&0&1&0&0&1&0&0&1&0&\bf{1}\\ 0&0&\bf{0}&0&\bf{0}&0&0&1&0&0&1&0&0&1&0&0&1&\bf{1}\\ \hline 0&0&\bf{1}&0&\bf{0}&1&1&1&0&0&0&0&0&0&0&0&0&\bf{0}\\ 0&0&\bf{0}&0&\bf{1}&0&0&0&1&1&1&0&0&0&0&0&0&\bf{0}\\ 0&1&\bf{0}&1&\bf{1}&0&0&0&0&0&0&1&1&1&0&0&0&\bf{0}\\ \end{array} \right]$$ By contradiction, this completes the proof. \[proof\] Let $\mathcal{M}=\mathcal{EX}(PG(3,2))$, and let $\mathcal{T}_{\mathcal{M}}=\linebreak\{\Phi_1,\dots\Phi_s,\Psi_1,\dots,\Psi_t\}$. By Lemma  \[conformonly\], for sufficiently large $r$, we have $h_{\mathcal{M}}(r)$ equal to the size of the largest simple matroid of rank $r$ that virtually conforms to any template in $\Phi\in\{\Phi_1,\dots\Phi_s\}$. If $\Phi\in\{\Phi_1,\dots\Phi_s\}$, then by Lemma  \[PG32Phi\] either $\Phi=\Phi_X$ or $\Phi$ is of the form $(\{1\},\emptyset,X,Y_0,Y_1,A_1,\{0\},\{0\})$, for some matrix $A_1$ and some sets $X$, $Y_0$, and $Y_1$. Moreover, by operation (5), we may assume that $A_1$ is of the following form, with $Y_0=V_0\cup V_1$ and with the stars representing arbitrary binary matrices: ----- ----- ----- ----- $I$ $*$ 0 $*$ 0 0 $I$ $*$ ----- ----- ----- ----- . The largest simple matroid of rank $r$ that virtually conforms to $\Phi$ is obtained by taking for the $\Gamma$-frame matrix a matrix representation of $M(K_{n+1})$, where $n=r-r(M(A_1[X,Y_1]))-|V_0|$. Thus, the largest simple matroid of rank $r$ that virtually conforms to $\Phi$ has size $\binom{n+1}{2}+|Y_1|n+|Y_1|+|Y_0|$. Substituting $r-r(M(A_1[X,Y_1]))-|V_0|$ for $n$, one sees that for sufficiently large $r$, this expression is less than $r^2-r+1$. Since the class of matroids virtually conforming to $\Phi_X$ is the class of even-cycle matroids, which has growth rate $r^2-r+1$, the result holds. 1-flowing Matroids {#1-flowing Matroids} ================== In this section, we prove Theorem  \[1flowing\]. The 1-flowing property is a generalization of the max-flow min-cut property of graphs. See Seymour  [@s81] or Mayhew  [@m15] for more of the background and motivation concerning 1-flowing matroids. We follow the notation and exposition of  [@m15]. Let $e$ be an element of a matroid $M$. Let $c_x$ be a non-negative integral capacity assigned to each element $x\in E(M)-e$. A flow is a function $f$ that assigns to each circuit $C$ containing $e$ a non-negative real number $f_C$ with the constraint that for each $x\in E-e$, the sum of $f_C$ over all circuits containing both $e$ and $x$ is at most $c_x$. We say that $M$ is *$e$-flowing* if, for every assignment of capacities, there is a flow whose sum over all circuits containing $e$ is equal to $$\min\{\sum_{x\in C^*-e}c_x | C^* \textnormal{ is a cocircuit containing }e \}.$$ If $M$ is $e$-flowing for each $e\in E(M)$, then $M$ is *1-flowing*. The matroid $T_{11}$ is the even-cycle matroid obtained from $K_5$ by adding a loop and making every edge odd, including the loop. In  [@s81], Seymour showed the following. The The class of 1-flowing matroids is minor-closed. Moreover, $AG(3,2)$, $U_{2,4}$, $T_{11}$, and $T^*_{11}$ are excluded minors for the class of 1-flowing matroids. Seymour  [@s81] conjectured that these are the only excluded minors. The set of excluded minors for the class of 1-flowing matroids consists of $AG(3,2)$, $U_{2,4}$, $T_{11}$, and $T^*_{11}$. Since $U_{2,4}$ is an excluded minor for the class of 1-flowing matroids, all such matroids are binary. Therefore, the results in this paper apply to 1-flowing matroids. We will now prove Theorem  \[1flowing\], which we restate below. There exist $k,l\in\mathbb{Z}_+$ such that every simple, vertically $k$-connected, 1-flowing matroid with at least $l$ elements is either graphic or cographic. The matroid $AG(3,2)$ conforms to $\Phi_{Y_1}$ since it is a restriction of $N_{12}$. Indeed, consider the matrix representing $N_{12}$ that virtually conforms to $\Phi_{Y_1}$. Add the rows labeled by $X$ in this matrix to one of the other rows. Then we can see the matrix representation $[I_4|J_4-I_4]$ of $AG(3,2)$ as a restriction of $N_{12}$. Also, it is not difficult to see that $AG(3,2)$ can be obtained from a matroid conforming to $\Phi_{Y_0}$ by contracting $Y_0$. Thus, $\mathcal{EX}(AG(3,2))$ contains neither $\mathcal{M}(\Phi_{Y_0})$ nor $\mathcal{M}(\Phi_{Y_1})$. Since $AG(3,2)$ is self-dual, $\mathcal{EX}(AG(3,2))$ does not contain $\mathcal{M}^*(\Phi_{Y_0})$, or $\mathcal{M}^*(\Phi_{Y_1})$ either. Therefore, by Corollary  \[Y0Y1\], $\mathcal{EX}(AG(3,2))$ is described by the trivial template. Thus, since $AG(3,2)$ is an excluded minor for the class of 1-flowing matroids, there exist $k,l\in\mathbb{Z}_+$ such that every simple, vertically $k$-connected, 1-flowing matroid with at least $l$ elements either conforms or coconforms to the trivial template. The result follows. Acknowledgements {#acknowledgements .unnumbered} ================ We thank the two anonymous referees for carefully reading the manuscript. In particular, we thank the first referee for giving many suggestions that improved the manuscript, including one that greatly simplified the proof of Lemma  \[graphicvscographic\]. [99]{} Jim Geelen, Bert Gerards, and Geoff Whittle, The highly connected matroids in minor-closed classes, [*Annals of Combinatorics*]{} [**19**]{} (2015), 107–123. Jim Geelen, and Kasper Kabell, Projective geometries in dense matroids, [*Journal of Combinatorial Theory, Series B*]{} [**99**]{} (2009), 1–8. Jim Geelen, and Geoff Whittle, Cliques in dense $\mathrm{GF}(q)$-representable matroids, [*Journal of Combinatorial Theory, Series B*]{} [**87**]{} (2003), 264–269. Jim Geelen, Joseph P.S. Kung, and Geoff Whittle, Growth rates of minor-closed classes of matroids, [*Journal of Combinatorial Theory, Series B*]{} [**99**]{} (2009), 420–427. Jim Geelen and Peter Nelson, Matroids denser than a clique, [*Journal of Combinatorial Theory, Series B*]{} [**114**]{} (2015), 51–69. Joseph P. S. Kung, Dillon Mayhew, Irene Pivotto, and Gordon F. Royle, Maximum size binary matroids with no $AG(3,2)$-minor Are graphic, [*SIAM Journal on Discrete Mathematics*]{} [**28**]{} (2014), 1559–1577. Dillon Mayhew, Seymour’s 1-flowing Conjecture III, The Matroid Union, WordPress, 19 Mar. 2015, Web, 19 May 2016. James Oxley, [*Matroid Theory*]{}, Second Edition, Oxford University Press, New York, 2011. Neil Robertson and P.D. Seymour, Graph Minors. XVI. Excluding a non-planar graph, [*Journal of Combinatorial Theory, Series B*]{} [**89**]{} (2003), 43–75. P.D. Seymour, Matroids and Multicommodity Flows. European Journal of Combinatorics [**2**]{} (1981), 257-290. [^1]: The first author was supported by a Huel D. Perkins Fellowship from the Louisiana State University Graduate School. The second author was supported by the National Science Foundation, grant 1500343. [^2]: The authors of  [@ggw15] divided our set $X$ into two separate sets which they called $X$ and $D$. Their set $X$ can be absorbed into $Y_0$, therefore we omit it.

--- abstract: '[We report a strong thickness dependence of the complex frequency-dependent optical dielectric function $\widetilde{\epsilon}(\omega)$ over a spectral range from 1.24 to 5 eV in epitaxial CaMnO$_3$(001) thin films on SrTiO$_3$(001), LaAlO$_3$(001), and SrLaAlO$_4$(001). A doubling of the peak value of the imaginary part of $\widetilde{\epsilon}(\omega)$ and spectral shifts of 0.5 eV for a given magnitude of absorption are observed. On the basis of experimental analyses and first-principles density functional theory calculations, contributions from both surface states and epitaxial strain to the optical dielectric function of CaMnO$_3$ are seen. Its evolution with thickness from 4 to 63 nm has several regimes. In the thinnest, strain-coherent films, the response is characterized by a significant contribution from the free surface that dominates strain effects. However, at intermediate and larger thicknesses approaching the bulk-like film, strain coherence and partial strain relaxation coexist and influence $\widetilde{\epsilon}(\omega)$.]{}' author: - Dominic Imbrenda$^1$ - Dongyue Yang$^2$ - Hongwei Wang$^2$ - 'Andrew R. Akbashev$^3$' - Leila Kasaei$^2$ - 'Bruce A. Davidson$^2$' - Xifan Wu$^2$ - Xiaoxing Xi$^2$ - 'Jonathan E. Spanier$^{1,3,4}$' bibliography: - 'bib\_file.bib' title: 'Surface- and strain-tuning of the optical dielectric function in epitaxially grown CaMnO$_3$' --- Strain engineering has long been used as an effective way to tune electronic, magnetic and optical properties of oxide thin films [@spaldin_strain_1; @bhattacharjee2009engineering]. The lattice distortions imposed by epitaxial strain can introduce dramatic changes in the properties of thin-film materials, e.g. allowing strong ferroelectric ordering in quantum paraelectrics [@haeni2004room], manipulation of transition temperatures in ferroelectrics [@strain_FE_1], tuning of magnetic and metal-insulator transitions in mixed-valence perovskite oxides [@strain_MIT_1; @strain_MIT_2], and controlling the volume of the magnetic phase in magnetically inhomogeneous media  [@phase_separ_1]. Owing to the direct relationship between the electronic structure and optical properties, epitaxial strain strongly influences dielectric constants, refractive indices and, ultimately, the bandgap of a thin film material  [@singh2014strain; @scafetta2014band; @liu2013strain; @scafetta2013optical]. However, such studies for oxides are still scarce, and the roles of chemistry, structure, native defects (oxygen vacancies), film thickness and surface effects (termination, admolecules, structural reconstruction) have yet to be clarified. In the bulk, electronic structure and optical properties of CaMnO$_3$ (CMO) are well studied [@jung1997determination; @loshkareva2004electronic; @molinari2014structural], yet little attention has been given to the optical properties of epitaxial CMO thin films. CMO has long been known as an archetypal mixed-valence manganite that exhibits colossal magnetoresistance (CMR). However, strain-induced multiferroicity has also been predicted for CMO  [@bhattacharjee2009engineering] with incipient ferroelectricity being later confirmed in tensile-strained films  [@CMO_FE]. Similar to other perovskite oxides, CMO can easily accommodate oxygen vacancies that can be introduced allowing the material to demonstrate modest electrocatalytic activity  [@CMO_catalyst_1; @CMO_catalyst_2]. Still, little is known about the changes in the electronic structure in thin and ultrathin CMO films despite a certain surge of interest to strain-mediated effects on the optical properties of perovskite oxides  [@dejneka2010tensile; @liu2013strain; @scafetta2014band; @singh2014strain; @roy2012effects; @chernova2015strain; @Choi2014LASTO]. In addition, because surface reconstruction and termination become crucial in a few unit cell thick perovskite oxides, ultrathin CMO films are expected to demonstrate optical properties that are distinct from those of thicker films  [@saldana2015structural]. Here we report pronounced thickness dependence of the complex frequency-dependent optical dielectric function $\widetilde{\epsilon}(\omega)$ in epitaxial CMO thin films. Using spectroscopic ellipsometry we perform detailed characterization of optical properties of CMO thin films, and we employ first-principles density functional theory (DFT) calculations to determine the contribution both from the surface states and epitaxial strain to the optical dielectric function of CMO. Epitaxial CMO thin films were grown by pulsed laser deposition (PLD) using a KrF excimer laser ($\lambda$ = 248 nm) on single-crystal (001)-oriented SrTiO$_3$ (STO), LaAlO$_3$ (LAO), and SrLaAlO$_4$ (SLAO) purchased from Crystec  [@crystec]. A laser repetition rate of 2.11 Hz and a laser energy density of 2.0 J/cm$^2$ were used. The substrate temperature was 650$^\circ$C and the background pressure was around 8.6$\times$10$^{-6}$ Torr. Films were deposited in oxygen environment at 30 mTorr and after the deposition, the sample was cooled to room temperature within an oxygen environment around 300 Torr to reduce/eliminate oxygen vacancies in the films. The thicknesses of the films were measured by X-ray reflectivity (XRR) (Fig. 1(d)) and the film deposition rate was determined to be approximately 0.74 Å/pulse. Samples ranging from 4.1 to 63 nm in thickness were each grown on SrTiO$_3$(001), and films ranging in thickness from 4.3 to 10 nm were grown on LAO(001) and SLAO(001). Grazing incidence X-ray diffraction (GIXRD)  [@Lee2011] (Fig. 1(a)) and reciprocal space mapping (RSM) were used to determine film lattice strain states by measuring the film in-plane lattice constant for thin ($<$20 nm) and thick ($>$20 nm) films respectively. As can be seen in Fig. 1(a), the thinnest film (4.1 nm) shows no substrate peak broadening and no extra peak, indicating the fully strained status of the film. In the 7.1 nm sample curve, the shoulder of the STO peak is evidence that the film starts to relax but is still mainly strained to the substrate. In the 10.4 nm sample curve, no shoulder at the STO peak position and the bump at the CMO(002) position shows a mainly relaxed state. X-ray diffraction (XRD) (Fig. 1(b)) was used to ensure the films were single phase crystals with no secondary phases and to measure out-of-plane lattice parameter. Atomic force microscopy (AFM) (Fig. 1(c)) was used to confirm film surface quality. Five CMO film samples of different thicknesses ranging from 4.1 nm to 62.9 nm were grown on STO, and two additional films of $\sim$10 nm and $\sim$4 nm were grown, each on LAO and SLAO, to probe and disentangle the effects of thickness, strain and strain relaxation on $\widetilde{\epsilon}(\omega)$ of CMO. The bulk in-plane lattice parameter of STO ($a_\text{STO}$ = 3.905 Å) is larger than that for both LAO ($a_\text{LAO}$ = 3.790 Å) and SLAO ($a_\text{SLAO}$ = 3.754 Å) and largest compared to that for bulk CMO ($a_\text{CMO}$ = 3.72 Å). As STO has the largest in-plane lattice parameter compared to bulk CMO, the in-plane CMO film strain coherency in CMO/STO persists to smaller thicknesses than for CMO films on LAO or SLAO. A summary of the parameters for each film, including substrate, thickness, measured in-plane lattice parameter $a_\parallel$, the corresponding in-plane strain, and surface roughness, is shown in Table I. Variable-angle spectroscopic ellipsometry (VASE) was performed at room temperature in ambient atmosphere with an electronically controlled rotating compensator and Glan-Taylor polarizers (J.A. Woollam, M2000). Measurements were performed at multiple angles between 65-75$^\circ$ and in the spectral range of 247 to 1000 nm with a resolution of 1.6 nm. Measurement of the components of linearly polarized reflectivity at each selected wavelength were used to obtain the ellipsometric parameters $\Psi$ and $\Delta$. To determine $\widetilde{\epsilon}(\omega)$ for CMO we assume a four-layer optical medium comprised of a homogeneous isotropic film layer on a semi-infinite bulk, incorporating surface roughness, in vacuum. The surface roughness layer is modeled using the Bruggeman effective medium approximation using a 50% film and 50% void at the surface [@tompkins1999spectroscopic] with thickness approximately equal to the rms roughness for each sample obtained from XRR. Our model accounting for surface effects is valid as long as the surface layers are thin compared to the bulk and have a refractive index lower than that of the bulk [@nelson2012dielectric]. Optical dielectric functions of each substrate from the same batch as those used for film growth were also determined, assuming a semi-infinite half-space in vacuum, including surface roughness, and incorporated as the bulk layer in the model. To fit the spectral dependence and calculate the complex index of refraction of each sample a combination of Gaussian curves and Lorentz oscillators [@tompkins1999spectroscopic] were used to model the experimental data for the CMO films. Both of these functions are Kramers-Kronig (K-K) consistent which ensures causality. Experimental data for the STO and SLAO substrates were modeled using Gaussian curves and Lorentz oscillators however, as the spectral range for measurement in our setup is below the bandgap of LAO [@lim2002dielectric] a Cauchy function was used to model its optical response. The number and type of oscillators used was sample dependent. Regression analysis using the Levenberg-Marquardt algorithm was performed until the weighted mean squared error (MSE) between the calculated and experimental data was minimized. Thickness of the film and surface roughness were determined from XRR and held constant in the model until a satisfactory fit was achieved. After obtaining an acceptable MSE the thickness and surface roughness were made free parameters to ensure good agreement of thickness between the model and XRR data. VASE- and XRR- determined thicknesses, along with the MSE for each film, are given in Table I. $\widetilde{\epsilon}(\omega)$ depends sensitively on film thickness (Fig. 2). For films grown on STO, increasing thickness is accompanied by progressive strain relaxation and evolution of $\widetilde{\epsilon}(\omega)$. For our thickest, bulk-like film (62.9 nm), $\widetilde{\epsilon}(\omega)$ is consistent with previously published data for bulk CMO crystals  [@loshkareva2004electronic; @nomerovannaya2006ellipsometric]. For thinner films and increasing thickness fraction of strain coherence, to fully strain-coherent (about 4 and 6 nm films), there is reduction by as much as 50% of both the real and imaginary parts of $\widetilde{\epsilon}(\omega)$. The linear portion of the optical absorption $\alpha(\omega)$ = 4$\pi k$/$\lambda$ shifts by as much as 0.5 eV with film thickness in the range of 2 $<$ $\hbar\omega$ $<$ 2.5 eV (Fig. 2(a)). This is due to the shift in energy of the first peak in the imaginary component ($\epsilon_{2}$) of $\widetilde{\epsilon}(\omega)$ and a reduction in transition strength as we discuss below. To further understand the origin of the thickness-dependence of $\widetilde{\epsilon}(\omega)$ we analyzed CMO films of comparable thicknesses, deposited on LAO and SLAO. The in-plane lattice parameter of SLAO is closest to CMO and the films deposited on SLAO are under a smaller in-plane strain compared to films on STO (Table I). The peak value $\epsilon_{2}$ is larger for a film deposited on SLAO than for a film of comparable thickness deposited on STO (Fig. 3). LAO has an in-plane lattice constant intermediate to SLAO and STO and films deposited on LAO are less strained than those deposited on STO but are under more strain than those on SLAO. This results in a peak value of $\epsilon_{2}$ intermediate to that of films of similar thickness deposited on SLAO and STO (Fig. 3). GIXRD data (Fig. 1(a)) indicate that both SLAO films and both LAO films are fully strained to the substrate, yet we still observe an evolution of $\epsilon_{2}$. If strain was the dominant effect, we would expect $\epsilon_{2}$ to be the same for a given strain state. First-principles calculations were performed by using DFT as implemented in the Vienna Ab Initio Simulation Package (VASP)[@kresse93; @kresse96]. The electron exchange and correlation were approximated by the generalized gradient approximation revised for solid (PBEsol)[@perdew08] and we adopted an effective on-site Coulomb repulsion $U-J = 3.0~{\rm eV}$ for the $d$ orbitals of Mn atoms [@Jiawang12]. The Kohn-Sham equations were expanded by plane-wave bases truncated at a cutoff energy of 500 eV. A $k$-point mesh of 6 $\times$ 6 $\times$ 1 was used in both the calculation of the (CMO)$_{4}$(STO)$_{4}$(CMO)$_{4}$ supercell and the calculation of the surface between this supercell and vacuum ((CMO)$_4$(STO)$_4$(CMO)$_4$-$surface$), in which vacuum is approximated by 10 [Å]{} distance along the \[001\] direction within the periodic boundary condition. For the studies of bulk CMO strain coherent with STO, a denser $k$-point mesh of 6 $\times$ 6 $\times$ 4 was used. The lattice constant $c$ along the \[001\] direction as well as all the atomic positions are fully relaxed with the remaining Hellman-Feynman force being less than 1 m[eV]{}/[Å]{}. In all the *surface* theoretical models, we consider the surface relaxation by including the vacuum in the model. The vacuum spans in the space for about 10 [Å]{} above the surface of the CMO under the periodic boundary condition. In order to determine the stability of different surface terminations, we performed the density functional theory total energy minimizations for both MnO$_2$ terminated and CaO terminated surface models, in which all the atomic positions are allowed to relax. It was found that the MnO$_2$ terminated surface is always more energetically stable than CaO in all the slabs. As a result, the MnO$_2$ terminated surface models are used in our theoretical simulations. Notably in the films of $\sim$5 nm, equilibrium surface atomic structure and electronic wavefunction mixing of the substrate with the bottom several unit cells of CMO can contribute to $\widetilde{\epsilon}(\omega)$, in addition to strain. DFT calculations were performed, considering several candidate structures. To provide further insight into the primary origin of the thickness-dependence we calculated $\widetilde{\epsilon}(\omega)$ in the in-plane direction for four and six unit-cell supercells of CMO. We first omitted contributions from strain from a substrate to determine the effect of energy-minimized surface truncation on $\widetilde{\epsilon}(\omega)$. Remarkably, the effect is very pronounced for a four unit-cell ($\sim$3.2 nm-thick) film, and considerably weaker for the 6 unit-cell ($\sim$4.8 nm-thick) film (Fig. 4(a)). Incorporation of epitaxial strain (Fig. 4(b)) by introducing STO into a four unit-cell supercell, (CMO)$_4$(STO)$_4$(CMO)$_4$, with no free surface, reduces $\epsilon_{2}$, but less than the reduction with a (CMO)$_4$ supercell with one free surface. Significantly, for the 4-layer CMO film with a free, vacuum-terminated surface, the effect of including substrate strain i.e., (CMO)$_4$(STO)$_4$(CMO)$_4$-$surface$ is relatively small (Fig. 4(b)). The first peak in the imaginary part of in-plane dielectric function is barely changed; while the second peak is slightly higher in the computed $\epsilon_{2}$ of (CMO)$_4$(STO)$_4$(CMO)$_4$-$surface$ than that of the 4-layer CMO film with a free, vacuum-terminated surface. This is due to the additional spectra signals from the STO component. Close inspection on the transition matrices involved in the calculation of dielectric function reveals that the first peak of the imaginary part of the in-plane dielectric function can be assigned to the hopping processes between equatorial oxygen *p* orbitals and the e$_g$ electrons of $d_{x^{2}-y^{2}}$ character. At the surface, a MnO$_2$ terminated slab is always found to be more energetically favorable than the SrO terminated one. As result, the Mn atom at MnO$_2$ terminated surface undergoes an abrupt reconstruction due to the missing of one apical oxygen atoms. Such a surface reconstruction in turn largely increases the $d_{x^{2}-y^{2}}$ levels originating from the shortened Mn-O bond length and therefore increases the Coulombic energy. The phonon frequencies due to the hopping processes between occupied equatorial oxygen *p* and the non-occupied e$_g$ ($d_{x^{2}-y^{2}}$) electrons also shift to higher energy. As a result, the surface reconstruction shifts the first peak of $\epsilon_{2}$ towards higher energy with a reduced magnitude. This optical effect from surface reconstruction will be more pronounced when the film goes to the thin limit as seen in both experiment and theory in Figs. 3 and 4 respectively. This analysis was repeated for thin films strained by a LAO substrate (Fig. 4(c)), producing the same results. According to the first-principles calculations, the difference in dielectric function between a large and a small CMO supercell originates from a structural reconstruction on the surface between the supercell and the vacuum, the energy associated with the reconstruction being $\sim$3 eV for the total surface model system. This value is much larger than 0.07 eV, which is the computed energy difference between bulk G-type antiferromagnetic CMO at zero temperature and paramagnetic bulk CMO at room temperature. These are modeled by the structural parameters relaxed to the ground state and using an approximate paramagnetic spin configuration [@Xiang2015], with structural parameters taken from the experiment performed at room temperature by Bozin [*et al.*]{} [@Bozin08]. Thus, our experimental results can be qualitatively interpreted by the first-principles calculations at 0 K assuming the G-type antiferromagnetic configuration. The thickness dependence of $\widetilde{\epsilon}(\omega)$ correlates well with strain relaxation and surface contribution. However, other possible contributions may take place. For example, CMO exhibits magnetic ordering, but the Néel temperature in bulk CMO, $T_\text{N, CMO}$, is $\sim$130 K ($\ll$ 300 K) [@nicastro2002exchange], and it is highly unlikely that strain and/or finiteness of the film allows magnetic ordering at 300 K. It has been demonstrated that under high tensile strain oxygen vacancy formation in CMO is favored  [@aschauer2013strain], which could alter band energies. These changes in band energies could alter the electronic and optical properties of thin films and their contribution, although not addressed in this work, cannot be ruled out. We have shown that $\widetilde{\epsilon}(\omega)$ decreases in magnitude for decreasing CMO film thickness, particularly as the strain transitions from partially relaxed to coherently strained. We observed that for the thinnest films this evolution of $\widetilde{\epsilon}(\omega)$ continues even under the same strain state. Taken together, our DFT results, combined with the spectroscopic ellipsometry analysis of the films, indicate that in the thinnest films the surface contribution is dominant, whereas in the thicker films progressive partial and full strain relaxation dominates the evolution of $\widetilde{\epsilon}(\omega)$. The interplay among thickness-driven large surface and substrate-induced strain contributions to $\widetilde{\epsilon}(\omega)$ in an epitaxial perovskite oxide thin film in its non-magnetic phase holds promise for a novel route to thickness-induced engineering of optical properties. [@ccccccc@]{} & **** ---------- *t*, XRR (nm) ---------- : Structural parameters of CMO films in this study. Thickness (*t*) was determined from XRR and VASE. Thickness from XRR data was determined first and used initially during VASE parameter fitting. Film in-plane lattice parameter ($a_\parallel$) is extracted from GIXRD (for films $<$20 nm) or RSM (for films $>$20 nm) characterization and data fitting. MSE is mean squared error from VASE fitting. Surface roughness was determined from XRR and VASE. The surface roughness as determined by XRR is given by the fittings of XRR data. Fitting of the XRR data provides a roughness value being the standard deviation of the rms error function. The VASE roughness error is determined from the 90% confidence limit of the value of the fitting parameter used to determine surface roughness. All error values are in parenthesis.[]{data-label="Table I."} & **** ----------- *t*, VASE (nm) ----------- : Structural parameters of CMO films in this study. Thickness (*t*) was determined from XRR and VASE. Thickness from XRR data was determined first and used initially during VASE parameter fitting. Film in-plane lattice parameter ($a_\parallel$) is extracted from GIXRD (for films $<$20 nm) or RSM (for films $>$20 nm) characterization and data fitting. MSE is mean squared error from VASE fitting. Surface roughness was determined from XRR and VASE. The surface roughness as determined by XRR is given by the fittings of XRR data. Fitting of the XRR data provides a roughness value being the standard deviation of the rms error function. The VASE roughness error is determined from the 90% confidence limit of the value of the fitting parameter used to determine surface roughness. All error values are in parenthesis.[]{data-label="Table I."} & **** ------------------- $a_\parallel$ (Å) ------------------- : Structural parameters of CMO films in this study. Thickness (*t*) was determined from XRR and VASE. Thickness from XRR data was determined first and used initially during VASE parameter fitting. Film in-plane lattice parameter ($a_\parallel$) is extracted from GIXRD (for films $<$20 nm) or RSM (for films $>$20 nm) characterization and data fitting. MSE is mean squared error from VASE fitting. Surface roughness was determined from XRR and VASE. The surface roughness as determined by XRR is given by the fittings of XRR data. Fitting of the XRR data provides a roughness value being the standard deviation of the rms error function. The VASE roughness error is determined from the 90% confidence limit of the value of the fitting parameter used to determine surface roughness. All error values are in parenthesis.[]{data-label="Table I."} & **** ---------- In-plane strain% ---------- : Structural parameters of CMO films in this study. Thickness (*t*) was determined from XRR and VASE. Thickness from XRR data was determined first and used initially during VASE parameter fitting. Film in-plane lattice parameter ($a_\parallel$) is extracted from GIXRD (for films $<$20 nm) or RSM (for films $>$20 nm) characterization and data fitting. MSE is mean squared error from VASE fitting. Surface roughness was determined from XRR and VASE. The surface roughness as determined by XRR is given by the fittings of XRR data. Fitting of the XRR data provides a roughness value being the standard deviation of the rms error function. The VASE roughness error is determined from the 90% confidence limit of the value of the fitting parameter used to determine surface roughness. All error values are in parenthesis.[]{data-label="Table I."} & **** ----- MSE ----- : Structural parameters of CMO films in this study. Thickness (*t*) was determined from XRR and VASE. Thickness from XRR data was determined first and used initially during VASE parameter fitting. Film in-plane lattice parameter ($a_\parallel$) is extracted from GIXRD (for films $<$20 nm) or RSM (for films $>$20 nm) characterization and data fitting. MSE is mean squared error from VASE fitting. Surface roughness was determined from XRR and VASE. The surface roughness as determined by XRR is given by the fittings of XRR data. Fitting of the XRR data provides a roughness value being the standard deviation of the rms error function. The VASE roughness error is determined from the 90% confidence limit of the value of the fitting parameter used to determine surface roughness. All error values are in parenthesis.[]{data-label="Table I."} & **** ----------------- Film roughness, VASE (nm) ----------------- : Structural parameters of CMO films in this study. Thickness (*t*) was determined from XRR and VASE. Thickness from XRR data was determined first and used initially during VASE parameter fitting. Film in-plane lattice parameter ($a_\parallel$) is extracted from GIXRD (for films $<$20 nm) or RSM (for films $>$20 nm) characterization and data fitting. MSE is mean squared error from VASE fitting. Surface roughness was determined from XRR and VASE. The surface roughness as determined by XRR is given by the fittings of XRR data. Fitting of the XRR data provides a roughness value being the standard deviation of the rms error function. The VASE roughness error is determined from the 90% confidence limit of the value of the fitting parameter used to determine surface roughness. All error values are in parenthesis.[]{data-label="Table I."} \ STO(001) & 4.1(0.18) & 4.1(0.79) & 3.905& 5.0 & 1.52 & 0.3(0.04)\ & 5.8(0.14) & 6.1(0.01) & 3.905& 5.0 & 2.08 & 0(0.02)\ & 7.1(0.19) & 7.0(0.19) & 3.90(0.002) & 4.8 & 1.93 & 0(0.01)\ & 10.4(0.19) & 10.3(0.97) & 3.76(0.002) & 1.1 & 3.61 & 0(0.05)\ & 62.9(0.17) & 60.8(0.18) & 3.75(0.001) & 0.8 & 2.71 & 0(0.12)\ LAO(001) & 9.6(0.85) & 9.6(0.21) & 3.790& 1.9 & 4.07 & 0.28(0.05)\ & 4.9(0.24) & 4.3(0.07) & 3.790& 1.9 & 2.99 & 0(0.02)\ SLAO(001) & 9.0(0.09) & 10.0(0.94) & 3.754& 0.91 & 5.16 & 0.1(0.05)\ & 4.3(0.37) & 4.2(0.04) & 3.754& 0.91 & 4.73 & 0(0.04)\ ![(*a*) GIXRD, (*b*) XRD,(*c*) AFM and (*d*) XRR data of CMO films with varying thickness grown on different substrates. (*a*) GIXRD spectra of a 10.4, 7.1, and 4.1 nm thick CMO film grown on STO substrates. The film peak is at 48.44$^\circ$. Fitting is shown (dashed line) for the 10.4 nm film. (*b*) The 2$\theta/\omega$ XRD scan of CMO thin films ($\sim$10 nm in thickness) on each substrate. Substrate peaks and film peaks (48.93$^\circ$) are identified. (*c*) AFM of the 10.4 nm thick CMO film on STO showing the underlying terraces from the annealed STO substrate, indicating an atomically flat film. (*d*) The XRR spectra for films with different thickness on STO, LAO and SLAO substrates. Fitting is given for each trace (dashed line).[]{data-label="fig:FIG_1"}](FIG_1.png) ![(*a*) real, $\epsilon_{1}$, and (*b*) imaginary part, $\epsilon_{2}$, of the experimentally determined complex frequency-dependent dielectric function as a function of photon energy for different CMO film thicknesses on STO(001). The arrow denotes films of decreasing thickness, as specified in Table I. The thinnest two films are strain coherent, as the films increase in thickness they begin to strain relax. The inset shows $\alpha$ for the 62.9 nm (solid) and 4.1 nm (dotted) films plotted on a semi-log scale over the range 1.2 - 3 eV. At a fixed value of absorption (solid line shows $\alpha$ = 2.5$\times$10$^5$ cm$^{-1}$) there is as much as a $\sim$0.5 eV spectral shift in the photon energy for decreasing film thickness.[]{data-label="fig:FIG_2"}](FIG_2.png) ![Imaginary part, $\epsilon_{2}$, of the experimentally determined complex frequency-dependent dielectric function as a function of photon energy for different CMO film thicknesses on three different substrates. The legend in the figure is the CMO film thickness on the indicated substrate. The dielectric function of the CMO/SLAO film (diamond marker) is closest of that of bulk CMO as compared to CMO grown on LAO (no marker) or STO (square marker). Thicker films are shown with a solid line, the thinner films are shown with a dashed line.[]{data-label="fig:FIG_3"}](FIG_3.png) ![(*a*) The effect of free energy-minimized surface termination on the imaginary part of the complex dielectric function ($\epsilon_{2}$) in the in-plane lattice direction for four- and six-unit cell structures with a free surface as compared to that for bulk CMO strain coherent with STO (CMO(STO)) or LAO (CMO (LAO)). (*b*) $\epsilon_{2}$ in the in-plane lattice direction incorporating epitaxial strain by introducing STO into the supercell both with no free surface ((CMO)$_4$(STO)$_4$(CMO)$_4$) and one free surface ((CMO)$_4$(STO)$_4$(CMO)$_4$-${surface}$). (*c*) $\epsilon_{2}$ in the in-plane lattice direction incorporating epitaxial strain by introducing LAO into the supercell both with no free surface ((CMO)$_4$(LAO)$_4$(CMO)$_4$) and one free surface ((CMO)$_4$(LAO)$_4$(CMO)$_4$-${surface}$).[]{data-label="fig:FIG_4"}](FIG_4.png) Acknowledgments --------------- The authors acknowledge the Air Force Office of Scientific Research under FA9550-13-1-0124. J.E.S. also acknowledges the U.S. Army Research Office for support of A.R.A. under W911NF-14-1-0500. J.E.S. thanks C. L. Schauer for access to the spectroscopic ellipsometer. [39]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}

--- abstract: 'Ubiquitous sensing devices frequently disseminate data among them. The use of a distributed event-based system that decouples publishers from subscribers arises as an ideal candidate to implement the dissemination process. In this paper, we present a network architecture that merges the network and overlay layers of typical structured event-based systems. Directional random walks are used for the construction of this merged layer. Our strategy avoids using a specific network protocol that provides point-to-point communication. This implies that the topology of the network is not maintained, so that nodes not involved in the system are able to save energy and computing resources. We evaluate the performance of the overlay layer using directional random walks and pure random walks for its construction. Our results show that directional random walks are more efficient because: (1) they use less nodes of the network for the establishment of the active path of the overlay layer and (2) they have a more reliable performance. Furthermore, as the number of nodes in the network increases, so do the number of nodes in the active path of the overlay layer for the same number of publishers and subscribers. Finally, we discard any correlation between the number of nodes that form the overlay layer and the maximum Euclidean distance traversed by the walkers.' author: - bibliography: - 'ref.bib' title: | Design of a Novel Network Architecture for Distributed Event-Based Systems\ Using Directional Random Walks in an Ubiquitous Sensing Scenario --- Distributed Event-Based Systems; Overlay Layer; Directional Random Walks; Pure Random Walks; Wireless Sensor Networks. Introduction {#sec:introduction} ============ Ubiquitous or pervasive computing [@FUTURECOMPUTING14DRW][@Cook:2012:RPC:2109687.2109848] uses many sources and destinations to gather and process data related to physical processes with the aim of making possible human-computer interaction. In the process of dissemination, some devices generate the data, while others are waiting for the sensing data. In this context, the use of a distributed event-based system [@Muhl:2006:DES:1162246] arises as an ideal candidate to implement the model of communication on the reception or transmission of events. The main characteristic of an event-based system is that publishers and subscribers are decoupled. This means that they do not have any information about each other. The element in charge of matching notifications with subscriptions is called the event notification service. In distributed networks, the event notification service is implemented using a network of brokers nodes (see Figure \[PubSub\]). It is considered that a broker is any node in the network that has information about any single or set of subscriptions. The complexity of designing this type of systems usually lies on the way to elect the nodes that act as brokers because of the decentralized nature of a distributed network. {width="2.75in"} \[PubSub\] In our research [@FUTURECOMPUTING14DRW], we assume that a node can be a publisher, a subscriber, a broker or a combination of these three possibilities. We also assume that all the nodes in the network are able to participate in it without the requirement to adopt the specific role of publisher or subscriber. Nodes that are actively participating in the network but do not take any specific role will be considered as part of the overlay layer. Those nodes of the overlay layer that are able to redirect messages will be considered as brokers. Event-based systems are classified as topic-based or content-based [@Muhl:2006:DES:1162246]. Topic-based systems take into account the subject of messages in order to match publications with subscriptions. Content-based systems use filters to specify the value of subscriptions attributes to redirect notifications. A filter is a boolean function that depends on the set of subscriptions. In our proposal, we plan to deal with a content-based system that uses Bloom filters at broker nodes in order to save memory resources and speed up routing decisions. Sensor networks frequently use tiny devices with limited battery capabilities that make unsuitable the use of a Global Positioning System (GPS) to disseminate information according to the coordinates of nodes. In addition to this, the use of virtual coordinates to substitute real coordinates requires the use of sinks or landmarks to structure the network. For these reasons, the use of coordinates in an unstructured sensing scenario is not recommended. We assume that we work in an unstructured scenario in which no routing protocol provides communication between the nodes of the network. The constraints of the network infrastructure lead us to the design of a network architecture for distributed event-based systems that must use as less resources as possible (i.e., battery, memory, etc.). In this paper, we present a solution that avoids implying all the nodes of the network in the dissemination process by using a distributed notification service defined by Directional Random Walks (DRWs). The rest of this paper is organized as follows: Section \[sec:state\] analyzes the state of the art. Section \[sec:methodology\] points out the approach to solve the problem specified in this section. Section \[sec:research\] presents the research efforts already done for the approach specified in Section \[sec:methodology\]. Section \[sec:design\] details the process of construction of the proposed architecture. Section \[sec:evaluation\] evaluates the performance of our solution using DRWs, comparing it with the use of Pure Random Walks (PRWs). Finally, Section \[sec:conclusion\] summarizes our proposal. State of the art {#sec:state} ================ Distributed and Structured Event-based Systems ---------------------------------------------- Distributed and structured event-based systems use three layers on the top of a bottom layer (see Figure \[3layersdistributedusualsystems\]), which provides data link functionalities, to facilitate topology control: 1. The network layer is in charge of providing data forwarding between the different nodes involved in the network. A network protocol, such as the Multicast Ad-hoc On-demand Distance Vector (MAODV) [@Roy05securingmaodv:] is needed to provide point-to-point communication. 2. The medium layer is called the overlay layer. It is a virtual layer that builds the event notification service by providing a network of brokers that redirect notifications to the corresponding subscribers. 3. Finally, on the top layer the event-based protocol is implemented. {width="2.45in"} \[3layersdistributedusualsystems\] One strategy to construct the overlay layer is to use a tree. In TinyMQ [@shi2011tinymq], which is designed specifically for wireless sensor networks, a multi-tree overlay layer is maintained. Another strategy is to clusterize the network and use cluster heads to manage messages as in Mires [@souto2006mires], which is a middleware for sensor networks. The Gradient Landmark-Based Distributed Routing (GLIDER) [@Fang05glider:gradient] organizes the network using some defined landmarks to compute the Delaunay graph for network partition. Then, the Landmark-Based Information Brokerage scheme (LBIB) [@Fang06landmark-basedinformation] uses an overlay layer based in GLIDER to match publishers with subscribers. A typical solution is to build the overlay layer using Distributed Hash Tables (DHTs). In these systems, a key is mapped to a particular node with storage location properties. In some DHT architectures, rendez-vous nodes depend on the node ID as in Pastry [@Pastry]. In others, as the Content Addressable Network (CAN) [@CAN], a region of the space is used to map a key. Some efforts have been made to apply this solution to sensor networks [@Fersi:2013:DHT:2429525.2429572]. When coordinates are available, sensor networks use Geographic Hash Tables (GHT) instead of a typical DHT. Currently, technology companies as Ericsson Research, are making an effort to develop applications that use GHTs in wireless sensor networks [@SENSORNETS14M3]. Distributed and Unstructured Event-based Systems ------------------------------------------------ The main characteristics of distributed and unstructured event-based systems is that they do not maintain an overlay layer. This fact makes easier to deal with network changes. The distributed notification service may be built using flooding, gossiping or random walks. Most of the algorithms proposed deal with the unstructured nature of wireless communications using flooding to build a tree. A typical solution is to use the On-Demand Multicast Routing Protocol (ODMRP) [@Lee:2002], which is based on the forwarding group concept. Groups are constructed and maintained periodically when a multicast source has data to send. This task is done by broadcasting the entire network with membership information. An extension for ODMRP has been proposed [@Yoneki:2004] to adapt a content-based system by adding subscriptions to Bloom filters. Trees also may be configured to self-repair themselves in base to brokers dynamicity [@Mottola:2008]. These solutions are reliable but increase the traffic of the network because they use flooding at some point. Flooding may also be used to continuously exchange subscription information clusterizing the network [@Voulgaris06]. Then, notifications are sent to the appropriate cluster, improving the efficiency of the network. Other mechanisms can be used as the combination of a DHT and random walks [@Tian:2005]. Cluster heads manage the DHTs while random walks help to connect the different cluster heads of the network. The cluster concept in the network of brokers can be improved in a dynamic scenario by enriching the topology management with predictions based on location [@Abdennadher:2013:APM:2508222.2508234]. ### Probabilistic approaches Probabilistic approaches are suitable to deal with dynamicity but they do not offer reliability. Some solutions propose that all the nodes in the network implement a broker that forwards messages to neighbors depending on the estimation of potential subscribers [@Haillot:2008]. Other solutions [@1437119], propose to flood subscriptions in a small area and then use random walks to reach these areas. In Quasar [@Wong:2008], subscriptions of a certain area are able to attract or reject notifications, that are propagated with a random walk, using an attenuated Bloom filter [@5751342]. A probabilistic solution that uses a random walk specifically designed to go deep into the network is CoQUOS (Continous Queries on Unstructured OverlayS) [@Ramaswamy:2011]. Continuous queries are launched to the network using random walks. Peers compute the overlap between their neighbor lists and use this information to forward the random walk to avoid remaining in a cluster. Then, some peers register the query with a probability that depends on the number of hops. Network architecture {#sec:methodology} ==================== Due to the unstructured nature of our network, we propose the development of a dissemination algorithm that merges the network and the overlay layers of a typical distributed and structured event-based system (see Figure \[3layersdistributedusualsystems\]). This means that no other network protocol is needed. The main advantage of not using another network protocol is that there is no necessity of maintaining a network topology. This implies that most nodes of the network, which do not actively participate in the process of dissemination, do not have to keep any information about topology. The main consequence is that nodes not involved in the system are able to save energy and computing resources. Our design (see Figure \[3layers\]) uses two layers on the top of a bottom layer that provides data link functionalities: 1. The overlay layer is in charge of providing the distributed network of brokers and, at the same time, provides point-to-point communication between publishers and subscribers. The main objective of this strategy is to avoid the use of global information of the network, which is costly to get and maintain. 2. As in Figure \[3layersdistributedusualsystems\], the event-based protocol is implemented at the top layer. {width="2.5in"} \[3layers\] As Section \[sec:introduction\] mentions, we assume that a node can be a publisher, a subscriber, a broker or a combination of these three possibilities. Moreover, our design takes advantage of some nodes in the network that want to collaborate. Nodes that participate in the system are considered as part of the overlay layer (blue path of Figure \[3layers\]). The overlay layer is formed by the intersection of different publishers and subscribers (blue nodes). Publishers and subscribers implement a DRW until intersecting other DRW. Broker nodes (yellow nodes) are the meeting point between two DRWs. A DRW is a probabilistic technique able to go forward into the network following a loop-free path. The principle assumed in this strategy is that two lines in a plane cross (see Figure \[sim\]). It is unclear how to construct a straight path of relaying nodes in ubiquitous unstructured networks without requiring global information and without making use of geo-coordinates. In this research, two different methods have been proposed in order to build DRWs [@ASCOMS13DRW][@SENSORNETS14DRW]. The strategy followed by the DRW is based on a tabu search [@gendreau2014tabu]. The tabu search is a technique used when difficult optimization problems arise. Unfortunately, the theoretical aspects related to a tabu search are so complicated that there is no formal proof of the convergence of the algorithm. By definition, a tabu search is an iterative procedure in which the next solution is defined by the current solution and a tabu list. A tabu list is a memory that keeps information about the previous iterations of the algorithm. It is used to select the optimal solution for the next iteration. In neighborhood search methods, the tabu list is referred to the set of neighbors of the actual solution. The design of a DRW uses a technique, which is similar to a tabu list based on a neighborhood search method. A DRW marks the closest nodes of nodes that are already part of the DRW. Afterwards, this information is used to go forward when adding more nodes to the DRW. The general algorithm for a tabu search based on a neighborhood method is the following: [p[0.43]{}p[0.0]{}]{} - Set initial solution $S_t$ where $t = 0$. Add $S_t$ to the tabu list. - Update the current number of iteration $t = t+1$. - Create the solution neighborhood $N(S_{t})$ discarding nodes that are part of the tabu list. - If $N(S_{t})=\emptyset$ then consider $S_{t-1}$ and return to Step 3. - If $N(S_{t})\neq \emptyset$ then evaluate the cost function for all $N(S_{t})$. - Select the best solution $S_{t+1}$ basing the choice on the minimum cost. Add $S_{t+1}$ to the tabu list. - Stop the algorithm if the stopping criterion is satisfied. Otherwise, return to Step 2. A tabu list needs a stopping condition. In our design, the condition is referred to an intersection with a node that is already part of another DRW in the network. {width="2.5in"} \[sim\] The matching of publishers and subscribers will be done using a special architecture of Bloom filters [@5751342] implemented at broker nodes. It is remarkable to mention that in our event-based system no advertisement table is required because filters only manage information about subscriptions. Bloom filters are probabilistic data structures that efficiently manage membership of a certain number of elements. The content related to membership is hashed using different hashing algorithms. Then, the positions of the Bloom structure corresponding to the hashes are set to one. The maximum number of elements to be inserted to the filter is fixed in order to maintain a certain probability of false positives. When searching for elements of a certain membership, the corresponding positions of the data structure are checked. The main advantage of Bloom filters is that they do not require much memory space and processing resources; so its use is very convenient in a sensing scenario in which devices have limited capabilities. In this research, we concentrate on the study of the properties of the overlay layer proposed. It is out of the scope of this work to study a more efficient architecture of Bloom filters at broker nodes for matching publications with subscriptions. Background {#sec:research} ========== In this section, we present the efforts already made in order to build DRWs. Based on [@ASCOMS13DRW], a first method to build DRWs is proposed. It is based on the addition of different nodes to the DRW by pre-computing different weights at each node that take into account the two hops path. A weight is defined as follows: $$\label{DRW Pierre} n_{xz}^y = \vert \: N(x)\cap N(z) \: \vert$$ where $y$ is the last node added to the DRW; $x$ is the penultimate node added to the DRW; $z$ can be any node of the set $N(y)$ and $N(a)$ is the set of neighbor nodes of node $a$. Furthermore, in this method, a penalty is added to the weight when a node is added to the DRW. Some properties about our heuristics were found using extensive simulations. The first property claims that DRWs decrease the time to intersection compared to pure random walks. The second property states that cooperation also decreases the time to intersection. Cooperation refers to synchronicity between publishers and subscribers. Finally, it is shown that DRWs are good at balancing the load of the network. Based on [@SENSORNETS14DRW], a second method to build DRWs is proposed. The main difference with the first design presented for DRWs is that nodes of the first and second neighborhoods of nodes added to the DRW are marked. In addition to this, the cost is not pre-computed, but it is computed when selecting a node as follows: $$\label{DRW Cristina} c(v) = \alpha \vert N(v) \cap N(DRW) \vert + \beta \vert N(v) \cap N^2(DRW) \vert$$ where $\alpha$ and $\beta$ are parameters used as weights; $v$ can be any node of the set related to the neighborhood of the last node added to the DRW; $DRW$ is the set of nodes that are part of the DRW; $N(a)$ is the set of neighbor nodes of node $a$ and $N^2(DRW)$ is the set of neighbor nodes of $N(DRW)$. In the first part of this research, the properties associated with a DRW were assessed. Implementations of DRWs of one or two branches were studied. The main results show that the use of one branch is as efficient as the use of two branches. Moreover, it is shown that the use of second neighborhoods to forward the DRW does not improve the Euclidean distance traversed in the network. It is also shown that shorter paths are obtained when using higher densities of nodes in the network. In the second part of this research, an information brokerage system was evaluated using a double ruling method. As in the first paper, it is shown that the algorithm is efficient at balancing the load using a few nodes of the network. In fact, we can state that the method proposed is as good as a traditional Rumor Routing algorithm [@Braginsky:2002:RRA:570738.570742] with an infinite memory. Design of the overlay layer {#sec:design} =========================== Network Model ------------- A DRW is defined in a graph $G=(V, E)$, where $V$ is the set of vertices and $E$ is the set of edges. $u, v \in V$ are connected $u \sim v$ if $(u, v) \in E$. The size of $G$ is denoted by $\mid V \mid = n$ and the number of edges is denoted by $\mid E \mid = m$. We denote the neighborhood of $v$ as $N(v)=\{ u \in V \hspace{0.1cm} | \hspace{0.1cm} u \sim v\}$. Implementation of the overlay layer ----------------------------------- In order to assess the architecture proposed, we have used a variation of the algorithm presented at [@SENSORNETS14DRW]. The set of edges and vertices associated to a DRW of ID $x$ are denoted by $E'_x$ and $V'_x$. Our technique consists of selecting the set of vertices $V'_x$ that are part of the DRW. Each vertex of $V'_x$ is denoted by $v'_{x, i}$. The current number of nodes in the active path of the DRW is denoted by $i$. Vertices are chosen consecutively until two DRWs intersect. A vertex $v$ is selected to be part of the DRW as $v'_{x,i}$ if it has the minimum cost at iteration $i$ between $N(v'_{x, i-1})$. The cost function used is a varitation of the cost function used in (\[DRW Cristina\]): $$\label{Cost} c(v) = \vert N(v) \cap N(DRW_{x, i}) \vert$$ where $N(DRW_{x, i})$ denotes the set of neighbors of $V'_x$ at iteration $i$. Formally, it is defined as: $$\label{DRW first neighborhood} N(DRW_{x, i}) = \left [\bigcup_{j=0}^{i} N(v'_{x, j}) \right ]$$ The use of $N(DRW_{x, i})$ is of particular interest to our research because it allows us to exploit the broadcast advantage of the wireless medium. This process can be seen as a repulsion mechanism to force a branch to keep moving forward. The result of this mechanism is that neighbors that are not part of $N(DRW_{x, i})$ have higher possibilities to be added to the DRW. Figure \[ConstructionofaDRW\] illustrates the selection of a node $z$ when nodes $x$ and $y$ have already been added to a DRW. Before adding $z$ to the DRW (see Figure \[ConstructionofaDRW\].a), neighbors of the penultimate node added to the DRW are marked as part of the neighborhood of the walker. At this stage, node $y$ has to select the next node to be added to the DRW between its neighbors $a$, $b$, $c$, $d$ and $z$ (see Figure \[ConstructionofaDRW\].b). In order to avoid remaining in the same zone, we are interested in selecting a candidate that helps to push the DRW to other zones of the network. Candidate $a$ has a cost of 3 because it has three neighbors marked as part of the neighborhood of the walker (see Figure \[ConstructionofaDRW\].b). Candidates $b$, $c$, $d$, and $z$; have a cost of 2, 3, 2 and 1, respectively. Candidate $z$ has the minimum cost, so that it is added to the DRW and neighbors of node $y$ are marked as part of the neighborhood of the walker. {width="\linewidth"}\ a) Two nodes added to the directional random walk\ {width="\linewidth"}\ b) Three nodes added to the directional random walk \[ConstructionofaDRW\] Publishers and subscribers are considered as initiators of a DRW. Algorithm \[alg:overlay\] shows how DRWs are intersected for a certain number of publishers and subscribers in the network. Algorithm \[alg:drw\] shows the pseudocode used for the construction of a DRW. The selection of a node is based on the computation of a cost (line 26). A candidate node is added to the DRW if it has the minimum cost between all the candidate nodes (line 30). A node is considered as candidate, if it is part of the neighborhood of the last node added to the branch (line 21). It is assumed that there are no candidate nodes when the neighborhood of the last node added is empty or all of them are already part of the DRW (line 18). In order to assure intersections our algorithm goes back in the DRW to search for the nearest non traversed neighbor. $Number\hspace{0.1cm} of\hspace{0.1cm} total\hspace{0.1cm} initiators: I$ Define Thread $drw0$ as a new <span style="font-variant:small-caps;">DRW($v'_{0, 1}$)</span> Define Thread $drw1$ as a new <span style="font-variant:small-caps;">DRW($v'_{1, 1}$)</span> Start Thread $drw0$ Start Thread $drw1$ **while**{(Intersection value of $drw0$ is 0) $\&\&$ (Intersection value of $drw1$ is 0)} **end while** Define Thread $drwi$ as a new <span style="font-variant:small-caps;">DRW($v'_{i, 1}$)</span> Start Thread $drwi$ **while**{(Intersection value of $drwi$ is 0)} **end while** $Initiator: v'_{x, 1} \in V$ $Intersection=0$ $v'_{x, 1}$ to $V'_x$ $Intersection=1$ $v'_{x, 2} \gets v$ $Intersection=1$ select $v'_{x, 2} \in N(v'_{x, 1})$ randomly $i=2;$ $v$ to $N(v'_{x, i})$ $i++;$ $i--;$ $v'_{x, i} \gets v$ $Intersection=1$ compute $c(v)$ as defined by (\[Cost\]) to the DRW $v$ $\hspace{0.1cm} | \hspace{0.1cm} v, min\{c(v)\}\in N(v'_{x, i})$ $Intersection=1$ Evaluation of the novel architecture {#sec:evaluation} ==================================== To assess the performance of the overlay layer we have implemented a Java simulator. The networks used for the numerical evaluation have been obtained by placing nodes randomly and uniformly in a squared area of side size $1\times1$ with a range of communication of $r = 0.05$. The communication model is defined by the range of communication. Two nodes that are closer than the range of communication can communicate. The graph we obtain in this way is often referred by Unit Disc Graph (UDG). Under these conditions, it is hard to obtain connected networks with less than 1.000 nodes, so we have conducted numerical validation for more dense networks assuring that they are completely connected. The total number of nodes considered has been 1.000, 2.000 and 3.000. As previously mentioned, for the implementation of the overlay layer we use different DRWs that intersect. Figure \[overlaylayer\] shows a simulation of the overlay layer in which yellow squares represent the distributed network of brokers while publishers and subscribers are represented using green circles. {width="2.5in"} \[overlaylayer\] The performance metric used is the $depth$ (see Figure \[depth\] and (\[eq:depth\])). The $depth$ compares the maximum Euclidean distance reached by all the nodes that are part of the list of relaying nodes of any DRW, with the maximum Euclidean distance that can be reached in the network. It is defined as: $$depth(Overlay\hspace{0.1cm}Layer) = \frac{max\{\{d(v'_i, v'_j) \hspace{0.1cm} | \hspace{0.1cm} v'_i, v'_j \in \bigcup_{x}V'_x\}}{max\{d(v_i, v_j) \hspace{0.1cm} | \hspace{0.1cm} v_i, v_j \in V\}} \label{eq:depth}$$ where $d$ denotes the Euclidean distance. {width="2.5in"} \[depth\] The evaluation of the design of the overlay layer is based on: - The number of nodes in the active path. - The $depth$. For this purpose, we assess the performance of the architecture proposed for different number of publishers and subscribers in the network. Our results, have been obtained by using extensive simulations for each network considered. Box and whisker plots are used to visualize data. As previously mentioned, we considered networks of 1.000, 2.000 and 3.000 nodes. The total number of networks simulated totals 126.000. The total number of simulations when evaluating networks of 1.000 nodes has been 20.000. We have evaluated the performance of the overlay layer for the following total number of publishers and subscribers: 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 75, 100, 250, 500, 625, 750 and 875. For each total number of initiators, we have simulated 100 different networks. Similarly, for networks of 2.000 nodes we have obtained 21.000 simulations using a total number of initiators of: 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 75, 100, 250, 500, 1.000, 1.250, 1.500 and 1.750. Finally, for networks of 3.000 nodes 22.000 simulations have been conducted for a total number of initiators of: 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 75, 100, 250, 500, 1.000, 1.500, 1.875, 2.250 and 2.625. Moreover, we have used PRWs for comparison with our method, which select the next node of the walker randomly. So that overlay layers formed by PRW have also been evaluated for 1.000, 2.000 and 3.000 nodes. Impact on the number of nodes in the active path ------------------------------------------------ Figure \[fig:activepath\] compares the results obtained by using DRWs and PRWs for networks of 1.000, 2.000 and 3.000 nodes. Interestingly, Figures \[fig:activepath\].a, \[fig:activepath\].c and \[fig:activepath\].e, that show the performance of the overlay layer when using DRWs are proportionally, almost identical for different densities of nodes in the network following the same logarithmic behavior. This behavior suggests that the base of the logarithm decreases as the density of the network is increased. Nevertheless, Figures \[fig:activepath\].b, \[fig:activepath\].d and \[fig:activepath\].f, that show the performance of the overlay layer when using PRWs, present a different logarithmic behavior. We observe that proportionally, less dense networks grow faster in terms of nodes in the active path; but still, the more nodes we have in the network, the more nodes we have in the active path. This means that as in the previous case, the base of the logarithm decreases as the density of the network is increased; but the change on the base is more dramatic for DRWs. This effect is produced because DRWs reduce the random component attached to the experiment, allowing to have a more predictable performance, that is traduced in more directionality or similarity to a straight line of the walker. Besides this, overlay layers constructed using PRWs present more outliers that overlay layers that use DRWs. The result is that DRWs present a more reliable performance in the construction of the active path of our architecture. In addition to this, it is obvious, that overlay layers that use PRWs present larger active paths. As previously mentioned, in all cases the more nodes we have in the network, the more nodes we have in the active path of the overlay layer for the same number of publishers and subscribers in the network. This consequence is reasonable, because the less density of nodes we have in the network, the less candidate nodes we have to construct the walker. The result of this performance is that less dense networks are saturated before. Impact on the depth ------------------- Figure \[fig:depth\] shows the resulting $depth$ for the different densities of networks considered using DRWs and PRWs. The main conclusion extracted is that $depths$ are very similar for all cases and that the maximum Euclidean distance that is going to be traversed in the network is reached very early. Figure \[fig:depth\_50\] shows in detail the performance when having a few number of publishers and subscribers in the network. In all cases, the depth is importantly increased when having three publishers and subscribers in the network. Furthermore, we observe that the directionality of DRWs leads to increase the $depth$ when having two publishers or subscribers compared to the $depth$ reached by PRWs. Finally, we can state that there exists no correlation between the number of nodes in the active path and the $depth$. So that other factors, as the density of the network, have more impact in the number of nodes of the active path. Conclusion {#sec:conclusion} ========== In this paper, a novel network architecture for distributed event-based systems that use sensing devices has been proposed. We present a network architecture that merges the network and overlay layers of typical structured event-based systems. Our results, validated through extensive simulations, show that DRWs are suitable for the construction of an overlay layer that provides point-to-point communication and a distributed notification service. Our strategy avoids using other network protocol to provide point-to-point communication. This implies that most nodes of the network, which do not actively participate in the process of dissemination, do not have to maintain any information about topology. The main consequence is that nodes not involved in the system are able to save energy and computing resources. We evaluate the performance of the overlay layer using DRWs and PRWs for its construction. Our results show that for our purpose DRWs are more efficient than PRWs. This is due, mainly to the good properties of DRWs, which use less nodes of the network for the establishment of the active path of the overlay layer. Moreover, we can state that overlay layers that use DRWs have a more reliable performance that overlay layers that use PRWs. Furthermore, it is remarkable to mention that, in all cases, the more nodes we have in the network, the more nodes we have in the active path of the overlay layer for the same number of publishers and subscribers in the network. Finally, it is interesting to discard any correlation between the number of nodes that form the overlay layer and the maximum Euclidean distance that is traversed by the walkers; mainly because the maximum $depth$ is quickly reached. Acknowledgment {#acknowledgment .unnumbered} ============== This work has been developed as part of the POPWiN project that is financially supported by the Swiss Hasler Foundation in its “SmartWorld - Information and Communication Technology for a Better World 2020” program.

--- abstract: 'Let $S_k$ be the set of separable states on $\B(\C^m \otimes \C^n)$ admitting a representation as a convex combination of $k$ pure product states, or fewer. If $m>1, n> 1$, and $k \le \max{(m,n)}$, we show that $S_k$ admits a subset $V_k$ such that $V_k$ is dense and open in $S_k$, and such that each state in $V_k$ has a unique decomposition as a convex combination of pure product states, and we describe all possible convex decompositions for a set of separable states that properly contains $V_k$. In both cases we describe the associated faces of the space of separable states, which in the first case are simplexes, and in the second case are direct convex sums of faces that are isomorphic to state spaces of full matrix algebras. As an application of these results, we characterize all affine automorphisms of the convex set of separable states, and all automorphisms of the state space of $\B(\C^m \otimes \C^n)$ that preserve entanglement and separability.' address: - 'Mathematics Department, University of Oslo, Blindern 1053, Oslo, Norway' - 'Mathematics Department, Wellesley College, Wellesley, Massachusetts 02481, USA' author: - Erik Alfsen - Fred Shultz date: 'September 17, 2009' title: 'Unique decompositions, faces, and automorphisms of separable states' --- Introduction ============ A state on the algebra $\B(\C^m \otimes \C^n)$ of linear operators is separable if it is a convex combination of product states. States that are not separable are said to be entangled, and are of substantial interest in quantum information theory. Easily applied conditions for separability are known only for special cases, e.g., if $m = n = 2$, then a state is separable iff its associated density matrix has positive partial transpose, cf. [@Peres; @Horodeckis]. Other necessary and sufficient conditions are known, e.g. [@Horodeckis], but are not easily applied in practice. An open question of great interest is to find a simple necessary and sufficient condition for a state to be separable. A product state $\omega \otimes \tau$ is a pure state iff $\omega$ and $\tau$ are pure states. Thus a separable state is precisely one that admits a representation as a convex combination of pure product states. It is natural to ask the extent to which this decomposition is unique. That is the main topic of this article. For the full state space $K$ of $\B(\C^m \otimes \C^n)$ each non-extreme point can be decomposed into extreme points in many different ways. But for the space $S$ of separable states the situation is totally different. While non-extreme points with many different decompositions exist (and are easy to find) in $S$ as well as in $K$, there are in $S$ also plenty of points for which the decomposition is unique. DiVincenzo, Terhal, and Thapliyal [@DiV] defined the *optimal ensemble cardinality* of a separable state $\rho$ to be $k$ if $k$ is the minimal number of pure product states required for a convex decomposition of $\rho$. Lockhart [@Lockhart] used the term “optimal ensemble length" for the same notion. For brevity, we will simply call this number the *length* of $\rho$, and we denote the set of separable states of length at most $k$ by $S_k$. We show in Theorem \[cor4\] that for $m>1, n> 1$ and $k \le \max(m,n)$, the set $S_k$ has a subset $V_k$ which is dense and open in $S_k$, with each $\sigma \in V_k$ admitting a unique decomposition into pure product states. Actually, we exhibit such a set $V_k$ consisting of states with the property that each generates a face of $S$ which is a simplex, from which the uniqueness follows. We remark that the sets $V_k$ are open and dense in the relative topology on $S_k$, but are not open or dense in $S$ or $K$ if $mn > 1$. (See the remarks after Theorem \[cor4\]). Indeed it would be surprising if a subset of low rank separable states were open and dense in the set of all states of that rank, since low rank states are almost surely entangled [@RuskaiWerner; @WalgateScott], and in general $S$ has measure which is a decreasingly small fraction of the measure of $K$ as $m, n$ increase, cf. [@AubrunSzarek; @Szarek]. While dimensions are too high to be able to accurately visualize the above results, the reader may be curious about the relationship to the well known tetrahedron/octahedron picture for $m = n = 2$, cf. [@HorodeckiTetra]. In that picture, there is a subset $\mathcal{T}$ of states which is a tetrahedron, and which has the property that for every state $\rho$ which restricts to the normalized trace on $\B(\C^2) \otimes I$ and on $I \otimes B(\C^2)$, there are unitaries $U$ and $V$ such that $(U\otimes V)^*\rho(U\otimes V) \in \mathcal{T}$. The midpoints of the six edges of this tetrahedron are the vertices of an octahedron that consists of the separable states in $\mathcal{T}$. Each vertex of the octahedron is a convex combination of two distinct pure product states (which of course are not in $\mathcal{T}$), cf. [@Ovrum eqn. (63)]. In fact, the vertices are the only states in the octahedron of length $\le \max(m,n)= 2$. It can be checked (e.g., by applying our Corollary \[cor3.9\]) that the decomposition of each of these vertices into pure product states is unique. Each state in the interior of this octahedron has rank $4 = mn$, so is an interior point of the full state space $K$, hence has a non-unique convex decomposition into pure product states (see the remarks after Theorem \[cor4\].) The tetrahedron also arises as a parameterization for a set of unital completely positive trace preserving maps from $M_2(\C)$ to $M_2(\C)$, with the octahedron consisting of the entanglement breaking maps in this set, cf. [@RuskaiKing Appendix B], [@Ruskai Thm. 4], and [@RuskaiWerner Fig. 2]. We also define a broader class of states that we show have a unique decomposition as a convex combination of product states $\rho_i\otimes\sigma_i$ that are not necessarily pure, but with the property that each of them generates a face of $S$ which is also a face of $K$ and is affinely isomorphic to the state space of $\B(\C^{p_i})$ for a suitable $p_i$. From this it follows that the ambiguity in decompositions for a given state in this class is restricted to the ambiguity in decompositions for points in the state space of the matrix algebras $\B(\C^{p_i})$. For a complete description of the possible decompositions of a state on $\B(\C^{p})$, see [@Kirkpatrick; @Schrodinger; @Wootters]. We use our results on the facial structure of $S$ to show that every affine automorphism of the space $S$ of separable states on $\B(\C^m \otimes \C^n)$ is given by a composition of the duals of the maps that are (i) conjugation by local unitaries (i.e., unitaries of the form $U_1 \otimes U_2$) (ii) the two partial transpose maps, or (iii) the swap automorphism that takes $A\otimes B$ to $B \otimes A$ (if $m = n$). A consequence is a description of the affine automorphisms $\Phi$ of the state space such that $\Phi$ preserves entanglement and separability. There is related work of Hulpke et al [@Hulpke]. They say a linear map $L$ on $\C^m \otimes \C^n$ preserves *qualitative entanglement* if $L$ sends separable (i.e., product) vectors to product vectors, and entangled vectors to entangled vectors. They show that a linear map $L$ preserves qualitative entanglement of vectors on $\C^m \otimes \C^n$ iff $L$ is a local operator (i.e. one of the form $L_1 \otimes L_2$), or if $L$ is a local operator composed with the swap map that takes $x\otimes y$ to $y \otimes x$. They then show that if $L$ preserves a certain *quantitative* measure of entanglement, then $L$ must be a local unitary. We thank Mary Beth Ruskai for helpful comments and references. Background: states on $\B(\C^n)$ ================================ We review basic facts about states on $\B(\C^n)$, and develop some facts about the relationship of independence of vectors $x$ in $\C^n$ and of the corresponding vector states $\omega_x$. In the following sections we will specialize to the case of interest: separable states. If $x$ is a vector in any vector space, $[x]$ denotes the subspace generated by $x$. $\C^n$ denotes the set of $n$-tuples of complex numbers viewed as an inner product space with the usual inner product (linear in the first factor). $\B(\C^n)$ denotes the linear transformations from $ \C^n$ into itself. For each unit vector $x \in \C^n $, we denote the associated vector state by $\omega_x $, so that $\omega_x(A) = (Ax, x)$. The convex set of states on $\B(\C^n)$ will be denoted by $K_n$. We recall that faces of the state space $K_n$ of $\B(\C^n)$ are in 1-1 correspondence with the projections in $\B(\C^n)$, and thus with the subspaces of $\C^n$ that are the ranges of these projections. If $Q$ is a projection in $\B(\C^n)$, then the associated face $F_Q$ of $K_n$ consists of all states taking the value $1$ on $Q$. The restriction map is an affine isomorphism from $F_Q$ onto the state space of $Q\B(\C^n )Q \cong \B(Q(\C^n))$. Thus $F_Q$ is affinely isomorphic to the state space of $\B(L)$, where $L = Q(\C^n)$. The set of extreme points of $K_n$ are the vector states, and it follows that the extreme points of $F_Q$ are the vector states $\omega _x$ with $x$ in the range of $Q$, and $F_Q$ is the convex hull of these vector states. For background, see [@Alfsen-Shultz Chapter 4] Recall that a convex set $C$ is said to be the *direct convex sum* of a collection of convex subsets $C_1, \ldots, C_p$ if each point $\omega \in C$ can be uniquely expressed as a convex combination [unique]{}= \_[i I]{} \_i \_i where $I \subset \{1, \ldots, p\}$, $\lambda_i > 0$ for all $i \in I$, $\omega_i \in C_i$ for all $i \in I$, and $\sum_{i \in I} \lambda_i = 1$. If $C$ is a convex subset of a real linear space and is located on an affine hyperplane which does not contain the origin (as is the case for our state spaces), then it is easily seen that $C$ is the direct convex sum of convex subsets $C_1, \ldots, C_p$ iff the span of $C$ is the direct sum of the real subspaces spanned by $C_1, \ldots, C_p$. A finite dimensional convex set is a *simplex* if it is the direct convex sum of a finite set of points. If the affine span of the points does not contain the origin, then their convex hull is a simplex iff the points are linearly independent (over $\R$). [directsumlemma]{}[Let $L$ be a subspace of $\C^n$ and suppose that $L$ is the direct sum of subspaces $L_1, \dots, L_p$. Let $F_1, \ldots, F_p$ be the corresponding faces of the state space of $\B(\C^n)$. Then the convex hull of $F_1, \ldots, F_p$ is the direct convex sum of those faces. In particular, if $x_1, \ldots, x_p$ are linearly independent unit vectors, then the corresponding vector states are linearly independent and the convex hull of the corresponding vector states is a simplex.]{} Let $I \subset \{0, \ldots, p\}$, and suppose $\{\omega_i \mid i \in I\}$ are nonzero functionals on $\B(\C^n)$ with $\omega_i \in \operatorname{span}_{\R} F_i$ for each $i$. To prove independence of $\{\omega_i \mid i \in I\}$, suppose that for scalars $\{\gamma_i\}_{i \in I}$ we have [dependence]{}\_[i I]{} \_i \_i = 0. Let $L_0$ be the orthogonal complement of $L$. Then $\C^n$ as a linear space is the direct sum of $L_0, L_1, \ldots, L_p$. For each $i \in I$, let $P_i$ be the projection associated with $F_i$. Then we can find $A_i \in P_i\B(\C^n)P_i$ such that $\omega_i(A_i) \not= 0$. Let $B_i \in \B(\C^n)$ be an operator such that $B_i$ is zero on $\sum_{j \not= i} L_j$, and such that $\omega_i(B_i) \not= 0$ (e.g., set $B_i= A_i$ on $L_i$). If $x \in L_j$ and $j \not= i$, then $\omega_x(B_i) = (B_ix,x) = 0$. Since every state in $F_j$ is a convex combination of vector states $\omega_x$ with $x \in L_j$, then $\omega_j(B_i) = 0$ if $j \not= i$. Now apply both sides of to $B_k$ to conclude that $\gamma_k\omega_k(B_k) = 0$ for all $k \in I$, so $\gamma_k = 0$ for all $k \in I$. Thus the set of vectors $\omega_1, \ldots, \omega_p$ is independent. We conclude that $\operatorname{co}(F_1, \ldots, F_p)$ is the direct convex sum of $F_1, \ldots, F_p$. If $x_1, \ldots, x_p$ are linearly independent unit vectors, applying the result above with $F_i = \{\omega_{x_i}\}$ shows that the convex hull of the vector states $\omega_{x_i}$ is a simplex. Hence the set $\{\omega_{x_1}, \ldots, \omega_{x_p}\}$ is linearly independent. Note that the converses of the statements above are not true. For example, while no set of more than two vectors in $\C^2$ is independent, it is easy to find a set of three linearly independent vector states on $\B(\C^2)$. Uniqueness of decompositions of separable states ================================================ We now turn to faces of the set of separable states on $\B(\C^m\otimes\C^n)$, and to the question of uniqueness of convex decompositions of such states. We identify $\B(\C^m\otimes\C^n)$ with $\B(\C^m) \otimes \B(\C^n)$ by $(A\otimes B)(x\otimes y) = Ax \otimes By$. We denote the convex set of all states on $\B(\C^m\otimes\C^n)$ by $K$, and the convex set of all separable states by $S$. [productvectorlemma]{} [ Let $e_1, e_2, \ldots, e_p$ and $f_1, f_2, \ldots, f_p$ be unit vectors in $\C^m$ and $\C^n$ respectively. We assume that $f_1, f_2, \ldots, f_p$ are linearly independent. If $e \in \C^m$ and $f\in \C^n$ are unit vectors such that $e\otimes f$ is in the linear span of $\{ e_i \otimes f_i \mid 1\le i \le p \}$, then there is an index $j$ such that $[e] = [e_j]$ and such that $f$ is in the span of those $f_i$ such that $[e_i] = [e_j]$. In the special case where $[e_1], \ldots, [e_p]$ are distinct, then $[e] = [e_j]$ and $[f] = [f_j]$ for some index $j$, and $\{ e_i \otimes f_i \mid 1\le i \le p \}$ is independent.]{} Extend $f_1 ,\ldots, f_p$ to a basis $f_1, \ldots, f_n$ of $\C^n$, and let $\widehat f_1, \ldots, \widehat f_n$ be the dual basis. For $1 \le k \le n$, let $T_k : \C^m \otimes \C^n \to \C^m$ be the linear map such that $T_k(x\otimes y) = \widehat f_k(y) x$ for $x \in \C^m$, $y \in \C^n$. Suppose that the product vector $e \otimes f$ is a linear combination $$e\otimes f = \sum_{i=1}^p \alpha_i e_i \otimes f_i.\label{comb}$$ For $j > p$, applying $T_j$ to both sides of gives $\widehat f_j(f)e = 0$, so $\widehat f_j(f) = 0$ for all such $j$. Now if $1 \le j \le p$, applying $T_j$ to both sides of gives $$\widehat f_j(f) e = \alpha_j e_j. \label{tj}$$ Since $\widehat f_j(f) $ can’t be zero for all $j$, then $e$ is a multiple of some $e_j$. Fix such an index $j$. If $1 \le i \le p$ and $[e_i] \not= [e_j]$, then $e_i$ can’t be a multiple of $e$, so $\widehat f_i(f) e = \alpha_i e_i$ implies $\alpha_i = 0$, and then also $\widehat f_i(f) = 0$. We have shown that $\widehat f_i(f) = 0$ if $i > p$, or if $i \le p$ and $[e_i] \not= [e_j]$. It follows that $f$ is in the linear span of those $f_i$ such that $[e_i] = [e_j]$. If it also happens that $[e_1], \ldots, [e_p]$ are distinct, and $[e] = [e_j]$, then $[f] = [f_j]$. Suppose now that $\sum_i \alpha_i e_i \otimes f_i = 0$. If $\alpha_k \not= 0$, then $e_k \otimes f_k$ is a linear combination of $\{e_i \otimes f_i \mid i \not= k\}$. Thus by the conclusion just reached, we must have $[e_k] =[e_i]$ for some $i \not = k$, contrary to the hypothesis that $[e_1], \ldots, [e_p]$ are distinct. We conclude that $\alpha_k = 0$ for all $k$, and we have shown that $\{ e_i \otimes f_i \mid 1\le i \le p \}$ is independent. [faceofKcase]{}[Let $e_1, \ldots, e_p \in \C^m$ and $f_1, \ldots, f_p \in \C^n$ be unit vectors. If $[e_1] = [e_2] = \ldots = [e_p]$, then the face $F$ of $S$ generated by the states $\{\omega_{e_i\otimes f_i} \mid 1 \le i \le p\}$ is also a face of $K$, and this face of $K$ is associated with the subspace $L = e_1 \otimes \operatorname{span}\{f_1, \ldots, f_p\}$ of $\C^m \otimes \C^n$, and $F$ is affinely isomorphic to the state space of $\B(L)$.]{} Let $G$ be the face of $K$ which is associated with the subspace $L$ of $\C^m \otimes C^n$. By assumption each $e_i$ is a multiple of $e_1$, so that $$L = \operatorname{span}\{e_1\otimes f_i \mid 1 \le i \le p\} = \operatorname{span}\{e_i\otimes f_i \mid 1 \le i \le p\}.$$ Hence $G$ is the face of $K$ generated by $\{\omega_{e_i \otimes f_i} \mid 1 \le i \le p\}$. We would like to show $G = F$. For brevity we denote the convex hull of the set $\{\omega_{e_i \otimes f_i} \mid 1 \le i \le p\}$ by $C$, and observe that $G$ and $F$ are the faces of $K$ and $S$ respectively generated by $C$. It follows easily from the definition of a face that the face generated by the convex set $C$ in either one of the two convex sets $S$ or $K$ consists of all points $\rho$ in $S$ or $K$ respectively which satisfy an equation $$\label{rho} \omega = \lambda \rho + (1-\lambda)\sigma$$ where $0 < \lambda < 1$, $\omega \in C$, and where $\sigma$ is in $S$ or $K$ respectively. It follows that $F = \operatorname{face}_S(C) \subset \operatorname{face}_K(C) = G$. Since each vector in $L$ is a product vector, the extreme points of $G$ are pure product states, so $G \subset S$. If $\rho$ is in the face $G$ of $K$ generated by $C$, then we can find $\sigma \in K$ and $\omega \in C$ such that holds. Then $\sigma$ is also in $G \subset S$, so both $\rho$ and $\sigma $ are in $S$. Hence $\rho$ is in the face $F$ of $S$ generated by $C$. Thus $G \subset F$, and so $F= G$ follows. So far we have considered collections of product vectors $\{e_i \otimes f_i\}$ with $\{f_1, \ldots, f_p\}$ linearly independent. In Lemma \[faceofKcase\] we have described the face $F$ of $S$ generated these states in the special case where all of the $e_i$ are multiples of each other. In this case $F$ is also a face of $K$. We now remove the restriction that all of the one dimensional subspaces $[e_i]$ coincide. We are going to partition the set of vectors $e_i \otimes f_i$ into subsets for which these subspaces coincide, and apply Lemma \[faceofKcase\] to each such subset. For simplicity of notation, we renumber the vectors in the fashion we now describe. [thm3]{}[ Let $e_1, e_2, \ldots, e_p$ and $f_1, f_2, \ldots, f_p$ be unit vectors in $\C^m$ and $\C^n$ respectively, and with $f_1, \ldots, f_p$ linearly independent. We assume that the vectors are ordered so that $[e_1] , \ldots, [e_q]$ are distinct, and so that for $i > q$ each $[e_i]$ equals one of $[e_1], \ldots, [e_q]$. For $1 \le i \le q$, let $F_i$ be the face of $S$ generated by the states $\{\omega_{e_j \otimes f_j} \mid [e_j] = [e_i]\} \text{ and $1 \le j \le p$}\}$. Then each $F_i$ is also a face of $K$, and the face $F$ of $S$ generated by $\{\omega_{e_i \otimes f_i} \mid 1\le i \le p \}$ is the direct convex sum of $F_1, \ldots, F_q$. Moreover, each $F_i$ is affinely isomorphic to the state space of $\B(L_i)$, where $L_i = e_i\otimes \operatorname{span}\{f_j \mid [e_i] = [e_j]\}$. In the special case when $[e_1], \ldots, [e_p]$ are distinct, then $F$ is the convex hull of $\{\omega_{e_i \otimes f_i}\mid 1 \le i \le p\}$, and $F$ is a simplex. ]{} By Lemma \[faceofKcase\], the face $F_i$ of $S$ is equal to the face of $K$ generated by $\{\omega_{e_j \otimes f_j}\mid [e_j] = [e_i]\}$, and is affinely isomorphic to the state space of $\B(L_i)$. We will show $L_1, \ldots, L_q$ are independent (i.e., that $L_1 + L_2 + \cdots L_q$ is a vector space direct sum). For $1\le i \le q$ let $e_i \otimes g_i$ be a nonzero vector in $L_i$. For $i \not= j$, $g_i$ and $g_j$ are linear combinations of disjoint subsets of $f_1, f_2, \ldots, f_p$, so by independence of $f_1, f_2, \ldots, f_p$, the subset $\{g_1, \ldots, g_q\}$ is independent. Thus by Lemma \[productvectorlemma\], $\{e_1\otimes g_1, \ldots, e_p \otimes g_p\}$ is independent, and hence the subspaces $L_1, \ldots, L_q$ are independent. Hence by Lemma \[directsumlemma\], the convex hull of the faces $F_i$ is a direct convex sum of those faces. Finally, we need to show that this convex hull coincides with the face $F$ of $S$. Extreme points of $F$ are extreme points of $S$, so are pure product states. Suppose that $\omega_{x\otimes y}$ is a pure product state in $F$. Then $\omega_{x\otimes y}$ is in the face of $K$ generated by $\{\omega_{e_i \otimes f_i}\mid 1 \le i \le p\}$, so $x\otimes y$ is in $\operatorname{span}\{e_i \otimes f_i \mid 1 \le i \le p\}$. By Lemma \[productvectorlemma\], $[x] = [e_j]$ for some $j$, and $y \in \operatorname{span}\{y_i \mid [e_i] = [e_j]\}$. Hence $\omega_{x\otimes y} \in F_j$. Thus each extreme point of $F$ is in some $F_j$, so $F$ is contained in the convex hull of $\{F_i \mid 1 \le i \le q\}$. Evidently $F$ contains every $F_j$, so this convex hull equals $F$. In Theorem \[thm3\] we showed that the face $F$ is the direct convex sum of faces that are affinely isomorphic to state spaces of full matrix algebras. Convex sets of this type were studied by Vershik (in both finite and infinite dimensions), who called them *block simplexes* [@Vershik]. Other examples are provided by state spaces of any finite dimensional C\*-algebra. Our Theorem \[thm3\] provides new examples of such block simplexes. [cor3.9]{} [Let $e_1, e_2, \ldots, e_p$ and $f_1, f_2, \ldots, f_p$ be unit vectors in $\C^m$ and $\C^n$ respectively. We assume that $[e_i] \not= [e_j]$ for $ i \not= j$, and that $f_1, f_2, \ldots, f_p$ are linearly independent. If $\lambda_1, \ldots, \lambda_k$ are nonnegative numbers with sum 1, then the separable state $\omega = \sum_i \lambda_i \omega_{e_i\otimes f_i}$ has a unique representation as a convex combination of pure product states. ]{} Suppose $\omega$ equals the convex combination $\sum_j \gamma_j \tau_j$ where each $\tau_j$ is a pure product state. Then each $\tau_j$ is in the face $F$ of $S$ generated by $\omega$. By Theorem \[thm3\], $F$ is a simplex, and the extreme points of $F$ are all of the form $\omega_{e_i \otimes f_i}$. Since each $\tau_j$ is a vector state, it is a pure state as well, so each state $\tau_j$ must be an extreme point of $F$, and thus must equal some $\omega_{e_i\otimes f_i}$. Uniqueness of the representation of $\omega$ follows from the uniqueness of convex decompositions into extreme points of a (finite dimensional) simplex. A separable state $\omega$ has *length* $k$ if $\omega$ can be expressed as a convex combination of $k$ pure product states and admits no decomposition into fewer than $k$ pure product states. We denote by $S_k$ the set of separable states of length at most $k$. A separable state $\omega$ has a *unique decomposition* if it can be written as a convex combination of pure product states in just one way By the above result, roughly speaking decompositions of separable states on $\B(\C^m \otimes \C^n)$ of length $\le \max(m,n)$ generically are unique. Here’s a more precise statement. Let $k \le \max(m,n)$, and let $V_k$ be the set of states $\omega$ admitting a convex decomposition $\omega = \sum_{i=1}^k \lambda_i \omega_{e_i\otimes f_i}$, where $e_1, \ldots, e_k$ and $f_1, \ldots, f_k$ are unit vectors in $\C^m$ and $\C^n$ respectively, $0 < \lambda_i$ for $1\le i \le k$, $[e_1], \ldots, [e_k]$ are distinct, and $\{f_1, \ldots, f_k\}$ is linearly independent. [cor4]{} [Let $m,n > 1$. For a given $k \le \max(m,n)$, the states in $V_k$ have length $k$, and have unique decompositions. The set $V_k$ is open and dense in the set $S_k$ of separable states of length at most $k$. ]{} Without loss of generality, we may assume $m\le n$. By Corollary \[cor3.9\], each $\omega \in V_k$ admits a unique representation as a convex combination of pure product states, and each state in $V_k$ has length $k$. We will show that $V_k$ is open and dense in $S_k$. To prove density, let $\omega\in S_k$ have a convex decomposition $\omega = \sum_{i=1}^k \lambda_i\omega_{x_i \otimes y_i}$. By slightly perturbing the coefficients $\lambda_i$ if necessary, we may assume that $\lambda_i > 0$ for all $i$. Given $\epsilon > 0$, by perturbing each $x_i$ and $y_i$ if necessary, we can find a second convex combination of pure product states $\omega' = \sum_{i=1}^k \lambda_i\omega_{e_i \otimes f_i}$ with $\|\omega - \omega'\|< \epsilon$, with $[e_1], \ldots, [e_k]$ distinct, and with $\{f_1, \ldots, f_k\}$ independent. (Indeed, to achieve independence we may append unit vectors $y_{k+1},\ldots, y_n$ to the vectors $y_1, \ldots, y_k$ to give the subset $\{y_1, y_2, \ldots, y_n\}$ of $\C^n$, and by small perturbations arrange that the determinant of the matrix with columns $y_1, \ldots, y_n$ is nonzero.) Thus $V_k$ is dense in $S_k$. Let $I_0 = \{(\lambda_1, \lambda_2, \ldots, \lambda_k) \in [0,1]^k\mid \sum_i \lambda_i = 1\}$. Let $U_m$ be the unit sphere of $\C^m$ and $U_n$ the unit sphere of $\C^n$. Let $X = I_0 \times (U_m)^k \times (U_n)^k$. Define $\psi:X \to S$ by $$\psi((\lambda_1, \ldots, \lambda_k), (x_1, \ldots, x_k), (y_1, \ldots, y_k)) = \sum_i \lambda_i \omega_{x_i \otimes y_i}.$$ Note that $\psi$ is continuous, that $X$ is compact with respect to the product topology, and that $\psi(X) = S_k$. Now let $X_0$ be the set $\{((\lambda_1, \ldots, \lambda_k), (x_1, \ldots, x_k), (y_1, \ldots, y_k))\}$ of members of $X$ of such that $[x_1], \ldots, [x_k]$ are distinct, such that $\{y_1, \ldots, y_k\}$ is linearly independent, and such that $\lambda_i>0$ for $1 \le i \le k$. By lower semicontinuity of the rank of a matrix whose columns are $y_1, \ldots, y_k$, the set of elements $((\lambda_1, \ldots, \lambda_k), (x_1, \ldots, x_m), (y_1, \ldots, y_k))$ of $X$ with $\{y_1, \ldots, y_k\}$ linearly independent is open in $X$, so it is clear that $X_0$ is an open subset of $X$. By construction, $\psi(X_0 )= V_k$. Since $X_0$ is open in $X$, then $X\setminus X_0$ is closed and hence compact. Since $\psi$ maps $X \setminus X_0$ onto $S_k \setminus \psi(X_0)$, then the latter is closed, so $V_k = \psi(X_0)$ is open in $S_k$. As remarked in the introduction, the sets $V_k$ are open and dense in the relative topology on $S_k$, but are not open or dense in $S$ or $K$ if $mn > 1$. To see this recall that a point $\sigma$ in a convex set $C$ is an algebraic interior point if for every point $\rho$ in $C$ there is a point $\tau$ in $C$ such that $\sigma$ lies on the open line segment from $\rho$ to $\tau$. Clearly for every algebraic interior point $\sigma$ of $S$ and every pure product state $\rho$, there is a convex decomposition of $\sigma$ that includes $\rho$ with positive weight. Since there are infinitely many pure product states, there are infinitely many convex decompositions for every algebraic interior point of $S$. Every nonempty subset which is open in $S$ contains an algebraic interior point of $S$ ([@SW pp. 88-91]), so contains points with nonunique decompositions. Thus $V_k$ is not open in $S$ or $K$. It is not dense in $S$ or $K$, since for any $m, n$ there exists $r > 0$ such that all states $\sigma$ within a distance $r$ from the normalized tracial state are separable, cf. [@Z Thm. 1]. Every such state $\sigma$ is an algebraic interior point of $S$, and so fails to have a unique decomposition. Observe that Theorem \[cor4\] implies that $V_k$ is also open and dense in the set of separable states of length equal to $k$. Description of convex decompositions ==================================== Let $e_1, e_2, \ldots, e_p$ and $f_1, f_2, \ldots, f_p$ be unit vectors in $\C^m$ and $\C^n$ respectively, with $f_1, \ldots, f_p$ linearly independent. Suppose $\omega$ is a convex combination of $\{\omega_{e_i \otimes f_i} \mid 1 \le i \le p\}$. In this section, we will describe all convex decompositions of $\omega$ into pure product states. Let $\omega = \sum_i \lambda_i \omega_i$ be any convex decomposition of $\omega$ into pure product states. Then following the notation of Theorem \[thm3\], each $\omega_i$ is in $\operatorname{face}_S(\omega) \subset F$. Since each $\omega_i$ is an extreme point of $S$, and $F$ is the direct convex sum of the faces $F_i$, then each $\omega_i$ must be in some $F_k$. If we define $\gamma_k = \sum_{\{i \mid \omega_i \in F_k\}}\lambda_i$ and $\sigma_k = \gamma_k^{-1}\sum_{\{i \mid \omega_i \in F_k\}}\lambda_i\omega_i$, then $\omega$ has the convex decomposition $$\label{decomp} \omega = \sum_k \gamma_k \sigma_k \text{ with $\sigma_k \in F_k$ for each $k$}.$$ Since $F$ is the direct convex sum of the $F_k$, the decomposition of $\omega$ in is unique. All possible convex decompositions of $\omega$ into pure product states can be found by starting with the unique decomposition $\omega = \sum_k \gamma_k \sigma_k$ with $\sigma_k \in F_k$, and then decomposing each $\sigma_k$ into pure states. (Every state in $F_k$ is separable, so pure states are pure product states). Since $F_k$ is affinely isomorphic to the state space of $\B(L_k)$, unless each $\sigma_k$ is itself a pure state, this can be done in many ways, as we discussed in the introduction. The possibilities have been described in [@Wootters; @Schrodinger; @Kirkpatrick]. A decomposition of a separable state $\omega$ as a convex combination of pure product states can be interpreted as a representation of $\omega$ as the barycenter of a probability measure on the extreme points of $S$. With this interpretation the statement above can be rephrased in terms of the concept of dilation of measures (as defined e.g. in [@Alfsen p. 25]). If $\omega$ is given as above, then the probability measures on pure product states that represent $\omega$ are precisely those which are dilations of the uniquely determined probability measure $\mu = \sum_k \gamma_k\mu_k$ obtained from (6) with $\mu_k=\delta_{\sigma_k}$. Affine automorphisms of the space $S$ of separable states ========================================================= Fix $m, n$. We denote the state space of $\B(\C^m)$ by $K_m$, the state space of $\B(\C^n)$ by $K_n$, and the state space of $\B(\C^m \otimes \C^n)$ by $K$ or $K_{m,n}$. The convex set of separable states in $K$ is denoted by $S$ or $S_{m,n}$. We will sometimes deal with a second algebra $\B(\C^{m'} \otimes \C^{n'})$, whose state space and separable state spaces we will denote by $K'$ or $S'$ respectively. From Theorem \[thm3\], the face of $S$ generated by two distinct pure product states $\omega_1 \otimes \sigma_1$ and $\omega_2 \otimes \sigma_2$ is a line segment (if $\omega_1 \not= \omega_2$ and $\sigma_1 \not= \sigma_2)$ or is isomorphic to the state space of $\B(\C^2)$ and hence is a 3-ball (when $\omega_1 = \omega_2$ but $\sigma_1 \not= \sigma_2$, or when $\sigma_1 = \sigma_2$ but $\omega_1 \not= \omega_2$). (By a 3-ball we mean a convex set affinely isomorphic to the closed unit ball of $\R^3$. The fact that the state space of $\B(\C^2)$ is a 3-ball can be found in many places, e.g., [@Alfsen-Shultz Thm. 4.4].) We define a relation $R$ on the pure product states of $K$ by $\rho \operatorname{\,R\,}\tau$ if $\operatorname{face}_S(\rho, \tau)$ is a 3-ball. By the remarks above, $(\omega_1 \otimes \sigma_1) \operatorname{\,R\,}(\omega_2 \otimes \sigma_2)$ iff ($\omega_1 = \omega_2$ but $\sigma_1 \not= \sigma_2$) or ($\sigma_1 = \sigma_2$ but $\omega_1 \not= \omega_2$). Note that an affine isomorphism $\Phi:S\to S'$ will take faces of $S$ to faces of $S'$, and will take 3-balls to 3-balls, so for pure product states $\rho, \tau$ we have $\rho \operatorname{\,R\,}\tau$ iff $\Phi(\rho) \operatorname{\,R\,}\Phi(\tau)$. The idea of the following lemmas is to show that if $\Phi(\omega\otimes \sigma) = \phi(\omega, \sigma) \otimes \psi(\omega, \sigma)$, then $\phi$ depends only on the first argument and $\psi$ depends only on the second argument, or possibly vice versa. Although we are interested in affine automorphisms of a single space of separable states, it will be easier to establish the needed lemmas in the context of affine isomorphisms from $S$ to $S'$. We use the notation $\partial_eC$ for the set of extreme points of a convex set $C$. For example, $\partial_eK$ is the set of pure states on $\B(\C^m \otimes \C^n)$. \[foureqns\] Let $\Phi:S_{m,n}\to S_{m', n'}$ be an affine isomorphism. Let $\omega_1$, $\omega_2 $ be distinct pure states in $K_m$ and $\sigma_1$, $\sigma_2$ distinct pure states in $K_n$. Then the following four equations cannot hold simultaneously. $$\begin{aligned} \label{four} \Phi(\omega_1 \otimes \sigma_1) &= \rho_1 \otimes \tau_1\cr \Phi(\omega_1 \otimes \sigma_2) &= \rho_1 \otimes \tau_2\cr \Phi(\omega_2 \otimes \sigma_1) &= \rho_2 \otimes \tau_3\cr \Phi(\omega_2 \otimes \sigma_2) &= \rho_3\otimes \tau_3\cr\end{aligned}$$ for $\rho_1, \rho_2, \rho_3 \in \partial_e K_{m'}$ and $\tau_1, \tau_2, \tau_3 \in \partial_e K_{n'}$. We assume for contradiction that all four equations hold. Since $(\omega_1 \otimes \sigma_1) \operatorname{\,R\,}(\omega_2 \otimes \sigma_1)$, then $(\rho_1 \otimes \tau_1) \operatorname{\,R\,}(\rho_2 \otimes \tau_3)$. Hence $$\label{e1} \rho_1 = \rho_2 \text{ or } \tau_1 = \tau_3.$$ Similarly $(\omega_1 \otimes \sigma_2) \operatorname{\,R\,}(\omega_2 \otimes \sigma_2)$, so $(\rho_1 \otimes \tau_2) \operatorname{\,R\,}(\rho_3 \otimes \tau_3)$. Hence $$\label{e2} \rho_1 = \rho_3 \text{ or } \tau_2 = \tau_3.$$ Since we are assuming that $\omega_1 \not= \omega_2$ and $\sigma_1 \not= \sigma_2$, the four states $\{\omega_i \otimes \sigma_j \mid 1\le i, j \le 2\}$ are distinct, so the four states on the right side of must be distinct. Combining and gives four possibilities, each contradicting the fact that the states on the right side of are distinct. Indeed: $$\begin{aligned} (\rho_1 = \rho_2 \text{ and } \rho_1 = \rho_3) \implies & \rho_2 \otimes \tau_3 = \rho_3 \otimes \tau_3\cr (\rho_1 = \rho_2 \text{ and } \tau_2 = \tau_3) \implies & \rho_1 \otimes \tau_2 = \rho_2 \otimes \tau_3\cr (\tau_1 = \tau_3 \text{ and } \rho_1 = \rho_3) \implies & \rho_1 \otimes \tau_1 = \rho_3 \otimes \tau_3\cr (\tau_1 = \tau_3 \text{ and } \tau_2 = \tau_3) \implies & \rho_1 \otimes \tau_1 = \rho_1 \otimes \tau_2.\cr\end{aligned}$$ We conclude that the four equations in cannot hold simultaneously. Recall that we identify $\B(\C^m \otimes \C^n)$ with $\B(\C^m) \otimes \B(\C^n)$. The *swap isomorphism* $(\alpha_{m,n})_*: \B(\C^n \otimes \C^m)\to \B(\C^m \otimes \C^n)$ is the \*-isomorphism that satisfies $(\alpha_{m,n})_*(A \otimes B) = B \otimes A$. If operators in $\B(\C^m \otimes \C^n)$ are identified with matrices, the swap isomorphism is the same as the “canonical shuffle" discussed in [@Paulsen Chapter 8]. The dual map $\alpha_{m,n}$ is an affine isomorphism from the state space of $\B(\C^m \otimes \C^n)$ to the state space of $ \B(\C^n \otimes \C^m)$, with $\alpha_{m,n}(\omega \otimes \sigma) = \sigma \otimes \omega$. This restricts to an affine isomorphism from $S_{m,n}$ to $S_{n,m}$, which we also refer to as the swap isomorphism. If $m = n$, then $(\alpha_{m,m})_*$ is a \*-automorphism of $ \B(\C^m \otimes \C^m)$, $\alpha_{m,m}$ is an affine automorphism of the state space $K$, and restricts to an affine automorphism of the space $S$ of separable states. \[factor\] Let $\Phi:S_{m,n}\to S_{m', n'}$ be an affine isomorphism. At least one of the following two possibilities occurs: 1. For every $\omega \in \partial_eK_m$ there exists $\rho \in \partial_e K_{m'}$ such that $\Phi(\omega \otimes K_n) = \rho \otimes K_{n'}$, and for every $\sigma \in \partial_e K_n$ there exists $\tau \in \partial_e K_{n'}$ such that $\Phi(K_m\otimes \sigma) = K_{m'} \otimes \tau$. 2. For each $\omega \in \partial_eK_m$ there exists $\tau \in \partial_e K_{n'}$ such that $\Phi(\omega \otimes K_n) = K_{m'} \otimes \tau$, and for every $\sigma \in \partial_e K_n$ there exists $\rho \in \partial_e K_{m'}$ such that $\Phi(K_m\otimes \sigma) = \rho \otimes K_{n'}$. If (i) occurs, then $m = m'$ and $n = n'$. If (ii) occurs, then $m = n'$ and $n = m'$. For fixed $\omega\in \partial_eK_m$ and distinct $\sigma_1, \sigma_2 \in \partial_e K_n$ we have $(\omega\otimes \sigma_1) \operatorname{\,R\,}(\omega \otimes \sigma_2)$, so $\Phi(\omega\otimes \sigma_1) \operatorname{\,R\,}\Phi(\omega \otimes \sigma_2)$. Thus either there exist $\rho_1 \in \partial_eK_{m'}$ and distinct $\tau_1, \tau_2 \in \partial_e K_{n'}$ such that $$\label{eq3} \Phi(\omega\otimes \sigma_i) = \rho_1 \otimes \tau_i \text{ for $i = 1, 2$},$$ or there exist distinct $\rho_1, \rho_2 \in \partial_e K_{m'}$ and $\tau_3 \in \partial K_{n'}$ such that $$\label{eq4} \Phi(\omega\otimes \sigma_i) = \rho_i \otimes \tau_3 \text{ for $i = 1, 2$}.$$ We will show that implies (i), and implies (ii). Suppose that holds. Let $\sigma \in \partial_e K_n$ with $\sigma \not= \sigma_1$ and $\sigma\not=\sigma_2$, and let $\Phi(\omega\otimes \sigma) = \rho \otimes \tau$. Since $(\omega\otimes \sigma) \operatorname{\,R\,}(\omega\otimes \sigma_i)$ for $i = 1, 2$, then $(\rho\otimes \tau) \operatorname{\,R\,}(\rho_1 \otimes \tau_i)$ for $i = 1, 2$. Hence ($\rho = \rho_1$ or $\tau = \tau_1$) and ($\rho = \rho_1$ or $\tau = \tau_2$). Since $\tau_1 \not= \tau_2$, then $\rho = \rho_1$. It follows that $\Phi(\omega \otimes K_n) \subset \rho_1 \otimes K_{n'}$. Thus $$\Phi(\omega\otimes \sigma_i) = \rho_1 \otimes \tau_i \text{ for $i = 1, 2$}\implies \Phi(\omega \otimes K_n)\subset \rho_1\otimes K_{n'}.\label{imp}$$ Now also implies $$\label{eq5} \Phi^{-1}(\rho_1 \otimes \tau_i) = \omega\otimes \sigma_i \text{ for $i = 1, 2$}.$$ If holds (and hence also , then applying the implication to with $\Phi^{-1}$ in place of $\Phi$ shows $ \Phi^{-1}(\rho_1\otimes K_{n'}) \subset \omega \otimes K_n$, so by equality holds. Hence we have shown $$\Phi(\omega\otimes \sigma_i) = \rho_1 \otimes \tau_i \text{ for $i = 1, 2$}\implies \Phi(\omega \otimes K_n)= \rho_1\otimes K_{n'}.\label{imp3}$$ Now suppose instead that holds. Let $\alpha_{m', n'}$ be the swap affine isomorphism defined above, so that $\alpha_{m', n'}: S_{m', n'} \to S_{n', m'}$. Then $$(\alpha_{m', n'}\circ\Phi)(\omega\otimes \sigma_i) = \alpha_{m', n'}(\rho_i \otimes \tau_3) = \tau_3 \otimes \rho_i \text{ for $i = 1, 2$}.$$ By the implication applied to $\alpha_{m', n'} \circ \Phi$ we conclude that $$(\alpha_{m', n'} \circ \Phi)(\omega \otimes K_n)= \tau_3\otimes K_{m'},$$ so $$\Phi(\omega \otimes K_n)= \alpha_{m', n'}^{-1}(\tau_3\otimes K_{m'}) = K_{m'} \otimes \tau_3.$$ Thus we have proven the implication $$\Phi(\omega\otimes \sigma_i) = \rho_i \otimes \tau_3 \text{ for $i = 1, 2$} \implies \Phi(\omega \otimes K_n)= K_{m'} \otimes \tau_3.\label{eq16}$$ By Lemma \[foureqns\] and the implications and , either must hold for all $\omega \in \partial_eK_m$ or must hold for all $\omega \in \partial_eK_m$. We conclude that either $$\label{choice1} \forall \omega \in \partial_eK_m \quad\exists \rho \in \partial_e K_{m'} \text{ such that } \Phi(\omega \otimes K_n) = \rho \otimes K_{n'}$$ or $$\label{choice2} \forall \omega \in \partial_eK_m \quad\exists \tau \in \partial_e K_{n'} \text{ such that } \Phi(\omega \otimes K_n) = K_{m'} \otimes \tau.$$ Similarly, either $$\label{choice3} \forall \sigma \in \partial_eK_n \quad\exists \tau' \in \partial_e K_{n'} \text{ such that } \Phi(K_m\otimes \sigma) = K_{m'}\otimes \tau'$$ or $$\label{choice4} \forall \sigma \in \partial_eK_n \quad\exists \rho' \in \partial_e K_{m'} \text{ such that } \Phi( K_m\otimes \sigma) = \rho'\otimes K_{n'}.$$ Suppose that and both held. For $\omega \in K_m$ and $\sigma \in K_n$ note that $\omega \otimes \sigma$ is in both $\omega \otimes K_n$ and $K_m \otimes \sigma$, so $\rho \otimes K_{n'}$ and $\rho' \otimes K_{n'}$ are not disjoint. This implies $\rho = \rho'$, so $\Phi(\omega\otimes K_n) = \Phi(K_m \otimes \sigma)$. Since $\Phi$ is bijective, $\omega \otimes K_n = K_m \otimes \sigma$ follows. This is possible only if $m = n = 1$. If $m = n = 1$, then all of , , , hold. Similarly if and both held then $m = n = 1$ is again forced. Thus the possibilities are that and both hold (which is the same as statement (i) of the lemma), or that and hold (equivalent to (ii)), or that $m = n = 1$, in which case both (i) and (ii) hold. Finally, since the affine dimensions of $K_p$ and $K_q$ are different when $p \not= q$, the statement in the last sentence of the lemma follows. If $\psi_1:K_m\to K_m$ and $\psi_2:K_n \to K_n$ are affine automorphisms, then we can extend each to linear maps on the linear span, and form the tensor product $\psi_1 \otimes \psi_2$. This will be bijective, but not necessarily positive. (A well known example of this phenomenon occurs when $\psi_1$ is the identity map and $\psi_2$ is the transpose map.) However, $\psi_1$ and $\psi_2$ will map pure states to pure states, and hence $\psi_1 \otimes \psi_2$ will map pure product states to pure product states. Thus $\psi_1 \otimes \psi_2$ will map $S$ onto $S$, and hence will be an affine automorphism of $S$. We will now see that all affine automorphisms of $S$ are either such a tensor product of automorphisms or such a tensor product composed with the swap automorphism. \[factoring\] If $m \not= n$, and $\Phi:S\to S$ is an affine automorphism, then there exist unique affine automorphisms $\psi_1:K_m\to K_m$ and $\psi_2: K_n \to K_n$ such that $\Phi = \psi_1 \otimes \psi_2$. If $m =n$ then either we can write $\Phi = (\psi_1 \otimes \psi_2)$ or $\Phi = \alpha_{m,m} \circ (\psi_1 \otimes \psi_2)$, where $\psi_1, \psi_2$ are again unique affine automorphisms of $K_m$ and $K_n$ respectively, and $\alpha_{m,m}:S \to S$ is the swap automorphism. We apply Lemma \[factor\]. For each $\omega \in \partial_eK_m$ and $\sigma \in \partial_eK_n$, define $\phi_\sigma:K_m \to K_m$ and $\psi_\omega: K_n \to K_n$ by $$\Phi(\omega \otimes \sigma) = \phi_\sigma(\omega) \otimes \psi_\omega(\sigma).$$ Suppose first that case (i) of Lemma \[factor\] occurs. Then $\psi_\sigma(\omega)$ is independent of $\sigma$ and $\psi_\omega(\sigma) $ is independent of $\omega$. Therefore there are functions $\psi_1:K_m \to K_m$ and $\psi_2: K_n \to K_n$ such that $$\Phi(\omega \otimes \sigma) = \psi_1(\omega) \otimes \psi_2(\sigma).$$ Since $\Phi$ is bijective and affine, so are $\psi_1$ and $\psi_2$. Suppose instead that case (ii) of Lemma \[factor\] occurs. Then $m = n$. If we define $\Phi'=\alpha_{m,m}\circ \Phi$, then $\Phi': S\to S$ satisfies case (i) of Lemma \[factor\]. Then from the first paragraph we can choose affine automorphisms $\psi_1:K_m \to K_m$ and $\psi_2: K_n \to K_n$ such that $\Phi' = \psi_1 \otimes \psi_2$. Since $\alpha_{m,m}^2 $ is the identity map, then $\Phi = \alpha_{m,m} \circ (\psi_1 \otimes \psi_2)$. We review some well known facts about affine automorphisms of state spaces and maps on the underlying algebra. Let $\psi$ be an affine automorphism of $K_m$. Then $\psi$ extends uniquely to a linear map on the linear span of $K_m$, which we also denote by $\psi$, and this map is the dual of a unique linear map $\psi_*$ on $\B(\C^m)$. By a result of Kadison [@Kad-Isom] $\psi_*$ will be a \*-isomorphism or a \*-anti-isomorphism. (Since the restriction of an affine automorphism to pure states preserves transition probabilities, this also follows from Wigner’s theorem [@Wigner]). The map $\psi_*$ will be a \*-isomorphism iff $\psi_*$ is completely positive, which is equivalent to $\psi$ being completely positive. If $\psi_*$ is a \*-isomorphism, then $\psi_*$ is implemented by a unitary, i.e., there is a unitary $U\in \B(\C^m)$ such that $\psi_*(A) = UAU^*$. If $\psi_*$ is a \*-anti-isomorphism, then the composition of $\psi_*$ with the transpose map (in either order) gives a \*-isomorphism, and the map $\psi_*$ is completely copositive. It follows that an affine automorphism $\psi$ of $K_m$ is either completely positive or completely copositive, and $\psi $ is completely positive iff $\psi ^{-1}$ is completely positive. If $t$ denotes the transpose map on $\B(\C^m)$ or $\B(\C^n)$, then $t$ is positive but $t \otimes id$ and $id \otimes t$ are not positive on $\B(\C^m) \otimes \B(\C^n)$ if $m, n > 1$. Background can be found in [@Alfsen-Shultz Chapters 4, 5]. Recall that a *local unitary* in $\B(\C^m \otimes \C^n)$ is a tensor product $U_1 \otimes U_2$ of unitaries. Every affine automorphism of the space $S$ of separable states on $\B(\C^m \otimes \C^n)$ is the dual of conjugation by local unitaries, one of the two partial transpose maps, the swap map (if $m = n$), or a composition of these maps. An affine automorphism $\Phi$ of $S$ extends uniquely to an affine automorphism of the full state space $K$ iff it can be expressed as one of the compositions just mentioned with both or neither of the partial transpose maps involved. We note first that if $m = 1$ or $n = 1$, the result is clear, so we assume hereafter that $m \ge 2$ and $n \ge 2$. We next show that if $\psi_1:K_m \to K_m$ and $\psi_2:K_n \to K_n$ are affine automorphisms, then $\Phi = \psi_1 \otimes \psi_2$ is an affine automorphism of $K$ iff $\psi_1$ and $\psi_2$ are both completely positive or both completely copositive. If $\psi_1$ and $\psi_2$ are completely positive, then $\Phi = \psi_1\otimes \psi_2 = (id \otimes \psi_2) \circ (\psi_1 \otimes id)$ is positive; hence $\Phi(K) \subset K$. Furthermore, $\psi_1^{-1}$ and $\psi_2^{-1}$ will be completely positive, so $\Phi^{-1}$ is positive, and hence $\Phi(K) = K$. If $\psi_1$ and $\psi_2$ are completely copositive, then $(t\circ \psi_1) \otimes (t\circ \psi_2)$ is positive. Composing with $t \otimes t$ shows $\psi_1 \otimes \psi_2$ is positive and as above we conclude that $\Phi(K) = K$. On the other hand, if $\psi_1$ is completely positive and $\psi_2$ is completely copositive, then $\psi_1 \otimes (t\circ \psi_2)$ is positive, so $(id \otimes t)\circ (\psi_1 \otimes \psi_2)$ is positive. If $(\psi_1 \otimes \psi_2)(K)=K$, then $id\otimes t$ would be positive, a contradiction since $m, n \ge 2$. Thus in this case $\psi_1 \otimes \psi_2$ is not an affine automorphism of $K$. If $\psi_1$ and $\psi_2$ are completely positive, then they are implemented by unitaries, so $\Phi = \psi_1\otimes \psi_2$ is implemented by a local unitary. If both are completely copositive, then $t\circ \psi_1$ and $t\circ \psi_2$ are implemented by unitaries, so $(t\otimes t) \circ (\psi_1 \otimes \psi_2)$ is implemented by a local unitary. Then $\Phi = (t\otimes t) \circ (t\otimes t) \circ (\psi_1 \otimes \psi_2)$ is the composition of the transpose map on $K$ and conjugation by local unitaries. The first statement of the theorem now follows from Theorem \[factoring\]. Uniqueness follows from the fact that the linear span of $S$ contains $K$. Let $\Phi:K\to K$ be an affine automorphism. We say $\Phi$ *preserves separability* if $\Phi$ takes separable states to separable states, i.e., if $\Phi(S) \subset S$. A state $\omega$ in $K$ is *entangled* if $\omega$ is not separable. $\Phi$ *preserves entanglement* if $\Phi$ takes entangled states to entangled states. Let $\Phi: K_{m,n} \to K_{m,n}$ be an affine automorphism. Then $\Phi$ preserves entanglement and separability iff $\Phi$ is a composition of maps of the types (i) conjugation by local unitaries, (ii) the transpose map, (iii) the swap automorphism (in the case that $m = n$). If $\Phi$ preserves entanglement and separability, then $\Phi$ maps $S$ into $S$ and $K\setminus S$ into $K\setminus S$, which is equivalent to $\Phi(S) = S$. \[one\] If $\Phi_t: S \to S$ is a one-parameter group of affine automorphisms, then there are one-parameter groups of unitaries $U_t$ and $V_t$ such that $\Phi_t(\omega(A)) = \omega((U_t \otimes V_t)A(U_t^* \otimes V_t^*))$. For each $t$, factor $\Phi_t = \phi_t \otimes \psi_t$ or $\Phi_t = \alpha \circ (\phi_t \otimes \psi_t)$, where $\alpha$ is the swap automorphism. In the latter case, $$\Phi_{2t} = \Phi_t \circ \Phi_t = \alpha \circ (\phi_t \otimes \psi_t) \circ \alpha \circ (\phi_t \otimes \psi_t)$$ $$= (\phi_t \otimes \psi_t) \circ (\phi_t \otimes \psi_t) = (\phi_t\circ \phi_t) \otimes (\psi_t \circ \psi_t).$$ It follows that the swap automorphism is not needed for $\Phi_{2t}$, and hence for $\Phi_t$ for any $t$. Uniqueness of the factorization $\Phi_t = \phi_t \otimes \psi_t$ shows that $\phi_t$ and $\psi_t$ are also one parameter groups of affine automorphisms. By a result of Kadison [@Kad], such automorphisms are implemented by one parameter groups of unitaries. If $\Phi_t: K \to K$ is a one-parameter group of entanglement preserving affine automorphisms, then there are one-parameter groups of unitaries $U_t$ and $V_t$ such that $\Phi_t(\omega(A)) = \omega((U_t \otimes V_t)A(U_t^* \otimes V_t^*))$. 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