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arXivTypical structure of sparse exponential random graph models Nicholas A. Cook, Amir Dembo 20220812 We consider general Exponential Random Graph Models (ERGMs) where the sufficient statistics are functions of homomorphism counts for a fixed collection of simple graphs $F_k$. Whereas previous work has shown a degeneracy phenomenon in dense ERGMs, we show this can be cured by raising the sufficient statistics to a fractional power. We rigorously establish the na\"ive meanfield approximation for the partition function of the corresponding Gibbs measures, and in case of ``ferromagnetic'' models with vanishing edge density show that typical samples resemble a typical Erd\H{o}sR\'enyi graph with a planted clique and/or a planted complete bipartite graph of appropriate sizes. We establish such behavior also for the conditional structure of the Erd\H{o}sR\'enyi graph in the large deviations regime for excess $F_k$homomorphism counts. These structural results are obtained by combining quantitative large deviation principles, established in previous works, with a novel stability form of a result of [5] on the asymptotic solution for the associated entropic variational problem. A technical ingredient of independent interest is a stability form of Finner's generalized H\"older inequality.
Subject 수학 Source arXiv URL https://arxiv.org/abs/2208.06397view Article Title Typical structure of sparse exponential random graph modelsAuthors Nicholas A. Cook; Amir DemboAbstract We consider general Exponential Random Graph Models (ERGMs) where the sufficient statistics are functions of homomorphism counts for a fixed collection of simple graphs $F_k$. Whereas previous work has shown a degeneracy phenomenon in dense ERGMs, we show this can be cured by raising the sufficient statistics to a fractional power. We rigorously establish the na\"ive meanfield approximation for the partition function of the corresponding Gibbs measures, and in case of ``ferromagnetic'' models with vanishing edge density show that typical samples resemble a typical Erd\H{o}sR\'enyi graph with a planted clique and/or a planted complete bipartite graph of appropriate sizes. We establish such behavior also for the conditional structure of the Erd\H{o}sR\'enyi graph in the large deviations regime for excess $F_k$homomorphism counts. These structural results are obtained by combining quantitative large deviation principles, established in previous works, with a novel stability form of a result of [5] on the asymptotic solution for the associated entropic variational problem. A technical ingredient of independent interest is a stability form of Finner's generalized H\"older inequality.Is Part Of 20220812 Identifier ISSN: Category math.PR math.COLicense 
arXiv$\mu$Hankel Operators on Compact Abelian Groups A. R. Mirotin 20220812 $(\mu;\nu)$Hankel operators between separable Hilbert spaces were introduced and studied recently (\textit{$\mu$Hankel operators on Hilbert spaces}, Opuscula Math., \textbf{41} (2021), 881899). This paper, is devoted to generalization of $(\mu;\nu)$Hankel operators to the (nonseparable in general) case of Hardy spaces over compact and connected Abelian groups. In this setting bounded $(\mu;\nu)$Hankel operators are fully described under some natural conditions. Examples of integral operators are considered.
Subject 수학 Source arXiv URL https://arxiv.org/abs/2208.06215view Article Title $\mu$Hankel Operators on Compact Abelian GroupsAuthors A. R. MirotinAbstract $(\mu;\nu)$Hankel operators between separable Hilbert spaces were introduced and studied recently (\textit{$\mu$Hankel operators on Hilbert spaces}, Opuscula Math., \textbf{41} (2021), 881899). This paper, is devoted to generalization of $(\mu;\nu)$Hankel operators to the (nonseparable in general) case of Hardy spaces over compact and connected Abelian groups. In this setting bounded $(\mu;\nu)$Hankel operators are fully described under some natural conditions. Examples of integral operators are considered.Is Part Of 20220812 Identifier ISSN: Category math.FALicense 
arXivSymmetries and conservation laws of the onedimensional shallow water magnetohydrodynamics equations in Lagrangian coordinates S. V. Meleshko, V. A. Dorodnitsyn, E. I. Kaptsov... more(3) 20220812 Symmetries of the onedimensional shallow water magnetohydrodynamics equations (SMHD) in Gilman's approximation are studied. The SMHD equations are considered in case of a plane and uneven bottom topography in Lagrangian and Eulerian coordinates. Symmetry classification separates out all bottom topographies which yields substantially different admitted symmetries. The SMHD equations in Lagrangian coordinates were reduced to a single second order PDE. The Lagrangian formalism and Noether's theorem are used to construct conservation laws of the SMHD equations. Some new conservation laws for various bottom topographies are obtained. The results are also represented in Eulerian coordinates. Invariant and partially invariant solutions are constructed.
Subject 수학 Source arXiv URL https://arxiv.org/abs/2208.06219view Article Title Symmetries and conservation laws of the onedimensional shallow water magnetohydrodynamics equations in Lagrangian coordinatesAuthors S. V. Meleshko; V. A. Dorodnitsyn; E. I. KaptsovAbstract Symmetries of the onedimensional shallow water magnetohydrodynamics equations (SMHD) in Gilman's approximation are studied. The SMHD equations are considered in case of a plane and uneven bottom topography in Lagrangian and Eulerian coordinates. Symmetry classification separates out all bottom topographies which yields substantially different admitted symmetries. The SMHD equations in Lagrangian coordinates were reduced to a single second order PDE. The Lagrangian formalism and Noether's theorem are used to construct conservation laws of the SMHD equations. Some new conservation laws for various bottom topographies are obtained. The results are also represented in Eulerian coordinates. Invariant and partially invariant solutions are constructed.Is Part Of 20220812 Identifier ISSN: Category physics.fludyn mathph math.MPLicense 
arXivZilberPink in a product of modular curves assuming multiplicative degeneration Christopher Daw, Martin Orr 20220812 We prove the ZilberPink conjecture for curves in $Y(1)^n$ whose Zariski closure in $(\mathbb{P}^1)^n$ passes through the point $(\infty, \ldots, \infty)$, going beyond the asymmetry condition of Habegger and Pila. Our proof is based on a height bound following Andr\'e's Gfunctions method. The principal novelty is that we exploit relations between evaluations of Gfunctions at unboundedly many nonarchimedean places.
Subject 수학 Source arXiv URL https://arxiv.org/abs/2208.06338view Article Title ZilberPink in a product of modular curves assuming multiplicative degenerationAuthors Christopher Daw; Martin OrrAbstract We prove the ZilberPink conjecture for curves in $Y(1)^n$ whose Zariski closure in $(\mathbb{P}^1)^n$ passes through the point $(\infty, \ldots, \infty)$, going beyond the asymmetry condition of Habegger and Pila. Our proof is based on a height bound following Andr\'e's Gfunctions method. The principal novelty is that we exploit relations between evaluations of Gfunctions at unboundedly many nonarchimedean places.Is Part Of 20220812 Identifier ISSN: Category math.NT math.AGLicense 
arXivThe Shrinking Target Problem for Matrix Transformations of Tori: revisiting the standard problem Bing Li, Lingmin Liao, Sanju Velani, ... more(5) 20220812 Let $T$ be a $d\times d$ matrix with real coefficients. Then $T$ determines a selfmap of the $d$dimensional torus ${\Bbb T}^d={\mathbb{R}}^d/{\Bbb Z}^d$. Let $ \{E_n \}_{n \in \mathbb{N}} $ be a sequence of subsets of ${\Bbb T}^d$ and let $W(T,\{E_n \})$ be the set of points $\mathbf{x} \in {\Bbb T}^d$ such that $T^n(\mathbf{x})\in E_n $ for infinitely many $n\in {\mathbb{N}}$. For a large class of subsets (namely, those satisfying the so called bounded property $ ({\boldsymbol{\rm B}}) $ which includes balls, rectangles, and hyperboloids) we show that the $d$dimensional Lebesgue measure of the shrinking target set $W(T,\{E_n \})$ is zero (resp. one) if a natural volume sum converges (resp. diverges). In fact, we prove a quantitative form of this zeroone criteria that describes the asymptotic behaviour of the counting function $R(x,N):= \# \big\{ 1\le n \le N : T^{n}(x) \in E_n \} $. The counting result makes use of a general quantitative statement that holds for a large class measurepreserving dynamical systems (namely, those satisfying the so called summablemixing property). We next turn our attention to the Hausdorff dimension of $W(T,\{E_n \})$. In the case the subsets $E_n$ are balls, rectangles or hyperboloids we obtain precise formulae for the dimension. These shapes correspond, respectively, to the simultaneous, weighted and multiplicative theories of classical Diophantine approximation. The dimension results for balls generalises those obtained in an earlier paper by Hill and the thirdnamed author for integer matrices to real matrices. In the final section, we discuss various problems that stem from the results proved in the paper.
Subject 수학 Source arXiv URL https://arxiv.org/abs/2208.06112view Article Title The Shrinking Target Problem for Matrix Transformations of Tori: revisiting the standard problemAuthors Bing Li; Lingmin Liao; Sanju Velani; Evgeniy Zorin; appendix by Baowei WangAbstract Let $T$ be a $d\times d$ matrix with real coefficients. Then $T$ determines a selfmap of the $d$dimensional torus ${\Bbb T}^d={\mathbb{R}}^d/{\Bbb Z}^d$. Let $ \{E_n \}_{n \in \mathbb{N}} $ be a sequence of subsets of ${\Bbb T}^d$ and let $W(T,\{E_n \})$ be the set of points $\mathbf{x} \in {\Bbb T}^d$ such that $T^n(\mathbf{x})\in E_n $ for infinitely many $n\in {\mathbb{N}}$. For a large class of subsets (namely, those satisfying the so called bounded property $ ({\boldsymbol{\rm B}}) $ which includes balls, rectangles, and hyperboloids) we show that the $d$dimensional Lebesgue measure of the shrinking target set $W(T,\{E_n \})$ is zero (resp. one) if a natural volume sum converges (resp. diverges). In fact, we prove a quantitative form of this zeroone criteria that describes the asymptotic behaviour of the counting function $R(x,N):= \# \big\{ 1\le n \le N : T^{n}(x) \in E_n \} $. The counting result makes use of a general quantitative statement that holds for a large class measurepreserving dynamical systems (namely, those satisfying the so called summablemixing property). We next turn our attention to the Hausdorff dimension of $W(T,\{E_n \})$. In the case the subsets $E_n$ are balls, rectangles or hyperboloids we obtain precise formulae for the dimension. These shapes correspond, respectively, to the simultaneous, weighted and multiplicative theories of classical Diophantine approximation. The dimension results for balls generalises those obtained in an earlier paper by Hill and the thirdnamed author for integer matrices to real matrices. In the final section, we discuss various problems that stem from the results proved in the paper.Is Part Of 20220812 Identifier ISSN: Category math.NTLicense 
arXivClassification of multistationarity for mass action networks with onedimensional stoichiometric subspace Casian Pantea, Galyna Voitiuk 20220812 We characterize completely the capacity for (nondegenerate) multistationarity of mass action reaction networks with onedimensional stoichiometric subspace in terms of reaction structure. Specifically, we show that networks with two or more source complexes have the capacity for multistationarity if and only if they have both patterns $(\to, \gets)$ and $(\gets, \to)$ in some 1D projections. Moreover, we specify the classes of networks for which only degenerate multiple steady states may occur. In particular, we characterize the capacity for nondegenerate multistationarity of small networks composed of one irreversible and one reversible reaction, or two reversible reactions
Subject 수학 Source arXiv URL https://arxiv.org/abs/2208.06310view Article Title Classification of multistationarity for mass action networks with onedimensional stoichiometric subspaceAuthors Casian Pantea; Galyna VoitiukAbstract We characterize completely the capacity for (nondegenerate) multistationarity of mass action reaction networks with onedimensional stoichiometric subspace in terms of reaction structure. Specifically, we show that networks with two or more source complexes have the capacity for multistationarity if and only if they have both patterns $(\to, \gets)$ and $(\gets, \to)$ in some 1D projections. Moreover, we specify the classes of networks for which only degenerate multiple steady states may occur. In particular, we characterize the capacity for nondegenerate multistationarity of small networks composed of one irreversible and one reversible reaction, or two reversible reactionsIs Part Of 20220812 Identifier ISSN: Category math.DS qbio.MNLicense 
arXivConnected Tur\'an number of trees Yair Caro, Balázs Patkós, Zsolt Tuza... more(3) 20220812 As a variant of the much studied Tur\'an number, $ex(n,F)$, the largest number of edges that an $n$vertex $F$free graph may contain, we introduce the connected Tur\'an number $ex_c(n,F)$, the largest number of edges that an $n$vertex connected $F$free graph may contain. We focus on the case where the forbidden graph is a tree. The celebrated conjecture of Erd\H{o}s and S\'os states that for any tree $T$, we have $ex(n,T)\le(T2)\frac{n}{2}$. We address the problem how much smaller $ex_c(n,T)$ can be, what is the smallest possible ratio of $ex_c(n,T)$ and $(T2)\frac{n}{2}$ as $T$ grows. We also determine the exact value of $ex_c(n,T)$ for small trees, in particular for all trees with at most six vertices. We introduce general constructions of connected $T$free graphs based on graph parameters as longest path, matching number, branching number, etc.
Subject 수학 Source arXiv URL https://arxiv.org/abs/2208.06126view Article Title Connected Tur\'an number of treesAuthors Yair Caro; Balázs Patkós; Zsolt TuzaAbstract As a variant of the much studied Tur\'an number, $ex(n,F)$, the largest number of edges that an $n$vertex $F$free graph may contain, we introduce the connected Tur\'an number $ex_c(n,F)$, the largest number of edges that an $n$vertex connected $F$free graph may contain. We focus on the case where the forbidden graph is a tree. The celebrated conjecture of Erd\H{o}s and S\'os states that for any tree $T$, we have $ex(n,T)\le(T2)\frac{n}{2}$. We address the problem how much smaller $ex_c(n,T)$ can be, what is the smallest possible ratio of $ex_c(n,T)$ and $(T2)\frac{n}{2}$ as $T$ grows. We also determine the exact value of $ex_c(n,T)$ for small trees, in particular for all trees with at most six vertices. We introduce general constructions of connected $T$free graphs based on graph parameters as longest path, matching number, branching number, etc.Is Part Of 20220812 Identifier ISSN: Category math.COLicense 
arXivA sufficient and necessary condition of generalized polynomial Li\'enard systems with global centers Hebai Chen, Zhijie Li, Rui Zhang... more(3) 20220812 The aim of this paper is to give a sufficient and necessary condition of the generalized polynomial Li\'enard system with a global center (including linear typer and nilpotent type). Recently, Llibre and Valls [J. Differential Equations, 330 (2022), 6680] gave a sufficient and necessary condition of the generalized polynomial Li\'enard system with a linear type global center. It is easy to see that our sufficient and necessary condition is more easy by comparison. In particular, we provide the explicit expressions of all the generalized polynomial Li\'enard differential systems of degree 5 having a global center at the origin and the explicit expression of a generalized polynomial Li\'enard differential system of indefinite degree having a global center at the origin.
Subject 수학 Source arXiv URL https://arxiv.org/abs/2208.06184view Article Title A sufficient and necessary condition of generalized polynomial Li\'enard systems with global centersAuthors Hebai Chen; Zhijie Li; Rui ZhangAbstract The aim of this paper is to give a sufficient and necessary condition of the generalized polynomial Li\'enard system with a global center (including linear typer and nilpotent type). Recently, Llibre and Valls [J. Differential Equations, 330 (2022), 6680] gave a sufficient and necessary condition of the generalized polynomial Li\'enard system with a linear type global center. It is easy to see that our sufficient and necessary condition is more easy by comparison. In particular, we provide the explicit expressions of all the generalized polynomial Li\'enard differential systems of degree 5 having a global center at the origin and the explicit expression of a generalized polynomial Li\'enard differential system of indefinite degree having a global center at the origin.Is Part Of 20220812 Identifier ISSN: Category math.DSLicense 
arXivCommunication network model of the immune system identifies the impact of interactions with SARSCoV2 proteins Swarnavo Sarkar 20220812 Interactions between SARSCoV2 and human proteins (SARSCoV2 PPIs) cause information transfer through biochemical pathways that contribute to the immunopathology of COVID19. Here, we present a communication network model of the immune system to compute the information transferred by the viral proteins using the available SARSCoV2 PPIs data. The amount of transferred information depends on the reference state of the immune system, or the state without SARSCoV2 PPIs, and can quantify how many variables of the immune system are controlled by the viral proteins. The information received by the immune system proteins from the viral proteins is useful to identify the biological processes (BPs) susceptible to dysregulation, and also to estimate the duration of viral PPIs necessary for the dysregulation to occur. We found that computing the drop in information from viral PPIs due to drugs provides a direct measure for the efficacy of therapies.
Subject 수학 Source arXiv URL https://arxiv.org/abs/2208.06355view Article Title Communication network model of the immune system identifies the impact of interactions with SARSCoV2 proteinsAuthors Swarnavo SarkarAbstract Interactions between SARSCoV2 and human proteins (SARSCoV2 PPIs) cause information transfer through biochemical pathways that contribute to the immunopathology of COVID19. Here, we present a communication network model of the immune system to compute the information transferred by the viral proteins using the available SARSCoV2 PPIs data. The amount of transferred information depends on the reference state of the immune system, or the state without SARSCoV2 PPIs, and can quantify how many variables of the immune system are controlled by the viral proteins. The information received by the immune system proteins from the viral proteins is useful to identify the biological processes (BPs) susceptible to dysregulation, and also to estimate the duration of viral PPIs necessary for the dysregulation to occur. We found that computing the drop in information from viral PPIs due to drugs provides a direct measure for the efficacy of therapies.Is Part Of 20220812 Identifier ISSN: Category qbio.MN cs.IT math.ITLicense 
arXivBicategories of Action Groupoids Carla Farsi, Laura Scull, Jordan Watts... more(3) 20220812 We prove that the 2category of action Lie groupoids localised in the following three different ways yield equivalent bicategories: localising at equivariant weak equivalences \`a la Pronk, localising using surjective submersive equivariant weak equivalences and anafunctors \`a la Roberts, and localising at all weak equivalences. This generalises the known result for representable orbifold groupoids. As an application, we show that any weak equivalence between action Lie groupoids is isomorphic to the composition of two special equivariant weak equivalences, again extending a result known for representable orbifold groupoids.
Subject 수학 Source arXiv URL https://arxiv.org/abs/2208.06281view Article Title Bicategories of Action GroupoidsAuthors Carla Farsi; Laura Scull; Jordan WattsAbstract We prove that the 2category of action Lie groupoids localised in the following three different ways yield equivalent bicategories: localising at equivariant weak equivalences \`a la Pronk, localising using surjective submersive equivariant weak equivalences and anafunctors \`a la Roberts, and localising at all weak equivalences. This generalises the known result for representable orbifold groupoids. As an application, we show that any weak equivalence between action Lie groupoids is isomorphic to the composition of two special equivariant weak equivalences, again extending a result known for representable orbifold groupoids.Is Part Of 20220812 Identifier ISSN: Category math.DGLicense