2019
http://publications.mfo.de/handle/mfo/1397
Wed, 17 Apr 2024 13:10:16 GMT2024-04-17T13:10:16ZGlobal Solutions to Stochastic Wave Equations with Superlinear Coefficients
http://publications.mfo.de/handle/mfo/3683
Global Solutions to Stochastic Wave Equations with Superlinear Coefficients
Millet, Annie; Sanz-Solé, Marta
We prove existence and uniqueness of a random field solution $(u(t,x);(t,x)\in [0,T]\times \mathbb{R}^d)$ to a stochastic wave equation in dimensions $d=1,2,3$ with diffusion and drift coefficients of the form $|x| \big(
\ln_+(|x|) \big)^a$ for some $a$>0. The proof relies on a sharp analysis of moment estimates of time and space increments of the corresponding stochastic wave equation with globally Lipschitz coefficients. We give examples of spatially correlated Gaussian driving noises where the results apply.
Wed, 13 Nov 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/36832019-11-13T00:00:00ZMillet, AnnieSanz-Solé, MartaWe prove existence and uniqueness of a random field solution $(u(t,x);(t,x)\in [0,T]\times \mathbb{R}^d)$ to a stochastic wave equation in dimensions $d=1,2,3$ with diffusion and drift coefficients of the form $|x| \big(
\ln_+(|x|) \big)^a$ for some $a$>0. The proof relies on a sharp analysis of moment estimates of time and space increments of the corresponding stochastic wave equation with globally Lipschitz coefficients. We give examples of spatially correlated Gaussian driving noises where the results apply.Matchings and Squarefree Powers of Edge Ideals
http://publications.mfo.de/handle/mfo/3682
Matchings and Squarefree Powers of Edge Ideals
Erey, Nursel; Herzog, Jürgen; Hibi, Takayuki; Saeedi Madani, Sara
Squarefree powers of edge ideals are intimately related to matchings of the underlying graph. In this paper we give bounds for the regularity of squarefree powers of edge ideals, and we consider the question of when such powers are linearly related or have linear resolution. We also consider the so-called squarefree Ratliff property.
Mon, 11 Nov 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/36822019-11-11T00:00:00ZErey, NurselHerzog, JürgenHibi, TakayukiSaeedi Madani, SaraSquarefree powers of edge ideals are intimately related to matchings of the underlying graph. In this paper we give bounds for the regularity of squarefree powers of edge ideals, and we consider the question of when such powers are linearly related or have linear resolution. We also consider the so-called squarefree Ratliff property.Reflective Prolate-Spheroidal Operators and the KP/KdV Equations
http://publications.mfo.de/handle/mfo/3680
Reflective Prolate-Spheroidal Operators and the KP/KdV Equations
Casper, W. Riley; Grünbaum, F. A.; Yakimov, Milen; Zurrián, Ignacio Nahuel
Commuting integral and differential operators connect the topics of Signal Processing, Random Matrix Theory, and Integrable Systems. Previously, the construction of such pairs was based on direct calculation and concerned
concrete special cases, leaving behind important families such as the operators associated to the rational solutions of the KdV equation. We prove a general theorem that the integral operator associated to every wave function in the infinite dimensional Adelic Grassmannian Gr $^{ad}$ of Wilson always reflects a differential operator (in the sense of Definition 1 below). This intrinsic property is shown to follow from the symmetries of Grassmannians of KP wave functions, where the direct commutativity property holds for operators associated to wave functions fixed by Wilson's sign involution but is violated in general. Based on this result, we prove a second main theorem that the integral operators in the computation of the singular values of the truncated generalized Laplace transforms associated to all bispectral wave functions of rank 1 reflect a differential operator. A 90$°$ rotation argument is used to prove a third main theorem that the integral operators in the computation of the singular values of the truncated generalized Fourier transforms associated to all such KP wave functions commute with a differential operator. These methods produce vast collections of integral operators with prolate-spheroidal
properties, including as special cases the integral operators associated to all rational solutions of the KdV and KP hierarchies considered by Airault-McKean-Moser and Krichever, respectively, in the late 70's. Many novel examples are presented.
Tue, 05 Nov 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/36802019-11-05T00:00:00ZCasper, W. RileyGrünbaum, F. A.Yakimov, MilenZurrián, Ignacio NahuelCommuting integral and differential operators connect the topics of Signal Processing, Random Matrix Theory, and Integrable Systems. Previously, the construction of such pairs was based on direct calculation and concerned
concrete special cases, leaving behind important families such as the operators associated to the rational solutions of the KdV equation. We prove a general theorem that the integral operator associated to every wave function in the infinite dimensional Adelic Grassmannian Gr $^{ad}$ of Wilson always reflects a differential operator (in the sense of Definition 1 below). This intrinsic property is shown to follow from the symmetries of Grassmannians of KP wave functions, where the direct commutativity property holds for operators associated to wave functions fixed by Wilson's sign involution but is violated in general. Based on this result, we prove a second main theorem that the integral operators in the computation of the singular values of the truncated generalized Laplace transforms associated to all bispectral wave functions of rank 1 reflect a differential operator. A 90$°$ rotation argument is used to prove a third main theorem that the integral operators in the computation of the singular values of the truncated generalized Fourier transforms associated to all such KP wave functions commute with a differential operator. These methods produce vast collections of integral operators with prolate-spheroidal
properties, including as special cases the integral operators associated to all rational solutions of the KdV and KP hierarchies considered by Airault-McKean-Moser and Krichever, respectively, in the late 70's. Many novel examples are presented.Groups with Spanier-Whitehead Duality
http://publications.mfo.de/handle/mfo/2518
Groups with Spanier-Whitehead Duality
Nishikawa, Shintaro; Proietti, Valerio
We introduce the notion of Spanier-Whitehead $K$-duality for a discrete group $G$, defined as duality in the KK-category between two $C*$-algebras which are naturally attached to the group, namely the reduced group $C*$-algebra and the crossed product for the group action on the universal example for proper actions. We compare this notion to the Baum-Connes conjecture by constructing duality classes based on two methods: the standard "gamma element" technique, and the more recent approach via cycles with property gamma. As a result of our
analysis, we prove Spanier-Whitehead duality for a large class of groups, including Bieberbach's space groups, groups acting on trees, and lattices in Lorentz groups.
Tue, 17 Sep 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25182019-09-17T00:00:00ZNishikawa, ShintaroProietti, ValerioWe introduce the notion of Spanier-Whitehead $K$-duality for a discrete group $G$, defined as duality in the KK-category between two $C*$-algebras which are naturally attached to the group, namely the reduced group $C*$-algebra and the crossed product for the group action on the universal example for proper actions. We compare this notion to the Baum-Connes conjecture by constructing duality classes based on two methods: the standard "gamma element" technique, and the more recent approach via cycles with property gamma. As a result of our
analysis, we prove Spanier-Whitehead duality for a large class of groups, including Bieberbach's space groups, groups acting on trees, and lattices in Lorentz groups.Group-Graded Rings Satisfying the Strong Rank Condition
http://publications.mfo.de/handle/mfo/2513
Group-Graded Rings Satisfying the Strong Rank Condition
Kropholler, Peter H.; Lorensen, Karl
A ring $R$ satisfies the $\textit{strong rank condition}$ (SRC) if, for every natural number $n$, the free $R$-submodules of $R^n$ all have rank $\leq n$. Let $G$ be a group and $R$ a ring strongly graded by $G$ such that the base ring $R_1$ is a domain. Using an argument originated by Laurent Bartholdi for studying cellular automata, we prove that $R$ satisfies SRC if and only if
$R_1$ satisfies SRC and $G$ is amenable. The special case of this result for group rings allows us to prove a characterization of amenability involving the group von Neumann algebra that was conjectured by Wolfgang Lück. In addition, we include two applications to the study of group rings and their modules.
Fri, 16 Aug 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25132019-08-16T00:00:00ZKropholler, Peter H.Lorensen, KarlA ring $R$ satisfies the $\textit{strong rank condition}$ (SRC) if, for every natural number $n$, the free $R$-submodules of $R^n$ all have rank $\leq n$. Let $G$ be a group and $R$ a ring strongly graded by $G$ such that the base ring $R_1$ is a domain. Using an argument originated by Laurent Bartholdi for studying cellular automata, we prove that $R$ satisfies SRC if and only if
$R_1$ satisfies SRC and $G$ is amenable. The special case of this result for group rings allows us to prove a characterization of amenability involving the group von Neumann algebra that was conjectured by Wolfgang Lück. In addition, we include two applications to the study of group rings and their modules.A Cheeger Type Inequality in Finite Cayley Sum Graphs
http://publications.mfo.de/handle/mfo/2512
A Cheeger Type Inequality in Finite Cayley Sum Graphs
Biswas, Arindam; Saha, Jyoti Prakash
Let $G$ be a finite group and $S$ be a symmetric generating set of $G$ with $|S| = d$. We show that if the undirected Cayley sum graph $C_{\Sigma}(G,S)$ is an expander graph and is non-bipartite, then the spectrum of its normalised adjacency operator is bounded away from $-1$. We also establish an explicit lower bound for the spectrum of these graphs, namely, the non-trivial eigenvalues of the normalised adjacency operator lies in the interval $\left(-1+\frac{h(G)^{4}}{\eta}, 1-\frac{h(G)^{2}}{2d^{2}}\right]$, where $h(G)$ denotes the (vertex) Cheeger constant of the $d$-regular graph $C_{\Sigma}(G,S)$ and $\eta = 2^{9}d^{8}$. Further, we improve upon a recently obtained bound on the non-trivial spectrum of the normalised adjacency operator of the non-bipartite Cayley graph $C(G,S)$.
Wed, 31 Jul 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25122019-07-31T00:00:00ZBiswas, ArindamSaha, Jyoti PrakashLet $G$ be a finite group and $S$ be a symmetric generating set of $G$ with $|S| = d$. We show that if the undirected Cayley sum graph $C_{\Sigma}(G,S)$ is an expander graph and is non-bipartite, then the spectrum of its normalised adjacency operator is bounded away from $-1$. We also establish an explicit lower bound for the spectrum of these graphs, namely, the non-trivial eigenvalues of the normalised adjacency operator lies in the interval $\left(-1+\frac{h(G)^{4}}{\eta}, 1-\frac{h(G)^{2}}{2d^{2}}\right]$, where $h(G)$ denotes the (vertex) Cheeger constant of the $d$-regular graph $C_{\Sigma}(G,S)$ and $\eta = 2^{9}d^{8}$. Further, we improve upon a recently obtained bound on the non-trivial spectrum of the normalised adjacency operator of the non-bipartite Cayley graph $C(G,S)$.On a Cheeger Type Inequality in Cayley Graphs of Finite Groups
http://publications.mfo.de/handle/mfo/2511
On a Cheeger Type Inequality in Cayley Graphs of Finite Groups
Biswas, Arindam
Let $G$ be a finite group. It was remarked by Breuillard-Green-Guralnick-Tao that if the Cayley graph $C(G,S)$ is an expander graph and is non-bipartite then the spectrum of the adjacency operator $T$ is bounded away from $-1$. In this article we are interested in explicit bounds for the spectrum of these graphs. Specifically, we show that the non-trivial spectrum of the adjacency operator lies in the interval $\left[-1+\frac{h(\mathbb{G})^{4}}{\gamma}, 1-\frac{h(\mathbb{G})^{2}}{2d^{2}}\right]$, where $h(\mathbb{G})$ denotes the (vertex) Cheeger constant of the $d$ regular graph $C(G,S)$ with respect to a symmetric set $S$ of generators and $\gamma = 2^{9}d^{6}(d+1)^{2}$.
Mon, 22 Jul 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25112019-07-22T00:00:00ZBiswas, ArindamLet $G$ be a finite group. It was remarked by Breuillard-Green-Guralnick-Tao that if the Cayley graph $C(G,S)$ is an expander graph and is non-bipartite then the spectrum of the adjacency operator $T$ is bounded away from $-1$. In this article we are interested in explicit bounds for the spectrum of these graphs. Specifically, we show that the non-trivial spectrum of the adjacency operator lies in the interval $\left[-1+\frac{h(\mathbb{G})^{4}}{\gamma}, 1-\frac{h(\mathbb{G})^{2}}{2d^{2}}\right]$, where $h(\mathbb{G})$ denotes the (vertex) Cheeger constant of the $d$ regular graph $C(G,S)$ with respect to a symmetric set $S$ of generators and $\gamma = 2^{9}d^{6}(d+1)^{2}$.On Co-Minimal Pairs in Abelian Groups
http://publications.mfo.de/handle/mfo/2509
On Co-Minimal Pairs in Abelian Groups
Biswas, Arindam; Saha, Jyoti Prakash
A pair of non-empty subsets $(W,W')$ in an abelian group $G$ is a complement pair if $W+W'=G$. $W'$ is said to be minimal to $W$ if $W+(W'\setminus \{w'\}) \neq G, \forall \,w'\in W'$. In general, given an arbitrary subset in a group, the existence of minimal complement(s) depends on its structure. The dual problem asks that given such a set, if it is a minimal complement to some subset. We study tightness property of complement pairs $(W,W')$ such that both $W$ and $W'$ are minimal to each other. These are termed co-minimal pairs and we show that any non-empty finite set in an arbitrary free abelian group belongs to some co-minimal pair. We also construct infinite sets forming co-minimal pairs. Finally, we remark that a result of Kwon on the existence of minimal self-complements in $\mathbb{Z}$, also holds in any abelian group.
Tue, 09 Jul 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25092019-07-09T00:00:00ZBiswas, ArindamSaha, Jyoti PrakashA pair of non-empty subsets $(W,W')$ in an abelian group $G$ is a complement pair if $W+W'=G$. $W'$ is said to be minimal to $W$ if $W+(W'\setminus \{w'\}) \neq G, \forall \,w'\in W'$. In general, given an arbitrary subset in a group, the existence of minimal complement(s) depends on its structure. The dual problem asks that given such a set, if it is a minimal complement to some subset. We study tightness property of complement pairs $(W,W')$ such that both $W$ and $W'$ are minimal to each other. These are termed co-minimal pairs and we show that any non-empty finite set in an arbitrary free abelian group belongs to some co-minimal pair. We also construct infinite sets forming co-minimal pairs. Finally, we remark that a result of Kwon on the existence of minimal self-complements in $\mathbb{Z}$, also holds in any abelian group.A Quantitative Analysis of the “Lion-Man” Game
http://publications.mfo.de/handle/mfo/2508
A Quantitative Analysis of the “Lion-Man” Game
Kohlenbach, Ulrich; López-Acedo, Genaro; Nicolae, Adriana
In this paper we analyze, based on an interplay between ideas and techniques from logic and geometric analysis, a pursuit-evasion game. More precisely, we focus on a discrete lion and man game with an $\varepsilon$-capture criterion. We prove that in uniformly convex bounded domains the lion always wins and, using ideas stemming from proof mining, we extract a uniform rate of
convergence for the successive distances between the lion and the man. As a byproduct of our analysis, we study the relation among different convexity properties in the setting of geodesic spaces.
Mon, 08 Jul 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25082019-07-08T00:00:00ZKohlenbach, UlrichLópez-Acedo, GenaroNicolae, AdrianaIn this paper we analyze, based on an interplay between ideas and techniques from logic and geometric analysis, a pursuit-evasion game. More precisely, we focus on a discrete lion and man game with an $\varepsilon$-capture criterion. We prove that in uniformly convex bounded domains the lion always wins and, using ideas stemming from proof mining, we extract a uniform rate of
convergence for the successive distances between the lion and the man. As a byproduct of our analysis, we study the relation among different convexity properties in the setting of geodesic spaces.On Residuals of Finite Groups
http://publications.mfo.de/handle/mfo/1423
On Residuals of Finite Groups
Aivazidis, Stefanos; Müller, Thomas
A theorem of Dolfi, Herzog, Kaplan, and Lev [DHKL07, Thm. C] asserts that in a finite group with trivial Fitting subgroup, the size of the soluble residual of the group is bounded from below by a certain power of the group order, and that the inequality is sharp. Inspired by this result and some of the arguments in [DHKL07], we establish the following generalisation: if ${\mathfrak{X}}$ is a subgroup-closed Fitting formation of full characteristic which does not contain all finite groups and $\overline{\mathfrak{X}}$ is the extension-closure of $\mathfrak{X}$, then there exists an (optimal) constant $\gamma$ depending only on $\mathfrak{X}$ such that, for all non-trivial finite groups G with trivial $\mathfrak{X}$-radical, ${\vert}G{\vert}^{\overline{\mathfrak{X}}} > {\vert}G{\vert}^\gamma$, where $G^{\overline{\mathfrak{X}}}$ is the ${\overline{\mathfrak{X}}}$-residual of $G$. When ${\mathfrak{X}}={\mathfrak{N}}$, the class of finite nilpotent groups, it follows that $\overline{\mathfrak{X}} = \mathfrak{S}$, the class of finite soluble groups, thus we recover the original theorem of Dolfi, Herzog, Kaplan, and Lev. In the last section of our paper, building on J. G. Thompson's classification of minimal simple groups, we exhibit a family of subgroup-closed Fitting formations X of full characteristic such that $\mathfrak{S}
\subset \overline{\mathfrak{X}} \subset \mathfrak{E}$, thus providing applications of our main result beyond the reach of [DHKL07, Thm. C]
Tue, 28 May 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14232019-05-28T00:00:00ZAivazidis, StefanosMüller, ThomasA theorem of Dolfi, Herzog, Kaplan, and Lev [DHKL07, Thm. C] asserts that in a finite group with trivial Fitting subgroup, the size of the soluble residual of the group is bounded from below by a certain power of the group order, and that the inequality is sharp. Inspired by this result and some of the arguments in [DHKL07], we establish the following generalisation: if ${\mathfrak{X}}$ is a subgroup-closed Fitting formation of full characteristic which does not contain all finite groups and $\overline{\mathfrak{X}}$ is the extension-closure of $\mathfrak{X}$, then there exists an (optimal) constant $\gamma$ depending only on $\mathfrak{X}$ such that, for all non-trivial finite groups G with trivial $\mathfrak{X}$-radical, ${\vert}G{\vert}^{\overline{\mathfrak{X}}} > {\vert}G{\vert}^\gamma$, where $G^{\overline{\mathfrak{X}}}$ is the ${\overline{\mathfrak{X}}}$-residual of $G$. When ${\mathfrak{X}}={\mathfrak{N}}$, the class of finite nilpotent groups, it follows that $\overline{\mathfrak{X}} = \mathfrak{S}$, the class of finite soluble groups, thus we recover the original theorem of Dolfi, Herzog, Kaplan, and Lev. In the last section of our paper, building on J. G. Thompson's classification of minimal simple groups, we exhibit a family of subgroup-closed Fitting formations X of full characteristic such that $\mathfrak{S}
\subset \overline{\mathfrak{X}} \subset \mathfrak{E}$, thus providing applications of our main result beyond the reach of [DHKL07, Thm. C]