We consider quasirandom properties for Cayley graphs of finite abelian groups. We show that having uniform edge-distribution (i.e., small discrepancy) and having large eigenvalue gap are equivalent properties for such Cayley graphs, even if they are sparse. This affirmatively answers a question of Chung and Graham (2002) for the particular case of Cayley graphs of abelian groups, while in general the answer is negative., Yoshiharu Kohayakawa, Vojtěch Rödl, Mathias Schacht., and Obsahuje seznam literatury
Let $G$ be a locally compact group and let $1 \le p < \infty.$ Recently, Chen et al. characterized hypercyclic, supercyclic and chaotic weighted translations on locally compact groups and their homogeneous spaces. There has been an increasing interest in studying the disjoint hypercyclicity acting on various spaces of holomorphic functions. In this note, we will study disjoint hypercyclic and disjoint supercyclic powers of weighted translation operators on the Lebesgue space $L^p(G)$ in terms of the weights. Sufficient and necessary conditions for disjoint hypercyclic and disjoint supercyclic powers of weighted translations generated by aperiodic elements on groups will be given., Liang Zhang, Hui-Qiang Lu, Xiao-Mei Fu, Ze-Hua Zhou., and Obsahuje bibliografické odkazy
The Bruhat order is defined in terms of an interchange operation on the set of permutation matrices of order n which corresponds to the transposition of a pair of elements in a permutation. We introduce an extension of this partial order, which we call the stochastic Bruhat order, for the larger class Ω_{n} of doubly stochastic matrices (convex hull of n×n permutation matrices). An alternative description of this partial order is given. We define a class of special faces of Ω_{n} induced by permutation matrices, which we call Bruhat faces. Several examples of Bruhat faces are given and several results are presented., Richard A. Brualdi, Geir Dahl, Eliseu Fritscher., and Obsahuje seznam literatury
Edge-colourings of graphs have been studied for decades. We study edge-colourings with respect to hereditary graph properties. For a graph G, a hereditary graph property P and l\geqslant 1 we define X{'_{P,l}} to be the minimum number of colours needed to properly colour the edges of G, such that any subgraph of G induced by edges coloured by (at most) l colours is in P. We present a necessary and sufficient condition for the existence of X{'_{P,l}} . We focus on edge-colourings of graphs with respect to the hereditary properties Ok and Sk, where Ok contains all graphs whose components have order at most k+1, and Sk contains all graphs of maximum degree at most k. We determine the value of X{'_{{S_k},l}}(G) for any graph G,k \geqslant 1, l\geqslant 1 and we present a number of results on X{'_{{O_k},l}}(G) ., Samantha Dorfling, Tomáš Vetrík., and Obsahuje seznam literatury
In this paper, we investigate a measure of similarity of graphs similar to the Ramsey number. We present values and bounds for g(n, l), the biggest number k guaranteeing that there exist l graphs on n vertices, each two having edit distance at least k. By edit distance of two graphs G, F we mean the number of edges needed to be added to or deleted from graph G to obtain graph F. This new extremal number g(n, l) is closely linked to the edit distance of graphs. Using probabilistic methods we show that g(n, l) is close to \frac{1} {2}\left( {\begin{array}{*{20}c} n // 2 // \end{array} } \right) for small values of l > 2. We also present some exact values for small n and lower bounds for very large l close to the number of non-isomorphic graphs of n vertices., Tomasz Dzido, Krzysztof Krzywdziński., and Obsahuje seznam literatury
In this paper, characterizations of the embeddings between weighted Copson function spaces ${\rm Cop}_{p_1,q_1}(u_1,v_1)$ and weighted Cesàro function spaces ${\rm Ces}_{p_2,q_2}(u_2,v_2)$ are given. In particular, two-sided estimates of the optimal constant $c$ in the inequality $ \align\biggl( \int_0^{\infty} &\biggl( \int_0^t f(\tau)^{p_2}v_2(\tau)\dd\tau\biggr)^{\!\!\frc{q_2}{p_2}} u_2(t)\dd t\biggr)^{\!\!\frc1{q_2}} $ \ $\le c \biggl( \int_0^{\infty} \biggl( \int_t^{\infty} f(\tau)^{p_1} v_1(\tau)\dd\tau\biggr)^{\!\!\frc{q_1}{p_1}} u_1(t)\dd t\biggr)^{\!\!\frc1{q_1}}, $ where $p_1,p_2,q_1,q_2 \in(0,\infty)$, $p_2 \le q_2$ and $u_1,u_2,v_1,v_2$ are weights on $(0,\infty)$, are obtained. The most innovative part consists of the fact that possibly different parameters $p_1$ and $p_2$ and possibly different inner weights $v_1$ and $v_2$ are allowed. The proof is based on the combination of duality techniques with estimates of optimal constants of the embeddings between weighted Cesàro and Copson spaces and weighted Lebesgue spaces, which reduce the problem to the solutions of iterated Hardy-type inequalities., Amiran Gogatishvili, Rza Mustafayev, Tuğçe Ünver., and Obsahuje bibliografii
Recently, the weak Triebel-Lizorkin space was introduced by Grafakos and He, which includes the standard Triebel-Lizorkin space as a subset. The latter has a wide applications in aspects of analysis. In this paper, the authors firstly give equivalent quasi-norms of weak Triebel-Lizorkin spaces in terms of Peetre's maximal functions. As an application of those equivalent quasi-norms, an atomic decomposition of weak Triebel-Lizorkin spaces is given., Wenchang Li, Jingshi Xu., and Seznam literatury
We study ergodic properties of stochastic geometric wave equations on a particular model with the target being the 2D sphere while considering only solutions which are independent of the space variable. This simplification leads to a degenerate stochastic equation in the tangent bundle of the 2D sphere. Studying this equation, we prove existence and non-uniqueness of invariant probability measures for the original problem and obtain also results on attractivity towards an invariant measure. We also present a structure-preserving numerical scheme to approximate solutions and provide computational experiments to motivate and illustrate the theoretical results., Ľubomír Baňas, Zdzisłlaw Brzeźniak, Mikhail Neklyudov, Martin Ondreját, Andreas Prohl., and Obsahuje seznam literatury
In this paper, we give theoretical results on Macaev ideal and Dixmier trace. Then we give a characterization of antiholomorphic symbols \overline f such that the Hankel operator {H_{\overline f }} on a Bergman weighted space is in an ideal of Macaev and we give the Dixmier trace. For this, we look at the behavior of Schatten’s norms \mathcal{S}^p when p tends to 1, using results of Engliš and Rochberg on Bergman space. We also give results on powers of such operators., Nous donnons des résultats théoriques sur l’idéal de Macaev et la trace de Dixmier. Ensuite, nous caractérisons les symboles antiholomorphes \overline f tels que l’opérateur de Hankel {H_{\overline f }} sur l’espace de Bergman à poids soit dans l’idéal de Macaev et nous donnons la trace de Dixmier. Pour cela, nous regardons le comportement des normes de Schatten \mathcal{S}^p quand p tend vers 1 et nous nous appuyons sur le résultat de Engliš et Rochberg sur l’espace de Bergman. Nous parlons aussi des puissances de tels opérateurs., Romaric Tytgat., and Obsahuje seznam literatury
In this paper, we give some estimates for the essential norm and a new characterization for the boundedness and compactness of weighted composition operators from weighted Bergman spaces and Hardy spaces to the Bloch space., Songxiao Li, Ruishen Qian, Jizhen Zhou., and Obsahuje bibliografické odkazy