A variational approach to the alternating projections method

The 2-sets convex feasibility problem aims at finding a point in the nonempty intersection of two closed convex sets A and B in a Hilbert space H. The method of alternating projections is the simplest iterative procedure for finding a solution and it goes back to von Neumann. In the present paper, we study some stability properties for this method in the following sense: we consider two sequences of closed convex sets $$\{A_n\}$$ { A n } and $$\{B_n\}$$ { B n } , each of them converging, with respect to the Attouch-Wets variational convergence, respectively, to A and B. Given a starting point $$a_0$$ a 0 , we consider the sequences of points obtained by projecting on the “perturbed” sets, i.e., the sequences $$\{a_n\}$$ { a n } and $$\{b_n\}$$ { b n } given by $$b_n=P_{B_n}(a_{n-1})$$ b n = P B n ( a n - 1 ) and $$a_n=P_{A_n}(b_n)$$ a n = P A n ( b n ) . Under appropriate geometrical and topological assumptions on the intersection of the limit sets, we ensure that the sequences $$\{a_n\}$$ { a n } and $$\{b_n\}$$ { b n } converge in norm to a point in the intersection of A and B. In particular, we consider both when the intersection $$A\cap B$$ A ∩ B reduces to a singleton and when the interior of $$A \cap B$$ A ∩ B is nonempty. Finally we consider the case in which the limit sets A and B are subspaces.

Collectively fixed point theorems in noncompact abstract convex spaces with applications

<abstract><p>In this paper, by using the KKM theory and the properties of $ \Gamma $-convexity and $ {\frak{RC}} $-mapping, we investigate the existence of collectively fixed points for a family with a finite number of set-valued mappings on the product space of noncompact abstract convex spaces. Consequently, as applications, some existence theorems of generalized weighted Nash equilibria and generalized Pareto Nash equilibria for constrained multiobjective games, some nonempty intersection theorems with applications to the Fan analytic alternative formulation and the existence of Nash equilibria, and some existence theorems of solutions for generalized weak implicit inclusion problems in noncompact abstract convex spaces are given. The results obtained in this paper extend and generalize many corresponding results of the existing literature.</p></abstract>

An Erdős-Ko-Rado Type Theorem via the Polynomial Method

A family F is an intersecting family if any two members have a nonempty intersection. Erdős, Ko, and Rado showed that | F | ≤ n − 1 k − 1 holds for a k-uniform intersecting family F of subsets of [ n ] . The Erdős-Ko-Rado theorem for non-uniform intersecting families of subsets of [ n ] of size at most k can be easily proved by applying the above result to each uniform subfamily of a given family. It establishes that | F | ≤ n − 1 k − 1 + n − 1 k − 2 + ⋯ + n − 1 0 holds for non-uniform intersecting families of subsets of [ n ] of size at most k. In this paper, we prove that the same upper bound of the Erdős-Ko-Rado Theorem for k-uniform intersecting families of subsets of [ n ] holds also in the non-uniform family of subsets of [ n ] of size at least k and at most n − k with one more additional intersection condition. Our proof is based on the method of linearly independent polynomials.

A Numerical Approach for the Filtered Generalized Čech Complex

In this paper, we present an algorithm to compute the filtered generalized Čech complex for a finite collection of disks in the plane, which do not necessarily have the same radius. The key step behind the algorithm is to calculate the minimum scale factor needed to ensure rescaled disks have a nonempty intersection, through a numerical approach, whose convergence is guaranteed by a generalization of the well-known Vietoris–Rips Lemma, which we also prove in an alternative way, using elementary geometric arguments. We give an algorithm for computing the 2-dimensional filtered generalized Čech complex of a finite collection of d-dimensional disks in R d , and we show the performance of our algorithm.

HYPERHROUPS AND Hv-GROUPS ASSOCIATED TO ELEMENTS WITH FOUR OXIDATION STATES

The theory of hyperstructures is of great importance due to its connections to various fields of Science. $H_v$-structures are hyperstructures where the equality is replaced by the nonempty intersection. This class of the hyperstructures is very large so one can use it in order to define several objects that they are not possible to be defined in the classical hyperstructure theory. This paper attempts an exposition of the connection between hyperstructure ($H_v$-structure) theory and certain type of chemical reactions. In this regard, we consider elements with four oxidation states and investigate their mathematical structures.

Twisted conjugacy classes in unitriangular groups

Let R be an integral domain of characteristic zero. In this note we study the Reidemeister spectrum of the group {{\rm UT}_{n}(R)} of unitriangular matrices over R. We prove that if {R^{+}} is finitely generated and {n>2|R^{*}|} , then {{\rm UT}_{n}(R)} possesses the {R_{\infty}} -property, i.e. the Reidemeister spectrum of {{\rm UT}_{n}(R)} contains only {\infty} , however, if {n\leq|R^{*}|} , then the Reidemeister spectrum of {{\rm UT}_{n}(R)} has nonempty intersection with {\mathbb{N}} . If R is a field and {n\geq 3} , then we prove that the Reidemeister spectrum of {{\rm UT}_{n}(R)} coincides with {\{1,\infty\}} , i.e. in this case {{\rm UT}_{n}(R)} does not possess the {R_{\infty}} -property.

Трансверсальное доминирование в двойных графах

Let \(G\) be any graph. A subset \(S\) of vertices in \(G\) is called a dominating set if each vertex not in \(S\) is adjacent to at least one vertex in \(S\). A dominating set \(S\) is called a transversal dominating set if \(S\) has nonempty intersection with every dominating set of minimum cardinality in \(G\). The minimum cardinality of a transversal dominating set is called the transversal domination number denoted by \(\gamma_{td}(G)\). In this paper, we are considering special types of graphs called double graphs obtained through a graph operation. We study the new domination parameter for these graphs. We calculate the exact value of domination and transversal domination number in double graphs of some standard class of graphs. Further, we also estimate some simple bounds for these parameters in terms of order of a graph.

On Upper Transversals in 3-Uniform Hypergraphs

A set $S$ of vertices in a hypergraph $H$ is a transversal if it has a nonempty intersection with every edge of $H$. The upper transversal number $\Upsilon(H)$ of $H$ is the maximum cardinality of a minimal transversal in $H$. We show that if $H$ is a connected $3$-uniform hypergraph of order $n$, then $\Upsilon(H) > 1.4855 \sqrt[3]{n} - 2$. For $n$ sufficiently large, we construct infinitely many connected $3$-uniform hypergraphs, $H$, of order~$n$ satisfying $\Upsilon(H) < 2.5199 \sqrt[3]{n}$. We conjecture that $\displaystyle{\sup_{n \to \infty} \, \left( \inf \frac{ \Upsilon(H) }{ \sqrt[3]{n} } \right) = \sqrt[3]{16} }$, where the infimum is taken over all connected $3$-uniform hypergraphs $H$ of order $n$.

Nonempty Intersection of Longest Paths in $2K_2$-Free Graphs

In 1966, Gallai asked whether all longest paths in a connected graph share a common vertex. Counterexamples indicate that this is not true in general. However, Gallai's question is positive for certain well-known classes of connected graphs, such as split graphs, interval graphs, circular arc graphs, outerplanar graphs, and series-parallel graphs. A graph is $2K_2$-free if it does not contain two independent edges as an induced subgraph. In this short note, we show that, in nonempty $2K_2$-free graphs, every vertex of maximum degree is common to all longest paths. Our result implies that all longest paths in a nonempty $2K_2$-free graph have a nonempty intersection. In particular, it strengthens the result on split graphs, as split graphs are $2K_2$-free.