Nonempty Intersection Research Articles

More on Externally <i>q</i>-Hyperconvex Subsets of <i>T</i><sub>0</sub>-Quasi-Metric Spaces

We continue earlier research on T0-quasi-metric spaces which are externally q-hyperconvex. We focus on external q-hyperconvex subsets of T0-quasi-metric spaces in particular. We demonstrate that a countable family of pairwise intersecting externally q-hyperconvex subsets has a non-empty intersection that is external q-hyperconvex under specific requirements on the underlying space (see Proposition 22). Last but not least, we demonstrate that if A is a subset of a supseparable and externally q-hyperconvex space Y, where Y ⊆ X, then A is also externally q-hyperconvex in X (Proposition 25).

Characterising clique convergence for locally cyclic graphs of minimum degree δ ≥ 6

The clique graph kG of a graph G has as its vertices the cliques (maximal complete subgraphs) of G, two of which are adjacent in kG if they have non-empty intersection in G. We say that G is clique convergent if knG≅kmG for some n≠m, and that G is clique divergent otherwise. We completely characterise the clique convergent graphs in the class of (not necessarily finite) locally cyclic graphs of minimum degree δ≥6, showing that for such graphs clique divergence is a global phenomenon, dependent on the existence of large substructures. More precisely, we establish that such a graph is clique divergent if and only if its universal triangular cover contains arbitrarily large members from the family of so-called “triangular-shaped graphs”.

On self-graphoidal graphs and their complements

The graphoidal graph G of graph H is the graph obtained by taking graphoidal cover Ψ of H as vertices and two vertices are adjacent if and only if the corresponding paths have a non-empty intersection. If G is isomorphic to one of its graphoidal graphs, then G is said to be a self-graphoidal graph. G is called self-complementary graphoidal graph if it is isomorphic to one of its complementary graphoidal graphs. In this article, we characterize self-graphoidal graphs and give a construction of self-graphoidal graphs from cycle and wheel graphs. Also, we give a characterization of self-complementary graphoidal graphs.

The alternating simultaneous Halpern–Lions–Wittmann–Bauschke algorithm for finding the best approximation pair for two disjoint intersections of convex sets

Given two nonempty and disjoint intersections of closed and convex subsets, we look for a best approximation pair relative to them, i.e., a pair of points, one in each intersection, attaining the minimum distance between the disjoint intersections. We propose an iterative process based on projections onto the subsets which generate the intersections. The process is inspired by the Halpern–Lions–Wittmann–Bauschke algorithm and the classical alternating process of Cheney and Goldstein, and its advantage is that there is no need to project onto the intersections themselves, a task which can be rather demanding. We prove that under certain conditions the two interlaced subsequences converge to a best approximation pair. These conditions hold, in particular, when the space is Euclidean and the subsets which generate the intersections are compact and strictly convex. Our result extends the one of Aharoni, Censor and Jiang [“Finding a best approximation pair of points for two polyhedra”, Computational Optimization and Applications 71 (2018), 509–23] who considered the case of finite-dimensional polyhedra.

Erdős-Ko-Rado theorem for bounded multisets

Let k,m,n be positive integers with k⩾2. A k-multiset of [n]m is a collection of k integers from the set {1,2,…,n} in which the integers can appear more than once but at most m times. A family of such k-multisets is called an intersecting family if every pair of k-multisets from the family have non-empty intersection. A finite sequence of real numbers (a1,a2,…,an) is said to be unimodal if there is some k∈{1,2,…,n}, such that a1⩽a2⩽…⩽ak−1⩽ak⩾ak+1⩾…⩾an. Given m,n,k, denote Ck,ℓ as the coefficient of xk in the generating function (∑i=1mxi)ℓ, where 1⩽ℓ⩽n. In this paper, we first show that the sequence of (Ck,1,Ck,2,…,Ck,n) is unimodal. Then we use this as a tool to prove that the intersecting family in which every k-multiset contains a fixed element attains the maximum cardinality for n⩾k+⌈k/m⌉. In the special case when m=1 and m=∞, our result gives rise to the famous Erdős-Ko-Rado Theorem and an unbounded multiset version for this problem given by Meagher and Purdy [11], respectively. The main result in this paper can be viewed as a bounded multiset version of the Erdős-Ko-Rado Theorem.

Reachable Set Estimation of Multiagent Systems: An Approximate Consensus Perspective

This paper is concerned with the estimation of reachable set for leaderless multi-agent systems (MASs) under fixed topology with bounded disturbances and nonzero initial conditions. In-neighbors' information is collected to design the protocol such that all agents' state trajectories are constrained within a compact set, and asymptotic consistency is regulated for disturbance-free MASs in the exponential stability sense while approximate consistency is regulated for noise-disturbed MASs in the input-to-state stability sense. The agreement dynamics are characterized by the consensus function with determined initial condition, whose estimated reachable set has non-empty intersection with that of MAS. The analytical framework is also applied to Markovian topology switching subject to completely and partially known transition rates, respectively. Numerical simulations confirm the theoretical results.

Feasibility problems via paramonotone operators in a convex setting

This paper is focussed on some properties of paramonotone operators on Banach spaces and their application to certain feasibility problems for convex sets in a Hilbert space and convex systems in the Euclidean space. In particular, it shows that operators that are simultaneously paramonotone and bimonotone are constant on their domains, and this fact is applied to tackle two particular situations. The first one, closely related to simultaneous projections, deals with a finite amount of convex sets with an empty intersection and tackles the problem of finding the smallest perturbations (in the sense of translations) of these sets to reach a nonempty intersection. The second is focussed on the distance to feasibility; specifically, given an inconsistent convex inequality system, our goal is to compute/estimate the smallest right-hand side perturbations that reach feasibility. We advance that this work derives lower and upper estimates of such a distance, which become the exact value when confined to linear systems.

Hyperideal-based zero-divisor graph of the general hyperring $ \mathbb{Z}_{n} $

<abstract><p>The aim of this paper is to introduce and study the concept of a hyperideal-based zero-divisor graph associated with a general hyperring. This is a generalized version of the zero-divisor graph associated with a commutative ring. For any general hyperring $ R $ having a hyperideal $ I $, the $ I $-based zero-divisor graph $ \Gamma^{(I)}(R) $ associated with $ R $ is the simple graph whose vertices are the elements of $ R\setminus I $ having their hyperproduct in $ I $, and two distinct vertices are joined by an edge when their hyperproduct has a non-empty intersection with $ I $. In the first part of the paper, we concentrate on some general properties of this graph related to absorbing elements, while the second part is dedicated to the study of the $ I $-based zero-divisor graph associated to the general hyperring $ \mathbb{Z}_n $ of the integers modulo $ n $, when $ n = 2p^mq $, with $ p $ and $ q $ two different odd primes, and fixing the hyperideal $ I $.</p></abstract>

A Countable Intersection Like Characterization of Star-Lindelöf Spaces

There have been various studies on star-Lindelöfness but they always explain it in terms of open coverings. So, we have demonstrated in this study a connection between star-Lindelöfness and the family of closed sets that resembles countable intersection property of Lindelöf space. We show that a topological space $X$ is star-Lindelöf if and only if every closed subset's family of $X$ not having the modified non-countable intersection property have non-empty intersection.

Economics as Religion and Christianity as oikonomia: Giorgio Agamben and the Homo Sacer

Among the contemporary thinkers who try to think of economics not just as having a non-empty intersection with religion but as being intrinsically religious, Giorgio Agamben occupies a singular place. Indeed, one of the main theses of his major work, Homo Sacer, is that the modern rupture between “sovereignty” and “government”—which lies at the heart of his political diagnosis of our contemporary situation—can be traced back to classical Trinitarian theology. Since this rupture is allegedly responsible for today’s Western understanding of economics, this implies that, according to the Italian philosopher, our current crisis has Christian theological roots. In this paper, I first discuss the argument put forward by Agamben to assert that, at least since the Trinitarian controversies of the second century, Christianity has become intrinsically oikonomia, that is, it understands history as the unfolding of a providential dynamic which, he claims, anticipates today’s celebrated “invisible hand” of decentralized markets. Next, I offer critical reflections upon this argument by questioning whether contemporary mainstream economics can be reduced to the “market fundamentalism” with which it is often confused. The article concludes by questioning, in turn, whether all Christian traditions boil down to the historical trend that Agamben characterizes as leading to today’s problematic mainstream economics.

A hybrid direct search and projected simplex gradient method for convex constrained minimization

We propose a new Derivative-free Optimization (DFO) approach for solving convex constrained minimization problems. The feasible set is assumed to be the non-empty intersection of a finite collection of closed convex sets, such that the projection onto each of these individual convex sets is simple and inexpensive to compute. Our iterative approach alternates between steps that use Directional Direct Search (DDS), considering adequate poll directions, and a Spectral Projected Gradient (SPG) method, replacing the real gradient by a simplex gradient, under a DFO approach. In the SPG steps, if the convex feasible set is simple, then a direct projection is computed. If the feasible set is the intersection of finitely many convex simple sets, then Dykstra's alternating projection method is applied. Convergence properties are established under standard assumptions usually associated to DFO methods. Some preliminary numerical experiments are reported to illustrate the performance of the proposed algorithm, in particular by comparing it with a classical DDS method. Our results indicate that the hybrid algorithm is a robust and effective approach for derivative-free convex constrained optimization.

A NOTE ON VISIBLE SETS

In this paper, we investigate some properties of visible sets and its characterization. Also we obtained relation between visible set and union of convex sets with nonempty intersection.

Numerical solution of singular Sylvester equations

We consider the numerical solution of the large scale singular Sylvester equation of the form AX−XD⊤=E (or AXB⊤−CXD⊤=E), where the spectra Λ(A) and Λ(D) (or, Λ(A−λC) and Λ(D−λB)) have a nonempty intersection. Using appropriate invariant subspaces, the singular Sylvester equation is rewritten as four Sylvester equations, three of which are nonsingular and one singular. When the invariant subspaces are small, so are three of the equations (including the singular one) which can be solved efficiently. The fourth is large but nonsingular with structures and may be solved using the projection method with Krylov subspaces or techniques involving hierarchical matrices. Some numerical examples for the subspace method are provided.

String graphs have the Erdős–Hajnal property

The following Ramsey-type question is one of the central problems in combinatorial geometry. Given a collection of certain geometric objects in the plane (e.g. segments, rectangles, convex sets, arcwise connected sets) of size n , what is the size of the largest subcollection in which either any two elements have a nonempty intersection, or any two elements are disjoint? We prove that there exists an absolute constant c>0 such that if \mathcal{C} is a collection of n curves in the plane, then \mathcal{C} contains at least n^{c} elements that are pairwise intersecting, or n^{c} elements that are pairwise disjoint. This resolves a problem raised by Alon, Pach, Pinchasi, Radoičić and Sharir, and Fox and Pach. Furthermore, as any geometric object can be arbitrarily closely approximated by a curve, this shows that the answer to the aforementioned question is at least n^{c} for any collection of n geometric objects.

GKZ hypergeometric systems of the three-loop vacuum Feynman integrals

We present the Gel’fand-Kapranov-Zelevinsky (GKZ) hypergeometric systems of the Feynman integrals of the three-loop vacuum diagrams with arbitrary masses, basing on Mellin-Barnes representations and Miller’s transformation. The codimension of derived GKZ hypergeometric systems equals the number of independent dimensionless ratios among the virtual masses squared. Through GKZ hypergeometric systems, the analytical hypergeometric series solutions can be obtained in neighborhoods of origin including infinity. The linear independent hypergeometric series solutions whose convergent regions have non-empty intersection can constitute a fundamental solution system in a proper subset of the whole parameter space. The analytical expression of the vacuum integral can be formulated as a linear combination of the corresponding fundamental solution system in certain convergent region.

Existence and Uniqueness of Polyhedra with Given Values of the Conditional Curvature

The theory of polyhedra and the geometric methods associated with it are interesting not only in their own right but also have a wide outlet in the general theory of surfaces. Certainly, it is only sometimes possible to obtain the corresponding theorem on surfaces from the theorem on polyhedra by passing to the limit. Still, the theorems on polyhedra give directions for searching for the related theorems on surfaces. In the case of polyhedra, the elementary-geometric basis of more general results is revealed. In the present paper, we study polyhedra of a particular class, i.e., without edges and reference planes perpendicular to a given direction. This work is a logical continuation of the author’s work, in which an invariant of convex polyhedra isometric on sections was found. The concept of isometry of surfaces and the concept of isometry on sections of surfaces differ from each other. Examples of isometric surfaces that are not isometric on sections and examples of non-isometric surfaces that are isometric on sections. However, they have non-empty intersections, i.e., some surfaces are both isometric and isometric on sections. In this paper, we prove the positive definiteness of the found invariant. Further, conditional external curvature is introduced for “basic” sets, open faces, edges, and vertices. It is proved that the conditional curvature of the polyhedral angle considered is monotonicity and positive definiteness. At the end of the article, the problem of the existence and uniqueness of convex polyhedra with given values of conditional curvatures at the vertices is solved.

GKZ-system of the 2-loop self energy with 4 propagators

Applying the system of linear partial differential equations derived from the Mellin–Barnes representation and the Miller transformation, we present the GKZ-system of the Feynman integral of the 2-loop self energy diagram with 4 propagators. The codimension of the derived GKZ-system equals the number of independent dimensionless ratios among the external momentum squared and virtual mass squared. In total 536 hypergeometric functions are obtained in the neighborhoods of the origin and infinity, in which 30 linearly independent hypergeometric functions whose convergent regions have nonempty intersection constitute a fundamental solution system in a proper subset of the whole parameter space.

Fine phase mixtures in one-dimensional non-convex elastodynamics

We show that fine phase mixtures arise in the initial-boundary value problem for a class of equations of non-convex elastodynamics in one space dimension. Specifically, we prove that there are infinitely many local-in-time Lipschitz weak solutions to such a problem that exhibit immediate fine-scale oscillations of the strain whenever the range of the initial strain has a nonempty intersection with the elliptic regime. Consequently, such solutions are nowhere C1 in the part of the space-time domain with fine phase mixtures, but are smooth in the other part of the domain.

Maximal norm Hankel operators

A Hankel operator Hφ on the Hardy space H2 of the unit circle with analytic symbol φ has minimal norm if ‖Hφ‖=‖φ‖2 and maximal norm if ‖Hφ‖=‖φ‖∞. The Hankel operator Hφ has both minimal and maximal norm if and only if |φ| is constant almost everywhere on the unit circle or, equivalently, if and only if φ is a constant multiple of an inner function. We show that if Hφ is norm-attaining and has maximal norm, then Hφ has minimal norm. If |φ| is continuous but not constant, then Hφ has maximal norm if and only if the set at which |φ|=‖φ‖∞ has nonempty intersection with the spectrum of the inner factor of φ. We obtain further results illustrating that the case of maximal norm is in general related to “irregular” behavior of log⁡|φ| or the argument of φ near a “maximum point” of |φ|. The role of certain positive functions coined apical Helson–Szegő weights is discussed in the former context.

All longest cycles in a 2‐connected partial 3‐tree share a common vertex

AbstractIn 1966 Gallai asked whether all longest paths in a connected graph have nonempty intersection. This is not true in general and various counterexamples have been found. In this paper, we study the related question on longest cycles instead of longest paths (to make the question valid, we require 2‐connectivity). The answer for cycles is also negative and not so much attention has been given to it. Classes for which the answer is known to be positive are dually chordal graphs and 3‐trees. Perhaps the main open class for the question of both paths and cycles is the class of chordal graphs. In this paper, we show that all longest cycles intersect in graphs with treewidth at most three. This class of graphs includes several important subclasses of graphs, such as series‐parallel graphs and ‐free chordal graphs.