Noncompact Sets Research Articles

Monomial asymptotic polynomials and applications to polynomial optimization problems

The problem of polynomial optimization plays an important role in many fields such as physics, chemistry, and economics. This problem has received research attention from many mathematicians recently. In this paper, we study the polynomial optimization problem over a non-compact semi-algebraic set, for which its constraint set of polynomials G is asymptotic with a finite family of monomials. By changing variables via a suitable monomial mapping, we transform the problem under consideration into the polynomial optimization problem over a compact semi-algebraic feasible set. We then apply the well-known result that the optimal value of a polynomial over a compact semi-algebraic set can be approximated as closely as desired by solving a hierarchy of semi-definite programs and the convergence is finite generically, to obtain results in the general case when the cone C(G) is unimodular. In particular, in the case of polynomials in two variables, we solve the problem quite completely without requiring C(G) to be unimodular

Nash’s Existence Theorem for Non-Compact Strategy Sets

In this paper, we apply the classical FKKM lemma to obtain the Ky Fan minimax inequality defined on nonempty non-compact convex subsets in reflexive Banach spaces, and then we apply it to game theory and obtain Nash’s existence theorem for non-compact strategy sets, which can be regarded as a new, simple but interesting application of the FKKM lemma and the Ky Fan minimax inequality, and we can also present another proof about the famous John von Neumann’s existence theorem in two-player zero-sum games. Due to the results of Li, Shi and Chang, the coerciveness in the conclusion can be replaced with the P.S. or G.P.S. conditions.

Bayesian equilibrium: From local to global

We study Bayesian games with a continuum of states which partition into a continuum of components, each of which is common knowledge, such that equilibria exist on each component. A canonical case is when each agent’s information consists of both public and private information, and conditional on each possible public signal, equilibria exist. We show that under some regularity conditions on the partition, measurable Bayesian equilibria exist for the game in its entirety. The results extend to pure equilibria, as well as to non-compact state-dependent action sets, uncommon priors, and non-bounded payoffs; the results also apply to several notions of approximate equilibria.

Metric Mean Dimension of Free Semigroup Actions for Non-Compact Sets

Metric Mean Dimension of Free Semigroup Actions for Non-Compact Sets

Hardy–Sobolev–Rellich, Hardy–Littlewood–Sobolev and Caffarelli–Kohn–Nirenberg Inequalities on General Lie Groups

In this paper, we establish a number of geometrical inequalities such as Hardy, Sobolev, Rellich, Hardy–Littlewood–Sobolev, Caffarelli–Kohn–Nirenberg, Gagliardo-Nirenberg inequalities and their critical versions for an ample class of sub-elliptic differential operators on general connected Lie groups, which include both unimodular and non-unimodular cases in compact and noncompact settings. We also obtain the corresponding uncertainty type principles.

Zeroes of Multifunctions with Noncompact Image Sets

In this article we consider zeroes for multifunctions with possibly noncompact image sets. We introduce the notion of multifunction with K-inf-compact support. We also establish three types of applications: the Bayesian approach for analysis of financial operational risk under certain constraints, occupational health and safety measures optimization, and transfer of space innovations and technologies.

Whitney maps and generalized continua

The use of Whitney maps in hyperspace theory is mostly restricted to the class of compact spaces. However, Whitney himself defined these maps in the class of separable metric spaces.Following the original approach of Whitney, we focus on Whitney properties in absence of compactness. We work within the class of generalized continua, and besides classical Whitney maps we will also consider a special kind of such maps termed compactwise proper Whitney maps.In this paper we observe that connectivity properties which are defined piecewise by continua (for instance, continuunwise connectedness) are Whitney properties also in the non-compact setting. These results can be regarded as extensions to generalized continua of well-known results in classical continuum theory.In contrast, ordinary connectedness fails to be a Whitney property in absence of compactness. Notwithstanding, it is a Whitney property for compactwise proper Whitney maps.A relevant feature of non-compact spaces is the connectivity at infinity measured by Freudenthal ends and strong ends. Here we prove that Freudenthal ends are preserved by compactwise proper Whitney maps but not by ordinary Whitney maps and also that the number of strong ends is not a compactwise Whitney property.

Freezing limits for Calogero–Moser–Sutherland particle models

AbstractOne‐dimensional interacting particle models of Calogero–Moser–Sutherland type with N particles can be regarded as diffusion processes on suitable subsets of like Weyl chambers and alcoves with second‐order differential operators as generators of the transition semigroups, where these operators are singular on the boundaries of the state spaces. The most relevant examples are multivariate Bessel processes and Heckman–Opdam processes in a compact and noncompact setting where in all cases, these processes are related to special functions associated with root systems. More precisely, the transition probabilities can be described with the aid of multivariate Bessel functions, Jack and Heckman–Opdam Jacobi polynomials, and Heckman–Opdam hypergeometric functions, respectively. These models, in particular, form dynamic eigenvalue evolutions of the classical random matrix models like β‐Hermite, β‐ Laguerre, and β‐Jacobi, that is, MANOVA, ensembles. In particular, Dyson's Brownian motions and multivariate Jacobi processes are included. In all cases, the processes depend on so‐called coupling parameters. We review several freezing limit theorems for these diffusions where, for fixed N, one or several of the coupling parameters tend to ∞. In many cases, the limits will be N‐dimensional normal distributions and, in the process case, Gauss processes. However, in some cases, normal distributions on half spaces and distributions related to some other ensembles appear as limits. In all cases, the limits are connected with the zeros of the classical one‐dimensional orthogonal polynomials of order N.

Solutions for zero‐sum two‐player games with noncompact decision sets and unbounded payoffs

AbstractThis article provides sufficient conditions for the existence of solutions for two‐person zero‐sum games with inf/sup‐compact payoff functions and with possibly noncompact decision sets for both players. Payoff functions may be unbounded, and we do not assume any convexity/concavity‐type conditions. For such games expected payoff may not exist for some pairs of strategies. The results of this article imply several classic facts. The article also provides sufficient conditions for the existence of a value and solutions for each player. The results of this article are illustrated with the number guessing game.

Polynomial Entropy of Amenable Group Actions for Noncompact Sets

Polynomial Entropy of Amenable Group Actions for Noncompact Sets

Semi-Algebraic Functions with Non-Compact Critical Set

Let Xsubset {mathbb {R}}^n be a closed semi-algebraic set, F:{mathbb {R}}^nrightarrow {mathbb {R}} be a {mathcal {C}}^2 semi-algebraic function and f=F_{|X}:Xrightarrow {mathbb {R}}^n be the restriction of F to X. We define the global index of a critical value c_i of f and prove an index formula for chi (X) that generalizes a result previously proved by the authors for the case of isolated critical points. We define also new indices at infinity and prove an alternative index formula for chi (X).

The Upper Capacity Topological Entropy of Free Semigroup Actions for Certain Non-compact Sets, II

This paper’s major purpose is to continue the work of Zhu and Ma in [J Stat Phys 182(1):19, 2021]. To begin, the $${\textbf{g}}$$ -almost product property, more general irregular and regular sets, and some new notions of the Banach upper density recurrent points and transitive points of free semigroup actions are introduced. Furthermore, under the $${\textbf{g}}$$ -almost product property and other conditions, we coordinate the Banach upper recurrence, transitivity with (ir)regularity, and obtain lots of generalized multifractal analyses for general observable functions of free semigroup actions. Finally, statistical $$\omega $$ -limit sets are used to consider the upper capacity topological entropy of the sets of Banach upper recurrent points and transitive points of free semigroup actions, respectively. Our analysis generalizes the results obtained by Huang et al. in [Nonlinearity 32(7):2721–2757, 2019] and Pfister and Sullivan in [Ergodic Theory Dynam Syst 27(3):929–956, 2007].

Constant rank theorems for curvature problems via a viscosity approach

An important set of theorems in geometric analysis consists of constant rank theorems for a wide variety of curvature problems. In this paper, for geometric curvature problems in compact and non-compact settings, we provide new proofs which are both elementary and short. Moreover, we employ our method to obtain constant rank theorems for homogeneous and non-homogeneous curvature equations in new geometric settings. One of the essential ingredients for our method is a generalization of a differential inequality in a viscosity sense satisfied by the smallest eigenvalue of a linear map Brendle et al. (Acta Math 219:1–16, 2017) to the one for the subtrace. The viscosity approach provides a concise way to work around the well known technical hurdle that eigenvalues are only Lipschitz in general. This paves the way for a simple induction argument.

Invariant measures in non-conformal fibered systems with singularities

We study a large class of transformations FT which are only piecewise differentiable on countably many pieces and also non-conformal. There exist random countably generated limit sets JT,ω in the fibers of FT. We prove a Global Volume Lemma implying that the push-forward measures of equilibrium measures from a countable shift space, are exact dimensional on the non-compact global basic set JT. A dimension formula of Ledrappier-Young type is obtained for these measures, by using their Lyapunov exponents and marginal entropies. Then, we study the geometric potentials ψT,s on ΣI, and we prove that the dimensions of the push-forward measures νsω in fibers are independent of ω, and they depend real-analytically on the parameter s from an interval F(T). Moreover, we establish a Variational Principle for dimension in fibers. Our results apply in particular to new types of multidimensional continued fractions.

Collocation approximation by deep neural ReLU networks for parametric and stochastic PDEs with lognormal inputs

We find the convergence rates of the collocation approximation by deep ReLU neural networks of solutions to elliptic PDEs with lognormal inputs, parametrized by $\boldsymbol{y}$ in the noncompact set ${\mathbb R}^\infty$. The approximation error is measured in the norm of the Bochner space $L_2({\mathbb R}^\infty, V, \gamma)$, where $\gamma$ is the infinite tensor-product standard Gaussian probability measure on ${\mathbb R}^\infty$ and $V$ is the energy space. We also obtain similar dimension-independent results in the case when the lognormal inputs are parametrized by ${\mathbb R}^M$ of very large dimension $M$, and the approximation error is measured in the $\sqrt{g_M}$-weighted uniform norm of the Bochner space $L_\infty^{\sqrt{g}}({\mathbb R}^M, V)$, where $g_M$ is the density function of the standard Gaussian probability measure on ${\mathbb R}^M$. Bibliography: 62 titles.

Random fixed points, systemic risk and resilience of heterogeneous financial network.

We consider a large random network, in which the performance of a node depends upon that of its neighbours and some external random influence factors. This results in random vector valued fixed-point (FP) equations in large dimensional spaces, and our aim is to study their almost-sure solutions. An underlying directed random graph defines the connections between various components of the FP equations. Existence of an edge between nodes i,j implies the i-th FP equation depends on the j-th component. We consider a special case where any component of the FP equation depends upon an appropriate aggregate of that of the random 'neighbour' components. We obtain finite dimensional limit FP equations in a much smaller dimensional space, whose solutions aid to approximate the solution of FP equations for almost all realizations, as the number of nodes increases. We use Maximum theorem for non-compact sets to prove this convergence.We apply the results to study systemic risk in an example financial network with large number of heterogeneous entities. We utilized the simplified limit system to analyse trends of default probability (probability that an entity fails to clear its liabilities) and expected surplus (expected-revenue after clearing liabilities) with varying degrees of interconnections between two diverse groups. We illustrated the accuracy of the approximation using exhaustive Monte-Carlo simulations.Our approach can be utilized for a variety of financial networks (and others); the developed methodology provides approximate small-dimensional solutions to large-dimensional FP equations that represent the clearing vectors in case of financial networks.

Blow-up versus global existence of solutions for reaction–diffusion equations on classes of Riemannian manifolds

It is well known from the work of Bandle et al. (J Differ Equ 251:2143–2163, 2011) that the Fujita phenomenon for reaction–diffusion evolution equations with power nonlinearities does not occur on the hyperbolic space {mathbb {H}}^N, thus marking a striking difference with the Euclidean situation. We show that, on classes of manifolds in which the bottom Lambda of the L^2 spectrum of -Delta is strictly positive (the hyperbolic space being thus included), a different version of the Fujita phenomenon occurs for other kinds of nonlinearities, in which the role of the critical Fujita exponent in the Euclidean case is taken by Lambda . Such nonlinearities are time-independent, in contrast to the ones studied in Bandle et al. (2011). As a consequence of our results we show that, on a class of manifolds much larger than the case M={mathbb {H}}^N considered in Bandle et al. (2011), solutions to (1.1) with power nonlinearity f(u)=u^p, p>1, and corresponding to sufficiently small data, are global in time. Though qualitative similarities with similar problems in bounded, Euclidean domains can be seen in the results, the methods are significantly different because of noncompact setting dealt with.

Some Inequalities of Hardy Type Related to Witten–Laplace Operator on Smooth Metric Measure Spaces

A complete Riemannian manifold equipped with some potential function and an invariant conformal measure is referred to as a complete smooth metric measure space. This paper generalizes some integral inequalities of the Hardy type to the setting of a complete non-compact smooth metric measure space without any geometric constraint on the potential function. The adopted approach highlights some criteria for a smooth metric measure space to admit Hardy inequalities related to Witten and Witten p-Laplace operators. The results in this paper complement in several aspect to those obtained recently in the non-compact setting.

Witten deformation on non-compact manifolds: heat kernel expansion and local index theorem

Asymptotic expansions of heat kernels and heat traces of Schrödinger operators on non-compact spaces are rarely explored, and even for cases as simple as {mathbb {C}}^n with (quasi-homogeneous) polynomials potentials, it’s already very complicated. Motivated by path integral formulation of the heat kernel, we introduced a parabolic distance, which also appeared in Li–Yau’s famous work on parabolic Harnack estimate. With the help of the parabolic distance, we derive a pointwise asymptotic expansion of the heat kernel for the Witten Laplacian with strong remainder estimate. When the deformation parameter of Witten deformation and time parameter are coupled, we derive an asymptotic expansion of trace of heat kernel for small-time t, and obtain a local index theorem. This is the second of our papers in understanding Landau–Ginzburg B-models on nontrivial spaces, and in subsequent work, we will develop the Ray–Singer torsion for Witten deformation in the non-compact setting.

Open/closed correspondence via relative/local correspondence

We establish a correspondence between the disk invariants of a smooth toric Calabi-Yau 3-fold X with boundary condition specified by a framed Aganagic-Vafa outer brane (L,f) and the genus-zero closed Gromov-Witten invariants of a smooth toric Calabi-Yau 4-fold X˜, proving the open/closed correspondence proposed by Mayr and developed by Lerche-Mayr. Our correspondence is the composition of two intermediate steps:•First, a correspondence between the disk invariants of (X,L,f) and the genus-zero maximally-tangent relative Gromov-Witten invariants of a relative Calabi-Yau 3-fold (Y,D), where Y is a toric partial compactification of X by adding a smooth toric divisor D. This correspondence can be obtained as a consequence of the topological vertex (Li-Liu-Liu-Zhou) and Fang-Liu where the all-genus open Gromov-Witten invariants of (X,L,f) are identified with the formal relative Gromov-Witten invariants of the formal completion of (Y,D) along the toric 1-skeleton. Here, we present a proof without resorting to formal geometry.•Second, a correspondence in genus zero between the maximally-tangent relative Gromov-Witten invariants of (Y,D) and the closed Gromov-Witten invariants of the toric Calabi-Yau 4-fold X˜=OY(−D). This can be viewed as an instantiation of the log-local principle of van Garrel-Graber-Ruddat in the non-compact setting.