Compactness In Spaces Research Articles

BPS/CFT Correspondence III: Gauge Origami Partition Function and qq-Characters

We study generalized gauge theories engineered by taking the low energy limit of the $Dp$ branes wrapping $X \times T^{p-3}$, with $X$ a possibly singular surface in a Calabi-Yau fourfold $Z$. For toric $Z$ and $X$ the partition function can be computed by localization, making it a statistical mechanical model, called the gauge origami. The random variables are the ensembles of Young diagrams. The building block of the gauge origami is associated with a tetrahedron, whose edges are colored by vector spaces. We show the properly normalized partition function is an entire function of the Coulomb moduli, for generic values of the $\Omega$-background parameters. The orbifold version of the theory defines the $qq$-character operators, with and without the surface defects. The analytic properties are the consequence of a relative compactness of the moduli spaces $M({\vec n}, k)$ of crossed and spiked instantons, demonstrated in arXiv:1608.07272.

Klein-Gordon-Schrödinger system: Dinucleon field

In this paper, we consider the 3-dimensional Klein-Gordon-Schrödinger system under the dinucleon interactions. By introducing the atomic spaces and establishing local Strichartz estimates for the perturbed Schrödinger equation to overcome the lack of compactness in whole space, we prove the unique solvability in the energy space. Additionally, we derive a Lipschitz estimate in a weak topology for the linearized equations to obtain the continuous dependence on the initial data.

Positive Solutions in the Space of Lipschitz Functions for Fractional Boundary Value Problems with Integral Boundary Conditions

In this paper, we study the existence of positive solutions for a class of nonlinear fractional boundary value problems with integral boundary conditions in Holder spaces. Our analysis relies on a sufficient condition for the relative compactness in Holder spaces and the classical Schauder fixed point theorem.

Existence of suitable weak solutions to the Navier–Stokes equations in time varying domains

We consider suitable weak solutions to the Navier–Stokes equations in time varying domains. We develop Schauder theory for the approximate Stokes equations in time varying domains whose solutions satisfy a uniform localized energy estimate including boundary. Existence of suitable weak solutions in time varying domains follows from compactness in Lebesgue and Sobolev spaces.

Existence of positive solutions in the space of Lipschitz functions to a class of fractional differential equations of arbitrary order

In this paper, we study sufficient conditions for the existence of positive solutions to a class of fractional differential equations of arbitrary order. Our solutions are placed in the space of Lipschitz functions and, perhaps, this is a part of the originality of the paper. For our study, we use a recent result about the relative compactness in Holder spaces and the classical Schauder fixed point theorem.

Tightness of Sums of Independent Identically Distributed Pseudo-Poisson Processes in the Skorokhod Space

We consider a pseudo-Poisson process of the following simple type. This process is a Poissonian subordinator for a sequence of i.i.d. random variables with finite variance. Further we consider sums of i.i.d. copies of a pseudo-Poisson process. For a family of distributions of these random sums, we prove the tightness (relative compactness) in the Skorokhod space. Under the conditions of the Central Limit Theorem for vectors, we establish the weak convergence in the functional Skorokhod space of the examined sums to the Ornstein–Uhlenbeck process.

An Integrated Framework for Allocation of New Emergency Response Facilities in Developing Nations

Developing countries are majorly resource-constrained societies which do not have the financial power to invest in meeting all infrastructure needs. Although there are multiple types of emergency response systems (ERS), the most important ones are the basic essentials—fire, medical and police. Along with emergency response planning for man-made emergencies in an urban scenario, there is a gap in the recent literature about integration of basic ERS into a single framework. We propose a novel framework that first evaluates the existing ERS in place and then provides solutions for improvement with the inclusion of additional facilities at optimal locations. Keeping in mind dynamic congestion patterns and land use resource constraints in different areas of the urban space, the proposed facilities are labeled as either permanent or floating. We also propose three different schemes, each with two competing strategies that cater to a specific type of constraint common to developing nations. Through comparison of system performance metrics like first response time, area coverage, and average response time, the ERS pre- and post-implementation of the framework are compared to check for degree of improvement. A case study of the framework is shown using the city of Chandigarh in India as an example. The framework proposed in the paper takes into account the growth and compactness of urban spaces and can be used to continually assess and improve ERS in mixed use urban spaces after fixed periods of time.

A fixed-point characterization of weak compactness in Banach spaces with unconditional Schauder basis

Let X be a Banach space with an 1-unconditional Schauder basis and without isomorphic copies of ℓ1⊕c0. We obtain an equivalent condition to weak compactness by means of a fixed-point theorem. Namely: a closed convex bounded subset C of X is weakly compact if and only if every cascading nonexpansive mapping T:C→C has a fixed point. We particularize our results when C is the closed unit ball of the Banach space X, obtaining a new characterization of reflexivity. Note that weak compactness is independent of the underlying equivalent norm and that every Banach space with an unconditional Schauder basis can be renormed to be 1-unconditional.

On arithmetic continuity in metric spaces

The main purpose of this article is to introduce the concept of arithmetic continuity and arithmetic compactness in metric spaces and prove some interesting results related to these notions.

Pre-g-bi-irresolute and pre-g-stable in ditopological texture spaces

We introduce and study new notions of continuity, compactness and stability in ditopological texture spaces based on the notions of pre-g-open and pre-g-closed sets and some of their characterizations are obtained.

Stability in locally L0-convex modules and a conditional version of James' compactness theorem

Locally L0-convex modules were introduced in Filipovic et al. (2009) [10] as the analytic basis for the study of conditional risk measures. Later, the algebra of conditional sets was introduced in Drapeau et al. (2016) [8]. In this paper we study locally L0-convex modules, and find exactly which subclass of locally L0-convex modules can be identified with the class of locally convex vector spaces within the context of conditional set theory. Second, we provide a version of the classical James' theorem of characterization of weak compactness for conditional Banach spaces. Finally, we state a conditional version of the Fatou and Lebesgue properties for conditional convex risk measures and, as application of the developed theory, we establish a version of the so-called Jouini–Schachermayer–Touzi theorem for robust representation of conditional convex risk measures defined on a L∞-type module.

On Fuzzy Rough Compactness

The notion of rough topological space has been well defined and studied by a number of researchers. Connectedness and compactness of fuzzy soft topological spaces have also been studied by some authors. In this paper we introduce the notion of compactness in fuzzy rough sets and establish some interesting theorems.

Light side of compactness in Lebesgue spaces: Sudakov theorem

Light side of compactness in Lebesgue spaces: Sudakov theorem

Walrasian economy and some properties of convexly compact sets

Abstract G. Žitković defined the notion of a convexly compact set in a topological space and, among other things, used it to give an extension of the Walrasian excess-demand theorem. We continue the study of convexly compactness in LCS spaces and prove a Krein-Milman theorem in this setting.

Fuzzy Almost e*-continuous Mappings and Fuzzy e*-compactness in Smooth Topological Spaces

In this paper, we introduce the concept of fuzzy almost * e -continuous, fuzzy weakly * e -continuous, r -fuzzy * e -compact, r -fuzzy almost * e -compactness and r - fuzzy near * e -compactness in S ⌣ ostak’s fuzzy topological spaces. We study some properties of them under several types of fuzzy * e -continuous mappings and fuzzy * e - regular spaces.

SPACE OF RICCI FLOWS (II)—PART A: MODULI OF SINGULAR CALABI–YAU SPACES

We establish the compactness of the moduli space of noncollapsed Calabi–Yau spaces with mild singularities. Based on this compactness result, we develop a new approach to study the weak compactness of Riemannian manifolds.

INTERVAL-VALUED INTUITIONISTIC SMOOTH TOPOLOGICAL SPACES

In this paper, we introduce the concepts of several types of interval-valued intuitionistic fuzzy mappings and several types of interval-valued intuitionistic fuzzy compactness in interval-valued intuitionistic smooth topological spaces and then investigate their properties.

Compactness of Fuzzy Cone Metric Spaces

Cone metric spaces may be considered as a generalization of determinist metric spaces or fuzzy metric spaces. In this paper, we will construct fuzzy cone metric space using different approaches, which is based on the definition of fuzzy points, and then study different types of compact subsets of such spaces, as well as, the relationship among them.

Characterizations of compact sets in fuzzy set spaces with Lp metric

Compactness criteria in fuzzy set spaces endowed with the Lp metric have been studied for several decades. Total boundedness is a key feature of compactness in metric spaces. However, comparing existing compactness criteria in fuzzy set spaces endowed with the Lp metric with the Arzelà–Ascoli theorem, the latter gives compactness criteria by characterizing totally bounded sets while the former does not characterize totally bounded sets. Currently, compactness criteria are only presented for three particular fuzzy set spaces under assumptions of convexity or star-shapedness. General fuzzy sets have become more important in both theory and applications. Therefore, this paper presents characterizations of totally bounded sets, relatively compact sets, and compact sets in general fuzzy set spaces equipped with the Lp metric, but which do not have any assumptions of convexity or star-shapedness. Subsets of these general sets include common fuzzy sets, such as fuzzy numbers, fuzzy star-shaped numbers with respect to the origin, fuzzy star-shaped numbers, and general fuzzy star-shaped numbers. Existing compactness criteria are stated for fuzzy numbers space, the space of fuzzy star-shaped numbers with respect to the origin, and the space of fuzzy star-shaped numbers endowed with the Lp metric, respectively. Constructing completions of fuzzy set spaces with respect to the Lp metric is a problem closely dependent on characterizing totally bounded sets. Based on characterizations of total boundedness and relatively compactness and some discussion of the convexity and star-shapedness of fuzzy sets, we show that the completions of fuzzy set spaces studied here can be obtained using the Lp extension. We also clarify relationships among the ten fuzzy set spaces studied here—the five pairs of original spaces and their corresponding completions. We show that the subspaces have parallel characterizations of totally bounded sets, relatively compact sets, and compact sets. Finally, we discuss properties of the Lp metric on fuzzy set space as an application of our results, and review compactness criteria proposed in previous work.

Calculus of directional coderivatives and normal cones in Asplund spaces

We study the directional Mordukhovich normal cones to nonsmooth sets, coderivatives of set-valued mappings in Asplund spaces and establish extensive calculus results on these constructions under various operations of sets and mappings. We also develop calculus of the directional sequential normal compactness both in general Banach spaces and in Asplund spaces.