Nonlinear potential theory in metric spaces

CHAPTER II - DIMENSION OF METRIC SPACES

Copula Theory and Its Applications

Purpose of the article: development of a methodology for the application of methods for analyzing big data based on topological constructions in relation to behavioral analytics systems to ensure corporate and cyber-physical security. Method: the technique is based on the algebraic theory of persistent homology. Along with algebraic topology, embedology (Takens-Mane embedding theory) and the theory of metric spaces are used. Result: the necessary concepts of algebraic topology are given, which underlie the analysis of user / entity behavior profiles: Vietoris-Rips simplicial complex, filtering by a set of cloud points, homology groups, persistence modules, topological characteristics and dependencies. At the first stage of the technique, the time series that describe the time-varying behavior of the user / entity are transformed into a cloud of points in the topological space. For this transformation, the methods of the Takens-Mane embedding theory and the algorithm of the method of false neighbors are used. At the subsequent stages of the methodology for the base and current point clouds, topological dependencies, diagrams (persistence, bar codes) characterizing the base and current behavior profiles, respectively, are built. At the final stage, the deviation of the current behavior profile from the baseline is revealed. To estimate the deviation, the Wasserstein, Chebyshev, bottleneck metrics and scaling based on the generalized Harrington desirability function are used. The results of practical testing of the proposed method of applying topological algorithms to the data of the monitoring system for the work of corporate network users with information resources are presented

Functional specifications have been used to specify and verify designs of a number of reactive, discrete systems. In this paper we extend this specification style to deal with real-time and hybrid systems. As mathematical foundation we employ Banach's fixed point theory in metric spaces. The goal is to show that the theory used for discrete functional specifications smoothly carries over to real-time and hybrid systems. An example of a thermostat specification illustrates the method.

The Euler characteristic is an invariant of a topological space that in a precise sense captures its canonical notion of size, akin to the cardinality of a set. The Euler characteristic is closely related to the homology of a space, as it can be expressed as the alternating sum of its Betti numbers, whenever the sum is well-defined. Thus, one says that homology categorifies the Euler characteristic. In his work on the generalisation of cardinality-like invariants, Leinster introduced the magnitude of a metric space, a real number that gives the number of of the space. Recently, Leinster and Shulman introduced a homology theory for metric spaces, called magnitude homology, which categorifies the magnitude of a space. When studying a metric space, one is often only interested in the metric space up to a rescaling of the distance of the points by a non-negative real number. The magnitude function describes how the effective number of points changes as one scales the distance, and it is completely encoded in the Euler characteristic of magnitude homology. When studying a finite metric space in topological data analysis using persistent homology, one approximates the space through a nested sequence of simplicial complexes so as to recover topological information about the space by studying the homology of this sequence. Here we relate magnitude homology and persistent homology as two different ways of computing homology of filtered simplicial sets.

The theory of metric spaces applied to infinitely differentiable functions

The theory of metric spaces is a convenient and very powerful way of examining the behavior of numerous mathematical models. In a previous paper, a new operation between functions on a compact real interval called fractal convolution has been introduced. The construction was done in the framework of iterated function systems and fractal theory. In this article we extract the main features of this association, and consider binary operations in metric spaces satisfying properties as idempotency and inequalities related to the distance between operated elements with the same right or left factor (side inequalities). Important examples are the logical disjunction and conjunction in the set of integers modulo 2 and the union of compact sets, besides the aforementioned fractal convolution. The operations described are called in the present paper convolutions of two elements of a metric space E. We deduce several properties of these associations, coming from the considered initial conditions. Thereafter, we define self-operators (maps) on E by using the convolution with a fixed component. When E is a Banach or Hilbert space, we add some hypotheses inspired in the fractal convolution of maps, and construct in this way convolved Schauder and Riesz bases, Bessel sequences and frames for the space.

In the previous chapter we studied a single metric space (X, d), and the various types of sets one could find in that space. While this is already quite a rich subject, the theory of metric spaces becomes even richer, and of more importance to analysis, when one considers not just a single metric space, but rather pairs \((X,d_X)\) and \((Y, d_Y)\) of metric spaces, as well as continuous functions \(f:X \rightarrow Y\) between such spaces.

This paper explores the applications of Fixed-point theory in metric spaces, focusing on the role of real numbers in matrix spaces and iterative methods. Using Banach's Contraction Principle, the study provides mathematical proofs to establish the existence and uniqueness of fixed points, while also analyzing the convergence and stability of iterative methods. In a complete metric space (𝑋, 𝑑), a contraction mapping 𝑓: 𝑋 → 𝑋 has a unique fixed point 𝑥∗ such that 𝑓 (𝑥∗) = 𝑥∗ and 𝑑 (𝑓 (𝑥), 𝑓 (𝑦)) ≤ 𝛼 𝑑 (𝑥, 𝑦), where 0 ≤ 𝛼 < 1. The Intermediate Value Theorem ensures that for a continuous function 𝑓on [𝑎, 𝑏] with 𝑓(𝑎) ≤ 𝐿 ≤ 𝑓(𝑏), there is a point 𝑐 ∈ [𝑎, 𝑏] such that 𝑓(𝑐) = 𝐿. The Mean Value Theorem states that for a differentiable function 𝑓 on (𝑎,), there exists a point 𝑐 ∈ (𝑎, 𝑏) such that 𝑓′(𝑐)=𝑓(𝑏) – 𝑓(𝑎) / 𝑏−𝑎. The Cauchy-Schwarz inequality for vectors 𝑢and 𝑣 gives ∣⟨𝑢, 𝑣⟩∣ ≤ ∥𝑢∥ ∥𝑣∥. For the function (𝑥) = ½ (𝑥+3), the fixed point is 𝑥∗ = 3. In matrix space, the operator (𝐴) = ½ 𝐴 has a fixed point 𝐴 = 0. For 𝑥 = cos (𝑥), the successive approximations method converges to 𝑥∗ ≈ 0.7391, and the Newton-Raphson method achieves faster convergence to the same fixed point.This research highlights the practical applications of fixed-point theory, supported by real numbers, as a foundation for accurate mathematical models. It enhances outcome prediction in optimization, stability analysis, and system modeling, emphasizing the critical role of real numbers in both theoretical and applied mathematics.

We present a framework for defining abstract interpreters for liveness properties, in particular program termination. The framework makes use of the theory of metric spaces to define a concrete semantics, relates this semantics with the usual order-theoretic semantics of abstract interpretation, and identifies a set of conditions for determining when an abstract interpreter is sound for analysing liveness properties. Our soundness proof of the framework is based on a novel relationship between unique fixpoints in metric semantics and post-fixpoints computed by abstract interpreters. We illustrate the power of the framework by providing an instance that can automatically prove the termination of programs with general (not necessarily tail) recursion.

Magnitude homology is an $\mathbf{R}^+$-graded homology theory of metric spaces that captures information on the complexity of geodesics. Here we address the question: when are two metric spaces magnitude homology equivalent, in the sense that there exist back-and-forth maps inducing mutually inverse maps in homology? We give a concrete geometric necessary and sufficient condition in the case of closed Euclidean sets. Along the way, we introduce the convex-geometric concepts of inner boundary and core, and prove a strengthening for closed convex sets of the classical theorem of Carathéodory.

Many topological spaces are simply sets of points (atoms) endowed with a topology. Some spaces, however, have elements that are functions, matrices, or other non-atomic items. Another special type of space has elements that are themselves subsets of another space; these spaces are called hyperspaces. Hyperspaces are metric spaces, and the metric defined on them is called the Hausdorff metric. Hyperspaces wllose points are the closed subsets and hyperspaces whose points are the closed connected subsets (of metric spaces) have been extensively studied. Also, hyperspaces of closed and convex subsets of a bounded convex set in Euclidean space are of great interest in geometry. See Lay [3]. In recent years, geometers and topological dynamicists have explored spaces of closed and bounded subsets of the plane in connection with the study of fractals. One of the major results in the theory is that the hyperspace of closed subsets of a closed interval of real numbers is homeomorphic with the Hilbert cube, and with the space IX, the countable product of unit intervals. For more general information about the Hausdorff metric and spaces of subsets, see Devaney [2] and Sieradski [6]. The main result of this paper is a topological characterization of the space of closed subsets of a convergent sequence of points. The proof given here provides a homneornorphic embedding of the space in the plane, E2. The result was first proved by Pelczynski [5]. He studied arbitrary compact zero-dimensional metric spaces, so his proofs are much more widely applicable, but they are also somewhat technical. Our proof depends only on some well-known results in the theory of metric spaces, and is therefore accessible to advanced undergraduate mathematics majors. We also prove that for no convergent sequence of real numbers is there an isometric embedding of the hyperspace in Euclidean space, E,1, for any n. For other results and discussion, see Nadler [4]. Let (X, d) denote a compact metric space. The hyperspace (2X, D) of X is the metric space whose points are the closed nonempty subsets of X and whose metric is the Hausdorff metric D given by

Structures in circumstellar matter reflect both fast processes and quasi-equilibrium states. A geometrical diversity of emitting circumstellar matter is observed around evolved massive stars, in particular around B[e] supergiants. We recapitulate classical analytical tools of linear and nonlinear potential theory, such as Cole–Hopf transformation and Grad–Shafranov theory, and develop them further to explain the occurrence of the circumstellar matter structures and their dynamics. We use potential theory to formulate the nonlinear hydrodynamical equations and test dilatations of the quasi-equilibrium initial conditions. We find that a wide range of flow patterns can basically be generated and the timescales can switch, based on initial conditions, and lead to eruptive processes, reinforcing that the nonlinear fluid environment includes both quasi-stationary structures and fast processes like finite-time singularities. Some constraints and imposed symmetries can lead to Keplerian orbits, while other constraints can deliver quasi-Keplerian ones. The threshold is given by a characteristic density at the stellar surface.

The classification theory of Riemann surfaces is generalized to Riemanniann-manifolds in the conformally invariant case. This leads to the study of the existence ofA-harmonic functions of typen with various properties and to an extension of the definition of the classical notions with inclusionsOG⊂OHP⊂OHB⊂OHD. In the classical case the properness of the inclusions were proved rather late, in the 50's by Ahlfors and Toki. Our main objective is to show that such inclusions are proper also in the generalized case.

Originally, harmonic morphisms were defined as continuous mappings φ:X → X' between harmonic spaces such that h'∘φ remains harmonic whenever h' is harmonic, see [1], p. 20. In general linear axiomatic potential theory, one has to replace harmonic function