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In set theory without the Axiom of Choice we prove that the assertion "For every metric space (X, d) with a Borel measure μ such that the measure of every open ball is positive and finite, (X, d) is separable.' is implied by the axiom of choice for countable collections of sets and implies the axiom of choice for countable collections of finite sets. We also show that neither implication is reversible in Zermelo–Fraenkel set theory weakend to permit the existence of atoms and that the second implication is not reversible in Zermelo–Fraenkel set theory. This gives an answer to a question of Dybowski and Górka (Arch Math Logic 62:735–749, 2023. https://doi.org/10.1007/s00153-023-00868-4). [ABSTRACT FROM AUTHOR]

The analogue of polar coordinates in the Euclidean space, a polar decomposition in a metric space, if well-defined, can be very useful in dealing with integrals with respect to a sufficiently regular measure. In this note we handle the technical details associated with such polar decompositions. Supplementary Information: The online version contains the Armenian language version of the article available at https://doi.org/10.1007/s10958-023-06674-w. [ABSTRACT FROM AUTHOR]

By using a priori estimates and the Riesz theorem, we investigate a boundary-value problem for nonuniformly 2b -elliptic equations with arbitrary power singularities in the coefficients of the equation and boundary conditions on a certain set of points. We establish the existence and obtain an integral representation of the unique solution of the formulated boundary-value problem in Hölder spaces with power weights whose order is determined via the orders of singularities in the coefficients of the equation and boundary conditions. [ABSTRACT FROM AUTHOR]

In the paper, we introduce the concepts of topological stability, expansivity and pseudo-orbit tracing property of Borel measures with respect to a given continuous map on a compact topological space from the viewpoint of open covers and prove that every expansive Borel measure with the pseudo-orbit tracing property is topologically stable. Furthermore, we obtain some properties of topologically stable measures. [ABSTRACT FROM AUTHOR]

We show that the Axiom of Countable Choice is necessary and sufficient to prove that the existence of a Borel measure on a pseudometric space such that the measure of open balls is positive and finite implies separability of the space. In this way a negative answer to an open problem formulated in Górka (Am Math Mon 128:84–86, 2020) is given. Moreover, we study existence of maximal δ -separated sets in metric and pseudometric spaces from the point of view the Axiom of Choice and its weaker forms. [ABSTRACT FROM AUTHOR]

In this article, we consider nondiffusive variational problems with mixed boundary conditions and (distributional and weak) gradient constraints. The upper bound in the constraint is either a function or a Borel measure, leading to the state space being a Sobolev one or the space of functions of bounded variation. We address existence and uniqueness of the model under low regularity assumptions, and rigorously identify its Fenchel pre-dual problem. The latter in some cases is posed on a nonstandard space of Borel measures with square integrable divergences. We also establish existence and uniqueness of solution to this pre-dual problem under some assumptions. We conclude the article by introducing a mixed finite-element method to solve the primal-dual system. The numerical examples illustrate the theoretical findings.

The regular open subsets of a topological space form a Boolean algebra, where the join of two regular open sets is the interior of the closure of their union. A content is a finitely additive measure on this Boolean algebra, or on one of its subalgebras. We develop a theory of integration for such contents. We then explain the relationship between contents, residual charges, and Borel measures. We show that a content can be represented by a normal Borel measure, augmented with a liminal structure, which specifies how two or more regular open sets share the measure of their common boundary. In particular, a content on a locally compact Hausdorff space can be represented by a normal Borel measure and a liminal structure on the Stone-Cech compactification of that space. We also show how contents can be represented by Borel measures on the Stone space of the underlying Boolean algebra of regular open sets. [ABSTRACT FROM AUTHOR]

We consider Borel measures on separable Banach spaces that are limits of their finite-dimensional images in the weak topology. The class of Banach spaces on which all measures have this property is introduced. The specified property is proved for all measures from the closure in variation of the linear span of the set of measures absolutely continuous with respect to Gaussian measures. Connections with the approximation property and the stochastic approximation property are considered. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notions of topological stability, shadowing, and weak shadowing properties for actions of some finitely generated groups on non-compact metric spaces which are dynamical properties and equivalent to the classical definitions in case of compact metric spaces. Also, we extend Walter's stability theorem to group actions on locally compact metric spaces. We show that action φ : F 2 × M → M of free group F2 =< a, b > on compact manifold M with dim(M) ≥ 2 has shadowing property if and only if it has weak shadowing property. This implies that if φ : F 2 × M → M is topologically stable, then it has shadowing property. Finally, we introduce a measure μ ∈ M (X) that is compatible with the weak shadowing property for the continuous action φ : G × X → X, denoted by μ ∈ M (X , φ) . We show that if φ can be approximated by μ-transitive actions for some μ ∈ M (X , φ) , then Ω(φ) = X. We give an example to show that it may be M (X , φ) ≠ ∅ while φ does not have weak shadowing property but we show that if there is a strictly measure μ ∈ M (X , φ) , then continuous action φ of G on compact metric space X has weak shadowing property. Also, we show that if G is a finitely generated abelian group and M (X , φ) ≠ ∅ , then closure of M (X , φ) has an φ-invariant measure. [ABSTRACT FROM AUTHOR]

Using the notion of topologically stable points, it is proved that every equicontinuous pointwise topologically stable homeomorphism of a compact metric space is persistent. Also, using the notion of strong topologically stable points of a Borel probability measure, it is shown that every pointwise strong topologically stable Borel probability measure with respect to an equicontinuous homeomorphism of a compact metric space is strong persistent. Further, it is established that any homeomorphism of [ 0 , 1 ] as well as that of (0,1) does not admit any uniformly expansive point. Finally, these results are used to show that the unit circle does not admit any expansive homeomorphism. [ABSTRACT FROM AUTHOR]

The meaning of weak solutions of a nonconservative hyperbolic system with discontinuous coefficients modeling traffic flows involving hysteresis is defined. An upwinding approximation scheme for the model is shown to be total variation diminishing. Vehicles' speeds and drivers' hysteresis states satisfy maximum and minimum principles. The limit of a convergent subsequence generated by the approximation scheme is shown to be a weak solution of the model if it is piecewise C1, establishing the existence of such solutions. [ABSTRACT FROM AUTHOR]

Dynamical high-dimensional quantum states can be tracked and manipulated in many cases. Using a new theoretical framework approach of manipulating quantum systems, we will show how one can manipulate and introduce parameters that allow tracking and descriptive insight in the dynamics of states. Using quantum topology and other novel mathematical representations, we will show how quantum states behave in critical points when the shift of probability distribution introduces changes. [ABSTRACT FROM AUTHOR]

Existence of expansivity for group action φ : G × X → X depends on algebraic properties of G and the topology of X. We give an expansive action of a solvable group on S 1 while there is no expansive action of a solvable group on a dendrite X. We prove that a continuous action φ : G × X → X on a compact metric space X is expansive if and only if there exists an open cover 𝒰 such that for any other open cover 𝒱 , ∧ g ∈ F φ (g − 1 , 𝒰) ≺ 𝒱 , for some finite set F ⊆ G. In this paper, we introduce the notion of topological expansivity of a group action φ : G × X → X on a T 1 -paracompact space X. If a T 1 -paracompact space X admits topologically expansive action, then X is Hausdorff space. We also show that a continuous action φ : G × X → X of a finitely generated group G on a compact Hausdorff uniform space (X , 𝒰) is expansive with an expansive neighborhood D ∈ 𝒰 if and only if for every E ∈ 𝒰 there is an entourage N ∈ 𝒰 such that for every two N -pseudo orbit F 0 , F 1 : G → X if (F 0 (g) , F 1 (g)) ∈ D for all g ∈ G , then (F 0 (g) , F 1 (g)) ∈ E for all g ∈ G. Finally, we introduce measure 1 -expansive actions on a uniform space. The set of all 1 -expansive measures with common expansive neighborhood for a group action φ : G × X → X is a convex, closed and φ ⋆ -invariant subset of the set of all Borel probability measures on X. Also, we show that a group action φ : G × X → X is expansive if all Borel probability measures are 1 -expansive or all Dirac measures m z , z ∈ X , have a common expansive neighborhood. [ABSTRACT FROM AUTHOR]

Let Ω⊆RN be either a bounded domain or the entire space. This paper concerns existence and global pointwise estimates in terms of Wolff potentials of solutions to quasilinear and Hessian equations of Lane‐Emden type: −Δpu=σP(u)+ω,Fk[−u]=σP(u)+ωunder minimal assumptions on σ and ω, where σ is an A∞ weight and ω is a nonnegative Borel measure, and P(u)∼expγuβ with γ>0, β ≥ 1. Our methods are based on systematic application of Wolff potentials and good‐λ inequality for fractional maximal potential. [ABSTRACT FROM AUTHOR]

Some questions are discussed concerning the existence of nonzero σ-finite invariant Borel measures in infinite-dimensional Polish vector spaces. [ABSTRACT FROM AUTHOR]

We obtain a sharp lower bound estimate for the approximation error of a continuous function by single hidden layer neural networks with a continuous activation function and weights varying on two fixed directions. We show that for a certain class of activation functions this lower bound estimate turns into equality. The obtained result provides us with a method for direct computation of the approximation error. As an application, we give a formula, which can be used to compute instantly the approximation error for a class of functions having second order partial derivatives. [ABSTRACT FROM AUTHOR]

Rules for calculating the densities of Borel measures which are absolutely continuous with respect to a positive nonatomic Radon measure are considered. The Borel measures are generated by composite functions which depend on continuous functions of bounded variation defined on an interval. Questions related to the absolute continuity of Borel measures generated by composite functions with respect to a positive Radon measure and rules for calculating the densities of Borel measures generated by composite functions with respect to a positive nonatomic Radon measure are studied. [ABSTRACT FROM AUTHOR]

In the present paper we determine, in terms of characters and additive functions, the solutions of the integral functional equation for the sine addition law ∫G f(xyt)dμ(t) = f (x)g(y) + g(x)f (y), x,y ∈ G, where G is a locally compact Hausdorff group and μ is a regular, compactly supported, complex-valued Borel measure on G. Some consequences of this result and an example are presented. [ABSTRACT FROM AUTHOR]

Explicit expressions and computational approaches are given for the Fortet–Mourier distance between a positively weighted sum of Dirac measures on a metric space and a positive finite Borel measure. Explicit expressions are given for the distance to a single Dirac measure. For the case of a sum of several Dirac measures one needs to resort to a computational approach. In particular, two algorithms are given to compute the Fortet–Mourier norm of a molecular measure, i.e. a finite weighted sum of Dirac measures. It is discussed how one of these can be modified to allow computation of the dual bounded Lipschitz (or Dudley) norm of such measures. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to provide the different types of Hardy-Littlewood Maximal Functions, the relationship between them and the corresponding extension of ℝn of the Hardy-Littlewood maximal function. We also give the generalization and the modification of Hardy-Littlewood maximal function.