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Given a compact space K, we denote by P(K) the space of all Radon probability measures on K, equipped with the weak* topology inherited from C(K)*. For nonmetrizable compacta K even basic properties of P(K) spaces depend of additional axioms of set theory. We discuss here older and quite recent results on the subject. [ABSTRACT FROM AUTHOR]

In this paper, we develop the Littman–Stampacchia–Weinberger duality approach to obtain global W 1 , p estimates for a class of elliptic problems involving Leray–Hardy operators and measure sources in a distributional framework associated to a dual formulation with a specific weight function. [ABSTRACT FROM AUTHOR]

Given a positive Borel measure μ on the one-dimensional Euclidean space R , consider the centered Hardy–Littlewood maximal function M μ acting on a finite positive Borel measure ν by M μ ν (x) : = sup r > r 0 (x) ν (B (x , r)) μ (B (x , r)) , x ∈ R , where r 0 (x) = inf { r > 0 : μ (B (x , r)) > 0 } and B(x, r) denotes the closed ball with centre x and radius r > 0 . In this note, we restrict our attention to Radon measures μ on the positive real line [ 0 , + ∞) . We provide a complete characterization of measures having weak-type asymptotic properties for the centered maximal function. Although we don’t know whether this fact can be extended to measures on the entire real line R , we examine some criteria for the existence of the weak-type asymptotic properties for M μ on R . We also discuss further properties, and compute the value of the relevant asymptotic quantity for several examples of measures. [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]

In this paper, we compare the Stone–Čech compactification of the space of Radon probability measures on a Tychonoff space , equipped with the weak topology, with the space of Radon probability measures on the Stone–Čech compactification of the space . It is shown that for any noncompact metric space , the compactification does not coincide with . We discuss the case of more general Tychonoff spaces and also the case of the Samuel compactification, for which the coincidence holds. [ABSTRACT FROM AUTHOR]

This paper is devoted to the invariance of time dependent sets under dynamical systems defined on a metric space. In the absence of vector structure, evolutions are described by the so-called mutational inclusions, that extend differential inclusions of the classical Euclidean framework to the one of general metric spaces. The main difficulty we have to overcome is the absence of local compactness of the constraints implying that the distance between a point and a closed set, in general, is not realized. The fairy technical proof we propose has a potential to be applicable also to problems of invariance of closed sets under various evolution laws. The reason to seek such generality lies in the desire to have universal results that can be applied to various settings, as for instance to classical differential inclusions, to continuity inclusions in the Wasserstein spaces, the controlled transport equation in the space of Radon measures or morphological systems in the space of closed subsets of a Banach space. The obtained results are illustrated by the example of dynamics described by a non-homogeneous controlled continuity equation on the space of Radon measures. [ABSTRACT FROM AUTHOR]

In this article, a simple linear method is established for designing optimal revolving objects with desired physical properties. First, the problem is converted to an optimal control problem by defining an artificial control related to an unknown generator curve. Next, considering a variational presentation and applying an embedding procedure, the problem is transferred into one whose unknown is an optimal Radon measure. Using two stages of approximation determines the optimal control, and the optimal generator rotating curve is obtained from the results of a finite linear programming problem. The accuracy and applications of the method are shown by presenting some classical numerical simulations and comparing the method with the level set method in those examples. [ABSTRACT FROM AUTHOR]

The extension of lattice linear space of continuous bounded functions on a completely regular space, generated by -Riemann-integrable functions on this space, is considered in the paper. To characterize this -Riemann extension some new functionally-analytical category of -latlineals with refinements ( -latlineals) is used. On its base the notion of -completion of some definite type is introduced. A functionally-analytical characterization of the -Riemann extension as some -completion of some definite type of the -latlineal of continuous bounded functions is given. [ABSTRACT FROM AUTHOR]

This study focuses on semilinear fractional elliptic problems with concave-convex type nonlinearities and measures as data. Suitable iteration techniques and embedding results are employed to ensure the existence and multiplicity of solutions. [ABSTRACT FROM AUTHOR]

The purpose of this work is to study a class of periodic parabolic equations having a critical growth nonlinearity with respect to the gradient and bounded Radon measure. By the main of the sub- and super-solution method, we employ some new technics to prove the existence of a nonnegative weak periodic solution to the studied problems. [ABSTRACT FROM AUTHOR]

We construct an example of a Hilbert -module which shows that Troitsky's theorem on the geometric essence of -compact operators between Hilbert -modules cannot be extended to modules which are not countably generated case (even in the case of a stronger uniform structure, which is also introduced). In addition, the constructed module admits no frames. [ABSTRACT FROM AUTHOR]

This article concerns about the existence of a positive SOLA (Solutions Obtained as Limits of Approximations) for the following singular critical Choquard problem involving fractional power of Laplacian and a critical Hardy potential. 0.1 (- Δ) s u - α u | x | 2 s = λ u + u - γ + β ∫ Ω u 2 b ∗ (y) | x - y | b d y u 2 b ∗ - 1 + μ in Ω , u > 0 in Ω , u = 0 in R N \ Ω. Here, Ω is a bounded domain of R N , s ∈ (0 , 1) , α , λ and β are positive real parameters, N > 2 s , γ ∈ (0 , 1) , 0 < b < min { N , 4 s } , 2 b ∗ = 2 N - b N - 2 s is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality and μ is a bounded Radon measure in Ω . [ABSTRACT FROM AUTHOR]

We study the existence of a solution to the mixed boundary value problem for Helmholtz and Poisson type equations in a bounded Lipschitz domain Ω ⊂ R N and in R N \ Ω for N ≥ 3 . The boundary ∂ Ω of Ω is the decomposition of Γ 1 , Γ 2 ⊂ ∂ Ω such that ∂ Ω = Γ = Γ ¯ 1 ∪ Γ 2 = Γ 1 ∪ Γ ¯ 2 and Γ 1 ∩ Γ 2 = ∅ . We have shown that if the Neumann data f 2 ∈ H - 1 2 (Γ 2) and the Dirichlet data f 1 ∈ H 1 2 (Γ 1) then the Helmholtz problem with mixed boundary data admits a unique solution. We have also shown the existence of a weak solution to a mixed boundary value problem governed by the Poisson equation with a measure data and the Dirichlet, Neumann data belongs to f 1 ∈ H 1 2 (Γ 1) , f 2 ∈ H - 1 2 (Γ 2) , respectively. [ABSTRACT FROM AUTHOR]

In this paper, we first prove the local L s norm estimate of composite operators △ k G m (u) by use of the L s norm of u. Then we establish the local and global higher norm inequalities of △ k G m (u) . Simultaneously, we also give a global higher norm estimate with Radon measure. Finally, as applications of these results, we give two examples to estimate the higher norm of △ k G m (u) . [ABSTRACT FROM AUTHOR]

This paper deals with properties of the ultraproducts for various structures. We introduce and study the concept of the ergodic action of a group with respect to a normal state on an abelian von Neumann algebra. In particular, we provide an example showing that the ultraproduct of ergodic states, generally speaking, is not ergodic. The ultraproduct of the Radon measures on a compact convex subset of a locally convex space is also investigated in the paper. As is well-known, the study of the extreme points in the state set for a algebra is a very interesting problem in itself. Considering the ultraproducts of -algebras and the states on these algebras, we get quite nontrivial results. [ABSTRACT FROM AUTHOR]

Let μ a nonnegative Radon measure on R d ; p , q , γ , k real numbers; M μ , k γ a fractional maximal operator; A p , q γ , k μ a Muckenhoupt condition associated to μ ; L p (·) (R d , μ) and F (q , p , α , μ) (R d) two generalized Lebesgue spaces. The purpose of the present work is double: First, a characterization of the Muckenhoupt A p , q γ , k μ is given. At the end, the fractional maximal M μ , k γ is applied on the two generalized Lebesgue spaces L p (·) (R d , μ) and F (q , p , α , μ) (R d) . [ABSTRACT FROM AUTHOR]

Regarding the some useful advantages of spherical coordinates for some special problems, in this paper, based on Radon measure properties, we present a new and basic solution method for general shape optimization problems defined in spherical coordinates. Indeed, our goal is to determine a bounded shape located over the (x, y)-plane, such that its projection in the (x, y)-plane and its volume is given and also it minimizes some given surface integral. To solve these kinds of problems, we somehow extend the embedding process in Radon measures space. First, the problem is converted into an infinite-dimensional linear programming one. Then, using approximation scheme and a special way for discretization in spherical region, this problem is reduced to a finite-dimensional linear programming one. Finally, the solution of this new problem is used to construct a nearly optimal smooth surface by applying an outlier detection algorithm and curve fitting. More than reducing the complexity, this approach in comparison with the other methods has some other advantages: linear treatment for even nonlinear problems, and the minimization is global and does not depend on initial shape and mesh design. Numerical examples are also given to demonstrate the effectiveness of the new method, especially for classical and obstacle problems. [ABSTRACT FROM AUTHOR]

We will establish the existence and uniqueness of measure-valued solutions of the continuity equation with a random potential. [ABSTRACT FROM AUTHOR]

In this paper, a new approach is proposed for designing the nearly-optimal three dimensional symmetric shapes with desired physical center of mass. Herein, the main goal is to find such a shape whose image in ( r, θ)-plane is a divided region into a fixed and variable part. The nearly optimal shape is characterized in two stages. Firstly, for each given domain, the nearly optimal surface is determined by changing the problem into a measure-theoretical one, replacing this with an equivalent infinite dimensional linear programming problem and approximating schemes; then, a suitable function that offers the optimal value of the objective function for any admissible given domain is defined. In the second stage, by applying a standard optimization method, the global minimizer surface and its related domain will be obtained whose smoothness is considered by applying outlier detection and smooth fitting methods. Finally, numerical examples are presented and the results are compared to show the advantages of the proposed approach. [ABSTRACT FROM AUTHOR]

We introduce two classes of Banach spaces $$F(q,p,\alpha )$$ and $$T^{p,\;\alpha }$$ ( $$1\le q\le \alpha \le p\le \infty $$ ) which are subspaces of Herz spaces and Morrey spaces of functions and measures respectively. We study the Riesz potential operators in these spaces and obtain norm inequalities on these operators from which we deduce a necessary condition for a nonnegative Radon measure to have its Riesz potential in a given Lebesgue space. [ABSTRACT FROM AUTHOR]

We consider a vector lattice $$\mathcal L $$ of bounded real continuous functions on a topological space $$X$$ that separates the points of $$X$$ and contains the constant functions. A notion of tightness for linear functionals is defined, and by an elementary argument we prove with the aid of the classical Riesz representation theorem that every tight continuous linear functional on $$\mathcal L $$ can be represented by integration with respect to a Radon measure. This result leads incidentally to an simple proof of Prokhorov’s existence theorem for the limit of a projective system of Radon measures. [ABSTRACT FROM AUTHOR]

We study optimal solutions to an abstract optimization problem for measures, which is a generalization of classical variational problems in information theory and statistical physics. In the classical problems, information and relative entropy are defined using the Kullback-Leibler divergence, and for this reason optimal measures belong to a one-parameter exponential family. Measures within such a family have the property of mutual absolute continuity. Here we show that this property characterizes other families of optimal positive measures if a functional representing information has a strictly convex dual. Mutual absolute continuity of optimal probability measures allows us to strictly separate deterministic and non-deterministic Markov transition kernels, which play an important role in theories of decisions, estimation, control, communication and computation. We show that deterministic transitions are strictly sub-optimal, unless information resource with a strictly convex dual is unconstrained. For illustration, we construct an example where, unlike non-deterministic, any deterministic kernel either has negatively infinite expected utility (unbounded expected error) or communicates infinite information. [ABSTRACT FROM AUTHOR]

We study algebraic and topological properties of the convolution semigroup of probability measures on a topological groups and show that a compact Clifford topological semigroup S embeds into the convolution semigroup P( G) over some topological group G if and only if S embeds into the semigroup $\exp(G)$ of compact subsets of G if and only if S is an inverse semigroup and has zero-dimensional maximal semilattice. We also show that such a Clifford semigroup S embeds into the functor-semigroup F( G) over a suitable compact topological group G for each weakly normal monadic functor F in the category of compacta such that F( G) contains a G-invariant element (which is an analogue of the Haar measure on G). [ABSTRACT FROM AUTHOR]

In the present paper, we propose a new approximation method in different function spaces. A specific feature of this method is that the choice of the basis approximating elements significantly depends on the topology of the given function space. Basis elements are constructed using the duality theory of locally convex spaces. A method of their exact calculation is presented. The approximating constructions are far-reaching generalizations of the classical Schoenberg splines and, by analogy with the latter, may be called topological splines. In the general case, such a definition of splines is not related to the choice of the grid. In this paper, we give many examples that are useful for practical applications. [ABSTRACT FROM AUTHOR]

We consider sums of exponentially stabilizing functionals (introduced by Penrose and Yukich) of Poisson point processes in d-dimensional Euclidean space. The asymptotic behavior of such sums is studied in terms of a random field (defined on the lattice ℤ d) each element of which is a certain sum of functionals with respect to the corresponding unit cube in ℝ d. For this random field, we obtain an exponential estimate of the decrease of the strong mixing coefficient and establish the law of the iterated logarithm. [ABSTRACT FROM AUTHOR]

We consider convex functions on infinite-dimensional spaces equipped with measures. Our main results give some estimates of the first and second derivatives of a convex function, where second derivatives are considered from two different points of view: as point functions and as measures. [ABSTRACT FROM AUTHOR]

We will explain a new method for obtaining the nearly optimal domain for optimal shape design problems associated with the solution of a nonlinear wave equation. Taking into account the boundary and terminal conditions of the system, a new approach is applied to determine the optimal domain and its related optimal control function with respect to the integral performance criteria, by use of positive Radon measures. The approach, say shape-measure, consists of two steps; first for a fixed domain, the optimal control will be identified by the use of measures. This function and the optimal value of the objective function depend on the geometrical variables of the domain. In the second step, based on the results of the previous one and by applying some convenient optimization techniques, the optimal domain and its related optimal control function will be identified at the same time. The existence of the optimal solution is considered and a numerical example is also given. [ABSTRACT FROM AUTHOR]

Results concerning the Stone--Weierstrass theorems for completely regular spaces are obtained. [ABSTRACT FROM AUTHOR]

It is proved that the space of configurations without multiple points on a smooth Riemannian manifold is Polish with respect to the weak topology; a criterion for a subspace of this space to be precompact is given. [ABSTRACT FROM AUTHOR]

Microlocal defect measures have been used to study the energy densities of semilinear wave equations with oscillatory initial data. An “open problem” in literature is solved. [ABSTRACT FROM AUTHOR]

We construct surface measures for surfaces of codimension n≥1 in Banach spaces, and in a wide class of locally convex spaces. It is assumed that the determining function has a continuous derivative along a subspace. [ABSTRACT FROM AUTHOR]

Let f be a bounded Pettis integrable function ranging in a Banach space X (the range of the indefinite Pettis integral is separable). We consider Pettis integrability conditions for the Stone transform of f and relate this problem to the regular oscillation condition for the family of functions { x f∶x∈B(X)}, where B(X) is the unit ball in X. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to give some generalizations of de Branges Lemma for weighted spaces to obtain different approximation theorems in weighted spaces for algebras, vector subspaces or convex cones. We recall that the (original) de Branges Lemma (Proc Am Math Soc 10(5):822–824, 1959) was demonstrated for continuous scalar function on a compact space while, the weighted spaces are classes of continuous scalar functions on a locally compact space (e.g. the space of function with compact support, the space of bounded functions, the space of functions vanishing at infinity, the space of functions rapidly decreasing at infinity). [ABSTRACT FROM AUTHOR]

In this paper, two approaches based on evolutionary algorithms are applied to solve a multi-objective optimal control problem governed by semilinear parabolic partial differential equations. In this approach, first, we change the problem into a measure-theoretical one, replace this with an equivalent infinite dimensional multi-objective nonlinear programming problem and apply approximating schemes. Finally, non-dominated sorting genetic algorithm and multi-objective particle swarm optimization are employed to obtain Pareto optimal solutions of the problem. Numerical examples are presented to show the efficiency of the given approach. [ABSTRACT FROM AUTHOR]