Vector Measures Research Articles

$p$-Regularity and Weights for Operators Between $L^p$-Spaces

We explore the connection between p -regular operators on Banach function spaces and weighted p -estimates. In particular, our results focus on the following problem. Given finite measure spaces \mu and \nu , let T be an operator defined from a Banach function space X(\nu) and taking values on L^p (v d \mu) for v in certain family of weights V\subset L^1(\mu)_+ we analyze the existence of a bounded family of weights W\subset L^1(\nu)_+ such that for every v\in V there is w \in W in such a way that T:L^p(w d \nu) \to L^p(v d \mu) is continuous uniformly on V . A condition for the existence of such a family is given in terms of p -regularity of the integration map associated to a certain vector measure induced by the operator T .

Optimality conditions for optimal impulsive control problems with multipoint state constraints

This paper addresses an optimal impulsive control problem whose trajectories are functions of bounded variation and impulsive controls are regular vector measures. This problem is characterized by two main features. First, the dynamical control system to be considered may not possess the so-called well-posedness property. Second, the constraints on the one-sided limits of states are presented. Such constraints are interpreted as multipoint state constraints. For this problem, we derive global optimality conditions based on using of compound Lyapunov type functions which possess strongly monotone properties with respect to the control system. As a motivating case, a model of advertising expenses optimization for mutually complementary products is considered. For this model, we propose four variants of resolving sets of Lyapunov type functions and explain the technique of applying the optimality conditions.

A large rotation finite element analysis of 3D beams by incremental rotation vector and exact strain measure with all the desirable features

Different strategies based on rotation vector and exact strain measure have been proposed over the years for analyzing flexible bodies undergoing arbitrary large rotations. To avoid the singularity of the vector-like parametrization, the interpolation of the incremental rotation vector is the most popular approach in this context, even if this leads to path dependence and numerical instability, i.e. error accumulation. It is also non objective, although both objectivity and path independence are recovered with h and p refinement. Corotational approaches do not have these drawbacks, even though the geometrically exact model is achieved by mesh refinement. In this work, we develop a novel strategy which uses the incremental nodal rotation vectors to define corotational nodal rotations, which are then interpolated for the evaluation of the nonlinear strains. This choice makes the approach singularity-free, allows for additive updates within each increment and preserves all the features of the theoretical problem for any mesh and interpolation: rotational variables, objectivity, exact strain measure, path independence and symmetric stiffness matrix for conservative loads. This last property is a consequence of the direct differentiation of the relation between local and global rotations, whose compact form also makes a simple and general definition of the internal forces and the tangent stiffness for any order of interpolation possible. In addition, we show how the common approach of interpolating incremental vectors can be made stable by a simple updating procedure based on local rotations carried out at the end of each increment in order to avoid cumulative errors. Geometrically exact 3D beams are considered as a demonstrative example. An iterative strategy based on mixed integration points is used to solve the nonlinear discrete equations efficiently.

Fourier analysis associated to a vector measure on a compact group

In this paper, we introduce and study the Fourier transform of functions which are integrable with respect to a vector measure on a compact group (not necessarily abelian). We also study the Fourier transform of vector measures. We also introduce and study the convolution of functions from $$L^p$$-spaces associated to a vector measure. We prove some analogues of the classical Young’s inequalities. Similarly, we also study convolution of a scalar measure and a vector measure.

PhenX: Vector measures for tobacco regulatory research

The PhenX (Phenotypes and eXposures) Toolkit provides researchers with recommended standard consensus measures for use in epidemiological, biomedical, clinical and translational studies. To expand the depth and breadth of measures...

Application of VMCM, to investigate, the dynamics of changes information society development

Abstract The article attempts to analyse the dynamics of change in information society development among European Union countries. Based on the seven selected indicators characterizing the digital economy area. Time and cross-sectional analyses of the analysed economies were compared and classified using the Vector Measure Construction Method. The analysis will be based on the data with Eurostat database.

Application of Vector Measure Construction Methods to estimate growth factors of fundamental importance for the economy on the example of nations in transition

Abstract The article attempts to analyse the fundamental factors of economic growth in the selected countries of Central and Eastern Europe and Central Asia, which in the early 1990s introduced a market system in their economies. The economies under studies were analysed by indicators characterizing fundamental growth factors, time and cross-sectional analyses and the economies were compared and classified using the Vector Measure Construction Method (VMCM). The empirical part of the study was preceded by a presentation of the nature and significance of fundamental factors of economic growth and a description of the relationships between them. The results of the analysis indicate that for various group of countries distinguished in terms of development another fundamental growth factors were important. For the richest countries international integration (economy openness) seems to be the most relevant element, while for the least developed countries - geographic location and the institution quality are the most important.

CONTROL AND ESTIMATION FOR A CLASS OF IMPULSIVE DYNAMICAL SYSTEMS

The nonlinear dynamical control system with uncertainty in initial states and parameters is studied. It is assumed that the dynamic system has a special structure in which the system nonlinearity is due to the presence of quadratic forms in system velocities. The case of combined controls is studied here when both classical measurable control functions and the controls generated by vector measures are allowed. We present several theoretical schemes and the estimating algorithms allowing to find the upper bounds for reachable sets of the studied control system. The research develops the techniques of the ellipsoidal calculus and of the theory of evolution equations for set-valued states of dynamical systems having in their description the uncertainty of set-membership kind. Numerical results of system modeling based on the proposed methods are included.

Interpolation between weighted Lorentz spaces with respect to a vector measure

In this paper, we consider weighted Lorentz spaces with respect to a vector measure and derive some of their properties. We describe the interpolation with a parameter function of these spaces. As an application, we get a type of the generalization of Steffensen's inequality for $L^p(\|m\|)$ and interpolation spaces for couples of Lorentz-Zygmund spaces.

How to Obtain Maximal and Minimal Subranges of Two-Dimensional Vector Measures

Abstract Let (X, ℱ) be a measurable space with a nonatomic vector measure µ =(µ 1, µ 2). Denote by R(Y) the subrange R(Y)= {µ(Z): Z ∈ ℱ, Z ⊆ Y }. For a given p ∈ µ(ℱ) consider a family of measurable subsets ℱ p = {Z ∈ ℱ : µ(Z)= p}. Dai and Feinberg proved the existence of a maximal subset Z* ∈ Fp having the maximal subrange R(Z*) and also a minimal subset M* ∈ ℱ p with the minimal subrange R(M*). We present a method of obtaining the maximal and the minimal subsets. Hence, we get simple proofs of the results of Dai and Feinberg.

Constrained minimum Riesz and Green energy problems for vector measures associated with a generalized condenser

For a finite collection $\mathbf{A}=(A_i)_{i \in I}$ of locally closed sets in $\mathbb R ^n$, $n \geqslant 3$, with the sign $s_i=\pm 1$ prescribed such that the oppositely charged plates are mutually disjoint, we consider the minimum energy problem relative to the $\alpha$-Riesz kernel $|x − y|^{\alpha-n}, \alpha \in (0,2]$, over positive vector Radon measures $\mathbf{μ}=(\mu^i)_{i \in I}$ such that each $\mu^i, i \in I$, is carried by $A_i$ and normalized by $\mu^1(A_i)=a_i \in (0, \infty)$, while the interaction between $\mu^i, i \in I$ , is determined by the matrix $(s_is_j)_{i,j\in I^\bullet}$. We show that, though the closures of oppositely charged plates may intersect each other even in a set of nonzero capacity, this problem has a solution $\mathbf{λ}_\mathbf{A}^\xi=(\lambda_\mathbf{A}^i)_{i \in I}$ (also in the presence of an external field) if we restrict ourselves to $\mathbf{μ}$ with $\mu^i\leqslant\xi^i, i \in I$, where the constraint $\mathbf{ξ}=(\xi^i)_{i \in I}$ is properly chosen. We establish the sharpness of the suffcient conditions on the solvability thus obtained, provide descriptions of the weighted vector $\alpha$-Riesz potentials of the solutions, single out their characteristic properties, and analyze the supports of the $\lambda_\mathbf{A}^i, i \in I$. Our approach is based on the simultaneous use of the vague topology and an appropriate semimetric structure defined in terms of the $\alpha$-Riesz energy on a set of vector measures associated with $\mathbf{A}$, as well as on the establishment of an intimate relationship between the constrained minimum $\alpha$-Riesz energy problem and a constrained minimum $\alpha$-Green energy problem, suitably formulated. The results are illustrated by examples.

A Vector Girsanov Result and its Applications to Conditional Measures via the Birkhoff Integrability

Some integration techniques for real-valued functions with respect to vector measures with values in Banach spaces (and viceversa) are investigated in order to establish abstract versions of classical theorems of Probability and Stochastic Processes. In particular the Girsanov Theorem is extended and used with the treated methods.

Range convergence monotonicity for vector measures and range monotonicity of the mass

We prove that the range of sequence of vector measures converging widely satisfies a weak lower semicontinuity property, that the convergence of the range implies the strict convergence (convergence of the total variation) and that the strict convergence implies the range convergence for strictly convex norms. In dimension 2 and for Euclidean spaces of any dimensions, we prove that the total variation of a vector measure is monotone with respect to the range.

Maximal Factorization of Operators Acting in Köthe–Bochner Spaces

Using some representation results for Kothe–Bochner spaces of vector valued functions by means of vector measures, we analyze the maximal extension for some classes of linear operators acting in these spaces. A factorization result is provided, and a specific representation of the biggest vector valued function space to which the operator can be extended is given. Thus, we present a generalization of the optimal domain theorem for some types of operators on Banach function spaces involving domination inequalities and compactness. In particular, we show that an operator acting in Bochner spaces of p-integrable functions for any $$1<p<\infty $$ having a specific compactness property can always be factored through the corresponding Bochner space of 1-integrable functions. Some applications in the context of the Fourier type of Banach spaces are also given.

Linear functionals on variable exponent Bochner–Lebesgue spaces

We characterize the linear functionals on variable exponent Bochner–Lebesgue spaces in terms of the variable exponent Riesz bounded variation spaces for vector measures, which are introduced in this paper.

Multi-Criteria Assessment of the Company's Location Selection: A Dynamic Approach

Purpose: The aim of the study is to present multi-criteria Vector Measure Construction Method (VMCM), which enables the construction of aggregate measures, the creation of rankings, classifications and the investigation of change dynamics. Design/Methodology/Approach: The VMCM is used to build the measure (in dynamic terms) for the purpose of assessing the friendliness of EU countries in the context of supporting investors in the selection of locations for newly established enterprises. Findings: Not all multi-criteria methods give the opportunity to study the dynamics of change. The VMCM allows taking into account objects from outside the sample, which are better than the pattern (the measure is not limited, neither from the bottom nor from the top). Practical Implications: The solution proposed allows to assess the investment friendliness of individual country and compare countries (objects) with each other in different sections of time. Originality/Value: The VMCM allows to eliminate the limitations of the other multi-criteria methods in making rankings, classifications of objects and the analysis of the change dynamics.

Generalized condensers and vector measures

Generalized condensers and vector measures

Correlations between the leading Lyapunov vector and pattern defects for chaotic Rayleigh-Bénard convection.

We probe the effectiveness of using topological defects to characterize the leading Lyapunov vector for a high-dimensional chaotic convective flow field. This is accomplished using large-scale parallel numerical simulations of Rayleigh-Bénard convection for experimentally accessible conditions. We quantify the statistical correlations between the spatiotemporal dynamics of the leading Lyapunov vector and different measures of the flow field pattern's topology and dynamics. We use a range of pattern diagnostics to describe the flow field structures which includes many of the traditional diagnostics used to describe convection as well as some diagnostics tailored to capture the dynamics of the patterns. We use the ideas of precision and recall to build a statistical description of each pattern diagnostic's ability to describe the spatial variation of the leading Lyapunov vector. The precision of a diagnostic indicates the probability that it will locate a region where the Lyapunov vector is larger than a threshold value. The recall of a diagnostic indicates its ability to locate all of the possible spatial regions where the Lyapunov vector is above threshold. By varying the threshold used for the Lyapunov vector magnitude, we generate precision-recall curves which we use to quantify the complex relationship between the pattern diagnostics and the spatiotemporally varying magnitude of the leading Lyapunov vector. We find that pattern diagnostics which include information regarding the flow history outperform pattern diagnostics that do not. In particular, an emerging target defect has the highest precision of all of the pattern diagnostics we have explored.

Lorentz spaces of vector measures and real interpolation of operators

Using the representation of the real interpolation of spaces of p-integrable functions with respect to a vector measure, we show new factorization theorems for p-th power factorable operators acting in interpolation couples of Banach function spaces. The recently introduced Lorentz spaces of the semivariation of vector measures play a central role in the resulting factorization theorems. We apply our results to analyze extension of operators from classical weighted Lebesgue Lp -spaces — in general with different weights — that can be extended to their q-th powers. This is the case, for example, of the convolution operators defined by Lp -improving measures acting in Lebesgue Lp -spaces or Lorentz spaces. A new representation theorem for Banach lattices with a special lattice geometric property, as a space of vector measure integrable functions, is also proved.

Extension and integral representation of the finite Hilbert transform in rearrangement invariant spaces

The finite Hilbert transform T is a classical (singular) kernel operator which is continuous in every rearrangement invariant space X over (–1, 1) having non-trivial Boyd indices. For X = Lp , 1 < p < ∞, this operator has been intensively investigated since the 1940’s (also under the guise of the “airfoil equation”). Recently, the extension and inversion of T : X → X for more general X has been studied in [6], where it is shown that there exists a larger space [T, X], optimal in a well defined sense, which contains X continuously and such that T can be extended to a continuous linear operator T : [T, X] → X. The purpose of this paper is to continue this investigation of T via a consideration of the X-valued vector measure mX : A.→ induced by T and its associated integration operator . In particular, we present integral representations of T: X →X based on the L 1-space of mX and other related spaces of integrable functions.