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We present existing results and properties for the solutions of some vector equilibrium problems with set-valued functions in the case of a vector space ordered by a cone with some "interiority" properties. Some applications concerning the existence of equilibrium for abstract economies and vector optimization problems are given. [ABSTRACT FROM AUTHOR]

This paper discusses a type of mixed-delay quaternion-valued neural networks (QVNNs) under impulsive and stochastic disturbances. The considered QVNNs model are treated as a whole, rather than as complex-valued neural networks (NNs) or four real-valued NNs. Using the vector Lyapunov function method, some criteria are provided for securing the mean-square exponential stability of the mixed-delay QVNNs under impulsive and stochastic disturbances. Furthermore, a type of chaotic QVNNs under stochastic and impulsive disturbances is considered using a previously established stability analysis method. After the completion of designing the linear feedback control law, some sufficient conditions are obtained using the vector Lyapunov function method for determining the mean-square exponential synchronization of drive–response systems. Finally, two examples are provided to demonstrate the correctness and feasibility of the main findings and one example is provided to validate the use of QVNNs for image associative memory. [ABSTRACT FROM AUTHOR]

The paper is devoted to the formal description of the running time of the user task on some virtual nodes in the computing network. Based on the probability theory framework, this time represents a random value with a finite mean and variance. For any class of user task, these moments are the functions of the node resources, task numerical characteristics, and the parameters of the current node state. These functions of the vector arguments can be treated as some surfaces in the multidimensional Euclidean spaces, so the proposed models are called the stochastic time complexity surfaces. The paper also presents a class of functions suitable for the description of both the mean and variance. They contain unknown parameters which should be estimated. The article includes the statement of the parameter identification problem given the statistical results of the node stress testing, recommendations concerning the test planning, and preprocessing of the raw experiment data. To illustrate the performance of the proposed model, the authors design it for an actual database application—the prototype of the passengers' personal data anonymization system. Its application functions are classified into two user task classes: the data anonymization procedures and fulfillment of the statistical queries. The authors identify the stochastic time complexity surfaces for both task types. The additional testing experiments confirm the high performance of the suggested model and its applicability to the solution of the practical providers' problems. [ABSTRACT FROM AUTHOR]

The article is a natural continuation of the systematic research of the properties of the generalized concept of differentiability for functions with a domain X ⊂ R n that is not necessarily open, at points that allow a neighbourhood ray in the domain. In the new context, the well-known Lagrange's mean value theorem for scalar functions is stated and proved, even for the case when the differential is not unique at all points of the observed segment in the domain. Likewise, it has been proven that its variant is valid for vector functions as well. Additionally, the paper provides a proof of the generalization of the mean value theorem for continuous scalar functions continuously differentiable in the interior of a compact domain. [ABSTRACT FROM AUTHOR]

We propose here a general framework covering a wide range of fractional operators for vector-valued functions. We indicate to what extent the case in which assumptions are expressed in terms of weak topology is symmetric to the case of norm topology. However, taking advantage of the differences between these cases, we emphasize the possibly less-restrictive growth conditions. In fact, we present a definition and a serious study of generalized Hilfer fractional derivatives. We propose a new version of calculus for generalized Hilfer fractional derivatives for vector-valued functions, which generalizes previously studied cases, including those for real functions. Note that generalized Hilfer fractional differential operators in terms of weak topology are studied here for the first time, so our results are new. Finally, as an application example, we study some n-point boundary value problems with just-introduced general fractional derivatives and with boundary integral conditions expressed in terms of fractional integrals of the same kind, extending all known cases of studies in weak topology. [ABSTRACT FROM AUTHOR]

In the framework of Hardy type inequalities and their applications to evolution problems, the paper deals with local and nonlocal weighted improved Hardy inequalities related to the study of Kolmogorov operators perturbed by singular potentials. The class of weights is wide enough. We focus our attention on weighted Hardy inequalities with potentials obtained by inverse square potentials adding a nonnegative correction term. The method used to get the results is based on the introduction of a suitable vector-valued function and on a generalized vector field method. The local estimates show some examples of this type of potentials and extend some known results to the weighted case. [ABSTRACT FROM AUTHOR]

The purpose of this article is to discuss some new aspects of the vector-valued metric space. The idea of an arbitrary binary relation along with the well-known F contraction is used to demonstrate the existence of fixed points in the context of a complete vector-valued metric space for both single- and multi-valued mappings. Utilizing the idea of binary relation, and with the help of F contraction, this work extends and complements some of the very recently established Perov-type fixed-point results in the literature. Furthermore, this work includes examples to justify the validity of the given results. During the discussion, it was found that some of the renowned metrical results proven by several authors using different binary relations, such as partial order, pre-order, transitive relation, tolerance, strict order and symmetric closure, can be weakened by using an arbitrary binary relation. [ABSTRACT FROM AUTHOR]

The problem of constructing and justifying the discrete algorithms of the support operator method for numerical modeling of differential repeated rotational operations of vector analysis ( c u r l c u r l ) in application to problems of magnetohydrodynamics is considered. Difference schemes of the support operator method on the unstructured meshes do not approximate equations in the local sense. Therefore, it is necessary to prove the convergence of these schemes to the exact solution, which is possible after analyzing the error structure of their approximation. For this analysis, a decomposition of the space of mesh vector functions into an orthogonal direct sum of subspaces of potential and vortex fields is introduced. Generalized centroid-tensor metric representations of repeated operations of tensor analysis ( d i v , g r a d , and c u r l ) are constructed. Representations have flux-circulation properties that are integrally consistent on spatial meshes of irregular structure. On smooth solutions of the model magnetostatic problem on a tetrahedral mesh with the first order of accuracy in the rms sense, the convergence of the constructed difference schemes is proved. The algorithms constructed in this work can be used to solve physical problems with discontinuous magnetic viscosity, dielectric permittivity, or thermal resistance of the medium. [ABSTRACT FROM AUTHOR]

In this paper we present an efficient Matlab computation of a 3-D electromagnetic scattering problem, in which a plane wave impinges with a generic inclination onto a conducting ellipsoid of revolution. This solid is obtained by the rotation of an ellipse around one of its axes, which is also known as a spheroid. We have developed a fast and ad hoc code to solve the electromagnetic scattering problem, using spheroidal vector wave functions, which are special functions used to describe physical problems in which a prolate or oblate spheroidal reference system is considered. Numerical results are presented, both for TE and TM polarization of the incident wave, and are validated by a comparison with results obtained by a commercial electromagnetic simulator. [ABSTRACT FROM AUTHOR]

Grüss-type inequalities have been widely studied and applied in different contexts. In this work, we provide and prove vectorial versions of Grüss-type inequalities involving vector-valued functions defined on R n for inner- and cross-products. [ABSTRACT FROM AUTHOR]

We review Pólya vector fields associated to holomorphic functions as an important pedagogical tool for making the complex integral understandable to the students, briefly mentioning its use in other dimensions. Techniques of differential geometry are then used to refine the study of holomorphic functions from a metric (Riemannian), affine differential or differential viewpoint. We prove that the only nontrivial holomorphic functions, whose Pólya vector field is torse-forming in the cannonical geometry of the plane, are the special Möbius transformations of the form f (z) = b (z + d) − 1 . We define and characterize several types of affine connections, related to the parallelism of Pólya vector fields. We suggest a program for the classification of holomorphic functions, via these connections, based on the various indices of nullity of their curvature and torsion tensor fields. [ABSTRACT FROM AUTHOR]

The Poisson surface reconstruction algorithm has become a very popular tool of reconstruction from point clouds. If we reconstruct each region separately in the process of multi-reconstruction, then the reconstructed objects may overlap with each other. In order to reconstruct multicomponent surfaces without self-intersections, we propose an efficient multi-reconstruction algorithm based on a modified vector-valued Allen–Cahn equation. The proposed algorithm produces smooth surfaces and closely preserves the original data without self-intersect. Based on operator splitting techniques, the numerical scheme is divided into one linear equation and two nonlinear equations. The linear equation is discretized using an implicit method, and the resulting discrete system of equation is solved by a fast Fourier transform. The two nonlinear equations are solved analytically due to the availability of a closed-form solution. The numerical scheme has merit in that it can be straightforwardly applied to a graphics processing unit, allowing for accelerated implementation that performs much faster than central processing unit alternatives. Various experimental, numerical results demonstrate the effectiveness and robustness of the proposed method. [ABSTRACT FROM AUTHOR]

Dyadic Green's function (DGF) is a powerful and elegant way of solving electromagnetic problems in the multilayered media. In this paper, we introduce the electric and magnetic DGFs in free space, respectively. Furthermore, the symmetry of different kinds of DGFs is proved. This paper focuses on the application of DGF in a two-layer model. By introducing the universal form of the vector wave functions in space rectangular, cylindrical and spherical coordinates, the corresponding DGFs are obtained. We derive the concise and explicit formulas for the electric fields represented by a vertical electric dipole source. It is expected that the proposed DGF can be extended to some more electric field problem in the two-layer model. [ABSTRACT FROM AUTHOR]

The fundamental goal of this study was to achieve the Ricci curvature inequalities for a skew CR-warped product (SCR W-P) submanifold isometrically immersed in a complex space form (CSF) in the expressions of the squared norm of mean curvature vector and warping functions (W-F). The equality cases were likewise examined. In particular, we also derived Ricci curvature inequalities for CR-warped product (CR W-P) submanifolds. To sustain this study, an example of these submanifolds is provided. [ABSTRACT FROM AUTHOR]

In this paper we carry out a reliability study of the consecutive-k1 and k2-out-of-n systems with independent and identically distributed components ordered in a line. More precisely, we obtain the generating function of the structure's reliability, while recurrence relations for determining its signature vector and reliability function are also provided. For illustration purposes, some numerical results and figures are presented and several concluding remarks are deduced. [ABSTRACT FROM AUTHOR]

In this paper, we show that, given a non-trivial concircular vector field u on a Riemannian manifold (M , g) with potential function f, there exists a unique smooth function ρ on M that connects u to the gradient of potential function ∇ f . We call the connecting function of the concircular vector field u. This connecting function is shown to be a main ingredient in obtaining characterizations of n-sphere S n (c) and the Euclidean space E n . We also show that the connecting function influences on a topology of the Riemannian manifold. [ABSTRACT FROM AUTHOR]

Hermite–Padé approximation has been a mainstay of approximation theory since the concept was introduced by Charles Hermite in his proof of the transcendence of e in 1873. This subject occupies a large place in the literature and it has applications in different subjects. Most of the studies of Hermite–Padé approximation have mainly concentrated on diagonal sequences. Recently, there were some significant contributions in the direction of row sequences of Type II Hermite–Padé approximation. Moreover, various generalizations of Type II Hermite–Padé approximation were introduced and studied on row sequences. The purpose of this paper is to reflect the current state of the study of Type II Hermite–Padé approximation and its generalizations on row sequences. In particular, we focus on the relationship between the convergence of zeros of the common denominators of such approximants and singularities of the vector of approximated functions. Some conjectures concerning these studies are posed. [ABSTRACT FROM AUTHOR]