Vector Space Of Functions Research Articles

Detective generalized multiscale hybridizable discontinuous Galerkin(GMsHDG) method for porous media

The Detective Generalized Multiscale Hybridizable Discontinuous Galerkin (Detective GMsHDG) method aims to further reduce the computational cost of the GMsHDG method. The GMsHDG method itself reduces the computational cost of the HDG method by employing an upscaled structure on a two-grid mesh. Given a PDE within a specified domain, we subdivide the domain into polygonal subdomains and transforms a HDG problem into globular and local problems. Globular problem concerns whether the solutions on smaller domains glue well to form a globular solution. The process involves generation of multiscale spaces, which is a vector space of functions defined on edges of the polygonal regions. A naive approximation by polynomials fails, especially in porous media, necessitating the generation of problem-specific spaces. The Detective GMsHDG method improves this process by replacing the generation of the multiscale space with the detective algorithm. The Detective GMsHDG method has two stages. First is called an offline stage. During the offline stage, we construct a detective function which, given a permeability distribution, it gives a multiscale space. Later stage is called the offline stage where, given the multiscale space, we use GMsHDG method to solve a given PDE numerically. We show numerical results to argue the liability of the solution using the detective GMsHDG method.

A minimax approach to duality for linear distributional sensitivity testing

We consider the dual formulation of the problem of finding the maximum of E ν [ f ( X ) ] , where ν is allowed to vary over all the probability measures on a Polish space X for which d c ( μ , ν ) ≤ r , with d c an optimal transport distance, f a real-valued function on X satisfying some regularity, μ a ‘baseline’ measure and r ≥ 0 . Whereas some derivations of the dual rely on Fenchel duality, applied on a vector space of functions in duality with a vector space of measures, we impose compactness on X to allow the use of the minimax theorem of Ky Fan, which does not require vector space structure.

Regularization of Solutions of One Class of Systems of Linear Volterra Integral Equations of the First Kind in the Space of Differentiable Vector Functions on the Axis

Regularizing operators according to Lavrentiev are constructed and sufficient conditions for the uniqueness of solutions to systems of linear Volterra integral equations of the first kind on the axis are established.

Noncompact uniform universal approximation

The universal approximation theorem is generalised to uniform convergence on the (noncompact) input space Rn. All continuous functions that vanish at infinity can be uniformly approximated by neural networks with one hidden layer, for all activation functions φ that are continuous, nonpolynomial, and asymptotically polynomial at ±∞. When φ is moreover bounded, we exactly determine which functions can be uniformly approximated by neural networks, with the following unexpected results. Let Nφl(Rn)¯ denote the vector space of functions that are uniformly approximable by neural networks with l hidden layers and n inputs. For all n and all l≥2, Nφl(Rn)¯ turns out to be an algebra under the pointwise product. If the left limit of φ differs from its right limit (for instance, when φ is sigmoidal) the algebra Nφl(Rn)¯ (l≥2) is independent of φ and l, and equals the closed span of products of sigmoids composed with one-dimensional projections. If the left limit of φ equals its right limit, Nφl(Rn)¯ (l≥1) equals the (real part of the) commutative resolvent algebra, a C*-algebra which is used in mathematical approaches to quantum theory. In the latter case, the algebra is independent of l≥1, whereas in the former case Nφ2(Rn)¯ is strictly bigger than Nφ1(Rn)¯.

Decomposition and Properties of a Spectrum

The study sought to show the relationships of the decomposed subsets of the spectrum by use of inclusions. In operator Theory, we study of linear operator on function spaces acting on vector spaces of functions. Since an element of s(A) (ie spectrum of A), is a spectrum value of A, we need to investigate more about these elements of A, by decomposing the spectrum to various subsets. From the decomposed subsets of the spectrum, we study the properties of spectrum to give an insight of deeper understanding of the spectrum. The deductions from the proofs of Theorems, propositions, Lemmas and corollaries will help to expose the properties being investigated.

On the Dual of a Weighted Space of Vector Functions and Some Approximation Results

On the Dual of a Weighted Space of Vector Functions and Some Approximation Results

Linear and nonlinear integral functional on the space of continuous vector functions

Linear and nonlinear integral functional on the space of continuous vector functions

Weakly symmetric functions on spaces of Lebesgue integrable functions

In this work, we present the notion of a weakly symmetric function. We show that the subset of all weakly symmetric elements of an arbitrary vector space of functions is a vector space itself. Moreover, the subset of all weakly symmetric elements of some algebra of functions is an algebra. Also we consider weakly symmetric functions on the complex Banach space $L_p[0,1]$ of all Lebesgue measurable complex-valued functions on $[0,1]$ for which the $p$th power of the absolute value is Lebesgue integrable. We show that every continuous linear functional on $L_p[0,1],$ where $p\in (1,+\infty),$ can be approximated by weakly symmetric continuous linear functionals.

Robustness of Solutions of Almost Every System of Equations

In mathematical modeling, it is common to have an equation , where the exact form of is not known. This article shows that there are large classes of where almost all share the same properties. The classes we investigate are vector spaces of functions that satisfy the following condition: has “almost constant rank" (ACR) if there is a constant integer such that rank for “almost every" and almost every . If the vector space is finite dimensional, then “almost every" is with respect to the Lebesgue measure on , and otherwise, it means almost every in the sense of prevalence, as described herein. Most function spaces commonly used for modeling purposes are ACR. In particular, we show that if all of the functions in are linear or polynomial or real analytic, or if is the set of all functions in a “structured system", then is ACR. For each and , the solution set of is SolSet . A solution set of is called robust if it persists despite small changes in and . The following two global results are proved for almost every in an ACR vector space : (1) Either the solution set SolSet is robust for almost every , or none of the solution sets are robust. (2) The solution set SolSet is a -manifold of dimension . In particular, is the same for almost every .

On Spaces of Vector Functions that Are Holomorphic in an Angular Domain

In this paper, we study spaces of vector functions that are holomorphic in the angular domain of the complex plane and with values in a separable Hilbert space. We show that, equipped with the appropriate norms, these spaces are Hilbert spaces.

Linear Operators and Equations with Partial Integrals

We consider linear operators and equations with partial integrals in Banach ideal spaces, spaces of vector functions, and spaces of continuous functions. We study the action, regularity, duality, algebras, Fredholm properties, invertibility, and spectral properties of such operators. We describe principal properties of linear equations with partial integrals. We show that such equations are essentially different compared to usual integral equations. We obtain conditions for the Fredholm alternative, conditions for zero spectral radius of the Volterra operator with partial integrals, and construct resolvents of invertible equations. We discuss Volterra–Fredholm equations with partial integrals and consider problems leading to linear equations with partial integrals.

On interpolation and 𝐾-monotonicity for discrete local Morrey spaces

A formula is given that makes it possible to reduce the calculation of an interpolation functor on a pair of local Morrey spaces to the calculation of this functor on pairs of vector function spaces constructed from the ideal spaces involved in the definition of the Morrey spaces in question. It is shown that a pair of local Morrey spaces is K K -monotone if and only if the pair of vector function spaces mentioned above is K K -monotone. This reduction makes it possible to obtain new interpolation theorems even for classical local spaces.

Invariant Operators in Some Space of Vector Functions

Invariant Operators in Some Space of Vector Functions

Invariant operators in some space of vector functions

Invariant operators in some space of vector functions

Approximation in weighted spaces of vector functions

In this paper, we present the duality theory for general weighted space of vector functions. We mention that a characterization of the dual of a weighted space of vector functions in the particular case $V \subset C^{+} (X)$ is mentioned by J. B. Prolla in [6]. Also, we extend de Branges lemma in this new setting for convex cones of a weighted spaces of vector functions (Theorem 4.2). Using this theorem, we find various approximations results for weighted spaces of vector functions: Theorems 4.2-4.6 as well as Corollary 4.3. We mention also that a brief version of this paper, in the particular case $V \subset C^{+} (X)$, is presented in [3], Chapter 2, subparagraph 2.5.

Transport operator in the space of vector functions

Transport operator in the space of vector functions

Necessary and Sufficient Conditions for the Solvability of the Complex Cauchy Problem in Classes of Functions of Vector-Exponential Type

We consider the Cauchy problem for general linear systems of complex partial differential equations in scales of Banach spaces of vector functions of the exponential type with an integral metric. Necessary and sufficient conditions for the well-posed solvability of this problem are obtained. Thus, we describe the structure of linear systems of complex partial differential equations for which the Cauchy problem is well posed.

Minimizers of curl prescribed full trace

This paper concerns the minimization problem of L2 norm of curl of vector fields prescribed full trace on the boundary of a multiconnected bounded domain. The existence of the minimizers in H1 are shown by orthogonal decompositions of vector function spaces and a constructed auxiliary variational problem. And the H2 estimate of the type II divergence-free part of the minimizers is established by div-curl-gradient type estimates of vector fields.

First order maximally dissipative singular differential operators

In this paper, using the Calkin-Gorbachuk method, the general form of all maximal dissipative extensions of the minimal operator generated by first order linear multipoint symmetric singular differential-operator expression in the direct sum of Hilbert space of vector-functions has been found. Later on, the structure of spectrum of these extensions is researched. Finally, the results are supported by an application.

A Generalization of the Gleason–Kahane–Zelazko Theorem to Topological Vector Spaces

In this paper, we give a version of the Gleason–Kahane–Zelazko theorem for continuous linear functionals on some topological algebras and topological vector spaces of functions.