Finite-dimensional Space Research Articles

Numerical approximation of the boundary control for the wave equation in a square domain with a spectral collocation method

We propose a spectral collocation method to approximate the exact boundary control of the wave equation in a square domain. The idea is to introduce a suitable approximate control problem that we solve in the finite-dimensional space of polynomials of degree N∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\in {\\mathbb {N}}$$\\end{document} in space. We prove that we can choose a sequence of controls fN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f^N$$\\end{document} associated with the approximate control problem in such a way that they converge, as N→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\rightarrow \\infty $$\\end{document}, to a control of the continuous wave equation. Unlike other numerical approximations tried in the literature, this one does not require regularization techniques and can be easily adapted to other equations and systems where the controllability of the continuous model is known. The method is illustrated with several examples in 1-d and 2-d in a square domain. We also give numerical evidence of the highly accurate approximation inherent to spectral methods.

On some generalizations of Hadamard's inversion theorem beyond differentiability

A recognized trend of research investigates generalizations of the Hadamard's inversion theorem to functions that may fail to be differentiable. In this vein, the present paper explores some consequences of a recent result about the existence of global Lipschitz continuous inverse by translating its assumptions of metric nature in terms of nonsmooth analysis constructions. This exploration focuses on continuous, but possibly not locally Lipschitz mappings, acting in finite-dimensional Euclidean spaces. As a result, sufficient conditions for global invertibility are formulated by means of strict estimators, ∗ -difference of convex compacta and regular/basic coderivatives. These conditions qualify the global inverse as a Lipschitz continuous mapping and provide quantitative estimates of its Lipschitz constant in terms of the above constructions.

Pseudoentropy for descendant operators in two-dimensional conformal field theories

We study the late-time behaviors of pseudo-(Rényi) entropy of locally excited states in rational conformal field theories. To construct the transition matrix, we utilize two nonorthogonal locally excited states that are created by the application of different descendant operators to vacuum. We show that when two descendant operators are generated by a single Virasoro generator acting on the same primary operator, the late-time excess of pseudoentropy and pseudo-Rényi entropy corresponds to the logarithm of the quantum dimension of the associated primary operator, in agreement with the case of entanglement entropy. However, for linear combination operators generated by the generic summation of Virasoro generators, we obtain a distinct late-time excess formula for the pseudo-(Rényi) entropy compared to that for (Rényi) entanglement entropy. As the mixing of holomorphic and antiholomorphic generators enhances the entanglement, in this case, the pseudo-(Rényi) entropy can receive an additional contribution. The additional contribution can be expressed as the pseudo-(Rényi) entropy of an effective transition matrix in a finite-dimensional Hilbert space. Published by the American Physical Society 2024

𝐶*-algebras associated to homeomorphisms twisted by vector bundles over finite dimensional spaces

In this paper we study Cuntz–Pimsner algebras associated to 𝐶*-correspondences over commutative C ∗ \mathrm {C}^* -algebras from the point of view of the C ∗ \mathrm {C}^* -algebra classification programme. We show that when the correspondence comes from an aperiodic homeomorphism of a finite dimensional infinite compact metric space X X twisted by a vector bundle, the resulting Cuntz–Pimsner algebras have finite nuclear dimension. When the homeomorphism is minimal, this entails classification of these C ∗ \mathrm {C}^* -algebras by the Elliott invariant. This establishes a dichotomy: when the vector bundle has rank one, the Cuntz–Pimsner algebra has stable rank one. Otherwise, it is purely infinite. For a Cuntz–Pimsner algebra of a minimal homeomorphism of an infinite compact metric space X X twisted by a line bundle over X X , we introduce orbit-breaking subalgebras. With no assumptions on the dimension of X X , we show that they are centrally large subalgebras and hence simple and stably finite. When the dimension of X X is finite, they are furthermore Z \mathcal {Z} -stable and hence classified by the Elliott invariant.

Level numbers of a bounded linear operator between normed linear spaces-II

We continue the study of level numbers and level vectors of a bounded linear operator between normed linear spaces. We prove that every linear operator on a finite-dimensional polyhedral Banach space possesses only finitely many level numbers. We also explore the level numbers of weighted permutation operators between ℓ p spaces. Furthermore, we introduce the concept of approximate level numbers and study its properties and relation with level numbers. In particular, our results indicate that the concept of level numbers can be potentially applied to the spectral theory of operators, in the setting of normed linear spaces.

An approximate fixed point property

A map of a metric space into itself has the approximate fixed point property (AFPP for short) if every nearly fixed point is close to some fixed point. It is proven that both linear operators acting on finite-dimensional Banach spaces and uniformly expansive linear homeomorphisms on Banach spaces exhibit the AFPP. Furthermore, an illustration is provided of a linear homeomorphism that does not satisfy the AFPP.

A Grammian matrix and controllability study of fractional delay integro-differential Langevin systems

<abstract><p>This study focused on introducing a fresh model of fractional operators incorporating multiple delays, termed fractional integro-differential Langevin equations with multiple delays. Additionally, the research evaluated the relative controllability of this model within finite-dimensional spaces. Employing fixed-point theory yields the desired outcomes, with the controllability assessment facilitated by Schauder's fixed point and the Grammian matrix defined through the Mittag-Leffler matrix function. Validation of the results was conducted through an application.</p></abstract>

ON THE CORRECTNESS OF ONE CLASS OF FOURTH-ORDERSINGULAR DIFFERENTIAL EQUATIONS

The differential equation of fourth-order with variable smooth coefficients, given on the real axis, is investigated. The coefficients of the equation can be unbounded, and the sign of the potential (the lowest coefficient) is not defined. The case is considered that the intermediate term containing the second derivative of the desired function, as an operator, does not obey the operator of the equation. It is known that fourth-order differential equations with variable coefficients are used in various problems of mathematical physics. The simplest biharmonic equation with minor terms is important for its applications in the theory of elasticity, the mechanics of elastic plates and the slow flow of viscous liquids. However, it is a very special case of a general equation with variable coefficients. Some problems of stochastic analysis, oscillation theory, biology and mathematical finance lead to the equation we are studying. In addition, fourth–order singular differential equations are often used as regularizers in the study of lower-order equations, for example, reaction-diffusion equations. In the paper, sufficient solvability conditions of the equation and the maximal regularity inequality of a strong generalized solution are obtained. The restrictions are formulated in terms of the coefficients themselves and rather weak. They do not contain conditions for any derivatives of coefficients and represent relations between the orders of growth of coefficients of different orders at infinity. The weight estimation of the norms of a solution established by us allows us to additionally apply the methods of the theory of functional spaces to the study of further properties of the solution, for example, the approximation of its elements by finite-dimensional spaces.

Component Intersection Graph of Finite Dimensional Vector Spaces

In this paper, we introduce a graph structure, called component intersection graph, on a finite dimensional vector space V. The connectivity, diameter, maximal independent sets, clique number, chromatic number of component intersection graph have been studied.

Revisiting Stochastic Realization Theory using Functional Itô Calculus

This paper considers the problem of constructing finite-dimensional state space realizations for stochastic processes that can be represented as the outputs of a certain type of a causal system driven by a continuous semimartingale input process. The main assumption is that the output process is infinitely differentiable, where the notion of differentiability comes from the functional Ito calculus introduced by Dupire as a causal (nonanticipative) counterpart to Malliavin's stochastic calculus of variations. The proposed approach builds on the ideas of Hijab, who had considered the case of processes driven by a Brownian motion, and makes contact with the realization theory of deterministic systems based on formal power series and Chen-Fliess functional expansions.

Staircasing effect for minimizers of the one-dimensional discrete Perona–Malik functional

We consider the one-dimensional Perona–Malik functional, that is the energy associated to the celebrated forward-backward equation introduced by P. Perona and J. Malik in the context of image processing, with the addition of a forcing term. We discretize the functional by restricting its domain to a finite dimensional space of piecewise constant functions, and by replacing the derivative with a difference quotient. We investigate the asymptotic behavior of minima and minimizers as the discretization scale vanishes. In particular, if the forcing term has bounded variation, we show that any sequence of minimizers converges in the sense of varifolds to the graph of the forcing term, but with tangent component which is a combination of the horizontal and vertical directions. If the forcing term is more regular, we also prove that minimizers actually develop a microstructure that looks like a piecewise constant function at a suitable scale, which is intermediate between the macroscopic scale and the scale of the discretization.

Fréchet Discrete Gradient and Hessian Operators on Infinite-Dimensional Spaces

Benefiting from the notion of Fréchet derivatives, we define Fréchet discrete operators, such as gradient and Hessian, on infinite-dimensional spaces. The Fréchet discrete gradient expands upon the concept of the discrete gradient of Gonzalez (1996) for finite-dimensional spaces. The Fréchet discrete Hessian elevates the property to second-order representations of the Fréchet derivative. By leveraging these operators, we offer an initial exploration of discrete gradient methods for convex optimization in infinite-dimensional spaces. Under mild conditions on the objective functional, we establish the convergence of any sequence generated by the proposed Fréchet discrete gradient method, regardless of the choice of the finite learning rate.

Rough set theory applied to finite dimensional vector spaces

In this paper, we study finite dimensional vector spaces using rough set theory (RST) by defining a Boolean information system IB associated with a vector space V for a given basis B. We define an indiscernibility relation on V and investigate different partitions induced on V. We identify that every reduct of the information system is a basis of V and the core is empty in the case of V over field F with |F|≥3. We study the measure of dependency between different subsets of V. Moreover, we define three posets and prove that these posets are isomorphic and complete. We study lower and upper approximations for different subsets of V and determine rough and exact sets. We give the general form of the entries of the discernibility matrix. We determine essential sets and the essential dimension of the information system, and prove that minimal entries of the discernibility matrix coincide with the essential sets. Finally, we demonstrate the use of the developed theory by providing two practical examples.

A DIFFERENTIABLE STRUCTURE ON A FINITE DIMENSIONAL REAL VECTOR SPACE AS A MANIFOLD

There are three conditions for a topological space to be said a topological manifold of dimension : Hausdorff space, second-countable, and the existence of homeomorphism of a neighborhood of each point to an open subset of or -dimensional locally Euclidean. The differentiable structure is given if the intersection of two charts is an empty chart or its transition map is differentiable. In this article, we study a differentiable manifold on finite dimensional real vector spaces. The aim is to prove that any finite-dimensional vector space is a differentiable manifold. First of all, it is proved that a finite dimensional vector space is a topological manifold by constructing a norm as its topology. Given a metric which is induced by a norm. Two norms on a finite dimensional vector space are always equivalent and they are determine the same topology. Secondly, it is proved that the transition map in the finite dimensional vector space is differentiable. As conclusion, we have that any finite dimensional vector space with independent norm topology choice is a differentiable manifold. As a matter of discussion, it can be studied that the vector space of all linear operators of a finite dimensional vector space has a differentiable manifold structure as well.

Modeling Infectious Disease Trend using Sobolev Polynomials

Trend analysis plays an important role in infectious disease control. An analysis of the underlying trend in the number of cases or the mortality of a particular disease allows one to characterize its growth. Trend analysis may also be used to evaluate the effectiveness of an intervention to control the spread of an infectious disease. However, trends are often not readily observable because of noise in data that is commonly caused by random factors, short-term repeated patterns, or measurement error. In this paper, a smoothing technique that generalizes the Whittaker-Henderson method to infinite dimension and whose solution is represented by a polynomial is applied to extract the underlying trend in infectious disease data. The solution is obtained by projecting the problem to a finite-dimensional space using an orthonormal Sobolev polynomial basis obtained from Gram-Schmidt orthogonalization procedure and a smoothing parameter computed using the Philippine Eagle Optimization Algorithm, which is more efficient and consistent than a hybrid model used in earlier work. Because the trend is represented by the polynomial solution, extreme points, concavity, and periods when infectious disease cases are increasing or decreasing can be easily determined. Moreover, one can easily generate forecast of cases using the polynomial solution. This approach is applied in the analysis of trends, and in forecasting cases of different infectious diseases.

New results on controllability analysis of nonlinear fractional order integrodifferential Langevin system with multiple delays

Nowadays, the study of fractional differential equations (FDEs) is one of the most interesting research topics. In this article, we study a novel nonlinear fractional order Langevin system involving delays. The primary focus of this article is to analyze the relative controllability of a nonlinear integrodifferential fractional Langevin dynamical system with multiple delays in control for finite-dimensional spaces. Fundamentals of the fixed-point theory are used to achieve the desired results. Using Schauder’s fixed theorem, sufficient conditions for the controllability are established and the Mittag-Leffler matrix function defines the controllability Grammian matrix. Additionally, the derived results are validated through an application.

On a Control Problem for a System of Implicit Differential Equations

We consider the differential inclusion F(t,x,x˙)∋0 with the constraint x˙(t)∈B(t), t∈[a,b], on the derivative of the unknown function, where F and B are set-valued mappings, F:[a,b]×Rn×Rn×Rm⇉ is superpositionally measurable, and B:[a,b]⇉Rn is measurable. In terms of the properties of ordered covering and the monotonicity of set-valued mappings acting in finite-dimensional spaces, for the Cauchy problem we obtain conditions for the existence and estimates of solutions as well as conditions for the existence of a solution with the smallest derivative. Based on these results, we study a control system of the form f(t,x,x˙,u)=0, x˙(t)∈B(t), u(t)∈U(t,x,x˙), t∈[a,b].

Raznostnye skhemy dekompozitsii na osnove rasshchepleniya resheniya i operatora zadachi

Domain decomposition methods are used for the approximate solution of boundary value problems for partial differential equations on parallel computing systems. The specifics of nonstationary problems is most completely taken into account when using noniterative domain decomposition schemes. Regionally additive schemes are constructed on the basis of various classes of splitting schemes. A new class of domain decomposition schemes with an additive representation of the solution on a new time level is distinguished that is based on splitting the domain into subdomains based on a partition of unity. An example of the Cauchy problem for first-order evolution equations with a positive self-adjoint operator in a finite-dimensional Hilbert space is considered. Unconditionally stable two- and three-level splitting schemes are constructed for the corresponding system of equations.

Lineynaya zadacha gruppovogo presledovaniya s drobnymi proizvodnymi, prostymi matritsami i raznymi vozmozhnostyami igrokov

In a finite-dimensional Euclidean space, we consider the problem of pursuit by a group of pursuers of one evader, which is described by a system of equations with a Caputo derivative of order a , where the sets of feasible controls are convex compact sets. We obtain sufficient conditions for the solvability of pursuit and evasion problems, in the study of which the method of resolving functions is used.

Estimates of Integrally Bounded Solutions of Linear Differential Inequalities

We study integrally bounded solutions of the differential equation A(x)=z, where A is a linear differential operator of order l defined on functions x:R→H (R=(−∞,∞), H () and H is a finite-dimensional Euclidean space). The right-hand side z is an integrally bounded function on R ranging in H and satisfying the inequality (ψ(t),z(t))≥δ|z(t)|, t∈R, δ0. Conditions are given on the operator A and the function ψ:R→H that guarantee an inverse inequality of the following form for the solutions x under consideration: ∫τ+1τ|x(l)(t)|dt≤c∫τ+2τ−1|x(t)|dt, where the constant is independent of the choice of a real number t and function x.