Finite-dimensional Space Research Articles

On Some Dynamics in Conceptual Spaces

AbstractIn this text, we consider Gärdenfors’ conceptual spaces that are separable Hilbert spaces. In particular, the results we obtained apply to finite-dimensional Euclidean spaces. Our main contribution can be formulated as a combination of the theory of opinion dynamics with the theory of conceptual spaces. This combination, in turn, leads us to propose a new model for the time evolution of conceptual spaces. To achieve this goal, we propose some extension of the multidimensional opinion dynamics model of Parsegov, Proskurnikov, Tempo and Friedkin to opinions with values in Hilbert spaces.

Some Lie Algebra Structures on Symmetric Powers

Let k be a field of any characteristic, V a finite-dimensional vector space over k, and S d ( V * ) be the d-th symmetric power of the dual space V * . Given a linear map φ on V and an eigenvector w of φ , we prove that the pair ( φ , w ) can be used to construct a new Lie algebra structure on S d ( V * ) . We prove that this Lie algebra structure is solvable, and in particular, it is nilpotent if φ is a nilpotent map. We also classify the Lie algebras for all possible pairs ( φ , w ) , when k = C and V is two-dimensional.

Mathematical and numerical analysis of reduced order interface conditions and augmented finite elements for mixed dimensional problems

In this paper, we are interested in the mathematical properties of methods based on a fictitious domain approach combined with reduced-order interface coupling conditions, which have been recently introduced to simulate 3D-1D fluid-structure or structure-structure coupled problems. To give insights on the approximation properties of these methods, we investigate them in a simplified setting by considering the Poisson problem in a two-dimensional domain with non-homogeneous Dirichlet boundary conditions on small inclusions. The approximated reduced problem is obtained using a fictitious domain approach combined with a projection on a Fourier finite-dimensional space of the Lagrange multiplier associated to the Dirichlet boundary constraints, obtaining in this way a Poisson problem with defective interface conditions. After analyzing the existence of a solution of the reduced problem, we prove its convergence towards the original full problem, when the size of the holes tends towards zero, with a rate which depends on the number of modes of the finite-dimensional space. In particular, our estimates highlight the fact that to obtain a good convergence on the Lagrange multiplier, one needs to consider more modes than the first Fourier mode (constant mode). This is a key issue when one wants to deal with real coupled problems, such as fluid-structure problems for instance. Next, the numerical discretization of the reduced problem using the finite element method is analyzed in the case where the computational mesh does not fit the small inclusion interface. As is standard for these types of problem, the convergence of the solution is not optimal due to the lack of regularity of the solution. Moreover, convergence exhibits a well-known locking effect when the mesh size and the inclusion size are of the same order of magnitude. This locking effect is more apparent when increasing the number of modes and affects the Lagrange multiplier convergence rate more heavily. To resolve these issues, we propose and analyze a stabilized method and an enriched method for which additional basis functions are added without changing the approximation space of the Lagrange multiplier. Finally, the properties of numerical strategies are illustrated by numerical experiments.

Theory and applications of the sum-of-squares technique

Abstract The sum-of-squares (SOS) approximation method is a technique used in optimization problems to derive lower bounds on the optimal value of an objective function. By representing the objective function as a sum of squares in a feature space, the SOS method transforms non-convex global optimization problems into solvable semidefinite programs. This note presents an overview of the SOS method. We start with its application in finite-dimensional feature spaces and, subsequently, we extend it to infinite-dimensional feature spaces using reproducing kernels (k-SOS). Additionally, we highlight the utilization of SOS for estimating some relevant quantities in information theory, including the log-partition function.

A novel explicit fast numerical scheme for the Cauchy problem for integro-differential equations with a difference kernel and its application

The present study focuses on designing a second-order novel explicit fast numerical scheme for the Cauchy problem incorporating memory associated with an evolutionary equation, where the integral term's kernel is a discrete difference operator. The Cauchy problem under consideration is related to a real finite-dimensional Hilbert space and includes a self-adjoint operator that is both positive and definite. We introduce a transformative technique for converting the Cauchy problem incorporating memory, into a local evolutionary system of equations by approximating the difference kernel using the sum of exponentials (SoE) approach. A second-order explicit scheme is then constructed to solve the local system. We thoroughly investigate the stability of this explicit scheme, and present the necessary conditions for the stability of the scheme. Moreover, we extended our investigation to encompass time-fractional diffusion-wave equations (TFDWEs) involving a fractional Caputo derivative with an order ranging between (1,2). Initially, we transform the main TFDWE model into a new model that incorporates the fractional Riemann-Liouville integral. Subsequently, we expand the applicability of our idea to develop an explicit fast numerical algorithm for approximating the model. The stability properties of this fast scheme for solving TFDWEs are assessed. Numerical simulations including a two-dimensional Cauchy problem as well as one-dimensional and two-dimensional TFDWE models are provided to validate the accuracy and experimental order of convergence of the schemes.

Characterization of continuous additive set-valued maps “modulo K” on finite dimensional linear spaces

Abstract Let Y be a real vector metric space and K ⊂ Y be a closed convex cone such that K ∩ (− K) = {0}. We prove that a convex compact-valued map F : ℝ → 2 Y ∖ {∅} is K-continuous and K-additive if and only if there are non-empty convex compact sets A, B ⊂ Y such that 0 ∈ A − B ⊂ K and F is equal “modulo K” to the continuous set-valued map G ( t ) = t A , t ≥ 0 , t B , t < 0. $$\begin{array}{} \displaystyle G(t)=\begin{cases} tA,&t\geq0,\\ tB,& t \lt 0. \end{cases} \end{array}$$ Next, we use this result to characterize convex compact-valued maps F : ℝ N → 2 Y ∖ {∅}.

Inversion problem for Radon transforms defined on pseudoconvex sets

This paper is devoted to some questions of inversion for the classical and generalized integral Radon transform. The main question is to determine information about the integrand functions if the values of some integrals are known. A feature of the work of the authors of this message is an analysis of the case when the function is integrated according to hyperplanes in finite-dimensional Euclidean space, and the integrands depend not only on the variables of integration, but also on some of the variables characterizing the hyperplanes. At the same time, the number of independent variables describing known integrals are smaller than those of the unknown integrand. We consider discontinuous integrands defined specifically introduced pseudo-convex sets. A Stefan-type problem is posed about finding surfaces discontinuities of the integrand function. The work provides formulas based on the application special integro-differential operators to known data and allowing you to solve the assigned tasks.

Capture of two coordinated evaders in a linear pursuit problem on time scales

In a finite-dimensional Euclidean space, we consider the problem of pursuit of two evaders by a group of pursuers, described by a linear system with a simple matrix on a given time scale. It is assumed that the evaders use the same control. The pursuers employ quasistrategies based on information about the initial positions and control history of the evaders. The set of admissible controls for each participant is a ball of unit radius centered at the origin, and the terminal sets are the origin. The goal of the group of pursuers is to capture the two evaders. In the study, we use the method of resolving functions as a base one, which allows us to obtain sufficient conditions for the solvability of the approach problem in a certain guaranteed time. In terms of the initial positions and parameters of the game, a sufficient condition for capturing the evaders is obtained.

Diversity and transition of periodic motion of a periodically excited soft-impacting machinery

Dynamics of a periodically excited vibro-impact system with soft impacts is investigated. Essential features of period-one multi-impact motion group and correlated transition characteristics in low-frequency range are discussed in detail by the way of two-parameter bifurcation space providing qualitative domains for different periodic motions. The main focus is given to the effect of sensitive parameters including constraint stiffness k 0, clearance threshold b, and damping parameter ζ on the system response. The low-frequency characteristics in the finite-dimensional parameter space are particularly explored. It is found that the increase of k 0 induces multi-type bifurcation of period-one double-impact symmetrical motion, which induces a rich variety of periodic motions, and period-one multi-impact motion group orbit primarily exist in the small-clearance b and low-frequency ω zone. Based on the evolution irreversibility of adjacent period-one multi-impact orbit, the mechanism of singularies appearing in pairs and two different transition zones (hysteresis and liguliform zones) is studied, the result of which provides a theoretical reference value for the common low-frequency vibration instability phenomenon in the field of mechanical engineering. For small-damping coefficient ζ, period-one multi-impact motion has a large quantity, and the main bridge for the transition of adjacent period-one multi-impact motion is liguliform zone, which embraces period-one multi-impact asymmetrical motion and period-n multi-impact subharmonic motion and a certain chaotic zone. For large-damping coefficient ζ, the amount of period-one multi-impact motion group is reduced, and the main bridge for the transition of adjacent period-one multi-impact motion is hysteresis zone, where adjacent period-one multi-impact orbits can coexist according to initial conditions. As designing and renovating impact mechanical equipment, the reasonable matching law of dynamic parameters can be determined through two-parameter bifurcation space, which is conducive to making the system work in stable periodic motion and obtaining larger instantaneous impact velocity.

A variational approach to stability relative to a set of single-valued and set-valued mappings

Based on the limiting normal cone relative to a set, we present in this paper the novel versions of the limiting coderivative relative to a set and subdifferentials relative to a set of multifunctions and singleton mappings, respectively. In addition to giving the necessary and sufficient conditions for the Aubin property relative to a set of multifunctions, the limiting coderivative relative to a set also provides a coderivative criterion for the metric regularity relative to a set of multifunctions. Besides, our study establishes sudifferential characteristics of the metric regularity and the locally Lipschitz continuity relative to a set for single-valued mappings. In finite dimensional spaces, our results are more general than the previous results. Furthermore, we also give examples to illustrate our results.

Frame representation of quantum systems with finite-dimensional Hilbert space

Abstract There exist many attempts to define a Wigner function for quantum systems with finite-dimensional Hilbert space, each of them coming with its advantages and limitations. The existing finite versions have simple definitions, but they are based only on the existence of a formal analogy with the continuous-variable Wigner function and do not allow an intuitive state analysis. The continuous versions have more complicated definitions, but they are closer to the original Wigner function and allow a visualization of the quantum states. The version based on the concept of tight frame we present is finite, but it has certain properties and applications similar to those of continuous versions. It allows us to present a new graphical representation of qubit states, and to define new parameters concerning them. An important advantage of frame representation follows from the use of redundant information. The values taken by the frame version of Wigner function are not independent. They have to satisfy a large number of mathematical relations, useful in error detection and correction.

Relative controllability for [formula omitted]Caputo fractional delay control system

This paper resolves around relative controllability of ψ−fractional delayed differential equations in finite dimensional space. The Mittag Leffler type of ψ−delayed perturbation matrix function with two parameters exhibits the Grammian matrix of fractional delay system. Based on this Grammian matrix we derived the necessary and sufficient conditions for the linear system which is relatively controllable. The fixed point approach is applied for obtaining a controllability result for semilinear system. This concept can be applied to some examples in order to illustrate the efficacy of our results.

Some Results about Acts over Monoid and Bounded Linear Operators

الأثر V بالنسبة إلى sinshT و خواصه قد تم دراسته في هذا البحث حيث تم دراسة علاقة الأثر المخلص والاثر المنتهى التولد والاثر المنفصل وربطها بالمؤثرات المتباينة حيث تم بهنة العلاقات التالية ان الاثر اذا وفقط اذا مقاس في حالة كون المؤثر هو عديم القوة وكذلك في حالة كون المؤثر شامل فان الاثر هو منتهي التولد اي ان الغضاء هو منتهي التولد وايضا تم برهن ان الاثر مخلص لكل مؤثر مقيد وك\لك قد تم التحقق من انه لاي مؤثر مقيد فان الاثر منفصل وفي حالة كون الاثر منفصل فان المؤثر سوف يكون متباين وايضا تم برهنة انه في حالة كون الفضاء يمتلك قاعدة فان الاثر سوف يكون كل عنصر فيه يمكن كتابته كتركيبة خطية ومن العلاقات المهمة التي قد تم برهانها انه اذا كان المؤثر مشابه لمؤثر اخر فان الاثر سوف يكون نوثيرين بوجد شرط على الفضاء .

A probabilistic approach to Lorentz balls [formula omitted]

We develop a probabilistic approach to study the volumetric and geometric properties of unit balls Bq,1n of finite-dimensional Lorentz sequence spaces ℓq,1n. More precisely, we show that the empirical distribution of a random vector X(n) uniformly distributed on its volume normalized unit ball converges weakly to a compactly supported symmetric probability distribution with explicitly given density; as a consequence we obtain a weak Poincaré-Maxwell-Borel principle for any fixed number k∈N of coordinates of X(n) as n→∞. Moreover, we prove a central limit theorem for the largest coordinate of X(n), demonstrating a quite different behavior than in the case of the ℓqn balls, where a Gumbel distribution appears in the limit. Finally, we prove a Schechtman-Schmuckenschläger type result for the asymptotic volume of intersections of volume normalized ℓq,1n and ℓpn balls.

A mixed-dimensional formulation for the simulation of slender structures immersed in an incompressible flow

We consider the simulation of slender structures immersed in a three-dimensional (3D) flow. By exploiting the special geometric configuration of the slender structures, this particular problem can be modeled by mixed-dimensional coupled equations. Taking advantage of the slenderness of the structure and thus considering 3D/1D coupled problems raise several challenges and difficulties. From a mathematical point of view, these include defining well-posed trace operators of co-dimension two. On the computational standpoint, the non-standard mathematical formulation makes it difficult to ensure the accuracy of the solutions obtained with the mixed-dimensional discrete formulation as compared to a fully resolved one. Here we proposed to circumvent theses issues by imposing the fluid–structure coupling conditions on the 2D fluid–structure interface but in a reduced way still taking advantage of the 1D dynamic of the slender structure. We consider the Navier–Stokes equations for the fluid and a Timoshenko beam model for the structure. We complement these models with a mixed-dimensional version of the fluid–structure interface conditions, based on the projection of kinematic coupling conditions on a finite-dimensional Fourier space on each beam cross section. Furthermore, we develop a discrete fictitious domain formulation within the framework of the finite element method, establish the energy stability of the scheme, provide extensive numerical evidence of the accuracy of the discrete formulation, notably with respect to a fully resolved (ALE based) model and a standard reduced modeling approach.

Fractional Hermite–Hadamard, Newton–Milne, and Convexity Involving Arithmetic–Geometric Mean-Type Inequalities in Hilbert and Mixed-Norm Morrey Spaces ℓq(·)(Mp(·),v(·)) with Variable Exponents

Function spaces play a crucial role in the study and application of mathematical inequalities. They provide a structured framework within which inequalities can be formulated, analyzed, and applied. They allow for the extension of inequalities from finite-dimensional spaces to infinite-dimensional contexts, which is crucial in mathematical analysis. In this note, we develop various new bounds and refinements of different well-known inequalities involving Hilbert spaces in a tensor framework as well as mixed Moore norm spaces with variable exponents. The article begins with Newton–Milne-type inequalities for differentiable convex mappings. Our next step is to take advantage of convexity involving arithmetic–geometric means and build various new bounds by utilizing self-adjoint operators of Hilbert spaces in tensorial frameworks for different types of generalized convex mappings. To obtain all these results, we use Riemann–Liouville fractional integrals and develop several new identities for these operator inequalities. Furthermore, we present some examples and consequences for transcendental functions. Moreover, we developed the Hermite–Hadamard inequality in a new and significant way by using mixed-norm Moore spaces with variable exponent functions that have not been developed previously with any other type of function space apart from classical Lebesgue space. Mathematical inequalities supporting tensor Hilbert spaces are rarely examined in the literature, so we believe that this work opens up a whole new avenue in mathematical inequality theory.

On function-on-function linear quantile regression

We present two innovative functional partial quantile regression algorithms designed to accurately and efficiently estimate the regression coefficient function within the function-on-function linear quantile regression model. Our algorithms utilize functional partial quantile regression decomposition to effectively project the infinite-dimensional response and predictor variables onto a finite-dimensional space. Within this framework, the partial quantile regression components are approximated using a basis expansion approach. Consequently, we approximate the infinite-dimensional function-on-function linear quantile regression model using a multivariate quantile regression model constructed from these partial quantile regression components. To evaluate the efficacy of our proposed techniques, we conduct a series of Monte Carlo experiments and analyze an empirical dataset, demonstrating superior performance compared to existing methods in finite-sample scenarios. Our techniques have been implemented in the ffpqr package in .

The Common Basis Complex and the Partial Decomposition Poset

Abstract For a finite-dimensional vector space $V$, the common basis complex of $V$ is the simplicial complex whose vertices are the proper non-zero subspaces of $V$, and $\sigma $ is a simplex if and only if there exists a basis $B$ of $V$ that contains a basis of $S$ for all $S\in \sigma $. This complex was introduced by Rognes in 1992 in connection with stable buildings. In this article, we prove that the common basis complex is homotopy equivalent to the proper part of the poset of partial direct sum decompositions of $V$. Moreover, we establish this result in a more general combinatorial context, including the case of free groups, matroids, vector spaces with non-degenerate sesquilinear forms, and free modules over commutative Hermite rings, such as local rings or Dedekind domains.

Vector valued piecewise continuous almost automorphic functions and some consequences

In the present work, for X a Banach space, the notion of piecewise continuous Z-almost automorphic functions with values in finite dimensional spaces is extended to piecewise continuous Z-almost automorphic functions with values in X. Several properties of this class of functions are provided, in particular it is shown that if X is a Banach algebra, then this class of functions constitute also a Banach algebra; furthermore, using the theory of Z-almost automorphic functions, a new characterization of compact almost automorphic functions is given. As consequences, with the help of Z-almost automorphic functions, it is presented a simple proof of the characterization of almost automorphic sequences by compact almost automorphic functions; the method permits us to give explicit examples of compact almost automorphic functions which are not almost periodic. Also, using the theory developed here, it is shown that almost automorphic solutions of differential equations with piecewise constant argument are in fact compact almost automorphic. Finally, it is proved that the classical solution of the 1D heat equation with continuous Z-almost automorphic source is also continuous Z-almost automorphic; furthermore, we comment applications to the existence and uniqueness of the asymptotically continuous Z-almost automorphic mild solution to abstract integro-differential equations with nonlocal initial conditions.

On the matricial truncated moment problem. II

We continue the study of truncated matrix-valued moment problems begun in [12]. Let q∈N. Suppose that (X,X) is a measurable space and E is a finite-dimensional vector space of measurable mappings of X into Hq, the Hermitian q×q matrices. A linear functional Λ on E is called a moment functional if there exists a positive Hq-valued measure μ on (X,X) such that Λ(F)=∫X〈F,dμ〉 for F∈E.In this paper a number of special topics on the truncated matricial moment problem are treated. We restate a result from [11] to obtain a matricial version of the flat extension theorem. Assuming that X is a compact space and all elements of E are continuous on X we characterize moment functionals in terms of positivity and obtain an ordered maximal mass representing measure for each moment functional. The set of masses of representing measures at a fixed point and some related sets are studied. The class of commutative matrix moment functionals is investigated. We generalize the apolar scalar product for homogeneous polynomials to the matrix case and apply this to the matricial truncated moment problem.