Finite-dimensional Vector Space Research Articles

Semigroups of linear transformations whose restrictions belong to a general linear group

Let V be a vector space and U a fixed subspace of V. We denote the semigroup of all linear transformations on V under composition of functions by L(V). In this paper, we study the semigroup of all linear transformations on V whose restrictions belong to the general linear group GL(U), denoted by L GL ( U ) ( V ) . More precisely, we consider the subsemigroup L GL ( U ) ( V ) = { α ∈ L ( V ) : α | U ∈ GL ( U ) } of L(V). In this work, Green’s relations and ideals of this semigroup are described. Then we also determine the minimal ideal and the set of all minimal idempotents of it. Moreover, we establish an isomorphism theorem when V is a finite dimensional vector space over a finite field. Finally, we find its generating set.

TQFTs and quantum computing

Quantum computing is captured in the formalism of the monoidal subcategory of VectC generated by C2 — in particular, quantum circuits are diagrams in VectC — while topological quantum field theories, in the sense of Atiyah, are diagrams in VectC indexed by cobordisms. We initiate a program to formalize this connection. In doing so, we equip cobordisms with machinery for producing linear maps by parallel transport along curves under a connection and then assemble these structures into a higher category. Finite-dimensional complex vector spaces and linear maps between them are given a suitable higher categorical structure which we call FVectC. We realize quantum circuits as images of cobordisms under monoidal functors from these modified cobordisms to FVectC, which are computed by taking parallel transports of vectors and then combining the results in a pattern encoded in the domain category.

Centrality measures-based sensitivity analysis and entropy of nonzero component graphs

The nonzero component graph of a finite-dimensional vector space over a finite field is a graph whose vertices are the nonzero vectors in the vector space, and any two vertices are adjacent if the corresponding linear combination contains a common basis vector. In this paper, we discuss the centrality measures and entropy of the nonzero component graph and also analyze the sensitivity of the graph using the centrality measures.

A general formulation of reweighted least squares fitting

We present a generalized formulation for reweighted least squares approximations. The goal of this article is twofold: firstly, to prove that the solution of such problem can be expressed as a convex combination of certain interpolants when the solution is sought in any finite-dimensional vector space; secondly, to provide a general strategy to iteratively update the weights according to the approximation error and apply it to the spline fitting problem. In the experiments, we provide numerical examples for the case of polynomials and splines spaces. Subsequently, we evaluate the performance of our fitting scheme for spline curve and surface approximation, including adaptive spline constructions.

Controllability and observability of conformable fractional finite dimensional linear systems

In this research paper, we investigate the controllability and observability of continuous-time fractional dynamical systems. The characterisation of all fractional continuous one-parameter semi groups on a finite dimensional vector space as fractional matrix-valued exponential functions is given. This characterisation is used to express the solution of continuous-time fractional dynamical system. Furthermore, we analyse controllability and observability of linear conformable fractional systems in the mathematical context of linear operators on a finite dimensional vector space. By defining the controllability and observability maps, we establish necessary and sufficient conditions for controllability and observability of this type of systems. We also present the Hautus test for controllability and observability, and introduce controllability and observability Gramians, both for finite and infinite time interval. Finally, we present several illustrative applications and examples to validate all the obtained results.

Robust Variational Physics-Informed Neural Networks

We introduce a Robust version of the Variational Physics-Informed Neural Networks method (RVPINNs). As in VPINNs, we define the quadratic loss functional in terms of a Petrov–Galerkin-type variational formulation of the PDE problem: the trial space is a (Deep) Neural Network (DNN) manifold, while the test space is a finite-dimensional vector space. Whereas the VPINN’s loss depends upon the selected basis functions of a given test space, herein, we minimize a loss based on the discrete dual norm of the residual. The main advantage of such a loss definition is that it provides a reliable and efficient estimator of the true error in the energy norm under the assumption of the existence of a local Fortin operator. We test the performance and robustness of our algorithm in several advection–diffusion problems. These numerical results perfectly align with our theoretical findings, showing that our estimates are sharp.

Quasitopological vector spaces

In this paper, motivated by a diffeological vector space with the D-topology, we introduce the concept of quasitopological vector space, which is a vector space with a topology satisfying the conditions that the vector addition is separately continuous in each variable, and the scalar multiplication is jointly continuous. The following results are proved: (1) Every quasitopological finite-dimensional vector space is a topological vector space. (2) For each infinite-dimensional vector space, we define a T1 non-Hausdorff topology on it such that it is a quasitopological vector space but is not a topological vector space.

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.

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.

The quadratic sum problem for symplectic pairs

Let (b,u) be a pair consisting of a symplectic form b on a finite-dimensional vector space V over a field F, and of a b-alternating endomorphism u of V (i.e. b(x,u(x))=0 for all x in V).Let p and q be arbitrary polynomials of degree 2 with coefficients in F. We characterize, in terms of the invariant factors of u, the condition that u splits into u1+u2 for some pair (u1,u2) of b-alternating endomorphisms such that p(u1)=q(u2)=0.

Lipschitz-free spaces and subsets of finite-dimensional spaces

We consider two questions on the geometry of Lipschitz-free $p$ -spaces $\mathcal {F}_p$ , where $0< p\leq 1$ , over subsets of finite-dimensional vector spaces. We solve an open problem and show that if $(\mathcal {M}, \rho )$ is an infinite doubling metric space (e.g. an infinite subset of an Euclidean space), then $\mathcal {F}_p (\mathcal {M}, \rho ^\alpha )\simeq \ell _p$ for every $\alpha \in (0,\,1)$ and $0< p\leq 1$ . An upper bound on the Banach–Mazur distance between the spaces $\mathcal {F}_p ([0, 1]^d, |\cdot |^\alpha )$ and $\ell _p$ is given. Moreover, we tackle a question due to Albiac et al. [4] and expound the role of $p$ , $d$ for the Lipschitz constant of a canonical, locally coordinatewise affine retraction from $(K, |\cdot |_1)$ , where $K=\cup _{Q\in \mathcal {R}} Q$ is a union of a collection $\emptyset \neq \mathcal {R} \subseteq \{ Rw + R[0,\,1]^d: w\in \mathbb {Z}^d\}$ of cubes in $\mathbb {R}^d$ with side length $R>0$ , into the Lipschitz-free $p$ -space $\mathcal {F}_p (V, |\cdot |_1)$ over their vertices.

Choi matrices revisited. II

In this paper, we consider all possible variants of Choi matrices of linear maps, and show that they are determined by non-degenerate bilinear forms on the domain space. We will do this in the setting of finite dimensional vector spaces. In case of matrix algebras, we characterize all variants of Choi matrices which retain the usual correspondences between k-superpositivity and Schmidt number ≤k as well as k-positivity and k-block-positivity. We also compare de Pillis’ definition [Pac. J. Math. 23, 129–137 (1967)] and Choi’s definition [Linear Algebra Appl. 10, 285–290 (1975)], which arise from different bilinear forms.

Set-independence graphs of vector spaces and partial quasigroups

As a generalization of independence graphs of vector spaces and groups, we introduce the notions of set-independence graphs of vector spaces and partial quasigroups. The former are characterized for finite-dimensional vector spaces over finite fields. Further, we prove that every finite simple graph is isomorphic to either the independence graph of a partial quasigroup or an induced subgraph of the latter. We also prove that isomorphic partial quasigroups give rise to isomorphic set-independence graphs. As an illustrative example, all finite graphs of order $n\leq 5$ are identified with the independence graph of a partial quasigroup of the same order.

A note on bireflectional elements of an algebra

A classical theorem of Wonenburger, Djokovič, Hoffmann and Paige [2, 3, 7] states that an element of the general linear group of a finite-dimensional vector space is the product of two involutions if and only if it is similar to its inverse. We give a very elementary proof of this result when the underlying field F is algebraically closed with characteristic other than 2. In that situation, the result is generalized to the group of invertibles of any finite-dimensional algebra over F .

Polyhedral compactifications, I

Abstract In this work we describe horofunction compactifications of metric spaces and finite-dimensional real vector spaces through asymmetric metrics and asymmetric polyhedral norms by means of nonstandard methods, that is, by ultrapowers of the spaces at hand. The polyhedral compactifications of the vector spaces carry the structure of stratified spaces with the strata indexed by dual faces of the polyhedral unit ball. Explicit neighborhood bases and descriptions of the horofunctions are provided.

Classification of Consistent Systems of Handlebody Group Representations

Abstract The classifying spaces of handlebody groups form a modular operad. Algebras over the handlebody operad yield systems of representations of handlebody groups that are compatible with gluing. We prove that algebras over the modular operad of handlebodies with values in an arbitrary symmetric monoidal bicategory $\mathcal{M}$ (we introduce for these the name ansular functor) are equivalent to self-dual balanced braided algebras in $\mathcal{M}$. After specialization to a linear framework, this proves that consistent systems of handlebody group representations on finite-dimensional vector spaces are equivalent to ribbon Grothendieck-Verdier categories in the sense of Boyarchenko-Drinfeld. Additionally, it produces a concrete formula for the vector space assigned to an arbitrary handlebody in terms of a generalization of Lyubashenko’s coend. Our main result can be used to obtain an ansular functor from vertex operator algebras subject to mild finiteness conditions. This includes examples of vertex operator algebras whose representation category has a non-exact monoidal product.

Realizations of unisingular representations by hyperelliptic Jacobians

A representation of a finite group G on a finite dimensional vector space V is called unisingular if every has 1 as an eigenvalue in its action on V. In this paper we show that certain unisingular representations can be realized as mod 2 representations of hyperelliptic Jacobians over Q. We additionally identify new unisingular representations of the symmetric and alternating groups.

Sums of two square-zero selfadjoint or skew-selfadjoint endomorphisms

Let V be a finite-dimensional vector space over a field F, equipped with a symmetric or alternating non-degenerate bilinear form b. When the characteristic of F is not 2, we characterize the endomorphisms u of V that split into u=a1+a2 for some pair (a1,a2) of b-selfadjoint (respectively, b-skew-selfadjoint) endomorphisms of V such that (a1)2=(a2)2=0. In the characteristic 2 case, we obtain a similar classification for the endomorphisms of V that split into the sum of two square-zero b-alternating endomorphisms of V when b is alternating (an endomorphism v is called b-alternating whenever b(x,v(x))=0 for all x∈V).Finally, if the field F is equipped with a non-identity involution, we characterize the pairs (h,u) in which h is a Hermitian form on a finite-dimensional space over F, and u is the sum of two square-zero h-selfadjoint endomorphisms.

A lower bound for the dimension of tetrahedral splines in large degree

Splines are piecewise polynomial functions which are continuously differentiable to some order r. For a fixed integer d the space of splines of degree at most d is a finite dimensional vector space, and a largely open problem in numerical analysis is to determine its dimension. While considerable attention has been given to this problem in the bivariate setting, the literature on trivariate splines is less conclusive. In particular, the dimension of generic trivariate splines is not known even in large degree when r>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r>1$$\\end{document}. In this paper we use a bound we previously derived for splines on vertex stars to compute a new lower bound on the dimension of trivariate splines in large enough degree. We illustrate in several examples that our formula gives the exact dimension of the spline space in large enough degree if vertex positions are generic. In contrast, for splines continuously differentiable of order r>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r>1$$\\end{document}, every lower bound in the literature diverges (often significantly) in large degree from the dimension of the spline space in these examples. We derive the bound using commutative and homological algebra.