Finite-dimensional Vector Space Research Articles

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.

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.

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.

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.

An Algorithm for Detecting Sentence Validity

In this paper, we introduce an algorithm for determining the grammatical validity of a sentence. We take a similar approach as in (Preller, 2007) and (Lambek, Type Grammar Revisited, 1999) (Lambek, From word to sentence: a computational algebraic approach to grammar, 2008) by encoding the English words based on word type which we call components. A sentence can be described both algebraically and geometrically. Our algorithm generates the geometric portion called underlinks from the generalized reductions of the algebraic portion. Underlinks uniquely determine the reduction of the components leading to the empty string. This is the mathematical basis for determining if a sentence is valid. We also provide a proof for the algorithm’s time complexity of O(n2) along with a Python implementation. This paper is part of a bigger project based on (Coecke, Mehrnoosh, Clarky, 2010) where we explore the combination of a sentence’s grammar and meaning. This is done by combining two compact closed categories; pregroups represent the grammar of a sentence, and finite dimensional vector spaces describe the meaning of a sentence. Together one compact closed category is created, representing both aspects of the sentence.

On the truncated matricial moment problem. I

This paper is about the general truncated matrix-valued moment problem. Let Hq denote the complex Hermitian q×q-matrices, 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. 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.We prove a matricial version of the Richter–Tchakaloff theorem which states that each moment functional on E has a finitely atomic representing measure. It is shown that strictly positive linear functionals on E are moment functionals. For a moment functional Λ, we study the set of atoms W(Λ) and define two Carathéodory numbers of Λ. Further, we define and investigate the core set V(Λ) which generalizes the core variety from the scalar case to the matricial setting. A main result of the paper is the equality W(Λ)=V(Λ).

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.

Generalized positive energy representations of groups of jets

Let V be a finite-dimensional real vector space and K a compact simple Lie group with Lie algebra \mathfrak{k} . Consider the Fréchet–Lie group G := J_0^\infty(V; K) of \infty -jets at 0\in V of smooth maps V \to K , with Lie algebra \mathfrak{g}=J_0^\infty(V; \mathfrak{k}) . Let P be a Lie group and write \mathfrak{p}:=\operatorname{Lie}(P) . Let \alpha be a smooth P -action on G . We study smooth projective unitary representations \bar{\rho} of G\rtimes_\alpha P that satisfy a so-called generalized positive energy condition. In particular, this class captures representations that are in a suitable sense compatible with a KMS state on the von Neumann algebra generated by \bar{\rho}(G) . We show that this condition imposes severe restrictions on the derived representation d\bar{\rho} of \mathfrak{g}\rtimes \mathfrak{p} , leading in particular to sufficient conditions for {\bar{\rho}}|_{G} to factor through J_0^2(V; K) , or even through K .

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.