Groups Preserving a Class of Bilinear Functions

1985 ◽  
Vol 28 (3) ◽  
pp. 267-271
Author(s):  
W. H. Greub ◽  
J. Malzan ◽  
J. R. Vanstone
Keyword(s):  
Vector Space ◽  
Bilinear Function ◽  
Finite Dimensional ◽  
Euclidean Vector Space ◽  
Bilinear Functions

AbstractGiven a finite dimensional Euclidean vector space V, ( , ) and an involution τ of V, one can form the bilinear function ( , )τ defined by (x, y)τ = (τ(x), y), x,y ∊ V.Let O(τ) = {ϕ ∊ GL(V)|(ϕx, ϕy)τ = (x, y)τ}.If t is self-adjoint the structure of O(t) is well known. The purpose of this paper is to detemine the structure of O(t) in the general case. This structure is also determined in the complex and quaternionic case, as well as the case when the condition on t is replaced by τ2 = ∊ι, ∊ ∈ ℝ.

scholarly journals Product rule for vector fractional derivatives

2012 ◽  
Vol 15 (3) ◽  
Author(s):  
Diogo Bolster ◽  
Mark Meerschaert ◽  
Alla Sikorskii
Keyword(s):  
Vector Space ◽  
Fractional Derivatives ◽  
Fourier Transforms ◽  
Product Rule ◽  
Finite Dimensional ◽  
Euclidean Vector Space ◽  
Derivatives Of

AbstractThis paper establishes a product rule for fractional derivatives of a realvalued function defined on a finite dimensional Euclidean vector space. The proof uses Fourier transforms.

scholarly journals FINITE-DIMENSIONAL ORDERED VECTOR SPACES WITH RIESZ INTERPOLATION AND EFFROS–SHEN’S UNIMODULARITY CONJECTURE

2016 ◽  
Vol 101 (2) ◽  
pp. 277-287
Author(s):  
AARON TIKUISIS
Keyword(s):  
Vector Space ◽  
Inductive Limit ◽  
Space Structure ◽  
Vector Spaces ◽  
Ordered Vector Space ◽  
Finite Dimensional ◽  
Ordered Vector Spaces

It is shown that, for any field $\mathbb{F}\subseteq \mathbb{R}$, any ordered vector space structure of $\mathbb{F}^{n}$ with Riesz interpolation is given by an inductive limit of a sequence with finite stages $(\mathbb{F}^{n},\mathbb{F}_{\geq 0}^{n})$ (where $n$ does not change). This relates to a conjecture of Effros and Shen, since disproven, which is given by the same statement, except with $\mathbb{F}$ replaced by the integers, $\mathbb{Z}$. Indeed, it shows that although Effros and Shen’s conjecture is false, it is true after tensoring with $\mathbb{Q}$.

scholarly journals Sets of idempotents that generate the semigroup of singular endomorphisms of a finite-dimensional vector space

1982 ◽  
Vol 25 (2) ◽  
pp. 133-139 ◽  
Author(s):  
R. J. H. Dawlings
Keyword(s):  
Vector Space ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Mathematical System

IfMis a mathematical system and EndMis the set of singular endomorphisms ofM, then EndMforms a semigroup under composition of mappings. A number of papers have been written to determine the subsemigroupSMof EndMgenerated by the idempotentsEMof EndMfor different systemsM. The first of these was by J. M. Howie [4]; here the case ofMbeing an unstructured setXwas considered. Howie showed that ifXis finite, then EndX=Sx.

On the Vanishing of a (G, σ) Product in a (G, σ) Space

1970 ◽  
Vol 22 (2) ◽  
pp. 363-371 ◽  
Author(s):  
K. Singh
Keyword(s):  
Vector Space ◽  
Multiplicative Group ◽  
Cartesian Product ◽  
Arbitrary Field ◽  
Necessary And Sufficient Condition ◽  
Tensor Space ◽  
Linear Character ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Necessary And Sufficient

In this paper, we shall construct a vector space, called the (G, σ) space, which generalizes the tensor space, the Grassman space, and the symmetric space. Then we shall determine a necessary and sufficient condition that the (G, σ) product of the vectors x1, x2, …, xn is zero.1. Let G be a permutation group on I = {1, 2, …, n} and F, an arbitrary field. Let σ be a linear character of G, i.e., σ is a homomorphism of G into the multiplicative group F* of F.For each i ∈ I, let Vi be a finite-dimensional vector space over F. Consider the Cartesian product W = V1 × V2 × … × Vn.1.1. Definition. W is called a G-set if and only if Vi = Vg(i) for all i ∊ I, and for all g ∊ G.

On Some Invariants of a Bilinear Form

1961 ◽  
Vol 4 (3) ◽  
pp. 261-264
Author(s):  
Jonathan Wild
Keyword(s):  
Vector Space ◽  
Bilinear Form ◽  
Arbitrary Field ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Special Cases ◽  
Orthogonal Subspace

Let E be a finite dimensional vector space over an arbitrary field. In E a bilinear form is given. It associates with every sub s pa ce V its right orthogonal sub space V* and its left orthogonal subspace *V. In general we cannot expect that dim V* = dim *V. However this relation will hold in some interesting special cases.

THE REGULAR PART OF A SEMIGROUP OF LINEAR TRANSFORMATIONS WITH RESTRICTED RANGE

2017 ◽  
Vol 103 (3) ◽  
pp. 402-419 ◽  
Author(s):  
WORACHEAD SOMMANEE ◽  
KRITSADA SANGKHANAN
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Dimensional Subspace ◽  
Finite Dimensional Subspace ◽  
Restricted Range ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Finite Dimensional ◽  
Idempotent Rank ◽  
Image Position

Let$V$be a vector space and let$T(V)$denote the semigroup (under composition) of all linear transformations from$V$into$V$. For a fixed subspace$W$of$V$, let$T(V,W)$be the semigroup consisting of all linear transformations from$V$into$W$. In 2008, Sullivan [‘Semigroups of linear transformations with restricted range’,Bull. Aust. Math. Soc.77(3) (2008), 441–453] proved that$$\begin{eqnarray}\displaystyle Q=\{\unicode[STIX]{x1D6FC}\in T(V,W):V\unicode[STIX]{x1D6FC}\subseteq W\unicode[STIX]{x1D6FC}\} & & \displaystyle \nonumber\end{eqnarray}$$is the largest regular subsemigroup of$T(V,W)$and characterized Green’s relations on$T(V,W)$. In this paper, we determine all the maximal regular subsemigroups of$Q$when$W$is a finite-dimensional subspace of$V$over a finite field. Moreover, we compute the rank and idempotent rank of$Q$when$W$is an$n$-dimensional subspace of an$m$-dimensional vector space$V$over a finite field$F$.

EMBEDDINGS OF FREE TOPOLOGICAL VECTOR SPACES

2019 ◽  
Vol 101 (2) ◽  
pp. 311-324
Author(s):  
ARKADY LEIDERMAN ◽  
SIDNEY A. MORRIS
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Dimensional Subspace ◽  
Vector Subspace ◽  
Compact Metric Space ◽  
Topological Vector Spaces ◽  
Finite Dimensional ◽  
Bernstein Set ◽  
Dimensional Cube ◽  
Open Question

It is proved that the free topological vector space $\mathbb{V}([0,1])$ contains an isomorphic copy of the free topological vector space $\mathbb{V}([0,1]^{n})$ for every finite-dimensional cube $[0,1]^{n}$, thereby answering an open question in the literature. We show that this result cannot be extended from the closed unit interval $[0,1]$ to general metrisable spaces. Indeed, we prove that the free topological vector space $\mathbb{V}(X)$ does not even have a vector subspace isomorphic as a topological vector space to $\mathbb{V}(X\oplus X)$, where $X$ is a Cook continuum, which is a one-dimensional compact metric space. This is also shown to be the case for a rigid Bernstein set, which is a zero-dimensional subspace of the real line.

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