scholarly journals The images of noncommutative polynomials evaluated on the quaternion algebra

2020 ◽  
pp. 2150074
Author(s):  
Sergey Malev
Keyword(s):  
Vector Space ◽  
Quaternion Algebra ◽  
Arbitrary Field ◽  
Multilinear Polynomial ◽  
Noncommutative Polynomials

Let [Formula: see text] be a multilinear polynomial in several noncommuting variables with coefficients in an arbitrary field [Formula: see text]. Kaplansky conjectured that for any [Formula: see text], the image of [Formula: see text] evaluated on the set [Formula: see text] of [Formula: see text] by [Formula: see text] matrices is a vector space. In this paper, we settle the analogous conjecture for a quaternion algebra.

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.

scholarly journals Right and Left Orthogonality

1961 ◽  
Vol 4 (2) ◽  
pp. 182-184
Author(s):  
Jonathan Wild
Keyword(s):  
Vector Space ◽  
Bilinear Form ◽  
Simple Proof ◽  
Arbitrary Field ◽  
Image Position

Let V be a vector space over an arbitrary field F. In V a bilinear formis given. If f is symmetric [(x, y) ≡ (y, x)] or skew-symmetric [(x, y) + (y, x) ≡ 0], then1Thus right and left orthogonality coincide. It is well known that (1) implies conversely that f is either symmetric or skew-symmetric in V. We wish to give a simple proof of this result.

scholarly journals On the Index of a Quadratic Form

1958 ◽  
Vol 1 (3) ◽  
pp. 180-180
Author(s):  
Jonathan Wild
Keyword(s):  
Quadratic Form ◽  
Vector Space ◽  
Bilinear Form ◽  
Direct Proof ◽  
Symmetric Bilinear Form ◽  
Arbitrary Field ◽  
Analytic Geometry ◽  
Fundamental Fact

Given a vector space V = {x, y, ...} over an arbitrary field. In V a symmetric bilinear form (x,y) i s given. A subspace W is called totally isotropic [t.i.] if (x,y) = 0 for every pair x W, y W.Let Vn and Vm be two t.i. subspaces of V; n < m. Lower indices always indicate dimensions. It is a well known and fundamental fact of analytic geometry that there exists a t.i. subspace Wm of V containing Vn [cf. Dieudonné: Les Groupes classiques , P. 18]. As no simple direct proof seems to be available, we propose to supply one.

The images of non-commutative polynomials evaluated on 2 × 2 matrices over an arbitrary field

2014 ◽  
Vol 13 (06) ◽  
pp. 1450004 ◽  
Author(s):  
Sergey Malev
Keyword(s):  
Partial Solution ◽  
Arbitrary Field ◽  
Multilinear Polynomial ◽  
Homogeneous Polynomials ◽  
Quadratic Extensions

Let p be a multilinear polynomial in several non-commuting variables with coefficients in an arbitrary field K. Kaplansky conjectured that for any n, the image of p evaluated on the set Mn(K) of n × n matrices is either zero, or the set of scalar matrices, or the set sl n(K) of matrices of trace 0, or all of Mn(K). This conjecture was proved for n = 2 when K is closed under quadratic extensions. In this paper, the conjecture is verified for K = ℝ and n = 2, also for semi-homogeneous polynomials p, with a partial solution for an arbitrary field K.

A Homomorphism in Exterior Algebra

1964 ◽  
Vol 16 ◽  
pp. 166-168 ◽  
Author(s):  
B. Kostant ◽  
A. Novikoff
Keyword(s):  
Vector Space ◽  
Exterior Algebra ◽  
Vector Spaces ◽  
Arbitrary Field ◽  
Dual Basis ◽  
Set Up ◽  
The Ideal

In the following, V is a vector space over an arbitrary field F, dimFV = n. Let {e1, . . . , en} be a basis for V, and {f1, . . . , fn} be the dual basis for and , then the operators *(u) and i(g) (exterior and inner multiplication by u and g respectively) set up an equivalence between the ideal = range of ∊(u) and the sub-algebra = range of i(g) considered as vector spaces. That is, ∊(u)i(g) is the identity on , i(g) ∊(u) is the identity on .

The Invariant Subspace Lattice of a Linear Transformation

1967 ◽  
Vol 19 ◽  
pp. 810-822 ◽  
Author(s):  
L. Brickman ◽  
P. A. Fillmore
Keyword(s):  
Vector Space ◽  
Linear Transformation ◽  
Invariant Subspaces ◽  
Arbitrary Field ◽  
Dimensional Vector ◽  
Subspace Lattice ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Algebraic Properties

The purpose of this paper is to study the lattice of invariant subspaces of a linear transformation on a finite-dimensional vector space over an arbitrary field. Among the topics discussed are structure theorems for such lattices, implications between linear-algebraic properties and lattice-theoretic properties, nilpotent transformations, and the conditions for the isomorphism of two such lattices. These topics correspond roughly to §§2, 3, 4, and 5 respectively.

On Semi - Pre irresolute Topological Vector Space

2018 ◽  
Vol 14 (3) ◽  
pp. 184-192
Author(s):  
Radhi Ali ◽  
◽  
Jalal Hussein Bayati ◽  
Suhad Hameed
Keyword(s):  
Vector Space ◽  
Topological Vector Space

Aplikasi Deteksi Kemiripan Tugas Paper

2017 ◽  
Vol 15 (2) ◽  
pp. 5
Author(s):  
Anthony Anggrawan ◽  
Azhari
Keyword(s):  
Information Retrieval ◽  
Vector Space ◽  
Vector Space Model ◽  
Mean Average Precision ◽  
Average Precision ◽  
Information Searching ◽  
Space Model ◽  
Model Method

Information searching based on users’ query, which is hopefully able to find the documents based on users’ need, is known as Information Retrieval. This research uses Vector Space Model method in determining the similarity percentage of each student’s assignment. This research uses PHP programming and MySQL database. The finding is represented by ranking the similarity of document with query, with mean average precision value of 0,874. It shows how accurate the application with the examination done by the experts, which is gained from the evaluation with 5 queries that is compared to 25 samples of documents. If the number of counted assignments has higher similarity, thus the process of similarity counting needs more time, it depends on the assignment’s number which is submitted.

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