scholarly journals Neutrosophic Quadruple Vector Spaces and Their Properties

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 758 ◽  
Author(s):  
Vasantha Kandasamy W.B. ◽  
Ilanthenral Kandasamy ◽  
Florentin Smarandache
Keyword(s):  
Finite Field ◽  
Final Section ◽  
Vector Spaces ◽  
Complex Numbers ◽  
Dimensional Vector ◽  
Characteristic P ◽  
Finite Dimensional ◽  
Classical Property ◽  
First Time

In this paper authors for the first time introduce the concept of Neutrosophic Quadruple (NQ) vector spaces and Neutrosophic Quadruple linear algebras and study their properties. Most of the properties of vector spaces are true in case of Neutrosophic Quadruple vector spaces. Two vital observations are, all quadruple vector spaces are of dimension four, be it defined over the field of reals R or the field of complex numbers C or the finite field of characteristic p, Z p ; p a prime. Secondly all of them are distinct and none of them satisfy the classical property of finite dimensional vector spaces. So this problem is proposed as a conjecture in the final section.

scholarly journals Multi-variable quaternionic spectral analysis

2021 ◽  
Vol 41 (3) ◽  
pp. 335-379
Author(s):  
Ilwoo Cho ◽  
Palle E.T. Jorgensen
Keyword(s):  
Spectral Analysis ◽  
Functional Equations ◽  
Linear Functional ◽  
Vector Spaces ◽  
Complex Numbers ◽  
Dimensional Vector ◽  
Finite Dimensional ◽  
Linear Functional Equations ◽  
Non Linear

In this paper, we consider finite dimensional vector spaces \(\mathbb{H}^n\) over the ring \(\mathbb{H}\) of all quaternions. In particular, we are interested in certain functions acting on \(\mathbb{H}^n\), and corresponding functional equations. Our main results show that (i) all quaternions of \(\mathbb{H}\) are classified by the spectra of their realizations under representation, (ii) all vectors of \(\mathbb{H}^n\) are classified by a canonical extended setting of (i), and (iii) the usual spectral analysis on the matricial ring \(M_n(\mathbb{C})\) of all \((n \times n)\)-matrices over the complex numbers \(\mathbb{C}\) has close connections with certain "non-linear" functional equations on \(\mathbb{H}^n\) up to the classification of (ii).

scholarly journals On nonzero component graph of vector spaces over finite fields

2017 ◽  
Vol 16 (01) ◽  
pp. 1750007 ◽  
Author(s):  
Angsuman Das
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Clique Number ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Nonzero Component ◽  
Component Graph

In this paper, we study nonzero component graph [Formula: see text] of a finite-dimensional vector space [Formula: see text] over a finite field [Formula: see text]. We show that the graph is Hamiltonian and not Eulerian. We also characterize the maximal cliques in [Formula: see text] and show that there exists two classes of maximal cliques in [Formula: see text]. We also find the exact clique number of [Formula: see text] for some particular cases. Moreover, we provide some results on size, edge-connectivity and chromatic number of [Formula: see text].

Plane polygons revisited

1982 ◽  
Vol 19 (A) ◽  
pp. 113-122 ◽  
Author(s):  
B. H. Neumann
Keyword(s):  
Vector Spaces ◽  
Complex Numbers ◽  
Dimensional Vector ◽  
Finite Dimensional ◽  
Electrical Engineers ◽  
Equilateral Triangles ◽  
Arbitrary Plane ◽  
Arbitrary Triangle ◽  
Napoleon’S Theorem ◽  
Current Systems

A method used by electrical engineers to analyse polyphase alternating current systems suggests a generalisation to arbitrary plane polygons of a theorem on triangles nowadays known, for obscure reasons, as ‘Napoleon's Theorem': the centroids of equilateral triangles erected on the sides of an arbitrary triangle form the vertices of an equilateral triangle. The generalisation to other polygons uses a construction first studied by C.-A. Laisant in 1877; results of Jesse Douglas (1940) and the author (1941) are re-derived by means of the elementary algebra of finite-dimensional vector spaces over the field of complex numbers.

scholarly journals On the Index of a Symmetric Form

1960 ◽  
Vol 3 (3) ◽  
pp. 293-295
Author(s):  
Jonathan Wild
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Bilinear Form ◽  
Symmetric Bilinear Form ◽  
Dimensional Vector ◽  
Symmetric Form ◽  
Characteristic P ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional

Let E be a finite dimensional vector space over a finite field of characteristic p > 0; dim E = n. Let (x,y) be a symmetric bilinear form in E. The radical Eo of this form is the subspace consisting of all the vectors x which satisfy (x,y) = 0 for every y ϵ E. The rank r of our form is the codimension of the radical.

Plane polygons revisited

1982 ◽  
Vol 19 (A) ◽  
pp. 113-122 ◽  
Author(s):  
B. H. Neumann
Keyword(s):  
Vector Spaces ◽  
Complex Numbers ◽  
Dimensional Vector ◽  
Finite Dimensional ◽  
Electrical Engineers ◽  
Equilateral Triangles ◽  
Arbitrary Plane ◽  
Arbitrary Triangle ◽  
Napoleon’S Theorem ◽  
Current Systems

A method used by electrical engineers to analyse polyphase alternating current systems suggests a generalisation to arbitrary plane polygons of a theorem on triangles nowadays known, for obscure reasons, as ‘Napoleon's Theorem': the centroids of equilateral triangles erected on the sides of an arbitrary triangle form the vertices of an equilateral triangle. The generalisation to other polygons uses a construction first studied by C.-A. Laisant in 1877; results of Jesse Douglas (1940) and the author (1941) are re-derived by means of the elementary algebra of finite-dimensional vector spaces over the field of complex numbers.

scholarly journals $q$-Exponential Families

10.37236/1789 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Kent E. Morrison
Keyword(s):  
Finite Field ◽  
Exponential Families ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Finite Sets ◽  
Finite Dimensional ◽  
Exponential Formula ◽  
Positive Integers

We develop an analog of the exponential families of Wilf in which the label sets are finite dimensional vector spaces over a finite field rather than finite sets of positive integers. The essential features of exponential families are preserved, including the exponential formula relating the deck enumerator and the hand enumerator.

scholarly journals An Introduction to $q$-Species

10.37236/1959 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Kent E. Morrison
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Vector Spaces ◽  
Combinatorial Theory ◽  
Dimensional Vector ◽  
Finite Sets ◽  
Enumerative Combinatorics ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Linear Maps

The combinatorial theory of species developed by Joyal provides a foundation for enumerative combinatorics of objects constructed from finite sets. In this paper we develop an analogous theory for the enumerative combinatorics of objects constructed from vector spaces over finite fields. Examples of these objects include subspaces, flags of subspaces, direct sum decompositions, and linear maps or matrices of various types. The unifying concept is that of a "$q$-species", defined to be a functor from the category of finite dimensional vector spaces over a finite field to the category of finite sets.

Hereditary undecidability of some theories of finite structures

1994 ◽  
Vol 59 (4) ◽  
pp. 1254-1262 ◽  
Author(s):  
Ross Willard
Keyword(s):  
Finite Field ◽  
Word Problem ◽  
Vertex Degree ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Finite Graphs ◽  
Finite Dimensional ◽  
Finite Semigroups ◽  
Finite Structures ◽  
Hereditary Undecidability

AbstractUsing a result of Gurevich and Lewis on the word problem for finite semigroups, we give short proofs that the following theories are hereditarily undecidable: (1) finite graphs of vertex-degree at most 3; (2) finite nonvoid sets with two distinguished permutations; (3) finite-dimensional vector spaces over a finite field with two distinguished endomorphisms.

A closer look at the stable completion

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Inverse Limit ◽  
Unit Vector ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Compact Sets ◽  
Linear Topology ◽  
Topological Embedding ◽  
Definable Set ◽  
Finite Dimensional ◽  
The One

This chapter introduces the concept of stable completion and provides a concrete representation of unit vector Mathematical Double-Struck Capital A superscript n in terms of spaces of semi-lattices, with particular emphasis on the frontier between the definable and the topological categories. It begins by constructing a topological embedding of unit vector Mathematical Double-Struck Capital A superscript n into the inverse limit of a system of spaces of semi-lattices L(Hsubscript d) endowed with the linear topology, where Hsubscript d are finite-dimensional vector spaces. The description is extended to the projective setting. The linear topology is then related to the one induced by the finite level morphism L(Hsubscript d). The chapter also considers the condition that if a definable set in L(Hsubscript d) is an intersection of relatively compact sets, then it is itself relatively compact.

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