scholarly journals A Hilton-Milner Theorem for Vector Spaces

10.37236/343 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
A. Blokhuis ◽  
A. E. Brouwer ◽  
A. Chowdhury ◽  
P. Frankl ◽  
T. Mussche ◽  
...  
Keyword(s):  
Vector Space ◽  
Chromatic Number ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Intersecting Family ◽  
Kneser Graphs

We show for $k \geq 2$ that if $q\geq 3$ and $n \geq 2k+1$, or $q=2$ and $n \geq 2k+2$, then any intersecting family ${\cal F}$ of $k$-subspaces of an $n$-dimensional vector space over $GF(q)$ with $\bigcap_{F \in {\cal F}} F=0$ has size at most $\left[{n-1\atop k-1}\right]-q^{k(k-1)}\left[{n-k-1\atop k-1}\right]+q^k$. This bound is sharp as is shown by Hilton-Milner type families. As an application of this result, we determine the chromatic number of the corresponding $q$-Kneser graphs.

The genus of graphs associated with vector spaces

2019 ◽  
Vol 19 (05) ◽  
pp. 2050086 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
K. Prabha Ananthi
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Undirected Graph ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Vertex Connectivity ◽  
Dimensional Vector Space ◽  
Nonzero Component ◽  
Vertex Set ◽  
Component Graph

Let [Formula: see text] be a k-dimensional vector space over a finite field [Formula: see text] with a basis [Formula: see text]. The nonzero component graph of [Formula: see text], denoted by [Formula: see text], is a simple undirected graph with vertex set as nonzero vectors of [Formula: see text] such that there is an edge between two distinct vertices [Formula: see text] if and only if there exists at least one [Formula: see text] along which both [Formula: see text] and [Formula: see text] have nonzero scalars. In this paper, we find the vertex connectivity and girth of [Formula: see text]. We also characterize all vector spaces [Formula: see text] for which [Formula: see text] has genus either 0 or 1 or 2.

scholarly journals On $k$-simplexes in $(2k-1)$-dimensional vector spaces over finite fields

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Le Anh Vinh
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Finite Fields ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Nous Montrons ◽  
International Audience

International audience We show that if the cardinality of a subset of the $(2k-1)$-dimensional vector space over a finite field with $q$ elements is $\gg q^{2k-1-\frac{1}{ 2k}}$, then it contains a positive proportional of all $k$-simplexes up to congruence. Nous montrons que si la cardinalité d'un sous-ensemble de l'espace vectoriel à $(2k-1)$ dimensions sur un corps fini à $q$ éléments est $\gg q^{2k-1-\frac{1}{ 2k}}$, alors il contient une proportion non-nulle de tous les $k$-simplexes de congruence.

scholarly journals Relative invariants and b-functions of prehomogeneous vector spaces

1985 ◽  
Vol 98 ◽  
pp. 139-156 ◽  
Author(s):  
Yasuo Teranishi
Keyword(s):  
Vector Space ◽  
Linear Algebraic Group ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Rational Representation ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Prehomogeneous Vector Spaces ◽  
Relative Invariants

Let G be a connected linear algebraic group, p a rational representation of G on a finite-dimensional vector space V, all defined over C.

scholarly journals On the Number of Orthogonal Systems in Vector Spaces over Finite Fields

10.37236/907 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Le Anh Vinh
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Finite Fields ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Graph Theoretic ◽  
Theoretic Proof ◽  
Orthogonal Systems

Iosevich and Senger (2008) showed that if a subset of the $d$-dimensional vector space over a finite field is large enough, then it contains many $k$-tuples of mutually orthogonal vectors. In this note, we provide a graph theoretic proof of this result.

Products of idempotent endomorphisms of an independence algebra of infinite rank

1993 ◽  
Vol 114 (2) ◽  
pp. 303-319 ◽  
Author(s):  
John Fountain ◽  
Andrew Lewin
Keyword(s):  
Vector Space ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Infinite Dimensional ◽  
Common Generalization ◽  
Finite Dimensional Vector Space ◽  
Infinite Rank ◽  
Finite Dimensional ◽  
Linear Maps

AbstractIn 1966, J. M. Howie characterized the self-maps of a set which can be written as a product (under composition) of idempotent self-maps of the same set. In 1967, J. A. Erdos considered the analogous question for linear maps of a finite dimensional vector space and in 1985, Reynolds and Sullivan solved the problem for linear maps of an infinite dimensional vector space. Using the concept of independence algebra, the authors gave a common generalization of the results of Howie and Erdos for the cases of finite sets and finite dimensional vector spaces. In the present paper we introduce strong independence algebras and provide a common generalization of the results of Howie and Reynolds and Sullivan for the cases of infinite sets and infinite dimensional vector spaces.

scholarly journals Orthogonal Systems in Vector Spaces over Finite Rings

10.37236/2147 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Thang Van Pham ◽  
Anh Vinh Le
Keyword(s):  
Vector Space ◽  
Expected Value ◽  
Finite Ring ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Finite Rings ◽  
Dimensional Vector Space ◽  
Orthogonal Systems

We prove that if a subset of the $d$-dimensional vector space over a finite ring is large enough, then the number of $k$-tuples of mutually orthogonal vectors in this set is close to its expected value.

scholarly journals Finite Vector Spaces and Certain Lattices

10.37236/1355 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Thomas W. Cusick
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Multiplicative Function ◽  
Integer Lattice ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Finite Vector ◽  
Galois Number ◽  
Finite Vector Spaces

The Galois number $G_n(q)$ is defined to be the number of subspaces of the $n$-dimensional vector space over the finite field $GF(q)$. When $q$ is prime, we prove that $G_n(q)$ is equal to the number $L_n(q)$ of $n$-dimensional mod $q$ lattices, which are defined to be lattices (that is, discrete additive subgroups of n-space) contained in the integer lattice ${\bf Z}^n$ and having the property that given any point $P$ in the lattice, all points of ${\bf Z}^n$ which are congruent to $P$ mod $q$ are also in the lattice. For each $n$, we prove that $L_n(q)$ is a multiplicative function of $q$.

scholarly journals N-Dimensional Binary Vector Spaces

2013 ◽  
Vol 21 (2) ◽  
pp. 75-81
Author(s):  
Kenichi Arai ◽  
Hiroyuki Okazaki
Keyword(s):  
Vector Space ◽  
Computer Science ◽  
Coding Theory ◽  
Binary Vector ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Binary Vectors ◽  
Dimensional Vector Space ◽  
Binary Field ◽  
Binary Fields

Summary The binary set {0, 1} together with modulo-2 addition and multiplication is called a binary field, which is denoted by F2. The binary field F2 is defined in [1]. A vector space over F2 is called a binary vector space. The set of all binary vectors of length n forms an n-dimensional vector space Vn over F2. Binary fields and n-dimensional binary vector spaces play an important role in practical computer science, for example, coding theory [15] and cryptology. In cryptology, binary fields and n-dimensional binary vector spaces are very important in proving the security of cryptographic systems [13]. In this article we define the n-dimensional binary vector space Vn. Moreover, we formalize some facts about the n-dimensional binary vector space Vn.

Simple and hyperhypersimple vector spaces

1978 ◽  
Vol 43 (2) ◽  
pp. 260-269 ◽  
Author(s):  
Allen Retzlaff
Keyword(s):  
Vector Space ◽  
Modular Lattice ◽  
Final Section ◽  
Early Research ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Infinite Dimensional ◽  
Recursively Enumerable ◽  
Algebraic Properties

AbstractLet V∞ be a fixed, fully effective, infinite dimensional vector space. Let be the lattice consisting of the recursively enumerable (r.e.) subspaces of V∞, under the operations of intersection and weak sum (see §1 for precise definitions). In this article we examine the algebraic properties of .Early research on recursively enumerable algebraic structures was done by Rabin [14], Frölich and Shepherdson [5], Dekker [3], Hamilton [7], and Guhl [6]. Our results are based upon the more recent work concerning vector spaces of Metakides and Nerode [12], Crossley and Nerode [2], Remmel [15], [16], and Kalantari [8].In the main theorem below, we extend a result of Lachlan from the lattice of r.e. sets to . We define hyperhypersimple vector spaces, discuss some of their properties and show if A, B ∈ , and A is a hyperhypersimple subspace of B then there is a recursive space C such that A + C = B. It will be proven that if V ∈ and the lattice of superspaces of V is a complemented modular lattice then V is hyperhypersimple. The final section contains a summary of related results concerning maximality and simplicity.

scholarly journals Orthogonal Systems in Vector Spaces over Finite Fields

10.37236/875 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Alex Iosevich ◽  
Steven Senger
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Finite Fields ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Orthogonal Systems

We prove that if a subset of the $d$-dimensional vector space over a finite field is large enough, then it contains many $k$-tuples of mutually orthogonal vectors.

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