scholarly journals EXISTENCE OF -ANALOGS OF STEINER SYSTEMS

2016 ◽  
Vol 4 ◽  
Author(s):  
MICHAEL BRAUN ◽  
TUVI ETZION ◽  
PATRIC R. J. ÖSTERGÅRD ◽  
ALEXANDER VARDY ◽  
ALFRED WASSERMANN
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Automorphism Groups ◽  
Steiner System ◽  
Dimensional Subspace ◽  
Steiner Systems ◽  
Cover Problem ◽  
Exact Cover

Let $\mathbb{F}_{q}^{n}$ be a vector space of dimension $n$ over the finite field $\mathbb{F}_{q}$. A $q$-analog of a Steiner system (also known as a $q$-Steiner system), denoted ${\mathcal{S}}_{q}(t,\!k,\!n)$, is a set ${\mathcal{S}}$ of $k$-dimensional subspaces of $\mathbb{F}_{q}^{n}$ such that each $t$-dimensional subspace of $\mathbb{F}_{q}^{n}$ is contained in exactly one element of ${\mathcal{S}}$. Presently, $q$-Steiner systems are known only for $t\,=\,1\!$, and in the trivial cases $t\,=\,k$ and $k\,=\,n$. In this paper, the first nontrivial $q$-Steiner systems with $t\,\geqslant \,2$ are constructed. Specifically, several nonisomorphic $q$-Steiner systems ${\mathcal{S}}_{2}(2,3,13)$ are found by requiring that their automorphism groups contain the normalizer of a Singer subgroup of $\text{GL}(13,2)$. This approach leads to an instance of the exact cover problem, which turns out to have many solutions.

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$F$.

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 THE INVARIANTS FIELD OF SOME FINITE PROJECTIVE LINEAR GROUP ACTIONS

2011 ◽  
Vol 85 (1) ◽  
pp. 19-25
Author(s):  
YIN CHEN
Keyword(s):  
Finite Field ◽  
Projective Space ◽  
Vector Space ◽  
Orthogonal Group ◽  
Homogeneous Polynomial ◽  
Classical Group ◽  
Projective Group ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Projective Linear Group

AbstractLet Fq be a finite field with q elements, V an n-dimensional vector space over Fq and 𝒱 the projective space associated to V. Let G≤GLn(Fq) be a classical group and PG be the corresponding projective group. In this note we prove that if Fq (V )G is purely transcendental over Fq with homogeneous polynomial generators, then Fq (𝒱)PG is also purely transcendental over Fq. We compute explicitly the generators of Fq (𝒱)PG when G is the symplectic, unitary or orthogonal group.

scholarly journals Weakly one-based geometric theories

2012 ◽  
Vol 77 (2) ◽  
pp. 392-422 ◽  
Author(s):  
Alexander Berenstein ◽  
Evgueni Vassiliev
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Geometric Theory ◽  
Common Generalization

AbstractWe study the class of weakly locally modular geometric theories introduced in [4], a common generalization of the classes of linear SU-rank 1 and linear o-minimal theories. We find new conditions equivalent to weak local modularity: “weak one-basedness”, absence of type definable “almost quasidesigns”, and “generic linearity”. Among other things, we show that weak one-basedness is closed under reducts. We also show that the lovely pair expansion of a non-trivial weakly one-based ω-categorical geometric theory interprets an infinite vector space over a finite field.

scholarly journals A Global Constraint for the Exact Cover Problem: Application to Conceptual Clustering

2020 ◽  
Vol 67 ◽  
pp. 509-547
Author(s):  
Maxime Chabert ◽  
Christine Solnon
Keyword(s):  
Scale Up ◽  
Global Constraint ◽  
Conceptual Clustering ◽  
Specific Data ◽  
Binary Constraints ◽  
Arc Consistency ◽  
Selected Subset ◽  
Cover Problem ◽  
Clustering Problems ◽  
Exact Cover

We introduce the exactCover global constraint dedicated to the exact cover problem, the goal of which is to select subsets such that each element of a given set belongs to exactly one selected subset. This NP-complete problem occurs in many applications, and we more particularly focus on a conceptual clustering application. We introduce three propagation algorithms for exactCover, called Basic, DL, and DL+: Basic ensures the same level of consistency as arc consistency on a classical decomposition of exactCover into binary constraints, without using any specific data structure; DL ensures the same level of consistency as Basic but uses Dancing Links to efficiently maintain the relation between elements and subsets; and DL+ is a stronger propagator which exploits an extra property to filter more values than DL. We also consider the case where the number of selected subsets is constrained to be equal to a given integer variable k, and we show that this may be achieved either by combining exactCover with existing constraints, or by designing a specific propagator that integrates algorithms designed for the NValues constraint. These different propagators are experimentally evaluated on conceptual clustering problems, and they are compared with state-of-the-art declarative approaches. In particular, we show that our global constraint is competitive with recent ILP and CP models for mono-criterion problems, and it has better scale-up properties for multi-criteria problems.

scholarly journals FUSION OVER A VECTOR SPACE

2006 ◽  
Vol 06 (02) ◽  
pp. 141-162 ◽  
Author(s):  
ANDREAS BAUDISCH ◽  
AMADOR MARTIN-PIZARRO ◽  
MARTIN ZIEGLER
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Vector Spaces

Let T1 and T2 be two countable strongly minimal theories with the DMP whose common theory is the theory of vector spaces over a fixed finite field. We show that T1 ∪ T2 has a strongly minimal completion.

Bivectors over a Finite Field

1981 ◽  
Vol 24 (4) ◽  
pp. 489-490
Author(s):  
J. A. MacDougall
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Symmetric Matrices ◽  
Dimensional Vector ◽  
Dimensional Vector Space

AbstractLet U be an n -dimensional vector space over a finite field of q elements. The number of elements of Λ2U of each irreducible length is found using the isomorphism of Λ2U with Hn, the space of n x n skew-symmetric matrices, and results due to Carlitz and MacWilliams on the number of skew-symmetric matrices of any given rank.

Direct Product of Derived Steiner Systems Using Inversive Planes

1981 ◽  
Vol 33 (6) ◽  
pp. 1365-1369 ◽  
Author(s):  
K. T. Phelps
Keyword(s):  
Direct Product ◽  
Triple System ◽  
Steiner Triple System ◽  
Steiner System ◽  
Steiner Systems

A Steiner system S(t, k, v) is a pair (P, B) where P is a v-set and B is a collection of k-subsets of P (usually called blocks) such that every t-subset of P is contained in exactly one block of B. As is well known, associated with each point x ∈ P is a S(t � 1, k � 1, v � 1) defined on the set Px = P\{x} with blocksB(x) = {b\{x}|x ∈ b and b ∈ B}.The Steiner system (Px, B(x)) is said to be derived from (P, B) and is called (obviously) a derived Steiner (t – 1, k – 1)-system. Very little is known about derived Steiner systems despite much effort (cf. [11]). It is not even known whether every Steiner triple system is derived.Steiner systems are closely connected to equational classes of algebras (see [7]) for certain values of k.

Strong map-symmetry of SL(3,K) and PSL(3,K) for every finite field K

2020 ◽  
pp. 2150048
Author(s):  
Antonio Breda d’Azevedo ◽  
Domenico A. Catalano
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Automorphism Groups ◽  
Group Automorphism ◽  
Regular Maps ◽  
Strong Map

In this paper, we show that for any finite field [Formula: see text], any pair of map-generators (that is when one of the generators is an involution) of [Formula: see text] and [Formula: see text] has a group automorphism that inverts both generators. In the theory of maps, this corresponds to say that any regular oriented map with automorphism group [Formula: see text] or [Formula: see text] is reflexible, or equivalently, there are no chiral regular maps with automorphism group [Formula: see text] or [Formula: see text]. As remarked by Leemans and Liebeck, also [Formula: see text] and [Formula: see text] are not automorphism groups of chiral regular maps. These two results complete the work of the above authors on simples groups supporting chiral regular maps.

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