scholarly journals Arcs with Large Conical Subsets in Desarguesian Planes of Even Order

10.37236/2907 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Kris Coolsaet ◽  
Heide Sticker
Keyword(s):  
Electronic Journal ◽  
Desarguesian Planes ◽  
Even Order

We give an explicit classification of the arcs in PG$(2,q)$ ($q$ even) with a large conical subset and excess 2, i.e., that consist of $q/2+1$ points of a conic and two points not on that conic. Apart from the initial setup, the methods used are similar to those for the case of odd $q$, published earlier (Electronic Journal of Combinatorics, 17, #R112).

scholarly journals The Lattice of Definable Equivalence Relations in Homogeneous $n$-Dimensional Permutation Structures

10.37236/5980 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Samuel Braunfeld
Keyword(s):  
Distributive Lattice ◽  
Equivalence Relations ◽  
Electronic Journal ◽  
Finite Distributive Lattice ◽  
Linear Orders ◽  
Finite Dimensional ◽  
Homogeneous Structures ◽  
Special Case

In Homogeneous permutations, Peter Cameron [Electronic Journal of Combinatorics 2002] classified the homogeneous permutations (homogeneous structures with 2 linear orders), and posed the problem of classifying the homogeneous $n$-dimensional permutation structures (homogeneous structures with $n$ linear orders) for all finite $n$. We prove here that the lattice of $\emptyset$-definable equivalence relations in such a structure can be any finite distributive lattice, providing many new imprimitive examples of homogeneous finite dimensional permutation structures. We conjecture that the distributivity of the lattice of $\emptyset$-definable equivalence relations is necessary, and prove this under the assumption that the reduct of the structure to the language of $\emptyset$-definable equivalence relations is homogeneous. Finally, we conjecture a classification of the primitive examples, and confirm this in the special case where all minimal forbidden structures have order 2. 

On the Frobenius spectrum of a finite group

2017 ◽  
Vol 16 (03) ◽  
pp. 1750051 ◽  
Author(s):  
Jiangtao Shi ◽  
Wei Meng ◽  
Cui Zhang
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Finite Groups ◽  
Prime Divisor ◽  
Complete Classification ◽  
Composition Factor ◽  
Frobenius Theorem ◽  
Even Order ◽  
A Finite Group

Let [Formula: see text] be a finite group and [Formula: see text] any divisor of [Formula: see text], the order of [Formula: see text]. Let [Formula: see text], Frobenius’ theorem states that [Formula: see text] for some positive integer [Formula: see text]. We call [Formula: see text] a Frobenius quotient of [Formula: see text] for [Formula: see text]. Let [Formula: see text] be the set of all Frobenius quotients of [Formula: see text], we call [Formula: see text] the Frobenius spectrum of [Formula: see text]. In this paper, we give a complete classification of finite groups [Formula: see text] with [Formula: see text] for [Formula: see text] being the smallest prime divisor of [Formula: see text]. Moreover, let [Formula: see text] be a finite group of even order, [Formula: see text] the set of all Frobenius quotients of [Formula: see text] for even divisors of [Formula: see text] and [Formula: see text] the maximum Frobenius quotient in [Formula: see text], we prove that [Formula: see text] is always solvable if [Formula: see text] or [Formula: see text] and [Formula: see text] is not a composition factor of [Formula: see text].

scholarly journals On Certain Edge-Transitive Bicirculants

10.37236/7588 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Robert Jajcay ◽  
Štefko Miklavič ◽  
Primož Šparl ◽  
Gorazd Vasiljević
Keyword(s):  
Infinite Family ◽  
Main Theme ◽  
Induced Subgraphs ◽  
Generalized Petersen Graphs ◽  
Symmetry Properties ◽  
Petersen Graphs ◽  
Even Order ◽  
Edge Transitive ◽  
Transitive Graphs

A graph $\Gamma$ of even order is a bicirculant if it admits an automorphism with two orbits of equal length. Symmetry properties of bicirculants, for which at least one of the induced subgraphs on the two orbits of the corresponding semiregular automorphism is a cycle, have been studied, at least for the few smallest possible valences. For valences $3$, $4$ and $5$, where the corresponding bicirculants are called generalized Petersen graphs, Rose window graphs and Tabač jn graphs, respectively, all edge-transitive members have been classified. While there are only 7 edge-transitive generalized Petersen graphs and only 3 edge-transitive Tabač jn graphs, infinite families of edge-transitive Rose window graphs exist. The main theme of this paper is the question of the existence of such bicirculants for higher valences. It is proved that infinite families of edge-transitive examples of valence $6$ exist and among them infinitely many arc-transitive as well as infinitely many half-arc-transitive members are identified. Moreover, the classification of the ones of valence $6$ and girth $3$ is given. As a corollary, an infinite family of half-arc-transitive graphs of valence $6$ with universal reachability relation, which were thus far not known to exist, is obtained.

scholarly journals Polynomials for hyperovals of Desarguesian planes

1991 ◽  
Vol 51 (3) ◽  
pp. 436-447 ◽  
Author(s):  
Christine M. O'keefe ◽  
Tim Penttila
Keyword(s):  
Projective Planes ◽  
Desarguesian Projective Planes ◽  
Desarguesian Planes ◽  
Even Order

AbstractThis paper studies o-polynomials, that is, polynomials which represent hyperovals in Desarguesian projective planes of even order. We present theoretical restrictions on the form that O-polynomials can have, and we determine the number of o-polynomials corresponding to each of the known classes of hyperovals (other than Cherowitzo's). We use this to give the number of known o-polynomials for the fields of orders 4, 8, 16 and 32. Exploratory computer searches for o-polynomials for fields of small orders greater than 16 are reported.

scholarly journals PERMUTATION-TWISTED MODULES FOR EVEN ORDER CYCLES ACTING ON TENSOR PRODUCT VERTEX OPERATOR SUPERALGEBRAS

2014 ◽  
Vol 25 (02) ◽  
pp. 1450018 ◽  
Author(s):  
KATRINA BARRON ◽  
NATHAN VANDER WERF
Keyword(s):  
Tensor Product ◽  
Significant Role ◽  
Vertex Operator ◽  
Cyclic Permutation ◽  
Odd Order ◽  
Order Case ◽  
Even Order ◽  
Vertex Operator Superalgebra

We construct and classify (1 2 ⋯ k)-twisted V⊗k-modules for k even and V a vertex operator superalgebra. In particular, we show that the category of weak (1 2 ⋯ k)-twisted V⊗k-modules for k even is isomorphic to the category of weak parity-twisted V-modules. This result shows that in the case of a cyclic permutation of even order, the construction and classification of permutation-twisted modules for tensor product vertex operator superalgebras are fundamentally different than in the case of a cyclic permutation of odd order, as previously constructed and classified by the first author. In particular, in the even order case it is the parity-twisted V-modules that play the significant role in place of the untwisted V-modules that play the significant role in the odd order case.

scholarly journals The Multipole Structure and Symmetry Classification of Even-Type Deviators Decomposed from the Material Tensor

Materials ◽  
2021 ◽  
Vol 14 (18) ◽  
pp. 5388
Author(s):  
Changxin Tang ◽  
Wei Wan ◽  
Lei Zhang ◽  
Wennan Zou
Keyword(s):  
Unit Vector ◽  
Symmetric Part ◽  
Order Tensor ◽  
Symmetry Classification ◽  
Direct Sum ◽  
Even Order ◽  
Back Calculation ◽  
Symmetry Problem ◽  
Unit Vectors

The number of distinct components of a high-order material/physical tensor might be remarkably reduced if it has certain symmetry types due to the crystal structure of materials. An nth-order tensor could be decomposed into a direct sum of deviators where the order is not higher than n, then the symmetry classification of even-type deviators is the basis of the symmetry problem for arbitrary even-order physical tensors. Clearly, an nth-order deviator can be expressed as the traceless symmetric part of tensor product of n unit vectors multiplied by a positive scalar from Maxwell’s multipole representation. The set of these unit vectors shows the multipole structure of the deviator. Based on two steps of exclusion, the symmetry classifications of all even-type deviators are obtained by analyzing the geometric symmetry of the unit vector sets, and the general results are provided. Moreover, corresponding to each symmetry type of the even-type deviators up to sixth-order, the specific multipole structure of the unit vector set is given. This could help to identify the symmetry types of an unknown physical tensor and possible back-calculation of the involved physical coefficients.

A Classification of Semi-Translation Planes

1969 ◽  
Vol 21 ◽  
pp. 1372-1387 ◽  
Author(s):  
Norman Lloyd Johnson
Keyword(s):  
General Type ◽  
Projective Planes ◽  
Finite Geometry ◽  
General Interest ◽  
Translation Planes ◽  
Desarguesian Planes ◽  
Finite Mathematics ◽  
Tremendous Amount ◽  
Dual Translation

The classification of certain types of projective planes has recently been of considerable interest to both geometers and group theorists. Due in part to the current general interest in finite mathematics and the developments connecting group theory and finite geometry, the Lenz-Barlotti classification of finite projective planes (2; 10), in particular, has generated a tremendous amount of research. A great deal of this research has been related to the construction of non-Desarguesian planes.Fryxell (6), Hughes (7), Luneburg (11), and Ostrom (13; 15; 18) have found examples of projective planes, all of which are of a general type that we call semi-translation planes. Many of these planes are of the same Lenz-Barlotti class I-1. (The Lùneburg planes are translation planes. However, the planes dual to the Luneburg planes are semi-translation planes as well as dual translation planes.)

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