scholarly journals Identities Generalizing the Theorems of Pappus and Desargues

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1382
Author(s):  
Roger D. Maddux
Keyword(s):  
Projective Plane ◽  
Vector Space ◽  
Invariant Theory ◽  
Lattice Theory ◽  
Three Dimensional ◽  
Dimensional Vector ◽  
Dimensional Vector Space

The Theorems of Pappus and Desargues (for the projective plane over a field) are generalized here by two identities involving determinants and cross products. These identities are proved to hold in the three-dimensional vector space over a field. They are closely related to the Arguesian identity in lattice theory and to Cayley-Grassmann identities in invariant theory.

scholarly journals THREE-DIMENSIONAL ISOLATED QUOTIENT SINGULARITIES IN EVEN CHARACTERISTIC

2017 ◽  
Vol 60 (2) ◽  
pp. 435-445
Author(s):  
VLADIMIR SHCHIGOLEV ◽  
DMITRY STEPANOV
Keyword(s):  
Finite Group ◽  
Vector Space ◽  
Three Dimensional ◽  
Finite Subgroup ◽  
Two Dimensional ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Quotient Singularities ◽  
Characteristic 2

AbstractThis paper is a complement to the work of the second author on modular quotient singularities in odd characteristic. Here, we prove that if V is a three-dimensional vector space over a field of characteristic 2 and G < GL(V) is a finite subgroup generated by pseudoreflections and possessing a two-dimensional invariant subspace W such that the restriction of G to W is isomorphic to the group SL2(𝔽2n), then the quotient V/G is non-singular. This, together with earlier known results on modular quotient singularities, implies first that a theorem of Kemper and Malle on irreducible groups generated by pseudoreflections generalizes to reducible groups in dimension three, and, second, that the classification of three-dimensional isolated singularities that are quotients of a vector space by a linear finite group reduces to Vincent's classification of non-modular isolated quotient singularities.

scholarly journals Noncommutative classical invariant theory

1988 ◽  
Vol 112 ◽  
pp. 153-169 ◽  
Author(s):  
Yasuo Teranishi
Keyword(s):  
Vector Space ◽  
Invariant Theory ◽  
Characteristic Zero ◽  
Tensor Algebra ◽  
Dimensional Vector ◽  
Classical Invariant Theory ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Classical Invariant

Let K be a field of characteristic zero, V a finite dimensional vector space and G a subgroup of GL(V). The action of G on V is extended to the symmetric algebra on V over K,and the tensor algebra on V over K,

scholarly journals Young diagrammatic methods in non-commutative invariant theory

1991 ◽  
Vol 121 ◽  
pp. 15-34
Author(s):  
Yasuo Teranishi
Keyword(s):  
Vector Space ◽  
Invariant Theory ◽  
Characteristic Zero ◽  
Symmetric Algebra ◽  
Tensor Algebra ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Invariant Ring

In this paper we will study some aspects of non-commutative invariant theory. Let V be a finite-dimensional vector space over a field K of characteristic zero and letK[V] = K⊕V⊕S2(V)⊕…, andK′V› = K⊕V⊕⊕2(V)⊕⊕3V⊕&be respectively the symmetric algebra and the tensor algebra over V. Let G be a subgroup of GL(V). Then G acts on K[V] and K′V›. Much of this paper is devoted to the study of the (non-commutative) invariant ring K′V›G of G acting on K′V›.In the first part of this paper, we shall study the invariant ring in the following situation.

Plastic Deformation Behavior of Mild Steel Along Orthogonal Trilinear Strain Trajectories in Three-Dimensional Vector Space of Strain Deviator

1981 ◽  
Vol 103 (4) ◽  
pp. 287-292 ◽  
Author(s):  
Y. Ohashi ◽  
E. Tanaka
Keyword(s):  
Plastic Deformation ◽  
Mild Steel ◽  
Vector Space ◽  
Three Dimensional ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
History Effects ◽  
Combined Loads ◽  
Plastic Deformation Behavior ◽  
Strain Deviator

An experiment was performed on the plastic deformation of mild steel S15C by applying combined loads to thin-walled tubular specimens so that the deformation developed along orthogonal trilinear strain trajectories in the three-dimensional vector space. Scalar and vectorial history effects on the stress vector σ in the deformation along the third branch are examined for 28 kinds of trajectory combined of various values of an angle θe between the first and third branches and a length sl of the second branch.

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.

Sign in / Sign up

Export Citation Format

Share Document