scholarly journals Projective Metric Geometry and Clifford Algebras

2021 ◽  
Vol 76 (4) ◽  
Author(s):  
Hans Havlicek
Keyword(s):  
Quadratic Form ◽  
Projective Space ◽  
Vector Space ◽  
Clifford Algebra ◽  
Clifford Algebras ◽  
Metric Geometry ◽  
Algebraic Description ◽  
Projective Metric ◽  
Point Set ◽  
Distinguished Group

AbstractEach vector space that is endowed with a quadratic form determines its Clifford algebra. This algebra, in turn, contains a distinguished group, known as the Lipschitz group. We show that only a quotient of this group remains meaningful in the context of projective metric geometry. This quotient of the Lipschitz group can be viewed as a point set in the projective space on the Clifford algebra and, under certain restrictions, leads to an algebraic description of so-called kinematic mappings.

Series expansions for monogenic functions in Clifford algebras and their application

2020 ◽  
Vol 17 (3) ◽  
pp. 365-371
Author(s):  
Anatoliy Pogorui ◽  
Tamila Kolomiiets
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Vector Space ◽  
Clifford Algebra ◽  
Clifford Algebras ◽  
Monogenic Function ◽  
Second Order ◽  
Series Expansions ◽  
Monogenic Functions ◽  
Partial Differential

This paper deals with studying some properties of a monogenic function defined on a vector space with values in the Clifford algebra generated by the space. We provide some expansions of a monogenic function and consider its application to study solutions of second-order partial differential equations.

scholarly journals REPRESENTATIONS OF ALTERNATIVE CLIFFORD ALGEBRAS OF QUADRATIC FORMS

2014 ◽  
Vol 57 (3) ◽  
pp. 579-590 ◽  
Author(s):  
STACY MARIE MUSGRAVE
Keyword(s):  
Quadratic Form ◽  
Clifford Algebra ◽  
Algebraic Structure ◽  
Quadratic Forms ◽  
Clifford Algebras ◽  
Future Research ◽  
Octonion Algebras

AbstractThis work defines a new algebraic structure, to be called an alternative Clifford algebra associated to a given quadratic form. I explored its representations, particularly concentrating on connections to the well-understood octonion algebras. I finished by suggesting directions for future research.

A Characterization of Projective Metric Spaces

1986 ◽  
Vol 29 (4) ◽  
pp. 450-455 ◽  
Author(s):  
Rolfdieter Frank
Keyword(s):  
Quadratic Form ◽  
Projective Space ◽  
Equivalence Relation ◽  
Metric Space ◽  
Metric Spaces ◽  
Projective Metric ◽  
Miquel’S Theorem

AbstractA projective metric space is a pappian projective space together with a quadric and a certain equivalence relation on the pairs of those points which do not belong to the quadric. This equivalence relation is defined by means of the corresponding quadratic form and satisfies a condition which is a projective version of Miquel's theorem. We characterize the projective metric spaces of dimension at least two over fields of order at least 13.

scholarly journals Explicit isomorphisms of real Clifford algebras

2006 ◽  
Vol 2006 ◽  
pp. 1-13
Author(s):  
N. Değırmencı ◽  
Ş. Karapazar
Keyword(s):  
Quadratic Form ◽  
Clifford Algebra ◽  
Matrix Algebra ◽  
Clifford Algebras ◽  
Explicit Construction ◽  
The Other ◽  
Matrix Algebras ◽  
Direct Sum ◽  
Other Hand ◽  
Nondegenerate Quadratic Form

It is well known that the Clifford algebraClp,qassociated to a nondegenerate quadratic form onℝn (n=p+q)is isomorphic to a matrix algebraK(m)or direct sumK(m)⊕K(m)of matrix algebras, whereK=ℝ,ℂ,ℍ. On the other hand, there are no explicit expressions for these isomorphisms in literature. In this work, we give a method for the explicit construction of these isomorphisms.

scholarly journals A note on Clifford algebras and central division algebras with involution

1985 ◽  
Vol 26 (2) ◽  
pp. 171-176 ◽  
Author(s):  
D. W. Lewis
Keyword(s):  
Quadratic Form ◽  
Clifford Algebra ◽  
Division Algebra ◽  
Clifford Algebras ◽  
Matrix Ring ◽  
Division Algebras ◽  
Central Division

In this note we consider the question as to which central division algebras occur as the Clifford algebra of a quadratic form over a field. Non-commutative ones other than quaternion division algebras can occur and it is also the case that there are certain central division algebras D which, while not themselves occurring as a Clifford algebra, are such that some matrix ring over D does occur as a Clifford algebra. We also consider the further question as to which involutions on the division algebra can occur as one of two natural involutions on the Clifford algebra.

scholarly journals Clifford algebras and isotropes

1987 ◽  
Vol 29 (2) ◽  
pp. 249-257
Author(s):  
P. L. Robinson
Keyword(s):  
Vector Space ◽  
Clifford Algebra ◽  
Clifford Algebras ◽  
Real Vector Space ◽  
Inner Product ◽  
Spin Representation ◽  
Real Vector ◽  
Arbitrary Signature

Isotropes play a distinguished rôle in the algebra of spinors. LetVbe an even-dimensional real vector space equipped with an inner productBof arbitrary signature. An isotrope of(V, B)is a subspace of the complexificationVcon whichBcis identically zero. Denote by ρ the spin representation of the complex Clifford algebraC(Vc, Bc) on a spaceSof spinors.

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 Projective metric number theory

2016 ◽  
Vol 2016 (712) ◽  
Author(s):  
Anish Ghosh ◽  
Alan Haynes
Keyword(s):  
Number Theory ◽  
Projective Space ◽  
Diophantine Approximation ◽  
Higher Dimension ◽  
Probabilistic Theory ◽  
Metric Number Theory ◽  
Projective Metric

AbstractIn this paper we consider the probabilistic theory of Diophantine approximation in projective space over a completion of ℚ. Using the projective metric studied in [Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 23 (1996), no. 2, 211–248] we prove the analogue of Khintchine's theorem in projective space. For finite places and in higher dimension, we are able to completely remove the condition of monotonicity and establish the analogue of the Duffin–Schaeffer conjecture.

scholarly journals THE PROCEDURE OF THE SLIP MODELS OF POLYCRYSTALLINE PLASTICITY IN THE GENERAL CASES

1992 ◽  
Vol 14 (2) ◽  
pp. 13-16
Author(s):  
Bui Huu Dan
Keyword(s):  
Vector Space ◽  
Clifford Algebra ◽  
Strain State ◽  
Stress And Strain ◽  
Slip Model ◽  
Dimension Vector ◽  
Computation Procedure ◽  
Five Dimension ◽  
The Many

The computation procedure of the slip model of polycrystalline plasticity was extended for the most general cases, when the stress and strain state are expressed in the five-dimension vector space this extent ion is based on the know ledges of Clifford algebra in the many-dimension (more than 3) vector space. The results would be reduced into the old results given in the more simple cases.

Sign in / Sign up

Export Citation Format

Share Document