scholarly journals Bowker's test for symmetry and modifications within the algebraic framework

2007 ◽  
Vol 51 (9) ◽  
pp. 4124-4142 ◽  
Author(s):  
Anne Krampe ◽  
Sonja Kuhnt
Keyword(s):  
Test For Symmetry ◽  
Algebraic Framework

scholarly journals Additive kinematic formulas for flag area measures

2021 ◽  
Author(s):  
Judit Abardia-Evéquoz ◽  
Andreas Bernig
Keyword(s):  
Recent Result ◽  
Flag Manifold ◽  
Unit Vector ◽  
Algebraic Framework

AbstractWe show the existence of additive kinematic formulas for general flag area measures, which generalizes a recent result by Wannerer. Building on previous work by the second named author, we introduce an algebraic framework to compute these formulas explicitly. This is carried out in detail in the case of the incomplete flag manifold consisting of all $$(p+1)$$ ( p + 1 ) -planes containing a unit vector.

scholarly journals Clifford Spinors and Root System Induction: $$H_4$$ and the Grand Antiprism

2021 ◽  
Vol 31 (3) ◽  
Author(s):  
Pierre-Philippe Dechant
Keyword(s):  
Root System ◽  
Root Systems ◽  
Invariant Sets ◽  
Simple Construction ◽  
Reflection Groups ◽  
Computational Work ◽  
Algebraic Framework ◽  
Underlying Geometry ◽  
Coxeter Plane ◽  
Insight Into

AbstractRecent work has shown that every 3D root system allows the construction of a corresponding 4D root system via an ‘induction theorem’. In this paper, we look at the icosahedral case of $$H_3\rightarrow H_4$$ H 3 → H 4 in detail and perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan–Dieudonné theorem, giving a simple construction of the $${\mathrm {Pin}}$$ Pin and $${\mathrm {Spin}}$$ Spin covers. Using this connection with $$H_3$$ H 3 via the induction theorem sheds light on geometric aspects of the $$H_4$$ H 4 root system (the 600-cell) as well as other related polytopes and their symmetries, such as the famous Grand Antiprism and the snub 24-cell. The uniform construction of root systems from 3D and the uniform procedure of splitting root systems with respect to subrootsystems into separate invariant sets allows further systematic insight into the underlying geometry. All calculations are performed in the even subalgebra of $${\mathrm {Cl}}(3)$$ Cl ( 3 ) , including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes, and are shared as supplementary computational work sheets. This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework.

An Algebraic Framework for the Many-Body Problem

1996 ◽  
pp. 357-399
Author(s):  
Werner O. Amrein ◽  
Anne Boutet Monvel ◽  
Vladimir Georgescu
Keyword(s):  
Body Problem ◽  
Many Body ◽  
Algebraic Framework ◽  
The Many

A modified runs test for symmetry

2013 ◽  
Vol 83 (5) ◽  
pp. 984-991 ◽  
Author(s):  
J. Corzo ◽  
G. Babativa
Keyword(s):  
Test For Symmetry

SYMMETRIES OF CERTAIN PHYSICAL THEORIES AND THE JORDAN ALGEBRAS

1992 ◽  
Vol 07 (15) ◽  
pp. 3623-3637 ◽  
Author(s):  
R. FOOT ◽  
G. C. JOSHI
Keyword(s):  
Jordan Algebra ◽  
Space Time ◽  
Jordan Algebras ◽  
Bosonic String ◽  
The Other ◽  
Structure Group ◽  
Division Algebras ◽  
Hermitian Matrices ◽  
Higher Dimensional ◽  
Algebraic Framework

It is shown that the sequence of Jordan algebras [Formula: see text], whose elements are the 3 × 3 Hermitian matrices over the division algebras ℝ, [Formula: see text], ℚ and [Formula: see text], can be associated with the bosonic string as well as the superstring. The construction reveals that the space–time symmetries of the first-quantized bosonic string and superstring actions can be related. The bosonic string and the superstring are associated with the exceptional Jordan algebra while the other Jordan algebras in the [Formula: see text] sequence can be related to parastring theories. We then proceed to further investigate a connection between the symmetries of supersymmetric Lagrangians and the transformations associated with the structure group of [Formula: see text]. The N = 1 on-shell supersymmetric Lagrangians in 3, 4 and 6-dimensions with a spin 0 field and a spin 1/2 field are incorporated within the Jordan-algebraic framework. We also make some remarks concerning a possible role for the division algebras in the construction of higher-dimensional extended objects.

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