scholarly journals Structure Transformations in Broken Symmetry Groups—Abstraction and Visualization

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 440 ◽  
Author(s):  
Valery Rau ◽  
Igor Togunov ◽  
Tamara Rau ◽  
Sergey Polyakov
Keyword(s):  
Symmetry Group ◽  
Oriented Graph ◽  
Structure Study ◽  
Broken Symmetry ◽  
Transformation Groups ◽  
Evolutionary Trees ◽  
Symmetry Groups ◽  
Structure Development ◽  
Computer Visualization ◽  
Development Direction

The work reports the finding and the study of transformation groups with two conditional elements (binary transformations of abstract structures of the finite numerical sets with broken symmetry). The term Broken Symmetry Group (BSG) is introduced. Transformation examples of relevant structures are studied with computer visualization and application in real structure study. A special type of BSG was discovered, which describes the subsets of “evolutionary trees” with convergent and divergent properties of the oriented graph (orgraph) with structure-development direction edges and “growth spirals”.

scholarly journals Влияние деформации на энергетический спектр и спектр оптического поглощения фуллерена C-=SUB=-20-=/SUB=- в модели Хаббарда

2019 ◽  
Vol 61 (2) ◽  
pp. 395
Author(s):  
А.В. Силантьев
Keyword(s):  
Group Theory ◽  
Hubbard Model ◽  
Symmetry Group ◽  
Energy Levels ◽  
Analytical Form ◽  
Symmetry Reduction ◽  
Energy Spectra ◽  
Symmetry Groups ◽  
Static Fluctuation Approximation ◽  
Static Fluctuation

Abstract —Anticommutator Green’s functions and energy spectra of fullerene C_20 with the I _ h , D _5 d , and D _3 d symmetry groups have been obtained in an analytical form within the Hubbard model and static fluctuation approximation. The energy states have been classified using the methods of group theory, and the allowed transitions in the energy spectra of fullerene C_20 with the I _ h , D _5 d , and D _3 d symmetry groups have been determined. It is also shown how the energy levels of fullerene C_20 with the I _ h symmetry group are split with the symmetry reduction.

scholarly journals Data transfer in the mathematical broken symmetry group model

2021 ◽  
Vol 2001 (1) ◽  
pp. 012007
Author(s):  
Kirill Gorshkov ◽  
Oleg Nikitin ◽  
Ekaterina Kuznetsova
Keyword(s):  
Symmetry Group ◽  
Data Transfer ◽  
Broken Symmetry ◽  
Group Model

scholarly journals Derivation of Asymptotic Dynamical Systems with Partial Lie Symmetry Groups

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Masatomo Iwasa
Keyword(s):  
Dynamical System ◽  
Asymptotic Behavior ◽  
Singular Perturbation ◽  
Symmetry Group ◽  
Group Analysis ◽  
Lie Symmetry ◽  
Symmetry Groups ◽  
Lie Group Analysis ◽  
Constraint Equations ◽  
Perturbation Problems

Lie group analysis has been applied to singular perturbation problems in both ordinary differential and difference equations and has allowed us to find the reduced dynamics describing the asymptotic behavior of the dynamical system. The present study provides an extended method that is also applicable to partial differential equations. The main characteristic of the extended method is the restriction of the manifold by some constraint equations on which we search for a Lie symmetry group. This extension makes it possible to find a partial Lie symmetry group, which leads to a reduced dynamics describing the asymptotic behavior.

scholarly journals Computing symmetry groups of polyhedra

2014 ◽  
Vol 17 (1) ◽  
pp. 565-581 ◽  
Author(s):  
David Bremner ◽  
Mathieu Dutour Sikirić ◽  
Dmitrii V. Pasechnik ◽  
Thomas Rehn ◽  
Achill Schürmann
Keyword(s):  
Linear Programming ◽  
Symmetry Group ◽  
Integer Linear Programming ◽  
Combinatorial Structure ◽  
Symmetry Groups ◽  
Graph Automorphism ◽  
Practical Experiences

AbstractKnowing the symmetries of a polyhedron can be very useful for the analysis of its structure as well as for practical polyhedral computations. In this note, we study symmetry groups preserving the linear, projective and combinatorial structure of a polyhedron. In each case we give algorithmic methods to compute the corresponding group and discuss some practical experiences. For practical purposes the linear symmetry group is the most important, as its computation can be directly translated into a graph automorphism problem. We indicate how to compute integral subgroups of the linear symmetry group that are used, for instance, in integer linear programming.

scholarly journals Cyclic p-groups of symmetries of surfaces

1991 ◽  
Vol 33 (2) ◽  
pp. 213-221 ◽  
Author(s):  
Ravi S. Kulkarni ◽  
Colin Maclachlan
Keyword(s):  
Symmetry Group ◽  
Finite Groups ◽  
Orientable Surface ◽  
Symmetry Groups ◽  
Compact Orientable Surface

Let Σg denote a compact orientable surface of genus g ≥ 2. We consider finite groups G acting effectively on Σg and preserving the orientation—for short, G acts on Σg or Gis a symmetry group of Σg. Each surface Σg admits only finitely many symmetry groups G and the orders of these groups are bounded by Wiman's bound of 84(g – 1). This bound is attained for infinitely many values of g [12], see also [9], and all values of g ≤ 104 for which it is attained are known [4].

Relativistic extension of non-compact symmetry groups

1966 ◽  
Vol 289 (1417) ◽  
pp. 177-191 ◽  
Keyword(s):  
Symmetry Group ◽  
Conventional Method ◽  
Space Time ◽  
Unitary Representations ◽  
Symmetry Groups ◽  
Infinite Dimensional ◽  
Induced Representations ◽  
Relativistic Extension ◽  
A Chain

The problem of relativistieally boosting the unitary representations of a non-compact spin-containing rest-symmetry group is solved by starting with non-unitary infinite-dimensional representations of a relativistic extension of this group, by adjoining to this extension four space-time translations and by the napplying Bargmann-Wigner equations to guarantee aunitary norm. The procedure has similarities to the conventional method of induced representations. The boosting problem considered here is the first step towards the solution of the problem of coupling of such infinite-dimensional representations which is also briefly investigated. Startin g from a rest-symmetry like U (6,6) a chain of subgroups GL (6), U (3,3), etc., is exhibited for collinear and coplanar processes, etc.

scholarly journals PROJECTIVE LINEAR GROUPS AS MAXIMAL SYMMETRY GROUPS

2008 ◽  
Vol 50 (1) ◽  
pp. 83-96 ◽  
Author(s):  
ANNA TORSTENSSON
Keyword(s):  
Symmetry Group ◽  
Hyperbolic Manifold ◽  
Three Dimensional ◽  
Maximal Order ◽  
Linear Groups ◽  
Symmetry Groups ◽  
Maximal Symmetry ◽  
Projective Linear Groups

AbstractA maximal symmetry group is a group of isomorphisms of a three-dimensional hyperbolic manifold of maximal order in relation to the volume of the manifold. In this paper we determine all maximal symmetry groups of the typesPSL(2,q) andPGL(2,q). Depending on the primepthere are one or two such groups withq=pkandkalways equals 1, 2 or 4.

scholarly journals Spatial symmetry groups as sensorimotor guidelines

2008 ◽  
Vol 17 (5-6) ◽  
pp. 347-359
Author(s):  
Gin McCollum
Keyword(s):  
Symmetry Group ◽  
Motor Neurons ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Symmetry Groups ◽  
Coordinate Systems ◽  
Spatial Function ◽  
Logical Structures ◽  
Vestibular Complex ◽  
Causal Logic

While some aspects of neuroanatomical organization are related to packing and access rather than to function, other aspects of anatomical/physiological organization are directly related to function. The mathematics of symmetry groups can be used to determine logical structure in projections and to relate it to function. This paper reviews two studies of the symmetry groups of vestibular projections that are related to the spatial functions of the vestibular complex, including gaze, posture, and movement. These logical structures have been determined by finding symmetry groups of two vestibular projections directly from physiological and anatomical data. Logical structures in vestibular projections are distinct from mapping properties such as the ability to maintain two- and three-dimensional coordinate systems; rather, they provide anatomical/physiological foundations for these mapping properties. The symmetry group of the direct projection from the semicircular canal primary afferents to neck motor neurons is that of the cube (O, the octahedral group), which can serve as a discrete skeleton for coordinate systems in three-dimensional space. The symmetry group of the canal projection from the secondary vestibular afferents to the inferior olive and thence to the cerebellar uvula-nodulus is that of the square (D8), which can support coordinates for the horizontal plane. While the mathematical relationship between these symmetry groups and functions of the vestibular complex are clear, these studies open a larger question: what is the causal logic by which neural centers and their intrinsic organization affect each other and behavior? The relationship of vestibular projection symmetry groups to spatial function make them ideal projections for investigating this causal logic. The symmetry group results are discussed in relationship to possible ways they communicate spatial structure to other neural centers and format spatial functions such as body movements. These two projection symmetry groups suggest that all vestibular projections may have symmetry groups significantly related to function, perhaps all to spatial function.

scholarly journals Automating Symmetry-Breaking Calculations

2004 ◽  
Vol 7 ◽  
pp. 101-119 ◽  
Author(s):  
P. C. Matthews
Keyword(s):  
Group Theory ◽  
Symmetry Breaking ◽  
Symmetry Group ◽  
Irreducible Representations ◽  
Symmetry Groups ◽  
Periodic Patterns ◽  
Alternating Groups ◽  
Spatial Symmetry

AbstractThe process of classifying possible symmetry-breaking bifurcations requires a computation involving the subgroups and irreducible representations of the original symmetry group. It is shown how this calculation can be automated using a group theory package such as GAP. This enables a number of new results to be obtained for larger symmetry groups, where manual computation is impractical. Examples of symmetric and alternating groups are given, and the method is also applied to the spatial symmetry-breaking of periodic patterns observed in experiments.

scholarly journals Approximate Hamiltonian Symmetry Groups and Recursion Operators for Perturbed Evolution Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
M. Nadjafikhah ◽  
A. Mokhtary
Keyword(s):  
Conservation Laws ◽  
Hamiltonian Systems ◽  
Evolution Equations ◽  
Comprehensive Analysis ◽  
Transformation Groups ◽  
Symmetry Groups ◽  
Recursion Operators ◽  
Gardner Equation ◽  
Small Parameters

The method of approximate transformation groups, which was proposed by Baikov et al. (1988 and 1996), is extended on Hamiltonian and bi-Hamiltonian systems of evolution equations. Indeed, as a main consequence, this extended procedure is applied in order to compute the approximate conservation laws and approximate recursion operators corresponding to these types of equations. In particular, as an application, a comprehensive analysis of the problem of approximate conservation laws and approximate recursion operators associated to the Gardner equation with the small parameters is presented.

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