scholarly journals Combinatorics of Edge Symmetry: Chiral and Achiral Edge Colorings of Icosahedral Giant Fullerenes: C80, C180, and C240

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1308
Author(s):  
Krishnan Balasubramanian
Keyword(s):  
Equivalence Classes ◽  
Group Symmetry ◽  
Irreducible Representations ◽  
Inversion Method ◽  
High Symmetry ◽  
Edge Colorings ◽  
Edge Group ◽  
Resonance Energies ◽  
Conjugated Circuits ◽  
Cycle Indices

We develop the combinatorics of edge symmetry and edge colorings under the action of the edge group for icosahedral giant fullerenes from C80 to C240. We use computational symmetry techniques that employ Sheehan’s modification of Pόlya’s theorem and the Möbius inversion method together with generalized character cycle indices. These techniques are applied to generate edge group symmetry comprised of induced edge permutations and thus colorings of giant fullerenes under the edge symmetry action for all irreducible representations. We primarily consider high-symmetry icosahedral fullerenes such as C80 with a chamfered dodecahedron structure, icosahedral C180, and C240 with a chamfered truncated icosahedron geometry. These symmetry-based combinatorial techniques enumerate both achiral and chiral edge colorings of such giant fullerenes with or without constraints. Our computed results show that there are several equivalence classes of edge colorings for giant fullerenes, most of which are chiral. The techniques can be applied to superaromaticity, sextet polynomials, the rapid computation of conjugated circuits and resonance energies, chirality measures, etc., through the enumeration of equivalence classes of edge colorings.

Aromaticity in heterocyclic molecules containing divalent sulfur

1988 ◽  
Vol 53 (9) ◽  
pp. 2023-2054 ◽  
Author(s):  
Milan Randić ◽  
Sonja Nikolić ◽  
Nenad Trinajstić
Keyword(s):  
Sulfur Atom ◽  
Positional Isomers ◽  
Heterocyclic Molecules ◽  
Resonance Energies ◽  
Conjugated Circuits ◽  
Divalent Sulfur

The conjugated circuits model is applied to heterocycles containing divalent sulfur. A novel parametrization is introduced for 4n + 2 and 4n conjugated circuits containing a single sulfur atom. The relative aromatic stabilities of a number of heterocyclic systems containing divalent sulfur are studied. Comparison is made whenever possible with earlier reported resonance energies of these compounds, obtained by using Huckel MO and SCF π-MO models, and appropriate reference structures. Special attention is given to positional isomers. An explanation of the differences amongst such isomers is given.

scholarly journals PROJECTIVE GROUP ALGEBRAS

1999 ◽  
Vol 14 (01) ◽  
pp. 129-146 ◽  
Author(s):  
R. CASALBUONI
Keyword(s):  
Algebraic Theory ◽  
Equivalence Classes ◽  
Irreducible Representations ◽  
Group Algebras ◽  
Projective Group ◽  
Noncommutative Tori ◽  
Noncommutative Spaces ◽  
Simple Generalization ◽  
Vector Representations ◽  
M Theory

In this paper we apply a recently proposed algebraic theory of integration to projective group algebras. These structures have received some attention in connection with the compactification of the M theory on noncommutative tori. This turns out to be an interesting field of applications, since the space [Formula: see text] of the equivalence classes of the vector unitary irreducible representations of the group under examination becomes, in the projective case, a prototype of noncommuting spaces. For vector representations the algebraic integration is equivalent to integrate over [Formula: see text]. However, its very definition is related only at the structural properties of the group algebra, therefore it is well defined also in the projective case, where the space [Formula: see text] has no classical meaning. This allows a generalization of the usual group harmonic analysis. Particular attention is given to Abelian groups, which are the relevant ones in the compactification problem, since it is possible, from the previous results, to establish a simple generalization of the ordinary calculus to the associated noncommutative spaces.

scholarly journals Characterizing the universal rigidity of generic tensegrities

2021 ◽  
Author(s):  
Ryoshun Oba ◽  
Shin-ichi Tanigawa
Keyword(s):  
Finite Groups ◽  
Stability Condition ◽  
Point Group ◽  
Group Symmetry ◽  
Irreducible Representations ◽  
Sufficient Condition ◽  
Point Group Symmetry ◽  
Block Diagonalization ◽  
Representations Of Finite Groups ◽  
Universal Rigidity

AbstractA tensegrity is a structure made from cables, struts, and stiff bars. A d-dimensional tensegrity is universally rigid if it is rigid in any dimension $$d'$$ d ′ with $$d'\ge d$$ d ′ ≥ d . The celebrated super stability condition due to Connelly gives a sufficient condition for a tensegrity to be universally rigid. Gortler and Thurston showed that super stability characterizes universal rigidity when the point configuration is generic and every member is a stiff bar. We extend this result in two directions. We first show that a generic universally rigid tensegrity is super stable. We then extend it to tensegrities with point group symmetry, and show that this characterization still holds as long as a tensegrity is generic modulo symmetry. Our strategy is based on the block-diagonalization technique for symmetric semidefinite programming problems, and our proof relies on the theory of real irreducible representations of finite groups.

Singular angular magnetoresistance in a magnetic nodal semimetal

Science ◽  
2019 ◽  
Vol 365 (6451) ◽  
pp. 377-381 ◽  
Author(s):  
T. Suzuki ◽  
L. Savary ◽  
J.-P. Liu ◽  
J. W. Lynn ◽  
L. Balents ◽  
...  
Keyword(s):  
Domain Walls ◽  
Transport Coefficients ◽  
Point Group ◽  
Group Symmetry ◽  
Partial Substitution ◽  
High Symmetry ◽  
Correlated Electron Systems ◽  
Strongly Coupled ◽  
Correlated Electron ◽  
Weyl Semimetal

Transport coefficients of correlated electron systems are often useful for mapping hidden phases with distinct symmetries. Here we report a transport signature of spontaneous symmetry breaking in the magnetic Weyl semimetal cerium-aluminum-germanium (CeAlGe) system in the form of singular angular magnetoresistance (SAMR). This angular response exceeding 1000% per radian is confined along the high-symmetry axes with a full width at half maximum reaching less than 1° and is tunable via isoelectronic partial substitution of silicon for germanium. The SAMR phenomena is explained theoretically as a consequence of controllable high-resistance domain walls, arising from the breaking of magnetic point group symmetry strongly coupled to a nearly nodal electronic structure. This study indicates ingredients for engineering magnetic materials with high angular sensitivity by lattice and site symmetries.

MODY: a program for calculation of symmetry-adapted functions for ordered structures in crystals

2004 ◽  
Vol 37 (6) ◽  
pp. 1015-1019 ◽  
Author(s):  
Wiesława Sikora ◽  
Franciszek Białas ◽  
Lucjan Pytlik
Keyword(s):  
Phase Transition ◽  
Magnetic Moments ◽  
Numerical Data ◽  
Irreducible Representations ◽  
High Symmetry ◽  
Ordered Structure ◽  
Ordered Structures ◽  
Symmetry Adapted Functions ◽  
Model Structures ◽  
Physical Quantities

This paper describes a computer program, based on the theory of groups and representations, which calculates symmetry-adapted functions used for the description of various ordered structures in crystals. It is assumed that the ordered structure, which is formed by a configuration of occupational probabilities, ion displacements, magnetic moments, quadrupolar moments or other local physical quantities, is obtained from a high-symmetry crystal structure with a given space groupG, as a result of a symmetry-lowering phase transition. The detailed characteristics of the phase transition are given by the specification of the irreducible representations of groupG, active in the transition. The symmetry-adapted functions obtained from the calculation are perfect tools for the construction of model structures, which can be used for comparison with experimental (e.g.neutron diffraction) data, and can be a great help in numerical data elaboration by reducing the number of adjustable parameters describing the structure of a given symmetry.

CORE SYMMETRIES OF A FLOW

2004 ◽  
Vol 16 (04) ◽  
pp. 479-507
Author(s):  
AKITAKA KISHIMOTO
Keyword(s):  
Equivalence Classes ◽  
Crossed Product ◽  
Irreducible Representations ◽  
Group Of Automorphisms ◽  
C Algebra ◽  
Kms State ◽  
Cocycle Perturbation

For a flow α on a C*-algebra one defines a symmetry as the group of automorphisms γ such that γαγ-1 is a cocycle perturbation of α. We propose to define a core of this symmetry, which acts trivially on the set of equivalence classes of KMS state representations, but may act non-trivially on the set of equivalence classes of covariant irreducible representations. In particular this core acts transitively on the set of those which induce faithful representations of the crossed product by α.

Opening, stellation and handle replacement of edges of the cube, tetrahedron and hexagonal prism: application to 3-connected three-dimensional polyhedra and (2, 3)-connected polyhedral units in three-dimensional nets

1994 ◽  
Vol 444 (1920) ◽  
pp. 217-238 ◽  
Keyword(s):  
Point Group ◽  
Three Dimensional ◽  
Group Symmetry ◽  
Hexagonal Prism ◽  
High Symmetry ◽  
Regular Tetrahedron ◽  
Vertex Connectivity ◽  
3D Framework ◽  
Low Symmetry

Opening, stellation or handle replacement of edges of the regular cube, the regular tetrahedron and the semi-regular hexagonal prism (minimum point group symmetry 2/ m ) yields totals of 144, 11 and 205 geometrically distinct configurations respectively. Edge opening is restricted to a vertex connectivity of two or three. Sixty-five of the (2, 3)-connected polyhedral units were found in real three-dimensional (3D) framework structures. Low-symmetry configurations are as abundant as high-symmetry ones. With minor exceptions, those (2, 3)-connected patterns with the fewest converted edges are observed in each point group. The 113 polyhedra derived by double-stellation or double-handle replacement are uniformly three-connected. The ones observed in frameworks have a high symmetry. This study extends the general theory of 3D polyhedra and can be further expanded to other polyhedra and geometrical transformations. It may provide insight on the nature and growth of some seemingly complex framework structures that cannot otherwise be easily described topologically. The new polyhedral units are useful for classification of known framework structures in zeolites and related materials. New hypothetical nets generated from linkage of the units may solve unknown framework structures.

Sign in / Sign up

Export Citation Format

Share Document