scholarly journals Fundamental Domains for Rhombic Lattices with Dihedral Symmetry of Order 8

10.37236/7802 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Joseph Ray Clarence G. Damasco ◽  
Dirk Frettlöh ◽  
Manuel Joseph C. Loquias
Keyword(s):  
Symmetry Group ◽  
Fundamental Domain ◽  
Point Group ◽  
Symmetry Groups ◽  
Fundamental Domains ◽  
Open Case ◽  
Point Groups ◽  
Index 2 ◽  
Rhombic Lattice ◽  
Planar Lattices

We show by construction that every rhombic lattice $\Gamma$ in $\mathbb{R}^{2}$ has a fundamental domain whose symmetry group contains the point group of $\Gamma$ as a subgroup of index $2$. This solves the last open case of a question raised in a preprint by the authors on fundamental domains for planar lattices whose symmetry groups properly contain the point groups of the lattices.  

scholarly journals CARTESIAN AXIS AND PLANE CONVENTIONS IN Cnv SYMMETRY GROUPS

Química Nova ◽  
2021 ◽  
Author(s):  
Lucas Dias ◽  
Roberto Faria
Keyword(s):  
Water Molecule ◽  
Point Group ◽  
Molecular Orbitals ◽  
Symmetry Groups ◽  
Long Time ◽  
Point Groups

In this work, we call the attention to the ambiguity found in the literature when labeling vibrations and molecular orbitals as B1 and B2 for molecules belonging to the C2v point group as, for example, the water molecule. A survey of several books and some articles shows that this ambiguity comes from a long time ago and persists today, being a source of misunderstanding and a waste of time for students and teachers. It means that, in the case of the point groups Cnv, Dn, and Dnh (n = 2, 4, 6), it is very important to draw students’ attention to this ambiguity that exists in the literature. It is unfortunate that the recommendation made by Mulliken, more than sixty years ago, to always place the water molecule in the yz plane, has not been followed.

scholarly journals Full Symmetry Groups and Exact Solutions to BKP and GKP Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Bo Ren ◽  
Jian-Yong Wang
Keyword(s):  
Exact Solutions ◽  
Symmetry Group ◽  
Symmetry Groups ◽  
Point Groups ◽  
Full Symmetry ◽  
The Relationship

We investigate the (2+1)-dimensional nonlinear BKP and GKP equations with the modified direct CK’s method. Then, we get its Lie point groups and the full symmetry group, and a relationship is constructed between the new solutions and the old one. Based on the relationship, the new solutions can be obtained by using a given solution of the equations.

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 The determination of point groups from imprecise molecular geometries

2021 ◽  
Author(s):  
Peter J. Knowles
Keyword(s):  
Symmetry Breaking ◽  
Point Group ◽  
Simple Function ◽  
Coordinate Frame ◽  
New Approach ◽  
Measure Of Symmetry ◽  
Point Groups

AbstractWe present a new approach for the assignment of a point group to a molecule when the structure conforms only approximately to the symmetry. It proceeds by choosing a coordinate frame that minimises a measure of symmetry breaking that is computed efficiently as a simple function of the molecular coordinates and point group specification.

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 Point-group symmetry detection in three-dimensional charge density of biomolecules

2019 ◽  
Vol 36 (7) ◽  
pp. 2237-2243
Author(s):  
Cyril F Reboul ◽  
Simon Kiesewetter ◽  
Dominika Elmlund ◽  
Hans Elmlund
Keyword(s):  
Charge Density ◽  
Single Particle ◽  
Statistical Tests ◽  
Point Group ◽  
Three Dimensional ◽  
Group Symmetry ◽  
Point Group Symmetry ◽  
Density Maps ◽  
Point Groups ◽  
Density Map

Abstract Motivation No rigorous statistical tests for detecting point-group symmetry in three-dimensional (3D) charge density maps obtained by electron microscopy (EM) and related techniques have been developed. Results We propose a method for determining the point-group symmetry of 3D charge density maps obtained by EM and related techniques. Our ab initio algorithm does not depend on atomic coordinates but utilizes the density map directly. We validate the approach for a range of publicly available single-particle cryo-EM datasets. In straightforward cases, our method enables fully automated single-particle 3D reconstruction without having to input an arbitrarily selected point-group symmetry. When pseudo-symmetry is present, our method provides statistics quantifying the degree to which the 3D density agrees with the different point-groups tested. Availability and implementation The software is freely available at https://github.com/hael/SIMPLE3.0.

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 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].

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