The Basis Functions and the Matrix Representations of the Single and Double Icosahedral Point Group

1992 ◽  
Vol 61 (8) ◽  
pp. 2735-2747 ◽  
Author(s):  
Koun Shirai
Keyword(s):  
Point Group ◽  
Basis Functions ◽  
Matrix Representations ◽  
The Matrix

scholarly journals Isomorphism and matrix representation of point groups

2019 ◽  
Vol 15 (1) ◽  
pp. 88-92
Author(s):  
Wan Heng Fong ◽  
Aqilahfarhana Abdul Rahman ◽  
Nor Haniza Sarmin
Keyword(s):  
Group Theory ◽  
Matrix Representation ◽  
Point Group ◽  
Multiplication Table ◽  
Matrix Representations ◽  
The Matrix ◽  
Point Groups ◽  
Complete Set ◽  
Symmetry Of Molecules ◽  
Symmetry Operations

In chemistry, point group is a type of group used to describe the symmetry of molecules. It is a collection of symmetry elements controlled by a form or shape which all go through one point in space, which consists of all symmetry operations that are possible for every molecule. Next, a set of number or matrices which assigns to the elements of a group and represents the multiplication of the elements is said to constitute representation of a group. Here, each individual matrix is called a representative that corresponds to the symmetry operations of point groups, and the complete set of matrices is called a matrix representation of the group. This research was aimed to relate the symmetry in point groups with group theory in mathematics using the concept of isomorphism, where elements of point groups and groups were mapped such that the isomorphism properties were fulfilled. Then, matrix representations of point groups were found based on the multiplication table where symmetry operations were represented by matrices. From this research, point groups of order less than eight were shown to be isomorphic with groups in group theory. In addition, the matrix representation corresponding to the symmetry operations of these point groups wasis presented. This research would hence connect the field of mathematics and chemistry, where the relation between groups in group theory and point groups in chemistry were shown.

Microdiffraction analysis of precipitate crystallography and morphology in Al alloys

1990 ◽  
Vol 48 (4) ◽  
pp. 954-955
Author(s):  
B.C. Muddle ◽  
G.R. Hugo
Keyword(s):  
Point Group ◽  
Hexagonal Lattice ◽  
Intersection Point ◽  
Saed Pattern ◽  
Point Symmetry ◽  
Hexagonal Symmetry ◽  
Precipitate Crystal ◽  
The Matrix ◽  
The Subject ◽  
Electron Microdiffraction

Electron microdiffraction has been used to determine the crystallography of precipitation in Al-Cu-Mg-Ag and Al-Ge alloys for individual precipitates with dimensions down to 10 nm. The crystallography has been related to the morphology of the precipitates using an analysis based on the intersection point symmetry. This analysis requires that the precipitate form be consistent with the intersection point group, defined as those point symmetry elements common to precipitate and matrix crystals when the precipitate crystal is in its observed orientation relationship with the matrix.In Al-Cu-Mg-Ag alloys with high Cu:Mg ratios and containing trace amounts of silver, a phase designated Ω readily precipitates as thin, hexagonal-shaped plates on matrix {111}α planes. Examples of these precipitates are shown in Fig. 1. The structure of this phase has been the subject of some controversy. An SAED pattern, Fig. 2, recorded from matrix and precipitates parallel to a <11l>α axis is suggestive of hexagonal symmetry and a hexagonal lattice has been proposed on the basis of such patterns.

scholarly journals EXTENSION OF THE POINCARÉ GROUP AND NON-ABELIAN TENSOR GAUGE FIELDS

2010 ◽  
Vol 25 (31) ◽  
pp. 5765-5785 ◽  
Author(s):  
GEORGE SAVVIDY
Keyword(s):  
Gauge Group ◽  
Gauge Transformation ◽  
Space Time ◽  
Gauge Fields ◽  
Poincaré Group ◽  
Matrix Representations ◽  
Poincare Group ◽  
Yang Mills ◽  
The Matrix ◽  
Transversal Plane

In the recently proposed generalization of the Yang–Mills theory, the group of gauge transformation gets essentially enlarged. This enlargement involves a mixture of the internal and space–time symmetries. The resulting group is an extension of the Poincaré group with infinitely many generators which carry internal and space–time indices. The matrix representations of the extended Poincaré generators are expressible in terms of Pauli–Lubanski vector in one case and in terms of its invariant derivative in another. In the later case the generators of the gauge group are transversal to the momentum and are projecting the non-Abelian tensor gauge fields into the transversal plane, keeping only their positively definite spacelike components.

scholarly journals Computing mu-values for Representations of Symmetric Groups in Engineering Systems

2018 ◽  
Vol 11 (3) ◽  
pp. 774-792
Author(s):  
Mutti-Ur Rehman ◽  
M. Fazeel Anwar
Keyword(s):  
Control Systems ◽  
Symmetric Groups ◽  
Singular Values ◽  
Linear Control ◽  
Linear Control Systems ◽  
Complex Numbers ◽  
Engineering Systems ◽  
Matrix Representations ◽  
The Matrix

In this article we consider the matrix representations of finite symmetric groups Sn over the filed of complex numbers. These groups and their representations also appear as symmetries of certain linear control systems [5]. We compute the structure singular values (SSV) of the matrices arising from these representations. The obtained results of SSV are compared with well-known MATLAB routine mussv.

scholarly journals Hashing Based Answer Selection

2020 ◽  
Vol 34 (05) ◽  
pp. 9330-9337
Author(s):  
Dong Xu ◽  
Wu-Jun Li
Keyword(s):  
Question Answering ◽  
Matrix Representation ◽  
Matrix Representations ◽  
Binary Matrix ◽  
Large Memory ◽  
The Matrix ◽  
Novel Method ◽  
Online Prediction ◽  
Memory Cost ◽  
Vector Representations

Answer selection is an important subtask of question answering (QA), in which deep models usually achieve better performance than non-deep models. Most deep models adopt question-answer interaction mechanisms, such as attention, to get vector representations for answers. When these interaction based deep models are deployed for online prediction, the representations of all answers need to be recalculated for each question. This procedure is time-consuming for deep models with complex encoders like BERT which usually have better accuracy than simple encoders. One possible solution is to store the matrix representation (encoder output) of each answer in memory to avoid recalculation. But this will bring large memory cost. In this paper, we propose a novel method, called hashing based answer selection (HAS), to tackle this problem. HAS adopts a hashing strategy to learn a binary matrix representation for each answer, which can dramatically reduce the memory cost for storing the matrix representations of answers. Hence, HAS can adopt complex encoders like BERT in the model, but the online prediction of HAS is still fast with a low memory cost. Experimental results on three popular answer selection datasets show that HAS can outperform existing models to achieve state-of-the-art performance.

scholarly journals Construction of generalized rotations and quasi-orthogonal matrices

2019 ◽  
Vol 7 (1) ◽  
pp. 107-113
Author(s):  
Luis Verde-Star
Keyword(s):  
Image Processing ◽  
Hadamard Matrices ◽  
Complex Numbers ◽  
Data Communications ◽  
Matrix Representations ◽  
Orthogonal Matrices ◽  
Orthogonal Designs ◽  
The Matrix ◽  
The One

Abstract We propose some methods for the construction of large quasi-orthogonal matrices and generalized rotations that may be used in applications in data communications and image processing. We use certain combinations of constructions by blocks similar to the one used by Sylvester to build Hadamard matrices. The orthogonal designs related with the matrix representations of the complex numbers, the quaternions, and the octonions are used in our construction procedures.

Energy Spectrum of a Particle Within a Confined Double Well Potential

2006 ◽  
Vol 3 (2) ◽  
pp. 257-262
Author(s):  
J. L. Marin ◽  
G. Campoy ◽  
R. Riera
Keyword(s):  
Energy Levels ◽  
Numerical Approach ◽  
Symmetric Case ◽  
Basis Functions ◽  
Hidden Symmetry ◽  
The Matrix ◽  
Jahn Teller ◽  
Free Case ◽  
Jahn Teller Effect ◽  
Double Well Potential

The energy levels of a particle within a confined double well potential are studied in this work. The spectrum of the particle can be obtained by solving the corresponding Schrödinger equation but, for practical purposes, we have used a numerical approach based in the diagonalization of the matrix related to the Hamiltonian when the wavefunction is represented as an expansion in terms of "a particle-in-a-box" basis functions. The results show that, in the symmetric confining case, the energy levels are degenerate and a regular pairwise association between them is observed, similarly as it occurs in the free case. Moreover, when the confining is asymmetric, the degeneration is partially lifted but the pairwise association of the energy levels becomes irregular. The lifting of the degeneration in the latter case is addressed to the lack of symmetry or distortion of the system, namely, to a sort of Jahn-Teller effect which is common in the energy levels of diatomic molecules, to which a double well potential can be crudely associated. In the symmetric case, the states with nodes at the origin are recognized to be the same as those of the harmonic oscillator confined by two impenetrable walls, in such a way that the system presented in this work would be interpreted as half the solution of the problem of a particle within a confined four well potential. The latter suggests the existence of a sort of hidden symmetry which remains to be studied in a more detailed way.

Symmetry adapted functions for double point groups II. Cubic point groups

1970 ◽  
Vol 67 (3) ◽  
pp. 647-656 ◽  
Author(s):  
A. P. Cracknell ◽  
S. J. Joshua
Keyword(s):  
Double Point ◽  
Point Group ◽  
Basis Functions ◽  
Kronecker Products ◽  
Symmetry Adapted Functions ◽  
Point Groups

AbstractA method of deriving the basis functions of the double-valued representations of a point group by the reduction of Kronecker products is described. The method has been used to derive expressions for these bases for cubic point groups for which the results are tabulated.

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