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.

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 Crystal Prediction using the Point Groups: an Application of Group Theory

2014 ◽  
Vol 70 (a1) ◽  
pp. C1427-C1427
Author(s):  
Gregory McColm
Keyword(s):  
Group Theory ◽  
Geometric Approach ◽  
Geometric Object ◽  
Topological Equivalence ◽  
Breadth First Search ◽  
Matrix Representations ◽  
The Core ◽  
Point Groups ◽  
Crystal Design ◽  
Crystal Net

As a blueprint for designing a crystal, a crystal net should be a geometric object. So a crystal design regime might be based on an algorithm that generates crystal nets using geometry. The "Crystal Turtlebug" is such a program, generating crystal nets as graphs embedded in 3-space. McColm et al (2011) described a "Maple" version; it generated crystal nets of one or two kinds of vertices and one or two kinds of edges. A new "Python" version can generate crystal nets of any number of kinds of vertices and edges. The algorithm employs matrix representations of point groups. The "core" of the program takes a fragment of the prospective crystal (e.g. for a crystal of 3 kinds of atoms and 2 kinds of bonds, a "transversal" of 3 vertices and 2 edges). The core assigns point groups to the vertices and isometries to the edges. The core then repeatedly applies groups to isometries and vice versa to generate a unit cell. The computational problem is finding a transversal that produces a chemically plausible crystal net, as most crystal nets found are implausible. We follow a breadth-first search and enumerate some transversals, screen them, generate crystal nets from the screened transversals, and screen the results. This process can entail generating millions of transversals to obtain a handful of crystal nets for users to view manually. Although such a geometric approach is somewhat different from the more popular directions of contemporary research in crystal prediction, there is precedent in the work of A. F. Wells, A. Le Bail, C. Wilmer et al, and in particular M. Treacy et al (2004). In principle, the Crystal Turtlebug will generate every crystal net up to topological equivalence. In practice, there is an exponential explosion in the number of transversals, so the time spent enumerating crystal nets within fixed parameters explodes exponentially with the number of kinds of atoms and bonds (McColm (2012)).

Matrix representations of right ample semigroups

2015 ◽  
Vol 08 (03) ◽  
pp. 1550042 ◽  
Author(s):  
Junying Guo ◽  
Xiaojiang Guo ◽  
K. P. Shum
Keyword(s):  
Matrix Representation ◽  
Matrix Representations ◽  
Ample Semigroup ◽  
The Matrix

The properties of right ample semigroups have been extensively considered and studied by many authors. In this paper, we concentrate on the matrix representations of right ample semigroups. The (left; right) uniform matrix representation is initially defined. After some properties of left uniform matrix representations of a right ample semigroup are given, we prove that any irreducible left uniform representations of a right ample semigroup can be obtained by using an irreducible left uniform representation of some primitive right ample semigroup. In particular, a construction theorem of prime left uniform representation of right ample semigroups is established.

Crystallography of embedded particles in Al–Mg–Zn alloys. Symmetry analysis

2014 ◽  
Vol 47 (5) ◽  
pp. 1736-1748 ◽  
Author(s):  
Gunnar Thorkildsen ◽  
Helge B. Larsen
Keyword(s):  
Proper Subgroup ◽  
Point Group ◽  
Alloy System ◽  
Group Symmetry ◽  
Symmetry Principle ◽  
Generating Sets ◽  
The Matrix ◽  
Embedded Particles ◽  
Point Groups ◽  
Zn Alloys

Following a proper heat treatment, the alloy system Al–Mg–Zn shows a great wealth of precipitate particles forming coherent (η′ crystals) and incoherent (η crystals) boundaries with the Al matrix. Both the matrix crystal and the precipitate crystals are holohedral, as their point groups correspond to their metric symmetries (m\overline{3}m and 6/mmm). On the basis of published orientational relationships for a principal variant of every known precipitate family, the full sets of orientational variants are deduced by the concepts of intersection groups,Hβ, and variant generating sets,Vβ. The intergrowth symmetry principle has been visualized by stereographic projections. Special attention has been given to patterns of superimposed lattice nodes in reciprocal space and the implications of overlap in terms of observable reflections. It has been uncovered, by artificially reducing the point group symmetry of the coherent η′ phase, whereVβis a proper subgroup of the matrix holohedral, that twin laws for merohedry are revealed whenVβis factorized into a weak direct product.

scholarly journals NUCLEI WITH TETRAHEDRAL SYMMETRY

2007 ◽  
Vol 16 (02) ◽  
pp. 516-532 ◽  
Author(s):  
J. DUDEK ◽  
J. DOBACZEWSKI ◽  
N. DUBRAY ◽  
A. GÓŹDŹ ◽  
V. PANGON ◽  
...  
Keyword(s):  
Rare Earth ◽  
Group Theory ◽  
Point Group ◽  
Tetrahedral Symmetry ◽  
Nuclear Stability ◽  
Point Groups ◽  
The Rare Earth

We discuss a point-group-theory based method of searching for new regions of nuclear stability. We illustrate the related strategy with realistic calculations employing the tetrahedral and the octahedral point groups. In particular, several nuclei in the Rare Earth region appear as excellent candidates to study the new mechanism.

scholarly journals Explicit expressions for totally symmetric spherical functions and symmetry-dependent properties of multipoles

2014 ◽  
Vol 470 (2171) ◽  
pp. 20140435
Author(s):  
Boris Zapol ◽  
Peter Zapol
Keyword(s):  
Point Group ◽  
Product Structure ◽  
Spherical Functions ◽  
Matrix Elements ◽  
Linear Combinations ◽  
Selection Rules ◽  
Charge Distributions ◽  
Symmetry Operators ◽  
Point Groups ◽  
Symmetry Operations

Closed expressions for matrix elements 〈 lm' | A (G)| lm 〉, where | lm 〉 are spherical functions and A (G) is the average of all symmetry operators of point group G, are derived for all point groups (PGs) and then used to obtain linear combinations of spherical functions that are totally symmetric under all symmetry operations of G. In the derivation, we exploit the product structure of the groups. The obtained expressions are used to explore properties of multipoles of symmetric charge distributions. We produce complete lists of selection rules for multipoles Q l and their moments Q lm , as well as of numbers of independent moments in a multipole, for any l and m and for all PGs. Periodicities and other trends in these properties are revealed.

scholarly journals Eigenvalues of matrices related to the octonions

4open ◽  
2019 ◽  
Vol 2 ◽  
pp. 16
Author(s):  
Rogério Serôdio ◽  
Patricia Beites ◽  
José Vitória
Keyword(s):  
Matrix Representation ◽  
Real Matrix ◽  
Matrix Representations ◽  
Estimation Algorithms ◽  
The Matrix

A pseudo real matrix representation of an octonion, which is based on two real matrix representations of a quaternion, is considered. We study how some operations defined on the octonions change the set of eigenvalues of the matrix obtained if these operations are performed after or before the matrix representation. The established results could be of particular interest to researchers working on estimation algorithms involving such operations.

scholarly journals Color-Coding Point Group & Space Group Diagrams and other Mathematical Journeys

2014 ◽  
Vol 70 (a1) ◽  
pp. C1275-C1275
Author(s):  
Maureen Julian
Keyword(s):  
General Position ◽  
Point Group ◽  
Three Dimensional ◽  
Irreducible Representations ◽  
Color Coding ◽  
Space Group ◽  
Point Groups ◽  
Symmetry Operations

Color clarifies diagrams point group and space group diagrams. For example, consider the general position diagrams and the symbol diagrams. Symmetry operations can be represented by matrices whose determinant is either plus one or minus one. In the former case there is no change of handedness and in the latter case there is a change of handedness. The general position diagrams emphasis this information by color-coding. The symbol diagrams are a little more complicated and will be demonstrated. The second topic is a comparison of the thirty-two three-dimensional point groups with their corresponding 18 abstract mathematical groups. The corresponding trees will be explored. This discussion leads into the topic of irreducible representations.

Irreducible Representations for Cyclic, Dihedral and Cubic Point Groups

1987 ◽  
Vol 40 (6) ◽  
pp. 1035
Author(s):  
A Rodger
Keyword(s):  
Multiplication Table ◽  
Irreducible Representations ◽  
The Matrix ◽  
Point Groups

A method for deriving the matrix irreducible representations for cyclic, dihedral and cubic groups purely from the structure of the group multiplication table is presented. Full irreducible representations for D∞h and O are given, from which those for all cyclic, dihedral and cubic groups can simply be derived.

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