scholarly journals From atoms to bonds, angles and torsions: molecular metrics from crystal space, and two Excel implementations

2020 ◽  
Vol 53 (4) ◽  
pp. 1101-1107
Author(s):  
Leslie Glasser
Keyword(s):  
Euclidean Space ◽  
Matrix Functions ◽  
Practical Application ◽  
Torsion Angles ◽  
Molecular Systems ◽  
The Matrix ◽  
Molecular Bond ◽  
Crystal Space ◽  
Vector Matrix ◽  
Computational Form

Values of molecular bond lengths, bond angles and (less frequently) bond torsion angles are readily available from databases, from crystallographic software, and/or from interactive molecular and crystal visualization programs such as Jmol. However, the methods used to calculate these values are less well known. In this paper, the computational methods are described in detail, and live Excel implementations, which permit readers to readily perform the calculations for their own molecular systems, are provided. The methods described apply to both fractional coordinates in crystal space and Cartesian coordinates in Euclidean space (space in which the geometric postulates of Euclid are valid) and are vector/matrix based. In their simplest computational form, they are applied as algebraic expansions which are summed. They are also available in matrix formulations, which are readily manipulated and calculated using the matrix functions of Excel. In particular, their general formulation as metric matrices is introduced. The methods in use are illustrated by a detailed example of the calculations. This contribution provides a significant practical application which can also act as motivation for the study of matrix mathematics with respect to its many uses in chemistry.

Matrix Elements of the Lorentzian Term

1976 ◽  
Vol 31 (6) ◽  
pp. 553-556 ◽  
Author(s):  
Ch. V. S. Ramachandrá Rao
Keyword(s):  
Harmonic Oscillator ◽  
Matrix Elements ◽  
Practical Application ◽  
The Matrix

Recursion formulae for the matrix elements of the Lorentzian term 1/(C2 + q2) as well as 1/(C2 + q2)2, on the basis of harmonic oscillator eigenfunctions, are obtained. A practical application where these formulae would be useful is discussed

scholarly journals QCD Factorization and PDFs from Lattice QCD Calculation

2015 ◽  
Vol 37 ◽  
pp. 1560041 ◽  
Author(s):  
Yan-Qing Ma ◽  
Jian-Wei Qiu
Keyword(s):  
Euclidean Space ◽  
Minkowski Space ◽  
Lattice Qcd ◽  
Correlation Functions ◽  
Parton Distribution ◽  
Distribution Functions ◽  
Matrix Elements ◽  
Parton Distribution Functions ◽  
Qcd Factorization ◽  
The Matrix

In this talk, we review a QCD factorization based approach to extract parton distribution and correlation functions from lattice QCD calculation of single hadron matrix elements of quark-gluon operators. We argue that although the lattice QCD calculations are done in the Euclidean space, the nonperturbative collinear behavior of the matrix elements are the same as that in the Minkowski space, and could be systematically factorized into parton distribution functions with infrared safe matching coefficients. The matching coefficients can be calculated perturbatively by applying the factorization formalism on to asymptotic partonic states.

scholarly journals Block Toeplitz operators with frequency-modulated semi-almost periodic symbols

2003 ◽  
Vol 2003 (34) ◽  
pp. 2157-2176 ◽  
Author(s):  
A. Böttcher ◽  
S. Grudsky ◽  
I. Spitkovsky
Keyword(s):  
Toeplitz Operator ◽  
Frequency Modulation ◽  
Real Line ◽  
Matrix Function ◽  
Toeplitz Operators ◽  
Matrix Functions ◽  
Almost Periodic ◽  
The Matrix ◽  
Matrix Symbol ◽  
Block Toeplitz Operators

This paper is concerned with the influence of frequency modulation on the semi-Fredholm properties of Toeplitz operators with oscillating matrix symbols. The main results give conditions on an orientation-preserving homeomorphismαof the real line that ensure the following: ifbbelongs to a certain class of oscillating matrix functions (periodic, almost periodic, or semi-almost periodic matrix functions) and the Toeplitz operator generated by the matrix functionb(x)is semi-Fredholm, then the Toeplitz operator with the matrix symbolb(α(x))is also semi-Fredholm.

Note on the Matrix Functions sin πA and cos πA

1966 ◽  
Vol 39 (5) ◽  
pp. 287
Author(s):  
Jerry C. South,
Keyword(s):  
Matrix Functions ◽  
The Matrix

scholarly journals Algebraic Characteristics of the Matrix Conversions Class and Its Hardware Implementation

2021 ◽  
Vol 4 (2) ◽  
Author(s):  
Stanislav V. Kudlai
Keyword(s):  
Algebraic System ◽  
Hardware Implementation ◽  
Matrix Transformations ◽  
Matrix Functions ◽  
Hardware Support ◽  
Algebra Isomorphism ◽  
Program Algebra ◽  
Matrix Operations ◽  
The Matrix ◽  
Recursive Matrix

This paper derives the algebraic characteristic of the matrix transformations class by the method of isomorphic mappings on the algebraic characteristic of the class of vector transformations using the primitive program algebras. The paper also describes the hardware implementation of the matrix operations accelerator based on the obtained results. The urgency of the work is caused by the fact that today there is a rapid integration of computer technology in all spheres of society and, as a consequence, the amount of data that needs to be processed per unit time is constantly increasing. Many problems involving large amounts of complex computation are solved by methods based on matrix operations. Therefore, the study of matrix calculations and their acceleration is a very important task. In this paper, as a contribution in this direction, we propose a study of the matrix transformations class using signature operations of primitive program algebra such as multi place superposition, branching, cycling, which are refinements of the most common control structures in most high-level programming languages, and also isomorphic mapping. Signature operations of primitive program algebra in combination with basic partial-recursive matrix functions and predicates allow to realize the set of all partial-recursive matrix functions and predicates. Obtained the result on the basis of matrix primitive program algebra. Isomorphism provides the reproduction of partially recursive functions and predicates for matrix transformations as a map of partially recursive vector functions and predicates. The completeness of the algebraic system of matrix transformations is ensured due to the available results on the derivation of the algebraic system completeness for vector transformations. A name model of matrix data has been created and optimized for the development of hardware implementation. The hardware implementation provides support for signature operations of primitive software algebra and for isomorphic mapping. Hardware support for the functions of sum, multiplication and transposition of matrices, as well as the predicate of equality of two matrices is implemented. Support for signature operations of primitive software algebra is provided by the design of the control part of the matrix computer based on the RISC architecture. The hardware support of isomorphism is based on counters, they allow to intuitively implement cycling in the functions of isomorphic mappings. Fast execution of vector operations is provided by the principle of computer calculations SIMD.

scholarly journals Asymptotic analysis for the electric field concentration with geometry of the core-shell structure

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Zhiwen Zhao
Keyword(s):  
Electric Field ◽  
Blow Up ◽  
Core Shell ◽  
Singular Behavior ◽  
Practical Application ◽  
Function Type ◽  
The Core ◽  
The Matrix ◽  
Perfect Conductivity

<p style='text-indent:20px;'>In the perfect conductivity problem arising from composites, the electric field may become arbitrarily large as <inline-formula><tex-math id="M1">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula>, the distance between the inclusions and the matrix boundary, tends to zero. In this paper, by making clear the singular role of the blow-up factor <inline-formula><tex-math id="M2">\begin{document}$ Q[\varphi] $\end{document}</tex-math></inline-formula> introduced in [<xref ref-type="bibr" rid="b27">27</xref>] for some special boundary data of even function type with <inline-formula><tex-math id="M3">\begin{document}$ k $\end{document}</tex-math></inline-formula>-order growth, we prove the optimality of the blow-up rate in the presence of <inline-formula><tex-math id="M4">\begin{document}$ m $\end{document}</tex-math></inline-formula>-convex inclusions close to touching the matrix boundary in all dimensions. Finally, we give closer analysis in terms of the singular behavior of the concentrated field for eccentric and concentric core-shell geometries with circular and spherical boundaries from the practical application angle.</p>

Computation of Fourier transform representations involving the generalized Bessel matrix polynomials

2021 ◽  
Author(s):  
mohamed abdalla
Keyword(s):  
Fourier Transform ◽  
Orthogonal Polynomials ◽  
Integral Transforms ◽  
General Character ◽  
Matrix Functions ◽  
Fourier Cosine ◽  
Matrix Polynomials ◽  
The Matrix ◽  
Special Matrix ◽  
Special Matrix Functions

Abstract Motivated by the recent studies and developments of the integral transforms with various special matrix functions, including the matrix orthogonal polynomials as kernels. In this article, we derive the formulas for Fourier cosine transforms and Fourier sine transforms of matrix functions involving generalized Bessel matrix polynomials. With the help of these transforms a number of results are considered which are extensions of the corresponding results in the standard cases. The results given here are of general character and can yield a number of (known and new) results in modern integral transforms.

Example of the Relationship Between the Matrix Functions and Modern Control Theory

2016 ◽  
Vol 14 (2) ◽  
pp. 511-516
Author(s):  
Eduardo Marcelo Seguin Batadi ◽  
Graciela Beatriz Ganyitano ◽  
Claudia Rosana Fernandez
Keyword(s):  
Control Theory ◽  
Matrix Functions ◽  
Modern Control Theory ◽  
The Matrix ◽  
The Relationship ◽  
Modern Control

Preparation and Study on Temperature-Resistance Environment-Friendly Ink Based on Natural Pigment and Soybean Oil

2013 ◽  
Vol 469 ◽  
pp. 13-16
Author(s):  
Zhuang Liu ◽  
Qi Liu ◽  
Liu Yang ◽  
Jing Yi Sun ◽  
Gui Yu Ma ◽  
...  
Keyword(s):  
High Temperature ◽  
Soybean Oil ◽  
Chain Structure ◽  
Practical Application ◽  
Application Process ◽  
Natural Pigment ◽  
Environment Friendly ◽  
Temperature Resistance ◽  
Molecular Bond ◽  
Practical Temperature

Traditional ink based on natural pigment and soybean oil has disadvantages of poor temperature adaptive in application process. This project broke the limitations that stocks printing with traditional ink based on natural pigment and soybean oil could not be used for heating. Developing practical temperature-resistance environment-friendly ink based on natural pigment and soybean oil is of great importance. The experiment introduced the silicone material, and made full use of its characteristic. The Si-O bond of silicone can be 121 kcal/mole, and its molecular bond can not be broken down at a high temperature (or exposure to radiation). The property of high temperature resistance mainly depends on its unique functional group or long polymer chain structure. The experiment studied temperature resistance environment-friendly ink's performance in color, in order to improve its practical application value.

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