Generalization of the continuous symmetry measure: the symmetry of vectors, matrices, operators and functions

Abstract

In this paper we generalize a method for evaluating the continuous symmetry measure, which is a quantitative estimate of the degree of symmetry of a given object. The generalization makes it possible to calculate the degree of symmetry content for any mathematical entity that is part of metric spaces such as vectors, matrices, operators and functions. Furthermore, by this new approach one can calculate the symmetry-content values for any compact symmetry groups either finite or infinite. An advantage of the new methodology is the ability to investigate analytically problems of symmetry changes. Examples of symmetry evaluation calculations are provided, including mixing of ideal gases, evaluation of the symmetry content of a Hamiltonian operator, the 2p(z) orbital of the hydrogen atom, and more.

Chirality Measures for Vectors, Matrices, Operators and Functions

“…[15]). Here we emphasize that Wi j , the closest symmetrical object, is a mathematical construct, usually without physical meaning.…”

“…Recently, we formulated a general methodology for calculating the CSM, [15] allowing one to calculate the symmetry content for variety of mathematical entities, including vectors, operators, matrices and functions. Herein we introduce a new general tool for calculating the chirality content of various mathematical descriptions.…”

Concurrent Symmetries: The Interplay Between Local and Global Molecular Symmetries

“…Needless to say, the Cp rings also retain their fivefold symmetry. It must be noted that the local symmetry is consistent with the equivalence of all the carbon atoms of the COT ring found in the 13 C NMR spectrum that appear as a multiplet at d = 72.0 ppm. [9] The aim of the present work is to analyze the existence of high degrees of local symmetry that cannot be accounted for by the global molecular symmetry.…”

“…Only acyclic substitutions were allowed in the aromatic rings and structures with disorder were disregarded. The relevant fragments of the studied molecules, devoid of substituents, were analyzed by calculating the symmetry measures with the algorithms reported earlier, [4,10,13] implemented in the program SYMGROUP.…”

Study of the Power Amplifier transfer function symmetry and its relation to dynamic effects

“…Using [7], the transfer function of the P A can be studied. In particular; we analyze and quantify its rotational symmetry [8]. The symmetry analysis of the P A transfer functions helps in significantly reducing the computational resources and storage requirements for the method in [7].…”