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A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (31 July 2023) | Viewed by 30766

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Prof. Dr. Evgeny Nikulchev

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Department of Digital Technologies of Data Processing, MIREA – Russian Technological University, 119454 Moscow, Russia
Interests: symmetry groups; lie groups; dynamic systems modeling; experimental processing; artificial intellectual technologies; information systems
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

In this Special Issue of Symmetry, we want to present research articles and review articles on the recent advances in the application of the symmetry group.

The theory of symmetry groups is one of the most interesting fields of mathematics, but it also plays an important role in different fields of modern science.

The theory of symmetry groups is one of the main mathematical tools in Galois theory, invariant theory, theory of combinatorics, and theory of Lie groups of differential equations.

The symmetric group on a set of size n is the Galois group of the general polynomial of degree n. This group plays an important role in the theory of finding solutions to the equations. In the invariant theory, the symmetric group acts on the variables of multivariable function, but the invariant functions are so-called symmetric functions. In the theory of combinatorics, symmetric groups defined on a permutations set provide a rich source of tasks and interesting solutions, in particular, to study group actions, homogeneous spaces, and the automorphism of graphs. This is an excellent mathematical subject, ranging from numbers theory and theory of combinatorics to geometry, theory of probability, quantum mechanics, and quantum field theory. Recently, it was used as a research tool in the theory of cooperative games.

In the application, the theory of symmetry groups of differential equations, developed by Sophus Lie, plays the most important role. Lie groups are defined by transformations of differential equations solutions. The symmetry group method provides a set of tools for analyzing differential equations and plays an important role in solution finding. One of the most important applications in the theory of differential is equations of recent years: Reduction, decrease of ordinary differential equations degree, invariant solution finding, mapping solutions to other solutions, calculation of linear transformations, and building signals filters. It has been used to simulate different objects and phenomena.

We also invite researchers to contribute with new research on the field of symmetry group applications.

Dr. Evgeny Nikulchev
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

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Keywords

Invariant theory and applications
Symmetry groups in game theory
Geometric theory and symmetry groups
Galois group
Lee groups and applications
Using symmetry groups for modeling and simulations
Analysis of signals and systems based on invariants and symmetries

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Published Papers (13 papers)

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Research

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17 pages, 297 KiB

Open AccessArticle

Exploring Roughness in Left Almost Semigroups and Its Connections to Fuzzy Lie Algebras

by Abdullah Assiry and Amir Baklouti

Symmetry 2023, 15(9), 1717; https://doi.org/10.3390/sym15091717 - 7 Sep 2023

Cited by 1 | Viewed by 895

Abstract

This paper explores the concept of Generalized Roughness in LA-Semigroups and its applications in various mathematical disciplines. We highlight the fundamental properties and structures of Generalized Roughness, examining its relationships with Fuzzy Lie Algebras, Order Theory, Lattice Structures, Algebraic Structures, and Categorical Perspectives. Moreover, we investigate the potential of mathematical modeling, optimization techniques, data analysis, and machine learning in the context of Generalized Roughness. Our findings reveal important results in Generalized Roughness, such as the preservation of roughness under the fuzzy equivalence relation and the composition of roughness sets. We demonstrate the significance of Generalized Roughness in the context of order theory and lattice structures, presenting key propositions and a theorem that elucidate its properties and relationships. Furthermore, we explore the applications of Generalized Roughness in mathematical modeling and optimization, highlighting the optimization of roughness measures, parameter estimation, and decision-making processes related to LA-Semigroup operations. We showcase how mathematical techniques can enhance understanding and utilization of LA-Semigroups in practical scenarios. Lastly, we delve into the role of data analysis and machine learning in uncovering patterns, relationships, and predictive models in Generalized Roughness. By leveraging these techniques, we provide examples and insights into how data analysis and machine learning can contribute to enhancing our understanding of LA-Semigroup behavior and supporting decision-making processes. Full article

(This article belongs to the Special Issue Recent Advances in the Application of Symmetry Group)

10 pages, 625 KiB

Open AccessArticle

A Modified Wei-Hua-He Digital Signature Scheme Based on Factoring and Discrete Logarithm

by Elumalai R and G. S. G. N. Anjaneyulu

Symmetry 2022, 14(11), 2443; https://doi.org/10.3390/sym14112443 - 17 Nov 2022

Cited by 1 | Viewed by 1773

Abstract

A symmetric cipher such as AES in cryptography is much faster than an asymmetric cipher but digital signatures often use asymmetric key ciphers because they provides the sender’s identity and data integrity. In this paper, a modified-He digital signature scheme is proposed using a one-way hash function. The proposed scheme, unlike the He signature technique, employs Euclid’s Division Lemma with large prime moduli p. Its security is built on large integer factoring, discrete logarithms and expanded root problems. The time complexity of the proposed scheme is

O (l o g^{3} p)

. The proposed modified-He scheme is efficient, as evidenced by the analytical results with key lengths greater than 512 bits. Full article

(This article belongs to the Special Issue Recent Advances in the Application of Symmetry Group)

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42 pages, 898 KiB

Open AccessArticle

Two-Dimensional Turbulent Thermal Free Jet: Conservation Laws, Associated Lie Symmetry and Invariant Solutions

by Erick Mubai and David Paul Mason

Symmetry 2022, 14(8), 1727; https://doi.org/10.3390/sym14081727 - 18 Aug 2022

Cited by 4 | Viewed by 1782

Abstract

The two-dimensional turbulent thermal free jet is formulated in the boundary layer approximation using the Reynolds averaged momentum balance equation and the averaged energy balance equation. The turbulence is described by Prandtl’s mixing length model for the eddy viscosity

ν_{T}

with mixing [...] Read more.

ν_{T}

with mixing length l and eddy thermal conductivity

κ_{T}

with mixing length

l_{θ}

. Since

ν_{T}

and

κ_{T}

are proportional to the mean velocity gradient the momentum and thermal boundaries of the flow coincide. The conservation laws for the system of two partial differential equations for the stream function of the mean flow and the mean temperature difference are derived using the multiplier method. Two conserved vectors are obtained. The conserved quantities for the mean momentum and mean heat fluxes are derived. The Lie point symmetry associated with the two conserved vectors is derived and used to perform the reduction of the partial differential equations to a system of ordinary differential equations. It is found that the mixing lengths l and

l_{θ}

are proportional. A turbulent thermal jet with

ν_{T} \neq 0

and

κ_{T} \neq 0

but vanishing kinematic viscosity

ν

and thermal conductivity

κ

is studied. Prandtl’s hypothesis that the mixing length is proportional to the width of the jet is made to complete the system of equations. An analytical solution is derived. The boundary of the jet is determined with the aid of a conserved quantity and found to be finite. Analytical solutions are derived and plotted for the streamlines of the mean flow and the lines of constant mean thermal difference. The solution differs from the analytical solution obtained in the limit

ν \to 0

and

κ \to 0

without making the Prandtl’s hypothesis. For

ν \neq 0

and

κ \neq 0

a numerical solution is derived using a shooting method with the two conserved quantities as targets instead of boundary conditions. The numerical solution is verified by comparing it to the analytical solution when

ν \to 0

and

κ \to 0

. Because of the limitations imposed by the accuracy of any numerical method the numerical solution could not reliably determine if the jet is unbounded when

ν \neq 0

and

κ \neq 0

but for large distance from the centre line,

ν > ν_{T}

and

κ > κ_{T}

and the jet behaves increasingly like a laminar jet which is unbounded. The streamlines of the mean flow and the lines of constant mean temperature difference are plotted for

ν = 0

and

κ = 0

. Full article

(This article belongs to the Special Issue Recent Advances in the Application of Symmetry Group)

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20 pages, 294 KiB

Open AccessArticle

Tripolar Picture Fuzzy Ideals of BCK-Algebras

by Ghulam Muhiuddin, Nabilah Abughazalah, Afaf Aljuhani and Manivannan Balamurugan

Symmetry 2022, 14(8), 1562; https://doi.org/10.3390/sym14081562 - 29 Jul 2022

Cited by 29 | Viewed by 1617

Abstract

In this paper, we acquaint new kinds of ideals of BCK-algebras built on tripolar picture fuzzy structures. In fact, the notions of tripolar picture fuzzy ideal, tripolar picture fuzzy implicative ideal (commutative ideal) of BCK-algebra are introduced, and related properties are studied. Also, a relation among tripolar picture fuzzy ideal, and tripolar picture fuzzy implicative ideal is well-known. Furthermore, it is shown that a tripolar picture fuzzy implicative ideal of BCK-algebra may be a tripolar picture fuzzy ideal, but the converse is not correct in common. Further, it is obtained that in an implicative BCK-algebra, the converse of aforementioned statement is true. Finally, the opinion of tripolar picture fuzzy commutative ideal is given, and some useful properties are explored. Many examples are constructed to sport our study. Full article

(This article belongs to the Special Issue Recent Advances in the Application of Symmetry Group)

16 pages, 303 KiB

Open AccessArticle

Hybrid Nil Radical of a Ring

by Kasi Porselvi, Ghulam Muhiuddin, Balasubramanian Elavarasan and Abdullah Assiry

Symmetry 2022, 14(7), 1367; https://doi.org/10.3390/sym14071367 - 3 Jul 2022

Cited by 13 | Viewed by 1723

Abstract

The nature of universe problems is ambiguous due to the presence of asymmetric data in almost all disciplines, including engineering, mathematics, medical sciences, physics, computer science, operations research, artificial intelligence, and management sciences, and they involve various types of uncertainties when dealing with them on various occasions. To deal with the challenges of uncertainty and asymmetric information, different theories have been developed, including probability, fuzzy sets, rough sets, soft ideals, etc. The strategies of hybrid ideals, hybrid nil radicals, hybrid semiprime ideals, and hybrid products of rings are introduced in this paper and hybrid structures are used to examine the structural properties of rings. Full article

(This article belongs to the Special Issue Recent Advances in the Application of Symmetry Group)

10 pages, 777 KiB

Open AccessArticle

A Generalization of Group-Graded Modules

by Mohammed Al-Shomrani and Najlaa Al-Subaie

Symmetry 2022, 14(4), 835; https://doi.org/10.3390/sym14040835 - 18 Apr 2022

Cited by 2 | Viewed by 1812

Abstract

In this article, we generalize the concept of group-graded modules by introducing the concept of

G

-weak graded R-modules, which are R-modules graded by a set

G

of left coset representatives, where R is a

G

-weak graded ring. Moreover, we [...] Read more.

In this article, we generalize the concept of group-graded modules by introducing the concept of

G

-weak graded R-modules, which are R-modules graded by a set

G

of left coset representatives, where R is a

G

-weak graded ring. Moreover, we prove some properties of these modules. Finally, results related to

G

-weak graded fields and their vector spaces are deduced. Many considerable examples are provided with more emphasis on the symmetric group

S_{3}

and the dihedral group

D_{6}

, which gives the group of symmetries of a regular hexagon. Full article

(This article belongs to the Special Issue Recent Advances in the Application of Symmetry Group)

21 pages, 363 KiB

Open AccessArticle

Heisenberg–Weyl Groups and Generalized Hermite Functions

by Enrico Celeghini, Manuel Gadella and Mariano A. del Olmo

Symmetry 2021, 13(6), 1060; https://doi.org/10.3390/sym13061060 - 12 Jun 2021

Cited by 4 | Viewed by 2285

Abstract

We introduce a multi-parameter family of bases in the Hilbert space

L^{2} (R)

that are associated to a set of Hermite functions, which also serve as a basis for

L^{2} (R)

. The Hermite functions are eigenfunctions [...] Read more.

We introduce a multi-parameter family of bases in the Hilbert space

L^{2} (R)

that are associated to a set of Hermite functions, which also serve as a basis for

L^{2} (R)

. The Hermite functions are eigenfunctions of the Fourier transform, a property that is, in some sense, shared by these “generalized Hermite functions”. The construction of these new bases is grounded on some symmetry properties of the real line under translations, dilations and reflexions as well as certain properties of the Fourier transform. We show how these generalized Hermite functions are transformed under the unitary representations of a series of groups, including the Heisenberg–Weyl group and some of their extensions. Full article

(This article belongs to the Special Issue Recent Advances in the Application of Symmetry Group)

15 pages, 3004 KiB

Open AccessArticle

Multisensory Gas Analysis System Based on Reconstruction Attractors

by Olga Cheremisina, Vladimir Kulagin, Suad El-Saleem and Evgeny Nikulchev

Symmetry 2020, 12(6), 964; https://doi.org/10.3390/sym12060964 - 5 Jun 2020

Cited by 1 | Viewed by 1839

Abstract

The paper describes the substance image formation based on the measurements by multisensor systems and the possibility of the development of a gas analysis device like an electronic nose. Classification of gas sensors and the need for their application for the recognition of difficult images of multicomponent air environments are considered. The image is formed based on stochastic transformations, calculations of correlation, and fractal dimensions of reconstruction attractors. The paper shows images created for substances with various structures that were received with the help of a multisensor system under fixed measurement conditions. Full article

(This article belongs to the Special Issue Recent Advances in the Application of Symmetry Group)

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18 pages, 7717 KiB

Open AccessArticle

A Study of Chaotic Maps Producing Symmetric Distributions in the Fish School Search Optimization Algorithm with Exponential Step Decay

by Liliya A. Demidova and Artyom V. Gorchakov

Symmetry 2020, 12(5), 784; https://doi.org/10.3390/sym12050784 - 8 May 2020

Cited by 30 | Viewed by 4526

Abstract

Inspired by the collective behavior of fish schools, the fish school search (FSS) algorithm is a technique for finding globally optimal solutions. The algorithm is characterized by its simplicity and high performance; FSS is computationally inexpensive, compared to other evolution-inspired algorithms. However, the premature convergence problem is inherent to FSS, especially in the optimization of functions that are in very-high-dimensional spaces and have plenty of local minima or maxima. The accuracy of the obtained solution highly depends on the initial distribution of agents in the search space and on the predefined initial individual and collective-volitive movement step sizes. In this paper, we provide a study of different chaotic maps with symmetric distributions, used as pseudorandom number generators (PRNGs) in FSS. In addition, we incorporate exponential step decay in order to improve the accuracy of the solutions produced by the algorithm. The obtained results of the conducted numerical experiments show that the use of chaotic maps instead of other commonly used high-quality PRNGs can speed up the algorithm, and the incorporated exponential step decay can improve the accuracy of the obtained solution. Different pseudorandom number distributions produced by the considered chaotic maps can positively affect the accuracy of the algorithm in different optimization problems. Overall, the use of the uniform pseudorandom number distribution generated by the tent map produced the most accurate results. Moreover, the tent-map-based PRNG achieved the best performance when compared to other chaotic maps and nonchaotic PRNGs. To demonstrate the effectiveness of the proposed optimization technique, we provide a comparison of the tent-map-based FSS algorithm with exponential step decay (ETFSS) with particle swarm optimization (PSO) and with the genetic algorithm with tournament selection (GA) on test functions for optimization. Full article

(This article belongs to the Special Issue Recent Advances in the Application of Symmetry Group)

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12 pages, 2308 KiB

Open AccessArticle

A General Principle of Isomorphism: Determining Inverses

by Vladimir S. Kulabukhov

Symmetry 2019, 11(10), 1301; https://doi.org/10.3390/sym11101301 - 15 Oct 2019

Cited by 3 | Viewed by 3130

Abstract

The problem of determining inverses for maps in commutative diagrams arising in various problems of a new paradigm in algebraic system theory based on a single principle—the general principle of isomorphism is considered. Based on the previously formulated and proven theorem of realization, the rules for determining the inverses for typical cases of specifying commutative diagrams are derived. Simple examples of calculating the matrix maps inverses, which illustrate both the derived rules and the principle of relativity in algebra based on the theorem of realization, are given. The examples also illustrate the emergence of new properties (emergence) in maps in commutative diagrams modeling (realizing) the corresponding systems. Full article

(This article belongs to the Special Issue Recent Advances in the Application of Symmetry Group)

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20 pages, 3497 KiB

Open AccessArticle

The Influence of a Network’s Spatial Symmetry, Topological Dimension, and Density on Its Percolation Threshold

by Dmitry O. Zhukov, Elena G. Andrianova and Sergey A. Lesko

Symmetry 2019, 11(7), 920; https://doi.org/10.3390/sym11070920 - 15 Jul 2019

Cited by 5 | Viewed by 3088

Abstract

Analyses of the processes of information transfer within network structures shows that the conductivity and percolation threshold of the network depend not only on its density (average number of links per node), but also on its spatial symmetry groups and topological dimension. The results presented in this paper regarding conductivity simulation in network structures show that, for regular and random 2D and 3D networks, an increase in the number of links (density) per node reduces their percolation threshold value. At the same network density, the percolation threshold value is less for 3D than for 2D networks, whatever their structure and symmetry may be. Regardless of the type of networks and their symmetry, transition from 2D to 3D structures engenders a change of percolation threshold by a value exp{−(d − 1)} that is invariant for transition between structures, for any kind of network (d being topological dimension). It is observed that in 2D or 3D networks, which can be mutually transformed by deformation without breaking and forming new links, symmetry of similarity is observed, and the networks have the same percolation threshold. The presence of symmetry axes and corresponding number of symmetry planes in which they lie affects the percolation threshold value. For transition between orders of symmetry axes, in the presence of the corresponding planes of symmetry, an invariant exists which contributes to the percolation threshold value. Inversion centers also influence the value of the percolation threshold. Moreover, the greater the number of pairs of elements of the structure which have inversion, the more they contribute to the fraction of the percolation threshold in the presence of such a center of symmetry. However, if the center of symmetry lies in the plane of mirror symmetry separating the layers of the 3D structure, the mutual presence of this group of symmetry elements do not affect the percolation threshold value. The scientific novelty of the obtained results is that for different network structures, it was shown that the percolation threshold for the blocking of nodes problem could be represented as an additive set of invariant values, that is, as an algebraic sum, the value of the members of which is stored in the transition from one structure to another. The invariant values are network density, topological dimension, and some of the elements of symmetry (axes of symmetry and the corresponding number of symmetry planes in which they lie, centers of inversion). Full article

(This article belongs to the Special Issue Recent Advances in the Application of Symmetry Group)

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Review

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13 pages, 1348 KiB

Open AccessReview

Prediction Intervals: A Geometric View

by Evgeny Nikulchev and Alexander Chervyakov

Symmetry 2023, 15(4), 781; https://doi.org/10.3390/sym15040781 - 23 Mar 2023

Cited by 4 | Viewed by 2074

Abstract

This article provides a review of the approaches to the construction of prediction intervals. To increase the reliability of prediction, point prediction methods are replaced by intervals for many aims. The interval prediction generates a pair as future values, including the upper and lower bounds for each prediction point. That is, according to historical data, which include a graph of a continuous and discrete function, two functions will be obtained as a prediction, i.e., the upper and lower bounds of estimation. In this case, the prediction boundaries should provide guaranteed probability of the location of the true values inside the boundaries found. The task of building a model from a time series is, by its very nature, incorrect. This means that there is an infinite set of equations whose solution is close to the time series for machine learning. In the case of interval use, the inverse problem of dynamics allows us to choose from the entire range of modeling methods, using confidence intervals as solutions, or intervals of a given width, or those chosen as a solution to the problems of multi-criteria optimization of the criteria for evaluating interval solutions. This article considers a geometric view of the prediction intervals and a new approach is given. Full article

(This article belongs to the Special Issue Recent Advances in the Application of Symmetry Group)

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29 pages, 4809 KiB

Open AccessReview

Theory of Projectors and Its Application to Molecular Symmetry

by Horace T. Crogman

Symmetry 2023, 15(2), 496; https://doi.org/10.3390/sym15020496 - 13 Feb 2023

Cited by 1 | Viewed by 1449

Abstract

Projector theory can serve as a powerful tool to perform the symmetric computation of molecular systems. The work of William Harter has long demonstrated the effectiveness of this theory in molecular spectroscopy; however, it seems its usefulness has not been realized by many in the field. We have described this methodology and have considered the D₃ symmetry system and the tetrahedral symmetry of methane molecules as concrete examples where the computed rotation tensors and vibrational wavefunction were derived for some symmetry states. Full article

(This article belongs to the Special Issue Recent Advances in the Application of Symmetry Group)

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