Application Of Group Theory Research Articles

Fractional quantum ferroelectricity

For an ordinary ferroelectric, the magnitude of the spontaneous electric polarization is at least one order of magnitude smaller than that resulting from the ionic displacement of the lattice vectors, and the direction of the spontaneous electric polarization is determined by the point group of the ferroelectric. Here, we introduce a new class of ferroelectricity termed Fractional Quantum Ferroelectricity. Unlike ordinary ferroelectrics, the polarization of Fractional Quantum Ferroelectricity arises from substantial atomic displacements that are comparable to lattice constants. Applying group theory analysis, we identify 28 potential point groups that can realize Fractional Quantum Ferroelectricity, including both polar and non-polar groups. The direction of polarization in Fractional Quantum Ferroelectricity is found to always contradict with the symmetry of the “polar” phase, which violates Neumann’s principle, challenging conventional symmetry-based knowledge. Through the Fractional Quantum Ferroelectricity theory and density functional calculations, we not only explain the puzzling experimentally observed in-plane polarization of monolayer α-In2Se3, but also predict polarization in a cubic compound of AgBr. Our findings unveil a new realm of ferroelectric behavior, expanding the understanding and application of these materials beyond the limits of traditional ferroelectrics.

The application of group theory behind modern cryptography

The importance of cryptography and securing data has become increasingly important, and the safety of previous cryptographic methods are also being questioned. In the past few decades, with the rise of modern mathematical tools, notably group theory, cryptography has quickly advanced to more complex and safer levels. This paper will begin by recalling the definition and some terminology about groups. Then, it will summarize two of the most used cryptography systems with underlying group theory: the Diffie-Hellman Key Exchange Protocol, together with one of its variants, the Ko-Lee-Cheon-Han-Kang-Park Key Agreement, and the RSA Protocol. Also, it examines what the safety of a cryptography system means, and methods to increase the security of these protocols. When talking about the two protocols, extensive group theory is used, both as a means of operation and as a method to prove the protocols validity. Finally, it mentions the possible directions of improvement in this field and whether these cryptographic methods are still reliable, even with the widespread use of quantum computers in the future.

Homomorphism, Isomorphism, and Their Applications in Group Theory

The main study of modern algebra is the so-called algebraic systems, i.e., sets with operations. Modern algebra has important applications in other branches of mathematics and natural science. In modern algebra, homomorphism and isomorphism are relatively elementary but extremely important concepts that are interrelated and different. A homomorphism is a mapping that preserves the constitution of a mathematical format and is a generalization of a homomorphism. it is a structurally invariant mapping between two algebraic structures in abstract algebra. The conditions for a homomorphism to be a homomorphism in different algebraic systems are different, and the conditions for a homomorphism to be a homomorphism are described here. Some applications of homomorphisms and homomorphisms to different algebraic systems are discussed, from which the importance of homomorphisms and homomorphisms is illustrated.

On the Proof of Sylow’s Theorems in Group Theory

The Sylow’s theorems are significant principles for analysis of special subgroups of a finite group, and they are significant theories in finite group research area. It is firmly established that every group owns more than one Sylow group pertaining to every prime factor of . The purpose of this article is to summarize and generalize the existing research progress of Sylow’s theorems. A summary of them will help more people comprehend and apply group theory to analyze problems. In this paper, the basic information of three distinct Sylow’s theorems are introduced, including their definitions and detailed proof procedures. After that, some examples and application that are related to the Sylow’s theorems are shown one by one. After that, the relationship between the Sylow’s theorems to that of the orbit-stabilizer theorem is discussed. This work will potentially stimulate more research efforts on the basis theorems in group theory.

Anharmonic Terms of the Potential Energy Surface: A Group Theoretical Approach

In the framework of density functional theory (DFT) simulations of molecules and materials, anharmonic terms of the potential energy surface are commonly computed numerically, with an associated cost that rapidly increases with the size of the system. Recently, an efficient approach to calculate cubic and quartic interatomic force constants in the basis of normal modes [Theor. Chem. Acc., 120, 23 (2008)] was implemented in the Crystal program [J. Chem. Theory Comput., 15, 3755-3765 (2019)]. By applying group theory, we are able to further reduce the associated computational cost, as the exploitation of point symmetry can significantly reduce the number of distinct atomically displaced nuclear configurations to be explicitly explored for energy and forces calculations. Our strategy stems from Wigner's theorem and the fact that normal modes are bases of the irreducible representations (irreps) of the point group. The proposed group theoretical approach is implemented in the Crystal program and its efficiency assessed on six test case systems: four molecules (methane, CH4; tetrahedrane, C4H4; cyclo-exasulfur, S6; cubane, C8H8), and two three-dimensional crystals (Magnesium oxide, MgO; and a prototypical Zinc-imidazolate framework, ZIF-8). The speedup imparted by this approach is consistently very large in all high-symmetry molecular and periodic systems, peaking at 76% for MgO.

Magnetic proximity enabled bulk photovoltaic effects in van der Waals heterostructures

The bulk photovoltaic (BPV) effect, a second-order nonlinear process that generates static current under light irradiation, requires centrosymmetric broken systems as its application platform. To realize measurable BPV photocurrent in spatially centrosymmetric materials, various schemes such as chemical doping, structural deformation, or electric bias have been developed. In this paper, we suggest that the magnetic proximity effect via van der Waals (vdW) interfacial interaction, a contact-free strategy, also breaks the centrosymmetry and generates large BPV currents. Using the ${\mathrm{Bi}}_{2}{\mathrm{Te}}_{3}$ quintuple layer as an exemplary material, we show that magnetic proximity effect from $\mathrm{Mn}{\mathrm{Bi}}_{2}{\mathrm{Te}}_{4}$ septuple layers yields a finite and tunable shift and injection photocurrents. We apply group theory and first-principles calculations to evaluate the layer-specific shift and injection current generations under linearly polarized light irradiation. We find that the magnetic injection photoconductivity that localized on the ${\mathrm{Bi}}_{2}{\mathrm{Te}}_{3}$ layer can reach $>70\ifmmode\times\else\texttimes\fi{}{10}^{8}\phantom{\rule{0.16em}{0ex}}\mathrm{A}/({\mathrm{V}}^{2}\phantom{\rule{0.16em}{0ex}}\mathrm{s})$, so that a one-dimensional linear current density on the order of 0.1 mA/nm can be achieved under an intermediate intensity light. In addition to charge current, we also extend our discussions into spin BPV current, giving pure photogenerated spin current. The vertical propagation direction between the charge and spin photocurrents suggests that they can be used individually in a single material. Compared with previously reported methods, the magnetic proximity effect via vdW interface does not significantly alter the intrinsic feature of the centrosymmetric material (e.g., ${\mathrm{Bi}}_{2}{\mathrm{Te}}_{3}$), and its manipulation can be easily achieved by the proximate magnetic configurations (of $\mathrm{Mn}{\mathrm{Bi}}_{2}{\mathrm{Te}}_{4}$), interlayer distance, and light polarization.

Applications of Group Theory

Groups play a fundamental role in Algebra: many algebraic structures, including rings, fields, and modules, can be seen as formed by adding new operations and axioms based on groups. Researchers often use group theory to explain many kinds of phenomena. In recent years, group theory has been introduced into crystallography to further explore the macroscopic symmetry of crystals from a mathematical point of view. In this paper, the applications of group theory in crystallography and magic cubic will be discussed. Basic definitions and models of these fields are demonstrated. A finite group is a group with a finite number of elements, which are the important contents of group theory. Besides, this paper proves that n−1 elements in a n order group can completely decide the nth element and gives a method of the nth element in a commutative group of order n. The analysis suggests that the research method of group theory has an important influence on other subjects.

A Genelization of Subgroups

Group theory is an important branch of mathematics. It has important applications in many disciplines, such as in physics, chemistry and quantum mechanics. The aim of this document is to provide readers with a greater understanding of the basic knowledge of group theory. Specifically, firstly, the concepts and properties of different types of groups, subgroups and the definition of coset are introduced, which will be used in the next section. Besides, several examples including groups, finite groups, infinite groups, subgroups and abelian groups are listed. Then, a discussion of the two simple theorems and some lemmas will be showed so that readers can be familiar with the properties of groups. This discussion will deepen the understanding of maximal subgroups. This paper investigates the basic definitions and theorems of group theory. In the future, more researchers can pay attention to group theory and the application of group theory in more areas.

GROUP THEORY CONCEPTS IN MOLECULAR SYSTEMS

In this work, we explore group theory ideas implemented to biology and in selective systems of biology. Symmetry performs a notable role in biology of bilaterians over radially symmetric bodies. The distinction between living and non-living organisms is best known. We concentrate on the group theory and abstract algebra covering the biology of molecular systems. We provide the applications of group theory to cancer cells. We review applications to the cells cycle and explore how cells growing and forming cycles. We applied group theory to Cancer Cells and present example to support the validity of the article.

Applications of group theory in crystallography

Group theory is a powerful tool for studying symmetric physical systems. Such systems include, in particular, molecules and crystals with symmetry. Group theory serves to explain the most important characteristics of atomic spectra. Group theory is also applied to the problems of atomic and nuclear physics. This paper gives examples of the use of the apparatus of group theory in research on crystallography, quantum mechanics, elementary particle physics. In particular, in these studies matrix groups and representations of unitary groups are actively used. For such groups we give an overview of the results on their recognition by the spectrum (by the orders of the elements of the group). This direction has been intensively developed in recent years both in our country and abroad. Recognition of finite simple non-Abelian groups by spectrum has been studied for last thirty years in Yekaterinburg at the Institute of Mathematics and Mechanics of the Ural Division of the Russian Academy of Sciences, in Chelyabinsk Federal University and in the Novosibirsk Institute of Mathematics of Siberian Division of the Russian Academy of Sciences. Some simple non-Abelian groups are not recognizable by their spectra. We have proposed an approach for recognizing groups by the bottom layer. The bottom layer of a group is the set of its elements of prime orders. A group is called recognizable by the bottom layer under additional conditions if it is uniquely restored by the bottom layer under these conditions. The paper considers some examples of simple non-Abelian finite groups that are not recognizable by spectra. For these examples, simultaneous recognition by spectrum and by the bottom layer is proved.

On Rayleigh-Taylor and Richtmyer-Meshkov Dynamics With Inverse-Quadratic Power-Law Acceleration

Rayleigh-Taylor (RT) and Richtmyer-Meshkov (RM) instabilities occur in many situations in Nature and technology from astrophysical to atomic scales, including stellar evolution, oceanic flows, plasma fusion, and scramjets. While RT and RM instabilities are sister phenomena, a link of RT-to-RM dynamics requires better understanding. This work focuses on the long-standing problem of RTI/RMI induced by accelerations, which vary as inverse-quadratic power-laws in time, and on RT/RM flows, which are three-dimensional, spatially extended and periodic in the plane normal to the acceleration direction. We apply group theory to obtain solutions for the early-time linear and late-time nonlinear dynamics of RT/RM coherent structure of bubbles and spikes, and investigate the dependence of the solutions on the acceleration’s parameters and initial conditions. We find that the dynamics is of RT type for strong accelerations and is of RM type for weak accelerations, and identify the effects of the acceleration’s strength and the fluid density ratio on RT-to-RM transition. While for given problem parameters the early-time dynamics is uniquely defined, the solutions for the late-time dynamics form a continuous family parameterised by the interfacial shear and include special solutions for RT/RM bubbles/spikes. Our theory achieves good agreement with available observations. We elaborate benchmarks that can be used in future research and in design of experiments and simulations, and that can serve for better understanding of RT/RM relevant processes in Nature and technology.

IrRep: Symmetry eigenvalues and irreducible representations of ab initio band structures

We present IrRep – a Python code that calculates the symmetry eigenvalues of electronic Bloch states in crystalline solids and the irreducible representations under which they transform. As input it receives bandstructures computed with state-of-the-art Density Functional Theory codes such as VASP, Quantum Espresso, or Abinit, as well as any other code that has an interface to Wannier90. Our code is applicable to materials in any of the 230 space groups and double groups preserving time-reversal symmetry with or without spin-orbit coupling included, for primitive or conventional unit cells. This makes IrRep a powerful tool to systematically analyze the connectivity and topological classification of bands, as well as to detect insulators with non-trivial topology, following the Topological Quantum Chemistry formalism: IrRep can generate the input files needed to calculate the (physical) elementary band representations and the symmetry-based indicators using the ▪ routine of the Bilbao Crystallographic Server. It is also particularly suitable for interfaces with other plane-waves based codes, due to its flexible structure. Program summaryProgram Title:IrRepCPC Library link to program files::https://doi.org/10.17632/yznd2ky9r6.1Developer's repository link:https://github.com/stepan-tsirkin/irrepLicensing provisions: GPLv3Programming language: PythonNature of problem: Symmetry properties of electronic band structures in solids are tightly related to their topological features. This relation is set mathematically by the formalisms of Topological Quantum Chemistry [1,2] and symmetry-based indicators of topology [3], but their application requires knowledge of the irreducible representations of the bands. Therefore, a code to calculate irreducible representations of ab initio bands is essential for a systematic theoretical search and classification of topological materials.Solution method:IrRep reads wave functions from files generated by VASP, Abinit, or Quantum Espresso (also files written as input for Wannier90), determines the space group and symmetry operations via the spglib library, evaluates the eigenvalues of the symmetry operations for the selected bands, and applies Group Theory to determine their irreducible representations.

The Conjugacy Class and Conjugacy Class Graph of Crystallographic Point Groups of Order 12 and Above

In chemistry, crystallographic point group is a set of algebraic groups that maintains at least one point in a fixed position while placing certain restrictions on the rotational symmetries. In this research, the application of group theory and graph theory to the symmetry study of a molecule is presented where the conjugacy classes and conjugacy class graphs of crystallographic point groups of order 12 and above are determined. Conjugacy class is a way of classifying the elements of a group such that two elements a and b are conjugate if there exists an element x and given that xax-1=b. The conjugacy classes obtained are then used to determine the conjugacy class graph in which their vertex set is the set of non-central classes of a group, that is the vertices are connected if the greatest common divisor of the cardinalities of the corresponding vertices is more than one

Further potential applications of group theory in information security

Group theory, specifically the combinatorial group theory of finitely presented groups,has been utilized effectively in cryptology. Several new public key cryptosystems have been developed and this has ushered a new area in cryptography called group based cryptography. Braid groups have been suggested as possible platforms and this has led to what is called braid group cryptography. This has also had a profound effect on theoretical group theory as techniques have been found to analyse these group-based cryptosystems.The basic idea is that a finitely presented group can be described by a finite amount of data.This provides techniques to enormously compress and hide information. This suggests that we have only barely scraped the surface of using finitely presented groups for data control, security and storage. For example, we describe a far-reaching extension for controlling access to files which could be relevant in medical records.

Breathing mode of nanoclusters: Definition and comparison to a continuous medium model

Breathing modes are closely related to many physical properties of nanoclusters. Decades of research, however, failed to formulate a general and unambiguous definition. Here we present a straightforward and widely applicable definition of breathing modes based on power spectra of geometric quantities, namely, surface area, volume, etc. Applying group theory, normal-mode analysis, and molecular dynamics simulations, we have explored breathing modes of several ${\mathrm{Al}}_{n}$ clusters with high and low symmetries. The results suggest that our definition is able to cover not only common breathing modes but also some hidden modes. Our consistent definition also allows us to make a comprehensive and in-depth comparison with Lamb's continuous medium model, which reveals some high-frequency breathing modes are explicable only at the atomic level.

Interfacing pure sciences and humanities – Non-traditional application of group theory and symmetry to a text

Interfacing pure sciences and humanities – Non-traditional application of group theory and symmetry to a text

On fundamental isomorphism theorems in soft subgroups

Molodsov initiated a novel concept of soft set theory, which is a completely new approach for modeling vagueness and uncertainty, which there is no limited condition to description of objects and is free from the difficulties affecting existing methods. This makes the theory very convenient and easy to apply in practice. After the pioneering work of Molodsov, there has been a great effort to obtain soft set analogues of classical theories. Among other fields, a progressive developments are made in the field of algebraic structure. To extend the soft set in group theory, many researchers introduced the notions of soft subgroup and investigated its applications in group theory and decision making. In this paper, by using the soft sets and their duality, we introduce new concepts on the soft sets, which are called soft quotient subgroup and quotient dual soft subgroup. We then derive their algebraic properties and, in sequel, investigate the fundamental isomorphism theorems in soft subgroups analogous to the group theory.

On the dynamics of Richtmyer–Meshkov bubbles in unstable three-dimensional interfacial coherent structures with time-dependent acceleration

Richtmyer–Meshkov instability (RMI) plays an important role in many areas of science and engineering, from supernovae and fusion to scramjets and nano-fabrication. Classical RMI is induced by a steady shock and impulsive acceleration, whereas in realistic environments, the acceleration is usually variable. We focus on RMI induced by acceleration with power-law time-dependence and apply group theory to study the dynamics of regular bubbles. For early time linear dynamics, we find the dependence of the growth rate on the initial conditions and show that it is independent of the acceleration parameters. For late-time nonlinear dynamics, we consider regular asymptotic solutions, find a continuous family of such solutions, including their curvature, velocity, Fourier amplitudes, and interfacial shear, and study their stability. For each solution, the interface dynamics is directly linked to the interfacial shear. The non-equilibrium velocity field has intense fluid motion near the interface and effectively no motion in the bulk. The quasi-invariance of the fastest stable solution suggests that the dynamics of nonlinear RM bubbles is characterized by two macroscopic length scales: the wavelength and the amplitude, in agreement with observations. The properties of a number of special solutions are outlined. These are the flat Atwood bubble, the curved Taylor bubble, the minimum shear bubble, the convergence limit bubble, and the critical bubble. We elaborate new theory benchmarks for future experiments and simulations.

Richtmyer–Meshkov dynamics with variable acceleration by group theory approach

Richtmyer–Meshkov instability (RMI) plays an important role in nature and technology, including supernovae, fusion, scramjets and nano-fabrication. Canonical RMI is induced by a steady shock and impulsive acceleration. In realistic environments the acceleration is often variable. We focus on the accelerations with power-law time-dependence, identify the values of the power-law exponent for which the dynamics is Richtmyer–Meshkov type, and apply group theory to solve the long-standing problem of RMI with variable acceleration. For early-time dynamics, we find that the RMI growth-rate depends on the initial conditions and not on the acceleration parameters. For late-time dynamics, we find a continuous family of regular asymptotic solutions, including their curvature, velocity, Fourier amplitudes, and interfacial shear function, and we study stability of these solutions. For each solution, the interface dynamics is set by the interfacial shear; the non-equilibrium velocity field has effectively no motion in the bulk and intense motion near the interface. The fastest stable solution in the family has the flattened front and the quasi-invariance property suggesting the multi-scale character of nonlinear dynamics in RMI. Our results agree with available observations and elaborate new benchmarks for future studies.

Application of n-Fold Rotational Symmetries to Eddy Currents Integral Model in the Time Domain

Group Theory is a powerful “tool” which has been extensively exploited to reduce the computational cost in the numerical analysis of several physical problems, including computational electromagnetics. This article proposes the extension of the application of group theory to an integral formulation of the eddy current problems in the presence of n-fold rotational symmetries. The proposed approach is applied to the eddy currents computation in toroidal fusion devices, where rotational symmetries play a key role.