Methods Of Group Theory Research Articles

Calculation of photofield emission current by using the projection operator method of group theory

A calculation of photofield emission is discussed in which initial state wave function has been deduced by using the projection operator method of group theory. A spatial dependent vector field is used to evaluate the matrix element for calculating the photofield emission current density.

Numerical solution of thermo-solutal mixed convective slip flow from a radiative plate with convective boundary condition

A mathematical model for mixed convective slip flow with heat and mass transfer in the presence of thermal radiation is presented. A convective boundary condition is included and slip is simulated via the hydrodynamic slip parameter. Heat generation and absorption effects are also incorporated. The Rosseland diffusion flux model is employed. The governing partial differential conservation equations are reduced to a system of coupled, ordinary differential equations via Lie group theory method. The resulting coupled equations are solved using shooting method. The influences of the emerging parameters on dimensionless velocity, temperature and concentration distributions are investigated. Increasing radiative-conductive parameter accelerates the boundary layer flow and increases temperature whereas it depresses concentration. An elevation in convection-conduction parameter also accelerates the flow and temperatures whereas it reduces concentrations. Velocity near the wall is considerably boosted with increasing momentum slip parameter although both temperature and concentration boundary layer thicknesses are decreased. The presence of a heat source is found to increase momentum and thermal boundary layer thicknesses but reduces concentration boundary layer thickness. Excellent correlation of the numerical solutions with previous non-slip studies is demonstrated. The current study has applications in bioreactor diffusion flows and high-temperature chemical materials processing systems.

Post-bifurcation and stability of a finitely strained hexagonal honeycomb subjected to equi-biaxial in-plane loading

The buckling and crushing mechanics of cellular honeycomb materials is an important engineering problem. Motivated by the pioneering experimental and numerical studies of Papka and Kyriakides (1994, 1999a,b), we review the literature on finitely strained honeycombs subjected to in-plane loading and identify two open questions: (i) How does the mechanical response of the honeycomb depend on the applied loading device? and (ii) What can the Bloch wave representation of all bounded perturbations contribute to our understanding of the stability of post-bifurcated equilibrium configurations? To address these issues we model the honeycomb as a two-dimensional infinite perfect periodic medium. We use analytical group theory methods (as opposed to the more common, but less robust, imperfection method) to study the honeycomb’s bifurcation behavior under three different far-field loadings that produce (initially) the same equi-biaxial contractive dilatation. Using an FEM discretization of the honeycomb walls (struts), we solve the equilibrium equations to find the principal and bifurcated equilibrium paths for each of the three loading cases. We evaluate the structure’s stability using two criteria: rank-one convexity of the homogenized continuum (long wavelength perturbations) and Bloch wave stability (bounded perturbations of arbitrary wavelength). We find that the post-bifurcation behavior is extremely sensitive to the applied loading device, in spite of a common principal solution. We confirm that the flower mode is always unstable, as previously reported. However, our (first ever) Bloch wave stability analysis of the post-bifurcated equilibrium paths shows that the flower mode is stable for all sufficiently short wavelength perturbations. This new result provides a realistic explanation for why this mode has been observed in the finite size specimen experiments of Papka and Kyriakides (1999a).

Conformational Analysis of [M(ABC)<sub>6</sub>] Complex

Enumeration of the conformers and conformational analysis were successfully conducted for an octahedral [M(ABC)6] complex. In this report the methods and results were briefly described. In the enumeration, 7173 conformers were found by the computational group theory method, and among them, 130 conformers were found to have stable M(AB)6 cores of S6 symmetry. The structures of the 130 conformers of [Zn(DMSO)6]2+ (DMSO: dimethylsulfoxide) were optimized by the DFT method, and the most stable conformer was obtained.

An uncountable homology group, where each element is an infinite product of commutators

In 1986 Umed Karimov constructed a space as a one-point compactification of a CW-complex, where each homotopy class of loops can be represented as an infinite product of commutators. We prove Karimov's conjecture that the first singular homology group of this space is uncountable. The proof uses methods of combinatorial group theory for one-relator groups and methods of geometric topology for smoothing singularities in van-Kampen diagrams corresponding to infinite words.

Manipulating pseudospin-polarized state of light in dispersion-immune photonic topological metacrystals

We proposed a group-theory method to calculate topological invariant in bi-isotropic photonic crystals invariant under crystallographic point group symmetries. Spin Chern number has been evaluated by the eigenvalues of rotation operators at high symmetry k-points after the pseudo-spin polarized fields are retrieved. Topological characters of photonic edge states and photonic band gaps can be well predicted by total spin Chern number. Nontrivial phase transition is found in large magnetoelectric coupling due to the jump of total spin Chern number. Light transport is also issued at the {\epsilon}/{\mu} mismatching boundary between air and the bi-isotropic photonic crystal. This finding presents the relationship between group symmetry and photonic topological systems, which enables the design of photonic nontrivial states in a rational manner.

Edge modes in a frustrated quantum Ising chain

The ground state properties of an Ising chain with nearest ($J_{1}$) and next-nearest neighbor ($J_{2}$) interactions in a transverse field are investigated using the density matrix renormalization group and cluster mean-field theory methods. Its quantum phase diagram has two ordered regions (separated by a quantum disordered phase), one of which has simple ferromagnetic or N\'eel order depending upon the sign of $J_{1}$, and the other is the double-staggered antiferromagnetic phase. The presence of the Majorana-like edge-modes in these ordered phases is inferred by calculating the end-to-end spin-spin correlations on open chains. It is found that there occur four edge modes in the double-staggered phase, while the ferromagnetic and N\'eel phases support two edge modes, except very near $J_{1}=0$ where it seems they have four edge modes like the exact case at $J_{1}=0$ itself.

Raman spectroscopic studies on p-terphenyl under high pressure

High-pressure Raman scattering studies are performed on p-terphenyl up to 5GPa. The Raman activities of different symmetric molecules were analyzed by means of group theory methods. A phase transition was detected at 1.3GPa from changes in the slope on plots of frequency versus pressure. The diminishing of internal modes indicated that the molecule symmetry transformed from C2 to D2h. This is an effective method for detecting planar molecular structure of p-terphenyl by ring–ring stretching vibration mode, which can provide a new spectroscopic evidence of planar conjugated polyphenyl molecular conformation.

Infinite Examples of Cancellative Monoids That Do Not Always Have Least Common Multiple

We will study the presentations of fundamental groups of the complement of complexified real affine line arrangements that do not contain two parallel lines. By Yoshinaga’s minimal presentation, we can give positive homogeneous presentations of the fundamental groups. We consider the associated monoids defined by the presentations. It turns out that, in some cases, left (resp. right) least common multiple does not always exist. Hence, the monoids are neither Garside nor Artin. Nevertheless, we will show that they carry certain particular elements similar to the fundamental elements in Artin monoids and that, by improving the classical method in combinatorial group theory, they are cancellative monoids. As a result, we will show that the word problem can be solved and the center of them is determined.

Monoids of O-type, subword reversing, and ordered groups

Abstract. We describe a simple scheme for constructing finitely generated monoids in which left-divisibility is a linear ordering, and for practically investigating these monoids. The approach is based on subword reversing, a general method of combinatorial group theory, and connected with Garside theory, here in a non-Noetherian context. As an application we describe several families of ordered groups whose space of left-invariant orderings has an isolated point, including torus knot groups and some of their amalgamated products.

Inter-valley mixing in Si/SiO2 heterostructures

We present a theory for heterostructures based on diamond structured semiconductors, with an emphasis of calculating inter-valley coupling. The theory allows taking into account microscopic symmetry of heterointerface within the envelope-function approach. To find out corrections to the effective Hamiltonian we use methods of group theory. We calculated valley-orbit splitting and electron wave functions for SiO2/Si/SiO2 finite square quantum wells. The results are in good agreement with those obtained by other authors.

Magnetoelectric effect and magnetic dynamics in the Gd2CuO4 antiferromagnet

The magnetoelectric effect and the magnetic dynamics in Gd2CuO4 have been studied using a phenomenological approach and group-theory methods. Vector order parameters are introduced based on four magnetic sublattices. Invariant products of the order parameters are determined, from which the thermodynamic potential density is constructed. Using the spin-wave representation, the calculations can be significantly simplified and the ground orientation magnetic state can be presumably determined. The magnetic dynamics is described by the Landau-Lifshitz equations, from which the antiferromagnetic resonance frequency and the dynamic susceptibilities, namely, magnetic, antiferromagnetic, magnetoelectric, and antiferroelectric susceptibilities are found. The frequency and the susceptibility are shown to be controlled by applied electric field.

Galois actions on regular dessins of small genera

Dessins d'enfants can be regarded as bipartite graphs embedded in compact orientable surfaces. According to Grothendieck and others, a dessin uniquely determines a complex structure on the surface, and even an algebraic structure (as a projective algebraic curve defined over a number field). The general problem of how to determine all properties of the curve from the combinatorics of the dessin is far from being solved. For regular dessins, which are those having an edge-transitive automorphism group, the situation is easier: currently available methods in combinatorial and computational group theory allow the determination of the fields of definition for all curves with regular dessins of genus 2 to 18.

Symmetry of the carbon nanotube modes and their origin from the phonon branches of graphene

A group-theory method to relate the spectra of tubular and parent planar structures is elaborated. The spectra correlation is obtained using a virtual intermediate planar periodic structure, which has a local symmetry coinciding with that of the tube surface. The method is applied to deduce the zone-center phonon modes of carbon nanotubes from the dispersion branches of the graphene monolayer. The results obtained open an additional way for experimental indexing of the carbon nanotubes by means of Raman spectroscopy, and the origin of the Raman-active modes in the achiral nanotubes is determined. We demonstrate that the number of modes in the $G$ band and the relation between their frequencies are governed by the nanotube type. The proposed theory can be easily explored to understand the phonon and electron spectra of many other tubular structures (boron nitride, transition metal dichalcogenides) which have emerged in the past decade.

Parity symmetry with respect to both [formula omitted] and [formula omitted] requires periodicity with period [formula omitted]: Connections between computer graphics, group theory and spectral methods for solving partial differential equations

A function is symmetric with respect to a point x=L if f(x+L)=f(-x+L) for all x and similarly is antisymmetric if f(x+L)=-f(-x+L). A function which is either symmetric or antisymmetric is said to be of “definite parity” with respect to L. The sines and cosines of a Fourier series have definite parity with respect to two points; all cosines are symmetric with respect to the origin while all sines are antisymmetric with respect to x=0;cos(2nx) and sin([2n+1]x) for integral n are also symmetric with respect to x=π/2 while all other Fourier basis functions are antisymmetric with respect to the same point. Such symmetries can be exploited in numerical calculations; for example, computing the angular Mathieu functions using N basis functions can be split into four uncoupled eigenproblems each of dimension N/4. It is natural to ask: Are there other classes of functions with similar symmetries? Using concepts from computer graphics, we prove that all functions which are symmetric with respect to two points separated by a distance L must be spatially periodic with period 4L. We also prove that the only function which is of definite parity with respect to three distinct points must be a constant. These theorems define parity in the usual sense of a global property such that even parity with respect to the origin means f(x)=f(-x) for all x∈[-∞,∞]. We construct counterexamples to both theorems that are functions with local parity, that is, symmetry which applies only for a finite interval in x.

Book Review: The geometry and topology of Coxeter groups; Topological methods in group theory

Book Review: The geometry and topology of Coxeter groups; Topological methods in group theory

Groups of Finite Morley Rank with Solvable Local Subgroups

We lay down the fundations of the theory of groups of finite Morley rank in which local subgroups are solvable and we proceed to the local analysis of these groups. We prove a main Uniqueness Theorem, analogous to the Bender method in finite group theory, and derive its corollaries. We also consider homogeneous cases and study torsion.

The kinetic model of the two dimensional cylindrical current sheet

In this paper we develop a one class of solutions of the steady Vlasov-Maxwell equations, which describes two dimensional cylindrical current sheets with current directed azimuthally j = j θ(ρ, z)e θ. Magnetic field of these sheets has two components B = B z (ρ, z)e z + B ρ(ρ, z)e ρ. From mathematical point of view, we find solutions of the nonlinear equation in partial derivatives for some function u(ρ, z): ∂2 u/∂x 2 + x −1∂2 u/∂z 2 = e −u , where x = ρ2. We apply methods of group theory to develop three-parameter class of solutions. We also derive asymptotic behavior of these solutions for large values of ρ and for ρ ∼ 0. We discuss applications of these solutions for description of current sheets in magnetospheres of planets with magnetic dipoles located near the ecliptic plane.

Sieve methods in group theory I: Powers in linear groups

A general sieve method for groups is formulated. It enables one to “measure” subsets of a finitely generated group. As an application we show that if Γ \Gamma is a finitely generated non-virtually solvable linear group of characteristic zero, then the set of proper powers in Γ \Gamma is exponentially small. This is a far-reaching generalization of a result of Hrushovski, Kropholler, Lubotzky, and Shalev.

Finite quantum models: constructive approach to description of quantum behavior

The universality of quantum mechanics — its applicability to physical systems of quite different nature and scales — indicates that quantum behavior can be a manifestation of general mathematical properties of systems containing indistinguishable elements, i.e., elements lying on the same orbit of some symmetry group. In this paper, we demonstrate that quantum behavior arises naturally in systems with finitely many elements connected by nontrivial symmetry groups. The “finite” approach allows one to see the peculiarities of a quantum description more distinctly without the need for concepts like “wave function collapse,” “Everett’s multiverses,” etc. In particular, under the finiteness assumption, any quantum dynamics is reduced to a simple permutation dynamics. The advantage of finite quantum models is that they can be studied constructively by means of computer algebra and computational group theory methods. Bibliography: 12 titles.