Group Theoretical Methods Research Articles

Exact solutions of nonlinear dynamical equations for large-amplitude atomic vibrations in arbitrary monoatomic chains with fixed ends

Intermode interactions in one-dimensional nonlinear periodic structures have been studied by many authors, starting with the classical work by Fermi, Pasta, Ulam, and Tsingou (FPUT) in the middle of the last century. However, symmetry selection rules for the energy transfer between nonlinear vibrational modes of different symmetry, which lead to the possibility of excitation of some bushes of such modes, were not revealed. Each bush determines an exact solution of nonlinear dynamical equations of the considered system. The collection of modes of a given bush does not change in time, while there is a continuous energy exchange between these modes. Bushes of nonlinear normal modes (NNMs) are constructed with the aid of group-theoretical methods and therefore they can exist for the case of large amplitude atomic vibrations and for any type of interatomic interactions. In most publications, bushes of NNMs or similar dynamical objects in one-dimensional systems are investigated under periodic boundary conditions. In this paper, we present a detailed study of the bushes of NNMs in monoatomic chains for the case of fixed boundary conditions, which sheds light on a series of new properties of the intermode interactions in such systems. We prove some theorems that justify a method for constructing bushes of NNMs by continuation of conventional normal modes to the case of large atomic oscillations. Our study was carried out for FPUT chains, for the chains with the Lennard-Jones interatomic potential, as well as for the carbon chains (carbynes) in the framework of the density functional theory. For one-dimensional bushes (Rosenberg nonlinear normal modes), the amplitude–frequency diagrams are presented and the possibility of their modulational instability is briefly discussed. We also argue in favor of the fact that our methods and main results are valid for any monoatomic chain.

A Group-Theoretic Approach to the Bifurcation Analysis of Spatial Cosserat-Rod Frameworks with Symmetry

We present a general approach to the bifurcation analysis of spatial framework structures with symmetry. While group-theoretic methods for bifurcation problems with symmetry are well known, their actual implementation in the context of geometrically exact frameworks is not straightforward. We consider spatial structures comprising assemblages of Cosserat rods; the main difficulty arises from the nonlinear configuration space, due to the presence of (cross-sectional) rotation fields. We avoid this via a single-rod formulation, developed earlier by one of the authors, whereby the governing equations are embedded in a slightly enlarged linear space. The field equations for a framework then comprise the assembly of all such rod equations, supplemented by compatibility and equilibrium conditions at the joints. We demonstrate their equivariance under the natural symmetry group action, and the implementation of group-theoretic methods is now clear within the enlarged linear space. All potential generic, symmetry-breaking bifurcations are predicted a-priori. We then employ an open-source path-following code, which can detect and compute simple, one-dimensional bifurcations; multiple bifurcation points are beyond its capabilities. For the latter, we construct symmetry-reduced problems implemented by appropriate substructures. Multiple bifurcations are rendered simple, and the path-following code is again applicable. We first analyze a simple tripod framework, providing all details of our methodology. We then treat a more complex hexagonal space frame via the same approach. Both structures exhibit simple and double bifurcation points.

Analyze the Flow Behavior for MHD Power-Law Fluids

Objective: This study investigates the Numerical solution of laminar boundary layer flow of Magnetohydrodynamics (MHD) model for power-law fluid over a continuous moving surface in the presence of a transverse magnetic. Methods: The governing partial differential equation for the flow was transformed into non-linear ordinary differential equation using the Group theoretic method. Firstly, we convert this non-linear ordinary differential equation (ODE) into linear by using quasilinearization process. This linear ODE was solved numerically by applying the Spline collocation method suggested by Bickley. Findings: The solution for displacement profile and velocity profile were obtained as functions of the magnetic parameters. The effect of the magnetic parameters was discussed graphically. We used MATLAB software for finding the outcomes. Novelty: The main goal of this article is to analyze boundary layer flow of Magneto hydrodynamics (MHD) model for power-law fluid over a continuous moving surface in the presence of a transverse magnetic. The conservation equations of mass, momentum and energy are converted into ordinary differential equations along with boundary conditions by appropriate similarity transformations and solved by applying Spline Collocation Method. The convergence of solutions is important for providing the developing linear functions of solutions, which is a benefit of the Spline Collocation Method. These research findings are applicable, for example, in predicting skin friction and heat transfer rate over a stretching sheet, which has implications in technological and manufacturing industries such as polymer extrusion. Comparisons with previously published works are made, and the results show a high level of agreement. This type of research is applicable to work in fire dynamics in insulation, solar collection systems, recovery of petroleum products, etc. Keywords: Power-Law Fluids; Magnetic Field; Nonlinear Differential Equation; Quasilinearization; Bickley’s Method; Linear Equations

Гиперполосное распределение, оснащенное полем сопряженных плоскостей

In this paper, we study a special class of hyperbands, i. e., a framed hyperband distribution. The study of hyperbands and their generalizations in spaces with different fundamental groups is of great interest in connec­tion with numerous applications in mathematics and physics. A special place is occupied by regular hyperstrips, for which the characteristic planes of families of principal tangent hyperplanes do not contain direc­tions tangent to the basal surface of the hyperstrip. In this paper, we use the method of external differential forms of E. Cartan and the group-theoretic method of G. F. Laptev. We consider a regular hyperband distribution of an affine space equipped with a field of conjugate planes with respect to an asymptotic bundle of tensors of the basic surface. The definition of the studied hy­perband distribution in the affine space with respect to the 1st order frame is given and the existence theorem is proved. A sequence of fundamental geometric objects of the 1st and 2nd order of a hyperband distribution equip­ped with a field of conjugate planes is constructed. Fields of qua­sitensors are constructed that define the fields of normals of the first kind of the distribution of the characteristics of the hyperband distribution. In a differential neighborhood of the 2nd order, the fields of Transon normals of the 1st and 2nd kind are constructed. The conditions for the coincidence of the Transon normal and the Blaschke normal are found.

Similarity solution using group theoretic method for unsteady flow behind shock wave in a self-gravitating dusty gas

In this article, the similarity solution using the Lie group theoretic method for unsteady isothermal and adiabatic flows behind cylindrical or spherical shock wave in a mixture of self-gravitating real gas and small solid particles is discussed. The Lie group theoretic method gives out different cases, i.e., exponential law and power-law shock paths. The similarity solution exists only when the shock path varies according to an exponential law. The dispersal of the flow variables with the variations of the solid particles’ mass concentration in the mixture kp, gravitational parameter g0, the ratio of the density of solid particles to the initial density of the gas μ1, non-idealness parameter b¯ and the geometry index ν are discussed graphically. It is found that an increase in the value of μ1 or g0 or ν leads to an increase in the shock strength, but the shock wave decay with an increase in b¯. A comparison between the solutions for cylindrical and spherical symmetry, and for adiabatic and isothermal flows is also made.

Group theoretical methods in solid state Physics of SnO2

Oxide semiconductor SnO2 shows excellent photoelectronic properties and sensitivities of gases. It is known that their special properties are originated by a stable large band-gap. In nano-SnO2, the above properties have been extensively explored, and nano-SnO2 will find wide applications in microelectronics, photoelectronics, sensor and compound function ceramics. According to our study using group theory, a single SnO2 crystal with rutile-type structure shows four Raman active modes, A 1g, B 1g, B 2g and E g. The additional A 2μ and E μ modes correspond to transverse-optical (TO) and longitudinal-optical (LO) vibrations. Moreover, we applied application of perturbation theory, consequently, the spectrum of commercial SnO2 sample showed the Raman bands in accordance with the theory.

Exploring symmetries in photoelectron holography with two-color linearly polarized fields

We investigate photoelectron holography in bichromatic linearly polarized fields of commensurate frequencies rω and sω, with emphasis on the existing symmetries and for which values of the relative phase between the two driving waves they are kept or broken. Using group-theoretical methods, we show that, additionally to the well-known half-cycle symmetry, which is broken for r + s odd, there are reflection symmetries around the field zero crossings and maxima, which may or may not be kept, depending on how both waves are dephased. The three symmetries are always present for monochromatic fields, while for bichromatic fields this is not guaranteed, even if r + s is even and the half-cycle symmetry is retained. Breaking the half-cycle symmetry automatically breaks one of the other two, while, if the half-cycle symmetry is retained, the other two symmetries are either both kept or broken. We analyze how these features affect the ionization times and saddle-point equations for different bichromatic fields. We also provide general expressions for the relative phases ϕ which retain specific symmetries. As an application, we compute photoelectron momentum distributions for ω − 2ω fields with the Coulomb quantum orbit strong-field approximation and assess how holographic structures such as the fan, the spider and interference carpets behave, focusing on the reflection symmetries. The features encountered can be traced back to the field gradient and amplitude affecting ionization probabilities and quantum interference in different momentum regions.

Propagation of shock wave in a non-ideal dusty gas in rotating medium using Lie group theoretic method: Isothermal flow

This paper concerns with the similarity solutions using Lie group theoretic method for one-dimensional unsteady isothermal flow behind the shock wave in the mixture of non-ideal gas and small solid particles in rotating medium. Group theoretic technique brings the different possible cases of potential solutions with the power law, particular case of power law and exponential law shock paths with the different choices of constants present in the generators of the Lie group of transformations. Similarity solutions for non-ideal dusty gas (a mixture of non-ideal gas and small solid dust particles) exist if the density of the ambient medium is constant. Solutions are obtained for both exponential law and power law shock paths. The effects of the variation of the physical parameters on the flow variables distribution behind the shock wave front, and on the shock are discussed. The shock strength increases with an increase in the ratio of the specific heat of the solid particles to the specific heat of the gas at constant volume [Formula: see text], the initial azimuthal velocity variation index [Formula: see text], and the ratio of the density of solid particles to the initial density of the gas [Formula: see text]. The shock wave decays with an increase in the value of gas non-idealness parameter [Formula: see text].

Similarity solutions for strong magnetogasdynamic cylindrical shock wave in rotating axisymmetric ideal gas with radiation heat flux using Lie group theoretic method

Similarity solutions for strong magnetogasdynamic cylindrical shock wave in rotating axisymmetric ideal gas with radiation heat flux using Lie group theoretic method

Investigation of wave solutions and conservation laws of generalized Calogero–Bogoyavlenskii–Schiff equation by group theoretic method

This work is focused to analyze the generalized Calogero–Bogoyavlenskii–Schiff equation (GCBSE) by the Lie symmetry method. GCBS equation has been utilized to explain the wave profiles in soliton theory. GCBSE was constructed by Bogoyavlenskii and Schiff in different ways (explained in the introduction section). With the aid of Lie symmetry analysis, we have computed the symmetry generators of the GCBSE and commutation relation. We observed from the commutator table, translational symmetries make an Abelian algebra. Then by using the theory of Lie, we have discovered the similarity variables, which are used to convert the supposed nonlinear partial differential equation (NLPDE) into a nonlinear ordinary differential equation (NLODE). Using the new auxiliary method (NAM), we have to discover some new wave profiles of GCBSE in the type of few trigonometric functions. These exits some parameters which we give to some suitable values to attain the different diagrams of some obtained solutions. Further, the GCBSE is presented by non-linear self-adjointness, and conserved vectors are discovered corresponding to each generator.

On computational analysis of highly nonlinear model addressing real world applications

This paper presents a numerical strategy for solving boundary value problems (BVPs) that is based on the Haar wavelets method (HWM). BVPs having high Prandtl numbers are discussed, Because they are very important in many practical problems of science and engineering. By using group-theoretic method, the considered model of partial differential equations (PDEs) are converted to system of nonlinear ordinary differential equations. By using HWM, the numerical results are established. Further, solutions obtained on a coarse resolution with low accuracy is refined towards higher accuracy by increasing the level of resolution. Superiority of the HWM has been established over the commercial software NDSolve and available numerical and approximated methods.

A generalization of the Blind Rotating Table game

We study a generalization of a well-known two player game with partial information. Using an approach different from those appearing in the literature, and introducing group-theoretic methods, a full characterization of the settings where a winning strategy exists is obtained.

On the incorporation of strontium into the crystal structure of bredigite: structural effects and phase transition

Sr-substitution in the crystal structure of bredigite has been studied in detail. Samples of a hypothetical solid-solution series with nominal composition Ca7-xSrxMg[SiO4]4 (x = 0, 2, …,7) have been prepared from sinter reactions in the temperature range between 1275 and 1325 °C and characterized using powder and single-crystal X-ray diffraction. Synthesis runs between x = 1 and x = 4 resulted in compounds with increasing Sr contents, for which single-crystal diffraction studies revealed the following Sr/(Sr + Ca) atomic ratios: 0.133,0.268, 0.409 and 0.559. They are isostructural to the pure calcium end-member (x = 0) and adopt the orthorhombic space group Pnnm. Evolution of the unit-cell parameters and cell volumes of the solid-solution series are defined by linear or nearly linear trends when plotted against the Sr/(Sr + Ca) atomic ratio. Replacement of calcium with strontium atoms on the different sites shows clear preferences for specific positions. For the experiment with x = 5, formation of bredigite-related single-crystals with Sr/(Sr + Ca) = 0.675 was observed. These samples, however, exhibited a halved c lattice parameter when compared with the corresponding value in the Pnnm structure, pointing to a compositionally induced phase transition somewhere in region between Sr/(Sr + Ca) = 0.559 and 0.675. The crystal structure of this new phase with composition Ca2.32Sr4.82Mg0.86[SiO4]4 was successfully determined in space group Pbam. Basic crystallographic data are as follows: a = 18.869(2) Å, b = 6.9445(8) Å, c = 5.5426(6) Å, V = 726.28(14) Å3, Z = 2. Structure determination was accomplished using charge flipping. Subsequent least-squares refinements resulted in a residual of R(|F|) = 2.70% for 822 independent reflections and 87 parameters. The Pbam- and the Pnnm-structures are in a group-subgroup relationship and topologically related. Both are based on so-called pinwheel-like MSi6O24 clusters consisting of a central magnesium-dominated [MO6]-octahedron as well as six attached [SiO4]-tetrahedra. The clusters are linked into chain-like elements running along [001]. Linkage between the chains is provided by mixed Sr/Ca positions with 6 to 10 oxygen ligands. Differences between the two phases result from changes in Sr-Ca site occupancies in combination with displacements of the atoms and tilts of the tetrahedra. The distortion pattern has been studied using group-theoretical methods including mode analysis. Notably, for the samples with x = 6 and x = 7—the latter corresponding to the hypothetical pure strontium end-member composition—no bredigite-type phases could be identified, indicating that there is an upper limit for the Sr-uptake.

A Variety of New Traveling Wave Packets and Conservation Laws to the Nonlinear Low-Pass Electrical Transmission Lines via Lie Analysis

This research is based on computing the new wave packets and conserved quantities to the nonlinear low-pass electrical transmission lines (NLETLs) via the group-theoretic method. By using the group-theoretic technique, we analyse the NLETLs and compute infinitesimal generators. The resulting equations concede two-dimensional Lie algebra. Then, we have to find the commutation relation of the entire vector field and observe that the obtained generators make an abelian algebra. The optimal system is computed by using the entire vector field and using the concept of abelian algebra. With the help of an optimal system, NLETLs convert into nonlinear ODE. The modified Khater method (MKM) is used to find the wave packets by using the resulting ODEs for a supposed model. To represent the physical importance of the considered model, some 3D, 2D, and density diagrams of acquired results are plotted by using Mathematica under the suitable choice of involving parameter values. Furthermore, all derived results were verified by putting them back into the assumed equation with the aid of Maple software. Further, the conservation laws of NLETLs are computed by the multiplier method.

Similarity solutions for magnetogasdynamic cylindrical shock wave in rotating non-ideal gas using Lie group theoretic method

Similarity solutions for magnetogasdynamic cylindrical shock wave in rotating non-ideal gas using Lie group theoretic method

Similarity solutions for cylindrical shock wave in rotating non-ideal gas using Lie group theoretic method

In this paper, the propagation of cylindrical shock wave in rotating non-ideal gas under adiabatic flow condition using Lie group of transformation method is investigated. The density is assumed to be constant and azimuthal fluid velocity is assumed to be varying in the undisturbed medium. The arbitrary constants appearing in the expressions for the infinitesimals of the Local Lie group of transformations bring about two different cases of solutions i.e. with a power-law and exponential-law shock paths. Numerical solutions are obtained for both the cases. Distribution of gasdynamical quantities is illustrated through figures. It is obtained that the reduced flow variables pressure and azimuthal fluid velocity decrease in general, whereas density and radial fluid velocity increase in case of power-law shock path. The reduced azimuthal fluid velocity decreases, whereas reduced density, pressure and radial fluid velocity increase in case of exponential-law shock path. Also, it is obtained that shock strength decreases with increase in value of adiabatic exponent or gas non-idealness parameter, whereas it increases due to increase in ambient azimuthal fluid velocity exponent.

Minimal Hopf-Galois Structures on Separable Field Extensions

In Hopf-Galois theory, every -Hopf-Galois structure on a field extension gives rise to an injective map from the set of -sub-Hopf algebras of into the intermediate fields of . Recent papers on the failure of the surjectivity of reveal that there exist many Hopf-Galois structures for which there are many more subfields than sub-Hopf algebras. In this paper we survey and illustrate group-theoretical methods to determine -Hopf-Galois structures on finite separable extensions in the extreme situation when has only two sub-Hopf algebras. This corresponds to the case when the lack of surjectivity is at its extreme.

James Philip Elliott. 27 July 1929—21 October 2008

James Philip Elliott made important contributions to improve our understanding of the structure of atomic nuclei in the second half of the twentieth century. In 1958 he proposed the SU(3) model, explaining rotational behaviour of nuclei in the context of the shell model. His idea, based on elegant and seminal group-theoretical concepts, reconciled the independent-particle with the liquid-drop model, which until then existed as disconnected views of the nucleus. In the 1960s and 1970s he developed methods to extract properties of the nuclear interaction from the phase shifts of nucleon–nucleon scattering. From 1980 until his death he contributed to the development of the interacting boson model of Arima and Iachello, and its microscopic understanding in terms of symmetries of the shell model. For his outstanding achievements in theoretical physics, in 2002 he and Francesco Iachello were awarded the Lise Meitner prize of the European Physical Society for ‘their innovative applications of group-theoretical methods to the understanding of atomic nuclei’. His achievements were also recognized by the award of the Rutherford Medal and Prize of the Institute of Physics in 1994.

Martensitic transformations of β-phase in zirconium

Martensitic transformations in the body-centered cubic β-phase (Im3¯m) of zirconium are studied using first-principles calculations, group-theoretical methods, and symmetry analysis. Phonon dispersion relations in the β-phase calculated within the harmonic approximation predicted an unstable phonon at wave vector 2πa[12,12,0](N) and a soft phonon at wave vector 2πa[23,23,23](Λ). The symmetry of the unstable phonon is the same as the symmetry of the N4− irreducible representation, and the symmetry of the soft phonon is the same as the symmetry of the Λ1 irreducible representation. Martensitic transformations are simulated considering two steps. Frozen phonon calculations are used to determine the first step, i.e., the transformation of the β-phase to an intermediate phase due to phonon motion. Structure relaxation is used to determine the second step, i.e., the transformation of the intermediate phase to the final phase. The unstable N4− phonon transforms the β-phase into an intermediate orthorhombic phase (Cmcm), which further transforms to a hexagonal close packed α-phase (P63/mmc) after structure relaxation. The soft Λ1 phonon transforms the β-phase into an intermediate trigonal phase (P3¯m1), which further transforms to a hexagonal close packed ω-phase (P6/mmm) after structure relaxation. The intermediate phase space group (Cmcm/P3¯m1) is a common subgroup of the parent phase (β) space group and the final phase (α/ω) space group. Therefore, the martensitic transformations in zirconium are reconstructive transformations of the second kind. Symmetry characterization of the martensitic transformations is also presented.

Discrete symmetry approach to exact bound-state solutions for a regular hexagon Dirac billiard

We consider solving the stationary Dirac equation for a spin-1/2 fermion confined in a two-dimensional quantum billiard with a regular hexagon boundary, using symmetry transformations of the point group C 6v . Closed-form bound-state solutions for this problem are obtained and the non-relativistic limit of our results are clearly discussed. Due to an adequate choice of confining boundary conditions the upper components of the planar Dirac-spinor eigenfunctions are shown to satisfy the corresponding hexagonal Schrödinger billiard, and the Dirac positive energy eigenvalues are proven to reduce directly to their Schrödinger counterparts in the non-relativistic limit. An illustrative application of our group theoretic method to the well-known square billiard problem has been explicitly provided. The success of our approach in solving equilateral-triangle, square and regular hexagon quantum billiards may well imply a possible applicability to other regular polygonal billiards. A quick look on nodal domains of the Schrödinger eigenfunctions for the hexagon billiard is also considered. Moreover, we have determined a number of distinct non-congruent polygonal billiards that have the same eigenvalue spectrum as that of the regular hexagon.