Symmetry Classes Research Articles

Szegö--Weinberger Type Inequalities for Symmetric Domains with Holes

Let $\mu_2(\Omega)$ be the first positive eigenvalue of the Neumann Laplacian in a bounded domain $\Omega\subset\mathbb{R}^N$. It was proved by Szeg\H{o} for $N=2$ and by Weinberger for $N \geq 2$ that among all equimeasurable domains $\mu_2(\Omega)$ attains its global maximum if $\Omega$ is a ball. In the present work, we develop the approach of Weinberger in two directions. Firstly, we refine the Szeg\H{o}-Weinberger result for a class of domains of the form $\Omega_{\text{out}}\setminus\overline{\Omega}_{\text{in}}$ which are either centrally symmetric or symmetric of order $2$ (with respect to every coordinate plane $(x_i,x_j)$) by showing that $\mu_{2}(\Omega_{\text{out}}\setminus\overline{\Omega}_{\text{in}})\leq\mu_2(B_\beta\setminus\overline{B}_\alpha)$, where $B_\alpha, B_\beta$ are balls centered at the origin such that $B_\alpha\subset\Omega_{\text{in}}$ and $|\Omega_{\text{out}}\setminus\overline{\Omega}_{\text{in}}|=|B_\beta\setminus\overline{B}_\alpha|$. Secondly, we provide Szeg\H{o}-Weinberger type inequalities for higher eigenvalues by imposing additional symmetry assumptions on the domain. Namely, if $\Omega_{\text{out}}\setminus\overline{\Omega}_{\text{in}}$ is symmetric of order $4$, then we prove $\mu_{i}(\Omega_{\text{out}}\setminus\overline{\Omega}_{\text{in}})\leq\mu_i(B_\beta\setminus\overline{B}_\alpha)$ for $i=3,\dots,N+2$, where we also allow $\Omega_{\text{in}}$ and $B_\alpha$ to be empty. If $N=2$ and the domain is symmetric of order $8$, then the latter inequality persists for $i=5$. Counterexamples to the obtained inequalities for domains outside of the considered symmetry classes are given. The existence and properties of nonradial domains with required symmetries in higher dimensions are discussed. As an auxiliary result, we obtain the non-radiality of the eigenfunctions associated to $\mu_{N+2}(B_\beta\setminus\overline{B}_\alpha)$.

Turbulence Statistics of Arbitrary Moments of Wall-Bounded Shear Flows: A Symmetry Approach.

The calculation of turbulence statistics is considered the key unsolved problem of fluid mechanics, i.e., precisely the computation of arbitrary statistical velocity moments from first principles alone. Using symmetry theory, we derive turbulent scaling laws for moments of arbitrary order in two regions of a turbulent channel flow. Besides the classical scaling symmetries of space and time, the key symmetries for the present work reflect the two well-known characteristics of turbulent flows: non-Gaussianity and intermittency. To validate the new scaling laws we made a new simulation at an unprecedented friction Reynolds number of 10 000, large enough to test the new scaling laws. Two key results appear as an application of symmetry theory, which allowed us to generate symmetry invariant solutions for arbitrary orders of moments for the underlying infinite set of moment equations. First, we show that in the sense of the generalization of the deficit law all moments of the streamwise velocity in the channel center follow a power-law scaling, with exponents depending on the first and second moments alone. Second, we show that the logarithmic law of the mean streamwise velocity in wall-bounded flows is indeed a valid solution of the moment equations, and further, all higher moments in this region follow a power law, where the scaling exponent of the second moment determines all higher moments. With this we give a first complete mathematical framework for all moments in the log region, which was first discovered about 100years ago.

Quantum quenches in an interacting field theory: Full quantum evolution versus semiclassical approximations

We develop a truncated Hamiltonian method to investigate the dynamics of the $(1+1)d~\phi^4$ theory following quantum quenches. The results are compared to two different semi-classical approaches, the self-consistent Gaussian approximation and the truncated Wigner approximation, and used to determine the range of validity of these widely used approaches. We show that the self-consistent approximation is strongly limited in comparison to the truncated Hamiltonian method which for larger cutoffs is practically exact for the parameter range studied. We find that the self-consistent approximation is only valid when the effective mass is in the vicinity of the renormalised mass. Similarly to the self-consistent approximation, the truncated Wigner approximation is not able to capture the correct mass renormalisation, and breaks down for strong enough interactions where the bare mass becomes negative. We attribute the failure of TWA to the presence of a classical symmetry broken fixed point. Besides establishing the truncated Hamiltonian approach as a powerful tool for studying the dynamics of the $\phi^4$ model, our results on the limitation of semi-classical approximations are expected to be relevant for modelling the dynamics of other quantum field theories.

Четырехволновое взаимодействие на фазово-амплитудных голографических решетках в фоторефрактивном пьезокристалле класса симметрии 43m

A system of coupled-wave equations for calculating the vector amplitudes of linearly polarized light waves at four-wave mixing by phase-amplitude holographic gratings in a cubic photorefractive semiconductor of an arbitrary cut belonging to the 4 ̅3m symmetry class is presented. The dependences of the intensities of the polarization components of the reversed light wave on the orientation angle for GaAs crystal of (110)-cut are calculated on the basis of the numerical solution of the system of coupled-wave equations. The obtained dependences are compared with the known theoretical and experimental data. It is shown that the best agreement between the results of theoretical modeling and experimental data at calculating the counterpropagating four-wave mixing in GaAs crystal of (110)-cut is achieved if formation of several phase-amplitude holographic gratings is allowed, and the contribution of the photoelastic and inverse piezoelectric effects are taken into account together with absorption of the crystal.

LIE SYMMETRIES, GROUP INVARIANT SOLUTIONS AND CONSERVATION LAWS OF IDEAL MHD EQUATIONS

In this paper, the method of Lie symmetry is used to study the incompressible ideal MHD equations. We derive the infinitesimal generators of MHD by using Lie symmetry method. Then, we get the group invariant solutions to MHD based on the group transformation and known solutions. An one-dimensional optimal system which only depends on the commutator of Lie algebras is constructed. Through the similarity transformation of the optimal system, we obtain the 1 + 1-dimensional reduced partial differential equations, and further obtain the exact solutions of these equations, such as Bell soliton solution, junction soliton solution and seed solution. In addition, we prove the nonlinear self-adjoint and conservation laws of the MHD equations using the Ibragimov method. Finally, the classical symmetry and solutions to incompressible ideal MHD equations with initial conditions left invariant are discussed.

An Alternative Algorithm for the Symmetry Classification of Ordinary Differential Equations

This is the first paper on symmetry classification for ordinary differential equations (ODEs) based on Wu’s method. We carry out symmetry classification of two ODEs, named the generalizations of the Kummer-Schwarz equations which involving arbitrary function. First, Lie algorithm is used to give the determining equations of symmetry for the given equations, which involving arbitrary functions. Next, differential form Wu’s method is used to decompose determining equations into a union of a series of zero sets of differential characteristic sets, which are easy to be solved relatively. Each branch of the decomposition yields a class of symmetries and associated parameters. The algorithm makes the classification become direct and systematic. Yuri Dimitrov Bozhkov, and Pammela Ramos da Conceição have used the Lie algorithm to give the symmetry classifications of the equations talked in this paper in 2020. From this paper, we can find that the differential form Wu’s method for symmetry classification of ODEs with arbitrary function (parameter) is effective, and is an alternative method.

OUP accepted manuscript

Abstract We develop a new approach to find asymptotic symmetries in general relativity as a modification of the Lie-algebra-based approach proposed in T. Tomitsuka et al. [Classical Quantum Gravity 38, 225007 (2021)]. Those authors proposed an algorithmic protocol to investigate asymptotic symmetries. In particular, their guiding principle helps us to find a non-vanishing charge that generates an infinitesimal diffeomorphism. However, in order to check the integrability condition for the charges, it is necessary to solve differential equations to identify the integral curve of vector fields, which is usually quite hard. In this paper, we provide a sufficient condition of the integrability condition that can be checked without solving any differential equations, avoiding the difficulties in the approach in the above reference. As a demonstration, we investigate the asymptotic symmetries on a Killing horizon and find a new class of asymptotic symmetries. In 4D spacetimes with a spherical Killing horizon, we show that the algebra of the corresponding charges is a central extension of the algebra of vector fields.

Finite element modelling of laminated composite: Effects of different ply orientations

The subject of analysis in this paper are laminated composites as an always popular topic in the field of composite materials. The introductory part of the paper presents a brief overview of the current state of research in this area and points out further directions of research related to this very interesting topic. The classification of laminated composite plates from the aspect of different criteria are described in detail. As the subject of the paper is stress-strain analysis of laminated composite, the expression for the generalized Hooke's law of orthotropic class of material symmetry is given. The terms of the constitutive matrix are expressed through engineering constants. At the end, the bending analysis of the rigidly supported laminated plate under a sinusoidal load was performed. The mentioned analysis was conducted using software based on the finite element method. A comparative analysis of the plate with the different ply orientation (0°/90°/90°/0° and -45°/45°/45°/-45°) was done. The obtained results were analyzed and appropriate conclusions were made.

Four-wave mixing by phase-amplitude holographic gratings in the photorefractive piezocrystal of 43m symmetry class

A system of coupled-wave equations for calculating the vector amplitudes of linearly polarized light waves at four-wave mixing by phase-amplitude holographic gratings in a cubic photorefractive semiconductor of an arbitrary cut belonging to the 43m symmetry class is presented. The dependences of the intensities of the polarization components of the reversed light wave on the orientation angle for GaAs crystal of (110)-cut are calculated on the basis of the numerical solution of the system of coupled-wave equations. The obtained dependences are compared with the known theoretical and experimental data. It is shown that the best agreement between the results of theoretical modeling and experimental data at calculating the counterpropagating four-wave mixing in GaAs crystal of (110)-cut is achieved if formation of several phase-amplitude holographic gratings is allowed, and the contribution of the photoelastic and inverse piezoelectric effects are taken into account together with absorption of the crystal. Keywords: four-wave mixing, photorefractive semiconductor, light wave, holographic grating, coupled wave equations.

Электрозвуковые волны в PT-симметричной структуре пьезоэлектриков вблизи особой точки

The spectral properties of electrosound waves in the gap of PT-symmetric piezoelectrics of symmetry class 6 are investigated theoretically. It is found that at a certain level of loss and gain in piezoelectrics, the symmetric and antisymmetric modes are coupled. The coupling point defines an exceptional point of PT-symmetric structure. It is shown that the frequency dependence of the amplitude at the exceptional point has an extremely narrow resonance peak, which opens up the possibility of creating supersensitive sensors based on PT-symmetric physical structures.

The "silent CO": a new technique for calculating transition metal carbonyl force fields.

The notion of a "silent CO group" (effectively an infinitely heavy CO group) is introduced to enable energy-factored force fields to be estimated accurately for molecules where there are fewer ν(CO) frequencies than force constants in the force field (viz. underdetermined force fields). The symmetry classes of molecules covered are the Cs tricarbonyls (e.g. Fe(CO)3(diene) and fac-Re(CO)3(L-L)X), C2v tricarbonyls (e.g. mer-M(CO)3(L)3 M = Cr, Mo, W), C3v tetracarbonyls (e.g. Fe(CO)4(L)), C2v tetracarbonyls (e.g. cis-M(CO)4(L)2 and Fe(CO)4(L)) and C4v pentacarbonyls (e.g. M(CO)5(L) M = Cr, Mo, W and M(CO)5(X) M = Mn, Re). It is a relatively simple matter to extend the method to types of molecules not directly considered in this paper.

Electrosound waves in a PT-symmetric piezoelectric structure near the exceptional point

The spectral properties of gap electroacoustic waves in PT-symmetric piezoelectric structures of symmetry class 6 are investigated theoretically. It was found out that, at a certain level of loss and gain in piezoelectrics, the symmetric and antisymmetric modes intersect. The intersection point defines an exceptional point of the PT-symmetric structure. It was shown that the frequency dependence of the amplitude at the exceptional point has an extremely narrow resonance peak, which opens up the possibility of creating supersensitive sensors based on PT-symmetric physical structures. Keywords: PT-symmetry, piezoelectrics, electroacoustic waves, gap structure.

Comment on “Distinct spin–lattice and spin–phonon interactions in monolayer magnetic CrI” by L. Webster, L. Liang and J.-A. Yan, Phys. Chem. Chem. Phys., 2018, 20, 23546

We point out four mistakes in the symmetry classification of vibrational modes of the CrI3 monolayer at the Γ point in Webster et al.'s paper [PCCP, 2018, 20, 23546]. The discrepancy of Raman activity of A2g mode as well as the misidentification of two optical modes A1u and A2u have been clarified.

Lattice Parameters, Electronic, and Magnetic Properties of Cubic Perovskite Oxides ARuO3 (A=Sr, Rb): A First‑Principles Study

Trioxides of rubidium, strontium, and ruthenium belong to the family of alkali and alkaline earth ruthenates. SrRuO3 crystallizes in various symmetry classes—orthorhombic, tetragonal, or cubic—whereas RbRuO3 is perovskite (cubic) structured and crystallizes only in the cubic space group Pm3¯¯¯m(No. 221). In this study, we investigated the structural stability as well as the electronic and magnetic properties of two cubic perovskites SrRuO3 and RbRuO3. We established the corresponding lattice parameters, magnetic moments, density of states (DOS), and band structures using ab‑initio density‑functional theory (DFT). Both compounds exhibited a metallic ferromagnetic ground state with lattice parameter values between 3.83 and 3.96 Å; RbRuO3 had magnetic moments between 0.29 and 0.34 µBwhereas SrRuO3 had magnetic moments between 1.33 and 1.66 µB. This study paves way for further RbRuO3 research.

The study of some nuclear properties of even-even114-120Cd isotopes using interacting boson model-1

Nuclear properties of even-even 114-120Cd are studied in the framework of the interacting boson model (IBM-1). In this study, we have determined the most appropriate Hamiltonian for the study of even-even Cd (114-120) to calculate, its energy levels. The values of parameters have been determined using the (IBM1) Hamiltonian which yield the best fit to the available experimental energy levels. In addition, the backbending (moment of inertia as a function of (ħω)) of the energy levels for each isotope was indicated through the experimental and calculated values of the energy levels. A simulation program with MATLAP-18 has been built to obtain the theoretical energy levels for Cadmium isotopes with neutron number N=16, 18, 20 and 22 up to spin-parity 〖14〗_1^+,〖14〗_1^+,〖14〗_1^+,〖and16〗_1^+, respectively. The ratio of excitation energies of the first 4+ and the first 2+ excited states, R4/2 is also studied for the classification of symmetry of the nuclei. The results show that the 114-120Cd isotopes are γ – unstable rotor nuclei and they are dynamical symmetry O(6).

Lie group symmetries' complete classification for a generalized Chazy equation and its equivalence group

In this work, a complete classification of the Lie group symmetries for a generalization of Chazy equation was carried out and the equivalence group for the generalized Chazy equation is calculated and used to present the principal algebra of the equation.

Universal ratio in random matrix theory and chaotic-to-integrable transition in type-I and type-II hybrid Sachdev-Ye-Kitaev models

We investigate chaotic to integrable transition in two types of hybrid SYK models, which contain both $q=4$ SYK with interaction $J$ and $q=2$ SYK with an interaction $K$ in type-I or ${(q=2)}^{2}$ SYK with an interaction $\sqrt{K}$ in type-II. These models include hybrid Majorana fermion, complex fermion, and bosonic SYK. For the Majorana fermion case, we discuss both $N$ even and $N$ odd case. We make exact symmetry analysis on the possible symmetry class of both types of hybrid SYK in the 10-fold way by random matrix theory (RMT) and also work out the degeneracy of each energy levels. We introduce a new universal ratio, which is the ratio of the next-nearest-neighbor (NNN) energy level spacing to characterize the RMT. We perform exact diagonalization to evaluate both the known NN ratio and the new NNN ratio, then use both ratios to study chaotic to integrable transitions (CIT) in both types of hybrid SYK models. Some preliminary results on possible quantum analog of Kolmogorov-Arnold-Moser (KAM) theorem and its dual version in the quantum chaotic side are given. We explore some intrinsic connections between the two complementary approaches to quantum chaos: the RMT and the Lyapunov exponent by the $1/N$ expansion in the large $N$ limit at a suitable temperature range. We stress the crucial differences between the quantum phase transition (QPT) characterized by renormalization groups at $N=\ensuremath{\infty}$, $1/N$ expansions at a finite $N$, and the CIT characterized by the RMT at a finite $N$: the former focus on the ground state and its low-energy excitations (edge states in the Fock space), the latter on excited states (bulk states in the Fock space). We also discuss eigenstate thermalization hypothesis (ETH)'s power on a quantum chaotic bulk state and its inability to encode the edge states. Comments on some previously related works are given. Some future perspectives, especially the failure of the Zamoloddchikov's c-theorem in 1d CFT RG flow are outlined.

A nonlocal Boussinesq equation: Multiple-soliton solutions and symmetry analysis

A nonlocal Boussinesq equation is deduced from the local one by using consistent correlated bang method. To study various exact solutions of the nonlocal Boussinesq equation, it is converted into two local equations which contain the local Boussinesq equation. From the N-soliton solutions of the local Boussinesq equation, the N-soliton solutions of the nonlocal Boussinesq equation are obtained, among which the (N = 2,3,4)-soliton solutions are analyzed with graphs. Some periodic and traveling solutions of the nonlocal Boussinesq equation are derived directly from the known solutions of the local Boussinesq equation. Symmetry reduction solutions of the nonlocal Boussinesq equation are also obtained by using the classical Lie symmetry method.

Generalized eigenproblem without fermion doubling for Dirac fermions on a lattice

The spatial discretization of the single-cone Dirac Hamiltonian on the surface of a topological insulator or superconductor needs a special ``staggered’’ grid, to avoid the appearance of a spurious second cone in the Brillouin zone. We adapt the Stacey discretization from lattice gauge theory to produce a generalized eigenvalue problem, of the form \bm{\mathcal H}\bm{\psi}=\bm{E}\bm{\mathcal P}\bm{\psi}ℋ𝛙=𝐄𝒫𝛙, with Hermitian tight-binding operators \bm{\mathcal H}ℋ, \bm{\mathcal P}𝒫, a locally conserved particle current, and preserved chiral and symplectic symmetries. This permits the study of the spectral statistics of Dirac fermions in each of the four symmetry classes A, AII, AIII, and D.

Minimal Functional Bases for Elasticity Tensor Symmetry Classes

Functional bases, synonymous with separating sets, are usually formulated for an entire vector space, such as the space Ela of elasticity tensors. We propose here to define functional bases limited to symmetry strata, i.e. sets of tensors of the same symmetry class. We provide such lowcardinal minimal bases for tetragonal, trigonal, cubic or transversely isotropic symmetry strata of the elasticity tensor.