Symmetry Conditions Research Articles

Linear filtration waves in a layered heterogeneous formation

Solutions to the problem of a filtration-wave pressure field in a layered inhomogeneous medium with a pressure pulse specified on the left boundary are obtained. The problem statement contains a two-dimensional wave equation in the central layer, a wave equation that takes into account the predominance of vertical motion in the environment, symmetry conditions in the center of the formation, equality of pressures and flows at the interfaces, the absence of disturbances of the filtration wave field at the initial moment of time and at an infinite distance from the source of disturbance. The article found an exact solution to the problem obtained using the Laplace–Carson and the Fourier transform of sine, an asymptotic solution and an exact solution for the special case of a homogeneous medium. It is shown that when the formal parameter tends to zero, the exact solution coincides with the main asymptotic approximation. The tendency of the thickness of the central layer to infinity reduces the solution for a layered inhomogeneous medium to an expression describing the pressure fields in a homogeneous medium. Spatiotemporal dependencies describing the dynamics of the pressure pulse in layered inhomogeneous and homogeneous media were constructed, and the general properties and features of linear filtration waves in layered inhomogeneous and homogeneous layers were studied.

The effect of an explosive shock wave on the plate of a protective structure of a critical infrastructure

The article presents the outcomes of an analysis on the stress-strain conditions of reinforced concrete structures subjected to an explosive shock wave resulting from the detonation of a combat unit from a kamikaze drone against a protective screen. When designing protective structures for critical infrastructure, employing computer simulation enables an assessment of the genuine impact of explosive loading on the structural elements' strength. The active phase of explosive loading is exceptionally brief, lasting only a fraction of a second. Under such circumstances, modeling is best performed using explicit methods of direct integration in time.
 The structure considered in this work is a reinforced concrete slab supported by a metal beam cage with I-beam cross-sections, topped with a sand backfill. The study was executed within the SIMULIA Abaqus software suite, incorporating models depicting nonlinear material behavior in a three-dimensional context. Discretely positioned reinforcement was considered for reinforced concrete structures, and the "Concrete Damage Plasticity" model was applied for concrete, accounting for damage accumulation. The devised computational scheme for the shelter represents a section of the protective structure's roof under conditions of cyclic symmetry.
 The article elucidates the core principles of incorporating explosive loading according to algorithm CONWEP.The results demonstrate that during the detonation of a kamikaze drone, an explosive wave created a crater in the sand backfill, exposing the slab. The study illustrates the development of damage in the reinforced concrete slab at various time intervals. Despite the identified damage to the slab, the protective structure overall withstood the explosive load. The intensity of the explosive shock wave diminishes significantly as it propagates away from the explosion site.

On the eigenvalue distribution of spatio-spectral limiting operators in higher dimensions

Prolate spheroidal wave functions are an orthogonal family of bandlimited functions on R that have the highest concentration within a specific time interval. They are also identified as the eigenfunctions of a time-frequency limiting operator (TFLO), and the associated eigenvalues belong to the interval [0,1]. Previous work has studied the asymptotic distribution and clustering behavior of the TFLO eigenvalues.In this paper, we extend these results to multiple dimensions. We prove estimates on the eigenvalues of a spatio-spectral limiting operator (SSLO) on L2(Rd), which is an alternating product of projection operators associated to given spatial and frequency domains in Rd. If one of the domains is a hypercube, and the other domain is convex body satisfying a symmetry condition, we derive quantitative bounds on the distribution of the SSLO eigenvalues in the interval [0,1].To prove our results, we design an orthonormal system of wave packets in L2(Rd) that are highly concentrated in the spatial and frequency domains. We show that these wave packets are “approximate eigenfunctions” of a spatio-spectral limiting operator. To construct the wave packets, we use a variant of the Coifman-Meyer local sine basis for L2[0,1], and we lift the basis to higher dimensions using a tensor product.

Parity time symmetry in two dimensional chiral optical lattice, via positive and negative refraction

The parity time symmetry and positive/negative refraction is investigated in chiral optical lattice. Significant parity time symmetry is reported in five level chiral optical lattice which satisfied the conditions Re(nr(±)(x,y))=Re(nr∗(±)(−x,−y) and Im(nr(±)(x,y))=−Im(nr∗(±)(−x,−y) for left and right circularly polarized beams. The related group indices, phase shifts and divergent angle are also satisfied the parity time symmetry conditions ng(±)(x,y))=ng∗(±)(−x,−y), ϕ(±)(x,y))=ϕ∗(±)(−x,−y) and θd,g(x,y)=θd,g∗(−x,−y) for left and right circularly polarized beams. Maximum negative group index is calculated to −25000. The maximum phase and group divergent angles are reported to 0.001 radian and −0.9 radian. The phase shifts in LCP and RCP beams are investigated to ±0.004 radian at a length L=10λ. The modified results have potential application in nanotechnology.

Electromagnetic Performance Analysis of a Multichannel Permanent Magnet Synchronous Generator

In this paper, we present an analysis of the properties of the prototype three-phase Multichannel Permanent Magnet Synchronous Generator (MCPMSG) prototype designed and constructed by the authors. Each channel of the generator has electrically separated windings, which allows us to create an island system of electricity generation. The analyzed MCPMSG is intended for critical applications, and it is designed for four-channel operation. The purpose of this work is to analyze various configurations of the generator channels to improve the redundancy of the electricity generation system. The MCPMSG operation with one or two independent sources of energy consumption in the case of a dual-channel or double dual-channel operation was investigated. For the analyzed cases, the original mathematical models of the three-phase MCPMSG were developed. On the basis of numerical and laboratory tests, the influence of individual configurations on the MCPMSG output parameters was determined. An original method for diagnosing the operation of the MCPMSG channels was developed. Numerical and laboratory tests of the proposed diagnostic method based on a single voltage signal were carried out. As part of the laboratory tests, selected operating states under conditions of full winding symmetry and internal asymmetry were analyzed. The advantage of the proposed diagnostic method is the control of the operating state of the channels both under load and in the de-energized state. The proposed diagnostic method for control of the individual channel requires measurement of only one voltage signal.

Unitary coupled-cluster for quantum computation of molecular properties in a strong magnetic field.

In truncated coupled-cluster (CC) theories, non-variational and/or generally complex ground-state energies can occur. This is due to the non-Hermitian nature of the similarity transformed Hamiltonian matrix in combination with CC truncation. For chemical problems that deal with real-valued Hamiltonian matrices, complex CC energies rarely occur. However, for complex-valued Hamiltonian matrices, such as those that arise in the presence of strong magnetic fields, complex CC energies can be regularly observed unless certain symmetry conditions are fulfilled. Therefore, in the presence of magnetic fields, it is desirable to pursue CC methods that are guaranteed to give upper-bound, real-valued energies. In this work, we present the first application of unitary CC to chemical systems in a strong magnetic field. This is achieved utilizing the variational quantum eigensolver algorithm applied to the unitary coupled-cluster singles and doubles (UCCSD) method. We benchmark the method on the H2 molecule in a strong magnetic field and then calculate UCCSD energies for the H4 molecule as a function of both geometry and field angle. We show that while standard CCSD can yield generally complex energies that are not an upper-bound to the true energy, UCCSD always results in variational and real-valued energies. We also show that the imaginary components of the CCSD energy are largest in the strongly correlated region. Last, the UCCSD calculations capture a large percentage of the correlation energy.

Lie symmetries for the cosmological field equations in brane-world gravitywith bulk scalar field

We address the group classification problem for gravitational field equations within the context of brane-world cosmology, considering the presence of a bulk scalar field. Our investigation revolves around a five-dimensional spacetime, with the four-dimensional Friedmann–Lemaître–Robertson–Walker geometry embedded within it. Additionally, we assume that the scalar field exists in this five-dimensional geometry (bulk) and possesses a nonzero mass. The resulting field equations constitute a system of nonlinear partial differential equations. We apply the Lie symmetry condition to identify all functional forms of the scalar field potential, ensuring that the field equations remain invariant under one-parameter point transformations. Consequently, we find that only the exponential potential exhibits Lie symmetries. Finally, the Lie invariants are used to construct similarity transformations, which enable us to derive exact solutions for the system.

Lyapunov function for interacting reinforced stochastic processes via Hopfield’s energy function

In 1984, Hopfield introduced an artificial neural network to understand the memory in living organisms. Under a condition of symmetry, he showed that there exists a Lyapunov function (known as energy function) for the network. In 2015, Budhiraja, Dupuis and Fischer introduced an idea to construct Lyapunov functions for nonlinear Markov processes via relative entropy. In this article, we introduce an approach based on the energy function of Hopfield networks to obtain Lyapunov functions for a class of interacting reinforced stochastic processes. Our result is an alternative to Budhiraja, Dupuis and Fischer’s approach and it works for the case of processes with finitely many 2-dimensional probability measures. We also bring from the neural network framework results about the total stability of differential equations. Finally, we include an application of the results to the study of differential equations associated to n≥2 vertex reinforced random walks on two-vertex graphs.

Combination Test for Mean Shift and Variance Change

This paper considers a new mean-variance model with strong mixing errors and describes a combination test for the mean shift and variance change. Under some stationarity and symmetry conditions, the important limiting distribution for a combination test is obtained, which can derive the limiting distributions for the mean change test and variance change test. As an application, an algorithm for a three-step method to detect the change-points is given. For example, the first step is to test whether there is at least a change-point. The second and third steps are to detect the mean change-point and the variance change-point, respectively. To illustrate our results, some simulations and real-world data analysis are discussed. The analysis shows that our tests not only have high powers, but can also determine the mean change-point or variance change-point. Compared to the existing methods of cpt.meanvar and mosum from the R package, the new method has the advantages of recognition capability and accuracy.

Existence and multiplicity of solutions for general quasi-linear elliptic equations with sub-cubic nonlinearities

We consider the following quasi-linear Schrödinger equation−∑i,j=1NDj(aij(u)Diu)+12∑i,j=1NDsaij(u)DiuDju+V(x)u=f(u),x∈RN,N≥3, which includes the modified nonlinear Schrödinger equations. Combining a p-Laplacian perturbation argument, we obtain the existence of positive solutions to the problem above and multiple solutions with additional symmetry conditions on aij and f. A new perturbation approach is used to treat the sub-cubic nonlinearity. In particular, for f(u)=|u|p−2u, the case p∈(2,4) is also considered.

Laser Pulse Interaction with Plasma under Conditions of Broken Axial Symmetry

Laser Pulse Interaction with Plasma under Conditions of Broken Axial Symmetry

Relationship between body condition and vertical movement symmetry in 109 riding school horses

Abstract For riding schools, the health and function of their horses is paramount. Unfortunately, lameness and overweight are common problems in this population, and a causal relationship whereby increased body weight causes lameness is also possible. This observational field study investigated the relationship between body condition score (BCS) and motion movement asymmetries in riding school horses and also explored the relationships between BCS, movement asymmetry and subjectively scored willingness to work. Eight riding schools within 2.5 h by car from Uppsala, Sweden, were convenience-sampled and 3-24 horses from each were included in the study. Body condition was scored (scale 1-9) and vertical movement asymmetry data were collected in straight-line trot, using an inertial measurement unit system. Asymmetry parameters were calculated as the mean difference between local vertical displacement minima/maxima for the head/pelvis (HD/PD) between the two halves of the stride. A questionnaire was used to assess perceived willingness to work and energy level of the horses. The data were analysed with linear mixed models, t-tests and Mann-Whitney U-tests. A correlation was found between BCS and PDmin (), an asymmetry parameter related to hindlimb weight-bearing lameness, where PDmin increased by 0.96 mm for each unit increase in BCS. Horses with vertical movement classified as symmetric had lower BCS than horses with movement classified as asymmetric (5.4 ± 0.4 vs 5.7 ± 0.6, ). Horses classed as overweight (BCS ≥ 6) scored lower on willingness to work () and energy level (). These results indicate that the body condition of riding school horses may influence their function and the health of their locomotor apparatus. Further studies are required to identify the underlying mechanisms.

Arbitrary (k, l) States-Solutions of the Dirac and Schrödinger Equations Interacting with Improved Spatially-Dependent Mass Coulomb Potential with an improved Coulomb-like tensor interaction Model for H-atoms from 3D-RNCS and 3D-NRNCS symmetries

The improved spatially dependent mass Coulomb potential with an improved Coulomb-like tensor interaction (ISDM(CP-CLTI)) model has been used to investigate the new bound state solutions of the deformed Dirac equation under the condition of spin symmetry and pseudospin symmetry in the 3D-relativistic noncommutative space (3D-RNCS) symmetries. The combination of spatially dependent mass Coulomb potential with Coulomb-like tensor interaction (ISDM(CP-CLTI)) and the terms (1/r^3 and 1/r^4 ) which are connected with the induced couplings (LΘ and L ̃Θ) produced from the topological defects of space-space gives ISDM(CP-CLTI) model. The new relativistic and non-relativistic energy eigenvalues for the hydrogen atoms (H-atoms) such as He+, Li+2, and Be3+ under the ISDM(CP-CLTI) model have been derived within the parameters of the parametric Bopp shift method and standard perturbation theory. The new values E_nc^sp(n,C,m_0,m_1,H,Θ,τ,χ,j,l,s,m) and E_nc^ps(n,C,m_0,m_1,H,Θ,τ,χ,j,l,l ̃,s ̃,m ̃) that we obtained appeared to be sensitive to the quantum atomic discrete quantum numbers (j,k,l,s,m,l ̃,s ̃,m ̃), the mixed potential depths (C/q,m_0,m_1,H), and noncommutativity parameters (NP) (Θ,σ,χ). Within the context of relativistic extended quantum mechanics, we have obtained certain significant special cases that, in our view, will be interesting to the specialized researcher. When we applied the three simultaneous limitations (Θ,σ,χ)→(0,0,0), NP we were able to retrieve the typical results of relativistic and non-relativistic cases in the literature. Our new model presented new physical features compared to other models known in the literature.

Arbitrary (k, l) States-Solutions of the Dirac and Schrödinger Equations Interacting with Improved Spatially-Dependent Mass Coulomb Potential with an improved Coulomb-like tensor interaction Model for H-atoms from 3D-RNCS and 3D-NRNCS symmetries

The deformed Dirac equation has been investigated, in the context of 3D-relativistic noncommutative space (3D-RNCS) symmetries, using the improved spatially dependent mass Coulomb potential with an improved Coulomb-like tensor interaction (ISDM(CP-CLTI)) model under the conditions of spin symmetry and pseudospin symmetry. The ISDM(CP-CLTI) model is the combining the spatially dependent mass Coulomb potential with the Coulomb-like tensor interaction (CP-CLTI) and the two central terms that are generated to the topological defects of spacespace. Within the confines of the parametric Bopp shift method and conventional perturbation theory, the new relativistic and non-relativistic energy eigenvalues for the hydrogen atoms ( H-atoms), such as He+,Li+2, and Be3+ under the ISDM(CP-CLTI) model have been derived. The novel values Encsp(n,C,m0,m1,H,Θ,τ,χ,j,l,s,m) and Encps(n,C,m0,m1,H,Θ,τ,χ,j,l,l̃,s̃,m̃) that we discovered examined to be dependent on the noncommutativity parameters (NP) (Θ,σ,χ), mixed potential depths C/q,m0,m1,H), and quantum atomic discrete quantum numbers (j,k,l,s,m,l̃,s̃,m̃). We have obtained several interesting special examples within the framework of relativistic extended quantum mechanics, which we believe will be of interest to the expert researcher. We were able to retrieve the typical results of relativistic and non-relativistic examples in the literature when we applied the three simultaneous constraints (Θ,σ,χ)→(0,0,0). Compared to previous models that are known from the literature, our new model had novel physical characteristics.

Generic ill-posedness of the energy–momentum equations and differential inclusions

We show that the energy–momentum equations arising from inner variations whose Lagrangian satisfies a generic symmetry condition are ill-posed. This is done by proving that there exists a subclass of Lipschitz solutions that are also solutions to a differential inclusion into the orthogonal group and in particular these solutions can be nowhere $C^1$ . We prove that these solutions are not stationary points if the Lagrangian $W$ is $C^1$ and strictly rank-one convex. In view of the Lipschitz regularity result of Iwaniec, Kovalev and Onninen for solution of the energy–momentum equation in dimension 2, we give a sufficient condition for the non-existence of a partial $C^1$ -regularity result even under the condition that the mappings satisfy a positive Jacobian determinant condition. Finally, we consider a number of well-known functionals studied in non-linear elasticity and geometric function theory and show that these do not satisfy this obstruction to partial regularity.

On stability estimations without any conditions of symmetry

Let X, X1, X2, ..., Xn be i.i.d. random variables. B. Ramachandran and C.R. Rao have proved that if distributions of sample mean ‾X = ‾X(n) = (X1 + ⋯ + Xn)/n and monomial X are coincident at least at two points n = j1 and n = j2 such that log j1/ log j2 is irrational, then X follows a Cauchy law. Assuming that condition of coincidence of \bar X(n) and X are fulfilled at least for two n values, but only approximately, with some error ε in metric λ, we prove (without any conditions of symmetry) that, in certain sense, characteristic function of X is close to the characteristic function of the Cauchy distribution.

Developing an immersed boundary method for compressible flow

AbstractDevelopment of a 3D sharp interface ghost node immersed boundary method (IBM) within a fluid solver based on the compressible Navier–Stokes equations is underway. The objective of the IBM is to accurately apply boundary conditions at fluid–solid interfaces. The Navier–Stokes solver is currently being verified using various test cases, including the classic cylinder in cross flow. As part of this verification process, particular attention was given to investigating the effects of different lateral boundary conditions. The results demonstrate that extrapolation boundary conditions exhibit better agreement with the literature compared to symmetry conditions, in cases with relatively narrow domains. These findings highlight the potential benefits of extrapolation boundary conditions in reducing confinement effects and removing nonphysical waves in external flow problems.

An analytical closed-form solution for the in-plane elastic modulus of plain-woven textile composites

Engineering application of textile composites needs simple relations for the estimation of their mechanical properties. Many homogenization and simplified models have been already proposed for the analytical characterization of the elastic moduli of this class of materials. Along this route, the current study introduces an analytical solution for the longitudinal elastic modulus of plain-woven textile composites. A hybrid beam-spring model of the representative unit cell (RUC) is postulated consisting of a horizontal beam connected to an inclined beam, two supporting springs beneath the free end of the horizontal beam and the common end of the horizontal and inclined beams, and a torsion spring at the free end of the inclined beam. The mechanical and geometrical properties of the beams and springs are calculated from the weave structure and the symmetry conditions. A reduction technique from the matrix analysis of structures is borrowed to consider the effect of internal degrees of freedom on the in-plane stiffness coefficient and a closed-form formula is extracted for the elastic modulus. The proposed analytical formula is finally verified against available experimental data and a wide range of FE predictions for the GFRP and CFRP plain-woven composites.

Linear and nonlinear spin current response of anisotropic spin-orbit coupled systems

We calculate the linear and the second harmonic (SH) spin current response of two anisotropic systems with spin–orbit (SO) interaction. General expressions of wide applicability for the these response functions are first derived for a generic two-band Hamiltonian. The first system is a two-dimensional (2D) electron gas in the presence of Rashba and k-linear Dresselhaus SO couplings. The calculations show how narrow or wide the response spectra can be, what is their overall shape and size, and frequency shiftings, depending on which crystal orientation is selected. The quantitative knowing of this makes possible a comparative study for several orientations, which would allow to select a spectrum with particular characteristic. We find that vanishing linear and second order response tensors are achievable under SU(2) symmetry conditions, characterized by a collinear SO vector field. Additional conditions under which specific tensor components vanish are possible, without having such collinearity. Thus, a proper choice of the growth direction and SO strengths allows to select the polarization of the linear and SH spin currents according to the direction of flowing. The second system is an anisotropic 2D free electron gas with anisotropic Rashba interaction, which has been employed to study the optical conductivity of 2D puckered structures with anisotropic energy bands. The presence of mass anisotropy and an energy gap open several distinct scenarios for the allowed optical interband transitions, which manifest in the linear and SH response contrastingly. The linear response displays only out-of-plane spin polarized currents, while the SH spin currents flow with spin orientation lying parallel to the plane of the system strictly. The models illustrate the possibility of the nonlinear spin Hall effect in systems with SO interaction, under the presence or absence of time-reversal symmetry. The results suggest different ways to manipulate the linear and nonlinear optical generation of spin currents which could find spintronic applications.

Generalized -dressing method for coupled nonlocal NLS equation

In this paper, we investigate a general integrable coupled nonlocal nonlinear Schrödinger (cnNLS) equation with reverse space and reverse time. First, we introduce a 4-component nonlocal nonlinear Schrödinger (nNLS) equation. Based on two 3 × 3 matrices -problems, we obtain the N-soliton solutions of the 4-component nNLS equation by constructing two spectral transformation matrices R and with some specific scattering data and . We have the symmetry condition for R and . We express the determinant in the N-soliton solution in the form of sums with the help of Cauchy matrix properties to facilitate follow-up reduction. The general nonlocal reduction of the 4-component nNLS equation to the cnNLS equation is discussed in detail. After obtaining the 1-soliton solution of cnNLS equation, we analyse the nonsingular region of the single soliton solution, with spectral parameters k 1 and λ 1 are conjugate. We also draw some typical images of 1-soliton solution. The 2-soliton solutions for cnNLS equation are derived and their asymptotic behaviours are discussed. After that we discuss the dynamic behaviour of the two waves in 2-soliton solution under the condition of different spectral parameters values.