Symmetry Conditions Research Articles

Constructing conditional symmetry in a chaotic map

Constructing conditional symmetry in a chaotic map

The equivalence principle is NOT a Noether symmetry

The connection between the equivalence principle and Noether’s theorem was discussed in Capozziello and Ferrara (Int J Geom Methods Mod Phys 21:2440014, 2024). However, it is known that the Noether symmetry condition is independent of the equations of motions as follows from Hamilton’s principle. In this paper, we critically examine the analysis presented in the aforementioned work, highlighting its flaws, and provide a detailed demonstration that there is no connection between Noether’s theorem and the equivalence principle. Furthermore, we offer various insights on symmetry analysis to prevent the perpetuation of inaccuracies regarding Noether’s work.

Transmission enhancement mechanism of PT symmetric photonic crystal coupled with grooved sub-wavelength grating

In order to enhance the sensing characteristics of the sensor, based on the periodic sub-wavelength grating(SWG), a groove structure is introduced to make the Fano resonance waveform sharper. At the same time, based on the Parity-Time(PT) symmetry principle, by making the periodic photonic crystal meet the PT symmetry condition, the structure can absorb the energy of the emitted light on the original basis, produce the transmission enhancement effect, and make the transmission peak of the structure exceed 1. The sensing is realized by using the property that the different refractive indices of the detected medium corresponds to different transmission. When the incident light is 1550 nm, the sensitivity and FOM (Figure of Merit) value of the proposed structure reach 5.702 × 105RIU−1and 1.9 × 107 respectively, which greatly improves the performance of the sensor.

Extension of Fejér's inequality to the class of sub-biharmonic functions

Abstract Fejér’s integral inequality is a weighted version of the Hermite-Hadamard inequality that holds for the class of convex functions. To derive his inequality, Fejér [Über die Fourierreihen, II, Math. Naturwiss, Anz. Ungar. Akad. Wiss. 24 (1906), 369–390] assumed that the weight function is symmetric w.r.t. the midpoint of the interval. In this study, without assuming any symmetry condition on the weight function, Fejér’s inequality is extended to the class of sub-biharmonic functions, namely, the set of functions f ∈ C 4 ( I ) f\in {C}^{4}\left({\mathbb{I}}) satisfying f ″ ″ ≤ 0 {f}^{^{\prime\prime} ^{\prime\prime} }\le 0 , where I {\mathbb{I}} is an interval of R {\mathbb{R}} . In the special case when the weight function is symmetric w.r.t. the midpoint of some interval, and the function f f is convex and sub-biharmonic, an interesting refinement of Fejér’s inequality is deduced. Moreover, this inequality is extended to a new class of functions defined on the plane, that we call the set of sub-biharmonic functions on the coordinates. Assuming in addition that the function is convex on the coordinates, a refinement of an inequality due to Dragomir [On the Hadamard’s inequality for convex functions on the co-ordinates in a rectangle from the plane, Taiwan. J. Math. 5 (2001), 775–788] is obtained.

Singular Abreu equations and linearized Monge–Ampère equations with drifts

We study the solvability of singular Abreu equations which arise in the approximation of convex functionals subject to a convexity constraint. Previous works established the solvability of their second boundary value problems either in two dimensions, or in higher dimensions under either a smallness condition or a radial symmetry condition. Here, we solve the higher-dimensional case by transforming singular Abreu equations into linearized Monge–Ampère equations with drifts. We establish global Hölder estimates for linearized Monge–Ampère equations with drifts under suitable hypotheses, and then apply them to prove the regularity and solvability of the second boundary value problem for singular Abreu equations in higher dimensions. Many cases with general right-hand side are also discussed.

Charging and Discharging Modeling of Inertial Sensors Based on Ultraviolet Charge Management

Inertial sensors act as inertial references in space gravitational wave detection missions, necessitating that their internal test mass (TM) maintains minimal residual acceleration noise. High-energy particles and cosmic rays in space, along with ion pumps in ground-based torsion pendulum experiments, can cause charge accumulation on the TM surface, leading to increased residual acceleration noise. Consequently, a charge management system was introduced to control the TM’s charge. The complex light path propagation within the electrode housing (EH) makes the TM’s charging and discharging process difficult to theoretically calculate and fully simulate. To address this issue, we propose a simulation method for charging and discharging inertial sensors within ultraviolet (UV) charge management systems. This method innovatively considers the impact of photoelectron emission angle and the TM’s position offset from symmetry on performance. The method also simulates charging and discharging rates over time under conditions of symmetry and preliminarily examines factors affecting the TM’s equilibrium potential. Simulation results indicate that this method effectively models the charge management system’s operation, providing a valuable reference for system design.

Identification and estimation of entry games under symmetry of unobservables

Abstract This paper provides a new point identification and estimation method for two-player entry games with complete information, based on symmetry of unobservables. Neither equilibrium selection nor parametric distributional assumptions are required. In addition, a weaker support condition is used relative to the existing literature. Unlike other semiparametric estimators, the estimator proposed here is $\sqrt{n}$-consistent. A test of the required symmetry condition is provided. Monte Carlo evidence shows that the estimator performs well with moderate sample sizes, and is robust to unimodal and multimodal error distributions. The estimator is applied to an entry game of discount retailers in Jia (2008). Our results suggest that researchers should apply the normality assumption with caution.

Conditional symmetry test based on empirical characteristic function

In this paper, we propose a method for testing the conditional symmetry of multivariate random variables, specifically testing whether or not the conditional distribution of one random variable is symmetric around zero given another. Both the conditional symmetry test for observable data and unobserved error terms in parametric models are considered. The proposed test statistics are built on the weighted L 2 -distance between empirical characteristic functions and require no distributional assumption about the data distribution. The asymptotic properties of the test statistics under the null and alternative hypotheses are investigated. Since the limiting null distributions are intractable, a nonparametric Monte Carlo resampling method is used to determine critical values. Simulation results show the good performance of the proposed method in various scenarios. Two real datasets are analysed to illustrate the practical usefulness of the proposed tests.

Mueller matrix symmetry for both reciprocal and nonreciprocal metamaterials

Mueller matrices relate the Stokes parameters of the incident and emerging light, providing useful information about the radiative properties and other characteristics of the medium. Determining all elements of the 4 × 4 Mueller matrix requires complete polarimetry, which is often challenging to perform. Partial polarimetry, on the other hand, uses simpler optical components in generating and/or analyzing states of polarization, thereby measuring only a subset of the Mueller matrix. However, it may determine the full Mueller matrix under specific symmetry conditions. The present study develops a symmetry classification scheme to categorize the Mueller matrix of materials. It is shown that the symmetry of the Mueller matrix is directly determined from the information of symmetries of the sample's optical properties. Numerical calculations of various measurement scenarios, structures, and materials (with or without Lorentz reciprocity) are carried out to validate the methodology. This study offers an insightful understanding of Mueller matrix symmetry and practical guidance for simplified ellipsometry measurements.

Spatiotemporal patterns in the active cyclic Potts model

Abstract The nonequilibrium dynamics of a cycling three-state Potts model is studied on a square lattice using Monte Carlo simulations and continuum theory. This model is relevant to chemical reactions on a catalytic surface and to molecular transport across a membrane. Several characteristic modes are formed depending on the flipping energies between successive states and the contact energies between neighboring sites. Under cyclic symmetry conditions, cycling homogeneous phases and spiral waves (SW) form at low and high flipping energies, respectively. In the intermediate flipping energy regime, these two modes coexist temporally in small systems and/or at low contact energies. Under asymmetric conditions, we observed small biphasic domains exhibiting amoeba-like locomotion and temporal coexistence of SW and a dominant non-cyclic one-state phase. An increase in the flipping energy between two successive states, say state 0 and state 1, while keeping the other flipping energies constant, induces the formation of the third phase (state 2), owing to the suppression of the nucleation of state 0 domains. Under asymmetric conditions regarding the contact energies, two different modes can appear depending on the initial state, due to a hysteresis phenomenon.

Obstacle-insensitive eigenfields due to boundary condition–symmetry compatibility

Obstacle-insensitive eigenfields due to boundary condition–symmetry compatibility

Polariton-polariton interaction induced by the hopping effect between two monolayers

Investigating the quantum many-body interaction in microcavity semiconductor systems is a significant objective in current research. Hence, in this study, Green’s function was employed to investigate the hopping effect between two monolayers denoted as L and R. The interaction between excitons in L monolayer and R monolayer, characterized by spin-like properties, leads to the generation of Josephson currents and emergence of two quasiparticle branches, known as LP+ and LP− polaritons (i.e., two low polariton states). And compared with other papers, such as reference 5, the research shows that presence of hopping interaction and many-body interaction between monolayers causes significant enhancement phenomenon of residue Z, which will further reduce the effective mass. This is more conducive to Bose–Einstein condensates (BEC). Furthermore, we investigated the Parity-Time symmetry (PT symmetry) of the system. The PT broken phase transition point will appear when the condition of PT symmetry is satisfied.

Canonical torsor bundles of prescribed rational functions on complex curves

The prescribed rational functions constitute a subset of rational functions satisfying certain symmetry and analyticity conditions. Over a smooth complex curve [Formula: see text], we construct explicitly a bundle [Formula: see text] with values in the prescribed rational functions. An intrinsic coordinate-independent formulation (the main result of the paper, Proposition 1) for such bundles is given. The construction presented in this paper is useful for studies of the canonical cosimplicial cohomology of infinite-dimensional Lie algebras on smooth manifolds, as well as for the purposes of conformal field theory and the theory of foliations.

Renormalization of conformal infinity as a stretched horizon

Abstract In this paper, we provide a comprehensive study of asymptotically flat spacetime in even dimensions d ⩾ 4 . We analyze the most general boundary condition and asymptotic symmetry compatible with Penrose’s definition of asymptotic null infinity I through conformal compactification. Following Penrose’s prescription and using a minimal version of the Bondi–Sachs gauge, we show that I is naturally equipped with a Carrollian stress tensor whose radial derivative defines the asymptotic Weyl tensor. This analysis describes asymptotic infinity as a stretched horizon in the conformally compactified spacetime. We establish that charge aspects conservation can be written as Carrollian Bianchi identities for the asymptotic Weyl tensor. We then provide a covariant renormalization for the asymptotic symplectic potential, which results in a finite symplectic flux and asymptotic charges. The renormalization scheme works even in the presence of logarithmic anomalies.

Dual Dirac Nodal Line in Nearly Freestanding Electronic Structure of β-Sn Monolayer.

Two-dimensional topological insulators (2D TIs) have distinct electronic properties that make them attractive for various applications, especially in spintronics. The conductive edge states in 2D TIs are protected from disorder and perturbations and are spin-polarized, which restrict current flow to a single spin orientation. In contrast, topological nodal line semimetals (TNLSM) are distinct from TIs because of the presence of a 1D ring of degeneracy formed from two bands that cross each other along a line in the Brillouin zone. These nodal lines are protected by topology and can be destroyed only by breaking certain symmetry conditions, making them highly resilient to disorder and defects. However, 2D TNLSMs do not possess protected boundary modes, which makes their investigation challenging. There have been several theoretical predictions of 2D TNLSMs, however, experimental realizations are rare. β-Sn, a metallic allotrope of tin with a superconducting temperature of 3.72 K, may be a candidate for a topological superconductor that can host Majorana Fermions for quantum computing. In this work, single layers of α-Sn and β-Sn on a Cu(111) substrate are successfully prepared and studied using scanning tunneling microscopy, angle-resolved photoemission spectroscopy, and density functional theory calculations. The lattice and electronic structure undergo a topological transition from 2D topological insulator α-Sn to 2D TNLSM β-Sn, with two types of nodal lines coexisting in monolayer β-Sn. Such a realization of two types of nodal lines in one 2D material has not been reported to date. Moreover, we also observed an unexpected phenomenon of freestanding-like electronic structures of β-Sn/Cu(111), highlighting the potential of ultrathin β-Sn films as a platform for exploring the electronic properties of 2D TNLSM and topological superconductors, such as few-layer superconducting β-Sn in lateral contact with topological nodal line single-layer β-Sn.

Tunable microwave circulator and amplifier in cavity magnonic system

An adjustable unidirectional amplifier is proposed in a hybrid cavity magnonic system. By eliminating the rapidly dissipative auxiliary mode, we obtain the effective Hamiltonian for a typical three-body circulator. Through analyzing the transmission coefficients of the system, we find that the nonreciprocal transmission of photon–magnon as well as the amplification of arbitrary signals can be both realized without requiring in the parity-time symmetry condition. In addition, the amplification factor of our system is highly adjustable. Our theoretical scheme has potential implication in realizing magnonic triodes and magnon-based quantum information processing.

Symmetries and Separation of Variables

In this paper, we look at the method of separation of variables of a PDE from its symmetry transformation point of view. Specifically, we discuss the relation between the existence of additively and multiplicatively separated variables of a PDE, and the form of its symmetry operators. We show that solutions in the form of separated variables are in fact, invariant solutions, i.e. solutions invariant under some subalgebra of the symmetry operators of the equation. For the case of two independent variables, we obtain the form of Lie point symmetry operators corresponding to additively and multiplicatively separated solutions, and generalize our results for the case when separated variables are any functions of independent variables. We also discuss the role of contact symmetry transformations and differential invariants for the existence of separated solutions, and outline the role of variational symmetries, as well as conditional (non-classical) symmetry operators. We demonstrate that the symmetry approach is a valuable tool for obtaining information regarding existence of solutions with separated variables.

Symmetric operator extensions of composites of higher order difference operators

In this paper we have considered two higher order difference operators generated by two higher order difference functions on the Hilbert space of square summable functions. By allowing the leading coefficients to be unbounded and the other coefficients as constant functions, we have shown that the composites of two symmetric difference operators are symmetric if the leading coefficients are scalar multiple of each other and the common divisor of their orders is 1. Using examples, we have shown that these conditions of symmetry cannot be weakened. Furthermore, We have shown that the deficiency indices of the composites is equal to the sum of the deficiency indices of the individual operators and that the spectra of the self-adjoint operator extensions is the whole of the real line.

An index for measuring degree of departure from symmetry for ordinal square contingency tables

AbstractFor the analysis of square contingency tables with the same row and column ordinal classifications, this study proposes an index for measuring the degree of departure from the symmetry model using new cumulative probabilities. The proposed index is constructed based on the Cressie and Read’s power divergence, or the weighted average of the Patil and Taillie’s diversity index. This study derives a plug-in estimator of the proposed index and an approximate confidence interval for the proposed index. The estimator of the proposed index is expected to reduce the bias more than the estimator of the existing index, even when the sample size is not large. The proposed index is identical to the existing index under the conditional symmetry model. Therefore, assuming the probability structure in which the conditional symmetry model holds, the performances of plug-in estimators of the proposed and existing indexes can be simply compared. Through numerical examples and real data analysis, the usefulness of the proposed index compared to the existing index is demonstrated.

Study on prestress distribution and structural performance of heptagonal six-five-strut alternated cable dome with inner hole

The cable dome structure is a highly efficient prestressed space structure with reasonable force, attractive shape, and large spanning capacity. It is increasingly widely used in stadiums, exhibition halls, terminals, and other large-scale public buildings and the roof structure of various transport hub projects. To solve the problem of weak overall structural stability of the traditional cable dome structure, a new type of prestressed space structure is proposed - heptagonal six-five-strut alternated cable dome with an inner hole. The composition of the structural system was given and a theoretical study of the distribution of structural prestress was carried out; based on the symmetry and periodicity conditions, the initial prestress distribution of the structure was accurately calculated using the node balance method, and the formulae for the geometric parameters of the structural cable-strut were derived; the numerical model was established by using the finite element software ANSYS 2022 R1, and the force characteristics of the structure under full-span uniform load, half span uniform live load, wind load, and temperature were studied. The effects of structural prestress, overall shape, and support stiffness, totaling three categories and six parameters, on the structural mechanical performance of the internal force of the member, node displacement, and support reaction force were analysed in detail and the sensitivity of the parameters is assessed quantitatively, accordingly, the values of the main design parameters such as thick-span ratio are given as suggestions. The analysis results show that the new structural system has a high load-carrying capacity and deformation resistance, and its proposal provides a new scheme and new ideas for the selection and design of the cable dome.