Symmetric Case Research Articles

Strong-coupling critical behavior in three-dimensional lattice Abelian gauge models with charged N-component scalar fields and SO(N) symmetry.

We consider a three-dimensional lattice Abelian Higgs gauge model for a charged N-component scalar field ϕ, which is invariant under SO(N) global transformations for generic values of the parameters. We focus on the strong-coupling regime, in which the kinetic Hamiltonian term for the gauge field is a small perturbation, which is irrelevant for the critical behavior. The Hamiltonian depends on a parameter v, which determines the global symmetry of the model and the symmetry of the low-temperature phases. We present renormalization-group predictions, based on a Landau-Ginzburg-Wilson effective description that relies on the identification of the appropriate order parameter and on the symmetry-breaking patterns that occur at the strong-coupling phase transitions. For v=0, the global symmetry group of the model is SU(N); the corresponding model may undergo continuous transitions only for N=2. For v≠0, i.e., in the SO(N) symmetric case, continuous transitions (in the Heisenberg universality class) are possible also for N=3 and 4. We perform Monte Carlo simulations for N=2,3,4,6, to verify the renormalization-group predictions. Finite-size scaling analyses of the numerical data are in full agreement.

Derivation, characterization, and application of complete orthonormal sequences for representing general three-dimensional states of residual stress

Residual stresses are self-equilibrated stresses on unloaded bodies. Owing to their complex origins, it is useful to develop functions that can be linearly combined to represent any sufficiently regular residual stress field. In this work, we develop orthonormal sequences that span the set of all square-integrable residual stress fields on a given three-dimensional region. These sequences are obtained by extremizing the most general quadratic, positive-definite functional of the stress gradient on the set of all sufficiently regular residual stress fields subject to a prescribed normalization condition; each such functional yields a sequence. For the special case where the sixth-order coefficient tensor in the functional is homogeneous and isotropic and the fourth-order coefficient tensor in the normalization condition is proportional to the identity tensor, we obtain a three-parameter subfamily of sequences. Upon a suitable parameter normalization, we find that the viable parameter space corresponds to a semi-infinite strip. For a further specialized spherically symmetric case, we obtain analytical expressions for the sequences and the associated Lagrange multipliers. Remarkably, these sequences change little across the entire parameter strip. To illustrate the applicability of our theoretical findings, we employ three such spherically symmetric sequences to accurately approximate two standard residual stress fields. Our work opens avenues for future exploration into the implications of different sequences, achieved by altering both the spatial distribution and the material symmetry class of the coefficient tensors, toward specific objectives.

Anisotropic stellar evolution and exotic matter

In this paper, we discuss the stability of isotropic pressure conditions for a spherically symmetric dissipative fluid distribution in the framework of f(G,T) gravity. We consider the fluid to be composed of seen and exotic matter particles. The Gauss Bonnet terms represent exotic matter particles. For the physical significance of the pressure isotropy conditions, we transform the system into the hydrostatic equilibrium approximation. We observe the presence of shear, dissipative fluxes, energy density inhomogeneities, and prominently, the exotic matter tends to abandon isotropy leading to pressure anisotropy. It is found that exotic matter and dissipation are major factors that induce matter anisotropy in the evolving stellar system. In order to discuss the physical significance of the stellar model in f(G,T) gravity, the behaviour of stellar structure is explored according to the observational data of a compact star 4U1820 − 30. It is found that f(G,T) terms indicating the exotic material in the stellar model plays a vital role in governing the dynamics of the evolution. To strengthen the results, we also incorporate some arguments by choosing an axially symmetric case.

Calculation of the (I-V) Characteristics of Corona Discharge in an Asymmetrical Model from Active Electrodes to Passive Electrodes

This research discusses the comparison of the formulation of the current-voltage (I-V) characteristics of symmetric and asymmetric direct current CCP equipment in the case of corona discharges in the air. The electrode configuration model taken for the symmetric CCP case is the coaxial cylinder and for the asymmetric CCP case is the plane-serrated knife. Both types of electrodes use a modified capacitance concept in the (I-V) characteristic calculations. This concept originates from a geometric approach solution (electrode sizes), where there is a corona electric current multiplier factor (corona current multiplication factor compared to conventional electric current) as well as the nature of the corona discharge which will collect at the sharp tip of the active electrode.

The Holonomy of Spherically Symmetric Projective Finsler Metrics of Constant Curvature

In this paper, we investigate the holonomy group of n-dimensional projective Finsler metrics of constant curvature. We establish that in the spherically symmetric case, the holonomy group is maximal, and for a simply connected manifold it is isomorphic to Diffo(Sn-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {D}}i\\!f \\hspace{-3pt} f_o({\\mathbb {S}}^{n-1})$$\\end{document}, the connected component of the identity of the group of smooth diffeomorphism on the n-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${n-1}$$\\end{document}-dimensional sphere. In particular, the holonomy group of the n-dimensional standard Funk metric and the Bryant–Shen metrics are maximal and isomorphic to Diffo(Sn-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {D}}i\\!f \\hspace{-3pt} f_o({\\mathbb {S}}^{n-1})$$\\end{document}. These results are the firsts describing explicitly the holonomy group of n-dimensional Finsler manifolds in the non-Berwaldian (that is when the canonical connection is non-linear) case.

Tame quivers and affine bases II: Nonsimply-laced cases

In [21], we give a Ringel-Hall algebra approach to the canonical bases in the symmetric affine cases. In this paper, we extend the results to general symmetrizable affine cases by using Ringel-Hall algebras of representations of a valued quiver. We obtain a bar-invariant basis B′={C(c,tλ)|(c,tλ)∈Ga} in the generic composition algebra C⁎ and prove that B′=B′⊔(−B′) coincides with Lusztig's signed canonical basis B. Moreover, in type B˜n,C˜n, B′ is the canonical basis B.

Profitable environmental corporate social responsibility under managers’ relative profit performance competition

AbstractWhen managers face relative profit performance competition, we formulate a green managerial delegation contract where the owners impose profit‐oriented environmental corporate social responsibility (ECSR) on their managers. We show that the owner adopts ECSR as a commitment device to reduce outputs under quantity competition if the degree of relative profit performance competition is sufficiently high, which can increase not only industry profits but also environmental quality. We also examine an endogenous choice of ECSR and find that the profitable level of ECSR in the asymmetric ECSR case is higher than that in the symmetric ECSR case while both firms undertake ECSR in equilibrium if the severity of competition is sufficiently high.

A REMARK ON THE N-INVARIANT GEOMETRY OF BOUNDED HOMOGENEOUS DOMAINS

Abstract Let $\mathbf {D}$ be a bounded homogeneous domain in ${\mathbb {C}}^n$ . In this note, we give a characterization of the Stein domains in $\mathbf {D}$ which are invariant under a maximal unipotent subgroup N of $Aut(\mathbf {D})$ . We also exhibit an N-invariant potential of the Bergman metric of $\mathbf {D}$ , expressed in a Lie theoretical fashion. These results extend the ones previously obtained by the authors in the symmetric case.

Parsing skin effect in a non-Hermitian spinless BHZ-like model

This work comprehensively investigates the non-Hermitian skin effect (NHSE) in a spinless Bernevig-Hughes-Zhang -like model in one dimension. It is generally believed that a system with non-reciprocal hopping amplitudes demonstrates NHSE. However, we show that there are exceptions, and more in-depth analyses are required to decode the presence of NHSE or its variants in a system. The fascinating aspects of our findings, depending on the inclusion of non-reciprocity in the inter-orbital hopping terms, concede the existence of conventional NHSE or NHSE at both edges and even a surprising absence of NHSE. The topological properties and the (bi-orthogonal) bulk-boundary correspondence, enumerated via computation of the (complex) Berry phase and spatial localization of the edge modes, highlight the topological phase transitions occurring therein. Further, to facilitate a structured discussion of the non-Hermitian model, we split the results into PT symmetric and non- PT symmetric cases with a view to comparing the two.

Interior solution of azimuthally symmetric case of Laplace equation in orthogonal similar oblate spheroidal coordinates

Curvilinear coordinate systems distinct from the rectangular Cartesian coordinate system are particularly valuable in the field calculations as they facilitate the expression of boundary conditions of differential equations in a reasonably simple way when the coordinate surfaces fit the physical boundaries of the problem. The recently finalized orthogonal similar oblate spheroidal (SOS) coordinate system can be particularly useful for a physical processes description inside or in the vicinity of the bodies or particles with the geometry of an oblate spheroid. The solution of the azimuthally symmetric case of the Laplace equation was found for the interior space in the orthogonal SOS coordinates. In the frame of the derivation of the harmonic functions, the Laplace equation was separated by a special separation procedure. A generalized Legendre equation was introduced as the equation for the angular part of the separated Laplace equation. The harmonic functions were determined as relations involving generalized Legendre functions of the first and of the second kind. Several lower-degree functions are reported. Recursion formula facilitating determination of the higher-degree harmonic functions was found. The general solution of the azimuthally symmetric Laplace equation for the interior space in the SOS coordinates is reported.

Exact closed forms for the transmittance of electromagnetic waves in one-dimensional anisotropic periodic media

In this work, we obtain closed expressions for the transfer matrix and the transmittance of electromagnetic waves propagating in finite one-dimensional anisotropic periodic stratified media with an arbitrary number of cells. By invoking the Cayley–Hamilton theorem on the transfer matrix for the electromagnetic field in a periodic stratified media formed by N cells, we obtain a fourth-order recursive relation for the matrix coefficients that defines the so-called Tetranacci polynomials (TPs). In the symmetric case, corresponding to a unit-cell transfer matrix with a characteristic polynomial where the coefficients of the linear and cubic terms are equal, closed expressions for the solutions to the recursive relation, known as symmetric TPs, have recently been derived, allowing us to write the transfer matrix and transmittance in a closed form. We show as sufficient conditions that the 4×4 differential propagation matrix of each layer in the binary unit cell, Δ , a) has eigenvalues of the form ±p1 , ±p2 , with p1≠p2 , and b) its off-diagonal 2×2 block matrices possess the same symmetric structure in both layers. Otherwise, the recursive relations are still solvable for any 4×4 matrix and provide an algorithm to compute the Nth power of the transfer matrix without carrying out explicitly the matrix multiplication of N matrices. We obtain analytical expressions for the dispersion relation and transmittance, in closed form, for two finite periodic systems: the first one consists of two birefringent uniaxial media with their optical axis perpendicular to the z-axis, and the second consists of two isotropic media subject to an external magnetic field oriented along the z-axis and exhibiting the Faraday effect. Our formalism applies also to lossy media, magnetic anisotropy or optical activity.

Dark energy stars in the context of black holes — The physical profiles

The dark energy star is a hypothetical model proposed as an alternative to address problems related to the structure of black holes (BHs), such as the presence of singularities. While dark energy star models have been previously explored, their application to BHs remains unexplored. This paper aims to investigate the concept of dark energy stars and compare their properties to BHs. The primary objective is to explore the physical profiles of dark energy stars and evaluate their similarity to the physical properties of BHs described by the Schwarzschild solution. To achieve this, specific properties of the dark energy star models need to be satisfied in the context of BHs, and their physical profiles are studied. The metric function [Formula: see text] proposed by M. R. Finch and J. E. F. Skea [Class. Quantum Grav. 6, 467 (1989)], as adopted by A. Banerjee, M. K. Jasim and A. Pradhan [Mod. Phys. Lett. A 35, 2050071 (2020), arXiv:1911.09546 [gr-qc]], is used and parametrized, making it close to BH spacetimes. The findings show that the model exhibits properties similar to BHs in terms of the stellar radius, compactness, surface redshift, and nature of gravity. Specifically, the dark energy star model behaves like BHs with a dark energy parameter [Formula: see text], satisfying all energy conditions. However, it should be noted that the investigation is limited to static spherically symmetric cases and further studies are required to explore the model in rotating cases. Overall, this study sheds light on the potential of dark energy star models in explaining BH properties and presents promising avenues for further research in understanding the nature of BHs and dark energy.

Quest for the golden ratio universality class.

Using mode-coupling theory, the conditions for all allowed dynamical universality classes for the conserved modes in one-dimensional driven systems are presented in closed form as a function of the stationary currents and their derivatives. With an eye on the search for the golden ratio universality class, the existence of some families of microscopic models is ruled out a priori by using an Onsager-type macroscopic current symmetry. In particular, if the currents are symmetric or antisymmetric under the interchange of the conserved densities, then at equal mean densities the golden modes can only appear in the antisymmetric case and if the conserved quantities are correlated, but not in the symmetric case where at equal densities one mode is always diffusive and the second may be either Kardar-Parisi-Zhang (KPZ), modified KPZ, 3/2-Lévy, or also diffusive. We also show that the predictions of mode-coupling theory for a noisy chain of harmonic oscillators are exact.

On the large solutions to a class of k-Hessian problems

Abstract In this paper, we consider the k-Hessian problem S k (D 2 u) = b(x)f(u) in Ω, u = +∞ on ∂Ω, where Ω is a C ∞-smooth bounded strictly (k − 1)-convex domain in R N ${\mathbb{R}}^{N}$ with N ≥ 2, b ∈ C∞(Ω) is positive in Ω and may be singular or vanish on ∂Ω, f ∈ C[0, ∞) ∩ C 1(0, ∞) (or f ∈ C 1 ( R ) $f\in {C}^{1}\left(\mathbb{R}\right)$ ) is a positive and increasing function. We establish the first expansions (equalities) of k-convex solutions to the above problem when f is borderline regularly varying and Γ-varying at infinity respectively. For the former, we reveal the exact influences of some indexes of f and principal curvatures of ∂Ω on the first expansion of solutions. For the latter, we find the principal curvatures of ∂Ω have no influences on the expansions. Our results and methods are quite different from the existing ones (including k = N). Moreover, we know the existence of k-convex solutions to the above problem (including k = N) is still an open problem when b possesses high singularity on ∂Ω and f satisfies Keller–Osserman type condition. For the radially symmetric case in the ball, we give a positive answer to this open problem, and then we further show the global estimates for all radial large solutions.

Coalescence-Induced Self-Propelled Particle Transport with Asymmetry Arrangement.

We experimentally investigated the coalescence-induced droplet-particle jumping phenomenon on a submillimeter scale in symmetric and asymmetric particle arrangements with poly(methyl methacrylate) (PMMA) particles and stainless steel (SS) particles. Coalescence-induced droplet-particle jumping exhibited excellent capability and interesting behavior for both droplet jumping enhancement and particle transport. The particle increased the normalized droplet jumping velocity from 0.250 for no particle case to 0.315 and 0.320 for symmetric and asymmetric particle cases. Compared with similar-sized macrostructures fixed between droplets, better jumping performance with particles may be attributed to avoiding the work of adhesion during droplet-macrostructure separation. Besides, all particles always sunk at the bottom in the symmetric cases, while the stick mode for PMMA particles and sink, wander, and jet modes for SS particles appeared in the asymmetry cases. We revealed that the asymmetric particle arrangement induces an unbalanced surface tension force, which may provide a driving force in the vertical direction. Additionally, a small enough resistive force caused by hydrophobic particles is another necessary condition for the wonder and jet mode. Finally, we realized a significantly superior particle transport in the asymmetric SS particle cases with maximum particle height reaching ∼2.1 mm, ∼12.4 times the particle radius, the most significant vertical self-propelled transport distance currently.

On the study of the pitch angular offset effects at various flapping frequencies for a two-dimensional asymmetric flapping airfoil in forward flight

This article explores the aerodynamic performance of a two-dimensional elliptical airfoil undergoing sinusoidal heaving and asymmetric pitching motions in forward-flight conditions for the Strouhal number (St) range of 0.1–0.6. The study employs numerical simulations and water tunnel experiments to investigate the effects of non-zero pitch angular offset angles (θoffset) while maintaining a fixed Reynolds number of 5000 and an effective angle of attack amplitude of 15° at the pivot location. The θoffset is varied from −15° to +15° at 5° intervals. The present research shows that these parameters significantly impact the leading-edge effective angle of attack, flow velocity, and the formation of high-pressure regions, which are crucial factors in thrust and lift generation throughout the flapping cycle. Moreover, the pitch angle determines whether the resultant force favors thrust or lift. It is observed that the cyclic time-averaged lift consistently increases with θoffset, surpassing symmetric cases (θoffset = 0°). Conversely, the cyclic time-averaged thrust is lower for non-zero θoffset values. Increasing St enhances both cyclic time-averaged thrust and lift up to the respective critical Sts, after which their performance declines. Notably, the critical St of cyclic time-averaged lift exceeds that of cyclic time-averaged thrust; interestingly, their values are invariant with θoffset. Moreover, in the conditions where thrust efficiency maximizes, lift efficiency attains a minimum value and vice versa. So, depending upon the application, one needs to suitably select the pitch angular offset and flapping frequency to maximize thrust or lift performance.

Models of Protocells Undergoing Asymmetrical Division.

The conditions that allow for the sustained growth of a protocell population are investigated in the case of asymmetrical division. The results are compared to those of previous studies concerning models of symmetrical division, where synchronization (between duplication of the genetic material and fission of the lipid container) was found under a variety of different assumptions about the kinetic equations and about the place where molecular replication takes place. Such synchronization allows a sustained proliferation of the protocell population. In the asymmetrical case, there can be no true synchronization, since the time to duplication may depend upon the initial size, but we introduce a notion of homogeneous growth that actually allows for the sustained reproduction of a population of protocells. We first analyze Surface Reaction Models, defined in the text, and we show that in many cases they undergo homogeneous growth under the same kinetic laws that lead to synchronization in the symmetrical case. This is the case also for Internal Reaction Models (IRMs), which, however, require a deeper understanding of what homogeneous growth actually means, as discussed below.

Dynamic Analysis of a Simple Cournot Duopoly Model Based on a Computed Cost

The paper is organized to study some mathematical properties and dynamics of a simple Cournot duopoly game based on a computed quadratic cost. The time evolution of this game is described by a two-dimensional noninvertible discrete time map using the bounded rationality mechanism. For this map, some dynamic characteristics such as multistability and synchronization are investigated. Its equilibrium points are obtained for the asymmetric case, and their conditions of stability are obtained. Our results investigate that the Nash equilibrium point may be unstable due to flip bifurcation and under certain parameter values, and Neimark–Sacker bifurcation is born after the period-4 cycle. Through some restrictions, the coordinate axes of the map construct an invariant manifold, and therefore, their dynamics can be analyzed by using a one-dimensional map. In the symmetric case, both firms behave identically, and this implies that the diagonal set forms an invariant manifold, and hence the synchronization phenomena take place. Furthermore, the global bifurcation of the map is confirmed through contact between critical curves and the boundaries of infeasible domains.

Dynamics of a bubble-pair between two parallel rigid walls

The dynamics of two gas bubbles placed side-by-side in liquid between parallel rigid plates is studied numerically using the multiphase volume of fluid (VOF) model in OpenFOAM. This model solves a single set of Navier–Stokes equations for both the gas in the bubbles and the surrounding liquid. Being initially stationary, the bubbles first expand to their maximum volume due to the high gas pressure inside them and then collapse. Both phases are treated as viscous and immiscible without any phase change. The model is validated against available experimental data of a single bubble oscillation between parallel plates and a bubble-pair oscillation near a single rigid wall. Parametric studies are then performed, with the main variables being the initial sizes of the bubbles and their standoff distances from the rigid plates. The bubbles exhibit symmetric or asymmetric behavior depending on these two variables. Three distinct scenarios are identified in the symmetric case based on the relative influence of bubble-bubble and bubble-wall interactions. The jet strength and direction, bubble oscillation period, coalescence, and early asymmetric bubble collapse are characterized by initial distances and radii. This study provides valuable insights into intricate hydrodynamic interactions and behaviors of bubble pairs bounded by rigid surfaces.

Exact fluctuation and long-range correlations in a single-file model under resetting.

Resetting is a renewal mechanism in which a process is intermittently repeated after a random or fixed time. This simple act of stop and repeat profoundly influences the behavior of a system as exemplified by the emergence of nonequilibrium properties and expedition of search processes. Herein we explore the ramifications of stochastic resetting in the context of a single-file system called random average process (RAP) in one dimension. In particular, we focus on the dynamics of tracer particles and analytically compute the variance, equal time correlation, autocorrelation, and unequal time correlation between the positions of different tracer particles. Our study unveils that resetting gives rise to rather different behaviors depending on whether the particles move symmetrically or asymmetrically. For the asymmetric case, the system for instance exhibits a long-range correlation which is not seen in absence of the resetting. Similarly, in contrast to the reset-free RAP, the variance shows distinct scalings for symmetric and asymmetric cases. While for the symmetric case, it decays (towards its steady value) as ∼e^{-rt}/sqrt[t], we find ∼te^{-rt} decay for the asymmetric case (r being the resetting rate). Finally, we examine the autocorrelation and unequal time correlation in the steady state and demonstrate that they obey interesting scaling forms at late times. All our analytical results are substantiated by extensive numerical simulations.