Symmetric Quantities Research Articles

Multipole moment description of passivants and complex pseudopotentials for semiconductor surfaces

From a theoretical point of view, surface passivation of semiconductors can be handled with the inclusion of hydrogen-like complex pseudopotentials. In this work, we show that the two components of such potentials are not spherical and that they correspond to different multipole electric moments of the passivant atom. In particular, we find that the real part of the complex pseudopotential behaves like a quadrupole and has an angular dependence. We show that disregarding the multipole description may lead to the occurrence of surface-localized virtual states between the conduction and valence bands. The multipole moments are identified according to their dependence upon the distance in real space. The effectiveness of our complex-pseudopotential passivation is demonstrated by calculating the band-gap energy of quantum wires and quantum slabs along different orientations for different materials (GaAs, AlAs, CdSe, Ge, and Si). Our derivation constitutes an improvement to the pseudopotential method and a more physically accurate description of passivants, whose real component has been treated as a spherically symmetric quantity.

Intrinsic metrics in ring domains

Three hyperbolic-type metrics including the triangular ratio metric, the j^*-metric, and the Möbius metric are studied in an annular ring. The Euclidean midpoint rotation is introduced as a method to create upper and lower bounds for these metrics, and their sharp inequalities are found. A new Möbius-invariant lower bound is proved for the conformal capacity of a general ring domain by using a symmetric quantity defined with the Möbius metric.

Exact Solutions for Interaction of Parallel Screw Dislocations with a Wedge Crack in One-Dimensional Hexagonal Quasicrystal with Piezoelectric Effects

The purpose of this paper is to consider the interaction between many parallel dislocations and a wedge-shaped crack and their collective response to the external applied generalized stress in one-dimensional hexagonal piezoelectric quasicrystal, employing the complex variable function theory and the conformal transformation method; the problem for the crack is reduced to the solution of singular integral equations, which can be further reduced to solving Riemann–Hilbert boundary value problems. The analytical solutions of the generalized stress field are obtained. The dislocations are subjected to the phonon field line force, phason field line force, and line charge at the core. The positions of the dislocations are arbitrary, but the dislocation distribution is additive. The dislocation is not only subjected to the external stress and the internal stress generated by the crack, but also to the force exerted on it by other dislocations. The closed-form solutions are obtained for field intensity factors and the image force on a screw dislocation in the presence of a wedge-shaped crack and a collection of other dislocations. Numerical examples are provided to show the effects of wedge angle, dislocation position, dislocation distribution containing symmetric configurations and dislocation quantities on the field intensity factors, energy release rate, and image force acting on the dislocation. The principal new physical results obtained here are (1) the phonon stress, phason stress, and electric displacement singularity occur at the crack tip and dislocations cores, (2) the increasing number of dislocations always accelerates the crack propagation, (3) the effect of wedge angle on crack propagation is related to the distribution of dislocations, and (4) the results of the image force on the dislocation indicate that the dislocations can either be attracted or rejected and reach stable positions eventually.

Statistical Approaches for the Analysis of Dependency Among Neurons Under Noise.

Neuronal noise is a major factor affecting the communication between coupled neurons. In this work, we propose a statistical toolset to infer the coupling between two neurons under noise. We estimate these statistical dependencies from data which are generated by a coupled Hodgkin–Huxley (HH) model with additive noise. To infer the coupling using observation data, we employ copulas and information-theoretic quantities, such as the mutual information (MI) and the transfer entropy (TE). Copulas and MI between two variables are symmetric quantities, whereas TE is asymmetric. We demonstrate the performances of copulas and MI as functions of different noise levels and show that they are effective in the identification of the interactions due to coupling and noise. Moreover, we analyze the inference of TE values between neurons as a function of noise and conclude that TE is an effective tool for finding out the direction of coupling between neurons under the effects of noise.

Quantifying high-order interdependencies via multivariate extensions of the mutual information.

This paper introduces a model-agnostic approach to study statistical synergy, a form of emergence in which patterns at large scales are not traceable from lower scales. Our framework leverages various multivariate extensions of Shannon's mutual information, and introduces the O-information as a metric that is capable of characterizing synergy- and redundancy-dominated systems. The O-information is a symmetric quantity, and can assess intrinsic properties of a system without dividing its parts into "predictors" and "targets." We develop key analytical properties of the O-information, and study how it relates to other metrics of high-order interactions from the statistical mechanics and neuroscience literature. Finally, as a proof of concept, we present an exploration on the relevance of statistical synergy in Baroque music scores.

Computational homogenization of nematic liquid crystal elastomers based on Landau‐de‐Gennes theory

AbstractNematic liquid crystal elastomers represent a group of materials, which combine elastic properties of rubber with orientational properties of liquid crystals. The present contribution proposes a variational homogenization principle for nematic liquid crystal elastomers. A symmetric and traceless tensorial quantity represents the nematic order parameter, which is used as a phase field. The model is implemented in a finite element framework and the validation is done with two as well as three dimensional nemato‐mechanical boundary value problems.

Flux-corrected transport algorithms preserving the eigenvalue range of symmetric tensor quantities

This paper presents a new approach to constraining the eigenvalue range of symmetric tensors in numerical advection schemes based on the flux-corrected transport (FCT) algorithm and a continuous finite element discretization. In the context of element-based FEM-FCT schemes for scalar conservation laws, the numerical solution is evolved using local extremum diminishing (LED) antidiffusive corrections of a low order approximation which is assumed to satisfy the relevant inequality constraints. The application of a limiter to antidiffusive element contributions guarantees that the corrected solution remains bounded by the local maxima and minima of the low order predictor.The FCT algorithm to be presented in this paper guarantees the LED property for the maximal and minimal eigenvalues of the transported tensor at the low order evolution step. At the antidiffusive correction step, this property is preserved by limiting the antidiffusive element contributions to all components of the tensor in a synchronized manner. The definition of the element-based correction factors for FCT is based on perturbation bounds for auxiliary tensors which are constrained to be positive semidefinite to enforce the generalized LED condition. The derivation of sharp bounds involves calculating the roots of polynomials of degree up to 3. As inexpensive and numerically stable alternatives, limiting techniques based on appropriate estimates are considered. The ability of the new limiters to enforce local bounds for the eigenvalue range is confirmed by numerical results for 2D advection problems.

Local Error Analysis and Comparison of the Swept- and Intersection-Based Remapping Methods

AbstractIn this paper, the numerical error of two widely used methods for remapping of discrete quantities from one computational mesh to another is investigated. We compare the intuitive, but resource intensive method utilizing intersections of computational cells with the faster and simpler swept-region-based method. Both algorithms are formally second order accurate, however, they are known to produce slightly different quantity profiles in practical applications. The second-order estimate of the error formula is constructed algebraically for both algorithms so that their local accuracy can be evaluated. This general estimate is then used to assess the dependence of the performance of both methods on parameters such as the second derivatives of the remapped distribution, mesh geometry or mesh movement. Due to the complexity of such analysis, it is performed on a set of simplified elementary mesh patterns such as cell corner expansion, rotation or shear. On selected numerical tests it is demonstrated that the swept-based method can distort a symmetric quantity distribution more substantially than the intersection-based approach when the computational mesh moves in an unsuitable direction.

Cheap talk and cooperation in Stackelberg games

Previous literature on cheap talk suggests that it is used to increase cooperation. We study cheap talk and the effect of the leader’s private payoff information in new repeated Stackelberg game settings. Our results confirm earlier studies that the players cooperate in repeated Stackelberg games with complete payoff information. In the cheap talk setting the follower has the actual first mover advantage and should in theory benefit from it, but we find that many followers cooperate instead. Similarly, many leaders do not use cheap talk for cheating but commit to symmetric joint-optimum quantities. The leader’s private payoff information results in a low frequency of cooperation but in the presence of cheap talk players do cooperate.

A Recipe for the Estimation of Information Flow in a Dynamical System

Information-theoretic quantities, such as entropy and mutual information (MI), can be used to quantify the amount of information needed to describe a dataset or the information shared between two datasets. In the case of a dynamical system, the behavior of the relevant variables can be tightly coupled, such that information about one variable at a given instance in time may provide information about other variables at later instances in time. This is often viewed as a flow of information, and tracking such a flow can reveal relationships among the system variables. Since the MI is a symmetric quantity; an asymmetric quantity, called Transfer Entropy (TE), has been proposed to estimate the directionality of the coupling. However, accurate estimation of entropy-based measures is notoriously difficult. Every method has its own free tuning parameter(s) and there is no consensus on an optimal way of estimating the TE from a dataset. We propose a new methodology to estimate TE and apply a set of methods together as an accuracy cross-check to provide a reliable mathematical tool for any given data set. We demonstrate both the variability in TE estimation across techniques as well as the benefits of the proposed methodology to reliably estimate the directionality of coupling among variables.

Geometric algebra description of polarization mode dispersion, polarization-dependent loss, and Stokes tensor transformations.

This paper demonstrates that numerous calculations involving polarization transformations can be condensed by employing suitable geometric algebra formalism. For example, to describe polarization mode dispersion and polarization-dependent loss, both the material birefringence and differential loss enter as bivectors and can be combined into a single symmetric quantity. Their frequency and distance evolution, as well as that of the Stokes vector through an optical system, can then each be expressed as a single compact expression, in contrast to the corresponding Mueller matrix formulations. The intrinsic advantage of the geometric algebra framework is further demonstrated by presenting a simplified derivation of generalized Stokes parameters that include the electric field phase. This procedure simultaneously establishes the tensor transformation properties of these parameters.

Choosing price or quantity? The role of delegation and network externalities

We consider a differentiated duopoly and endogenise the firm choice of the strategy variable (price or quantity) to play on the product market in the presence of network externalities. We model this choice by assuming both competition between entrepreneurial (owner-managed) firms and competition between managerial firms in which market decisions are delegated from owners to revenue-concerned managers. While network externalities are shown not to alter the symmetric equilibrium quantity choice arising in the no-delegation case, sufficiently strong network effects allow us to eliminate the multiplicity of equilibria under delegation and lead to a unique equilibrium in which both firms choose price.

Discerning “Indistinguishable” Quantum Systems

In a series of recent papers, Simon Saunders, Fred Muller, and Michael Seevinck have collectively argued, against the folklore, that some nontrivial version of Leibniz’s principle of the identity of indiscernibles is upheld in quantum mechanics. They argue that all particles—fermions, paraparticles, anyons, even bosons—may be weakly discerned by some physical relation. Here I show that their arguments make illegitimate appeal to nonsymmetric, that is, permutation-noninvariant, quantities and that therefore their conclusions do not go through. However, I show that alternative, symmetric quantities may be found to do the required work. I conclude that the Saunders-Muller-Seevinck heterodoxy can be saved.

Competitiveness, cooperation, and strategic interaction. A classroom experiment on oligopoly

We run a classroom experiment on oligopoly with students enrolled on basic and medium level microeconomics courses. Students compete in a symmetric quantity setting environment. The experiment runs over an entire academic semester and is divided into 20 one-week rounds. We want to explore whether the effect of knowledge and social interaction between players modifies the cooperative and competitive behavior observed in similar experiments run in a lab. Our hypothesis is that players are socially influenced. Hence, individuals adjust behavior in a dynamic way aimed at maximizing profits, but also according to social pressures. Overall, we obtain different learning processes across academic levels and also slightly different behavior from that predicted by economic theory. We argue that students’ utility function depends not only on profit levels but also on social relationships. Moreover, we believe that the effect of reputation plays an important role in our framework.

Symmetry between partially polarised light and partial polarisers in the vectorial Pauli algebraic formalism

The problem of the gain of dichroic devices is analysed in a framework based on a vectorial pure operatorial (non-matrix) Pauli algebraic approach to polarisation optics. We show that the partially polarised light and the partial polarisers can be described in this framework by absolutely similar, symmetric quantities. In this sense, a physically essential device parameter, the degree of dichroism, is defined. This symmetry between the description of the polarised light and of the polarisation devices leads to an expressive form of the generalised Malus’ law, the consequences of which are analysed in detail. Most importantly, one can for the first time describe and graphically illustrate the generalised structure of the gain of a dichroic device.

Non-adiabatic perturbations in multi-component perfect fluids

The evolution of non-adiabatic perturbations in models with multiple coupled perfect fluids with non-adiabatic sound speed is considered. Instead of splitting the entropy perturbation into relative and intrinsic parts, we introduce a set of symmetric quantities, which also govern the non-adiabatic pressure perturbation in models with energy transfer. We write the gauge invariant equations for the variables that determine on a large scale the non-adiabatic pressure perturbation and the rate of changes of the comoving curvature perturbation. The analysis of evolution of the non-adiabatic pressure perturbation has been made for several particular models.

Minimal renormalization without ϵ-expansion: Three-loop amplitude functions of the 0( n) symmetric Φ4 theory in three dimensions below Tc

We present an analytic three-loop calculation for thermodynamic quantities of the O(n) symmetric Φ 4 theory below T c within the minimal subtraction scheme at fixed dimension d = 3. Goldstone singularities arising at an intermediate stage in the calculation of O( n) symmetric quantities cancel among themselves leaving a finite result in the limit of zero external field. From the free energy we calculate the three-loop terms of the amplitude functions ƒ Φ, F + and F − of the order parameter and the specific heat above and below Tc, respectively, without using the e = 4- d expansion. A Borel resummation for the case n = 2 yields resummed amplitude functions f Φ and F − that are slightly larger than the one-loop results. Accurate knowledge of these functions is needed for testing the renormalization-group prediction of critical-point universality along the λ −line of superfluid 4He. Combining the three-loop result for F − with a recent five-loop calculation of the additive renormalization constant of the specific heat yields excellent agreement between the calculated and measured universal amplitude ratio A +/ A - of the specific heat of 4He. In addition we use our result for f Φ to calculate the universal combination Rc of the amplitudes of the order parameter, the susceptibility and the specific heat for n = 2 and n = 3. Our Borel-resummed three-loop result for R c is significantly more accurate than the previous result obtained from the ϵ-expansion up to O( ϵ 2.

Cournot Competition Under Knightian Uncertainty

This article applies a theorem of Nash equilibrium under uncertainty (Dow & Werlang, 1994) to the classic Cournot model of oligopolistic competition. It shows, in particular, how one can map all Cournot-Nash equilibria (which includes the cartel and the null solutions) to only a function of the uncertainty aversion coefficients of the producers. The effects of these parameters on the symmetric equilibrium quantities and output are examined in a comparative statics analysis, under two alternative assumptions: a closed market with an exogenous number of firms and a free-entry/exit regime. In both cases, a collusive effect of the uncertainty aversion on the production is obtained. Under rather few restrictive assumptions, there is a symmetric uncertainty aversion level for the producers at which their optimal quantities and the industry output become equal to the optimal counterpart cartel's outcomes. These results improve upon the literature on collusion since, in contrast to other analogous findings, they enhance that a cooperative cartel may be endogenously generated in a one-shot (noncooperative) game played by uncertainty averse producers. For the competitive case (under free-entry/exit) the paper shows that Cournotian competition among weakely or moderately uncertainty averse producers entails a higher industry output (if the market is large and/or entry is easy) and surely entails lower optimal quantities for the firms than those achieved under uncertainty neutrality. Thus, competition under free-entry/exit in a Knightian uncertainty environment should act to prevent monopoly power for the individual firms.

Minimal renormalization without ϵ-expansion: Amplitude functions for O( n) symmetric systems in three dimensions below Tc

Massive field theory at fixed dimension d < 4 is combined with the minimal subtraction scheme to calculate the amplitude functions of thermodynamic quantities for the O( n) symmetric Φ 4 model below T c in two-loop order. Goldstone singularities arising at an intermediate stage in the calculation of O( n) symmetric quantities are shown to cancel among themselves leaving a finite result in the limit of zero external field. From the free energy we calculate the amplitude functions in zero field for the order parameter, specific heat and helicity modulus (superfluid density) in three dimensions. We also calculate the q 2 part of the inverse of the wavenumber-dependent transverse susceptibility χT( q) which provides an independent check of our result for the helicity modulus. The two-loop contributions to the superfluid density and specific heat below T c turn out to be comparable in magnitude to the one-loop contributions, indicating the necessity of higher-order calculations and Padé-Borel type resummations.

Plane symmetric higher dimensional space-times

In this paper the general form of plane symmetric line element and the plane symmetric geometric quantities in five dimensions have been discussed.