A condition on the chiral symmetry breaking solution of the Dyson–Schwinger equation in three-dimensional QED

In three-dimensional QED, which is analyzed in the 1/N expansion, we obtain a sufficient and necessary condition for a nontrivial solution of the Dyson–Schwinger equation to be chiral symmetry breaking solution. In the derivation, a normalization condition of the Goldstone bound state is used. It is showed that the existent analytical solutions satisfy this condition.     

Geometric scaling behavior of the scattering amplitude for DIS with nuclei

The main question, that we answer in this paper, is whether the initial condition can influence on the geometric scaling behavior of the amplitude for DIS at high energy. We re-write the non-linear Balitsky-Kovchegov equation in the form which is useful for treating the interaction with nuclei. Using the simplified BFKL kernel, we find the analytical solution to this equation with the initial condition given by the McLerran-Venugopalan formula. This solution does not show the geometric scaling behavior of the amplitude deeply in the saturation region. On the other hand, the BFKL Pomeron calculus with the initial condition at xA=1/mRA given by the solution to Balitsky-Kovchegov equation, leads to the geometric scaling behavior. The McLerran-Venugopalan formula is the natural initial condition for the Color Glass Condensate (CGC) approach. Therefore, our result gives a possibility to check experimentally which approach: CGC or BFKL Pomeron calculus, is more satisfactory.     

An iterative solution of the rough-surface scattering problem

Abstract

An iterative solution to the problem of scattering from a one-dimensional rough surface is obtained for the Dirichlet boundary condition. The advantages of this method are that bounds for convergence of the solution can be established and that the solution may readily be iterated to sufficiently high order in the interaction to examine the rate at which it converges. Absolute convergence of the iterative solution is also a sufficient condition for the convergence of the operator expansion method for surfaces on which the slope is everywhere less than unity. A numerical example of scattering from an echelette grating is considered, and bounds for convergence established. It is found that for scattering from such surfaces the rate at which the iterative solution converges decreases as the surface slope is increased. Corresponding results are found for the operator expansion method.     

Mean spherical approximation for a model liquid metal potential

The solution of the Ornstein-Zernike equation for a direct correlation function c(x) with damped oscillations and a hard core condition imposed upon the total correlation function h(x) has been proposed by Cummings as a means of treating a simple model potential for liquid metals in the mean spherical approximation [1]. Here some numerical results are given for this model and their significance is discussed. The solution of the Ornstein-Zernike equation is also extended; the hard core condition is generalized to a soft core condition, and Yukawa terms are added to the oscillatory c(x). Ways in which these extensions can be incorporated into more accurate liquid metal models, as well as into more accurate approximations for these models, are discussed. Finally, it is shown that our solution of the Ornstein-Zernike equation, after a change in the core condition, yields the structure of a spin glass model considered by Høye and Stell in the MSA-like approximation they propose [22].     

Effect of finite temperature on compositeness condition

The effect of finite temperature on compositeness condition in the Yukawa theory is investigated in 2 and 3 space-time dimensions. It is shown that this condition remains satisfied at all temperatures in these dimensions. These findings are contrary to the results obtained by Kikuchi and Takahashi. According to them the compositeness condition has a solution at all temperatures in 2 dimensions while in 3 dimensions it has a solution at low temperature and there is no solution at high temperature.     

Effects of condition number on preconditioning for low Mach number flows

The effects of the condition number on convergence characteristics and solution quality for the preconditioned Navier–Stokes equations are studied. A general approach to the construction of preconditioning parameters is proposed to account for the effects of the condition number on these parameters. To verify this technique, laminar flows past a circular cylinder at Reynolds numbers of 20 and 40, and laminar flows past a NACA0012 airfoil at Reynolds numbers of 2500 and 5000 are solved. It is shown that the condition number has effects on the convergence characteristics and solution qualities, and also that a condition number exists that optimizes the convergence characteristics and solution quality.     

An elastodynamic solution of finite long orthotropic hollow cylinder under torsion impact

The paper presents an analytical method to solve the elastodynamic problem of a finite-length orthotropic hollow cylinder subjected to a torsion impact often occurring in engineering fields. The elastodynamic solution is composed of a quasi-static solution of homogeneous equation satisfied with the non-homogeneous boundary condition and a dynamic solution of non-homogeneous equation satisfied with homogeneous boundary condition. The quasi-static solution can be obtained by directly solving the quasi-static equation satisfied with the non-homogeneous boundary condition. The solution of a non-homogeneous dynamic equation is obtained by means of a finite Hankel transform to a radial variable r, Laplace transform to a time variable t and finite Fourier transform to an axial variable z. Thus, the elastodynamic solution of the finite length of an orthotropic hollow cylinder subjected to a torsion impact is obtained. On the other hand, a dynamic finite element for the same problem is also carried out by applying the ANSYS finite-element analysis system. Comparing the theoretical solution with finite-element solution, it can be found that two kinds of results obtained by making use of two different solving methods are suitably approached. Therefore, it is further concluded that the methods and computing processes of the theoretical solution are effective and accurate.     

On the steady motions of a flat domain wall in a ferromagnet

A new derivation is presented of Walker's exact solution to Gilbert equation, a solution which mimicks the travelling-wave motion of a flat domain wall at 180^°. It is shown that a process during which the working of the applied magnetic field exactly compensates dissipation (the Walker condition) exists both under the constitutive circumstances considered in the standard Gilbert equation and when either the internal free-energy or the dissipation, or both, are generalized by the introduction of higher-gradient terms; but that such a process cannot solve the generalized Gilbert equation. It is also shown that, when dry-friction dissipation is considered and a suitable magnetic field is applied, the associated Gilbert equation has a Walker-type solution mimicking a flat wall, at 90^° this time, which however does not satisfy the Walker condition. Received 16 November 2001     

A new class of spherically symmetric interior solution with cosmological constant Λ

In this paper, a new class of spherically symmetric interior solution with cosmological constant is obtained, to which Florides' solution is extended. As a special case, the metric in uniform proper density is discussed. The Robson junction condition is also extended to the case with nonvanishing, which is a set of necessary and sufficient conditions for a spherically symmetric metric to join smoothly onto the vacuum metric at a nonnull boundary surface. The new solution satisfies the extended Robson condition.     

Solution of the thermoelastoplastic problem for a thin disk of plastically compressible material subjected to thermal loading

Elastoplastic solutions for thin plates and disks are sensitive to loading and plasticity conditions [1–5]. The plasticity condition for a number of metal materials depends on the mean stress [6–8]. In this case, when using the associated flow rule, plastic deformations do not satisfy the incompressibility condition, which is commonly accepted in statements of boundary-value problems for thin elastoplastic plates and disks [9–13]. It is of interest to determine the effect of plastic compressibility on the behavior of solutions for such structures. In this paper, a hollow disk in a rigid container subjected to a uniform temperature field is considered. The plasticity condition proposed in [14] is accepted. A general study of the set of equations including this plasticity condition and the associated flow rule was performed in [15]. The solution under the Mises plasticity condition was obtained in [1].     

Réalisabilité d'un schéma non linéaire de retour à l'isotropie de la corrélation pression-gradient d'un scalaire

The study of the equilibrium states of a homogeneous turbulence, in the absence of mean gradients, leads to an equilibrium solution with zero values for the turbulent scalar flows. Stability of this equilibrium solution is achieved by a simple condition on one of the coefficients of the model. A realisibility study of the turbulent scalar flow has been carried out an has led to a general realisibility condition imposed on the model coefficients. Moreover, it has been established that, if the return to isotropy is compatible with two supplementary constraints on the model coefficients, it is possible to substitute a sufficient yet much simpler condition for the general realisability condition. Finally, a numerical optimization on the basis of the experimental results and the direct simulation results, has proven that the proposed model ensures a better prediction of the scalar turbulence with respect to Rotta's model.     

Partial Exactness for the Penalty Function of Biconvex Programming

Biconvex programming (or inequality constrained biconvex optimization) is an important model in solving many engineering optimization problems in areas like machine learning and signal and information processing. In this paper, the partial exactness of the partial optimum for the penalty function of biconvex programming is studied. The penalty function is partially exact if the partial Karush–Kuhn–Tucker (KKT) condition is true. The sufficient and necessary partially local stability condition used to determine whether the penalty function is partially exact for a partial optimum solution is also proven. Based on the penalty function, an algorithm is presented for finding a partial optimum solution to an inequality constrained biconvex optimization, and its convergence is proven under some conditions.     

An efficient method for solving highly anisotropic elliptic equations

Solving elliptic PDEs in more than one dimension can be a computationally expensive task. For some applications characterized by a high degree of anisotropy in the coefficients of the elliptic operator, such that the term with the highest derivative in one direction is much larger than the terms in the remaining directions, the discretized elliptic operator often has a very large condition number – taking the solution even further out of reach using traditional methods. This paper will demonstrate a solution method for such ill-behaved problems. The high condition number of the D-dimensional discretized elliptic operator will be exploited to split the problem into a series of well-behaved one and (D − 1)-dimensional elliptic problems. This solution technique can be used alone on sufficiently coarse grids, or in conjunction with standard iterative methods, such as Conjugate Gradient, to substantially reduce the number of iterations needed to solve the problem to a specified accuracy. The solution is formulated analytically for a generic anisotropic problem using arbitrary coordinates, hopefully bringing this method into the scope of a wide variety of applications.     

An anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems

Heterogeneous anisotropic diffusion problems arise in the various areas of science and engineering including plasma physics, petroleum engineering, and image processing. Standard numerical methods can produce spurious oscillations when they are used to solve those problems. A common approach to avoid this difficulty is to design a proper numerical scheme and/or a proper mesh so that the numerical solution validates the discrete counterpart (DMP) of the maximum principle satisfied by the continuous solution. A well known mesh condition for the DMP satisfaction by the linear finite element solution of isotropic diffusion problems is the non-obtuse angle condition that requires the dihedral angles of mesh elements to be non-obtuse. In this paper, a generalization of the condition, the so-called anisotropic non-obtuse angle condition, is developed for the finite element solution of heterogeneous anisotropic diffusion problems. The new condition is essentially the same as the existing one except that the dihedral angles are now measured in a metric depending on the diffusion matrix of the underlying problem. Several variants of the new condition are obtained. Based on one of them, two metric tensors for use in anisotropic mesh generation are developed to account for DMP satisfaction and the combination of DMP satisfaction and mesh adaptivity. Numerical examples are given to demonstrate the features of the linear finite element method for anisotropic meshes generated with the metric tensors.     

On the applicability of a spherical basis for spheroidal layered scatterers

The applicability of an analog of the extended boundary condition method, which is popular in light-scattering theory, is studied in combination with the standard spherical basis for the solution of an electrostatic problem appearing for spheroidal layered scatterers the sizes of which are small as compared to the incident radiation wavelength. In the case of two or more layers, polarizability and other optical characteristics of particles in the far zone are shown to be undeterminable if the condition under which the appearing systems of linear equations for expansion coefficients of unknown fields are Fredholm systems solvable by the reduction method is broken. For two-layer spheroids with confocal boundaries, this condition is transformed into a simple restriction on the ratio of particle semiaxes a/b< $\sqrt 2 $ + 1. In the case of homogeneous particles, the solvability condition is that the radius of convergence of the internal-field expansion must exceed that of the expansion of an analog of the scattering field. Since homogeneous spheroids (ellipsoids) are unique particles inside which the electrostatic field is homogeneous, it is shown that the solution can be always found in this case. The obtained results make it possible to match in principle the results of theoretical and numerical determinations of the domain of applicability for the extended boundary condition method with a spherical basis for spheroidal scatterers.     

The FKG inequality for the Yukawa2 quantum field theory

We establish the FKG correlation inequality for the Euclidean scalar Yukawa₂ quantum field model and, when the Fermi mass is zero, for pseudoscalar Yukawa₂. To do so we approximate the quantum field model by a lattice spin system and show that the FKG inequality for this system follows from a positivity condition on the fundamental solution of the Euclidean Dirac equation with external field. We prove this positivity condition by applying the Vekua-Bers theory of generalized analytic functions.Research partially supported by the National Research Council of Canada.Alfred P. Sloan Foundation Fellow.     

关于干涉反演光谱技术能否扩大F-P干涉仪光谱范围的探讨

本文从物理学的角度提出了Fredholm第一类积分方程数值解的可靠性概念,证明了在被称为Fabry-Perot干涉反演光谱技术中,当△σ=2/x,△σ=2/σ时,若取样点数为一个适当的奇数,那么积分方程的数值解是稳定的.但是进一步的计算机模拟实验表明,该数值解不是原积分方程的可靠解,因此,干涉反演光谱技术不能扩大Fabry-Perot干涉仪的光谱范围.理论分析表明,在前述条件下,积分方程数值解不可靠的根本原因在于该积分方程本身没有唯一解.     

Pressure boundary conditions for computing incompressible flows with SPH

In Smoothed Particle Hydrodynamics (SPH) methods for fluid flow, incompressibility may be imposed by a projection method with an artificial homogeneous Neumann boundary condition for the pressure Poisson equation. This is often inconsistent with physical conditions at solid walls and inflow and outflow boundaries. For this reason open-boundary flows have rarely been computed using SPH. In this work, we demonstrate that the artificial pressure boundary condition produces a numerical boundary layer that compromises the solution near boundaries. We resolve this problem by utilizing a “rotational pressure-correction scheme” with a consistent pressure boundary condition that relates the normal pressure gradient to the local vorticity. We show that this scheme computes the pressure and velocity accurately near open boundaries and solid objects, and extends the scope of SPH simulation beyond the usual periodic boundary conditions.     

Initial-Boundary Value Problem for the Camassa�CHolm Equation with Linearizable Boundary Condition

We present a Riemann?CHilbert problem formalism for the initial boundary value problem for the Camassa?CHolm equation on the half-line x > 0 with homogeneous Dirichlet boundary condition at x = 0. We show that, similarly to the problem on the whole line, the solution of this problem can be obtained in parametric form via the solution of a Riemann?CHilbert problem determined by the initial data via associated spectral functions. This allows us to apply the non-linear steepest descent method and to describe the large-time asymptotics of the solution.     

Existence and Uniqueness of Stationary Solutions for 3D Navier–Stokes System with Small Random Forcing via Stochastic Cascades

We consider the 3D Navier–Stokes system in the Fourier space with regular forcing given by a stationary in time stochastic process satisfying a smallness condition. We explicitly construct a stationary solution of the system and prove a uniqueness theorem for this solution in the class of functions with Fourier transform majorized by a certain function h. Moreover we prove the following “one force—one solution” principle: the unique stationary solution at time t is presented as a functional of the realization of the forcing in the past up to t. Our explicit construction of the solution is based upon the stochastic cascade representation.