Equations of motion in fourth approximation

The equations of motion in fourth approximation for a system of massive bodies of finite size moving in the gravitational field of the system are obtained.     

The closure approximation in the hierarchy equations

The expectation of the solution process in a stochastic operator equation can be obtained from averaged equations only under very special circumstances. Conditions for validity are given and the significance and validity of the approximation in widely used hierarchy methods and the self-consistent field approximation in nonequilibrium statistical mechanics are clarified. The error at any level of the hierarchy can be given and can be avoided by the use of the iterative method.Supported by the National Aeronautics and Space Administration (Grant NGR 11-003-020) and partially supported by the Office of Naval Research (Contract N 00014-69-A-0423 Themis).     

Transport equations of amplitudes in the eikonal approximation of dynamical diffraction equations

An approach of the eikonal approximation of the dynamical diffraction equations of X-rays in deformed crystals, based on the second-order differential equations for the transmitted and diffracted waves, is presented. By analogy with usual optics, this approach allows one not only to obtain the eikonal equation and to study the behavior of the amplitude in zero-order approximation, which usually is performed in the eikonal dynamical diffraction theory, but also to establish for all orders of the amplitude asymptotic expansion the corresponding transport equations and to present their solutions as integrals over the amplitude propagation trajectory. Summarizing the transport equations, an equation for the total amplitude, analogous with the parabolic diffraction equation in optics, is obtained.     

Renormalization group equations in local approximation

For a class of models with Ginzburg-Landau-Wilson functions of a local form it is proved that the spectrum of a renormalization group operator which is linearized near a fixed point is discrete, real, and limited from above. In the framework of a local model, critical exponents for the limitn= are calculated.     

The geodesic approximation for the Yang-Mills-Higgs equations

In this paper we consider the dynamics of the monopole solutions of Yang-Mills-Higgs theory on Minkowski space. The monopoles are solutions of the Yang-Mills-Higgs equations on three dimensional Euclidean space. It is of interest to understand how they evolve in time when considered as solutions of the Yang-Mills-Higgs equations on Minkowski space-i.e. the time dependent equations. It was suggested by Manton that in certain situations the monopole dynamics could be understood in terms of geodesics with respect to a certain, metric on the space of guage equivalence classes of monopoles-the moduli space. The metric is defined by taking theL ² inner product of tangent vectors to this space. In this paper we will prove that Manton's approximation is indeed valid in the right circumstances, which correspond to the slow motion of monopoles. The metric on the moduli space of monopoles was analysed in a book by Atiyah and Hitchin, so together with the results of this paper a detailed and rigorous understanding of the low energy dynamics of monopoles in Yang-Mills-Higgs theory is obtained. The strategy of the proof is to develop asymptotic expansions using appropriate gauge conditions, and then to use energy estimates to prove their validity. For the case of monopoles to be considered here there is a technical obstacle to be overcome-when the equations are linearised about the monopole the continuous spectrum extends all the way to the origin. This is overcome by using a norm introduced by Taubes in a discussion of index, theory for the Yang-Mills-Higgs functional.Supported by grant DMS-9214067 from the National Science Foundation.     

Analytical approximation schemes for solving exact renormalization group equations in the local potential approximation

The relation between the Wilson–Polchinski and the Litim optimized ERGEs in the local potential approximation is studied with high accuracy using two different analytical approaches based on a field expansion: a recently proposed genuine analytical approximation scheme to two-point boundary value problems of ordinary differential equations, and a new one based on approximating the solution by generalized hypergeometric functions. A comparison with the numerical results obtained with the shooting method is made. A similar accuracy is reached in each case. Both two methods appear to be more efficient than the usual field expansions frequently used in the current studies of ERGEs (in particular for the Wilson–Polchinski case in the study of which they fail).     

Solutions of a fourth order Bethe Salpeter equation

A Bethe Salpeter equation with interaction up to the fourth order is solved numerically for a bound state with energy equal to zero. It is found that the fourth order term is important even for very heavy exchanged mesons, and for angular momentum equal to 1. The correction found is to be considered the highest limit for the corrections with energy different from zero.     

Complex temperature plane zeros in the mean-field approximation

We derive asymptotic expressions for the complex temperature plane zeros of the infinite-range Ising model in the scaling regime. The results also apply to high-dimensional, short-range Ising systems. For thenth zero in a system ofN spins, the leading asymptotic result ist _n (n/N)^1/2(–1 ±i).     

Solutions of the exact time-dependent microbending equations

Bandwidth improvement factor and losses of two-dimensional fibers with power-law index profiles, submitted to microbends with power-law curvature spectra, are given here in a graphical form. The domain of validity of the analytical solutions is discussed.     

Solutions of coupled KdV-type equations

We consider a family of coupled (Ito type) KdV equations in 1+1 dimensions and use Hlavatý's technique to obtain a class of explicit wave-type solutions.     

Solutions of gravitational field equations

In this paper we use iterative methods to generate series solutions of the gravitational field equations in a cosmological model with heat flow.     

Asymptotic approximation of the moving spike-type contrast structure in reaction-diffusion equations

The asymptotic approximation of a moving spike-type contrast structure for the reaction-diffusion equation is constructed. The results were applied for an important case of cubic nonlinearity.     

Analytical approximation of the thermal and caloric equations of state for real gases over a wide density and temperature range

The thermal and caloric equations of state of the main components of gas mixtures normally used in calculations of gas dynamic processes, including combustion and detonation, as well as in problems of internal ballistics, are presented in an analytical form. The formulas obtained for the equations of state contain a relatively small number of parameters and provide an average error of approximation of less than 1% at temperatures from 500 to 2000–2500 K and densities up to the critical.     

Solutions of the Einstein equations with zero coupling

A study is made of the algebraic properties and geometric structure of solutions of the Einstein equations the metric tensors of which differ by the product of two identical isotropic vectors. Proof is offered for a theorem which states that when the congruence of isotropic lines with a tangent vector field used for the coupling is geodesic, both spaces are algebraically special in the Petrov-Penrose sense. A noncoordinate transformation of this type can be used to find new exact solutions from known solutions.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii Fizika, No. 10, pp. 32–37, October, 1971.The author thanks Professor V. I. Rodichev for interest in the study and for valuable discussions.     

Derivation of a quasiclassical scattering-phase approximation from the phase equations

A new radial phase equation is obtained for particles in a spherically symmetric potential field; the equation allows the quasiclassical asymptotes of the scattering phases _l() to be investigated.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 64–69, August, 1978.     

Kernel approximation for solving few-body integral equations

This paper investigates an approximate method for solving integral equations that arise in few-body problems. The method is to replace the kernel by a degenerate kernel defined on a finite dimensional subspace of piecewise Lagrange polynomials. Numerical accuracy of the method is tested by solving the two-body Lippmann-Schwinger equation with non-separable potentials, and the three-body Amado-Lovelace equation with separable two-body potentials.     

Fokker-Planck approximation for N-dimensional nonmarkovian langevin equations

The Fokker-Planck approximation for n-dimensional nonmarkovian Langevin equations is discussed through an expansion in powers of the correlation time of the noise. Exact cases are considered and an application to brownian motion is presented.