Emission of Cherenkov radiation as a mechanism for Hamiltonian friction

We study the motion of a heavy tracer particle weakly coupled to a dense, weakly interacting Bose gas exhibiting Bose–Einstein condensation. In the so-called mean-field limit, the dynamics of this system approaches one determined by nonlinear Hamiltonian evolution equations. We prove that if the initial speed of the tracer particle is above the speed of sound in the Bose gas, and for a suitable class of initial states of the Bose gas, the particle decelerates due to emission of Cherenkov radiation of sound waves, and its motion approaches a uniform motion at the speed of sound, as time t tends to ∞.     

Obtaining equations of motion for charged particles in the (<Emphasis Type="Italic">v/c</Emphasis>)<Superscript>3</Superscript>-approximation by the Einstein-Infeld-Hoffmann method

We consider some principal methodological problems that appear when the Einstein-Infeld-Hoffmann method is used to find approximate solutions of the general relativity equations and to obtain information about the motion of particles whose interaction force is much greater than the gravitational attraction force. Among these problems are normalizing approximate expressions by expanding exact solutions written in the same coordinate conditions used in the Einstein-Infeld-Hoffmann method, assigning the smallness orders depending on relations between the smallness parameters in play, and verifying cancellations of divergent terms arising in surface integrals. Solving these questions in accordance with the internal logic of the Einstein-Infeld-Hoffmann method results in new tools and techniques for applying the method. We demonstrate these tools and techniques in the example of the problem of the motion of two electrically charged pointlike particles in the (v/c)³-approximation.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 1, pp. 160–176, January, 2005.     

R2中非局部椭圆型方程基态解的存在性

本文考虑了一类非局部椭圆型方程-△u+V(x)u=(1/|x|μ*Q(x)F(u)/|x|β)Q(x)f(u)|x|β,x∈R^x,其中V是正的连续位势函数,0<μ<2,0≤β<1/2,2β+μ≤2,F(s)是f(s)的原函数.假设非线性项f(s)满足Trudinger-Moser型次临界指数增长,利用变分方法证明了该方程基态解的存在性.     

On a Strong Solution in the Method of Spherical Harmonics for a Nonstationary Transport Equation

We prove existence of a strong generalized solution in the Sobolev space W ₂ ¹ to the nonstationary problem for the system of the method of spherical harmonics (MSH) corresponding to the radiation transport problem.     

Preservation of Oscillations in the Runge-Kutta Method for a Type of Advanced Differential Equation

This article is concerned with the preservation of oscillations for differential equations with piecewise constant arguments of advanced type. By using the Runge-Kutta method, new oscillation conditions for numerical solution are established. We prove that oscillations of the analytic solution are preserved by the numerical solution in the Runge-Kutta method under some conditions. Some experiments are given.     

Integral Equation for the Spinor Amplitude of a Dirac Particle in a Curved Space–Time

We obtain a system of integral equations for the spinor amplitude of a wave packet describing a massive neutral Dirac particle in a curved space–time with an arbitrary geometry. This equation permits describing the spin dynamics of fermions in gravitational fields adequately to the quantum nature of spin. We consider a specific example of the Kerr–Schild metric. We also discuss the problem of massive neutrino oscillations in an external gravitational field.     

A model and simulation for homeorhesis in the motion of a single individual

In contrast to nonliving systems, all living systems perform homeorhesis. The system state tends to the so-called necessary path, or creode, when the exogenous signals are in a certain system-relevant range. The present work develops the homeorhesis-aware dynamical model for the motion of a single individual (e.g., human). The model allows for the purposeful behaviour of the individual, the creode, the exogenous forces, and the individual-specific sensitivity to their influences. The model also describes the homeorhetic-dysfunction movements. The transparency of the model is such that it allows a physical analogue in the form of electronic circuits. The model is a first step towards the construction of sociologically relevant models for the prediction of human behaviour. It is indispensable for the analyses of dangerous scenarios where the experiments are impossible, for example when predicting the behaviour of panic-stricken crowds in life-threatening situations. The work illustrates the corresponding numerical-simulation results with a series of figures and suggests topics for future research.     

抽象空间中一类带奇异性的混合型积分-微分方程边值问题的正解

利用不动点指数理论,研究了Banach空间中一类带奇性的混合型积分——微分方程边值问题正解的存在性,得到了多个正解存在的充分条件,并给出了相应的例子以说明所得结果的合理性和应用性.     

具有转向点超曲面的奇摄动椭圆型方程边值问题

本文利用多重尺度法和比较定理，研究了n维空间中一类具有转向点超曲面的奇摄动椭圆方程边值问题，研究了该值问题解的渐近性态。     

On Explicit Formulas for the Number of Solutions to the Equation (x1+...+xn)2 =ax1...xn in a Finite Field

We consider the equation of the title in a finite field of q elements. Assuming certain relations between n and q, we obtain explicit formulas for the number of solutions to this equation.     

A boundary value problem for a PDE model in mass transfer theory: Representation of solutions and applications

The system of partial differential equations