Quantum stationary states in the Bianchi universes

A path integral formulation is used in the superspace of geometries leading to a model of quantum gravity. It is shown that this method agrees with the standard path integral technique in the special case of Friedmann universes. A Schrödinger-like equation is set up, leading to the stationary states. Some solutions to these equations are presented.     

A note on moments of limit log-infinitely divisible stochastic measures of Bacry and Muzy

A multiple integral representation of single and joint moments of the total mass of the limit log-infinitely divisible stochastic measure of Bacry and Muzy (Commun Math Phys 236:449–475, 2003) is derived. The covariance structure of the total mass of the measure is shown to be logarithmic. A generalization of the Selberg integral corresponding to single moments of the limit measure is proposed and shown to satisfy a recurrence relation. The joint moments of the limit lognormal measure, classical Selberg integral with \(\lambda _1=\lambda _2=0,\) and Morris integral are represented in the form of multiple binomial sums. For application, low moments of the limit log-Poisson measure are computed exactly and low joint moments of the limit lognormal measure are considered in detail.     

Reduction of radiative transfer problems in semi-infinite media to linear Fredholm integral equations

Linear Fredholm integral equations are derived for the Stokes vector of polarized radiation, emergent from a scattering plane parallel semi-infinite medium, by means of the full range orthogonality and completeness properties of Case's eigensolutions. A renormalization concerning the eigenmode with the greatest discrete eigenvalue is applied, which permits us to obtain a new integral equation for the zeroth Fourier component of the radiation field. The kernel of the integral equations is given in terms of Case's eigenfunctions or of the Green's function matrix for an infinite medium. For isotropic scattering, it is shown that the integral equation can be solved by means of a very rapidly convergent Neumann series. Physical arguments lead to the conclusion that the renormalized Fredholm integral equations are well suited also for arbitrary phase matrices.     

A relationship between Gel'fand-Levitan and Marchenko kernels

A new integral equation which relates the output kernels of the Gel'fand-Levitan and Marchenko inverse scattering equations in a continuous range of their variables is specified. Structural details of this integral equation are studied when theS-matrix is a rational function, and the output kernels are separable in terms of Bessel, Hankel and Jost solutions.     

The cross-correlation integral: Its features and application to nonstationarity detection in time series

A generalization of the correlation integral, called the cross-correlation integral, is suggested. Features of the cross-correlation integral are studied, and a new attribute of a time series is defined. It is viewed as some kind of dimension and is associated with the fill rate of the attractor. It is demonstrated that the cross-correlation integral is calculated in much the same way as the wavelet transform of the density of points in the attractor. The cross-correlation integral is applied to detection of nonstationarity in time series. A comparison with statistical methods is made.     

Space-time transformations in radial path integrals

Nonlinear space-time transformations in the radial path integral are discussed. A transformation formula is derived, which relates the original path integral to the Green function of a new quantum system with an effective potential containing an observable quantum correction ≈?². As an example the formula is applied to spherical brownian motion.     

Analytical expressions for the correlation function of a hard sphere dimer fluid

A closed form expression is given for the correlation function of a hard sphere dimer fluid. A set of integral equations is obtained from Wertheim's multidensity Ornstein-Zernike integral equation theory with Percus-Yevick approximation. Applying the Laplace transformation method to the integral equations and then solving the resulting equations algebraically, the Laplace transforms of the individual correlation functions are obtained. By the inverse Laplace transformation, the radial distribution function (RDF) is obtained in closed form out to 3D (D is the segment diameter). The analytical expression for the RDF of the hard dimer should be useful in developing the perturbation theory of dimer fluids.     

Three-dimensional radiative transfer in an anisotropically scattering, plane-parallel medium: generalized reflection and transmission functions

The problem of spatially varying, collimated radiation incident on an anisotropically scattering, plane-parallel medium is considered. A very general phase function is allowed. An integral transform is used to reduce the three-dimensional radiative transport equation to a one-dimensional form, and a modified Ambarzumian's method is applied to derive nonlinear integral and integro-differential equations for the generalized reflection and transmission functions. The integration is over the polar and azimuthal angles—this formulation is referred to as the double-integral formulation. The integral equations are used to illustrate symmetry relationships and to obtain single- and double-scattering approximations. The generalized reflection and transmission functions are important in the construction of the solutions to many multidimensional problems. Coupled integral equations for the interior and emergent intensities are developed and, for the case of two identical homogeneous layers, used to formulate a doubling procedure. Results for an isotropic and Rayleigh scattering medium are presented to illustrate the computational characteristics of the formulation.     

广义Birkhoff系统的积分不变量

研究广义Birkhoff系统的积分不变量.给出系统存在积分不变量的条件,在此条件下导出系统的线性积分不变量、通用积分不变量和二阶绝对积分不变量.举例说明结果的应用. 关键词： 广义Birkhoff方程 线性积分不变量 通用积分不变量 二阶绝对积分不变量     

Burton-Miller改进型边界方程的多频计算方法

提出了一种采用Burton-Miller改进型边界积分方程进行多频计算的方法。将Burton-Miller方程中的高奇异积分转化为弱奇异积分形式,获得Burton-Miller改进型边界积分方程;将方程中格林函数进行Taylor级数展开,并把波数从方程中分离出来,从而使随波数变化的计算矩阵表示为波数的矩阵级数形式。数值分析表明,本方法不仅保证了解在全波数范围内的唯一性,并且计算频率点数较多时可以节约大量时间,提高计算效率。      

Generalized lattice model of liquid state and its relation to Ginzburg–Landau theory

Basic principles of the generalized lattice model of multicomponent condensed systems are formulated. Short-range parts of interatomic interactions are taken into account by means of the geometric constraints method. Long-range parts of the interactions are taken into account in mean field approximation. The expression for Helmholtz free energy is obtained. A system of integral equations for the equilibrium distributions of components is derived. The asymptotic properties of its solutions are investigated. Moment expansion of interatomic interactions and localization of integral terms in free energy is obtained. A Ginzburg–Landau-like functional of free energy is derived.     

Symmetry in extended phase space for singular Lagrangian with subsidiary constraints

A generalization of the Noether theorem to a singular nonholonomic system in the canonical formalism is given and its inverse theorem is presented. Based on the canonical action integral, a generalization of the Poincaré-Cartan integral invariant of a singular nonholonomic system is obtained. It is shown that this invariant is equivalent to the canonical equations of a singular constrained system. Some confusions in the literature are corrected. An example is given.     

On the theory of temporal aberrations for cathode lenses

A new approach to the theory of temporal aberration for cathode lenses is given in the present paper. A definition of temporal aberration is given in which a certain initial energy of electron emission along the axial direction ε_z1 (0ε_z1ε_0max) is considered. A new method to calculate the temporal aberration coefficients of cathode lenses named “direct integral method” is also presented. The “direct integral method” gives new expressions of the temporal aberration coefficients which are expressed in integral forms. The difference between “direct integral method” and “τ-variation method” is that the “τ-variation method” needs to solve the differential equations for the three of temporal geometrical aberration coefficients of second order, while the “direct integral method” only needs to carry out the integral calculation for all of these temporal aberration coefficients of second order.All of the formulae of the temporal aberration coefficients deduced from “direct integral method” and “τ-variation method” have been verified by an electrostatic concentric spherical system model, and contrasted with the analytical solutions. Results show that these two methods have got identical solutions and the solutions of temporal aberration coefficients of the first and second order are the same as with the analytical solutions. Although some forms of the results seem different, but they can be transformed into the same form. Thus, it can be concluded these two methods given by us are equivalent and correct, but the “direct integral method” is related to solve integral equations, which is more convenient for computation and could be suggested for use in practical design.     

Some general problems of fast electrons transport

A generalization is given of the segments method in the form of a multistep method with generalized time for computing the transport of fast particles. The integral equation for a flow with generalized time in the phase space of variables is written under the assumption that the flow cuts the generalized time surface at right angles. The Green's function for the differential flow operator is the kernel of the integral equation. It is also shown that such an integral equation which can be obtained from a nonstationary kinetic equation provides a uniform consistent algorithm for solving either nonstationary or stationary problems. Examples of Green's functions are given for an operator of differential flow of fast electrons.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 110–114, August, 1974.The author would like to express his thanks to A. A. Vorob'ev and B. A. Kononov for their encouragement, to A. P. Yalovets for discussing the work with him, and to A. M. Kol'chuzhkin for going through the text.     

Path Integral Solution for the Coulomb Potential in a Curved Space of Constant Positive Curvature

A new path integral treatment of a hydrogen-like atom in a uniformly curved space with a constant positive curvature is presented. By converting the radial path integral into a path integral for the modified Pöschl-Teller potential with the help of the space-time transformation technique, the radial Green’s function is expressed in closed form, from which the energy spectrum and the corresponding normalized wave functions of the bound states are extracted. In the limit of vanishing curvature, the Green’s function, the energy spectra and the correctly normalized wave functions of bound and scattering states for a standard hydrogen-like atom are found.     

A General Deterministic Treatment of Derivatives in Particle Methods

A unified approach to approximating spatial derivatives in particle methods using integral operators is presented. The approach is an extension of particle strength exchange, originally developed for treating the Laplacian in advection–diffusion problems. Kernels of high order of accuracy are constructed that can be used to approximate derivatives of any degree. A new treatment for computing derivatives near the edge of particle coverage is introduced, using “one-sided” integrals that only look for information where it is available. The use of these integral approximations in wave propagation applications is considered and their error is analyzed in this context using Fourier methods. Finally, simple tests are performed to demonstrate the characteristics of the treatment, including an assessment of the effects of particle dispersion, and their results are discussed.     

A unitary model for three-body collisions

A connected 3 → 3 formalism for three-body collision processes is reduced to a hierarchy of three on-energy-shell integral equations and one off-energy-shell integral equation. Only the on-energy-shell equations, which involve only on-energy-shell three-body and two-body amplitudes, need be solved exactly in order to obtain elastic and break-up amplitudes satisfying the unitarity constraints exactly. Applied to n-d break-up, the on-energy-shell equations ensure that the n-d initial-state interaction, the nucleon-nucleon final-state interactions, and more complicated 3 → 3 processes are correctly described. After angular momentum analysis the on-energy-shell equations are one-dimensional integral equations, even in the case of local two-body potentials. This unitary model provides a practical scheme for calculating approximate three-body elastic and break-up amplitudes when two-body local potentials are used to describe the two-body subsystems.     

Operator ordering schemes and covariant path integrals of quantum and stochastic processes in Curved space

A method is given for the derivation of covariant path integral solutions of quantum and stochastic processes in curved space.The correspondence between operator ordering schemes and ordinary functions in phase space is studied and applied to the explicit construction of a class of equivalent lattice representations of path integrals. It is shown that this class is uniquely related to a covariant functional integral. Some arguments forR/12 as the quantum mechanical curvature potential are given. Simple rules for nonlinear point transformations are stated. The connection with previous works is discussed.     

直接积分法研究电子光学成像系统的时间像差理论

提出了计算动态电子光学成像系统时间像差系数的新方法——直接积分法.以阴极面逸出的轴向电子初能为ε_z1（0≤ε_z1≤ε_0max）的近轴电子 轨迹为比较 基准,给出了时间像差的定义,详细叙述了直接积分法并给出求解动态电子光学成像系统时 间像差系数的积分表达式.τ变分法求得的二级几何时间像差系数必须求解微分方程, 而直接积分法求得的二级几何时间像差系数全部以积分形式表示,仅需进行积分运算,更适 用于成像系统的实际计算与设计. 关键词： 电子光学成像系统 阴极透镜 动态电子光学 时间像差理论     

Aluminum integral foams with tailored density profile by adapted blowing agents

The goal of the present work is the variation of the structure of aluminum integral foams regarding the thickness of the integral solid skin as well as the density profile. A modified die casting process, namely integral foam molding, is used in which an aluminum melt and blowing agent particles (magnesium hydride MgH₂) are injected in a permanent steel mold. The high solidification rates at the cooled walls of the mold lead to the formation of a solid skin. In the inner region, hydrogen is released by thermal decomposition of MgH₂ particles. Thus, the pore formation takes place parallel to the continuing solidification of the melt. The thickness of the solid skin and the density profile of the core strongly depend on the interplay between solidification velocity and kinetics of hydrogen release. By varying the melt and blowing agent properties, the structure of integral foams can be systematically changed to meet the requirements of the desired field of application of the produced component.