Analog of the Liouville Equation and BBGKY Hierarchy for a System of Hard Spheres with Inelastic Collisions

Dynamics of a system of hard spheres with inelastic collisions is investigated. This system is a model for granular flow. The map induced by a shift along the trajectory does not preserve the volume of the phase space, and the corresponding Jacobian is different from one. A special distribution function is defined as the product of the usual distribution function and the squared Jacobian. For this distribution function, the Liouville equation with boundary condition is derived. A sequence of correlation functions is defined for canonical and grand canonical ensemble. The generalized BBGKY hierarchy and boundary condition are deduced for correlation functions. __________ Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 6, pp. 818–839, June, 2005.     

On constructive distribution functions

The concept is introduced of a distribution function given by means of a sequence of finite-step functions that converges in the integral metric. (As shown in [1], the pointwise representation of a constructive distribution function at all points is not possible.) It is shown that there is a correspondence between constructive distributive functions and constructive characteristic functions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 60, pp. 59–64, 1976. Results announced December 20, 1973 and June 12, 1975.The author thanks N. A. Shanin for prolonged attention to the subject of this article and for critical remarks leading to the appearance of Secs. 2 and 3, as well as A. O. Slisenko, who pointed out isolated errors.     

A class of filled functions for finding global minimizers of a function of several variables

This paper is concerned with filled function methods for finding global minimizers of a function of several variables. A class of filled functions is defined. The advantages and disadvantages of every filled function in the class are analyzed. The best one in this class is pointed out. The idea behind constructing a better filled function is given and employed to construct the class of filled functions. A method is also explored on how to locate minimizers or saddle points of a filled function through only the use of the gradient of a function.The authors are indebted to Dr. L. C. W. Dixon for stimulating discussions.     

Note on new solutions of LR fuzzy linear systems using ranking functions and ABS algorithms

Recently, Ghanbari and Mahdavi-Amiri focused on solving LR fuzzy linear systems by use of ranking functions. They applied a ranking function introduced by Cheng, which is based on the centroid point, to illustrate their method. Also, they presented an important lemma using the centroid formulae provided by Cheng, to determine the centroid point for a class of fuzzy numbers. Unfortunately, they didn’t consider that the formulae are incorrect and have led to some misapplications as pointed out by Wang and his colleagues. Therefore, in this paper, we first show that Lemma 19 of Ghanbari and Mahdavi-Amiri’s paper is not true and then correct it using the centroid formulae suggested by Wang. Finally, we correct the results obtained in Ghanbari and Mahdavi-Amiri’s paper for a special example.     

An approach for calculating correlation functions in the six-vertex model with domain wall boundary conditions

We address the problem of calculating correlation functions in the six-vertex model with domain wall boundary conditions by considering a particular nonlocal correlation function, called the row configuration probability. This correlation function can be used as a building block for computing various (both local and nonlocal) correlation functions in the model. We calculate the row configuration probability using the quantum inverse scattering method, giving the final result in terms of multiple integrals. We also discuss the relation to the emptiness formation probability, another nonlocal correlation function, which was previously computed using similar methods.     

Cost Functions in the Location of Depots for Multiple-Delivery Journeys

Most wholesale distribution is performed during multiple-delivery journeys. Mathematical methods of locating depots utilize simple functions of delivery data, e.g. weight and distance from the depot, to measure the delivery "cost"; the total "cost" is minimized to find the depot location.It is pointed out that the cost of a delivery is influenced by the occurrence of other deliveries. It is shown that, in a few examples, simple functions of the delivery data are not always good measures of variable cost, as measured by the length of journeys planned to carry out the same deliveries, and that the minimum points of the simple functions rarely coincide with the point of minimum variable cost. It is concluded that, subject to reservations, which are discussed, about the experiments, the use of simple cost functions to locate wholesale distribution depots will probably give misleading results.     

一种特殊函数的算法研究

由于有些函数的计算量巨大,因此通过算法的研究减少计算量是计算机发展的一个重要方向.本文通过对一个函数算法的研究,提出一种减少计算量的新算法.     

The effective conductivity of an infinitely interchangeable mixture

We are concerned with the estimation of the effective electrical conductivity of random heterogenous materials. The purpose of this paper is to discuss a property of “statistical symmetry” verified by the symmetric cell materials of Miller. This property will be referred to as infinite interchangeability. The usual way to approach cell materials is through n-point correlation functions. The property of infinite interchangebility permits us to approach cell materials from a completely different point of view. Our main result is a simple algorithm, based on this symmetry property, for computing any coefficient of the perturbation expansion in terms of information from the dilute limit. Specifically, knowledge of the coefficients of the expansion in powers of the volume fraction up to order r allows for computation of the perturbation expansion coefficients up to order (2r + 1). This result, which was previously known for r = 2 in the isotropic case and for r = 1 in the anisotropic case, can also be obtained from the standard correlation function approach, as pointed out by Milton.     

Monotonicity properties of functions related to the psi function

In this paper, the monotonicity properties of functions related to the psi function are obtained, a double inequality for the Euler-Mascheroni constant is established. Relevant connections of the results presented here with those derived in earlier works are also pointed out.     

P-凸函数及其性质

给出P-凸函数的定义及其性质,将一个有关凸函数的命题推广到P-凸函数情形,并指出赵海清等人关于该命题的证明过程是错误的.     

Stochastic equilibrium assignment with variable demand: Theoretical and implementation issues

Recently, it has been pointed out that transport models should reflect all significant traveler choice behavior. In particular, trip generation, trip distribution, modal split as well as route choice should be modeled in a consistent process based on the equilibrium between transport supply and travel demand. In this paper a general fixed-point approach that allows dealing with multi-user stochastic equilibrium assignment with variable demand is presented. The main focus was on investigating the effectiveness of internal and external approaches and of different algorithmic specifications based on the method of successive averages within the internal approach. The vector demand function was assumed non-separable, non-symmetric cost functions were adopted and implementation issues, such updating step and convergence criterion, were investigated. In particular the aim was threefold: (i) compare the internal and the external approaches; (ii) investigate the effectiveness of different algorithmic specifications to solve the variable demand equilibrium assignment problem through the internal approach; (iii) investigate the incidence of the number of the links with non-separable and/or asymmetrical cost functions. The proposed analyses were carried out with respect to two real-scale urban networks regarding medium-size urban contexts in Italy.     

Additive relations and function tables

We present work with a second grade classroom where we carried out a teaching experiment that attempted to bring out the algebraic character of arithmetic. In this paper, we specifically illustrate our work with the second graders on additive relations, through the children’s work with function tables. We explore the different ways in which the children represented the information of a problem in the form of a self-designed function table. We argue that the choices children make about the kind of information to represent or not, as well as the way in which they constructed their tables, highlight some of the issues that children may find relevant in their construction of function tables. This open-ended format pointed to how they were understanding and appropriating tables into their thinking about additive relations.     

Ordinary differential operator and some of its applications to certain meromorphically p-valent functions

By making use of the well-known assertions given in Miller and Mocanu (1978) [13] and Nunokawa (1993) [14], certain theorems concerning p-valently meromorphic (strongly) starlike and (strongly) convex functions obtained in this investigation are firstly proved and then their certain consequences which will be interesting or important for analytic and geometric function theory are pointed out.     

非线性l1问题的光滑近似算法

为非线性l1问题的求解构造了光滑逼近函数.首先将非线性l1问题转化为等价的不可微优化问题;其次通过两步提出光滑逼近函数的一般性构造方法;最后进行了数值仿真.文中介绍了光滑逼近函数的有关性质,指出相关文献已有的光滑函数方法是本文的特例,并证明了方法的收敛性及有效性.     

Dependence structure between LIBOR rates by copula method

This paper discusses the correlation structure between London Interbank Offered Rates (LIBOR) by using the copula function. We start from one simplified model of A. Brace, D. Gatarek, and M. Musiela (1997) and find out that the copula function between two LIBOR rates can be expressed as a sum of an infinite series, where the main term is a distribution function with Gaussian copula. Partial differential equation method is used for deriving the copula expansion. Numerical results show that the copula of the LIBOR rates and Gaussian copula are very close in the central region and differ in the tail, and the Gaussian copula approximation to the copula function between the LIBOR rates provides satisfying results in the normal situation.     

Eigenanalysis on a bivariate covariance kernel

Certain constructions of copulas can be interpreted as an eigendecomposition of a kernel. We study some properties of the eigenfunctions and their integrals of a covariance kernel related to a bivariate distribution. The covariance between functions of random variables in terms of the cumulative distribution function is used. Some bounds for the trace of the kernel and some inequalities for a continuous random variable concerning a function and its derivative are obtained. We also obtain relations to diagonal expansions and canonical correlation analysis and, as a by-product, series of constants for some particular distributions.     

Approximation by Entire Functions on a Countable Union of Segments on the Real Axis: 3. Further Generalization

In this paper, an approximation of functions of extensive classes set on a countable unit of segments of a real axis using the entire functions of exponential type is considered. The higher the type of the approximating function is, the higher the rate of approximation near segment ends can be made, compared with their inner points. The general approximation scale, which is nonuniform over its segments, depending on the type of the entire function, is similar to the scale set out for the first time in the study of the approximation of the function by polynomials. For cases with one segment and its approximation by polynomials, this scale has allowed us to connect the so-called direct theorems, which state a possible rate of smooth function approximation by polynomials, and the inverse theorems, which give the smoothness of a function approximated by polynomials at a given rate. The approximations by entire functions on a countable unit of segments for the case of Hölder spaces have been studied by the authors in two preceding papers. This paper significantly expands the class of spaces for the functions, which are used to plot an approximation that engages the entire functions with the required properties.     

Stationary and multi-self-similar random fields with stochastic volatility

This paper introduces stationary and multi-self-similar random fields which account for stochastic volatility and have type G marginal law. The stationary random fields are constructed using volatility modulated mixed moving average (MA) fields and their probabilistic properties are discussed. Also, two methods for parameterizing the weight functions in the MA representation are presented: one method is based on Fourier techniques and aims at reproducing a given correlation structure, the other method is based on ideas from stochastic partial differential equations. Moreover, using a generalized Lamperti transform we construct volatility modulated multi-self-similar random fields which have type G distribution.