Random sets. A survey of results and applications

A survey of current directions in the theory of random closed sets is presented; these include: the central limit theorem, the law of large numbers for Minkowski sums and unions of random sets, semi-Markov random closed sets, Boolean models and statistical estimation of their parameters, specification of distributions and associated problems of capacity theory. Weak convergence of random closed sets is defined and its application to limit theorems for graphs and epi-graphs of random processes and problems of stochastic optimization is described. Other connections with the theory of random processes (level sets, multivalued and controllable random processes) are also discussed.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 12, pp. 1587–1599, December, 1991.     

Markov processes with creation of mass. II

Summary A theory of Markov processes for which the sample paths have random starting and terminal times was first considered by Hunt [4, 5]. Such processes require that the classical notion of a probability measure space be modified so as to allow -finite probability measures. Such an extension necessitates only minor adjustments of the language of probability theory; e.g., the phrase absolute probability of an event E does not admit the usual interpretation and is not used. Only conditional probabilities are used in the extended theory and do admit the usual interpretation. The purpose of this paper is to establish regularity properties of Markov processes having sample paths with random creation and terminal times. An existence theorem for such processes appears in [3].This research was supported by the National Science Foundation.     

Application of the nonlinear theory of heredity to the description of time effects in polymeric materials

A relation is established between the heredity theory with time-invariant nonlinearity and fractional-exponential kernels and the Volterra-Fréchet theory for uniaxial tension. A constitutive equation is proposed for processes accompanied by decreasing strain. A procedure for determining the necessary material characteristics from creep and recovery data is considered.     

Boson fock representations of stochastic processes

A classification theory of quantum stationary processes similar to the corresponding theory for classical stationary processes is presented. Our main result is the classification of those pairs of classical stationary processes that admit a joint boson Fock canonical representation. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 3–14, January, 2000.     

Gas-dynamic equations for spatially inhomogeneous gas mixtures with internal degrees of freedom. I. General theory

A general algorithm for building a uniform asymptotic solution of the kinetic equations for spatially inhomogeneous reactive gas mixtures is proposed. It solves the problem of irregular asymptotic solution arising in the ordinary Chapman–Enskog method, providing expressions for chemical reaction rates that agree with the mono-molecular reaction theory. We study a quasi-stationary behavior of the system, characterized by the slowly varying gas-dynamic variables which number is greater than the number of integral invariants of the collision operator. The gas-dynamic equations for reacting and relaxing gas mixtures are derived in general form. It is shown that accurate treatment of non-equilibrium processes gives rise to additional terms caused by the strong influence of small perturbations of quasi-equilibrium distribution functions on the kinetics of high-threshold physical and chemical processes. These terms are describing the influence of inelastic collisions, expansion/compression processes and spatial non-uniformity of gas-dynamic variables.     

A covariant method for calculating amplitudes of processes involving polarized spin 1/2 particles. Calculation of the interference terms in their cross sections

A covariant method is proposed for calculating the amplitudes of processes involving polarized spin 1/2 particles. It is suitable for calculating the interference terms in the cross sections of such processes. As an illustration, expressions are given for the amplitudes of electron-electron scattering in the lowest order of perturbation theory and expressions for the electron current in the case of emission of two bremsstrahlung photons in the ultrarelativistic limit.Research Institute of Nuclear Problems at the Belorussian State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 96, No. 1, pp. 96–108, July, 1993.     

维纳和布朗运动

布朗运动,作为一种特殊的随机过程,在随机过程理论处于一个中心地位.布朗运动理论在其他许多领域也有重要应用.在布朗运动理论的发展和完善过程中,布朗,爱因斯坦和维纳等人都作出了重要贡献.通过解读原始文献,考察了维纳建立布朗运动数学理论的过程.揭示了维纳在布朗运动的数学理论严格化进程中的重要作用.     

Stochastic processes indexed by hypergroups. I

For many applications it is desirable to have extensions of the classical theory of weakly stationary processes to certain classes of nonstationary ones. A large family for which Fourier methods still play a major role is the harmonizable class and some related processes. The purpose of this paper is to initiate the study of a class of stochastic processes with a stationarity condition based on the notion of hypergroups.     

Wiener过程分布密度的矩阵算法

在随机过程的理论及其应用中．Ｗｉｅｎｅｒ过程起着基本的作用．本文结出用矩阵的初等变换，建立Ｗｉｅｎｅｒ过程ｐ．ｄ，ｆ的一个崭新的方法；此方法，对理解过程的统计特性，掌握过程的ｐ，ｄ，ｆ的结构等，均是有益的．     

Branching processes. I

The main results obtained from 1968 to 1983 in the theory of Markov branching processes and processes with transformations depending on the age of particles are reflected in this article. Along with the traditional sections (integral and local theorems, stationary measures), the survey includes sections devoted to statistics of branching processes. The bibliography contains mainly works reviewed in RZh Matematika.Translated from Itogi Nauki i Tekhniki, Seriya Teoriya Veroyatnostei, Matematicheskaya Stati tika, Teoreticheskaya Kibernetika, Vol. 23, pp. 3–67, 1985.     

Evaluation of the mathematical and economic basis for conversion processes in the LEAP energy-economy model

An evaluation was made of the mathematical and economic basis for conversion processes in the LEAP energy-economy model. Conversion processes are the main modelling subunit in LEAP used to represent energy conversion industries and are supposedly based on the classical economic theory of the firm. The study arose out of questions about the uniqueness and existence of LEAP solutions and their relation to classical equilibrium economic theory. An analysis of classical theory and LEAP model equations was made to determine their exact relationship. The conclusions drawn from this analysis were that LEAP theory is not consistent with the classical theory of the firm. Specifically, the capacity for factor formalism used by LEAP does not support a classical interpretation in terms of a technological production function for energy conversion processes. The economic implications of this inconsistency are suboptimal process operation and short term negative profits in years where plant operation should be terminated. A new capacity factor formalism, which retains the behavioural features of the original model, is proposed to resolve these discrepancies.     

Itô’s excursion theory and its applications

A presentation of It?’s excursion theory for general Markov processes is given, with several applications to Brownian motion and related processes.     

Approximation of stochastic equations driven by predictable processes

Summary A theory of stochastic differential equations driven by predictable processes in Stratonovich sense is developed. These driving processes include a large class of discontinuous semimartingales. The theory of stochastic differential equations driven by continuous semimartingales in Stratonovich sense is extended without involving Lebesgue-Stieltjes integrals as done by Meyer. Moreover, a change of variables formula without extra terms involving the jumps of the processes holds for this theory. Results on approximation of driving processes are preserved.Research partially supported by the Institute for Mathematics and its Applications at the University of Minnesota, Minneapolis, USA; by the AFOSR under contract #AFOSR-85-0315, the ARO under contract #DAAG 29-84-K-0082, and #DAAL 03-86-K-0171.     

Markov调制风险模型的轨道刻划和概率构造

用随机过程的轨道,严格地刻划了Markov调制风险模型U=(Q,G,F;J,s,X),它是已有的Markov调制风险模型的一般化.基于模型U,分别给出带保费率向量C和带税率向量γ的Markov调制风险过程R~u={R~u(t),t≥0}和R~u(γ)={R~u(γ,t),t≥0}.给定特征组A=(Q,G,F),用概率方法构造了模型U.从而为用随机过程理论和方法研究Markov调制风险模型和过程,奠定了严实的随机过程基础.     

From high-energy physics to the relativistic theory of gravity

We give a brief review of the most important results on inclusive processes and the relativistic theory of gravity obtained by A. A. Logunov.     

On theA-stability of implicit Runge-Kutta processes

A new technique to calculate the characteristic functions and to examine theA-stability of implicit Runge-Kutta processes is presented. This technique is based on a direct algebraic approach and an application of theC-polynomial theory of Nørsett. New processes are suggested. These processes can be exponentially fitted in anA-stable manner.