基于落影表现理论下的模糊子群

本文利用落影表现理论给出了一个群的模糊子群的两种定义，这两种模糊子群与现有的模糊子群有着十分密切的关系，从而为现有的模糊子群的定义提供了新的理论根据。通过本文的讨论便知：可以把落影表现理论应用到模糊代数的研究中去。     

Random walks on finitely structured transfinite networks

A general theory for random walks on transfinite networks whose ranks are arbitrary natural numbers is established herein. In such networks, nodes of higher ranks connect together transfinite networks of lower ranks. The probabilities for transitions through such nodes are obtained as extensions of the Nash-Williams rule for random walks on ordinary infinite networks. The analysis is based on the theory of transfinite electrical networks, but it requires that the transfinite network have a structure that generalizes local-finiteness for ordinary infinite networks. The shorting together of nodes of different ranks are allowed; this complicates transitions through such nodes but provides a considerably more general theory. It is shown that, with respect to any finite set of nodes of any ranks, a transfinite random walk can be represented by an irreducible reversible Makov chain, whose state space is that set of nodes.This work was supported by the National Science Foundation under the grants DMS-9200738 and MIP-9200748.     

On a Class of Covariance Operators

This paper is the continuation of [1] in which complex symmetries of distributions and their covariance operators are investigated. Here we also study the most general quaternion symmetries of random vectors. Complete classification theorems on these symmetries are proved in terms of covariance operator spectra.     

Maximally Equidistributed Combined Tausworthe Generators

Tausworthe random number generators based on a primitive trinomial allow an easy and fast implementation when their parameters obey certain restrictions. However, such generators, with those restrictions, have bad statistical properties unless we combine them. A generator is called maximally equidistributed if its vectors of successive values have the best possible equidistribution in all dimensions. This paper shows how to find maximally equidistributed combinations in an efficient manner, and gives a list of generators with that property. Such generators have a strong theoretical support and lend themselves to very fast software implementations.

     

Diffusion controlled multiplicative process: typical versus average behaviour

We investigate the dynamics of the number of particles diffusing in a multiplicative medium. We show that the typical behaviour of the growth process is different from the average. We develop a new formalism to study the average growth process and extend it to the calculation of higher moments and finally of the probability distribution. We show that the fluctuations of the growth process increase exponentially with time. We describe the interesting features of the distribution.     

On the convergence of the Baba and Dorea random optimization methods

The Baba and Dorea global minimization methods have been applied to two physical problems. The first one is that of finding the global minimum of the transformer design function of six variables subject to constraints. The second one is the problem of fitting the orbit of a satellite using a set of observations. The latter problem is reduced to that of finding the global minimum of the sum of the squares of the differences between the observed values of the azimuth, elevation, and range at certain intervals of time from the epoch and the computed values of the azimuth, elevation, and range at the same intervals of time. Baba and Dorea established theoretically that the random optimization methods converge to the global minimum with probability one. The numerical experiments carried out for the above two problems show that convergence is very slow for the first problem and is even slower for the second problem. In both cases, it has not been possible to reach the global minimum if the search domains of the variables are wide, even after a very large number of function evaluations.The author thanks the referee for his suggestions on improving the presentation of the paper.     

Large-n limit of the Heisenberg model: Random external field and random uniaxial anisotropy

The thermodynamic equivalence of the large-n limit of then-vector model in a random external field and the corresponding disordered spherical model is proved. An analytic expression for the free energy and a phase diagram of the large-n limit of then-vector model with random uniaxial anisotropy are obtained by rigorous argument. The ferromagnetic order in the large-n limit is proved to be stable against the switching on of an arbitrarily small random anisotropy.     

Scaling exponents of random walks in random sceneries

We compute the exact asymptotic normalizations of random walks in random sceneries, for various null recurrent random walks to the nearest neighbours, and for i.i.d., centered and square integrable random sceneries. In each case, the standard deviation grows like n with . Here, the value of the exponent is determined by the sole geometry of the underlying graph, as opposed to previous examples, where this value reflected mainly the integrability properties of the steps of the walk, or of the scenery. For discrete Bessel processes of dimension d[0;2[, the exponent is . For the simple walk on some specific graphs, whose volume grows like n^d for d[1;2[, the exponent is =1−d/4. We build a null recurrent walk, for which without logarithmic correction. Last, for the simple walk on a critical Galton–Watson tree, conditioned by its nonextinction, the annealed exponent is . In that setting and when the scenery is i.i.d. by levels, the same result holds with .     

Random external fields

The various theoretical considerations for the effects of quenched random fields (RF) on second-order transitions as well as the experimental situation are briefly reviewed. Some of the physical realizations of the RF models are discussed, with an emphasis on solid-state first-order transitions in impure systems. The physical arguments for the RF effects in the bulk as well as on phase interfaces are discussed. In the latter case it is suggested that scattering experiments can probe the details of the interface fluctuations. The role of long relaxation times and metastability in Ising RF systems is emphasized.Research at Tel Aviv University partially supported by the U.S.-Israel Binational Science Foundation (BSF), Jerusalem, Israel.