Fubini theorem for anticipating stochastic integrals in Hilbert space

Let (X,l,) be a measure space, letW be a cylindrical Hilbert-Wiener process, and let be an anticipating integrable process-valued function onX. We prove, under natural assumptions on, that there exists a measurable version Y_x,x X, of the anticipating integral of(x) such that the integral _x Y_x(dx) is a version of the anticipating integral of _X (x)(dx). We apply this anticipating Fubini theorem to study solutions of a class of stochastic evolution equations in Hilbert space.     

SOME BLENDING INTERPOLATION METHODS IN CAGD

Some of the blending interpolation methods (which) stated by a convex combination technique are presented. This includes C1 ,C2 blending triangular interpolanls and C1 -blending quadrilateral inter polanls for surfaces. The mentioned methods can be generalized to the construction of C" blending interpolants.     

Stochastic dominance with nonadditive probabilities

Choquet expected utility which uses capacities (i.e. nonadditive probability measures) in place of-additive probability measures has been introduced to decision making under uncertainty to cope with observed effects of ambiguity aversion like the Ellsberg paradox. In this paper we present necessary and sufficient conditions for stochastic dominance between capacities (i.e. the expected utility with respect to one capacity exceeds that with respect to the other one for a given class of utility functions). One wide class of conditions refers to probability inequalities on certain families of sets. To yield another general class of conditions we present sufficient conditions for the existence of a probability measureP with f dC= f dP for all increasing functionsf whenC is a given capacity. Examples includen-th degree stochastic dominance on the reals and many cases of so-called set dominance. Finally, applications to decision making are given including anticipated utility with unknown distortion function.     

Global optimization and stochastic differential equations

Let ⁿ be then-dimensional real Euclidean space,x=(x ₁,x ₂, ...,x _n)^{T n} , and letf: ⁿ R be a real-valued function. We consider the problem of finding the global minimizers off. A new method to compute numerically the global minimizers by following the paths of a system of stochastic differential equations is proposed. This method is motivated by quantum mechanics. Some numerical experience on a set of test problems is presented. The method compares favorably with other existing methods for global optimization.This research has been supported by the European Research Office of the US Army under Contract No. DAJA-37-81-C-0740.The third author gratefully acknowledges Prof. A. Rinnooy Kan for bringing to his attention Ref. 4.     

Anomalous Density and Permittivity Effects on the Structure of Water

The relationship between the structural peculiarities of the hydrogen bond net and the anomalous behavior of density and dielectric permittivity of water is investigated. The degree of ordering in the network of hydrogen bonds is described in terms of the structural functions, among which the most important ones are tetrahedricity and the mean number of hydrogen bonds per molecule. The temperature dependence of the number of hydrogen bonds per molecule is discussed in terms of the analysis of the experimental temperature dependences of density and permittivity of water on the saturation line. The estimated density of hexagonal ice at the melting point is reproduced in terms of the concepts suggested for analysis; the estimate is rather close to the experimental value. Possible applications of the new approach are discussed.     

The effects of the nature of catalyst and of the solvent on the stereoselectivity in amine-catalyzed asymmetric synthesis of substituted cyclohexa-1,3-dienes from prenal and monoesters of ylidenemalonic acids

In the amine-catalyzed reactions of prenal with (Z)-5-methyl-2-(methoxycarbonyl)hexa-2,4-dienoic or (Z)-3-phenyl-2-(ethoxycarbonyl)prop-2-enoic acid chiral β-amino alcohols provide for higher enantiomeric purity of the resulting alkyl 4-methyl-6-(2-methylprop-1-enyl)-and 4-methyl-6-phenylcyclohexa-1,3-dienoates than that provided by related chiral amines without hydroxy group. The values ofee attained in nonpolar solvents are higher than those observed in the polar ones. Substituting stoichiometric amounts of a chiral 1-amino-3-methylbuta-1,3-diene for a combination of prenal with 0.1 equiv. of the corresponding chiral amine results in the products of much lower enantiomeric purity. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 1, pp. 84–92, January, 1998.     

Statistical mechanics of a dynamical system based on Conway's game of Life

We study a discrete dynamical system whose evolution is governed by rules similar to those of Conway's game of Life but also include a stochastic element (parametrized by a temperature). Statistical properties that are examined are density as a function of temperature and entropy (suitably defined). A phase transition and a certain thermodynamic constant of the motion are observed.Lady Davis Visiting Scientist at the Technion 1974–75.     

Lower bounds on the cluster size distribution

We rigorously prove that the probabilityP _n that the origin of ad-dimensional lattice belongs to a cluster of exactlyn sites satisfiesP _n > exp(–n ^(d–1)/d ) whenever percolation occurs. This holds for the usual (noninteracting) percolation models for any concentrationp > p _c , as well as for the equilibrium states of lattice spin systems with quite general interactions. Such a lower bound applies also if no percolation occurs, but if it appears in some other phase of the system.     

Fractional noise

Fractional noiseN(t),t 0, is a stochastic process for every , and is defined as the fractional derivative or fractional integral of white noise. For = 1 we recover Brownian motion and for = 1/2 we findf ^–1-noise. For 1/2 1, a superposition of fractional noise is related to the fractional diffusion equation.