The issue of reliability of computer predictions of physical events is examined as the goal of verification and validation processes. It is argued that verification, both solution and code verification, can be carried out with a high degree of confidence, even though much remains to be done to improve and advance verification procedures. It is validation of mathematical models that stands as th...

in this paper, a model of an optimal control problem with chance constraints is introduced. the parametersof the constraints are fuzzy, random or fuzzy random variables. todefuzzify the constraints, we consider possibility levels.  bychance-constrained programming the chance constraints are converted to crisp constraints which are neither fuzzy nor stochastic and then the resulting classical op...

We prove that, for a two-dimensional Riemannian manifold, the Ricci flow is obtained by a Wiener process.

Existing definitions for optimal orbit determination are not satisfactory. Any useful definition must explicitly address questions relating to sequential processing, linearization, performance function and its extremalization, state estimate structure completeness, use of physics in applicable stochastic processes, and criteria for validation. Such a definition is presented.

Suppose we are given a multi-dimensional Itô process, which can be regarded as a model for an underlying asset price together with related stochastic processes, e.g., volatility. The drift and diffusion terms for this Itô process are permitted to be arbitrary adapted processes. We construct a weak solution to a diffusion-type equation that matches the distribution of the Itô process at each fix...

In this paper, the problem of stability analysis for a class of impulsive stochastic fuzzy neural networks with timevarying delays and reaction-diffusion is considered. By utilizing suitable Lyapunov-Krasovskii funcational, the inequality technique and stochastic analysis technique, some sufficient conditions ensuring global exponential stability of equilibrium point for impulsive stochastic fu...

In this paper, a model of an optimal control problem with chance constraints is introduced. The parametersof the constraints are fuzzy, random or fuzzy random variables. Todefuzzify the constraints, we consider possibility levels.  Bychance-constrained programming the chance constraints are converted to crisp constraints which are neither fuzzy nor stochastic and then the resulting classical op...

Let N(T, d, δ) denote the covering number, i.e., for every δ > 0, let N(T, d, δ) denote the minimal number of d-balls of radius δ required to cover T . The supremum of a stochastic process Xt defined on T , supt∈T Xt can be quantified in terms of N(T, d, δ) (see (Talagrand, 2005, Chapter 1) for instance) under various assumptions on the process Xt. Department of Mathematical Sciences, Universit...

A new, wide class of relativistic stochastic processes is introduced. All relativistic processes considered so far in the literature are members of this class. For each process, the stochastic equations of motion are obtained in an arbitrary Lorentz frame. The associated Kolmogorov equation is also derived for the first time, in both (3 + 1)and manifestly covariant forms.