On completely equidistributed numbers

We analyzed a new method for proving equidistribution of numbers. The proposed method is simple and can be used to prove equidistribution of all known classes of numbers such as Weyl\u27s numbers and Koksma\u27s numbers. Emphasis of this approach is put on the complete equidistribution of numbers and a non-trivial result in this direction is obtained

Explicit stable methods for second order parabolic systems

We show that it is possible to construct stable, explicit finite difference approximations for the classical solution of the initial value problem for the parabolic systems of the form $partial_tu=A(t,{bf x})u+f$ on $R^d$, where $A(t,{bf x}) = sum_{ij} a_{ij}(t,{bf x}) partial_ipartial_j + sum_i b_i(t,{bf x}) partial_i + c(t,{bf x})$. The numerical scheme relies on an approximation of the elliptic operator $A(t,{bf x})$ on an equidistant mesh by matrices that possess structure of a generator of Markov jump process. In the case of ${R}^2$ scaling of second difference operators can be applied to get the necessary structure of approximations, while in the case of $R^d, : d > 2$, rotations at grid-knots are performed in order to get the mentioned structure. Numerical experiments illustrate the theory

Solving 2^{nd} order parabolic system by simulations of Markov jump processes

There are known methods of approximating the solution of parabolic 2^{nd} order systems by solving stochastic differential equations instead. The main idea is based on the fact that a stochastic differential equation defines a diffusion process, generated by an elliptic differential operator on R^{d}. We propose a difference scheme for the elliptic operator, which possesses the structure of Markov (jump) process. The existence of such a scheme is proved, the proof relying on the choice of new coordinates in which the elliptic operator is "almost\u27\u27 Laplacian, and has the properties necessary for discretization. Time discretization, which involves difference schemes for parabolic equations with known stability difficulties, can thus be replaced by space discretization and simulation of the associated Markov (jump) process

Solving 2^{nd} order parabolic system by simulations of Markov jump processes

There are known methods of approximating the solution of parabolic 2^{nd} order systems by solving stochastic differential equations instead. The main idea is based on the fact that a stochastic differential equation defines a diffusion process, generated by an elliptic differential operator on R^{d}. We propose a difference scheme for the elliptic operator, which possesses the structure of Markov (jump) process. The existence of such a scheme is proved, the proof relying on the choice of new coordinates in which the elliptic operator is "almost\u27\u27 Laplacian, and has the properties necessary for discretization. Time discretization, which involves difference schemes for parabolic equations with known stability difficulties, can thus be replaced by space discretization and simulation of the associated Markov (jump) process