Abstract:
Filtering of multi-band bandlimited signals by means of a linear digital filter with one or more stopbands is explored. The main goal of the paper is to demonstrate that such a task can be accomplished using sampling rates lower than Landau rate, where the Landau rate is defined as the total bandwidth of the input signal. In order to reach such low rates Periodic Nonuniform Sampling is employed. We show that the proposed filtering method is most efficient when bandpass and multiband filtering is required. Necessary and sufficient conditions for filtering are derived, and an algorithm for designing PNS grids that allow sub-Landau sampling and filtering is proposed. Reconstruction systems are discussed and experimental examples are presented, which confirm the feasibility of the approach.

Abstract:
In the paper are studied the deformations of the planetary orbits caused by the time dependent gravitational potential in the universe. It is shown that the orbits are not axially symmetric and the time dependent potential does not cause perihelion precession. It is found a simple formula for the change of the orbit period caused by the time dependent gravitational potential and it is tested for two binary pulsars.

Abstract:
In the paper [4] is presented a theory which unifies the gravitation theory and the mechanical effects, which is different from the Riemannian theories like GTR. Moreover it is built in the style of the electomagnetic field theory. This paper is a continuation of [4] such that the complex variant of that theory yields to the required unification of gravitation and electromagnetism. While the gravitational field is described by a scalar potential $\mu $, taking a complex value of $\mu $ we obtain the unification theory. For example the electric field appears to be imaginary 3-vector field of acceleration, the magnetic field appears to be imaginary 3-vector of angular velocity and the imaginary part of a complex mass is just electric charge of the particle.

Abstract:
In this paper an alternative theory about space-time is given. First some preliminaries about 3-dimensional time and the reasons for its introduction are presented. Alongside the 3-dimensional space (S) the 3-dimensional space of spatial rotations (SR) is considered independently from the 3-dimensional space. Then it is given a model of the universe, based on the Lie groups of real and complex orthogonal 3x3 matrices in this 3+3+3-dimensional space. Special attention is dedicated for introduction and study of the space SxSR, which appears to be isomorphic to SO(3,R)xSO(3,R) or S^3xS^3. The influence of the gravitational acceleration to the spinning bodies is considered. Some important applications of these results about spinning bodies are given, which naturally lead to violation of Newton's third law in its classical formulation. The precession of the spinning axis is also considered.

Abstract:
We study Gibbs measures invariant under the flow of the NLS on the unit disc of $\R^2$. For that purpose, we construct the dynamics on a phase space of limited Sobolev regularity and a wighted Wiener measure invariant by the NLS flow. The density of the measure is integrable with respect to the Wiener measure for sub cubic nonlinear interactions. The existence of the dynamics is obtained in Bourgain spaces of low regularity. The key ingredient are bilinear Strichartz estimates for the free evolution. The bilinear effect in our analysis results from simple properties of the Bessel functions and estimates on series of Bessel functions.

Abstract:
We prove the existence and the invariance of a Gibbs measure associated to the defocusing sub-quintic Nonlinear Schroedinger equations on the disc of the plane $\R^2$. We also prove an estimate giving some intuition to what may happen in 3 dimensions.

Abstract:
These notes are devoted to the notion of well-posedness of the Cauchy problem for nonlinear dispersive equations. We present recent methods for proving ill-posedness type results for dispersive PDE's. The common feature in the analysis is that the proof of such results requires the construction of high frequency approximate solutions on small time intervals (possibly depending on the frequency).

Abstract:
We define a finite Borel measure of Gibbs type, supported by the Sobolev spaces of negative indexes on the circle. The measure can be seen as a limit of finite dimensional measures. These finite dimensional measures are invariant by the ODE's which correspond to the projection of the Benjamin-Ono equation, posed on the circle, on the first N>>1 modes in the trigonometric bases.

Abstract:
We prove the quasi-invariance of gaussian measures (supported by functions of increasing Sobolev regularity) under the flow of one dimensional Hamiltonian PDE's such as the regularized long wave (BBM) equation.

Abstract:
We introduce a new approach to the fractional derivatives of the analytical functions using the Taylor series of the functions. In order to calculate the fractional derivatives of f, it is not sufficient to know the Taylor expansion of f, but we should also know the constants of all consecutive integrations of f. For example, any fractional derivative of ex is ex only if we assume that the nth consecutive integral of ex is ex for each positive integer n. The method of calculating the fractional derivatives very often requires a summation of divergent series, and thus, in this note, we first introduce a method of such summation of series via analytical continuation of functions.