Abstract:
Using the Walsh-Fourier transform, we give a construction of compactly supported nonstationary dyadic wavelets on the positive half-line. The masks of these wavelets are the Walsh polynomials defined by finite sets of parameters. Application to compression of fractal functions are also discussed.

Abstract:
An explicit description of all Walsh polynomials generating tight wavelet frames is given. An algorithm for finding the corresponding wavelet functions is suggested, and a general form for all wavelet frames generated by an appropriate Walsh polynomial is described. Approximation properties of tight wavelet frames are also studied. In contrast to the real setting, it appeared that a wavelet tight frame decomposition has an arbitrary large approximation order whenever all wavelet functions are compactly supported.

Abstract:
The lack of Birkhoff
theorem in finite-range gravitation reveals nonzero acceleration of the test
body inside the massive spherical shell, as well as breakdown of screening
inside the charged conductor gives rise to acceleration of the test charge. An
application of this effect to the motion of galaxies in Local Group allows to
constraint quintessence parameter in some massive gravitational theories.

Abstract:
Hydrodynamic equations for ideal incompressible fluid are written in terms of generalized stream function. Two-dimensional version of these equations is transformed to the form of one dynamic equation for the stream function. This equation contains arbitrary function which is determined by inflow conditions given on the boundary. To determine unique solution, velocity and vorticity (but not only velocity itself) must be given on the boundary. This unexpected circumstance may be interpreted in the sense that the fluid has more degrees of freedom than it was believed. Besides, the vorticity is a less observable quantity as compared with the velocity. It is shown that the Clebsch potentials are used essentially at the description of vortical flow.

Abstract:
A geometric conception is a method of a geometry construction. The Riemannian geometric conception and a new T-geometric one are considered. T-geometry is built only on the basis of information included in the metric (distance between two points). Such geometric concepts as dimension, manifold, metric tensor, curve are fundamental in the Riemannian conception of geometry, and they are derivative in the T-geometric one. T-geometry is the simplest geometric conception (essentially, only finite point sets are investigated) and simultaneously, it is the most general one. It is insensitive to the space continuity and has a new property: the nondegeneracy. Fitting the T-geometry metric with the metric tensor of Riemannian geometry, we can compare geometries, constructed on the basis of different conceptions. The comparison shows that along with similarity (the same system of geodesics, the same metric) there is a difference. There is an absolute parallelism in T-geometry, but it is absent in the Riemannian geometry. In T-geometry, any space region is isometrically embeddable in the space, whereas in Riemannian geometry only convex region is isometrically embeddable. T-geometric conception appears to be more consistent logically, than the Riemannian one.

Abstract:
We present two versions of the Egorov theorem for orbifolds. The first one is a straightforward extension of the classical theorem for smooth manifolds. The second one considers an orbifold as a singular manifold, the orbit space of a Lie group action, and deals with the corresponding objects in noncommutative geometry.

Abstract:
Discrete cosine transform dct-i, unlike dct-ii, does not concentrate the energy of a transformed vector sufficiently well, so it is not used practically for the purposes of digital image compression. By performing regular normalization of the basic cosine transform matrix, we obtain a discrete cosine transform which has the same cosine basis as dct-i, coincides as dct-i with its own inverse transform, but unlike dct-i, it does not reduce the proper ability of cosine transform to the energy concentration. In this paper we consider briefly the properties of this transform, its possible integer implementation for the case of 8x8-matrix, its applications to the image itself and to the preliminary rgb colour space transformations, further more we investigate some models of quantization, perform an experiment for the estimation of the level of digital images compression and the quality achieved by use of this transform. This experiment shows that the transform can be sufficiently effective for practical use, but the question of its comparative effectiveness with respect to dct-ii remains open.