Vol. 44 No. 3 (223) (2010)

The mathematical investigation of large-scale biomolecular sequences is being carried out by analyzing the properties of events arising in such sequences. This survey is devoted to discussion of results in this field. Based on general empirical facts being fulfilled for all frequency distributions, we discuss the axiomatics suggested by J. Astola and E. Danielyan. The axiomatics postulates the regular variation of frequency distribution with asymptotically constant slowly varying component, the form of its shape, and the stability by parameters. The verification of the axiomatics fulfillment for wellknown frequency distributions is done. The paper describes also methods of construction of new parametric families of frequency distributions. These methods are: usage of stationary distributions of birth-death process, special functions, stable densities, etc. The problem of stability by parameters is formulated the results on stability by parameters in terms of various classical metrics are given. The conditions of regular variation for different families of frequency distributions are formulated.

In the present paper the Urysohn type nonlinear integral equation with noncompact operator on the half-line is considered. It is assumed that the Wiener–Hopf–Hankel type operator is a local minorant for the initial Urysohn operator. The existence of a positive and bounded solution is proved. The limit of constructed solution at infinity is calculated. At the end of the work a list of examples is given.

The paper studies differential equations of fractional order of the form $D^{1/\rho}y(z)+\lambda y(z)=f(z)$ in the complex domain, where$\rho\geq 1$, $\lambda$ is an arbitrary parameter, $D^{1/\rho}$ is the Riemann–Liouville differential operator. For functions of some classes Cauchy type problems are considered.

We study some properties of algebras of bounded continuous functions on a completely regular space, these algebras being equipped with the strong topology defined by of family multiplication operators (strict uniform algebras). We prove an analog of a theorem due to M. Sheinberg for strict uniform algebras (see [1–3]).

A group $G$ is called unitarizable, if every uniformly bounded representation $\pi:G \rightarrow B(H)$ of $G$ on a Hilbert space $H$ is unitarizable. N. Monod and N. Ozawa in [6] prove that free Burnside groups $B(m, n)$ are non unitarizable for arbitrary composite odd number $n=n_1n_2$ , where $n_1\geq 665$ . We prove that for the same $n$ the groups $B(4, n)$ have continuum many non-isomorphic factor-groups, each one of which is non-unitarizable and uniformly non-amenable.

In solution of plane problems of mechanics there are several elements used in finite element software. In ANSYS – one of the best finite element software – there are about six type of elements. We consider different Plane Models in simple bending problem and compare the various results to distinguish the type of elements, which are suitable for solving the problem under consideration. Comparing results with analytical solutions shows that the Plane 42 and 82 models are the most suitable ones among the others. Also results in Plane 42 with usual mesh is closer to the same problem solution with fine mesh than Plane 82 in the same case.

The behavior of aerosol particles was studied at filtration through super-thin basalt fibers. Diffusion and inertial effects were investigated at the particle sedimentation using theoretical models for fiber filters taking into account the aerosol gas capture factor. Composition of radioactive aerosols was analyzed over the period of Armenian NPP operation.

We present the development results of a new method of aerial (on a helicopter or airplane) infrared (IR) scanning of extensive spaces with the purpose of detecting weak heat sources (fire centers at an early stage of their development) to prevent the occurrence of large-scale forest fires. The paper presents the description of the IR radiometer as well as the measurement method of point and extended thermal sources with wavelength range of 2.5 to 5.5 microns.

A method for measuring of any composition films and tapes thickness with a nanometer resolution is suggested and validated experimentally. That operates on the base of a single-layer flat-coil-oscillator technique. A laboratory prototype of a device is designed and created, based on this method. Besides, PC operation in a “NI LabVIEW” software environment, as well as preliminary tests and calibration of the created device is implemented. It may find variety of applications in a research and in high-tech technology.

The paper generalizes one formula of traces, established by J. Neumann for square matrices, for nuclear operators.