On Chebyshev polynomials and torus knots

In this work we demonstrate that the q-numbers and their two-parameter generalization, the q,p-numbers, can be used to obtain some polynomial invariants for torus knots and links. First, we show that the q-numbers, which are closely connected with the Chebyshev polynomials, can also be related with the Alexander polynomials for the class T(s,2) of torus knots, s being an odd integer, and used for finding the corresponding skein relation. Then, we develop this procedure in order to obtain, with the help of q,p-numbers, the generalized two-variable Alexander polynomials, and prove their direct connection with the HOMFLY polynomials and the skein relation of the latter.Comment: 6 pages (two-column UJP style

Analysis of the Efficiency PETSc and PETIGA Libraries in Solving the Problem of Crystal Growth

We present an analysis of high performance computational method for solving the problem of crystal grows. The method uses PETSc and PETIGA C-language based libraries and supports parallel computing. The evolution of calculation process was studied in series of special computations are obtained on innovative mobile cluster platform, which provides exclusive system tuning abilities. The results of research confirm the high efficiency of the proposed algorithm on multi-core computer systems and allow us to recommend the use of PETSc and PETIGA for solving high order differential equations

ЗНАЧЕННЯ ОСОБИСТОСТІ ПЕДАГОГА У ФОРМУВАННІ ДУХОВНО-МОРАЛЬНОГО ПОТЕНЦІАЛУ УЧНІВ ( The Value of a Teacher’s Personality in Forming the Spiritual and Moral Potential of the Students)

Метою статті є дослідження значення особистості педагога у формуванні духовного потенціалу учнів. Розглядається роль та основні завдання сучасного педагога (The purpose of paper is to investigate the value of the teacher in shaping of students’ moral and intellectual potential. The role and main tasks of the modern educator are examined

Transients in a circulating fluidized bed boiler

Transients in a circulating fluidized bed boiler firing biomass are considered. An attempt is made to describe transients with the use of concepts applied in the automatic control theory. The parameters calculated from an analysis of unsteady heat balance equations are compared with the experimental data obtained in the 12-MW boiler of the Chalmers University of Technology. It is demonstrated that these equations describe the transient modes of operation with good accuracy. Dependences for calculating the time constants of unsteady processes are obtained. © 2013 Pleiades Publishing, Inc