Vol 23, No 2 (2017)

In the article we consider structure problems in the theory of modular forms. The phenomenon of the exact cutting for the spaces Sk(Γ0(N), χ), where χ is a quadratic character with the condition χ(−1) = = (−1)k . We prove that for the levels N ̸= 3, 17, 19 the cutting function is a multiplicative eta–product of an integral weight. In the article we give the table of the cutting functions. We prove that the space of an cutting function is one–dimensional. Dimensions of the spaces are calculated by the Cohen-Oesterle formula, the orders in cusps are calculated by the Biagioli formula.

In this paper we investigated the qualitative properties of the operator of fractional differentiation in Kipriyanov sense. Based on the concept of multidimensional generalization of operator of fractional differentiation in Marchaud sense we have adapted earlier known techniques of proof theorems of one-dimensional theory of fractional calculus for the operator of fractional differentiation in Kipriyanov sense. Along with the previously known definition of the fractional derivative in the direction we used a new definition of multidimensional fractional integral in the direction of allowing you to expand the domain of definition of formally adjoint operator. A number of theorems that have analogs in one-dimensional theory of fractional calculus is proved. In particular the sufficient conditions of representability of a fractional integral in the direction are received. Integral equality the result of which is the construction of the formal adjoint operator defined on the set of functions representable by the fractional integral in direction is proved.

In the work transition to discrete time in the equation of movement of van der Pol – Dyuffing generator is described. The transition purpose—to create mappings of the generator as subjects of the theory of nonlinear oscillations (nonlinear dynamics) in discrete time. The method of sampling is based on the use of counting of the pulse characteristic of an oscillatory contour as the sampling series for a signal in a self-oscillating ring ”active nonlinearity – the resonator – feedback”. The choice of the consecutive scheme of excitement of a contour allows to receive the iterated displays in the form of recurrent formulas. Two equivalent forms of discrete displays of the generator of van der Pol – Dyuffing—complex and valid are presented. In approximation of method of slow-changing amplitudes it is confirmed that the created discrete mappings have dynamic properties of an analog prototype. Also within the numerical experiment it is shown that in case of the high power of generation the effect of changing of frequencies of harmonicas of the generated discrete signal significantly influence dynamics of the self-oscillators. In particular, in the discrete generator of van der Pol – Dyuffing the chaotic self-oscillations are observed.