Vol 49, No 1 (2016)

The first Jacobi–Trudi identity expresses Schur polynomials as determinants of matrices, the entries of which are complete homogeneous polynomials. The Schur polynomials were defined by Cauchy in 1815 as the quotients of determinants constructed from certain partitions. The Schur polynomials have become very important because of their close relationship with the irreducible characters of the symmetric groups and the general linear groups, as well as due to their numerous applications in combinatorics. The Jacobi–Trudi identity was first formulated by Jacobi in 1841 and proved by Nicola Trudi in 1864. Since then, this identity and its numerous generalizations have been the focus of much attention due to the important role which they play in various areas of mathematics, including mathematical physics, representation theory, and algebraic geometry. Various proofs of the Jacobi–Trudi identity, which are based on different ideas (in particular, a natural combinatorial proof using Young tableaux), have been found. The paper contains a short simple proof of the first Jacobi–Trudi identity and discusses its relationship with other well-known polynomial identities.

The separation of signal components is an important problem of time-series analysis. For example, the solution of this problem allows one to extract a trend and to separate harmonic signals with different frequencies. In the paper, the modification of the singular spectrum analysis (SSA) method is considered for improving the separability of time-series components. The new method is called SSA-AMUSE, since it is based on the AMUSE method, which is used to apply independent component analysis to signal separation. The suggested modification weakens the conditions of the so-called strong separability and, thus, improves the quality of the separation of time-series components by comparing similar methods. The paper contains proof of the algorithm, as well as the conditions of separability for the considered modification. Besides the exact separability, the asymptotic separability is also considered. The separability conditions are applied to the case of two harmonic time series. It appears that separability by SSA-AMUSE does not depend on the amplitudes of the separated harmonics, while the Basic SSA method requires different amplitudes. A numerical example demonstrates the advantage of the SSA-AMUSE method compared with a similar modification.

The paper presents a method of finding optimal control of generalized deterministic abstract automaton, the structure of which is given by an arbitrary finite graph in a fuzzy environment. The control is found in order to achieve a fuzzy goal, which is given as a fuzzy set in any fixed finite vertex of the automaton structural graph. The problem solution is divided into two stages. The first stage provides the greatest possible degree of achieving the fuzzy goal depending on the path from the initial graph vertex to the fixed one, while the second stage makes it possible to construct a set of input words that ensure the achievement of this goal on the selected path. The conclusion presents an example of the application of the proposed method for constructing a regular expression of control sequences for the given abstract finite-nonstationary deterministic automaton.

For an arbitrary local field K (a finite extension of the field Q_p) and an arbitrary formal group law F over K, we consider an analog c_F of the classical Hilbert pairing. A theorem by S.V. Vostokov and I.B. Fesenko says that if the pairing c_F has a certain fundamental symbol property for all Lubin–Tate formal groups, then c_F = 0. We generalize the theorem of Vostokov–Fesenko to a wider class of formal groups. Our first result concerns formal groups that are defined over the ring O_K of integers of K and have a fixed ring O₀ of endomorphisms, where O₀ is a subring of O_K. We prove that if the symbol c_F has the above-mentioned symbol property, then c_F = 0. Our second result strengthens the first one in the case of Honda formal groups. The paper consists of three sections. After a short introduction in Section 1, we recall basic definitions and facts concerning formal group laws in Section 2. In Section 3, we state and prove two main results of the paper (Theorems 1 and 2). Refs. 8.

The longitudinal impact on a thin elastic rod, which generates a periodic system of longitudinal waves in it, is considered. At definite values of the parameters of the problem in the linear approximation, these waves induce parametric resonances, which are accompanied by an unlimited increase in the amplitude of the transverse vibrations. To obtain finite values of the amplitudes, a quasilinear system is considered in which the effect of the transverse vibrations on the longitudinal vibrations is taken into account. This system was previously solved using the Bubnov–Galerkin method and beats accompanied by energy transfer between the transverse and longitudinal vibrations were obtained. In this work, an approximate analytical solution of the system has been derived that is based on double-scale expansions. A qualitative analysis of this solution has been carried out. An estimate of the maximum transverse bending has been obtained for various methods of loading. Both shortand long-term pulses have been considered. It has been shown that, in the case of a spontaneously applied long-term pulse that is lower than the Euler critical load, intensive transverse vibrations can occur.

The paper discusses the impact of the material properties of transversely isotropic circular plates on its natural frequencies. Two refined theories of plates have been used to analyze the free vibration behavior of homogeneous plates. Both theories take into account normal and rotary inertias. Fundamental frequencies for plates with radial inhomogeneity have been obtained with the help of finite element package Comsol Multiphysics 5.0. It has been shown that the inhomogeneity of the plate have a profound impact on the first (lowest) frequency of the plate, while the plate orthotropy has a greater influence on the second and higher vibration mode [2] (Fig. 1, Table 1).

Based on modified Flügge equations and nonlocal elasticity theory, free axisymmetric oscillations of a long double-walled carbon nanotube embedded into an inhomogeneous elastic medium is studied. The ambient medium is simulated by the Winkler foundation. Van der Waals forces are introduced in order to take into account the interaction between the nanotube walls. Using Tovstik’s asymptotic method, eigenmodes are constructed in the form of functions that decay far from the line on the surface of the outer wall, on which the modulus of subgrade reaction has a local minimum. Eigenmodes and eigenfrequencies corresponding to the coand counterdirected wall motions are found. It has been found that introducing a nonlocality parameter into the model results in eigenmodes that are not inherent in macroscale shells. In particular, an increase in the stretching force leads first to greater localization of vibrations and increase in the amplitudes of tangential atomic oscillations and, second, to reduction in the frequencies in the case when the tube lies in a sufficiently stiff medium.