On an integral equation of the problem for an elastic strip with a slit

A new singular integral equation is obtained that describes the elastic equilibrium of a strip with both an inner and an edge slit (crack) and has a considerable advantage over existing equations /1–9/, etc.) from the viewpoint of a numerical realization and clarification of the analytical relationship with an analogous equation for a half-plane. Numerical results are given of a computation of the stress intensity coefficients at the tips of the inner and edge cracks that refine data in the literature.     

Lieb-thirring integral inequalities and sharp bounds for the dimension of the attractor of the Navier-Stokes equations with friction

A two-dimensional Navier-Stokes system with friction is considered in a large rect-angular periodic domain with area on the order of α^?1, α → 0. Bounds for the dimension of the attractor are obtained, which are sharp both as α → 0 and v → 0, where v is the viscosity coefficient.     

Influence of barrier parameters on tunneling in ferromagnet-insulator-ferromagnet contacts

The contribution of electrons moving at large angles to the barrier junction plane to the tunnel current is calculated. This contribution turns out to be small only if the Fermi energy of the electrons equals several electron volts. Otherwise, specifically, when the Fermi energy is no higher than 1–2 eV, this contribution dominates in high and thin potential barriers. It is found that the tunnel magnetic resistance in ferromagnet-insulator-ferromagnet contacts correlates with this contribution. It is this correlation that is responsible for a decrease in this contribution as the potential barriers get lower and thicker.     

Method of linear prediction in the ultrasonic spectroscopy of rock

The possibility of using the resonant acoustic spectroscopy method for testing the internal state of rock is considered. This method can be used for the determination of the concentration of defects and their spatial distribution. A rock specimen with defects is characterized by a high loss of vibrational energy, which leads to the overlapping of the resonant responses. In this case, the complex resonant frequencies are determined as the parameters of a model that describes the measured transfer function of the response of the mechanical system. The linearity of the vibratory system is used as the a priori information for presetting the functional dependences of the model. An algorithm is developed for the determination of the parameters of the resonances in conditions of their partial overlapping. Examples are presented to demonstrate the possibility of analyzing the complex elastic constant tensor of rock by the acoustic spectroscopy method. The proposed method of experimental data processing provides the possibility for the determination of the weak anisotropy that occurs in the velocity of sound because of a partial fracture of rock.     

Boundary Problems for the Quantum Bose Gas

A relaxation kinetic equation that describes the behavior of a Bose gas is derived. The Kramers half-space problem on isothermal slip is treated. An analytical solution and the number-of-particle distribution function for particles flying toward a wall, in explicit form, are obtained. The dependence of the slip velocity on the parameter that is the ratio of the chemical potential to the product of Boltzmann's constant by the absolute temperature is analyzed. The influence of the quantum effects on the isothermal slip coefficient is evaluated for He⁴.     

The effect of geoacoustic characteristics of the bottom on the long-range sound propagation in an inhomogeneous ocean

The long-range sound propagation from a deep ocean to a receiving system located on the shelf is modeled. The waveguide model is constructed on the basis of the data of an acoustic-oceanographic experiment carried out in the northwestern Pacific. The sensitivity and the frequency dependence of the difference in the sound field levels at the crossing of the frontal zone on the geoacoustic characteristics of the bottom of the shelf and the continental slope are investigated. It is shown that the level difference decreases by 8.2 dB as the velocity of longitudinal waves increases by 100 m/s in the range within 1490–1820 m/s.     

Factor and term ranks for matrix union over semirings

We consider the arithmetic properties of the factor-rank and term-rank functions for matrices over semirings. In particular, we investigate the sets of matrices that satisfy the extremes of inequalities for these rank functions of matrix union. The classification of the linear transformations that keep these sets invariant is obtained. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 3, pp. 175–197, 2003.     

Prognostication of aging of polymeric composite materials

Experimental data demonstrate that a study of accelerated aging can enable prognostication of the working capacity and storage life of polymeric formulations used in aircraft components on the basis of changes in thermogravimetric constants of these polymeric formulations.     

A Symmetric-Impulse Approximation Applied to the Faddeev Theory on the Charge Transfer Amplitude

Off-the-shell anomalous factors of the two-body Coulomb transition matrices appear in the integral form of the Faddeev second-order nuclear-electronic amplitude, for proton-hydrogen charge transfer scattering in a typical nlm → n′l′m′ transition. A symmetric-impulse approximation (SIA) is applied to eliminate these factors and an induction method is proposed to analytically calculate the remaining integrals. The nuclear-electronic amplitude is derived for the general case, and for totally symmetric collisions, in terms of generalized hypergeometric functions of two variables, F ₄, and of one variable, ₃ F ₂, respectively. The angular distribution of the second-order nuclear-electronic charge transfer amplitude shows the Thomas mechanism as a peak or a hump for symmetric and asymmetric collisions. There also exists a peak in the forward angular distribution of the second-order nuclear-electronic amplitude, which partly cancels the kinematic peak in the angular distribution of the charge transfer differential cross sections.