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.     

Dual Characterizations of Set Containments with Strict Convex Inequalities

Characterizations of the containment of a convex set either in an arbitrary convex set or in the complement of a finite union of convex sets (i.e., the set, described by reverse-convex inequalities) are given. These characterizations provide ways of verifying the containments either by comparing their corresponding dual cones or by checking the consistency of suitable associated systems. The convex sets considered in this paper are the solution sets of an arbitrary number of convex inequalities, which can be either weak or strict inequalities. Particular cases of dual characterizations of set containments have played key roles in solving large scale knowledge-based data classification problems where they are used to describe the containments as inequality constraints in optimization problems. The idea of evenly convex set (intersection of open half spaces), which was introduced by W. Fenchel in 1952, is used to derive the dual conditions, characterizing the set containments.     

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.