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.     

Synthesis of cationic molybdenum and cobalt diene complexes from neutral allyl compounds by ligand exchange in a strong protic acid

Conclusions The allyl ligand in neutral molybdenum and cobalt complexes is readily exchanged by dienes in the presence of a strong protic acid with the formation of cationic complexes. This exchange is facilitated by an increase in the basicity of the starting compounds.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 7, pp. 1655–1657, July, 1987.     

The Pathwise Numerical Approximation of Stationary Solutions of Semilinear Stochastic Evolution Equations

Under a one-sided dissipative Lipschitz condition on its drift, a stochastic evolution equation with additive noise of the reaction-diffusion type is shown to have a unique stochastic stationary solution which pathwise attracts all other solutions. A similar situation holds for each Galerkin approximation and each implicit Euler scheme applied to these Galerkin approximations. Moreover, the stationary solution of the Euler scheme converges pathwise to that of the Galerkin system as the stepsize tends to zero and the stationary solutions of the Galerkin systems converge pathwise to that of the evolution equation as the dimension increases. The analysis is carried out on random partial and ordinary differential equations obtained from their stochastic counterparts by subtraction of appropriate Ornstein-Uhlenbeck stationary solutions.     

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.     

Morphological and electronic properties of the thin film phase of pentacene investigated by AFM and STM/STS

We investigated the morphological, structural and electronic properties of Pentacene thin films grown by vacuum thermal evaporation on different inert substrates at room temperature. The results of our AFM and STM analysis give an interplanar spacing of 1.54 nm corresponding to the (0 0 1) distance of the so-called “thin film phase”. The STS measurements show an HOMO-LUMO gap of 2.2 eV.     

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.