Some Special Tauberian Theorems for Fourier Type Integrals and Eigenfunction Expansions of Sturm-Liouville Equations on the Whole Line

We offer a new proof of a special Tauberian theorem for Fourier type integrals. This Tauberian theorem was already considered by us in the papers [1] and [2]. The idea of our initial proof was simple, but the details were complicated because we used Bochner's definition of generalized Fourier transform for functions of polynomial growth. In the present paper we work with L. Schwartz's generalization. This leads to significant simplification. The paper consists of six sections. In Section 1 we establish an integral representation of functions of polynomial growth (subjected to some Tauberian conditions), in Section 2 we prove our main Tauberian theorems (Theorems 2.1 and 2.2.), using the integral representation of Section 1, in Section 3 we study the asymptotic behavior of M. Riesz's means of functions of polynomial growth, in Sections 4 and 5 we apply our Tauberian theorems to the problem of equiconvergence of eigenfunction expansions of Sturm-Liouville equations and expansion in ordinary Fourier integrals, and in Section 6 we compare our general equiconvergence theorems of Sections 4 and 5 with the well known theorems on eigenfunction expansions in classical orthogonal polynomials. In some sense this paper is a re-made survey of our results obtained during the period 1953-58. Another proof of our Tauberian theorem and some generalization can be found in the papers [3] and [4].     

Effect of a longitudinal magnetic field on the characteristics of a stabilized channel arc

The results of calculations of the temperature profiles and volt-ampere characteristics of a long cylindrical argon arc in a longitudinal uniform magnetic field are presented. The calculation was made for the following parameters: pressure p =0.1–10.0 atm; temperatures T = 1000-20,000°K; magnetic field induction B =0-10 T; diameter of cylindrical channel d = 1.0 cm. It is shown that for strongly radiating arcs (p1.0 atm) the temperature profiles become more inflated with an increase in the magnetic field, while for weakly radiating arcs (p 0.1 atm) the appearance of loops in the volt-ampere characteristics is typical for certain conditions (14,000T20,000°K, B1.0 T), indicating the impossibility of arcing under these conditions.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 147–153, March–April, 1975.     

Problem of focusing and the asymptotics of the spectral function of the Laplace-Beltrami operator. I

The connection between the asymptotics of the spectral function and the formal shortwave expansion of the solution of the problem of the asymptotics of the Green function near a geodesically concave boundary of a two-dimensional surface is considered in the paper.     

Ostwald Ripening in Two Dimensions: Treatment with Pairwise Interactions

Ostwald ripening is the last stage of the evolution of a system with two coexisting phases. It is a relatively simple nonequilibrium phenomenon with several interesting features. For example, as the system coarsens it goes through a scaling state, one which looks the same (up to an overall length scale, which grows) at all times. The dynamics of the problem can be mapped, in two dimensions, onto an evolving Coulomb system. In this work we present a brief summary of a novel theoretical approach to this problem, based on an analytic derivation (using a mean-field approach) of an effective two-body interaction between droplets of the minority phase. The resulting interacting many-body dynamics is solved by a very efficient numerical algorithm, allowing us to follow the evolution of more than 10⁶ droplets on a simple workstation. The results are in excellent agreement with recent experiments.     

Formation of dislocations with local mechanical damage to a silicon surface at room temperature

A study is made of the formation of dislocations in silicon with local damage to the surface at room temperature and subsequent annealing within the range 1073–1473 K. The damages to the surface are modeled with the use of micro-hardness indentations. Measurements of mean linear dislocation density in a ray of the indentation rosette show that the number of dislocations in the rosette is independent of both the temperature and duration of isothermal annealing. It was found that annealing at 573–773 K leads to partial relaxation of elastic stresses from the indentation due to the formation of sections of silicon with a hexagonal structure near the indentation. Further annealing at high temperatures leads to the disappearance of these sections and the formation of a normal dislocation rosette, with the number of dislocations in the rays corresponding to the case of one-stage annealing. The results are empirical confirmation of the hypothesis of incomplete shear. In accordance with the latter, dislocations are formed during deformation at room temperature, not during subsequent annealing.Zaporozh'e University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 78–82, May, 1992.