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.     

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.     

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.     

THE VECTORIAL PHOTO-EXCITATION IN SOLAR-TRACKING LEAVES OF Lavatera cretica (MALVACEAE)*,†

Abstract— The diaphototropic responses of the solar-tracking leaves of Lavatera cretica were studied under constant levels of vectorial photo-excitation (negligible variations in angle of incidence and fluence rates). The results showed the following:
(a) The photoreceptors, that are associated with the (major) veins, can equally perceive vectorial excitation (by an oblique light-beam in the plane of symmetry of the vein), directed either towards its tip (tip-oriented: TO), or towards its base (base-oriented: BO).
(b) The mechanism of photo-perception is apparently qualitatively the same for TO and BO excitation.
(c) The response to TO and BO vectorial excitation, which takes place in a circular sheath of motor-cell tissue in the pulvinus (a2–3 mm long segment at the top of the petiole), is also highly directional and results in bending in the vertical plane of the oblique beam. Each sector of the motor tissue can expand longitudinally in response to TO excitation, presumably by influx of solutes, and contract longitudinally in response to BO excitation of the same vein, presumably by efflux of solutes.
(d) The angular velocity of the vectorial response to a constant level of vectorial excitation remains constant over large angular displacements and long periods, and was similar for TO and BO over a wide range of fluence rates (30-400 μmol m⁻²s⁻¹) of white light (400-700 nm).
(e) Reversal of the direction of vectorial excitation (TO to BO, and vice versa) results in reversal of the direction of the vectorial response (laminar reorientation). The change in the direction of laminar reorientation involved a larger overshoot in the original direction when the change was from BO to TO, than in the reverse case.