Diffusion approximation of branching migration processes

We study the behavior of a Galton-Watson process with homogeneous migration component stopped at zero (i.e., the state zero is absorbing). Assuming that the process is initiated at time zero by a large number of particles, we find a diffusion approximation for this process in the case where the average number of offspring per individual is close to one. Supported by the Russian Foundation for Fundamental Research (grant Nos. 96-01-00338 and 96-15-96092) and INTAS-RFBR (grant No. 95-0099). Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part III.     

On the stability of characterization by the distribution of any order statistic

Some upper and lower estimates of stability of characterization of distribution by the distribution of any order statistic are obtained. Supported by the Russian Foundation for Fundamental Research (grant Nos. 96-01-01919 and 97-01-00273) and INTAS-RFBR (grant No. IR-98-0537). Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part I.     

On the rate of convergence in a limit theorem for random symmetric polynomials

In terms of the mean metric, a convergence rate estimate is obtained in a limit theorem for random symmetric polynomials. Supported by the Russian Foundation for Fundamental Research (grant No. 96-01-01920) and by the Hungarian National Foundation for Scientific Research (grant No. T 16665). Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, Russia, 1996, Part 11.     

The exponential growth of codimensions for Capelli identities

By applying the theorem that every positive integer is a sum of four squares, we calculate the exponential growth of the codimensions for the relatively free algebra satisfying Capelli identities. Work partially supported by RFFI grants 96-01-00146 and 98-01-01020. Work partially supported by ISF grant 6629/1. Work partially supported by RFFI grants 96-01-00146 and 96-15-96050.     

Extension by zero of functions of several variables

We obtain sufficient conditions on a domainG ⊂ ℝn for functions defined onG to be extendable by zero to the entire space ℝn with smoothness preserved in an integral norm. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 351–365, September, 1998. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00243 and by program “Leading Science Schools” under grant No. 96-15-96102.     

Solutions of the heat equation in domains with singularities

We study the asymptotics of solutions to the Dirichlet problem for the heat equation in time-dependent domains with singular points.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 163–179, August, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00504 and by INTAS under grant No. 93-351.     

The theory of the numerical-analytic method: Achievements and new trends of development. IV

We analyze the application of the numerical-analytic method proposed by Samoilenko in 1965 to autonomous systems of differential equations and impulsive equations. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 12, pp. 1656–1672, December, 1998. This work was partially supported by INTAS (grant No. 96-0915) and OTKA (grant No. TO19095).     

Reconstruction of Sturm-Liouville differential operators with singularities inside the interval

The inverse spectral problem for Sturm-Liouville differential operators on a finite interval is studied for an arbitrary and finite number of regular singular points inside the interval. A uniqueness theorem is proved; necessary and sufficient conditions and a procedure for the solution of the inverse problem are obtained.Translated fromMatematicheskie Zametki, Vol. 64, No. 1, pp. 143–156, July, 1998.This research was supported by the Ministry of Education (KTsFE) under grant No. 96-1.7-4 and by the Russian Foundation for Basic Research under grant No. 97-01-00566.     

Asymptotic behavior of compound nonordinary Cox processes with nonzero means

We present necessary and sufficient conditions for the weak convergence of compound, nonordinary, doubly stochastic Poisson processes (also called compound nonordinary Cox processes) when the jumps have nonzero means and finite variances, describe the class of limit laws, give the convergence rate estimate, and construct the asymptotic expansions for the distributions of compound nonordinary Cox processes and the estimates for their concentration functions. Supported by the Russian Foundation for Fundamental Research (grant Nos. 96-01-01919 and 97-01-00271) and by the Russian Humanitarian Scientific Foundation (grant No. 97-02-02235). Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part I.     

The asymptotic behavior of extrema of compound Cox processes with nonzero means

We give necessary and sufficient conditions for the weak convergence of one-dimensional distributions of extrema of compound Cox processes with jumps having finite variances and nonzero expectations. No moment-type restrictions are imposed on the controlling process. As corollaries we obtain criteria of the normal convergence of extrema of compound Cox processes with nonzero means. Convergence rate estimates are also presented. Supported by the Russian Foundation for Basic Research (grant Nos. 96-01-01919 and 97-01-00271), the Russian Humanitarian Sciences Foundation (grant No. 97-02-02235), and the Committee on Knowledge Extension Research (CKER) of the American Society of Actuaries. Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part I.     

Standard position for objects in hyperbolic space

It is shown that, by applying hyperbolic isometries, one may arrangen points in hyperbolic (n–1)-space in a certain standard position. Similar results are developed for complex hyperbolic space, as well as hyperbolic spaces defined over nearly arbitrary fields. The algebraic basis of the paper is the determination of the structure of a double coset space which occurs in representation theory.Partially supported by NSA grant # MDA-904-96-0018.Partially supported by NSA grant # MDA904-95-1-1089.     

Integral characteristics in the inverse problem of magnetotelluric sounding

This article examines some aspects of the one-dimensional inverse problem of magnetotelluric sounding. A uniqueness theorem is proved in the presence ofS-surfaces. A numerical algorithm based on transformation formulas is proposed. This research was partially supported by Russian Foundation for Basic Research (grant No. 96-01-00410) and by the State Scientific-Technical Program “Future Information Technologies” (grant No. 0201.06.010). Translated from Chislennye Metody v Matematicheskoi Fizike, Moscow State University, pp. 53–66, 1998.     

Convergence rate estimates in local limit theorems for poisson random sums

The asymptotic properties of Poisson random sums are investigated. A series of convergence-rate estimates in local limit theorems for those sums are obtained. Supported by the Russian Foundation for Basic Research (grant Nos. 96-01-01919, 97-01-00271, and 97-01-00273). Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part II.     

Mixed-integer programming—1968 and thereafter

We present some personal reflections on the developments in integer and mixed-integer programming from 1968 up to around 1980. Professor Emeritus, NYU. Work supported by a grant from the Office of Naval Research (N00014-96-0327)     

Solvable elimination of ramification in extensions of discretely valued fields

An analog of the Abhyankar lemma is proved for an arbitrary finite separable extensions of discretely valued fields with a functional residue field. Supported by RFFR grants Nos. 96-01-01675 and 96-01-01678, and by RF Goskomvuz grant No. 2. Translated fromAlgebra i Logika, Vol. 37, No. 1, pp. 63–87., January–February, 1998.     

Critical cases of stability. Converse implicit function theorem for dynamical systems with cosymmetry

Stability criteria of boundary equilibria for dynamical systems in the three critical cases, (n, k)=(3, 0), (2, 1), and (1, 1), are obtained.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 572–578, April, 1998.The author wishes to thank V. I. Yudovich for useful discussions.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01791 and by the International Science Foundation under grant NRQ000.     

Natural estimates of convergence rate in the central limit theorem

The concept of natural convergence rate estimates in the central limit theorem is proposed connecting convergence criteria and convergence rate. Supported by the Russian Foundation for Fundamental Research (grant No. 96-01-01920). Proceedings of the Seminar on Stability Problems for Stochastic Models. Hajdúszoboszló. Hungary. 1997, Part II.     

Some conic bundless

A representation of the anticanonical K3 surface of a singular pencil of conics is described. This generalizes the well-known Shokurov theorem.Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 903–910, June, 1998.The author is greatly indebted to V. A. Iskovskikh, Yu. G. Prokhorov, and I. A. Chel'tsov for fruitful discussions.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00820 and by INTAS under grant No. 93-2805.00-00-00.     

Strong law of large numbers for weighted sums

The law of large numbers for weighted sums of independent identically distributed random variables due to Chow and Lai is generalized to nonidentically distributed or pairwise independent summands. Supported by the Russian Foundation for Fundamental Research (grant No. 96-01-01920). Proceedings of the Seminar on Stability Problems for Stochastic, Models, Moscow, Russia, 1996, Part. II.     

Body forces equivalent to jumps of stresses and displacements in the Biot model of an anisotropic porous medium

Formulas for the body forces equivalent to given jumps of stresses and displacements in the anisotropic Biot model are obtained. Similarly to the case of elastic media, Betti's identity, the reciprocity relations, and the representation integrals for a porous media are derived first. The equivalent body forces are obtained by applying the representation integrals to a volume with surfaces of discontinuity. Some physical consequences are indicated, and examples of equivalent body forces for the isotropic Biot model are provided.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 33–44.The author wishes to express his sincere gratitude to V. M. Babich and L. A. Molotkov for helpful discussions. This work was supported in part by the Soros Foundation under grant A96-378 and the Russian Foundation for Basic Research under grant 96-05-65904.