On a multitype Galton-Watson process with state-dependent immigration

We study the behavior of a multitype Galton-Watson process with state-dependent immigration. A limit theorem is obtained in the near-critical case. 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, Vologda, Russia, 1998, Part I.     

Complementarity and nondegeneracy in semidefinite programming

Primal and dual nondegeneracy conditions are defined for semidefinite programming. Given the existence of primal and dual solutions, it is shown that primal nondegeneracy implies a unique dual solution and that dual nondegeneracy implies a unique primal solution. The converses hold if strict complementarity is assumed. Primal and dual nondegeneracy assumptions do not imply strict complementarity, as they do in LP. The primal and dual nondegeneracy assumptions imply a range of possible ranks for primal and dual solutionsX andZ. This is in contrast with LP where nondegeneracy assumptions exactly determine the number of variables which are zero. It is shown that primal and dual nondegeneracy and strict complementarity all hold generically. Numerical experiments suggest probability distributions for the ranks ofX andZ which are consistent with the nondegeneracy conditions. Supported in part by the U.S. National Science Foundation grant CCR-9625955. Supported in part by U.S. National Science Foundation grant CCR-9501941 and the U.S. Office of Naval Research grant N00014-96-1-0704. Supported in part by U.S. National Science Foundation grant CCR-9401119.     

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.     

Uniformity of stably integral points on principally polarized abelian varieties of dimension ≤2

The purpose of this paper is to prove, assuming that the conjecture of Lang and Vojta holds true, that there is a uniform bound on the number of stably integral points in the complement of the theta divisor on a principally polarized abelian surface defined over a number field. Most of our argument works in arbitrary dimension and the restriction on the dimension ≤2 is used only at the last step, where we apply Pacelli’s stronger uniformity results for elliptic curves. Partially supported by NSF grant DMS-9700520 and by an Alfred P. Sloan research fellowship. Partially supported by NSA grant MDA904-96-1-0008.     

Abstract isomorphisms of quasiminimal groups

A connected solvable algebraic group is called minimal if its center is trivial and it has no proper connected subgroups with trivial center. We show that abstract isomorphisms of minimal groups defined over fields of characteristic zero are standard. Supported by RFFR grants Nos. 96-01-01678 and 96-01-01675, and by the State Committee for Higher Education of Russia, grant No. 2. Translated fromAlgebra i Logika, Vol. 35, No. 5, pp. 567–586, September–October, 1996.     

On the weak ergodicity of nonhomogeneous continuous-time Markov chains

The conditions of weak ergodicity for nonhomogeneous continuous-time Markov chains are obtained in terms of intensities. The birth and death process on a finite state space is studied as a special case. Supported by the Russian Foundation for Fundamental Research (grant No. 96-01-01369). Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part III.     

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.     

Groups of formal power series are fully orderable

We construct an example of a fully orderable group that is not locally solvable. It is also shown that a free group is embedded in a fully orderable group. To meet these ends, use is made of a group of invertible formal power series with zero free term under composition. Supported by RFFR grant No. 96-01-00088. Translated fromAlgebra i Logika, Vol. 37, No. 3, pp. 301–319, May–June, 1998.     

On representation varieties of Artin groups, projective arrangements and the fundamental groups of smooth complex algebraic varieties

We prove that for any affine variety S defined overQ there exist Shephard and Artin groups G such that a Zariski open subset U of S is biregular isomorphic to a Zariski open subset of the character variety X(G, PO(3)) = Hom(G, PO(3))//PO(3). The subset U contains all real points of S. As an application we construct new examples of finitely-presented groups which are not fundamental groups of smooth complex algebraic varieties. This research was partially supported by NSF grant DMS-96-26633 at University of Utah. This research was partially supported by NSF grant DMS-95-04193 at University of Maryland.     

Zero energy asymptotics of the resolvent for a class of slowly decaying potentials

We prove a limiting absorption principle at zero energy for two-body Schrödinger operators with long-range potentials having a positive virial at infinity. More precisely, we establish a complete asymptotic expansion of the resolvent in weighted spaces when the spectral parameter tends to zero in cones which are adjacent to the positive real axis. The principal tools are absence of eigenvalue at zero, singular Mourre theory and microlocal estimates.in final form: 14 November 2003S. Fournais was supported by a grant from the Carlsberg Foundation (before 31.12.02) and by a Marie Curie Fellowship of the European Community Programme Improving the Human Research Potential and the Socio-Economic Knowledge Base under contract number HPMF-CT-2002-01822 (from 01.01.03).E. Skibsted is (partially) supported by MaPhySto – A Network in Mathematical Physics and Stochastics funded by The Danish National Research Foundation.     

Torsion sections of elliptic surfaces

Given a torsion section of a semistable elliptic surface we prove equidistribution results for the components of singular fibers which are hit by the section and for the root of unity (identifying the zero component withC) which is hit by the section in case the section hits the zero component Research supported in part by the NSF under grant DMS 9104058 and the NSA under grant MDA 904-92 H 3022     

System stability and optimal control

Stochastic models arising in inventory, insurance, queueing theory, and other areas are considered. The underlying process is a random walk with two absorbing boundaries, 0 and n. It is proved that the introduction of a multi-level control radically changes the asymptotic behavior (as n→∞) of normalized absorption time and simultaneously stabilizes the system. Some optimization problems are solved as well. Supported by the Russian Foundation for Fundamental Research (grant No. 96-01-01092). Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, 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.     

Heavy traffic analysis of M/G/1 type queueing systems with Markov-modulated arrivals

M/G/1 type queueing systems whose arrival rate is a function of an independent continuous time Markov chain are considered. We suggest a simple analytical approach which allows rigorous mathematical analysis of the stationary characteristics under heavy traffic. Their asymptotic behaviour is described in terms of characteristics of the modulating process (defined as a solution of a set of linear algebraic equations). The analysis is based on certain “semi-explicit” formulas for the performance characteristics. This research was supported by INTAS under grant No. 96-0828.     

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.     

Elliptic boundary value problems in hybrid domains

We derive asymptotic transmission conditions at points where segments are attached to a three-dimensional body. These conditions result in a formally self-adjoint problem on a hybrid set with properties similar to those of standard boundary value problems. In particular, the problem has a zero index and possesses a variational statement. If the systems of differential equations have a special form, then the operator of the problem is realized as a self-adjoint extension of the decoupled operators of the problems on the body and the segments. From this viewpoint, we interpret the results of asymptotic analysis of coupled thin and solid bodies.Dedicated to Victor Borisovich Lidskii on the occasion of his jubileeTranslated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 55–72, 2004Original Russian Text Copyright © by S. A. NazarovSupported by the Russian Foundation for Basic Research (grant No. 00-01-00835).     

Legendre transforms on a triangular lattice

We show that the condition of invariance with respect to generalized Legendre transforms effectively singles out a class of integrable difference equations on a triangular lattice; these equations are discrete analogs of relativistic Toda lattices. Some of these equations are apparently new. For one of them, higher symmetries are written out and the zero curvature representation is obtained. The research was supported by RFBR grant No. 96-01-00128. Mathematical Institute, Russian Academy of Sciences. Ufa. Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 34, No. 1, pp. 1–11, January–March, 2000. Translated by V. E. Nazaikinskii     

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.     

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).     

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.