Topological complexity and real roots of polynomials

The topological complexity of an algorithm is the number of its branchings. In the paper we prove that the minimal topological complexity of an algorithm that approximately computes a root of a real polynomial of degreed equalsd/2 for evend, is greater than or equal to 1 for oddd>–3, and equals 1 ford=3 or 5.Translated fromMatematicheskie Zametki, Vol. 60, No. 5, pp. 670–680, November, 1996.This research was supported by the Russian Foundation for Basic Research under grant No. 95-01-00846a and by the INTAS under grant No. 4373.     

Estimation of the Mean of a Wiener Sheet

A shifted Wiener sheet is observed above a decreasing curve Γ. By the help of a direct discrete approach and under weaker assumptions than in the paper of Arató [Comput. Math. Appl. 33 (1997), 13–25], an explicit formula is derived for the maximum likelihood estimator of the shift parameter. This estimator is a weighted linear combination of the values at the endpoints of the curve Γ and weighted integrals of the observed process and its normal derivative along the curve Γ. This revised version was published online in June 2006 with corrections to the Cover Date.     

On images of Borel measures under Borel mappings

Let and be metric spaces. We show that the tight images of a (fixed) tight Borel probability measure on , under all Borel mappings , form a closed set in the space of tight Borel probability measures on with the weak-topology. In contrast, the set of images of under all continuous mappings from to may not be closed. We also characterize completely the set of tight images of under Borel mappings. For example, if is non-atomic, then all tight Borel probability measures on can be obtained as images of , and as a matter of fact, one can always choose the corresponding Borel mapping to be of Baire class 2.

     

Generalized Fredholm property and a priori estimates for linear operators on tensor products of Hilbert spaces

For a linear operator acting in a Hilbert space, the generalized Fredholm property (invertibility modulo a certain ideal) is proved to be equivalent to certaina priori estimates. This result is applied to establish a connection between properties of linear operators on tensor products of Hilbert spaces, such asn- andd-normality, the (generalized and ordinary) Fredholm property, and appropriatea priori estimates.Translated fromMatematicheskie Zametki, Vol. 64, No. 6, pp. 902–912, December, 1998.The author is grateful to V. M. Deundyak for useful discussion of this work.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01195.     

An Adjacency Criterion for the Prime Graph of a Finite Simple Group

For every finite non-Abelian simple group, we give an exhaustive arithmetic criterion for adjacency of vertices in a prime graph of the group. For the prime graph of every finite simple group, this criterion is used to determine an independent set with a maximal number of vertices and an independent set with a maximal number of vertices containing 2, and to define orders on these sets; the information obtained is collected in tables. We consider several applications of these results to various problems in finite group theory, in particular, to the recognition-by-spectra problem for finite groups. Supported by RFBR grant No. 05-01-00797; by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1; by the RF Ministry of Education Developmental Program for Scientific Potential of the Higher School of Learning, project No. 8294; by FP “Universities of Russia,” grant No. UR.04.01.202; and by Presidium SB RAS grant No. 86-197. __________ Translated from Algebra i Logika, Vol. 44, No. 6, pp. 682–725, November–December, 2005.     

On Global Properties of Polynomials Guaranteed by Their Behavior on a Subset

In this paper, we establish the order of growth of different norms of polynomials as a function of their degree by the given estimates of their values on a subset of the closed interval under consideration.     

Center of some solvable groups with one defining relation

Necessary and sufficient conditions for the center of a metabelian group with one defining relation to be nontrivial are found. The center of such a group is described. The center of a group of the formF/Ng ^F is studied under certain conditions. By means of a new technique, the recent result of A. F. Krasnikov and the author on the center of a group of the above form is sharpened.Translated fromMatematickeskie Zametki, Vol. 64, No. 6, pp. 925–931, December, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01948.     

近似凸集的某些性质

主要研究了两类近似凸集的关系和性质.首先,举例说明两类近似凸集没有相互包含关系.其次,在近似凸集(nearly convex)条件下,证明了在一定条件下函数上图是近似凸集与凸集的等价关系.同时,考虑了近似凸函数与函数上图是近似凸集的等价刻画、近似凸函数与函数水平集是近似凸集的必要性,并用例子说明近似凸函数与函数水平集是...     

A moving-boundary problem for concrete carbonation: Global existence and uniqueness of weak solutions

This paper deals with a one-dimensional coupled system of semi-linear parabolic equations with a kinetic condition on the moving boundary. The latter furnishes the driving force for the moving boundary. The main result is a global existence and uniqueness theorem of positive weak solutions. The system under consideration is modelled on the so-called carbonation of concrete - a prototypical chemical-corrosion process in a porous solid - concrete - which incorporates slow diffusive transport, interfacial exchange between wet and dry parts of the pores and, in particular, a fast reaction in thin layers, here idealized as a moving-boundary surface in the solid. We include simulation results showing that the model captures the qualitative behaviour of the carbonation process.     

Stability of generalized solutions to equations of one-dimensional motion of viscous heat-conducting gases

Nonhomogeneous initial boundary value problems for a specific quasilinear system of equations of composite type are studied. The system describes the one-dimensional motion of a viscous perfect polytropic gas. We assume that the initial data belong to the spacesL () orL ₂() and the problems under consideration have generalized solutions only. For such solutions, a theorem on strong stability is proved, i.e., estimates for the norm of the difference of two solutions are expressed in terms of the sums of the norms of the differences of the corresponding data. Uniqueness of generalized solutions is a simple consequence of this theorem.Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 835–846, June, 1998.This research was supported by the Russian Foundation for Basic Research under grants No. 96-01-00621 and No. 97-01-00214, and by the INTAS Foundation under grant No. 93-27-16.     

Regular sets in residual classes

Sets regular modulo a fixed odd prime power are explicitly constructed under the condition that their cardinalities do not exceed an arbitrarily small positive power of the modulus.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 224–228, August, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 97-01-00721 and by the Professor B. Novak grant (Karlov University, Prague).     

On four-sheeted polynomial mappings of ℂ2. I. The case of an irreducible ramification curve

The paper is devoted to the Jacobian Conjecture: a polynomial mappingf²² with a constant nonzero Jacobian is polynomially invertible. The main result of the paper is as follows. There is no four-sheeted polynomial mapping whose Jacobian is a nonzero constant such that after the resolution of the indeterminacy points at infinity there is only one added curve whose image is not a point and does not belong to infinity.Translated fromMatematicheskie Zametki, Vol. 64, No. 6, pp. 847–862, December, 1998.The authors are grateful to A. G. Vitushkin and P. Cassou-Nogues for useful discussions.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01218. The work of the second author was done under the financial support of DGICYT (Spain), grant No. SAB95-0502.     

The Weierstrass semigroup of a pair and moduli inM 3

We classify all the Weierstrass semigroups of a pair of points on a curve of genus 3, by using its canonical model in the plane. Moreover, we count the dimension of the moduli of curves which have a pair of points with a specified Weierstrass semigroup.This work has been supported by the Japan Society for the Promotion of Science and the Korea Science and Engineering Foundation (Project No. 976-0100-001-2). Also the first author is partially supported by Korea Research Foundation Grant (KRF-99-005-D00003).     

Newton's method for a class of nonsmooth functions

This paper presents and justifies a Newton iterative process for finding zeros of functions admitting a certain type of approximation. This class includes smooth functions as well as nonsmooth reformulations of variational inequalities. We prove for this method an analogue of the fundamental local convergence theorem of Kantorovich including optimal error bounds.The research reported here was sponsored by the National Science Foundation under Grants CCR-8801489 and CCR-9109345, by the Air Force Systems Command, USAF, under Grants AFOSR-88-0090 and F49620-93-1-0068, by the U. S. Army Research Office under Grant No. DAAL03-92-G-0408, and by the U. S. Army Space and Strategic Defense Command under Contract No. DASG60-91-C-0144. The U. S. Government has certain rights in this material, and is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon.     

On a Problem Concerning k-Subdomination Numbers of Graphs

One of numerical invariants concerning domination in graphs is the k-subdomination number of a graph G. A conjecture concerning it was expressed by J.H. Hattingh, namely that for any connected graph G with n vertices and any k with the inequality holds. This paper presents a simple counterexample which disproves this conjecture. This counterexample is the graph of the three-dimensional cube and k = 5.     

On the Existence of Periodic Solutions for a Nonlinear System of Ordinary Differential Equations

Abstract This paper is concerned with the existence of periodic solutions for a nonlinear system of ordinary differential equations. We obtain a Nagumo-type a priori bound for the periodic solutions and then by using this a priori bound we prove the existence of at least one T-periodic solution under some general conditions Research supported by the NNSF of China and the RFDP of China.     

The coupled problem of a solid oscillating in a viscous fluid under the action of an elastic force

The torsional oscillations are studied of a solid of revolution under the action of elastic torque inside a container with a viscous incompressible fluid. We prove the asymptotic stability of the static equilibrium. We use the two approaches: the direct Lyapunov and linearization methods. The global asymptotic stability is established using a one-parameter family of Lyapunov functionals. Then small oscillations are studied of the fluid-solid system. The linearized operator of the problem of a solid oscillating in a fluid can be realized as an operator matrix obtained by appending two scalar rows and two columns to the Stokes operator. This operator is therefore a two-dimensional bordering of the Stokes operator and inherits many properties of the latter; in particular, the spectrum is discrete. The eigenvalue problem for the linearized operator is reduced to solving a dispersion equation. Inspection of the equation shows that all eigenvalues lie inside the right (stable) half-plane. Basing on this, we justify the linearization. Using an abstract theorem of Yudovich, we prove the asymptotic stability in a scale of function spaces, the infinite differentiability of solutions, and the decay of all their derivatives in time.     

Co-rectangular bands and cosheaves in categories of algebras

Conditions on a categoryC are studied which imply that every structure of rectangular band on an objectS ofC arises from a unique product decompositionS=S ₁×S ₂, especially in the case whereC is the opposite of a category of algebras.Sheaves on Stone spaces with values in opposites of categories of algebras are examined.The analog of the bounded Boolean power constructionR[B]^* forR an object of a general category is described.This work was done while the author was partly supported by NSF contract DMS 85-02330.Presented by R. S. Pierce.     

Restrictions of bounded linear operators: Closed range

Let be a bounded linear operator on a Banach space and let be a subspace of which is a Banach space and invariant. Denote by the restriction of to This paper explores the questions:

If the range of is closed, under what conditions is the range of closed?

If the range of is closed, under what conditions is the range of closed?