Hilbert and Hilbert—Samuel polynomials and partial differential equations

Systems of linear partial differential equations with constant coefficients are considered. The spaces of formal and analytic solutions of such systems are described by algebraic methods. The Hilbert and Hilbert—Samuel polynomials for systems of partial differential equations are defined.Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 141–151.Original Russian Text Copyright © 2005 by A. G. Khovanskii, S. P. Chulkov.This revised version was published online in April 2005 with a corrected issue number.     

Cauchy problem for \left\{ {\vec p;\vec h} \right\}-parabolic equations with time-dependent coefficients

We establish the existence of a unique solution continuously depending on the initial data to the Cauchy problem for -parabolic equations with time-dependent coefficients for which the initial data are generalized functions (distributions) of slow growth. For a particular class of equations, we state necessary and sufficient conditions for the existence of a unique solution of the Cauchy problem with properties of its spatial variable which are characteristic of its fundamental solution.Translated from Matematicheskie Zametki, vol. 77, no. 3, 2005, pp. 395–411.Original Russian Text Copyright © 2005 by V. A. Litovchenko.This revised version was published online in April 2005 with a corrected issue number.     

Concerning an Example of Paskiewich

We generalize the construction proposed by A. Paskiewich of an example of an orthonormal system which establishes the sharpness of the Men’shov-Rademacher theorem. The connection of his example with that of Men’shov is elucidated.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 286–291.Original Russian Text Copyright © 2005 by A. P. Solodov.     

On locally free Abelian groups

The following result is proved in the paper. An Abelian group A is Lw₁, w-equivalent to the free Abelian group of countable rank if and only if it is a countably free Abelian group.Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 121–126.Original Russian Text Copyright © 2005 by E. G. Sklyarenko.This revised version was published online in April 2005 with a corrected issue number.     

On the Geometry of Point-Transformation Invariant Class of Third-Order Ordinary Differential Equations

Applying geometric methods, we study a class of third-order ordinary differential equations closed with respect to point transformations. We associate with such an equation the pseudovector fields formed by its coefficients. The equation possesses a maximal algebra of point-transformation symmetries if five pseudovector fields vanish.__________Translated from Matematicheskie Zametki, vol. 77, no. 5, 2005, pp. 719–726.Original Russian Text Copyright ©2005 by V. V. Kartak.     

Some aspects of the nontrivial solvability of homogeneous Dirichlet problems for linear equations of arbitrary even order in the disk

In this paper, we obtain a necessary and sufficient condition for the nontrivial solvability of homogeneous Dirichlet problems in the disk for linear equations of arbitrary even order 2m with constant complex coefficients and homogeneous nondegenerate symbol in general position. The cases m=1, 2, 3 are studied separately. For the case m=2, we consider examples of real elliptic systems reducible to single equations with constant complex coefficients for which the homogeneous Dirichlet problem in the disk has a countable set of linearly independent polynomial solutions.Translated from Matematicheskie Zametki, vol. 77, no. 4, 2005, pp. 498–508.Original Russian Text Copyright © 2005 by V. P. Burskii, E. A. Buryachenko.This revised version was published online in April 2005 with a corrected issue number.     

Stepwise Gauge Equivalence of Differential Operators

In this paper, we study the relation between the notion of gauge equivalence and solutions of certain systems of nonlinear partial differential equations. This relation is based on stepwise gauge equivalence.__________Translated from Matematicheskie Zametki, vol. 77, no. 6, 2005, pp. 917–929.Original Russian Text Copyright ©2005 by S. P. Khekalo.     

Algebraic splines in locally convex spaces

In a vector space of continuous functions, a variational solution of a finite system of linear functional equations is found. The locally convex topology on the vector space and the properties of the objective functional required for obtaining the solution in the form of a decomposition in the basis dual to the family of functionals of the system are determined. The basis elements are calculated exactly and called basis algebraic splines; their linear span is called the space of algebraic splines in the corresponding locally convex space.Translated from Matematicheskie Zametki, vol. 77, no. 3, 2005, pp. 339–353.Original Russian Text Copyright © 2005 by A. P. Kolesnikov.This revised version was published online in April 2005 with a corrected issue number.     

The description of pairs of compatible first-order differential geometric poisson brackets

We show that bi-Hamiltonian structures of systems of hydrodynamic type can be described in terms of solutions of nonlinear equations. These equations can be integrated by the inverse scattering transform for arbitrary metrics as well as for flat ones. In particular, if one metric is flat and the other has a constant curvature, then the corresponding integrable system is a reduction of the Cherednik chiral-field model.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 2, pp. 293–309, February, 2005.     

Constructive Solvability Conditions for the Riemann-Hilbert Problem

Sufficient and necessary conditions for the solvability of the Riemann-Hilbert problem are studied. These conditions consist in the possibility of constructing stable and semistable pairs (of bundles and connections) for a given monodromy. The obtained results make it possible to develop algorithms for testing the solvability conditions for the Riemann-Hilbert problem.__________Translated from Matematicheskie Zametki, vol. 77, no. 5, 2005, pp. 643–655.Original Russian Text Copyright ©2005 by I. V. V’yugin.I dedicate this work to the blessed memory of my teacher Andrei Andreevich Bolibrukh     

Some Limit Theorems for a Particle System of Single Point Catalytic Branching Random Walks

We study the scaling limit for a catalytic branching particle system whose particles perform random walks on Z and can branch at 0 only. Varying the initial （finite） number of particles, we get for this system different limiting distributions. To be more specific, suppose that initially there are n^β particles and consider the scaled process Zt^n（·） = Znt（√n·）, where Zt is the measure-valued process 1 and to a representing the original particle system. We prove that Ztn converges to 0 when β 〈1/4 and to a nondegenerate discrete distribution when β=1/4.In addition,if 1/4〈β〈1/2 then n-^（2β-1/2）Zt^n converges to a random limit,while if β 〉21then n^-βZtn converges to a deterministic limit.     

General scheme of maxima of sums of independent random variables and its applications

A general scheme of maxima of sums of independent random variables is introduced. We prove a theorem on the convergence of the maxima in probability. We study its applications to large jumps in random walks and to extrema of shot-noise fields in the case of regularly varying tails. Nondegenerate limiting laws are obtained.Translated from Matematicheskie Zametki, vol. 77, no. 4, 2005, pp. 544–550.Original Russian Text Copyright © 2005 by A. V. Lebedev.This revised version was published online in April 2005 with a corrected issue number.     

Estimates of the Entropy of the Set of Means for Some Classes of Stationary and Quasistationary Sequences

We consider the set of arithmetic means for some classes of stationary and quasistationary sequences. Order-sharp estimates of the entropy of this set for the classes under consideration are given.__________Translated from Matematicheskie Zametki, vol. 78, no. 1, 2005, pp. 52–58.Original Russian Text Copyright © 2005 by V. F. Gaposhkin.     

Equi-Attraction and Continuous Dependence of Strong Attractors of Set-Valued Dynamical Systems on Parameters

Equi-attraction properties of the global attractors of a family of set-valued dynamical systems {G_λ, λ∈Λ} with respect to a parameter λ∈Λ, where Λ is a compact metric space, are investigated. In particular, under appropriate conditions, it is shown that the equi-attraction of the family is equivalent to the continuity of in λ with respect to the Hausdorff distance. An example involving differential inclusions is given. Mathematics Subject Classifications (2000) Primary 47H17, 58F03, 34C20. Desheng Li: Supported by NNSF of China (10251002) and NNSF of Gansu province (3ZS041-A25-006).     

Homotopy Theories of Algebras over Operads

Homotopy theories of algebras over operads, including operads over “little n-cubes,” are defined. Spectral sequences are constructed and the corresponding homotopy groups are calculated.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 278–285.Original Russian Text Copyright © 2005 by V. A. Smirnov.     

Widths of classes of finitely smooth functions in Sobolev spaces

We describe the weak asymptotics of the behavior of the Kolmogorov, Gelfand, linear, Aleksandrov, and entropy widths of the unit ball of the space W _p ^l H^w (I ^d) in the space W _q ^m (I ^d).Translated from Matematicheskie Zametki, vol. 77, no. 4, 2005, pp. 535–539.Original Russian Text Copyright © 2005 by S. N. Kudryavtsev.This revised version was published online in April 2005 with a corrected issue number.     

Asymptotics of the eigenvalues and the formula for the trace of perturbations of the Laplace operator on the sphere \mathbb{S}^2

In this paper, we study the asymptotics of the eigenvalues of the Laplace operator perturbed by an arbitrary bounded operator on the sphere . For the first time, for the partial differential operator of second order, the leading term of the second correction of perturbation theory is obtained. A connection between the coefficient of the second term of the asymptotics of the eigenvalues and the formula for the traces of the operator under consideration is established.Translated from Matematicheskie Zametki, vol. 77, no. 3, 2005, pp. 434–448Original Russian Text Copyright © 2005 by V. A. Sadovnichii, Z. Yu. Fazullin.This revised version was published online in April 2005 with a corrected issue number.     

Soluble Groups and Varieties of <Emphasis Type="Italic">l</Emphasis>-Groups

A sufficient condition is given under which factors of a system of normal convex subgroups of a linearly ordered (l.o.) group are Abelian. Also, a sufficient condition is specified subject to which factors of a system of normal convex subgroups of an l.o. group are contained in a group variety . In particular, for every soluble l.o. group G of solubility index n, n ⩾ 2, factors of a system of normal convex subgroups are soluble l.o. groups of solubility index at most n − 1. It is proved that the variety of all lattice-ordered groups, approximable by linearly ordered groups, does not coincide with a variety generated by all soluble l.o. groups. It is shown that if is any o-approximable variety of l-groups, and if every identity in the group signature is not identically true in , then contains free l.o. groups.Supported by FP “Universities of Russia” grant UR. 04. 01. 001.__________Translated from Algebra i Logika, Vol. 44, No. 3, pp. 355–367, May–June, 2005.     

Exact boundary controllability of unsteady flows in a network of open canals

By means of the general results on the exact boundary controllability for quasilinear hyperbolic systems, the author establishes the exact boundary controllability of unsteady flows in both a single open canal and a network of open canals with star configuration respectively. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)