A new proof of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> estimates for the parabolic polyharmonic equations

In this paper we obtain local L ^p estimates for the parabolic polyharmonic equations by a straightforward approach. Yao was supported by the Innovation Foundation of Shanghai University (Grant No. A10-0101-08-905), Shanghai Leading Academic Discipline Project (Grant No. J50101) and Key Disciplines of Shanghai Municipality (Grant No. S30104). Zhou was supported by the National Basic Research Program of China (Grant No. 2006CB705700), National Natural Science Foundation of China (Grant No. 60532080), and the Key Project of Chinese Ministry of Education (Grant No. 306017)     

A note on a theorem of Yamamuro

Conditions are presented which are necessary for the existence of a regular fixed point of aC ¹ map.Work supported in part by NSF Grant No. MCS 77-15509.Work supported in part by ONR Grant No. N0014-75-C-0495 and NSF Grant No. Eng. 76-81058.     

Accurate SVDs of weakly diagonally dominant M-matrices

Summary. We present a new O(n³) algorithm which computes the SVD of a weakly diagonally dominant M-matrix to high relative accuracy. The algorithm takes as an input the offdiagonal entries of the matrix and its row sums.Mathematics Subject Classification (1991): 65F15Revised version received September 19, 2003This material is based in part upon work supported by the LLNL Memorandum Agreement No. B504962 under DOE Contract No. W-7405-ENG-48, DOE Grants No. DE-FG03-94ER25219, DE-FC03-98ER25351 and DE-FC02-01ER25478, NSF Grant No. ASC-9813362, and Cooperative Agreement No. ACI-9619020.     

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.     

Isomorphism types of Rogers semilattices for families from different levels of the arithmetical hierarchy

We investigate differences in isomorphism types for Rogers semilattices of computable numberings of families of sets lying in different levels of the arithmetical hierarchy. Supported by RFBR grant No. 05-01-00819 and by INTAS grant No. 00-499. Supported by NSFC grant No. 60310213. __________ Translated from Algebra i Logika, Vol. 45, No. 6, pp. 637–654, November–December, 2006.     

Nonlinear Kolmogorov widths

We investigate a generalization of width in the sense of Kolmogorov suitable for estimating bestm-term approximations. We generalize Carl's inequality which gives lower estimate of Kolmogorov widths in terms of entropy numbers. Application of these new inequalities gives some progress in the problem of estimating bestm-term trigonometric approximations of multivariate functions.Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 891–902, June, 1998.This research was supported by the National Science Foundation under grant No. DMS 9622925 and by the ONR foundation under grant No. N00014-96-1-1003.     

The number of periodic solutions of polynomial differential equations

We estimate the number of periodic solutions for special classes ofnth-order ordinary differential equations with variable coefficients. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 720–727, November, 1998. The author thanks Yu. S. Il'yashenko for setting the problems, permanent advice, and overall support. The author is also thankful to D. A. Panov for numerous discussions. This research was supported by the CRDF Foundation under grant MR1-220, by the INTAS Foundation under grant No. 93-05-07, and by the Russian Foundation for Basic Research under grant No. 95-01-01258.     

Fuzzy logics with modalities

We explore the basic fuzzy logic BL as well as propositional fuzzy logics with modalities □ and ◊ and a total accessibility relation. Formulations and proofs are given to replacement theorems for BL. A basic calculus of modal fuzzy logic is introduced. For this calculus and its extensions, we prove replacement and deduction theorems. Supported by RFBR grant No. 06-01-00358, by INTAS grant No. 04-77-7080, and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-4787.2006.1. __________ Translated from Algebra i Logika, Vol. 45, No. 6, pp. 731–757, November–December, 2006.     

Automatic recognition of interpolation in modal calculi

We deal with some issues on automatic recognition of interpolation properties in modal calculi extending the logics S5 and S4.3. Supported by RFBR grant No. 06-01-00358 and by INTAS grant No. 04-77-7080. __________ Translated from Algebra i Logika, Vol. 46, No. 1, pp. 103–119, January–February, 2007.     

The Projective Beth Property and Interpolation in Positive and Related Logics

We look at the interplay between the projective Beth property in non-classical logics and interpolation. Previously, we proved that in positive logics as well as in superintuitionistic and modal ones, the projective Beth property PB2 follows from Craig's interpolation property and implies the restricted interpolation property IPR. Here, we show that IPR and PB2 are equivalent in positive logics, and also in extensions of the superintuitionistic logic KC and of the modal logic Grz.2. Supported by RFBR grant No. 06-01-00358, by INTAS grant No. 04-77-7080, and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1. __________ Translated from Algebra i Logika, Vol. 45, No. 1, pp. 85–113, January–February, 2006.     

Finite subgroups of hyperbolic groups

Let G be a hyperbolic group which is -thin w.r.t. a finite generating set X. We show that every finite subgroup of G is conjugate to a subgroup each element of which has length at most 2 + 1relative to X. Translated fromAlgebra i Logika, Vol. 34, No. 6, pp. 619-622, November-December, 1995.Supported by RFFR grant No. 93-11-1501 and by ISF grant RPC000.Supported by RFFR grant No. 93-011-16171.     

Coedge regular graphs without 3-stars

We describe coedge regular graphs such that antineighborhoods of their vertices are coedge regular graphs with the same value of the parameterμ. As a consequence of the main theorem, we obtain a classification of coedge regular graphs without 3-stars. Translated fromMatematicheskie Zametki, Vol. 60, No. 4, pp. 495–503, October, 1996. This research was supported by the Russion Foundation for Basic Research under grants No. 93-01-01529 and No. 94-01-00802a.     

Intertwining operators and soliton equations

The fermionic approach to the Kadomtsev-Petviashvili hierarchy, suggested by the Kyoto school (Sato, Date, Jimbo, Kashiwara, and Miwa) in 1981–4, is generalized on the basis of the idea that, in a sense, the components of intertwining operators are a generalization of free fermions forgl _∞. Integrable hierarchies related to symmetries of Kac-Moody algebras are described in terms of intertwining operators. The bosonization of these operators for various choices of the Heisenberg subalgebra is explicity written out. These various realizations result in distinct hierarchies of soliton equations. For example, forsl _N -symmetries this gives the hierarchies obtained by the (n ₁,...,n _s )-reduction from thes-component KP hierarchy introduced by Kac and van de Leur. The research of both authors was supported in part by RFBR grant No. 96-02-18046, grant No. 96-15-96455 for the support of leading scientific schools, and RFBR-CNRS grant No. 98-01-22033. International Center for Nonlinear Science at the Landau Institute of Theoretical Physics. Institute of Theoretical and Experimental Physics. Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 33, No. 4, pp. 1–24, October–December, 1999. Translated by M. I. Golenishcheva-Kutuzova     

Freedom from the interpolation property for tense calculi associated with Ershov spaces

We study into the question whether calculi associated with Ershov topological spaces possess Craig’s interpolation property. Supported by RFBR grant No. 06-01-00358, by INTAS grant No. 04-77-7080, and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-4787.2006.1. __________ Translated from Algebra i Logika, Vol. 46, No. 6, pp. 745–762, November–December, 2007.     

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.     

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.     

Asymptotics ast → ∞ of the solutions of nonlinear equations with nonsmall initial perturbations

A new method is proposed for finding asymptotics ast → ∞ of the solutions of the Cauchy problem for nonlinear evolution equations with nonsmall initial data. Translated fromMatematicheskie Zametki, Vol. 59, No. 6, pp. 855–864, June, 1996. This research was supported by the Russian Foundation for Basic Research under grant No. 93-011-134 and by the International Science Foundation under grant No. BX000.     

Modules over quantum polynomials

We describe the structure of finitely generated modules over general quantum Laurent polynomials and prove that Artin modules over general quantum Laurent polynomials are cyclic. Translated fromMatematicheskie Zametki, Vol. 59, No. 4, pp. 497–503, April, 1996. This research was partially supported by the Russian Foundation for the Basic Research under grant No. 93-011-1544 and by the INTAS program under grant No. 93-2618.     

Classification of lattice-regular lattice convex polytopes

We completely describe lattice convex polytopes in ℝ ⁿ (for any dimension n) that are regular with respect to the group of affine transformations preserving the lattice. Supported in part by the RFBR (Grant Nos. SS-1972.2003.1 and 05-01-01012a) and the NWO-RFBR (Grant No. 047.011.2004.026/RFBR No. 05-02-89000-NWO_a).     

Conditions for the spatial flatness and spatial injectivity of an indecomposable CSL algebra of finite width

This paper is concerned with the connection between the geometric properties of the latticeL of subspaces of a Hilbert spaceH and homological properties (flatness and injectivity) ofH regarded as a natural module over the reflexive algebra AlgL that consists of all operators leaving invariant each element of the latticeL. It follows from these results that the cohomology groups with coefficients inB(H) are trivial for a broad class of reflexive algebras. Translated fromMatemalicheskie Zametki, Vol. 63, No. 1, pp. 9–20, January, 1998. The author gladly expresses his sincere gratitude to A. Ya. Khelemskii and Yu. V. Selivanov for their assistance in this work. In particular, A. Ya. Khelemskii indicated how to simplify the proof of Theorem 1. This research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00156, by the International Science Foundation under grant No. M95000, and by the INTAS fund under grant No. 93-1376.