Properties of algebras of primary order with one ternary Mal'tsev operation

We study properties of free solvable p-algebras and of p-algebras of primary orders. It is shown that a free solvable p-algebra is embeddable in a subdirect product of algebras with these orders. An example of a simple non-Abelian p-algebra of a primary order is constructed, and we also give an example of a solvable non-nilpotent p-algebra of any finite order not less than 6.Translated fromAlgebra i Logika, Vol. 34, No. 2, pp. 132-141, March-April, 1995.Partially supported by the Russian Foundation for Fundamental Research, the Program Universities of Russia, and the Russian Ministry of Science, Higher Schools of Learning, and Technical Policy, grant No. 2-11-1-2.     

Extremal problems of Bloch function spaces

Letf be analytic in a hyperbolic region . The Bloch constant _f off is defined by , where (z)|dz| is the Poincaré metric in . Suppose is hyperbolic and where . Then for allf withf() , we have _f 1/(). In this paper we study the extremal functions defined by _f =1/() and the existence of those functions.Supported by the National Natural Science Foundation of China.     

On the growth of sums of independent random variables

Some estimates of the growth of sums of independent random variables almost surely are established without any moment conditions. Bibliography: 6 titles.Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 294, 2002, pp. 158–164.This research was partially supported by the Russian Foundation for Basic Research, grant 02-01-00779, and by the Program Leading Scientific Schools, grant 00-15-96019.Translated by V. V. Petrov.     

A globally and quadratically convergent affine scaling method for linearℓ 1 problems

Recently, various interior point algorithms related to the Karmarkar algorithm have been developed for linear programming. In this paper, we first show how this interior point philosophy can be adapted to the linear ₁ problem (in which there are no feasibility constraints) to yield a globally and linearly convergent algorithm. We then show that the linear algorithm can be modified to provide aglobally and ultimatelyquadratically convergent algorithm. This modified algorithm appears to be significantly more efficient in practise than a more straightforward interior point approach via a linear programming formulation: we present numerical results to support this claim.This paper was presented at the Third SIAM Conference on Optimization, in Boston, April 1989.Research partially supported by the Applied Mathematical Sciences Research Program (KC-04-02) of the Office of Energy Research of the U.S. Department of Energy under grant DE-FG02-86ER25013.A000, by the U.S. Army Research Office through the Mathematical Sciences Institute, Cornell University, and by the Computational Mathematics Program of the National Science Foundation under grant DMS-8706133.Research partially supported by the U.S. Army Research Office through the Mathematical Sciences Institute, Cornell University and by the Computational Mathematics Program of the National Science Foundation under grant DMS-8706133.     

Thermodynamics and Hydrodynamics (Statistical Foundations): 3. Hydrodynamic Equations

We show that all the hydrodynamic equations can be obtained from the BBGKY hierarchy. The theory is constructed by expanding the distribution functions in series in a small parameter = R/L 10^–8, where R 10^–7cm is the radius of the correlation sphere and L is the characteristic macroscopic dimension. We also show that in the zeroth-order approximation with respect to this parameter, the BBGKY hierarchy implies the local equilibrium and the transport equations for the ideal Euler fluid; in the first-order approximation with respect to , the BBGKY hierarchy implies the hydrodynamic equations for viscous fluids. Moreover, we prove that the intrinsic energy flux must include both the kinetic energy flux proportional to the temperature gradient and the potential energy flux proportional to the density gradient. We show that the hydrodynamic equations hold for t 10^–12s and L R 10^–7cm.     

Recognizing Berge Graphs

A graph is Berge if no induced subgraph of G is an odd cycle of length at least five or the complement of one. In this paper we give an algorithm to test if a graph G is Berge, with running time O(|V (G)|⁹). This is independent of the recent proof of the strong perfect graph conjecture.* Currently this author is a Clay Mathematics Institute Research Fellow.** Supported by NSF grant DMI-0352885 and ONR grant N00014-97-1-0196. Supported by ONR grant N00014-01-1-0608, and NSF grant DMS-0070912. Supported by EPSRC grant GR/R35629/01.     

A class of formally normal operators

A densely closed operator N given in Hilbert space is called formally normal if D(N) D(N*)and Nf = N*f for allf D(N). In the present work the necessary and sufficient conditions for a formally normal operator, possessing a bounded inverse, to have a normal extension in the original Hilbert space are given. The result obtained is analogous to a result of M. I. Vishik [1], relating to the case of a symmetric operator [7 References].Translated from Matematicheskie Zametki, Vol. 2, No. 6, pp. 605–614, December, 1967.     

Worst case asymptotics for some classical optimization problems

We provide a simple and natural method for obtaining the worst case asymptotics of some of the classical problems in combinatorial optimization and operations research. Worst case asymptotics for the minimal spanning tree, shortest tour, and minimal matching onn points are found. The key simplifying idea involves consideration of the associated boundary processes. The general approach considered here also handles the case of power weighted edges.Research supported in part by NSA grant MDA904-95-H-1005.     

A new smoothing-regularization approach for a maximum-likelihood estimation problem

We consider the problem min _i=1 ^m (aⁱ,x–b_iloga ⁱ, z) subject tox 0 which occurs as a maximum-likelihood estimation problem in several areas, and particularly in positron emission tomography. After noticing that this problem is equivalent to mind(b, Ax) subject tox 0, whered is the Kullback-Leibler information divergence andA, b are the matrix and vector with rows and entriesa ⁱ,b _i, respectively, we suggest a regularized problem mind(b, Ax) + d(v, Sx), where is the regularization parameter,S is a smoothing matrix, andv is a fixed vector. We present a computationally attractive algorithm for the regularized problem, establish its convergence, and show that the regularized solutions, as goes to 0, converge to the solution of the original problem which minimizes a convex function related tod(v, Sx). We give convergence-rate results both for the regularized solutions and for their functional values.The research of A. N. Iusem was partially supported by CNPq Grant No. 301280/86-MA.     

Large deviations for the invariant measure of a reaction-diffusion equation with non-Gaussian perturbations

Summary In this paper we establish a large deviations principle for the invariant measure of the non-Gaussian stochastic partial differential equation (SPDE) _t v =v +f(x,v )+(x,v ) . Here is a strongly-elliptic second-order operator with constant coefficients, h:=DH _xx-h, and the space variablex takes values on the unit circleS ¹. The functionsf and are of sufficient regularity to ensure existence and uniqueness of a solution of the stochastic PDE, and in particular we require that 0<mM wherem andM are some finite positive constants. The perturbationW is a Brownian sheet. It is well-known that under some simple assumptions, the solutionv ² is aC ^k (S ¹)-valued Markov process for each 0<1/2, whereC (S ¹) is the Banach space of real-valued continuous functions onS ¹ which are Hölder-continuous of exponent . We prove, under some further natural assumptions onf and which imply that the zero element ofC (S ¹) is a globally exponentially stable critical point of the unperturbed equation _t ⁰ = ⁰ +f(x,⁰), that has a unique stationary distributionv ^K, on (C (S ¹), (C ^K (S ¹))) when the perturbation parameter is small enough. Some further calculations show that as tends to zero,v ^K, tends tov ^K,0, the point mass centered on the zero element ofC (S ¹). The main goal of this paper is to show that in factv ^K, is governed by a large deviations principle (LDP). Our starting point in establishing the LDP forv ^K, is the LDP for the process , which has been shown in an earlier paper. Our methods of deriving the LDP forv ^K, based on the LDP for are slightly non-standard compared to the corresponding proofs for finite-dimensional stochastic differential equations, since the state spaceC (S ¹) is inherently infinite-dimensional.This work was performed while the author was with the Department of Mathematics, University of Maryland, College Park, MD 20742, USA     

Dualities for the Domany–Kinzel Model

We study the Domany–Kinzel model, which is a class of discrete-time Markov processes in one-dimension with two parameters (p ₁,p ₂)[0,1]². When p ₁= and p ₂=(2– ²) with (,)[0,1]², the process can be identified with the mixed site-bond oriented percolation model on a square lattice with probabilities of a site being open and of a bond being open. This paper treats dualities for the Domany–Kinzel model _t ^A and the DKdual _t ^A starting from A. We prove that , as long as one of A,B is finite and p ₂p ₁.     

Approximation algorithms for indefinite quadratic programming

We consider-approximation schemes for indefinite quadratic programming. We argue that such an approximation can be found in polynomial time for fixed andt, wheret denotes the number of negative eigenvalues of the quadratic term. Our algorithm is polynomial in 1/ for fixedt, and exponential int for fixed. We next look at the special case of knapsack problems, showing that a more efficient (polynomial int) approximation algorithm exists.Part of this work was done while the author was visiting Sandia National Laboratories, Albuquerque, New Mexico, supported by the U.S. Department of Energy under contract DE-AC04-76DP00789. Part of this work was also supported by the Applied Mathematical Sciences Program (KC-04-02) of the Office of Energy Research of the U.S. Department of Energy under grant DE-FG02-86ER25013.A000 and in part by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS 8920550.     

Quantum Mechanical Algorithms for the Nonabelian Hidden Subgroup Problem

We provide positive and negative results concerning the standard method of identifying a hidden subgroup of a nonabelian group using a quantum computer.* Supported in part by NSF grants CCR-9820931 and CCR-0208929. Supported in part by NSF CAREER grant 0049092, the NSF Institute for Quantum Information, the Charles Lee Powell Foundation, and the Mathematical Sciences Research Institute. Supported in part by an NSF Mathematical Sciences Postdoctoral Research Fellowship and NSF grant DMS-0301320.§ Supported in part by DARPA QUIST grant F30602-01-2-0524, ARO grant DAAD19-03-1-0082, and NSF ITR grant CCR-0121555.     

The geometric structure of Chebyshev sets in ℓ<Superscript>∞</Superscript>(<Emphasis Type="Italic">n</Emphasis>)

A subset M of a normed linear space X is called a Chebyshev set if each x X has a unique nearest point in M. We characterize Chebyshev sets in (n) in geometric terms and study the approximative properties of sections of Chebyshev sets, suns, and strict suns in (n) by coordinate subspaces.__________Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 39, No. 1, pp. 1–10, 2005Original Russian Text Copyright © by A. R. AlimovSupported by RFBR grant No.02-01-00248.Translated by A. R. Alimov     

Non-intersection exponents for Brownian paths

Summary LetX andY be independent 3-dimensional Brownian motions,X(0)=(0,0,0),Y(0)=(1,0,0) and letp _r =P(X[0,r] Y[0,r]=). Then the non-intersection exponent exists and is equal to a similar non-intersection exponent for random walks. Analogous results hold inR ² and for more than 2 paths.Supported in part by NSF grant DMS 8702620Supported by NSF grant DMS 8702879 and an Alfred P. Sloan Research Fellowship     

RUC-decompositions in symmetric operator spaces

We show that ifX is a Banach space of type 2 andG is a compact Abelian group, then any system of eigenvectors {x }_G (with respect to a strongly continuous representation ofG onX) is an RUC-system. As an application, we exhibit new examples of RUC-bases in certain symmetric spaces of measurable operators.Research supported by the Australian Research Council     

Simultaneous approximation of algebraic irrationalities

This paper proves three theorems concerning the simultaneous approximation of numbers from a totally real algebraic number field. It is shown that for two given numbers ₁ and ₂ from a totally real algebraic number field, the constant ₁₂ can be explicitly calculated, this being the upper limit of the numbers c₁₂ such that the inequality max (q₁, q₂)(qc₁₂)^–1/2 holds for infinitely many natural numbers q; likewise for the constant a₁₂ such that the inequality q₁·q₂< a₁₂(q^logq) holds for infinitely many natural numbers q. It is shown that there exist n –1 numbers ₁, ..., _n–1 in an algebraic number field of degree n and discriminant d such that the inequality holds only for finitely many natural numbers q if 2^{ - \left[ {\tfrac{{n - 1}}{2}} \right]} \sqrt d $$ " align="middle" border="0"> . is fixed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 116, pp. 142–154, 1982.     

The best one-sided approximation of the classes WrHω

In this paper we calculate the upper bounds of the best one-sided approximations, by trigonometric polynomials and splines of minimal defect in the metric of the space L, of the classes W^rH (r = 2, 4, 6, ...) of all 2-periodic functions f(x) that are continuous together with their r-th derivative f^r(x) and such that for any points x and x we have ¦f ^r (x) f^r (x) ¦ (x–x¦), where (t) is a modulus of continuity that is convex upwards.Translated from Matematicheskie Zametki, Vol. 21, No. 3, 313–327, March, 1977.     

Some inequalities about the covering radius of Reed-Muller codes

Let R(r, m) be the rth order Reed-Muller code of length 2 ^m , and let (r, m) be its covering radius. We prove that if 2 k m - r - 1, then (r + k, m + k) (r, m + 2(k - 1). We also prove that if m - r 4, 2 k m - r - 1, and R(r, m) has a coset with minimal weight (r, m) which does not contain any vector of weight (r, m) + 2, then (r + k, m + k) (r, m) + 2k(. These inequalities improve repeated use of the known result (r + 1, m + 1) (r, m).This work was supported by a grant from the Research Council of Wright State University.     

Bericht über κ-affine Geometrien

Ohne ZusammenfassungMeinem verehrten Lehrer, Herrn Prof. Dr. HELMUT KARZEL in Dankbarkeit gewidmet.Diesem Bericht liegt meine Habilitationsschrift -affine RÄume zugrunde. Auszugsweise habe ich über den Gegenstand dieser Arbeit auf der Conference on Geometry in Waterloo/Ont. 1971 vorgetragen.