Distribution of the shape of Markovian random words

The distribution of the shape of the semi-standard tableau of a random word in k letters is asymptotically given by the distribution of the spectrum of a random traceless k×k Gaussian Unitary Ensemble (GUE) matrix provided that these letters are independent with uniform distribution. Kuperberg (2002) conjectured that this result by Johansson (2001) remains valid if the letters of the word are generated by an irreducible Markov chain on the alphabet with cyclic transition matrix. In this paper we give a proof of this conjecture for an alphabet with k=2 letters.Research supported by DFG GO-420/3-3 in Bielefeld.Research supported by INTAS 99-00317, RFBR–DFG 99-01-04027.Mathematics Subject Classification (2000): 82B41, 60C05, 60F05, 60F10     

Maximal resolvability of totally bounded and ℵ0-bounded groups

The maximal resolvability of totally bounded groups (and, under the assumption that the generalized continuum hypothesis holds, of ₀-bounded groups) is proved.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 593–598, April, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 94-01-01374.     

Variable stepsize continuous two-step Runge-Kutta methods for ordinary differential equations

A general class of variable stepsize continuous two-step Runge-Kutta methods is investigated. These methods depend on stage values at two consecutive steps. The general convergence and order criteria are derived and examples of methods of orderp and stage orderq=p orq=p–1 are given forp5. Numerical examples are presented which demonstrate that high order and high stage order are preserved on nonuniform meshes with large variations in ratios between consecutive stepsizes.The work of the first author was supported by the National Science Foundation under grant NSF DMS-9208048. The work of the second author was supported by the Italian Government.     

Convergence of linear multistep methods for differential equations with discontinuities

Summary A new stability functional is introduced for analyzing the stability and consistency of linear multistep methods. Using it and the general theory of [1] we prove that a linear multistep method of design orderqp1 which satisfies the weak stability root condition, applied to the differential equationy (t)=f (t, y (t)) wheref is Lipschitz continuous in its second argument, will exhibit actual convergence of ordero(h ^p–1) ify has a (p–1)th derivativey ^(p–1) that is a Riemann integral and ordero(h ^p) ify ^(p–1) is the integral of a function of bounded variation. This result applies for a functiony taking on values in any real vector space, finite or infinite dimensional.This work was supported by Grant GJ-938 from the National Science Foundation     

Semicontinuous solutions for Hamilton-Jacobi equations and theL ∞control problem

We prove uniqueness for extended real-valued lower semicontinuous viscosity solutions of the Bellman equation forL -control problems. This result is then used to prove uniqueness for lsc solutions of Hamilton-Jacobi equations of the form –u _t +H(t, x, u, –Du)=0, whereH(t, x, r, p) is convex inp. The remaining assumptions onH in the variablesr andp extend the currently known results.Supported in part by Grant DMS-9300805 from the National Science Foundation.     

Numerical Simulation of Nanopulse Penetration of Biological Matter Using the <Emphasis Type="Italic">z</Emphasis>-Transform

Short duration, fast rise time ultra-wideband (UWB) electromagnetic pulses (nanopulses) are generated by numerous electronic devices in use today. Moreover, many new technologies involving nanopulses are under development and expected to become widely available soon. Study of nanopulse bioeffects is needed to probe their useful range in possible biomedical and biotechnological applications, and to ensure human safety. In this work we develop a computational approach to investigate electromagnetic fields in biological cells exposed to nanopulses. The simulation is based on a z-transformation of the electric displacement and a second-order Taylor approximation of a Cole–Cole expression for the frequency dependence of the dielectric properties of tissues, useful for converting from the frequency domain to the time domain. Maxwells equations are then calculated using the finite difference time domain method (FDTD), coupled with a perfectly matched layer to eliminate reflections from the boundary. Numerical results for a biological cell model are presented and discussed.Mathematics Subject Classifications (2000) 65M06, 92C05.     

Novel approach to accelerating Newton's method for sup-norm optimization arising inH ∞-control

We present algorithms for solving general sup-norm minimization problems over spaces of analytic functions, such as those arising inH control. We also give an analysis and some theory of these algorithms. Part of this is specific to analytic optimization, while part holds for general sup-norm optimization. In particular, we are proposing a type of Newton-type algorithm which actually uses very high-order terms. The novel feature is that higher-order terms can be chosen in many ways while still maintaining a second-order convergence rate. Then, a clever choice of higher-order terms greatly reduces computation time. Conceivably this technique can be modified to accelerate Newton algorithms in some other circumstances. Estimates of order of convergence as well as results of numerical tests are also presented.This work was partially supported by the Air Force Office of Scientific Research and the National Science Foundation.     

On <Emphasis Type="Italic">p</Emphasis>-nilpotency of finite groups*

Let be a class of groups. A subgroup H of a group G is called -s-supplemented in G, if there exists a subgroup K of G such that G = HK and K/K ∩ H_G belongs to where H_G is the maximal normal subgroup of G which is contained in H. The main purpose of this paper is to study some subgroups of Fitting subgroup and generalized Fitting subgroup -s-supplemented and some new criterions of p-nilpotency of finite groups are obtained. *This research is supported by the grant of NSFC and TianYuan Fund of Mathematics of China (Grant #10626047).     

On the Neighbourhood of a Subclass of Univalent Functions Related to Complex Order

We consider the class of functions R(A, B) introduced by Dixit and Pal, where b 0 is a complex number and A, B are fixed members –1 B < A 1. We will study the -neighbourhoods for functions belonging to R^b(A, B), by using convolution techniques.AMS Mathematics Classification (2000): 30C55     

Multivariate nonhomogeneous refinement equations

We give necessary and sufficient conditions for the existence and uniqueness of compactly supported distribution solutionsf=(f ₁,...,f _r)^T of nonhomogeneous refinement equations of the form

On a new class of multiplicative pseudo-random number generators

In the computing literature, there are few detailed analytical studies of the global statistical characteristics of a class of multiplicative pseudo-random number generators.We comment briefly on normal numbers and study analytically the approximately uniform discrete distribution or (j,)-normality in the sense of Besicovitch for complete periods of fractional parts {x _{0 1} ⁱ /p} on [0, 1] fori=0, 1,..., (p–1)p^–1–1, i.e. in current terminology, generators given byx _{n+1 1} x _n mod p wheren=0, 1,..., (p–1)p ^–1–1,p is any odd prime, (x ₀,p)=1, ₁ is a primitive root modp ², and 1 is any positive integer.We derive the expectationsE(X, ),E(X ², ),E(X _nX_n+k); the varianceV(X, ), and the serial correlation coefficient k. By means of Dedekind sums and some results of H. Rademacher, we investigate the asymptotic properties of k for various lagsk and integers 1 and give numerical illustrations. For the frequently used case =1, we find comparable results to estimates of Coveyou and Jansson as well as a mathematical demonstration of a so-called rule of thumb related to the choice of ₁ for small k.Due to the number of parameters in this class of generators, it may be possible to obtain increased control over the statistical behavior of these pseudo-random sequences both analytically as well as computationally.     

Julia sets for the gamma recursion in nonlinear psychophysics

Julia sets for the map za(z–ie)(1–z)(z+ie) are illustrated for some attractors of interest. This work extends previous analyses of the cubic complex polynomial and considers dynamics in regions which may be associated with the modelling of the results of overload in sensory inputs.Work supported by grant No. A7881403 (1988) from the Australian Research Council.     

Adaptive polynomial preconditioning for hermitian indefinite linear systems

This paper explores the use of polynomial preconditioned CG methods for hermitian indefinite linear systems,Ax=b. Polynomial preconditioning is attractive for several reasons. First, it is well-suited to vector and/or parallel architectures. It is also easy to employ, requiring only matrix-vector multiplication and vector addition. To obtain an optimum polynomial preconditioner we solve a minimax approximation problem. The preconditioning polynomial,C(), is optimum in that it minimizes a bound on the condition number of the preconditioned matrix,C(A)A. We also characterize the behavior of this minimax polynomial, which makes possible a thorough understanding of the associated CG methods. This characterization is also essential to the development of an adaptive procedure for dynamically determining the optimum polynomial preconditioner. Finally, we demonstrate the effectiveness of polynomial preconditioning in a variety of numerical experiments on a Cray X-MP/48. Our results suggest that high degree (20–50) polynomials are usually best.This research was supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Dept. of Energy, by Lawrence Livermore National Laboratory under contract W-7405-ENG-48.This research was supported in part by the Dept. of Energy and the National Science Foundation under grant DMS 8704169.This research was supported in part by U.S. Dept. of Energy grant DEFG02-87ER25026 and by National Science Foundation grant DMS 8703226.     

On V. Fleischer's characterization of absolutely flat monoids

In his paperCompletely flat monoids (Uh. Zap. Tartu Un-ta610 (1982), 38–52 (Russian)) V. Fleischer gives a characterization of the absolute flatness of a monoidS in terms of certain one-sided ideals and one-sided congruences ofS. In the present work an alternative, more direct proof of Fieischer's theorem is provided, and the result is used to show that the multiplicative monoid of any semisimple Artinian ring is absolutely flat.Research supported by Natural Sciences and Engineering Research Council grant A4494.Research supported by Natural Sciences and Engineering Research Council grant A9241.Presented by Boris M. Schein.     

Local structure of Rogers semilattices of Σ<Subscript>n</Subscript><Superscript>0</Superscript>-computable numberings

We deal in specific features of the algebraic structure of Rogers semilattices of _n ⁰-computable numberings, for n 2. It is proved that any Lachlan semilattice is embeddable (as an ideal) in such every semilattice, and that over an arbitrary non 0-principal element of such a lattice, any Lachlan semilattice is embeddable (as an interval) in it.Supported by INTAS grant No. 00-499, by FP Universities of Russia grant UR.04.01.013, and by the Grant Center for Fundamental Research (GCFR), project PD02-1.1-475.__________Translated from Algebra i Logika, Vol. 44, No. 2, pp. 148–172, March–April, 2005.     

Continuity in G

For a discrete group G, we consider βG, the Stone– ech compactification of G, as a right topological semigroup, and G^*=βGG as a subsemigroup of βG. We study the mappings λ_p^* :G^*→G^*and μ^* :G^*→G^*, the restrictions to G^* of the mappings λ_p :βG→βG and μ :βG→βG, defined by the rules λ_p(q)=pq, μ(q)=qq. Under some assumptions, we prove that the continuity of λ_p^* or μ^* at some point of G^* implies the existence of a P-point in ω^*.     

A note on the recursive calculation of dominant singular subspaces

A recursive procedure for computing an approximation of the left and right dominant singular subspaces of a given matrix is proposed in [1]. The method is particularly suited for matrices with many more rows than columns. The procedure consists of a few steps. In one of these steps a Householder transformation is multiplied to an upper triangular matrix. The following step consists in recomputing an upper triangular matrix from the latter product. In [1] it is said that the latter step is accomplished in O(k³) operations, where k is the order of the triangular matrix. In this short note we show that this step can be accomplished in O(k²) operations. This research was partially supported by MIUR, grant number 2002014121 (first author) and by the Research Council K.U.Leuven, project OT/00/16 (SLAP: Structured Linear Algebra Package), by the Fund for Scientific Research–Flanders (Belgium), projects G.0078.01 (SMA: Structured Matrices and their Applications), G.0176.02 (ANCILA: Asymptotic aNalysis of the Convergence behavior of Iterative methods in numerical Linear Algebra), G.0184.02 (CORFU: Constructive study of Orthogonal Functions) and G.0455.0 (RHPH: Riemann–Hilbert problems, random matrices and Padé–Hermite approximation), and by the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Ministers Office for Science, Technology and Culture, project IUAP V-22 (Dynamical Systems and Control: Computation, Identification & Modelling) (second and third author). The scientific responsibility rests with the authors.AMS subject classification 15A15, 15A09, 15A23     

Generalized Neuwirth Groups and Seifert Fibered Manifolds

The topological properties of the generalized Neuwirth groups, _n^k are discussed. For examp, we demonstrate that the group, _n^k is the fundamental group of the Seifert fibered space _n^k. Moreover, discuss some other invariants and algebraic properties of the above groups.This work was supported by Polish grant (BW-5100–5–0259–9) and the Russian Foundation for Basic Research (grant number 98–01–00699).2000 Mathematics Subject Classification: 20F34, 57M05, 57M60