On sequences of Dirichlet polynomials uniformly bounded on half planes

Given and a sequence of Dirichlet polynomials estimates for the coefficientsa _n are proved if {_n} is uniformly bounded on a region containing a half plane. Thereby a result is obtained which is an analogue of a known result for polynomials, that is for theA-transforms of the geometric sequence; moreover a Jentzsch type theorem for {_n(z)} is derived.     

Para-orthogonal Laurent polynomials and associated sequences of rational functions

On the space, , of Laurent polynomials (L-polynomials) we consider a linear functional which is positive definite on (0, ) and is defined in terms of a given bisequence, { _k } _– . Two sequences of orthogonal L-polynomials, {Q _n (z) ₀ and , are constructed which span in the order {1,z ^–1,z,z ^–2,z ²,...} and {1,z,z ^–1,z ²,z ^–2,...} respectively. Associated sequences of L-polynomials {P _n (z) ₀ , and are introduced and we define rational functions , wherew is a fixed positive number. The partial fraction decomposition and integral representation of,M _n (z, w) are given and correspondence of {M _n (z, w)} is discussed. We get additional solutions to the strong Stieltjes moment problem from subsequences of {M _n (z, w)}. In particular when { _k } _– is a log-normal bisequence, {M _2n (z, w)} and {M _2n+1 (z, w)} yield such solutions.Research supported in part by the National Science Foundation under Grant DMS-9103141.     

On the acceleration of limit periodic continued fractions

Summary A continued fraction (c.f.)K(a _n /1) is called limit periodic if . Fora anda(–,–1/4],a0, Thron-Waadeland (1980) examined a modification of a limit periodic c.f. for accelerating the convergence. This acceleration remains modest if thea _n converge only logarithmically. Thus it is proposed to apply an Euler summability method to the series equivalent to the c.f. Properties of the equivalent function are derived. These properties are used for choosing appropriate parameters for the summability method such that a considerable acceleration can be expected even if thea _n converge logarithmically.Dedicated to Prof. F.L. Bauer on the occasion of his 60th birthday     

On the Function w( x)=|{1= s= k : x= a s (mod n s)}|

For a finite system of arithmetic sequences the covering function is w(x) = |{1 s k : x a_s (mod n_s)}|. Using equalities involving roots of unity we characterize those systems with a fixed covering function w(x). From the characterization we reveal some connections between a period n₀ of w(x) and the moduli n₁, . . . , n_k in such a system A. Here are three central results: (a) For each r=0,1, . . .,n_k/(n₀,n_k)–1 there exists a Jc{1, . . . , k–1} such that . (b) If n₁ ···n_k–l <n_k–l+1 =···=n_k (0 < l < k), then for any positive integer r < n_k/n_k–l with r 0 (mod n_k/(n₀,n_k)), the binomial coefficient can be written as the sum of some (not necessarily distinct) prime divisors of n_k. (c) max_(xw(x) can be written in the form where m₁, . . .,m_k are positive integers.The research is supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, and the National Natural Science Foundation of P. R. China.     

Hypersurfaces of restricted type in Minkowski space

A submanifold M _n ^r of Minkowski space is said to be of restricted type if its shape operator with respect to the mean curvature vector is the restriction of a fixed linear transformation of to the tangent space of M _n ^r at every point of M _n ^r . In this paper we completely classify hypersurfaces of restricted type in . More precisely, we prove that a hypersurface of is of restricted type if and only if it is either a minimal hypersurface, or an open part of one of the following hypersurfaces: S ^k × , S _k ¹ × , H ^k × , S _n ¹ , H ⁿ , with 1kn–1, or an open part of a cylinder on a plane curve of restricted type.This work was done when the first and fourth authors were visiting Michigan State University.Aangesteld Navorser N.F.W.O., Belgium.     

Divergence of a Random Walk Through Deterministic and Random Subsequences

Let {S _n} _n0 be a random walk on the line. We give criteria for the existence of a nonrandom sequence n _i for which respectively We thereby obtain conditions for to be a strong limit point of {S _n} or {S _n /n}. The first of these properties is shown to be equivalent to for some sequence a _i , where T(a) is the exit time from the interval [–a,a]. We also obtain a general equivalence between and for an increasing function fand suitable sequences n _i and a _i. These sorts of properties are of interest in sequential analysis. Known conditions for and (divergence through the whole sequence n) are also simplified.     

Truncation error bounds for modified continued fractions with applications to special functions

Much recent work has been done to investigate convergence of modified continued fractions (MCF's), following the proof by Thron and Waadeland [35] in 1980 that a limit-periodic MCFK(a _n , 1;x ₁), with andnth approximant

The Homology of Partitions with an Even Number of Blocks

Let denote the subposet obtained by selecting even ranks in the partition lattice . We show that the homology of has dimension , where is the tangent number. It is thus an integral multiple of both the Genocchi number and an André or simsun number. Using the general theory of rank-selected homology representations developed in [22], we show that, for the special case of , the character of the symmetric group S _2n on the homology is supported on the set of involutions. Our proof techniques lead to the discovery of a family of integers b _i(n), 2 i n, defined recursively. We conjecture that, for the full automorphism group S _2n, the homology is a sum of permutation modules induced from Young subgroups of the form , with nonnegative integer multiplicity b _i(n). The nonnegativity of the integers b _i(n) would imply the existence of new refinements, into sums of powers of 2, of the tangent number and the André or simsun number a _n(2n).Similarly, the restriction of this homology module to S _2n–1 yields a family of integers d _i(n), 1 i n – 1, such that the numbers 2^–i d _i(n) refine the Genocchi number G _2n . We conjecture that 2^–i d _i(n) is a positive integer for all i.Finally, we present a recursive algorithm to generate a family of polynomials which encode the homology representations of the subposets obtained by selecting the top k ranks of , 1 k n – 1. We conjecture that these are all permutation modules for S _2n .     

Solution of the Truncated Hyperbolic Moment Problem

Let Q(x, y) = 0 be an hyperbola in the plane. Given real numbers β ≡ β ⁽²ⁿ⁾={ β _ij } _{i,j ≥ 0,i+j ≤ 2n} , with β₀₀ > 0, the truncated Q-hyperbolic moment problem for β entails finding necessary and sufficient conditions for the existence of a positive Borel measure μ, supported in Q(x, y) = 0, such that We prove that β admits a Q-representing measure μ (as above) if and only if the associated moment matrix is positive semidefinite, recursively generated, has a column relation Q(X,Y) = 0, and the algebraic variety associated to β satisfies card In this case, if then β admits a rank -atomic (minimal) Q-representing measure; if then β admits a Q-representing measure μ satisfying      

List Colouring When The Chromatic Number Is Close To the Order Of The Graph

Ohba has conjectured [7] that if G has 2 (G)+1 or fewer vertices then the list chromatic number and chromatic number of G are equal. In this short note we prove the weaker version of the conjecture obtained by replacing 2 (G)+1 by * This research was partially supported by DIMACS and by CNRS/NSF collaboration grant. Research supported in part by NSF grants DMS-0106589, CCR-9987845 and by the State of New Jersey.     

Tail Estimation Based on Numbers of Near m-Extremes

Let be a sequence of independent random variables with common continuous distribution function F having finite upper endpoint. A new tail index estimator ^ _n is defined based on only two numbers of near m-extremes where X _(i:n) denotes the -th order statistic and a ₂ > a ₁ > 0. The weak and almost sure convergence of ^ _n to the tail index , as well as the asymptotic distribution is given. Moreover, the asymptotic distribution of K _n (a _n , m) for a _n 0 is derived.     

Lower Bounds For Multidimensional Zero Sums

Let f(n, d) denote the least integer such that any choice of f(n, d) elements in contains a subset of size n whose sum is zero. Harborth proved that (n-1)2 ^d +1 f(n,d) (n-1)n ^d +1. The upper bound was improved by Alon and Dubiner to c _d n. It is known that f(n-1) = 2n-1 and Reiher proved that f(n-2) = 4n-3. Only for n = 3 it was known that f(n,d) > (n-1)2 ^d +1, so that it seemed possible that for a fixed dimension, but a sufficiently large prime p, the lower bound might determine the true value of f(p,d). In this note we show that this is not the case. In fact, for all odd n 3 and d 3 we show that .     

The maximum number of balancing sets

Leta ₁, ...,a _n be a sequence of nonzero real numbers with sum zero.A subsetB of {1, 2,...,n} is called a balancing set if a _b = 0 (b B). S. Nabeya showed that the number of balancing sets is bounded above by and this bound achieved forn even witha _j =(–1) ^j . Here his conjecture is verified, showing a tight upper bound whenn = 2k + 1. The essentially unique extremal configuration is:a ₁ = 2,a ₂ = ... =a _k = 1,a _k+1 = ... =a _2k+1 = -1.     

Convergence acceleration of limit periodic continued fractions under asymptotic side conditions

Summary We present a method of convergence acceleration for limitk-periodic continued fractionsK(a _n /1) orK(1/b _n ) satisfying certain asymptotic side conditions. The method represents an improvement of the fixed point modification considered by Thron and Waadeland [8], under these conditions. The regularC-fraction expansions of hypergeometric functions₂ F ₁(a, 1;c; z) and₂ F ₁(a, b; c; z)/₂ F ₁(a, b+1;c+1;z) are examples of continued fractions satisfying these conditions.     

Law of the iterated logarithm for perturbed empirical distribution functions evaluated at a random point for nonstationary random variables

We consider perturbed empirical distribution functions , where {Ginn, n1} is a sequence of continuous distribution functions converging weakly to the distribution function of unit mass at 0, and {X _{i, i}1} is a non-stationary sequence of absolutely regular random variables. We derive the almost sure representation and the law of the iterated logarithm for the statistic whereU _n is aU-statistic based onX ₁,...,X _n . The results obtained extend or generalize the results of Nadaraya,⁽⁷⁾ Winter,⁽¹⁶⁾ Puri and Ralescu,^(9,10) Oodaira and Yoshihara,⁽⁸⁾ and Yoshihara,⁽¹⁹⁾ among others.Research supported by the Office of Naval Research Contract N00014-91-J-1020.     

A functional limit theorem for Erdös and Rényi's law of large numbers

Summary We use the theory of large deviations on function spaces to extend Erdös and Rényi's law of large numbers. In particular, we show that with probability 1, the double-indexed set of paths {W _{N, n} } defined by where , {X _i : _i 1} is an iid sequence of random variables, andh(N)=[clogN] is relatively compact; the limit set is given by the set [xI ^*(x)1/c] whereI ^*(x) = ₀ ¹ I(x(t))dt andI is Cramér's rate function.     

On Bergman-Toeplitz operators with commutative symbol algebras

Let be the unit disk in, be the Bergman space, consisting of all analytic functions from , and be the Bergman projection of onto . We constructC ^*-algebras , for functions of which the commutator of Toeplitz operators [T _a ,T _b ]=T _a T _b –T _b T _a is compact, and, at the same time, the semi-commutator [T _a ,T _b )=T _a T _b –T _ab is not compact.It is proved, that for each finite set =n ₀,n ₁, ...,n _m , where 1=n _{0 1} <... _m , andn _k {}, there are algebras of the above type, such that the symbol algebras Sym of Toeplitz operator algebras arecommutative, while the symbol algebras Sym of the algebras , generated by multiplication operators and , haveirreducible representations exactly of dimensions n ₀,n ₁,..., n _m .This work was partially supported by CONACYT Project 3114P-E9607, México.     

Spectral estimates of Cauchy's transform inL 2(Ω)

We prove that the singular numbers of the Cauchy transform onL ²(D) are asymptotically , whiles _n (C _| L _a ² (D))1/n (whereL _a ² (D) is the subspace of analytic functions inL ²(D)). Also, the singular numbers of the logarithmic potential onL ²(D) are asympoticallys _n (L)1/n, whiles _n(L _{|L a} ² (D))1/n ². Our methods yield the asymptotic behavior of the singular numbers of the Cauchy Transform fromL _L ² () intoL ²() where and are rotation-invariant measures on .The author was partly supported by a grant from the national Science Foundation.     

Isoperimetric inequalities in potential theory

Given a non-empty bounded domainG in ⁿ ,n2, letr ₀(G) denote the radius of the ballG ₀ having center 0 and the same volume asG. The exterior deficiencyd _e (G) is defined byd _e (G)=r _e (G)/r ₀(G)–1 wherer _e (G) denotes the circumradius ofG. Similarlyd _i (G)=1–r _i (G)/r ₀(G) wherer _i (G) is the inradius ofG. Various isoperimetric inequalities for the capacity and the first eigenvalue ofG are shown. The main results are of the form CapG(1+cf(d _e (G)))CapG ₀ and ₁(G)(1+cf(d _i (G)))₁(G ₀),f(t)=t ³ ifn=2,f(t)=t ³/(ln 1/t) ifn=3,f(t)=t ^(n+3)/2 ifn4 (for convex G and small deficiencies ifn3).     

A central limit theorem for solutions of the porous medium equation

We study the large–time behavior of the second moment (energy) for the flow of a gas in a N-dimensional porous medium with initial density v₀(x) 0. The density v(x, t) satisfies the nonlinear degenerate parabolic equation v_t = v^m where m > 1 is a physical constant. Assuming that for some > 0, we prove that E(t) behaves asymptotically, as t , like the energy E_B(t) of the Barenblatt-Pattle solution B(|x|, t). This is shown by proving that E(t)/E_B(t) converges to 1 at the (optimal) rate t^{–2/(N(m-1)+2)}. A simple corollary of this result is a central limit theorem for the scaled solution E(t)^N/2v(E(t)^1/2x, t).