Simultaneous reduction of a lattice basis and its reciprocal basis

Given a latticeL we are looking for a basisB=[b ₁, ...b _n ] ofL with the property that bothB and the associated basisB ^*=[b ₁ ^* , ...,b _n ^* ] of the reciprocal latticeL ^* consist of short vectors. For any such basisB with reciprocal basisB ^* let . Håstad and Lagarias [7] show that each latticeL of full rank has a basisB withS(B)exp(c ₁·n ^1/3) for a constantc ₁ independent ofn. We improve this upper bound toS(B)exp(c ₂·(lnn)²) withc ₂ independent ofn.We will also introduce some new kinds of lattice basis reduction and an algorithm to compute one of them. The new algorithm proceeds by reducing the quantity . In combination with an exhaustive search procedure, one obtains an algorithm to compute the shortest vector and a Korkine-Zolotarev reduced basis of a lattice that is efficient in practice for dimension up to 30.     

Fourier and Hermite series estimates of regression functions

Summary In the paper we estimate a regressionm(x)=E {Y|X=x} from a sequence of independent observations (X ₁,Y ₁),…, (X _n, Y_n) of a pair (X, Y) of random variables. We examine an estimate of a type , whereN depends onn andϕ _N is Dirichlet kernel and the kernel associated with the hermite series. Assuming, that E|Y|<∞ and |Y|≦γ≦∞, we give condition for to converge tom(x) at almost allx, provided thatX has a density. if the regression hass derivatives, then converges tom(x) as rapidly asO(nC^{−(2s−1)/4s}) in probability andO(n ^{−(2s−1)/4s} logn) almost completely.     

Hölder Regularity for Solutions of Ultraparabolic Equations in Divergence Form

We consider the second order differential equation , where (x,t) ^N+1, 0<m ₀N, the coefficients a _i,j belong to a suitable space of vanishing mean oscillation functions VMO _L and B=(b _i,j ) is a constant real matrix. The aim of this paper is to study interior regularity for weak solutions to the above equation assuming that F _j belong to a function space of Morrey type.     

Non-commutative Clarkson Inequalities for n-Tuples of Operators

Let A ₀, ... , A _n−1 be operators on a separable complex Hilbert space , and let α₀,..., α _n−1 be positive real numbers such that 1. We prove that for every unitarily invariant norm,

An algorithm for a linear diophantine equation and a problem of Frobenius

Summary We solve the diophantine equation for nonnegative variablesx _j , wherea _j andL are positive integers. We characterize both the values ofL that lead to solutions and those that do not lead to solutions. We solve the Frobenius problem of finding the largest value ofL for which no solution exists.     

Inequalities for absolute maxima of a polynomial and its derivative

Suppose z ₁, z ₂, ... z _n are complex numbers with absolute values more than 1 and Arg z _j Arg z _k for j k where Arg w stands for the argument of the complex number w in [0,2). In this note we show that

Square function inequalities for non-commutative martingales

We prove a non-commutative version of the weak-type (1,1) boundedness of square functions of martingales. More precisely, we prove that there is an absolute constantK with the following property: ifM is a semifinite von Neumann algebra with a faithful normal traceτ and (M _n ) _n=1 ^∞ is an increasing filtration of von Neumann subalgebras of (M then for any martingalex= _n=1 ^∞ inL ¹(M,τ), adapted to (M _n ) _n=1 ^∞ , there is a decomposition into two sequences (x _n ) _n=1 ^∞ and (z _n ) _n=1 ^∞ withx _n=y _n+z _nfor everyn≥1 and such that . This generalizes a result of Burkholder from classical martingale theory to non-commutative martingales. We also include some applications to martingale Hardy spaces. Supported in part by NSF grant DMS-0096696.     

Polynomials of minimalL α-deviation, α>0

Let _n–1 be the linear space of algebraic polynomials of degreen–1. We prove that the extremal problem

p-Sets for random walks

Summary Let (,,P) be a probability space and let {itX _n ()} _n=1 be a sequence of i.i.d. random vectors whose state space isZ ^m for some positive integerm, where Z denotes the integers. Forn = 1, 2,... letS _n () be the random walk defined by . ForxZ ^m andU ^m, them-dimensional torus, let . Finally let be the characteristic function of the X's.In this paper we show that, under mild restrictions, there exists a set withP{ ₀ } = 1 such that for ₀ we have for all aU ^m,le0.As a consequence of this theorem, we obtain two corollaries. One is concerned with occupancy sets form-dimensional random walks, and the other is a mean ergodic theorem.Research supported by N.S.F. Grant # MCS 77-26809     

Zur Interpolation und Integration differenzierbarer periodischer Funktionen

Summary Letx ₀<x ₁<...<x _n–1<x ₀+2 be nodes having multiplicitiesv ₀,...,v _n–1, 1v _k r (0k<n). We approximate the evaluation functional ,x fixed, and the integral respectively by linear functionals of the form and determine optimal weights for the Favard classesW _r C ₂. In the even case of optimal interpolation these weights are unique except forr=1,x(x _k +x _k–1)/2 mod 2. Moreover we get periodic polynomial splinesw _{k, j} (0k<n, 0j<v _k ) of orderr such that are the optimal weights. Certain optimal quadrature formulas are shown to be of interpolatory type with respect to these splines. For the odd case of optimal interpolation we merely have obtained a partial solution.

Bojanov hat in [4, 5] ähnliche Resultate wie wir erzielt. Um Wiederholungen zu vermeiden, werden Resultate, deren Beweise man bereits in [4, 5] findet, nur zitiert     

Un nouveau type d'inégalités pour les martingales discrètes

Summary We prove the following extension of classical Burkholder-Davis-Gundy inequalities: let (X _n ) _nN be a martingale; for p1, in order that and belong to L ^p, it is sufficient that Inf(X ^*, S(X)) belong to L ^p. For «regular» martingales this result holds for p>0.     

The maximal accuracy of stable difference schemes for the wave equation

We consider three time-level difference schemes, symmetric in time and space, for the solution of the wave equation,u _tt =c ² u _xx , given by

Asymptotic property of approximation to x αsgn x by Newman Type Operators

Approximation to the function ｜x｜ plays an important role in approximation theory. This paper studies the approximation to the function xαsgn x, which equals ｜x｜ if α = 1. We construct a Newman Type Operator rn（x） and prove max ｜x｜≤1｜xαsgn x-rn（x）｜～Cn1/4e-π1/2（1/2）αn.     

Estimates for Sums of Coefficients of Dirichlet Series with Functional Equation

Suppose we have a Dirichlet series L(s) = _{n = 1} a _n n ^–s such that it, and its twists by Dirichlet characters have analytic continuation and a functional equation of a specific kind. Suppose also that the root numbers of the twists are equidistributed on the unit circle. The purpose of this note is to get an estimate for the quantity for a prime modulus p.We use a modification of the method of Chandrasekharan and Narasimhan and we use in an essential way a Rankin-Selberg type estimate for the average of |a _n|².     

Realization and Smoothness Related to the Laplacian

For f L _n (T ^d ) and , the modulus of smoothness

Some properties for a class of symmetric functions with applications

For x = (x ₁, x ₂, ..., x _n ) ∈ ℝ₊ ⁿ , the symmetric function ψ _n (x, r) is defined by $\psi _n (x,r) = \psi _n \left( {x_1 ,x_2 , \cdots ,x_n ;r} \right) = \sum\limits_{1 \leqslant i_1 < i_2 \cdots < i_r \leqslant n} {\prod\limits_{j = 1}^r {\frac{{1 + x_{i_j } }} {{x_{i_j } }}} } ,$\psi _n (x,r) = \psi _n \left( {x_1 ,x_2 , \cdots ,x_n ;r} \right) = \sum\limits_{1 \leqslant i_1 < i_2 \cdots < i_r \leqslant n} {\prod\limits_{j = 1}^r {\frac{{1 + x_{i_j } }} {{x_{i_j } }}} } ,     

On the uniqueness of canonical points in the Hobby-Rice theorem

According to the Hobby-Rice theorem for anyn-dimensional subspaceU _n ofL ₁([a, b], ) ( positive, finite, nonatomic) there exist points =s ₀x ₁x _m+1=b, where 0mn, such that

A general result on precise asymptotics for linear processes of positively associated sequences

Let {εt; t ∈ Z^＋} be a strictly stationary sequence of associated random variables with mean zeros, let 0〈Eε1^2〈∞ and σ^2=Eε1^2＋1∑j=2^∞ Eε1εj with 0〈σ^2〈∞.{aj;j∈Z^＋} is a sequence of real numbers satisfying ∑j=0^∞｜aj｜〈∞.Define a linear process Xt=∑j=0^∞ ajεt-j,t≥1,and Sn=∑t=1^n Xt,n≥1.Assume that E｜ε1｜^2＋δ′〈 for some δ′〉0 and μ（n）=O（n^-ρ） for some ρ〉0.This paper achieves a general law of precise asymptotics for {Sn}.     

Monotony in interpolatory quadratures

Summary Let , be holomorphic in an open disc with the centrez ₀=0 and radiusr>1. LetQ _n (n=1, 2, ...) be interpolatory quadrature formulas approximating the integral . In this paper some classes of interpolatory quadratures are considered, which are based on the zeros of orthogonal polynomials corresponding to an even weight function. It is shown that the sequencesQ _n 9f] (n=1, 2, ...) are monotone. Especially we will prove monotony in Filippi's quadrature rule and with an additional assumption onf monotony in the Clenshaw-Curtis quadrature rule.     

Zeros of H ∞-Functions on Hyperplanes in \mathbb{B} n

Let ⁿ be the unit ball in ℂⁿ, n ≥ 2. Let T_α = {z ∈ ⁿ : (z, a) = |a|²} for a ∈ ⁿ and denote for a discrete set A in ⁿ. We find a sharp necessary condition for a set A to be a part of the zero-set for a function in H^∞( ⁿ). Bibliography 4 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 303, 2003, pp. 272–278.