Approximation of Densities of Absolutely Continuous Components of Measures in a Hilbert Space Using the Ornstein-Uhlenbeck Semigroup

We study the behavior of measures obtained as a result of the action of the Ornstein-Uhlenbeck semigroup T _t associated with the Gaussian measure μ on an arbitrary probability measure ν in a separable Hilbert space as t → 0+. We prove that the densities of the parts of T _t ν absolutely continuous with respect to μ converge in the measure μ to the density of the part of ν absolutely continuous with respect to μ. For a finite-dimensional space, we prove the convergence of these densities μ-almost everywhere. In the infinite-dimensional case, we give sufficient conditions for almost-everywhere convergence. We also consider conditions on the absolute continuity of T _t ν with respect to μ in terms of the coefficients of the expansion of T _t ν in a series in Hermite polynomials (an analog of the Ito- Wiener expansion) and the connection with finite absolute continuity.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 12, pp. 1654 – 1664, December, 2004.     

ν-perfect groups

Let F_∞ be a free group on a countable set {x₁, x₂, …} and ν be a variety of groups, defined by the set of outer commutators V, in the free generators x_i's.The paper is devoted to give the complete structure of a ν-covering of ν-perfect groups. Fur thermore necessary and sufficient conditions for the universality of a ν-central extension by a group and its ν-covering group will be presented.     

Weak Limits for Multivariate Random Sums

Let {Xi, i1} be a sequence of i.i.d. random vectors inRd, and letνp, 0<p<1, be a positive, integer valued random variable, independent ofXis. Theν-stable distributions are the weak limits of properly normalized random sums ∑νpi=1 Xiasνp ∞ andpνp ν. We study the properties ofν-stable laws through their representation via stable laws. In particular, we estimate their tail probabilities and provide conditions for finiteness of their moments.     

Trace cocharacters and the Kronecker products of Schur functions

It follows from the theory of trace identities developed by Procesi and Razmyslov that the trace cocharacters arising from the trace identities of the algebra M_r(F) of r×r matrices over a field F of characteristic zero are given by TC_r,n=∑_λΛr(n)χ^λχ^λ where χ^λχ^λ denotes the Kronecker product of the irreducible characters of the symmetric group associated with the partition λ with itself and Λ_r(n) denotes the set of partitions of n with r or fewer parts, i.e. the set of partitions λ=(λ₁λ_k) with kr. We study the behavior of the sequence of trace cocharacters TC_r,n. In particular, we study the behavior of the coefficient of χ^(ν,n−m) in TC_r,n as a function of n where ν=(ν₁ν_k) is some fixed partition of m and n−mν_k. Our main result shows that such coefficients always grow as a polynomial in n of degree r−1.     

Realizations of coupled vectors in the tensor product of representations of and

Using the realization of positive discrete series representations of in terms of a complex variable z, we give an explicit expression for coupled basis vectors in the tensor product of ν+1 representations as polynomials in ν+1 variables z₁,…,z_ν+1. These expressions use the terminology of binary coupling trees (describing the coupled basis vectors), and are explicit in the sense that there is no reference to the Clebsch–Gordan coefficients of . In general, these polynomials can be written as (terminating) multiple hypergeometric series. For ν=2, these polynomials are triple hypergeometric series, and a relation between the two binary coupling trees yields a relation between two triple hypergeometric series. The case of is discussed next. Also here the polynomials are determined explicitly in terms of a known realization; they yield an efficient way of computing coupled basis vectors in terms of uncoupled basis vectors.     

q-Bessel, arbres, et chemins valués

We give interpretations for quotient J_ν+1/J_ν of q-Bessel functions. These q-analogs are related to generating function of weighted complete binary trees according to the number of leaves and to multichains on a partially ordered set, corresponding to weighted paths in the plane.Nous donnons des interprétations combinatoires du rapport J_ν+1/J_ν de q-fonctions de Bessel. Ces q-analogues énumèrent des classes d'arbres binaires complets valués suivant le nombre de feuilles et des multichaînes d'un ensemble partiellement ordonné, correspondant à des chemins valués dans le plan.     

Geometric interpretations of two branching theorems of D.E. Littlewood

D.E. Littlewood proved two branching theorems for decomposing the restriction of an irreducible finite-dimensional representation of a unitary group to a symmetric subgroup. One is for restriction of a representation of U(n) to the rotation group SO(n) when the given representation τ_λ of U(n) has nonnegative highest weight λ of depth n/2. It says that the multiplicity in τ_λ|_SO(n) of an irreducible representation of SO(n) of highest weight ν is the sum over μ of the multiplicities of τ_λ in the U(n) tensor product τ_μτ_ν, the allowable μ's being all even nonnegative highest weights for U(n). Littlewood's proof is character-theoretic. The present paper gives a geometric interpretation of this theorem involving the tensor products τ_μτ_ν explicitly. The geometric interpretation has an application to the construction of small infinite-dimensional unitary representations of indefinite orthogonal groups and, for each of these representations, to the determination of its restriction to a maximal compact subgroup. The other Littlewood branching theorem is for restriction from U(2r) to the rank-r quaternion unitary group Sp(r). It concerns nonnegative highest weights for U(2r) of depth r, and its statement is of the same general kind. The present paper finds an analogous geometric interpretation for this theorem also.     

Quadrature formulae connected to σ-orthogonal polynomials

Let dλ(t) be a given nonnegative measure on the real line , with compact or infinite support, for which all moments exist and are finite, and μ₀>0. Quadrature formulas of Chakalov–Popoviciu type with multiple nodes

where σ=σ_n=(s₁,s₂,…,s_n) is a given sequence of nonnegative integers, are considered. A such quadrature formula has maximum degree of exactness d_max=2∑_ν=1ⁿs_ν+2n−1 if and only if

The proof of the uniqueness of the extremal nodes τ₁,τ₂,…,τ_n was given first by Ghizzetti and Ossicini (Rend. Mat. 6(8) (1975) 1–15). Here, an alternative simple proof of the existence and the uniqueness of such quadrature formulas is presented. In a study of the error term R(f), an influence function is introduced, its relevant properties are investigated, and in certain classes of functions the error estimate is given. A numerically stable iterative procedure, with quadratic convergence, for determining the nodes τ_ν, ν=1,2,…,n, which are the zeros of the corresponding σ-orthogonal polynomial, is presented. Finally, in order to show a numerical efficiency of the proposed procedure, a few numerical examples are included.     

Analysis of Carry Propagation in Addition: An Elementary Approach

Our goal in this paper is to analyze carry propagation in addition using only elementary methods (that is, those not involving residues, contour integration, or methods of complex analysis). Our results concern the length of the longest carry chain when two independent uniformly distributed n-bit numbers are added. First, we show using just first- and second-moment arguments that the expected length C_n of the longest carry chain satisfies C_n = log₂n + O(1). Second, we use a sieve (inclusion–exclusion) argument to give an exact formula for C_n. Third, we give an elementary derivation of an asymptotic formula due to Knuth, C_n = log₂n + Φ(log₂ n) + O((logn)⁴/n), where Φ(ν) is a bounded periodic function of ν, with period 1, for which we give both a simple integral expression and a Fourier series. Fourth, we give an analogous asymptotic formula for the variance V_n of the length of the longest carry chain: V_n = Ψ(log₂ n) + O((logn)⁵/n), where Ψ(ν) is another bounded periodic function of ν, with period 1. Our approach can be adapted to addition with the “end-around” carry that occurs in the sign-magnitude and 1s-complement representations. Finally, our approach can be adapted to give elementary derivations of some asymptotic formulas arising in connection with radix-exchange sorting and collision-resolution algorithms, which have previously been derived using contour integration and residues.     

Removable Singularities for Lu=Ψ(u) and Orlicz Capacities

Suppose L is a second order elliptic differential operator in ^d and let α>1. Baras and Pierre have proved in 1984 that Γ is removable for Lu=u^α if and only if its Bessel capacity Cap_2, α′(Γ)=0. We extend this result to a general equation Lu=Ψ(u) where Ψ(u) is an increasing convex function subject to Δ₂ and ₂ conditions. Namely, we prove that Γ is removable for Lu=Ψ(u) if and only if its Orlicz capacity is zero, that is, the integral ∫_B dx Ψ(∫_Γ |x−y|^2−d ν(dy)) is equal to 0 or ∞ for every measure ν concentrated on Γ, where B stands for any ball containing Γ.     

Routing through a generalized switchbox

We present an algorithm for the routing problem for two-terminal nets in generalized switchboxes. A generalized switchbox is any subset R of the planar rectangular grid with no nontrivial holes, i.e., every finite face has exactly four incident vertices. A net is a pair of nodes of nonmaximal degree on the boundary of R. A solution is a set of edge-disjoint paths, one for each net. Our algorithm solves standard generalized switchbox routing problems in time O(n(log n)²) where n is the number of vertices of R, i.e., it either finds a solution or indicates that there is none. A problem is standard if deg(ν) + ter(ν) is even for all vertices ν where deg(ν) is the degree of ν and ter(ν) is the number of nets which have ν as a terminal. For nonstandard problems we can find a solution in time O(n(log n)² + |U|²) where U is the set of vertices ν with deg(ν) + ter(ν) is odd.     

Orthogonal Polynomial Solutions of Spectral Type Differential Equations: Magnus' Conjecture

Let τ=σ+ν be a point mass perturbation of a classical moment functional σ by a distribution ν with finite support. We find necessary conditions for the polynomials {Q_n(x)}^∞_n=0, orthogonal relative to τ, to be a Bochner–Krall orthogonal polynomial system (BKOPS); that is, {Q_n(x)}^∞_n=0 are eigenfunctions of a finite order linear differential operator of spectral type with polynomial coefficients: L_N[y](x)=∑^N_i=1 ℓ_i(x) y⁽ⁱ⁾(x)=λ_ny(x). In particular, when ν is of order 0 as a distribution, we find necessary and sufficient conditions for {Q_n(x)}^∞_n=0 to be a BKOPS, which strongly support and clarify Magnus' conjecture which states that any BKOPS must be orthogonal relative to a classical moment functional plus one or two point masses at the end point(s) of the interval of orthogonality. This result explains not only why the Bessel-type orthogonal polynomials (found by Hendriksen) cannot be a BKOPS but also explains the phenomena for infinite-order differential equations (found by J. Koekoek and R. Koekoek), which have the generalized Jacobi polynomials and the generalized Laguerre polynomials as eigenfunctions.     

Banach function subspaces of L of a vector measure and related Orlicz spaces

Given a vector measure ν with values in a Banach space X, we consider the space L¹(ν) of real functions which are integrable with respect to ν. We prove that every order continuous Banach function space Y continuously contained in L¹(ν) is generated via a certain positive map related to ν and defined on X* x M, where X* is the dual space of X and M the space of measurable functions. This procedure provides a way of defining Orlicz spaces with respect to the vector measure ν.     

Compensated Compactness, Paracommutators, and Hardy Spaces

LetB₁: ⁿ× ^N₁→ ^m₁,B₂: ⁿ× ^N₂→ ^m₂andQ: ^m₂→ ^m₁be bilinear forms which are related as follows: ifμandνsatisfyB₁(ξ, μ)=0 andB₂(ξ, ν)=0 for someξ≠0, thenμ^τQν=0. Supposep⁻¹+q⁻¹=1. Coifman, Lions, Meyer and Semmes proved that, ifuL^p( ⁿ) andvL^q( ⁿ), and the first order systemsB₁(D, u)=0,B₂(D, v)=0 hold, thenu^τQvbelongs to the Hardy spaceH¹( ⁿ), provided that both (i)p=q=2, and (ii) the ranks of the linear mapsB_j(ξ, ·) : ^N_j→ ^m₁are constant. We apply the theory of paracommutators to show that this result remains valid when only one of the hypotheses (i), (ii) is postulated. The removal of the constant-rank condition whenp=q=2 involves the use of a deep result of Lojasiewicz from singularity theory.     

The Geometry of the First Non-zero Stekloff Eigenvalue

Let (Mⁿ, g) be a compact Riemannian manifold with boundary and dimensionn2. In this paper we discuss the first non-zero eigenvalue problem \begin{align}\Delta\varphi & = & 0\qquad & on\quad M,\\ \frac{\partial\varphi}{\partial \eta} & = & \ u_1\varphi\qquad & on\quad\partial M.\end{align}\eqno (1) Problem (1) is known as the Stekloff problem because it was introduced by him in 1902, for bounded domains of the plane. We discuss estimates of the eigenvalueν₁in terms of the geometry of the manifold (Mⁿ, g). In the two-dimensional case we generalize Payne's Theorem [P] for bounded domains in the plane to non-negative curvature manifolds. In this case we show thatν₁k₀, wherek_gk₀andk_grepresents the geodesic curvature of the boundary. In higher dimensionsn3 for non-negative Ricci curvature manifolds we show thatν₁>k₀/2, wherek₀is a lower bound for any eigenvalue of the second fundamental form of the boundary. We introduce an isoperimetric constant and prove a Cheeger's type inequality for the Stekloff eigenvalue.     

Multiplying Schur functions

A new combinatorial rule for expanding the product of Schur functions as a sum of Schur functions is formulated. The rule has several advantages over the Littlewood-Richardson rule (D. E. Littlewood and A. R. Richardson, Philos. Trans. Roy. Soc. London Ser. A233 (1934), 49–141). First this rule allows for direct computation of the expansion of the product of any number of Schur functions, not just the product of two Schur functions. Also, the rule is easily stated and is well suited to computer implementation. It is shown that the rule implies the Littlewood-Richardson rule and gives a combinatorial proof that the coefficient of S_λ in the product S_μS_ν equals the coefficient of S_ν in the expansion of the skew Schur function S_λ/μ. The rule is derived from some results proved independently by A. P. Hillman and R. M. Grassl (J. Combin. Comput. Sci. Systems5 (1980), 305–316) and by D. White (J. Combin. Theory Ser. A30 (1981), 237–247) on the Robinson-Schensted-Knuth correspondence.     

Rook-by-rook rook theory: Bijective proofs of rook and hit equivalences

Suppose μ and ν are integer partitions of n, and N>n. It is well known that the Ferrers boards associated to μ and ν are rook-equivalent iff the multisets [μ_i+i:1iN] and [ν_i+i:1iN] are equal. We use the Garsia–Milne involution principle to produce a bijective proof of this theorem in which non-attacking rook placements for μ are explicitly matched with corresponding placements for ν. One byproduct is a direct combinatorial proof that the matrix of Stirling numbers of the first kind is the inverse of the matrix of Stirling numbers of the second kind. We also prove q-analogues and p,q-analogues of these results. We also use the Garsia–Milne involution principle to show that for any two rook boards B and B^′, if B and B^′ are bijectively rook-equivalent, then B and B^′ are bijectively hit-equivalent.     

An Error Expansion for some Gauss–Turán Quadratures and <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-Estimates of the Remainder Term

Our aim in this paper is to obtain error expansions in the Gauss–Turán quadrature formula ∫₋₁¹f(t)w(t) dt=∑_ν=1ⁿ∑_i=0^2sA_i,νf⁽ⁱ⁾(τ_ν)+R_n,s(f), in the case when f is an analytic function in some region of the complex plane containing the interval [−1,1] in its interior. Using a representation of the remainder term R_n,s(f) in the form of contour integral over confocal ellipses, we obtain R_n,1(f) for the four Chebyshev weights and R_n,2(f) for the Chebyshev weight of the first kind. Also, we get a few new L¹-estimates of the remainder term, which are stronger than the previous ones. Some numerical results, illustrations and comparisons are also given. AMS subject classification (2000) 41A55, 65D30, 65D32.Received January 2004. Accepted October 2004. Communicated by Lothar Reichel.M. M. Spalević: This work was supported in part by the Serbian Ministry of Science and Environmental Protection (Project: Applied Orthogonal Systems, Constructive Approximation and Numerical Methods, grant number 2002).     

Advanced topology on the multiscale sequence spaces

We pursue the study of the multiscale spaces S^ν introduced by Jaffard in the context of multifractal analysis. We give the necessary and sufficient condition for S^ν to be locally p-convex, and exhibit a sequence of p-norms that defines its natural topology. The strong topological dual of S^ν is identified to another sequence space depending on ν, endowed with an inductive limit topology. As a particular case, we describe the dual of a countable intersection of Besov spaces.     

Some Properties of Nonstar Steps in Addition Chains and New Cases Where the Scholz Conjecture Is True

Let ℓ(n) be the smallest possible length of addition chains for a positive integer n. Then Scholz conjectured that ℓ(2ⁿ − 1) ≤ n + ℓ(n) − 1, which still remains open. It is known that the Scholz conjecture is true when ν(n) ≤ 4, where ν(n) is the number of 1's in the binary representation of n. In this paper, we give some properties of nonstar steps in addition chains and prove that the Scholz conjecture is true for infinitely many new integers including the case where ν(n) = 5.