Wavelet expansions of fractal measures

Let μ be a measure on ℝⁿ that satisfies the estimate μ(B _r(x))≤cr ^α for allx ∈ ℝⁿ and allr ≤ 1 (B _r(x) denotes the ball of radius r centered atx. Let ϕ _j,k ^(ɛ) (x)=2 ^nj2ϕ^(ɛ)(2 ^j x-k) be a wavelet basis forj ∈ ℤ, κ ∈ ℤⁿ, and ∈ ∈E, a finite set, and letP _j (T)=Σ_ɛ,k <T,ϕ _j,k ^(ɛ) >ϕ _j,k ^(ɛ) denote the associated projection operators at levelj (T is a suitable measure or distribution). Iff ∈Ls ^p(dμ) for 1 ≤p ≤ ∞, we show thatP _j(f dμ) ∈ L^p(dx) and ||P _j (fdμ)||L _p(dx)≤c2 ^{j((n-α)/p′))}||f||L _p(dμ) for allj ≥ 0. We also obtain estimates for the limsup and liminf of ||P _j (fdμ)||L _p(dx) under more restrictive hypotheses. Communicated by Guido Weiss     

The Johnson–Lindenstrauss Lemma Almost Characterizes Hilbert Space, But Not Quite

Let X be a normed space that satisfies the Johnson–Lindenstrauss lemma (J–L lemma, in short) in the sense that for any integer n and any x ₁,…,x _n ∈X, there exists a linear mapping L:X→F, where F⊆X is a linear subspace of dimension O(log n), such that ‖x _i −x _j ‖≤‖L(x _i )−L(x _j )‖≤O(1)⋅‖x _i −x _j ‖ for all i,j∈{1,…,n}. We show that this implies that X is almost Euclidean in the following sense: Every n-dimensional subspace of X embeds into Hilbert space with distortion 2^2O(log*n)2^{2^{O(\log^{*}n)}} . On the other hand, we show that there exists a normed space Y which satisfies the J–L lemma, but for every n, there exists an n-dimensional subspace E _n ⊆Y whose Euclidean distortion is at least 2^Ω(α(n)), where α is the inverse Ackermann function.     

Exact laws for sums of ratios of order statistics from the Pareto distribution

Consider independent and identically distributed random variables {X _nk, 1 ≤ k ≤ m, n ≤ 1} from the Pareto distribution. We select two order statistics from each row, X _n(i) ≤ X _n(j), for 1 ≤ i < j ≤ = m. Then we test to see whether or not Laws of Large Numbers with nonzero limits exist for weighted sums of the random variables R _ij = X _n(j)/X _n(i).     

Asymptotic separation for independent trajectories of Markov processes

We say that n independent trajectories ξ₁(t),…,ξ _n (t) of a stochastic process ξ(t)on a metric space are asymptotically separated if, for some ɛ > 0, the distance between ξ _i (t _i ) and ξ _j (t _j ) is at least ɛ, for some indices i, j and for all large enough t ₁,…,t _n , with probability 1. We prove sufficient conitions for asymptotic separationin terms of the Green function and the transition function, for a wide class of Markov processes. In particular,if ξ is the diffusion on a Riemannian manifold generated by the Laplace operator Δ, and the heat kernel p(t, x, y) satisfies the inequality p(t, x, x) ≤ Ct ^−ν/2 then n trajectories of ξ are asymptotically separated provided . Moreover, if for some α∈(0, 2)then n trajectories of ξ^(α) are asymptotically separated, where ξ^(α) is the α-process generated by −(−Δ)^α/2. Received: 10 June 1999 / Revised version: 20 April 2000 / Published online: 14 December 2000 RID="*" ID="*" Supported by the EPSRC Research Fellowship B/94/AF/1782 RID="**" ID="**" Partially supported by the EPSRC Visiting Fellowship GR/M61573     

Dependent percolation in two dimensions

For a natural number k, define an oriented site percolation on ℤ² as follows. Let x _i , y _j be independent random variables with values uniformly distributed in {1, …, k}. Declare a site (i, j) ∈ℤ² closed if x _i = y _j , and open otherwise. Peter Winkler conjectured some years ago that if k≥ 4 then with positive probability there is an infinite oriented path starting at the origin, all of whose sites are open. I.e., there is an infinite path P = (i ₀, j ₀)(i ₁, j ₁) · · · such that 0 = i ₀≤i ₁≤· · ·, 0 = j ₀≤j ₁≤· · ·, and each site (i _n , j _n ) is open. Rather surprisingly, this conjecture is still open: in fact, it is not known whether the conjecture holds for any value of k. In this note, we shall prove the weaker result that the corresponding assertion holds in the unoriented case: if k≤ 4 then the probability that there is an infinite path that starts at the origin and consists only of open sites is positive. Furthermore, we shall show that our method can be applied to a wide variety of distributions of (x _i ) and (y _j ). Independently, Peter Winkler [14] has recently proved a variety of similar assertions by different methods. Received: 4 March 1999 / Revised version: 27 September 1999 / Published online: 21 June 2000     

Smooth convex bodies with proportional projection functions

For a convex body K ⊂ ℝⁿ and i ∈ {1, …, n − 1}, the function assigning to any i-dimensional subspace L of ℝⁿ, the i-dimensional volume of the orthogonal projection of K to L, is called the i-th projection function of K. Let K, K ₀ ⊂ ℝⁿ be smooth convex bodies with boundaries of class C ² and positive Gauss-Kronecker curvature and assume K ₀ is centrally symmetric. Excluding two exceptional cases, (i, j) = (1, n − 1) and (i, j) = (n − 2, n − 1), we prove that K and K ₀ are homothetic if their i-th and j-th projection functions are proportional. When K ₀ is a Euclidean ball this shows that a convex body with C ² boundary and positive Gauss-Kronecker with constant i-th and j-th projection functions is a Euclidean ball. The second author was supported in part by the European Network PHD, FP6 Marie Curie Actions, RTN, Contract MCRN-511953.     

Partitioning Posets

Given a poset P = (X, ≺ ), a partition X ₁, ..., X _k of X is called an ordered partition of P if, whenever x ∈ X _i and y ∈ X _j with x ≺ y, then i ≤ j. In this paper, we show that for every poset P = (X, ≺ ) and every integer k ≥ 2, there exists an ordered partition of P into k parts such that the total number of comparable pairs within the parts is at most (m − 1)/k, where m ≥ 1 is the total number of edges in the comparability graph of P. We show that this bound is best possible for k = 2, but we give an improved bound, , for k ≥ 3, where c(k) is a constant depending only on k. We also show that, given a poset P = (X, ≺ ) and an integer 2 ≤ k ≤ |X|, we can find an ordered partition of P into k parts that minimises the total number of comparable pairs within parts in time polynomial in the size of P. We prove more general, weighted versions of these results. Supported by an EPSRC doctoral training grant.     

Approximations for a Three Dimensional Scan Statistic

Let X _ijk ,1 ≤ i ≤ N ₁,1 ≤ j ≤ N ₂, 1 ≤ k ≤ N ₃ be a sequence of independent and identically distributed 0 − 1 Bernoulli trials. X _ijk  = 1 if an event has occurred at the i,j,k ^th location in a three dimensional rectangular region and X _ijk  = 0, otherwise. For 2 ≤ m _j  ≤ N _j  − 1,1 ≤ j ≤ 3, a three dimensional discrete scan statistic is defined as the maximum number of events in any m ₁×m ₂×m ₃ rectangular sub-region in the entire N ₁×N ₂×N ₃ rectangular region. In this article, a product-type approximation and three Poisson approximations are derived for the distribution of this three dimensional scan statistic. Numerical results are presented to evaluate the accuracy of these approximations and their use in testing for randomness.     

Gersgorin variations II: On themes of Fan and Gudkov

Assume F={f₁,. . .,f_n} is a family of nonnegative functions of n−1 nonnegative variables such that, for every matrix A of order n, |a_ii|>f_i (moduli of off-diagonal entries in row i of A) for all i implies A nonsingular. We show that there is a positive vector x, depending only on F, such that for all A=(a_ij), and all i, f_i≥∑_j|a_ij|{x_j}/x_i. This improves a theorem of Ky Fan, and yields a generalization of a nonsingularity criterion of Gudkov. Dedicated to Charles A. Micchelli, in celebration of his 60th birthday and our 30 years of friendship     

Carleson type estimates for second order elliptic equations with unbounded drift

We derive a Carleson type estimate for positive solutions of non-divergence second order elliptic equations Lu = a _ij D _ij u + b _i D _i u = 0 in a bounded domain Ω ⊂ ℝ ⁿ . We assume that b _i ∈ L ⁿ (Ω) and Ω is a twisted H?lder domain of order α ∈ (1/2, 1] which satisfies a weak regularity condition. We also provide an example which shows that the main result fails in general if α ∈ (0, 1/2]. Bibliography: 18 titles.     

Construction of multivariate compactly supported orthonormal wavelets

We propose a constructive method to find compactly supported orthonormal wavelets for any given compactly supported scaling function φ in the multivariate setting. For simplicity, we start with a standard dilation matrix 2I_2×2 in the bivariate setting and show how to construct compactly supported functions ψ₁,. . .,ψ_n with n>3 such that {2^kψ_j(2^kx−ℓ,2^ky−m), k,ℓ,m∈Z, j=1,. . .,n} is an orthonormal basis for L₂(ℝ²). Here, n is dependent on the size of the support of φ. With parallel processes in modern computer, it is possible to use these orthonormal wavelets for applications. Furthermore, the constructive method can be extended to construct compactly supported multi-wavelets for any given compactly supported orthonormal multi-scaling vector. Finally, we mention that the constructions can be generalized to the multivariate setting. Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday Mathematics subject classifications (2000) 42C15, 42C30.     

Clustering of linearly interacting diffusions and universality of their long-time limit distribution

Let K⊂ℝ ^d (d≥ 1) be a compact convex set and Λ a countable Abelian group. We study a stochastic process X in K ^Λ, equipped with the product topology, where each coordinate solves a SDE of the form dX _i (t) = ∑ _j a(j−i) (X _j (t) −X _i (t))dt + σ (X _i (t))dB _i (t). Here a(·) is the kernel of a continuous-time random walk on Λ and σ is a continuous root of a diffusion matrix w on K. If X(t) converges in distribution to a limit X(∞) and the symmetrized random walk with kernel a _S (i) = a(i) + a(−i) is recurrent, then each component X _i (∞) is concentrated on {x∈K : σ(x) = 0 and the coordinates agree, i.e., the system clusters. Both these statements fail if a _S is transient. Under the assumption that the class of harmonic functions of the diffusion matrix w is preserved under linear transformations of K, we show that the system clusters for all spatially ergodic initial conditions and we determine the limit distribution of the components. This distribution turns out to be universal in all recurrent kernels a _S on Abelian groups Λ. Received: 10 May 1999 / Revised version: 18 April 2000 / Published online: 22 November 2000     

On antipodal Euclidean tight (2e + 1)-designs

Neumaier and Seidel (1988) generalized the concept of spherical designs and defined Euclidean designs in ℝ ⁿ . For an integer t, a finite subset X of ℝ ⁿ given together with a weight function w is a Euclidean t-design if holds for any polynomial f(x) of deg(f)≤ t, where {S _i , 1≤ i ≤ p} is the set of all the concentric spheres centered at the origin that intersect with X, X _i = X∩ S _i , and w:X→ ℝ_{> 0}. (The case of X⊂ S ⁿ⁻¹ with w≡ 1 on X corresponds to a spherical t-design.) In this paper we study antipodal Euclidean (2e+1)-designs. We give some new examples of antipodal Euclidean tight 5-designs. We also give the classification of all antipodal Euclidean tight 3-designs, the classification of antipodal Euclidean tight 5-designs supported by 2 concentric spheres.     

Near-interpolation

Summary.  A parametric curve fL ₂ ^(m) ([a,b]ℝ ^d ) is a ``near-interpolant' to prescribed data z <sub>ij</sub> ℝ <sup>d</sup> at data sites t <sub>i</sub> [a,b] within tolerances 0<ɛ <sub>ij</sub> ≤∞ if |f <sup>(j−1)</sup> (t <sub>i</sub> )−z <sub>ij</sub> |≤ɛ <sub>ij</sub> for i=1:n and j=1:m, and a ``best near-interpolant' if it also minimizes ∫ _a ^b |f ^(m) |². In this paper optimality conditions are derived for these best near-interpolants. Based on these conditions it is shown that the near-interpolants are actually smoothing splines with weights that appear as Lagrange multipliers corresponding to the constraints. The optimality conditions are applied to the computation of near-interpolants in the last sections of the paper. Received September 4, 2001 / Revised version received July 22, 2002 / Published online October 29, 2002 Mathematics Subject Classification (1991): 41A05, 41A15, 41A29     

A counterexample to the dominating set conjecture

The metric polytope met _n is the polyhedron associated with all semimetrics on n nodes and defined by the triangle inequalities x _ij − x _ik − x _jk ≤ 0 and x _ij + x _ik + x _jk ≤ 2 for all triples i, j, k of {1,..., n}. In 1992 Monique Laurent and Svatopluk Poljak conjectured that every fractional vertex of the metric polytope is adjacent to some integral vertex. The conjecture holds for n ≤ 8 and, in particular, for the 1,550,825,600 vertices of met₈. While the overwhelming majority of the known vertices of met₉ satisfy the conjecture, we exhibit a fractional vertex not adjacent to any integral vertex.     

Analytic properties of conditional curvatures of convex hypersurfaces and the dirichlet problem for the Monge-Ampère equation

The existence and uniqueness of a surface with given geometric characteristics is one of the important topical problems of global differential geometry. By stating this problem in terms of analysis, we arrive at second-order elliptic and parabolic partial differential equations. In the present paper we consider generalized solutions of the Monge-Ampère equation ||z _ij || = ϕ(x, z, p) in Λ ⁿ , wherez = z(x ₁,...,z _n ) is a convex function,p = (p ₁,...,P _n) = (∂z/∂x ₁,...,ϖz/ϖx _n), andz _ij =ϖ ² z/ϖx _i ϖx _j. We consider the Cayley-Klein model of the space Λ ⁿ and use a method based on fixed point principle for Banach spaces. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 763–768, November, 1998.     

Order of magnitude bounds for expectations of Δ2-functions of generalized random bilinear forms

Let Φ be a symmetric function, nondecreasing on [0,∞) and satisfying a Δ₂ growth condition, (X ₁,Y ₁), (X ₂,Y ₂),…,(X _n ,Y _n ) be arbitrary independent random vectors such that for any given i either Y _i =X _i or Y _i is independent of all the other variates. The purpose of this paper is to develop an approximation of

Numerical solutions of AXB = C for centrosymmetric matrix X under a specified submatrix constraint

We say that X = [x_ij] is centrosymmetric if x_ij = x_{n ? j + 1, n ? i + 1}, 1?i, j?n. In this paper, we present an efficient algorithm for minimizing ∥AXB ? C∥ where ∥·∥ is the Frobenius norm, A∈?^{m × n}, B∈?^{n × s}, C∈?^{m × s} and X∈?^{n × n} is centrosymmetric with a specified central submatrix [x_ij]_{p?i, j?n ? p}. Our algorithm produces a suitable X such that AXB = C in finitely many steps, if such an X exists. We show that the algorithm is stable in any case, and we give results of numerical experiments that support this claim. Copyright © 2011 John Wiley & Sons, Ltd.     

Consistency of a nonparametric conditional quantile estimator for random fields

Given a stationary multidimensional spatial process (Z _i = (X _i , Y _i ) ∈ ℝ ^d × ℝ, i ∈ ℤ ^N ), we investigate a kernel estimate of the spatial conditional quantile function of the response variable Y _i given the explicative variable X _i . Almost complete convergence and consistency in L ^2r norm (r ∈ ℕ*) of the kernel estimate are obtained when the sample considered is an α-mixing sequence.     

Large antipodal families

A family {A _i | i ∈ I} of sets in ℝ ^d is antipodal if for any distinct i, j ∈ I and any p ∈ A _i , q ∈ A _j , there is a linear functional ϕ:ℝ ^d → ℝ such that ϕ(p) ≠ ϕ(q) and ϕ(p) ≤ ϕ(r) ≤ ϕ(q) for all r ∈ ∪ _i∈I A _i . We study the existence of antipodal families of large finite or infinite sets in ℝ³. The research was supported by the Hungarian-South African Intergovernmental Scientific and Technological Cooperation Programme, NKTH Grant no. ZA-21/2006 and South African National Research Foundation Grant no. UID 61853, as well as Hungarian National Foundation for Scientific Research Grants no. NK 67867, no. T47102, and no. K72537.