Sum-Difference Sequences and Catalan Numbers

 Let be a sequence of natural numbers > 1, and set . The sequence is called admissible if a _i divides for all i. It is known that the admissible sequences are counted by the Catalan numbers. We present a proof of this fact which, in turn, leads to some interesting combinatorial and number-theoretic questions. Received 12 May 1997; in revised form 9 June 1997     

Limits of Teichmüller mappings on trajectories

A classical Teichmüller sequence is a sequence of quasiconformal mapsf _i with complex dilatations of the form , where φ is a quadratic differential and 0≤k _i<1 are numbers such thatk _i→1 asi→∞. This situation occurs in the Teichmüller theory when one moves along a Teichmüller geodesic toward the boundary. The central result is that if τ is a vertical trajectory associated to φ, then there is often, for instance if the sequence is normalized so thatf _i fix 3 points, a subsequence such thatf _i tend either toward a constant or an injective map of τ (Theorem 4.1). If the limit is injective, it is an embedding of τ if τ does not contain points such that τ returns infinitely often to every neighborhood of the point. The main idea is to composef _i locally with a map ϱ_i so that the composed mapf _iϱ_i is conformal and coincides withf _i on τ. Normal family arguments are applied to the sequencef _iϱ_i. Various extensions are presented. The research for this paper has been supported by the project 51749 of the Academy of Finland.     

Poincaré-Type inequalities and reconstruction of Paley-Wiener functions on manifolds

The main goal of the article is to show that Paley-Wiener functions ƒ ∈ L ₂(M) of a fixed band width to on a Riemannian manifold of bounded geometry M completely determined and can be reconstructed from a set of numbers Φ_i (ƒ), i ∈ ℕwhere Φ_i is a countable sequence of weighted integrals over a collection of “small” and “densely” distributed compact subsets. In particular, Φ_i, i ∈ ℕ,can be a sequence of weighted Dirac measures δ_xi, x_i ∈M. It is shown that Paley-Wiener functions on M can be reconstructed as uniform limits of certain variational average spline functions. To obtain these results we establish certain inequalities which are generalizations of the Poincaré-Wirtingen and Plancherel-Polya inequalities. Our approach to the problem and most of our results are new even in the one-dimensional case.     

Large Deviation Principles for Moving Average Processes of Real Stationary Sequences

Let X _k =∑ _i=−∞^∞ a _i ξ _k−i ,k≥1, be the moving average processes, where (ξ _i ) _i∈ℤ is a sequence of real stationary random variables. Under the assumptions that the large deviation principle (LDP) for real stationary sequence holds, LDP for the moving average processes of real stationary sequence is established.      

Teichmüller sequences on trajectories invariant under a Kleinian group

We extend the results of [T2] to the situation where there is a compatibility with the action of a Kleinian group. A classical Techmüller sequence is a sequence of quasiconformal mapsf _i with complex dilatations of the form , where ϕ is a quadratic differential and 0<-k _i<1 are numbers such thatk _i→1 asi→∞. We proved in [T2] that if τ is a vertical trajectory associated to ϕ, then there is often, for instance if the sequence is normalized so thatf _i fix 3 points, a subsequence such thatf _i tend either toward a constant or an injective map of τ. If there is compatibility with the action of a non-elementary finitely generated Kleinian groupG, we can given a precise characterization which of these cases occurs. Suppose thatf _i induce isomorphisms ϕ_i ofG onto another Kleinian group and that ϕ_i have algebraic limit ϕ. If the quadratic differential is defined on a component of the ordinary set ofG, if there are no parabolic elements, and if τ is extended maximally so that all branches coming together at a singular point are included, then we can state the main result as follows. The limit is a constantc if the stabilizerG _τ of τ is elementary; and, if it is non-elementary, then the limit is injective. In the first case, ϕ(g) is parabolic with fixpointc wheneverg∈G _τ is of infinite order; and in the latter case, the limitf is an embedding of τ in a natural topology of τ, andf embeds τ into a component of the limit set of ϕG whose stabilizer is ϕG _τ. Various extensions and generalizations are presented. The research for this paper has been supported by the project 51749 of the Academy of Finland.     

Analysis of Alternative Digit Sets for Nonadjacent Representations

It is known that every positive integer n can be represented as a finite sum of the form ∑_ia_i2ⁱ, where a_i ∈ {0, 1,−1} and no two consecutive a_i’s are non-zero (“nonadjacent form”, NAF). Recently, Muir and Stinson [14, 15] investigated other digit sets of the form {0, 1, x}, such that each integer has a nonadjacent representation (such a number x is called admissible). The present paper continues this line of research. The topics covered include transducers that translate the standard binary representation into such a NAF and a careful topological study of the (exceptional) set (which is of fractal nature) of those numbers where no finite look-ahead is sufficient to construct the NAF from left-to-right, counting the number of digits 1 (resp. x) in a (random) representation, and the non-optimality of the representations if x is different from 3 or −1. This paper was written while the first author was a visitor at the John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, University of the Witwatersrand, Johannesburg. He thanks the centre for its hospitality. He was also supported by the grant S8307-MAT of the Austrian Science Fund. This author is supported by the grant NRF 2053748 of the South African National Research Foundation. The research of this author was done while he was with the John Knopfmacher Centre for Applicable Analysis and Number Theory, University of the Witwatersrand, Johannesburg.     

On a metric generalization of ramsey’s theorem

An increasing sequence of realsx=〈x _i :i<ω〉 is simple if all “gaps”x _i +1−x _i are different. Two simple sequencesx andy are distance similar ifx _i +1−x _i <x _j +1−x _j if and only ify _i +1−y _i <y _j +1−y _j for alli andj. Given any bounded simple sequencex and any coloring of the pairs of rational numbers by a finite number of colors, we prove that there is a sequencey distance similar tox all of whose pairs are of the same color. We also consider many related problems and generalizations. Partially supported by OTKA-4269. Partially supported by NSF grant STC-91-19999. Partially supported by OTKA-T-020914, NSF grant CCR-9424398 and PSC-CUNY Research Award 663472.     

Adaptive density deconvolution with dependent inputs

In the convolution model Z _i = X _i + ε _i , we give a model selection procedure to estimate the density of the unobserved variables (X _i )_1≤i≤n , when the sequence (X _i ) _i≥1 is strictly stationary but not necessarily independent. This procedure depends on whether the density of the ɛ _i is supersmooth or ordinary smooth. The rates of convergence of the penalized contrast estimators are the same as in the independent framework, and are minimax over most regularity classes on ℝ. Our results apply to mixing sequences, but also to many other dependent sequences. When the errors are supersmooth, the condition on the dependence coefficients is the minimal condition of that type ensuring that the sequence (X _i ) _i≥1 is not a long-memory process.      

On the Complete Convergence of Moving Average Process with Banach Space Valued Random Elements

Let {Y _i ;−∞<i<∞} be a doubly infinite sequence of independent random elements taking values in a separable real Banach space and stochastically dominated by a random variable X. Let {a _i ;−∞<i<∞} be an absolutely summable sequence of real numbers and set V _i =∑ _k=−∞^∞ a _i+k Y _i ,i≥1. In this paper, we derive that if and E|X| ^μ log  ^ρ |X|<0, for some μ (0<μ<2, μ≠1) and ρ>0 then for all ε>0. This work was partially supported by the Korean Research Foundation Grant funded by the Korean Government (KRF-2006-353-C00006, KRF-2006-251-C00026).     

Joint Distributions of Numbers of Runs of Specified Lengths in a Sequence of Markov Dependent Multistate Trials

Let Z ₀, Z ₁,...,Z _n be a sequence of Markov dependent trials with state space Ω = {F ₁,...,F _λ, S ₁,...,S _ν}, where we regard F ₁,...,F _λ as failures and S ₁,...,S _ν as successes. In this paper, we study the joint distribution of the numbers of S _i -runs of lengths k _ij (i = 1,2,...,ν, j = 1,2,...,r _i ) based on four different enumeration schemes. We present formulae for the evaluation of the probability generating functions and the higher order moments of this distribution. In addition, when the underlying sequence is i.i.d. trials, the conditional distribution of the same run statistics, given the numbers of success and failure is investigated. We give further insights into the multivariate run-related problems arising from a sequence of the multistate trials. Besides, our results have potential applications to problems of various research areas and will come to prominence in the future. This research was partially supported by the ISM Cooperative Research Program (2004-ISM·CRP-2007).     

On moments of the maximum of partial sums of moving average processes under dependence assumptions

Let {Y i;∞ < i < ∞} be a doubly infinite sequence of identically distributed-mixing random variables and let {a i;∞ < i < ∞} be an absolutely summable sequence of real numbers.In this paper we study the moments of sup(1 ≤ r < 2,p > 0) under the conditions of some moments.     

A bounded N-tuplewise independent and identically distributed counterexample to the CLT

Summary. A sequence of random variables X ₁,X ₂,X ₃,… is said to be N-tuplewise independent if X i ₁,X i ₂,…,X i _N are independent whenever (i ₁,i ₂,…,i _N ) is an N-tuple of distinct positive integers. For any fixed N∈ℤ⁺, we construct a sequence of bounded identically distributed N-tuplewise independent random variables which fail to satisfy the central limit theorem. Received: 17 May 1996 / In revised form: 28 January 1998     

Small deviations of the maximal element of a sequence of weighted independent random variables

Consider a sequence {X _i } of independent copies of a nonnegative random variable X and let M = sup _{j ≥ 1}λ _j X _j , where {λ _j } is a nonincreasing sequence of positive numbers for which P(M < ∞) = 1. The asymptotic behavior of -logP(M < r) as r → 0 is studied.     

A survey of Skolem-type sequences and Rosa’s use of them

Let D be a set of positive integers. A Skolem-type sequence is a sequence of i ∈ D such that every i ∈ D appears exactly twice in the sequence at positions a _i and b _i , and |b _i − a _i | = i. These sequences might contain empty positions, which are filled with null elements. Thoralf A. Skolem defined and studied Skolem sequences in order to generate solutions to Heffter’s difference problems. Later, Skolem sequences were generalized in many ways to suit constructions of different combinatorial designs. Alexander Rosa made the use of these generalizations into a fine art. Here we give a survey of Skolem-type sequences and their applications. Supported by an NSERC Graduate fellowship. This work is in partial fulfillment of an M.Sc.     

On the ternary goldbach problem with primes in independent arithmetic progressions

We show that for every fixed A > 0 and θ > 0 there is a ϑ = ϑ(A, θ) > 0 with the following property. Let n be odd and sufficiently large, and let Q ₁ = Q ₂:= n ^1/2(log n)^−ϑ and Q ₃:= (log n) ^θ . Then for all q ₃ ≦ Q ₃, all reduced residues a ₃ mod q ₃, almost all q ₂ ≦ Q ₂, all admissible residues a ₂ mod q ₂, almost all q ₁ ≦ Q ₁ and all admissible residues a ₁ mod q ₁, there exists a representation n = p ₁ + p ₂ + p ₃ with primes p _i ≡ a _i (q _i ), i = 1, 2, 3.      

The logic of pseudo-S-integers

Let {n _i } be a sequence of natural numbers and let {p _i } be a listing of rational primes. Then an abelian groupG={x ∈ √| ord _pi x ≥ −n _i } is called a group of pseudo-integers. We investigate the logical properties of such groups of pseudo-integers and the counterparts of such groups in global fields in the case the number of primes allowed to appear in the denominator is infinite. We show that, while the addition problem of any recursive group of pseudo-integers is decidable, the Diophantine problem for some recursive groups of pseudo-integers with infinite number of primes allowed in the denominator, is not decidable. More precisely, there exist recursive groups of pseudo-integers, where infinite number of primes are allowed to appear in the denominator, such that there is no uniform algorithm to decide whether a polynomial equation over ℤ in several variables has solutions in the group. This result is obtained by giving a Diophantine definition of ℤ over these groups. The proof is based on the strong Hasse norm principal. The research for this paper has been partially supported by NSA grant MDA904-96-1-0019.     

Relative entropy and mixing properties of infinite dimensional diffusions

Summary. Let η be a diffusion process taking values on the infinite dimensional space T ^Z , where T is the circle, and with components satisfying the equations dη _i =σ _i (η) dW _i +b _i (η) dt for some coefficients σ _i and b _i , i∈Z. Suppose we have an initial distribution μ and a sequence of times t _n →∞ such that lim _{n →∞}μS _tn =ν exists, where S _t is the semi-group of the process. We prove that if σ _i and b _i are bounded, of finite range, have uniformly bounded second order partial derivatives, and inf _{i ,η}σ _i (η)>0, then ν is invariant. Received: 12 September 1996 / In revised form: 10 November 1997     

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.     

Slicing Convex Sets and Measures by a Hyperplane

Given convex bodies K ₁,…,K _d in ℝ ^d and numbers α ₁,…,α _d ∈[0,1], we give a sufficient condition for existence and uniqueness of an (oriented) halfspace H with Vol (H∩K _i )=α _i ⋅Vol K _i for every i. The result is extended from convex bodies to measures.     

On the influence of parameters determining anisotropic spaces of smooth functions on characteristics of the embedding theorems

An anisotropic Sobolev and Nikol'skii-Besov space on a domain G is determined by its integro-differential (shortly, ID) parameters. On the other hand, the geometry of G is characterized by the set Λ(G) of all vectors λ=(λ₁,..., λ_n) such that G satisfies the λ-horn condition. We study the dependence of the totality of possible embeddings upon the set Λ(G) and theID-parameters of the space. We consider only embeddings with q≥pⁱ, where pⁱ are the integral parameters of the space and q is the integral embedding parameter. For a given space, we introduce its initial matrix A₀ determined by theID-parameters. A₀ turns out to be a Z-matrix. On the basis of a natural classification of Z-matrices, a classification of anisotropic spaces is introduced. This classification allows one to restate the existence of an embedding with q≥pⁱ in terms of certain specific properties of A₀. Let A₀ be a nondegenerate M-matrix. Any vector λ∈Λ(G) gives rise to a certain set of admissible values of the embedding parameters. We call λ optimal if this set is the largest possible. It turns out that the optimal vector λ _G ^* is determined by Λ(G) and A₀, and may be found by a linear optimization procedure. The following cases are possible: a) , b) , c) λ _G ^* does not exist. In case a) the set of admissible values of the embedding parameters is the biggest, while in case c) no embeddings with q≥pⁱ exist. In case b) the so-called saturation phenomenon occurs, i.e., certain variations of some differential parameters of the space do not change the set of admissible values of the embedding parameters. The latter fact has some applications to the problem of extension of all functions belonging to the given space from G to Eⁿ. Bibliography: 20 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 201, 1992, pp. 22–94. Translated by A. A. Mekler.