Equations fonctionnelles et estimations de normes de matrices

We solve independently the equations 1/θ(x)θ(y)=ψ(x)−ψ(y)+φ(x−y)/θ(x−y) and 1/θ(x)θ(y)=σ(x)−σ(y)/θ(x−y)+τ(x)τ(y), τ(0)=0. In both cases we find θ²=aθ⁴+bθ²+c. We deduce estimates for the spectral radius of a matrix of type(1/θ(x _r −x _s )) (the accent meaning that the coefficients of the main diagonal are zero) and we study the case where thex _r are equidistant.

Dédié to à Monsieur le Professeur Otto Haupt à l'occasion de son cententiare avec les meilleurs voeux     

Estimation of Parameters for Diffusion Processes with Jumps from Discrete Observations

In this paper, we consider a multidimensional diffusion process with jumps whose jump term is driven by a compound Poisson process. Let a(x,θ) be a drift coefficient, b(x,σ) be a diffusion coefficient respectively, and the jump term is driven by a Poisson random measure p. We assume that its intensity measure q^θ has a finite total mass. The aim of this paper is estimating the parameter α = (θ,σ) from some discrete data. We can observe n + 1 data at t_iⁿ = ih_n, . We suppose h_n → 0, nh_n → ∞, nh_n² → 0. Final version 20 December 2004     

Almost poised basic hypergeometric series

Given a basic hypergeometric series with numerator parametersa ₁,a ₂, ...,a _r and denominator parametersb ₂, ...,b _r, we say it isalmost poised ifb _i, =a ₁ q ^δ,i a _i,δ_i = 0, 1 or 2, for 2 ≤i ≤r. Identities are given for almost poised series withr = 3 andr = 5 when a₁, =q ⁻²ⁿ. Partially supported by N.S.F. Grant No. DMS-8521580.     

Error bounds for asymptotic expansion of the scale mixtures of the normal distribution

Summary LetX be a standard normal random variable and let σ be a positive random variable independent ofX. The distribution of η=σX is expanded around that ofN(0, 1) and its error bounds are obtained. Bounds are given in terms of E(σ ²V−σ ²−1) ^k whereσ ²Vσ ⁻² denotes the maximum of the two quantitiesσ ² andσ ⁻², andk is a positive integer, and of E(σ ²−1) ^k , ifk is even. The Institute of Statistical Mathematics     

Inclusion sets for the singular values of a square matrix

The paper presents a general approach to deriving inclusion sets for the singular values of a matrix A = (a_ij) ∈ ℂ ^n×n. The key to the approach is the following result: If σ is a singular value of A, then a certain matrix C(σ, A) of order 2n, whose diagonal entries are σ² − | a_ii|², i = 1, …, n, is singular. Based on this result, we use known diagonal-dominance type nonsingularity conditions to obtain inclusion sets for the singular values of A. Scaled versions of the inclusion sets, allowing one, in particular, to obtain Ky Fan type results for the singular values, are derived by passing to the conjugated matrix D⁻¹C(σ, A)D, where D is a positive-definite diagonal matrix. Bibliography: 16 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 52–77.     

A characterization of limiting distributions of estimators in an autoregressive process

Summary Let the random variablesX ₁,X ₂, ...,X _n be generated by the first-order autoregressive modelX _i =θX _i−1 +e _i wheree _i ,i=1, 2, ...,n, are i.i.d. random variables with mean zero, variance σ², and with unspecified density functiong(·). In the present paper we obtain a characterization of limiting distributions of nonparametric and parametric estimators of θ as well as a local asymptotic minimax bound of the risks of estimators.     

Analytic Continuation of Dirichlet Series with Almost Periodic Coefficients

We consider Dirichlet series z_g,a(s)=?_n=1^￥ g(na) e^{-l_n s}{\zeta_{g,\alpha}(s)=\sum_{n=1}^\infty g(n\alpha) e^{-\lambda_n s}} for fixed irrational α and periodic functions g. We demonstrate that for Diophantine α and smooth g, the line Re(s) = 0 is a natural boundary in the Taylor series case λ _n  = n, so that the unit circle is the maximal domain of holomorphy for the almost periodic Taylor series ?_n=1^￥ g(na) zⁿ{\sum_{n=1}^{\infty} g(n\alpha) z^n}. We prove that a Dirichlet series z_g,a(s) = ?_n=1^￥ g(n a)/n^s{\zeta_{g,\alpha}(s) = \sum_{n=1}^{\infty} g(n \alpha)/n^s} has an abscissa of convergence σ ₀ = 0 if g is odd and real analytic and α is Diophantine. We show that if g is odd and has bounded variation and α is of bounded Diophantine type r, the abscissa of convergence σ ₀ satisfies σ ₀ ≤ 1 − 1/r. Using a polylogarithm expansion, we prove that if g is odd and real analytic and α is Diophantine, then the Dirichlet series ζ _g,α (s) has an analytic continuation to the entire complex plane.     

On the Lower Tail Probabilities of Some Random Sequences in <Emphasis Type="Italic">l</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

We investigate the behaviour of the logarithmic small deviation probability of a sequence (σ _n θ _n ) in l _p , 0<p≤∞, where (θ _n ) are i.i.d. random variables and (σ _n ) is a decreasing sequence of positive numbers. In particular, the example σ _n ∼n ^−μ (1+log n)^−ν is studied thoroughly. Contrary to the existing results in the literature, the rate function and the small deviation constant are expressed expli- citly in the present treatment. The restrictions on the distribution of θ ₁ are kept to an absolute minimum. In particular, the usual variance assumption is removed. As an example, the results are applied to stable and Gamma-distributed random variables.     

Local convexity of the growth function of finitely generated groups and question 5.2 in the Kourovka Notebook

Using the group <a,b|a³=b³=(ab)³=1>, we refute the conjecture dubbed in 1976 by V. Belyaev and N. Sesekin, which maintained that the growth function σ(n) of a finitely generated group satisfies the inequality σ(n)≤(σ(n−1)+σ(n+1))/2 for all sufficiently large n. Supported by the National Research Foundation of Switzerland, and by RFFR grant No. 96-01-00974. Translated fromAlgebra i Logika, Vol. 37, No. 6, pp. 621–626, November–December, 1998.     

On mappings of the Galois space

LetV be a metric vector space over a fieldK, dimV=n<∞, and let δ:V×V→K denote the corresponding distance function. Given a mappingσ:V→V such that δ(p,q) = 1⇒ δ(p ^σ ,q ^ς) = 1, ifn=2, indV=1 and charK≠2, 3, 5, thenσ is semilinear [5], [11]; ifn≧3,K=R and the distance function is either Euclidean or Minkowskian, thenσ is linear [3], [10]. Here the following is proved: IfK=GF(p ^m ),p>2 andn≧3, thenσ is semilinear (up to a translation), providedn≠0, −1, −2 (modp) or the discriminant ofV satisfies a certain condition. The proof is based on the condition for a regular simplex to exist in a Galois space, which may be of interest for its own sake.     

Beurling primes with large oscillation

A Beurling generalized number system is constructed having integer counting function N_B(x) = κx +O(x^θ) with κ>0 and 1/2 <θ <1, whose prime counting function satisfies the oscillation estimate π_B(x) =li(x) + Ω(xexp(-c)), and whose zeta function has infinitely many zeros on the curve σ=1−a/logt, t≥2, and no zero to the right of this curve, where a is chosen so that a>(4/e)(1−θ). The construction uses elements of classical analytic number theory and probability. The author was supported in part by NSF grants DMS-0070720 and DMS-0244660. The author was supported in part by NSF grant DMS-0244660.     

On counting twists of a character appearing in its associated Weil representation

Consider an irreducible, admissible representation π of GL(2,F) whose restriction to GL(2,F)⁺ breaks up as a sum of two irreducible representations π ₊ + π −. If π = r _θ , the Weil representation of GL(2,F) attached to a character θ of K ^* does not factor through the norm map from K to F, then c ? [^(K^*)]\chi\in \widehat{K^*} with (c. q^-1)| _{F ^*} =w _K/F(\chi . \theta ^{-1})\vert _{ F^{ * }}=\omega _{ {K/F}} occurs in r _{θ  +} if and only if e(qc^-1,y₀)=e([`(q)]c^-1,y₀)=1\epsilon(\theta\chi^{-1},\psi_0)=\epsilon(\overline \theta\chi^{-1},\psi_0)=1 and in r _θ− if and only if both the epsilon factors are − 1. But given a conductor n, can we say precisely how many such χ will appear in π? We calculate the number of such characters at each given conductor n in this work.     

The distribution of the root degree of a random permutation

Given a permutation ω of {1, …, n}, let R(ω) be the root degree of ω, i.e. the smallest (prime) integer r such that there is a permutation σ with ω = σ ^r . We show that, for ω chosen uniformly at random, R(ω) = (lnlnn − 3lnlnln n + O ^p (1))⁻¹ lnn, and find the limiting distribution of the remainder term. Research supported in part by NSF grants CCR-0225610, DMS-0505550 and ARO grant W911NF-06-1-0076. Research supported by NSF grant DMS-0406024.     

A rate of convergence for the set compound estimation in a family of certain retracted distributions

Summary This paper is concerned with the set compound squared-error loss estimation problem. Here, the author obtains Lévy consistent estimate of the empiric distributionG _n of the parameters θ₁,...,θ_n for a more general family of retracted distributions on the interval [θ, θ+1) than the uniform on [θ, θ+1) as in R. Fox (1970,Ann. Math. Statist.,41, 1845–1852; 1978,Ann. Statist.,6, 846–853) and exhibits a decision procedure based on with a convergence rateO((n ⁻¹ logn)^1/4) for the mofified regret uniformly in (θ₁, θ₂, ..., θ_n ∈ Ωⁿ with bounded Ω. The author also gives a counterexample to the convergence of the modified regret for Ω=(−∞, ∞). This is part of the author's Ph. D. Thesis at Michigan State University.     

The smallest degree sum that yields potentially Kr,r-graphic sequences

We consider a variation of a classical Turán-type extremal problem as follows: Determine the smallest even integer σ(K_r,r,n) such that every n-term graphic sequence π = (d₁,d₂,...,d_n) with term sum σ(π) = d₁ + d₂ + ... + d_n ≥ σ(K_r,r,n) is potentially K_r,r-graphic, where K_r,r is an r × r complete bipartite graph, i.e. π has a realization G containing K_r,r as its subgraph. In this paper, the values σ(K_r,r,n) for even r and n ≥ 4r² - r - 6 and for odd r and n ≥ 4r² + 3r - 8 are determined.     

Pseudo 1-homogeneous distance-regular graphs

Let Γ be a distance-regular graph of diameter d≥2 and a ₁≠0. Let θ be a real number. A pseudo cosine sequence for θ is a sequence of real numbers σ ₀,…,σ _d such that σ ₀=1 and c _i σ _i−1+a _i σ _i +b _i σ _i+1=θ σ _i for all i∈{0,…,d−1}. Furthermore, a pseudo primitive idempotent for θ is E _θ =s ∑ _i=0 ^d σ _i A _i , where s is any nonzero scalar. Let be the characteristic vector of a vertex v∈VΓ. For an edge xy of Γ and the characteristic vector w of the set of common neighbours of x and y, we say that the edge xy is tight with respect to θ whenever θ≠k and a nontrivial linear combination of vectors , and Ew is contained in . When an edge of Γ is tight with respect to two distinct real numbers, a parameterization with d+1 parameters of the members of the intersection array of Γ is given (using the pseudo cosines σ ₁,…,σ _d , and an auxiliary parameter ε). Let S be the set of all the vertices of Γ that are not at distance d from both vertices x and y that are adjacent. The graph Γ is pseudo 1-homogeneous with respect to xy whenever the distance partition of S corresponding to the distances from x and y is equitable in the subgraph induced on S. We show Γ is pseudo 1-homogeneous with respect to the edge xy if and only if the edge xy is tight with respect to two distinct real numbers. Finally, let us fix a vertex x of Γ. Then the graph Γ is pseudo 1-homogeneous with respect to any edge xy, and the local graph of x is connected if and only if there is the above parameterization with d+1 parameters σ ₁,…,σ _d ,ε and the local graph of x is strongly regular with nontrivial eigenvalues a ₁ σ/(1+σ) and (σ ₂−1)/(σ−σ ₂).     

Reciprocity transformations for radial nonlinear heat equations

Nonlocal transformations of some quasilinear parabolic equations which describe spherically symmetric heat conduction and diffusion processes are considered. One of them transforms the equationr ⁿ⁻¹θ _t =(r ⁿ⁻¹|θ _r | ^l θ _r ) _r to an equation of the same type but with a different value of the exponent n. Another transformation reduces the equationr ⁿ⁻¹θ _t =(r ⁿ⁻¹θ⁻²θ _r ) _r to an equation with coefficients which do not depend on the space variable. The third nonlocal transformation preserves the equationrθ _t =(rθ ⁻¹θ _r ) _r . Some exact solutions of the mentioned equations are analyzed. Bibliography: 15 titles. Dedicated to V. A. Solonnikov on his sixtieth anniversary Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 213, 1994, pp. 151–163. Translated by S. Yu. Pilyugin     

On the growth of an entire dirichlet series

We establish the relation between the increase of the quantityM(σ,F) = ∣a ₀∣ + ∑ _n=1 ^∞ ∣a _n ∣ exp (σλ _n ) and the behavior of sequences (|a _n |) and (λ _n ), where (λ _n ) is a sequence of nonnegative numbers increasing to + ∞, andF(s) =a ₀ + ∑ _n=1 ^∞ a _n e ^sλn ,s=σ+it, is the Dirichlet entire series. Lviv University, Lviv. Translated from Ukrainskii Matematicheskii Zhurmal, Vol. 51, No. 8, pp. 1149–1153, August, 1999.     

Nonuniqueness ing-functions

We give an example of two distinct stationary processes {X _n} and {X′ _n} on {0, 1} for whichP[X₀=1|X₋₁=a₋₁,X₋₂=a₋₂, …]=P[X′₀=1|X′₋₁=a₋₁,X′₋₂=a₋₂, …] for all {a _i},i=−1, −2, …, even though these probabilities are bounded away from 0 and 1, and are continuous in {a _i}. Supported in part by NSF Grant DMS 89-01545. Supported in part by the US Army Research Office.     

Large deviations for the Ginzburg–Landau ∇φ interface model

Hydrodynamic large scale limit for the Ginzburg-Landau ∇φ interface model was established in [6]. As its next stage this paper studies the corresponding large deviation problem. The dynamic rate functional is given by