Asymptotic formulas for the hyperġeometric function2 F 1 of matrix argument,useful in multivariate analysis

Summary LetS _i have the Wishart distributionW _p(∑_i,n_i) fori=1,2. An asymptotic expansion of the distribution of for large n=n₁+n₂ is derived, when∑ ₁∑ ₂ ⁻¹ =I+n^−1/2θ, based on an asymptotic solution of the system of partial differential equations for the hypergeometric function₂ F ₁, obtained recently by Muirhead [2]. Another asymptotic formula is also applied to the distributions of −2 log λ and −log|S ₂(S ₁+S ₂)⁻¹| under fixed∑ ₁∑ ₂ ⁻¹ , which gives the earlier results by Nagao [4]. Some useful asymptotic formulas for₁ F ₁ were investigated by Sugiura [7].     

Decomposition of the conjugacy representation of the symmetric groups

Consider the two natural representations of the symmetric groupS _n on the group algebra ℂ[S _n ]: the regular representation and the conjugacy representation (acting on the basis by conjugation). Letm(λ) be the multiplicity of the irreducible representationS ^λ in the conjugacy representation and letf ^λ be the multiplicity ofS ^λ in the regular representation. By the character estimates of [R1] and [Wa] we prove

Overcompleteness of Sequences of Reproducing Kernels in Model Spaces

We give necessary conditions and sufficient conditions for sequences of reproducing kernels (k_Θ(·, λ_n))_{n ≥ 1} to be overcomplete in a given model space K_Θ^p where Θ is an inner function in H^∞, p ∈ (1, ∞), and where (λ_n)_{n ≥ 1} is an infinite sequence of pairwise distinct points of Under certain conditions on Θ we obtain an exact characterization of overcompleteness. As a consequence we are able to describe the overcomplete exponential systems in L² (0, a).     

Pseudo unitary conformal groups and Clifford algebras for standard pseudo hermitian spaces

This paper, self-contained, deals with pseudo-unitary spin geometry. First, we present pseudo-unitary conformal structures over a 2n-dimensional complex manifold V and the corresponding projective quadrics for standard pseudo-hermitian spaces H_p,q. Then we develop a geometrical presentation of a compactification for pseudo-hermitian standard spaces in order to construct the pseudo-unitary conformal group of H_p,q. We study the topology of the projective quadrics and the “generators” of such projective quadrics. Then we define the space S of spinors canonically associated with the pseudo-hermitian scalar product of signature (2ⁿ⁻¹, 2ⁿ⁻¹). The spinorial group Spin U(p,q) is imbedded into SU(2ⁿ⁻¹, 2ⁿ⁻¹). At last, we study the natural imbeddings of the projective quadrics      

The Existence of BIB Designs

Abstract Given any positive integers k≥ 3 and λ, let c(k, λ) denote the smallest integer such that v∈B(k, λ) for every integer v≥c(k, λ) that satisfies the congruences λv(v− 1) ≡ 0(mod k(k− 1)) and λ(v− 1) ≡ 0(mod k− 1). In this article we make an improvement on the bound of c(k, λ) provided by Chang in [4] and prove that . In particular, . Supported by NSFC Grant No. 19701002 and Huo Yingdong Foundation     

Nonexistence of Ground States of -△u = u^p - u^q

The authors use the method of moving spheres to prove the nonexistence of ground states of -△u = u^p - u^q for n≥3,-∞〈p〈（n＋2）/（n-2） and q〉max （1,p）,
In fact this conclusion is a special case of -△u =f（u） for n≥2.     

Asymptotic behavior on the Hénon equation with supercritical exponent

The main purpose of this paper is to analyze the asymptotic behavior of the radial solution of Hénon equation −Δu = |x| ^α u ^p−1, u > 0, x ∈ B _R (0) ⊂ ℝ ⁿ (n ⩾ 3), u = 0, x ∈ ∂B _R (0), where $ p \to p(\alpha ) = \frac{{2(n + \alpha )}} {{n - 2}} $ p \to p(\alpha ) = \frac{{2(n + \alpha )}} {{n - 2}} from left side, α > 0.     

The Feichtinger Conjecture for Reproducing Kernels in Model Subspaces

We obtain two results concerning the Feichtinger conjecture for systems of normalized reproducing kernels in the model subspace K _Θ=H ²⊖ΘH ² of the Hardy space H ², where Θ is an inner function. First, we verify the Feichtinger conjecture for the kernels [(k)\tilde]_ln=k_ln/||k_ln||\tilde{k}_{\lambda_{n}}=k_{\lambda_{n}}/\|k_{\lambda _{n}}\| under the assumption that sup  _n |Θ(λ _n )|<1. Second, we prove the Feichtinger conjecture in the case where Θ is a one-component inner function, meaning that the set {z:|Θ(z)|<ε} is connected for some ε∈(0,1).     

Compact embedded rotation hypersurfaces of <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis>+1</Superscript>

In this paper, we prove that and round geodesic spheres are the only n-dimensional compact embedded rotation hypersurfaces with H_m = 0 (1 ≤ m ≤ n − 1) in a unit sphere Sⁿ⁺¹(1). When m = 1, our result reduces to the result of T. Otsuki [O1], [O2], Brito and Leite [BL]. The project is supported by the grant No. 10531090 of NSFC.     

Generalized self-duality of 2-forms

Self-dual 2–forms on play a fundamental role in gauge theory. For generalized Seiberg-Witten theory (and for some other purposes in mathematical physics) a notion of self-duality of 2–forms on is needed. There are several definitions, but the one given by [Bilge, Dereli, Ko?ak ; JMP 38(9), 1997] is intimately related with Clifford algebras. They defined a 2–form to be self-dual if the anti-symmetric matrix Ω = (ω_ij) satisfies Ω² = λ I for a scalar λ and proved that the space of such forms is non-linear with dimension n² − n + 1, but contains maximal linear subspaces with dimension the Radon-Hurwitz number of (2n). It is important to have an algorithm for construction of such maximal linear subspaces and we give an explicit one with the help of representations of Clifford algebras on whereby we show that the representations given by the standart recursion formulas are anti-symmetric.     

Precise asymptotics in the law of the iterated logarithm*

Let X₁, X₂, ... be i.i.d. random variables with EX₁ = 0 and positive, finite variance σ², and set S_n = X₁ + ... + X_n. For any α > −1, β > −1/2 and for κ_n(ε) a function of ε and n such that κ_n(ε) log log n → λ as n ↑ ∞ and , we prove that

Invariant Subspaces for Banach Space Operators with a Multiply Connected Spectrum

We consider a multiply connected domain where denotes the unit disk and denotes the closed disk centered at with radius r _j for j = 1, . . . , n. We show that if T is a bounded linear operator on a Banach space X whose spectrum contains ∂Ω and does not contain the points λ₁, λ₂, . . . , λ _n , and the operators T and r _j (T − λ _j I)⁻¹ are polynomially bounded, then there exists a nontrivial common invariant subspace for T ^* and (T − λ _j I)^*-1.     

The independence of initial vectors in the subdivision schemes

Starting with an initial vector λ = (λ(κ))κ∈z ∈ ep(Z), the subdivision scheme generates asequence (Snaλ)∞n=1 of vectors by the subdivision operator Saλ(κ) = ∑λ(j)a(k - 2j), k ∈ Z. j∈zSubdivision schemes play an important role in computer graphics and wavelet analysis. It is very interesting tounderstand under what conditions the sequence (Snaλ)∞n=1 converges to an Lp-function in an appropriate sense.This problem has been studied extensively. In this paper we show that the subdivision scheme converges forany initial vector in ep(Z) provided that it does for one nonzero vector in that space. Moreover, if the integertranslates of the refinable function are stable, the smoothness of the limit function corresponding to the vectorλ is also independent of λ.     

On estimation of a density and its derivatives

Summary Letf _n ^(p) be a recursive kernel estimate off ^(p) thepth order derivative of the probability density functionf, based on a random sample of sizen. In this paper, we provide bounds for the moments of and show that the rate of almost sure convergence of to zero isO(n ^−α), α<(r−p)/(2r+1), iff ^(r),r>p≧0, is a continuousL ₂(−∞, ∞) function. Similar rate-factor is also obtained for the almost sure convergence of to zero under different conditions onf. This work was supported in part by the Research Foundation of SUNY.     

On the synthesis of oriented contact circuits with certain restrictions on adjacent contacts

Realization of Boolean functions in the class of oriented contact circuits (OCCs) with certain restrictions on the weight, number, and types of adjacent contacts is studied. Oriented contact circuits are considered in which, from an arbitrary vertex, at most λ arcs issue and at most ν different Boolean variables are used in the marks of the issuing arcs. The weight of a vertex of an OCC is defined as being equal to λ if one arc enters a vertex and equal to λ(1 + ω), where ω > 0, otherwise. Then, as usual, the weight of an OCC is defined as the sum of the weights of its vertices; the weight of a Boolean function, as the minimum weight of OCCs realizing it; and Shannon function W _{λ, ν, ω}(n), as the maximum weight of the Boolean function of n variables. For this Shannon function, the so-called high-accuracy bound

Nearly flat almost monotone measures are big pieces of lipschitz graphs

A concentrated (ξ, m) almost monotone measure inR ⁿ is a Radon measure Φ satisfying the two following conditions: (1) Θ ^m (Φ,x)≥1 for every x ∈spt (Φ) and (2) for everyx ∈R ⁿ the ratioexp [ξ(r)]r^−mΦ(B(x,r)) is increasing as a function of r>0. Here ξ is an increasing function such thatlim _r→0-ξ(r)=0. We prove that there is a relatively open dense setReg (Φ) ∋spt (Φ) such that at each x∈Reg(Φ) the support of Φ has the following regularity property: given ε>0 and λ>0 there is an m dimensional spaceW ⊂R ⁿ and a λ-Lipschitz function f from x+W into x+W^‖ so that (100-ε)% ofspt(Φ) ∩B (x, r) coincides with the graph of f, at some scale r>0 depending on x, ε, and λ.     

Limitations to Fréchet’s metric embedding method

Fréchet’s classical isometric embedding argument has evolved to become a major tool in the study of metric spaces. An important example of a Fréchet embedding is Bourgain's embedding [4]. The authors have recently shown [2] that for every ε>0, anyn-point metric space contains a subset of size at leastn ^1−ε which embeds into ℓ₂ with distortion . The embedding used in [2] is non-Fréchet, and the purpose of this note is to show that this is not coincidental. Specifically, for every ε>0, we construct arbitrarily largen-point metric spaces, such that the distortion of any Fréchet embedding into ℓ_p on subsets of size at leastn ^1/2+ε is Ω((logn)^1/p ). Supported in part by a grant from the Israeli National Science Foundation. Supported in part by a grant from the Israeli National Science Foundation. Supported in part by the Landau Center.     

On approximation of periodic functions by Fourier sums

Let L _p , 1 ≤ p< ∞, be the space of 2π-periodic functions f with the norm || f ||_p = ( ò _{- p}^p | f |^p )^{1 \mathord}

Lower bound in the Bernstein inequality for the first derivative of algebraic polynomials

We prove that max |p′(x)|, where p runs over the set of all algebraic polynomials of degree not higher than n ≥ 3 bounded in modulus by 1 on [−1, 1], is not lower than ( n - 1 ) \mathord

Growth and Hölder conditions for the sample paths of Feller processes

Let (A,D(A)) be the infinitesimal generator of a Feller semigroup such that C _c ^∞(ℝ ⁿ )⊂D(A) and A|C _c ^∞(ℝ ⁿ ) is a pseudo-differential operator with symbol −p(x,ξ) satisfying |p(•,ξ)|_∞≤c(1+|ξ|²) and |Imp(x,ξ)|≤c ₀Rep(x,ξ). We show that the associated Feller process {X _t } _{t ≥0} on ℝ ⁿ is a semimartingale, even a homogeneous diffusion with jumps (in the sense of [21]), and characterize the limiting behaviour of its trajectories as t→0 and ∞. To this end, we introduce various indices, e.g., β_∞ ^x :={λ>0:lim _{|ξ|→∞ | x − y |≤2/|ξ|}|p(y,ξ)|/|ξ|^λ=0} or δ_∞ ^x :={λ>0:liminf _{|ξ|→∞ | x − y |≤2/|ξ| |ε|≤1}|p(y,|ξ|ε)|/|ξ|^λ=0}, and obtain a.s. (ℙ ^x ) that lim _{t →0} t ^−1/λ _{s ≤ t} |X _s −x|=0 or ∞ according to λ>β_∞ ^x or λ<δ_∞ ^x . Similar statements hold for the limit inferior and superior, and also for t→∞. Our results extend the constant-coefficient (i.e., Lévy) case considered by W. Pruitt [27]. Received: 21 July 1997 / Revised version: 26 January 1998