Estimates for the closeness of successive convolutions of multidimensional symmetric distributions

Summary Let ξ₁, ξ₂,... be i.i.d random vectors in ℝ ^k with a common distribution ℒ(ξ_i),... = F, i = 1, 2,.... Let S _n = ξ₁+...+ξ _n . We investigate how small is the difference between ℒ(S _n ) and ℒ(S _{n+ m} ) in the case when ξ _i have symmetric distributions.     

A characteristic property of spheres

Summary We prove: Let S be a closed n-dimensional surface in an(n+1)-space of constant curvature (n ≥ 2); k₁ ≥ ... ≥ k_n denote its principle curvatures. Let φ(ξ₁, ..., ξ_n) be such that . Then if φ(k₁, ..., k_n)=const on S and S is subject to some additional general conditions (those(II ₀) or(II) n^o 1), S is a sphere. To Enrico Bompiani on his scientific Jubilee     

On subgroups of the general linear group that contain a maximal nonsplit torus

The paper deals with the structure of intermediate subgroups of the general linear group GL(n, k) of degree n over a field k of odd characteristic that contain a nonsplit maximal torus related to a radical extension of degree n of the ground field k. The structure of ideal nets over a ring that determine the structure of intermediate subgroups containinga transvection is given. Let K = k( ⁿ?{d} ) K = k\left( {\sqrt[n]{d}} \right) be a radical degree-n extension of a field k of odd characteristic, and let T =(d) be a nonsplit maximal torus, which is the image of the multiplicative group of the field K under the regular embedding in G =GL(n, k). In the paper, the structure of intermediate subgroups H, T ≤ H ≤ G, that contain a transvection is studied. The elements of the matrices in the torus T = T (d) generate a subring R(d) in the field k.Let R be an intermediate subring, R(d) ⊆ R ⊆ k, d ∈ R. Let σ_R denote the net in which the ideal dR stands on the principal diagonal and above it and all entries of which beneath the principal diagonal are equal to R. Let σ^R denote the net in which all positions on the principal diagonal and beneath it are occupied by R and all entries above the principal diagonal are equal to dR. Let E(σ_R) be the subgroup generated by all transvections from the net group G(σ_R). In the paper it is proved that the product TE(σ_R) is a group (and thus an intermediate subgroup). If the net σ associated with an intermediate subgroup H coincides with σ_R,then TE(σ_R) ≤ H ≤ N(σ_R),where N(σ_R) is the normalizer of the elementary net group E(σ_R) in G. For the normalizer N(σ_R),the formula N(σ_R)= TG(σ^R) holds. In particular, this result enables one to describe the maximal intermediate subgroups. Bibliography: 13 titles.     

On the first pontrjagin class of homotopy complex projective spaces

Let M ²ⁿ be a closed smooth manifold homotopy equivalent to the complex projective space ℂP(n). It is known that the first Pontrjagin class p ₁(M) of M ²ⁿ has the form (n+1+24α(M))u ² for some integer α(M) where u is a generator of H ²(M; ℤ). We prove that α(M) is even when n is even but not divisible by 64.     

Multipliers for logarithmic Cauchy integrals in the ball

Let B _n denote the unit ball in \mathbbC \mathbb{C} ⁿ , n ≥ 1. Let K \mathcal{K} ₀(n) denote the class of functions defined for z ∈ B _n as a constant plus the integral of the kernel log(1/(1 −〈z, ζ〉)) against a complex Borel measure on the sphere {ζ ∈ \mathbbC \mathbb{C} ⁿ ,: |ζ| = 1}. Properties of holomorphic functions g such that fg ∈ K \mathcal{K} ₀(n) for all f ∈ K \mathcal{K} ₀(n) are studied. The extended Cesàro operators are investigated on the spaces K \mathcal{K} ₀(n), n ≥ 1. Bibliography: 15 titles.     

On samples of independent random vectors in spaces of infinitely increasing dimension

We study the asymptotic behavior of a set of random vectors ξ_i, i = 1,..., m, whose coordinates are independent and identically distributed in a space of infinitely increasing dimension. We investigate the asymptotics of the distribution of the random vectors, the consistency of the sets M _m⁽ⁿ⁾ = ξ₁,..., ξ_m and X _n^λ = x ∈ X _n: ρ(x) ≤ λn, and the mutual location of pairs of vectors. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 12, pp. 1706–1711, December, 1998.     

Hardy spaces via distribution spaces

Let ℐ(ℝⁿ) be the Schwartz class on ℝⁿ and ℐ_∞(ℝⁿ) be the collection of functions ϕ ∊ ℐ(ℝⁿ) with additional property that

Universal PI-algebras and algebras of generic matrices

Let Ω[ξ] denote the polynomial algebra (with 1) in commutative indeterminates {ie65-1}, 1 ≦i, j ≦n, 1 ≦k < ∞, over a commutative ring Ω. Thealgebra of generic matrices Ω [Y] is defined to be the Ω-subalgebra ofM _n (Ω[ξ]) generated by the matricesY _k=({ie65-2}), 1 ≦i, j ≦n, 1 ≦k < ∞. This algebra has been studied extensively by Amitsur and by Procesi in particular Amitsur has used it to construct a finite dimensional, central division algebra Ω (Y) which is not a crossed product. In this paper we shall prove, for Ω a domain, that Ω(Y) has exponentn in the Brauer group (Amitsur may already know this fact); consequently, for Ω an infinite field andn a multiple of 4, iff(X ₁, …,X _m) is a polynomial linear in all theX _i but one (similar to Formanek’s central polynomials for matrix rings) andf ² is central forM _n (Ω), thenf is central forM _n (Ω). (The existence of a polynomial not central forM _n (Ω), but whose square is central forM _n(Ω) is equivalent to every central division algebra of degreen containing a quadratic extension of its center; well-known theory immediately shows this is the case of 4‖n and 8χn.) Also, information is obtained about Ω(Y) for arbitary Ω, most notably that the Jacobson radical is the set of nilpotent elements. Partial support for this work was provided by National Science Foundation grant NSF-GP 33591.     

On conditions for Dirichlet series absolutely convergent in a half-plane to belong to the class of convergence

Let M(σ) = sup{|F(σ + it)|: t ∈ ℝ} and μ(σ) = max {|a _n |exp(σλ_n): n ≥ 0}, σ < 0, for a Dirichlet series {fx995-01} with abscissa of absolute convergence σ_a = 0. We prove that the condition ln ln n = o(ln λ_n), n → ∞, is necessary and sufficient for the equivalence of the relations {fx995-02}, for each series of this type. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 6, pp. 851–856, June, 2008.     

Eigenvalue distributions of random unitary matrices

 Let U be an n × n random matrix chosen from Haar measure on the unitary group. For a fixed arc of the unit circle, let X be the number of eigenvalues of M which lie in the specified arc. We study this random variable as the dimension n grows, using the connection between Toeplitz matrices and random unitary matrices, and show that (X -E [X])/(\Var (X))^1/2 is asymptotically normally distributed. In addition, we show that for several fixed arcs I ₁ , ..., I _m , the corresponding random variables are jointly normal in the large n limit. Received: 15 November 2000 / Revised version: 27 September 2001 / Published online: 17 May 2002     

Uncountable admissibles II: Compactness

AssumeV=L. Let κ be a cardinal and forX⊆κ, n<ω let α _n (X) denote the least ordinal α such thatL _α[X] is Σ _n admissible. In our earlier paperUncountable admissibles I: forcing, we characterized those ordinals of the form σ _n (X) when κ is regular. This paper treats the singular case using Barwise compactness, an effective version of Jensen's covering lemma and β-recursion theory.     

Ordered overlaps in disordered mean-field models

Let M(N) be a sequence of integers with M→∞ as N→∞ and M=o(N). For bounded i.i.d. r.v. ξ _i ^k and bounded i.i.d. r.v. σ _i , we study the large deviation of the family of (ordered) scalar products X ^k =N ⁻¹∑ _{i =1} ^N σ _i ξ _i ^k ,k≤M, under the distribution conditioned on the ξ _i ^k 's. To get a full large deviation principle, it is necessary to specify also the total norm(∑ _{k ≤ M} (X ^k )²)^1/2, which turns to be associated with some extra Gaussian distribution. Our results apply to disordered, mean-field systems, including generalized Hopfield models in the regime of a sublinear number of patterns. We build also a class of examples where this norm is the crucial order parameter. Received: 6 April 1999 / Revised version: 29 May 2000 /?Published online: 24 July 2001     

Minimum ideal triangulations of hyperbolic 3-manifolds

Let σ(n) be the minimum number of ideal hyperbolic tetrahedra necessary to construct a finite volumen-cusped hyperbolic 3-manifold, orientable or not. Let σ_or(n) be the corresponding number when we restrict ourselves to orientable manifolds. The correct values of σ(n) and σ_or(n) and the corresponding manifolds are given forn=1,2,3,4 and 5. We then show that 2n−1≤σ(n)≤σ_or(n)≤4n−4 forn≥5 and that σ_or(n)≥2n for alln. Both authors were supported by NSF Grants DMS-8711495, DMS-8802266 and Williams College Research Funds.     

On quasi-similarity of subnormal operators

For a compact subset K in the complex plane, let Rat(K) denote the set of the rational functions with poles off K. Given a finite positive measure with support contained in K, let R2(K,v) denote the closure of Rat(K) in L2(v) and let Sv denote the operator of multiplication by the independent variable z on R2(K, v), that is, Svf = zf for every f∈R2(K, v). SupposeΩis a bounded open subset in the complex plane whose complement has finitely many components and suppose Rat(Ω) is dense in the Hardy space H2(Ω). Letσdenote a harmonic measure forΩ. In this work, we characterize all subnormal operators quasi-similar to Sσ, the operators of the multiplication by z on R2(Ω,σ). We show that for a given v supported onΩ, Sv is quasi-similar to Sσif and only if v/■Ω■σ and log(dv/dσ)∈L1(σ). Our result extends a well-known result of Clary on the unit disk.     

On Operators Satisfying T＊｜T2｜T≥ T＊｜T＊｜2T

Let T be a bounded linear operator on a complex Hilbert space H. In this paper we introduce a new class denoted by l-＊-A, of operators satisfying T＊｜T2｜T≥ T＊｜T＊｜2T, and we prove the basic properties of these operators. Using these results, we also prove that if T or T＊ ∈l-＊-A, then w（f（T）） = f（w（T））, σea（f（T）） = f（σea（T）） for every f C H（σ（T））, where g（σ（T）） denotes the set of all analytic functions on an open neighborhood of σ（T）.     

On Optimal Orientations of Cartesian Products of Trees

. Let d(D) (resp., d(G)) denote the diameter and r(D) (resp., r(G)) the radius of a digraph D (resp., graph G). Let G×H denote the cartesian product of two graphs G and H. An orientation D of G is said to be (r, d)-invariant if r(D)=r(G) and d(D)=d(G). Let {T _i }, i=1,…,n, where n≥2, be a family of trees. In this paper, we show that the graph ∏ _{i =1} ⁿ T _i admits an (r, d)-invariant orientation provided that d(T ₁)≥d(T ₂)≥4 for n=2, and d(T ₁)≥5 and d(T ₂)≥4 for n≥3. Received: July 30, 1997 Final version received: April 20, 1998     

Summability of Fourier-Laplace Series with the Method of Lacunary Arithmetical Means at Lebesgue Points

Let ∑ _{n −1} be the unit sphere in the n-dimensional Euclidean space ℝ ⁿ . For a funcion ƒ∈L(∑ _{n −1}) denote by σ^δ _N (ƒ) the Cesàro means of order δ of the Fourier-Laplace series of ƒ. The special value of δ is known as the critical index. In the case when n is even, this paper proves the existence of the ‘rare’ sequence {n _k } such that the summability takes place at each Lebesgue point satisfying some antipole conditions. Received June 28, 1999, Revised August 11, 1999, Accepted February 16, 2000     

Permutation Matrices, Wreath Products, and the Distribution of Eigenvalues

We consider a class of random matrix ensembles which can be constructed from the random permutation matrices by replacing the nonzero entries of the n×n permutation matrix matrix with M×M diagonal matrices whose entries are random Kth roots of unity or random points on the unit circle. Let X be the number of eigenvalues lying in a specified arc I of the unit circle, and consider the standardized random variable (X–E[X])/(Var(X))^1/2. We show that for a fixed set of arcs I ₁,...,I _N , the corresponding standardized random variables are jointly normal in the large n limit, and compare the covariance structures which arise with results for other random matrix ensembles.     

On the ratio of the expected maximum of a martingale and theL p-norm of its last term

For eachp>1, the supremum,S, of the absolute value of a martingale terminating at a random variableX inL _p, satisfiesES≦(Γ(q))^1/q‖X‖_p (q=p(p-1)^-1).The maximum,M, of a mean-zero martingale which starts at zero and terminates atX, satisfiesES≦(Γ(q))^1/q‖X‖_p (q=p(p-1)^-1), whereσ _q is the unique solution of the equationt = ‖Z −t ‖ _q for an exponentially distributed random variableZ with mean 1.σ _p has other characterizations and satisfies lim _p‖ q ^{− 1} σ _q =c withc determined byce ^c+1 = 1. Equalities in (1) and (2) are attainable by appropriate martingales which can be realized as stopped segments of Brownian motion. A presumably new property of the exponential distribution is obtained en route to inequality (2).