On the zeros of a class of generalised Dirichlet series-XIV

We prove a general theorem on the zeros of a class of generalised Dirichlet series. We quote the following results as samples. Theorem A.Let 0<θ<1/2and let {a _n }be a sequence of complex numbers satisfying the inequality for N = 1,2,3,…,also for n = 1,2,3,…let α _n be real and |α_n| ≤ C(θ)where C(θ) > 0is a certain (small)constant depending only on θ. Then the number of zeros of the function in the rectangle (1/2-δ⩽σ⩽1/2+δ,T⩽t⩽2T) (where 0<δ<1/2)is ≥C(θ,δ)T logT where C(θ,δ)is a positive constant independent of T provided T ≥T ₀(θ,δ)a large positive constant. Theorem B.In the above theorem we can relax the condition on a _n to and |a_N| ≤ (1/2-θ)^-1.Then the lower bound for the number of zeros in (σ⩾1/3−δ,T⩽t⩽2T)is > C(θ,δ) Tlog T(log logT)^-1.The upper bound for the number of zeros in σ⩾1/3+δ,T⩽t⩽2T) isO(T)provided for every ε > 0. Dedicated to the memory of Professor K G Ramanathan     

Extension of Floquet''''s theory to nonlinear quasiperiodic differential equations

In this paper, we consider the following autonomous system of differential equations: x = Ax f(x,θ), θ = ω, where θ∈Rm, ω = (ω1,…,ωm) ∈ Rm, x ∈ Rn, A ∈ Rn×n is a constant matrix and is hyperbolic, f is a C∞ function in both variables and 2π-periodic in each component of the vector e which satisfies f = O(||x||2) as x → 0. We study the normal form of this system and prove that under some proper conditions this system can be transformed to an autonomous system: x = Ax g(x), θ = ω. Additionally, the proof of this paper naturally implies the extension of Chen's theory in the quasi-periodic case.     

On Maps Preserving Zero Jordan Products

Let R be a ring, A = M _n (R) and θ: A → A a surjective additive map preserving zero Jordan products, i.e. if x,y ∈ A are such that xy + yx = 0, then θ(x)θ(y) + θ(y)θ(x) = 0. In this paper, we show that if R contains and n ≥ 4, then θ = λϕ, where λ = θ(1) is a central element of A and ϕ: A → A is a Jordan homomorphism. The third author is Corresponding author.     

Construction of <Emphasis Type="Italic">σ</Emphasis>-orthogonal polynomials and gaussian quadrature formulas

Let dα be a measure on R and let σ = (m ₁, m ₂,...,m _n ), where m _k ≥ 1, k = 1,2,...,n, are arbitrary real numbers. A polynomial ω _n (x) := (x − x ₁)(x − x ₂)...(x − x _n ) with x ₁ ≤ x ₂ ≤ ... ≤ x _n is said to be the nth σ-orthogonal polynomial with respect to dα if the vector of zeros (x ₁, x ₂, ..., x _n)^T is a solution of the extremal problem

Diffusion Equations and Geometric Inequalities

Let =(₀, ₁) be a fixed vector in R ² with strictly positive components and suppose ₀, ₁ > 0. Set = _{0 0} + _{1 1} and, if x ₀, x ₁ R ⁿ , set x = ₀ x ₀ + ₁ x ₁. Moreover, for any j {0, 1, }, let c _j : R ⁿ R be a continuous, bounded function and denote by p _j , c _j (t, x, y) the fundamental solution of the diffusion equation

Small zeros of additive forms in several variables

Letf(X) be an additive form defined by

ℓ<Superscript>1</Superscript>-spreading models in mixed Tsirelson spacemodels in mixed Tsirelson space

Suppose that (F _n ) _n=1 ^∞ is a sequence of regular families of finite subsets of ℝ and (θ _n ) _n=1 ^∞ is a nonincreasing null sequence in (0,1). The mixed Tsirelson spaceT[(θ _n ,F _n ) _n=1 ^∞ ] is the completion ofc ₀₀ with respect to the implicitly defined norm , where the last supremum is taken over all sequences (E _i ) _i=1 ^k in [ℕ]^<∞ such that maxE _i<minE _i +1 and . Necessary and sufficient conditions are obtained for the existence of higher order ℓ¹-spreading models in every subspace generated by a subsequence of the unit vector basis ofT[(θ _n ,F _n ) _n=1 ^∞ ].     

Implicatively selector sets

Let A⊆N={0,1,2,...} and β be an n-ary Boolean function. We call A a β-implicatively selector (β-IS) set if there exists an n-ary selector general recursive function f such that (∀x₁,...,x_n)(β(χ(x₁),...,χ(x_n))=1⟹f(x₁,...,x_n)∈A), where χ is the characteristic function of A. Let F^(m), m≥1, be the family of all d _m+1 ^* -IS sets, where , F⁽⁰⁾=N, and F^(∞) is the class of all subsets in N. The basic result of the article says that the family of all β-IS sets coincides with one of F^(m), m≥0, or F^(∞), and, moreover, the inclusions F⁽⁰⁾⊂F⁽¹⁾⊂...⊂F^(∞) hold. Translated fromAlgebra i Logika, Vol. 35, No. 2, pp. 145–153, March–April, 1996.     

The roots of <Emphasis Type="Italic">σ</Emphasis>-polynomials

Let G be a connected graph. We denote by σ(G,x) and δ(G) respectively the σ-polynomial and the edge-density of G, where . If σ(G,x) has at least an unreal root, then G is said to be a σ-unreal graph. Let δ(n) be the minimum edgedensity over all n vertices graphs with σ-unreal roots. In this paper, by using the theory of adjoint polynomials, a negative answer to a problem posed by Brenti et al. is given and the following results are obtained: For any positive integer a and rational number 0≤c≤1, there exists at least a graph sequence {G _i}_1≤i≤a such that G _i is σ-unreal and δ(G _i)→c as n→∞ for all 1 ≤i≤a, and moreover, δ(n)→0 as n→∞. Supported by the National Natural Science Foundation of China (10061003) and the Science Foundation of the State Education Ministry of China.     

p-Energia in spazi metrici generalizzati

Given a setS and a function σ:S x S→[0, +∞[ such that σ(x, x)=0, we define the generalizedp-energy of a curve γ: [a, b]→S, so that, ifS is a Hilbert space and σ(x, y)=‖x−y‖ we obtain . Sufficient conditions for the equality of the defined energies are also given. Moreover the case in whichS is a set of measurable parts of ℝⁿ and σ_r is a family of functions used in order to study the generalized minimizing motions, is discussed.

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Conferenza tenuta il 30 ottobre 1995     

Algébres topologiquement uniformes et algébres fortement topologiquement uniformes

We characterize locally convex topological algebrasA satisfying: a sequence (x _n) inA converges to 0 if, and only if, (x _n ²) converges to 0. We also show that a real Banach algebra such thatx _n ²+y _n ²→0 if, and only if,x _n → 0 andy _n → 0, for every sequences (x _n) and (y _n) inA, is isomorphic to, whereX is a compact space.      

Maximal deviation theory of some estimators of prior distribution functions

Summary Let be a sequence of independent identically distributed random variables withθ ₁∼G and the conditional distribution ofx ₁ givenθ ₁=θ given by . HereG is unknown andF _θ(·) is known. This paper provides estimators ofG based onx ₁, …,x _n such that the random variable sup has an asymptotic distribution asn→∞ under certain on conditionsG and for certain choices ofF _θ. A simulation model has been discussed involving the uniform distribution on (0, θ) forF _θ and an exponential distribution forG. Research supported by the National Science Foundation under Grant #MCS77-26809.     

On zeros of functions of given proximate formal order analytic in a half-plane

We describe sequences of zeros of functionsf≢0 that are analytic in the half-plane ℂ₊={z:Rez> and satisfy the condition

Asymptotic property of approximation to x αsgn x by Newman Type Operators

Approximation to the function ｜x｜ plays an important role in approximation theory. This paper studies the approximation to the function xαsgn x, which equals ｜x｜ if α = 1. We construct a Newman Type Operator rn（x） and prove max ｜x｜≤1｜xαsgn x-rn（x）｜～Cn1/4e-π1/2（1/2）αn.     

Problems similar to the additive divisor problem

For multiplicative functions ƒ(n), let the following conditions be satisfied: ƒ(n)≥0 ƒ(p ^r)≤A ^r,A>0, and for anyε>0 there exist constants ,α>0 such that and Σ _p≤x ƒ(p) lnp≥αx. For such functions, the following relation is proved:

On Maps Preserving Zero Jordan Products

Let R be a ring, A = M _n (R) and θ: A → A a surjective additive map preserving zero Jordan products, i.e. if x,y ∈ A are such that xy + yx = 0, then θ(x)θ(y) + θ(y)θ(x) = 0. In this paper, we show that if R contains \frac12\frac{1}{2} and n ≥ 4, then θ = λϕ, where λ = θ(1) is a central element of A and ϕ: A → A is a Jordan homomorphism.     

A strong law of large numbers for nonexpansive vector-valued stochastic processes

A mapT: X→X on a normed linear space is callednonexpansive if ‖Tx-Ty‖≤‖x-y‖∀x, y∈X. Let (Ω, Σ,P) be a probability space, an increasing chain of σ-fields spanning Σ,X a Banach space, andT: X→X. A sequence (x_n) of strongly -measurable and stronglyP-integrable functions on Ω taking on values inX is called aT-martingale if . LetT: H→H be a nonexpansive mapping on a Hilbert spaceH and let (x_n) be aT-martingale taking on values inH. If then x _n /n converges a.e. LetT: X→X be a nonexpansive mapping on ap-uniformly smooth Banach spaceX, 1<p≤2, and let (x_n) be aT-martingale (taking on values inX). If then there exists a continuous linear functionalf∈X ^* of norm 1 such that If, in addition, the spaceX is strictly convex, x _n /n converges weakly; and if the norm ofX ^* is Fréchet differentiable (away from zero), x _n /n converges strongly. This work was supported by National Science Foundation Grant MCS-82-02093     

Asymptotic expansion of the log-likelihood function based on stopping times defined on a Markov process

Consider the parameter space Θ which is an open subset of ℝ ^k ,k≧1, and for each θ∈Θ, let the r.v.′sY _n ,n=0, 1, ... be defined on the probability space (X,A,P _θ) and take values in a Borel setS of a Euclidean space. It is assumed that the process {Y _n },n≧0, is Markovian satisfying certain suitable regularity conditions. For eachn≧1, let υ _n be a stopping time defined on this process and have some desirable properties. For 0 < τ _n → ∞ asn→∞, set h _n →h ∈R ^k , and consider the log-likelihood function of the probability measure with respect to the probability measure . Here is the restriction ofP _θ to the σ-field induced by the r.v.′sY ₀,Y ₁, ..., . The main purpose of this paper is to obtain an asymptotic expansion of in the probability sense. The asymptotic distribution of , as well as that of another r.v. closely related to it, is obtained under both and . This research was supported by the National Science Foundation, Grant MCS77-09574. Research supported by the National Science Foundation, Grant MCS76-11620.     

A central limit theorem for convex sets

We show that there exists a sequence for which the following holds: Let K⊂ℝⁿ be a compact, convex set with a non-empty interior. Let X be a random vector that is distributed uniformly in K. Then there exist a unit vector θ in ℝⁿ, t₀∈ℝ and σ>0 such that

Discretization of Compact Riemannian Manifolds Applied to the Spectrum of Laplacian

For κ ⩾ 0 and r₀ > 0 let ℳ(n, κ, r₀) be the set of all connected, compact n-dimensional Riemannian manifolds (Mⁿ, g) with Ricci (M, g) ⩾ −(n−1) κ g and Inj (M) ⩾ r₀. We study the relation between the kth eigenvalue λ_k(M) of the Laplacian associated to (Mⁿ,g), Δ = −div(grad), and the kth eigenvalue λ_k(X) of a combinatorial Laplacian associated to a discretization X of M. We show that there exist constants c, C > 0 (depending only on n, κ and r₀) such that for all M ∈ ℳ(n, κ, r₀) and X a discretization of for all k < |X|. Then, we obtain the same kind of result for two compact manifolds M and N ∈ ℳ(n, κ, r₀) such that the Gromov–Hausdorff distance between M and N is smaller than some η > 0. We show that there exist constants c, C > 0 depending on η, n, κ and r₀ such that for all . Mathematics Subject Classification (2000): 58J50, 53C20 Supported by Swiss National Science Foundation, grant No. 20-101 469