Boundary functions on a bounded balanced domain

We solve the following Dirichlet problem on the bounded balanced domain with some additional properties: For p > 0 and a positive lower semi-continuous function u on ∂Ω with u(z) = u(λ z) for |λ| = 1, z ∈ ∂Ω we construct a holomorphic function f ∈ (Ω) such that for z ∈ ∂Ω, where = {λ ∈ ℂ: |λ| < 1}.      

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     

A bound for Wilson's general theorem

Given any set K of positive integers and positive integer 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 β(K) and λ(v-1)≡0 (mod α(K)). Let K0 be an equivalent set of K, k and k* be the smallest and the largest integers in K0. We prove that c(K,λ)≤exp exp{Q0}Qo=max{2(2p(ko)2-k2kk)p(ko)4,(Kk242y-k-2)(y2)}, whereand y=k*+k(k-1)+1.     

Existence and Multiplicity Results for a Degenerate Elliptic Equation

The goal of this paper is to study the multiplicity result of positive solutions of a class of degenerate elliptic equations. On the basis of the mountain pass theorems and the sub- and supersolutions argument for p-Laplacian operators, under suitable conditions on the nonlinearity f（x, s）, we show the following problem：-△pu=λu^α-a（x）u^q in Ω,u│δΩ=0 possesses at least two positive solutions for large λ, where Ω is a bounded open subset of R^N, N ≥ 2, with C^2 boundary, λ is a positive parameter, Ap is the p-Laplacian operator with p 〉 1, α, q are given constants such that p - 1 〈α 〈 q, and a（x） is a continuous positive function in Ω^-.     

Discrete spectrum in the gaps of a perturbed pseudorelativistic hamiltonian

The pseudorelativistic Hamiltonian is considered under wide conditions on potentials A(x), W(x). It is assumed that a real point λ is regular for G_1/2. Let G_1/2(α)=G_1/2−αV, where α>0, V(x)≥0, and V ∈L _d(ℝ^d). Denote by N(λ, α) the number of eigenvalues of G_1/2(t) that cross the point λ as t increases from 0 to α. A Weyl-type asymptotics is obtained for N(λ, α) as α→∞. Bibliography: 5 titles. To O. A. Ladyzhenskaya Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 249, 1997. pp. 102–117. Translated by A. B. Pushnitskii.     

Nodal solutions for a nonlinear fourth-order eigenvalue problem

We are concerned with determining the values of λ, for which there exist nodal solutions of the fourth-order boundary value problem y″″=λa（x）f（y）,0〈x〈1,y（0）=y（1）=y″（0）=y″（1）=0where λ is a positive parameter, a ∈ C（[0, 1], （0, ∞）, f ∈C（R,R） satisfies f（u）u 〉 0 for all u ≠ 0. We give conditions on the ratio f（s）/s, at infinity and zero, that guarantee the existence of nodal solutions.The proof of our main results is based upon bifurcation techniques.     

Sugli zeri delle funzioni zeta di Dedekind

Riassunto Scopo di questo lavoro è dare una formula asintotica per il numero degli zeri di ReF _K(λ+it) e di ImF _K(λ+it), dove eζ _K(8) è la funzione zeta di Dedekind associata al campo numericoK, con 0<t<T e λ numero reale fissato tale che 1−1/n<λ<1 doven è il grado diK.

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Summary The aim of this paper is to give an asymptotic formula for the number of zeros of ReF _K(λ+it) and ImF _K(λ+it), where andζ _K(8) is the Dedekind zeta function for a number fieldK, with 0<t<T and λ fixed real number such that 1−1/n<λ<1, wheren is the degree ofK.

     

Periodic parabolic problems with concave and convex nonlinearities

Let be a smooth bounded domain, let a, b be two functions that are possibly discontinuous and unbounded with a ≥ 0 in and b > 0 in a set of positive measure and let 0 < p < 1 < q. We prove that there exists some 0 < Λ < ∞ such that the nonlinear Dirichlet periodic parabolic problem in has a positive solution for all 0 < λ < Λ and that there is no positive solution if λ > Λ. In some cases we also show the existence of a minimal solution for all 0 < λ < Λ and that the solution u _λ can be chosen such that λ → u _λ is differentiable and increasing. We also give some upper and lower estimates for such a Λ. All results remain true for the analogous elliptic problems. Partially supported by CONICET, Secyt-UNC, ANPCYT and Agencia Cordoba Ciencia     

Nonlinear differential equations with asymptotically stable solutions

We establish conditions of asymptotic stability of all solutions of the equation , t≥0in a Banach space E in the case where σ(A(x)⊂{λ:Reλ<0}∀x∈E. We give an example of an equation with unstable solutions. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 2, pp. 264–273, February, 1998.     

Representation of Contractive Solutions of a Class of Algebraic Riccati Equations as Characteristic Functions of Maximal Dissipative Operators

Let

Vectorspacelike representation of absolute planes

The pointset E of an absolute plane can be provided with a binary operation "+" such that (E, +) becomes a loop and for each a E \ {o} the line [a] through o and a is a commutative subgroup of (E, +). Two elements a, b E \ {o} are called independent if [a] ∩ [b] = {o} and the absolute plane is called vectorspacelike if for any two independent elements we have E = [a] + [b] := {x + y | x [a], y [b]}. If is singular then (E, +) is a commutative group and is vectorspacelike iff is Euclidean. If is a hyperbolic plane then is vectorspacelike and in the continous case if a, b are independent, each point p has a unique representation as a quasilinear combination p = α · a + μ · b where α · a [a]and β · b [b] are points, α, β real numbers such that λ (o, λ · a) = |λ|· λ (o, a) and λ (o, μ · b) = |μ|. λ(o, b) and λ is the distance function. This work was partially supported by the Research Project of MIUR (Italian Ministery of Education and University) “Geometria combinatoria e sue applicazioni” and by the research group GNSAGA of INDAM. Dedicated to Walter Benz on the occasion of his 75 ^th birthday, in friendship     

Lyapunov exponents for products of matrices and multifractal analysis. Part I: Positive matrices

Let (Σ,σ) be a full shift space on an alphabet consisting ofm symbols and letM: Σ→L ⁺(ℝ ^d , ℝ ^d ) be a continuous function taking values in the set ofd×d positive matrices. Denote by λ _M (x) the upper Lyapunov exponent ofM atx. The set of possible Lyapunov exponents is just an interval. For any possible Lyapunov exponentα, we prove the following variational formula, , where dim is the Hausdorff dimension or the packing dimension,P _M(q) is the pressure function ofM, μ is aσ-invariant Borel probability measure on Σ,h(μ) is the entropy ofμ, and . The author was partially supported by a HK RGC grant in Hong Kong and the Special Funds for Major State Basic Research Projects in China.     

Sharp Jackson-Stechkin inequality inL2 for multidimensional spheres

In this paper we prove the Jackson-Stechkin inequalityE _n−1(f)<ω _n (f, 2τ _n ,λ),n≥1,m≥5,r≥1, f ∈L2( ),f ≢ const, which is sharp for eachn=2, 3, ...; hereE _n−1 (f) is the best approximation of a functionf by spherical polynomials of degree ≤n−1, ω _n (f, τ) is theτth modulus of continuity off based on the translations ,t ∈ ℝ,x ∈ , , is the measure of the unit Euclidean sphere , , andτ _n ,λ is the first positive zero of the Gegenbauer cosine polynomial (cost). Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 333–355, September, 1996. The present paper was discussed at Ural State University in a seminar headed by Professor Arestov. The author is grateful to Professor Arestov and Associate Professor Popov for useful conversations. This research was supported by the State Commission for Higher Education of the Russian Federation under grant No. 2-16-5-31 and by the Russian Foundation for Basic Research under grant No. 93-011-196.     

Absolute continuity of the distribution of some Markov geometric series

Let (∈n)≥0 be the Markov chain of two states with respect to the probability measure of the maximal entropy on the subshift space ∑A defined by Fibonacci incident matrix A.We consider the measure μλ of the probability distribution of the random series ∑∞n=0 εnλn (0 ＜λ＜ 1).It is proved that μλ is singular if λ∈ (0,√5-1/2) and that μλ is absolutely continuous for almost all λ∈ (√5-1/2,0.739).     

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.     

Inverse problem for zeros of certain koebe-related functions

If w₁,…,w _N is a finite sequence of nonzero points in the unit disk, then there are distinct points λ₁,…, λ_N on the unit circle and positive numbers Μ₁,…,Μ _N such that is the zero sequence of the function 1 — . The points λ₁,…, λ_N and numbers Μ₁,…,Μ_N are unique (except for reorderings).     

Completeness of systems of entire functions in spaces of holomorphic functions

Letf be an entire function in . For a broad class of distribution densities of the set Λ, a scale of sufficient conditions for the completeness of the system of functions {f(λ×z):λ∈Λ},z∈E, where , in the spaceH(E) of holomorphic functions onE with respect to the topology of uniform convergence on compact subsets is given in terms of the mutual indicator of the functionf and the setE. These conditions are new already forn=1 even ifE is a disk. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 603–616, October, 1999.     

On cardinal interpolation by Gaussian radial-basis functions: Properties of fundamental functions and estimates for Lebesgue constants

Suppose λ is a positive number. Basic theory of cardinal interpolation ensures the existence of the Gaussian cardinal functionL _λ(x)

On the estimation of the regression coefficients of a continuous parameter process with stationary residual

The asymptotic expressions of the covariance matrices for both the least square estimates _L α _T and Markov (best linear) estimates are obtained, based on a sample in a finite interval (0, T) of the regression co-efficients α = (α ₁, …, α _{m 0})′ of a parameter-continuous process with a stationary residual. We assume that the regression variables φ _ν(t), t ⩾ 0, ν = 1, …, m ₀, are continuous in t, and satisfy conditions (3.1)–(3.3). For the residual, we assume that it is a stationary process that possesses a bounded continuous spectral density f(λ). Under these assumptions, it is proven that

Directional derivatives for the value-function in semi-infinite programming

For the problemP(λ): Maximizec ^T z subject toz∈Z(λ), whereZ(λ) is defined by an in general infinite set of linear inequalities, it is shown that the value-function has directional derivatives at every point such thatP( ) and its dual are both superconsistent. To compute these directional derivatives a min-max-formula, well-known in convex programming, is derived. In addition, it is shown that derivatives can be obtained more easily by a limit-process using only convergent selections of solutions ofP(λ _n ), λ _n → and their duals.